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+} bind def + +/GrayDirectClassPacket +{ + % + % Get a DirectClass packet; convert to grayscale. + % + % Parameters: + % red + % green + % blue + % length: number of pixels minus one of this color (optional). + % + currentfile color_packet readhexstring pop pop + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/GrayPseudoClassPacket +{ + % + % Get a PseudoClass packet; convert to grayscale. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + color_packet 0 get 0.299 mul + color_packet 1 get 0.587 mul add + color_packet 2 get 0.114 mul add + cvi + /gray_packet exch def + compression 0 eq + { + /number_pixels 1 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add def + } ifelse + 0 1 number_pixels 1 sub + { + pixels exch gray_packet put + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassPacket +{ + % + % Get a PseudoClass packet. + % + % Parameters: + % index: index into the colormap. + % length: number of pixels minus one of this color (optional). + % + currentfile byte readhexstring pop 0 get + /offset exch 3 mul def + /color_packet colormap offset 3 getinterval def + compression 0 eq + { + /number_pixels 3 def + } + { + currentfile byte readhexstring pop 0 get + /number_pixels exch 1 add 3 mul def + } ifelse + 0 3 number_pixels 1 sub + { + pixels exch color_packet putinterval + } for + pixels 0 number_pixels getinterval +} bind def + +/PseudoClassImage +{ + % + % Display a PseudoClass image. + % + % Parameters: + % class: 0-PseudoClass or 1-Grayscale. + % + currentfile buffer readline pop + token pop /class exch def pop + class 0 gt + { + currentfile buffer readline pop + token pop /depth exch def pop + /grays columns 8 add depth sub depth mul 8 idiv string def + columns rows depth + [ + columns 0 0 + rows neg 0 rows + ] + { currentfile grays readhexstring pop } image + } + { + % + % Parameters: + % colors: number of colors in the colormap. + % colormap: red, green, blue color packets. + % + currentfile buffer readline pop + token pop /colors exch def pop + /colors colors 3 mul def + /colormap colors string def + currentfile colormap readhexstring pop pop + systemdict /colorimage known + { + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { PseudoClassPacket } false 3 colorimage + } + { + % + % No colorimage operator; convert to grayscale. + % + columns rows 8 + [ + columns 0 0 + rows neg 0 rows + ] + { GrayPseudoClassPacket } image + } ifelse + } ifelse +} bind def + +/DisplayImage +{ + % + % Display a DirectClass or PseudoClass image. + % + % Parameters: + % x & y translation. + % x & y scale. + % label pointsize. + % image label. + % image columns & rows. + % class: 0-DirectClass or 1-PseudoClass. + % compression: 0-none or 1-RunlengthEncoded. + % hex color packets. + % + gsave + /buffer 512 string def + /byte 1 string def + /color_packet 3 string def + /pixels 768 string def + + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + x y translate + currentfile buffer readline pop + token pop /x exch def + token pop /y exch def pop + currentfile buffer readline pop + token pop /pointsize exch def pop + x y scale + currentfile buffer readline pop + token pop /columns exch def + token pop /rows exch def pop + currentfile buffer readline pop + token pop /class exch def pop + currentfile buffer readline pop + token pop /compression exch def pop + class 0 gt { PseudoClassImage } { DirectClassImage } ifelse + 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+22422A21412921412921412921412A22422B21412A22422B22422B22422D20402B1F3F2A20402B +21412C1F3F281F3F281F3F281F3F281F3F281F3F281F3F281F3F28 + +end +%%PageTrailer +%%Trailer +%%EOF diff --git a/doc/LectureNotes/make.sh b/doc/LectureNotes/make.sh new file mode 100644 index 000000000..ab4fa6733 --- /dev/null +++ b/doc/LectureNotes/make.sh @@ -0,0 +1,43 @@ +#!/bin/sh +set -x + +function system { + "$@" + if [ $? -ne 0 ]; then + echo "make.sh: unsuccessful command $@" + echo "abort!" + exit 1 + fi +} + +if [ $# -eq 0 ]; then +echo 'bash make.sh slides1|slides2' +exit 1 +fi + +name=$1 +rm -f *.tar.gz + +opt="--encoding=utf-8" +# Note: Makefile examples contain constructions like ${PROG} which +# looks like Mako constructions, but they are not. Use --no_mako +# to turn off Mako processing. +opt="--no_mako" + +rm -f *.aux + +# IPython notebook +system doconce format ipynb $name $opt + + +# Ordinary plain LaTeX document +rm -f *.aux # important after beamer +system doconce format pdflatex $name --minted_latex_style=trac --latex_admon=paragraph $opt +system doconce ptex2tex $name envir=minted +# Add special packages +doconce subst "% Add user's preamble" "\g<1>\n\\usepackage{simplewick}" $name.tex +doconce replace 'section{' 'section*{' $name.tex +pdflatex -shell-escape $name +pdflatex -shell-escape $name +mv -f $name.pdf ${name}-minted.pdf +cp $name.tex ${name}-plain-minted.tex diff --git a/doc/Projects/2019/Project2/html/._Project2-bs000.html b/doc/Projects/2019/Project2/html/._Project2-bs000.html new file mode 100644 index 000000000..b542ddbd5 --- /dev/null +++ b/doc/Projects/2019/Project2/html/._Project2-bs000.html @@ -0,0 +1,448 @@ + + + + + + + + +Project 2 on Machine Learning, deadline November 8 + + + + + + + + + + + + + + + + + + + + + + + +

+
+ + Project 2 on Machine Learning, deadline November 8 +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + + + +
+

Project 2 on Machine Learning, deadline November 8

+ +

+ + +

+Data Analysis and Machine Learning FYS-STK3155/FYS4155 +
+ +

+ + +

Department of Physics, University of Oslo, Norway
+
+

+

Oct 17, 2019

+
+

+

+ +

Classification and Regression, from linear and logistic regression to neural networks

+ +

+The main aim of this project is to study both classification and +regression problems, where we can reuse the regression algorithms studied +in project 1. We will include logistic regresion for classification +problems and write our own multilayer perceptron code for studying +both regression and classification problems. The codes developed in +project 1, including bootstrap and/or cross-validation as well as the +computation of the mean-squared error and/or the \( R2 \) or the accuracy score (classification problems) functions can +also be utilized in the present analysis. + +

+The data sets that we propose here are (the default sets) + +

    +
  • Regression (fitting a continuous function). In this part you will need to bring up your results from project 1 and compare these with what you get from you Neural Network code to be developed here. The data sets could be + +
      +
    1. Either the Franke function or the terrain data from project 1, or data sets your propose.
      2. +
    + +
  • Classification. Here you will also need to develop a Logistic regression code that you will use to compare with the Neural Network code. The data set we propose are + +
      +
    1. The credit card data set from UCI. This data set links to a recent scientific article. Furthermore, in the lecture slides on Logistic Regression, you will find an example code for the Credit Card data. You could use this code as an example on how to read the data and use Scikit-Learn to run a classification problem.
      2. +
    + +
+ +However, if you would like to study other data sets, feel free to +propose other sets. What we listed here are mere suggestions from our +side. If you opt for another data set, consider using a set which +has been studied in the scientific literature. This makes it easier +for you to compare and analyze your results. It is also an essential +elements of the scientific discussion. + +

+In particular, when developing your own Logistic Regression code for classification problems, the so-called Wisconsin Cancer data (which is a binary problem, benign or malignant tumors) may be studied. You find more information about this at the Scikit-Learn site or at the University of California at Irvine. The lecture slides on dimensionality reduction have several code examples on this data set. + +

Part a): Write your Logistic Regression code, first step

+ +

+If you opt for the credit card data, your first task is to familiarize yourself with the data set and the scientific article. +We recommend also that you study the code example in the Logistic Regression. + +

+Write the part of the code which reads in the data and sets up the relevant data sets. + +

Part b): Write your Logistic Regression code, second step

+ +

+Now you should write your Logistic Regression code with the aim to +reproduce the Logistic Regression analysis of the scientific +article. + +

+Define your cost function and the design matrix before you start writing your code. + +

+In order to find the optimal parameters of your logistic regressor you +should include a gradient descent solver, as discussed in the +gradient descent +lectures. +Since we don't have so many data points, you may just code the +standard gradient descent with a given learning rate, or even attempt +to use the Newton-Raphson method. Alternatively, it may be useful for +the next part on neural networks to implement a stochastic gradient +descent with and without mini-batches. Stochastic gradient with +mini-batches may give the best results. You could finally compare your +code with the output from scikit-learn's toolbox for optimization +methods applied to logistic regression. + +

+To measure the performance of our classification problem we use the +so-called accuracy score. The accuracy is as you would expect just +the number of correctly guessed targets \( t_i \) divided by the total +number of targets. A perfect classifier will have an accuracy score of +\( 1 \). + +$$ +\text{Accuracy} = \frac{\sum_{i=1}^n I(t_i = y_i)}{n} , +$$ + +

+where \( I \) is the indicator function, \( 1 \) if \( t_i = y_i \) and \( 0 \) +otherwise if we have a binary classifcation problem. Here \( t_i \) +represents the target and \( y_i \) the outputs of your Logistic +Regression code. + +

+You can compare your own results with those obtained using +scikit-learn. + +

+As stated in the introduction, it can also be useful to study other datasets. In particular, when developing your own Logistic Regression code for classification problems, the so-called \ +Wisconsin Cancer data (which is a binary problem, benign or malignant tumors) may be studied. You find more \ +information about this at the Scikit-Learn site or at the "University of California at Irvine":"https://archive.ics.uci.e\ +du/ml/datasets/breast+cancer+wisconsin+(original)". The lecture slides on dimensionality reduction have sev\ +eral code examples on this data set. + +

Part c): Writing your own Neural Network code

+ +

+Your aim now, and this is the central part of this project, is to +write to your own Feed Forward Neural Network code implementing the back +propagation algorithm discussed in the lecture +slides. We +start with the Logistic Regression case and the data set discussed in parts a) and b) but train +now the network to find the optimal weights and biases. You are free +to use the codes in the above lecture slides as starting points. + +

+Discuss again your choice of cost function. + +

+Train your network and compare the results with those from your Logistic Regression code. +You should test your results against a similar code using Scikit-Learn (see the examples in the above lecture notes) or tensorflow/keras. + +

+Comment your results and give a critical discussion of the results +obtained with the Logistic Regression code and your own Neural Network +code. Make an analysis of the regularization parameters and the learning rates employed to find the optimal accurary score. + +

+A useful reference on the back progagation algorithm is Nielsen's +book. It is an excellent +read. + +

Part d): Regression analysis using neural networks

+ +

+Here we will change the cost function for our neural network code +developed in part c) in order to perform a regression (fitting a +function or some data set) analysis. As stated above, our default data +sets could be either the Franke function or the terrain data from +project 1. + +

+Compare you results from the neural network regression analysis (with a discussion of learning rates and regularization parameters) +with those you obtained in project 1. Alternatively, if you opt for other data sets, you would need to run your standard ordinary least squares, Ridge and Lasso calculations using your codes from project 1. + +

+Again, we strongly recommend that you compare your own neural Network code and results against a similar code using Scikit-Learn (see the examples in the above lecture notes) or tensorflow/keras. + +

Part e) Critical evaluation of the various algorithms

+ +

+After all these glorious calculations, you should now summarize the +various algorithms and come with a critical evaluation of their pros +and cons. Which algorithm works best for the regression case and which +is best for the classification case. These codes can also be part of +your final project 3, but now applied to other data sets. + +

Background literature

+ +
    +
  1. The text of Michael Nielsen is highly recommended, see Nielsen's book. It is an excellent read.
    2. +
  2. The textbook of Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer, chapters 3 and 7 are the most relevant ones for the analysis here.
    3. +
  3. Mehta et al, arXiv 1803.08823, A high-bias, low-variance introduction to Machine Learning for physicists, ArXiv:1803.08823.
    4. +
+ +

Introduction to numerical projects

+ +

+Here follows a brief recipe and recommendation on how to write a report for each +project. + +

    +
  • Give a short description of the nature of the problem and the eventual numerical methods you have used.
    • +
  • Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.
    • +
  • Include the source code of your program. Comment your program properly.
    • +
  • If possible, try to find analytic solutions, or known limits in order to test your program when developing the code.
    • +
  • Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes.
    • +
  • Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc.
    • +
  • Try to give an interpretation of you results in your answers to the problems.
    • +
  • Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts you've made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it.
    • +
  • Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don't properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning.
    • +
+ +

Format for electronic delivery of report and programs

+ +

+The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report: + +

    +
  • Use Devilry to hand in your projects, log in at http://devilry.ifi.uio.no with your normal UiO username and password and choose either 'fysstk3155' or 'fysstk4155'. There you can load up the files within the deadline.
    • +
  • Upload only the report file! For the source code file(s) you have developed please provide us with your link to your github domain. The report file should include all of your discussions and a list of the codes you have developed. Do not include library files which are available at the course homepage, unless you have made specific changes to them.
    • +
  • In your git repository, please include a folder which contains selected results. These can be in the form of output from your code for a selected set of runs and input parameters.
    • +
  • In this and all later projects, you should include tests (for example unit tests) of your code(s).
    • +
  • Comments from us on your projects, approval or not, corrections to be made etc can be found under your Devilry domain and are only visible to you and the teachers of the course.
    • +
+ +Finally, +we encourage you to collaborate. Optimal working groups consist of +2-3 students. You can then hand in a common report. + +

Software and needed installations

+ +

+If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages, +we recommend that you install the following Python packages via pip as + +

    +
  1. pip install numpy scipy matplotlib ipython scikit-learn tensorflow sympy pandas pillow
    2. +
+ +For Python3, replace pip with pip3. + +

+See below for a discussion of tensorflow and scikit-learn. + +

+For OSX users we recommend also, after having installed Xcode, to install brew. Brew allows +for a seamless installation of additional software via for example + +

    +
  1. brew install python3
    2. +
+ +For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution +you can use pip as well and simply install Python as + +
    +
  1. sudo apt-get install python3 (or python for python2.7)
    2. +
+ +etc etc. + +

+If you don't want to install various Python packages with their dependencies separately, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely + +

    +
  1. Anaconda Anaconda is an open source distribution of the Python and R programming languages for large-scale data processing, predictive analytics, and scientific computing, that aims to simplify package management and deployment. Package versions are managed by the package management system conda
    2. +
  2. Enthought canopy is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.
    3. +
+ +Popular software packages written in Python for ML are + + + +These are all freely available at their respective GitHub sites. They +encompass communities of developers in the thousands or more. And the number +of code developers and contributors keeps increasing. + +
    +
  1. For project 3, you should feel free to use your own codes from projects 1 and 2, eventually write your own for SVMs and/or Decision trees and random forests' or use the available functionality of scikit-learn, tensorflow, etc.
    2. +
  2. The estimates you used and tested in projects 1 and 2 should also be included, that is the \( R2 \)-score, MSE, cross-validation and/or bootstrap if these are relevant.
    3. +
  3. If possible, you should link the data sets with exisiting research and analyses thereof. Scientific articles which have used Machine Learning algorithms to analyze the data are highly welcome. Perhaps you can improve previous analyses and even publish a new article?
    4. +
  4. A critical assessment of the methods with ditto perspectives and recommendations is also something you need to include.
    5. +
+ + +

+ +

    +
  • 1
    • +
+ + +
+ + + + + + + +
+ © 1999-2019, "Data Analysis and Machine Learning FYS-STK3155/FYS4155":"http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html". Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/Projects/2019/hw2/.DS_Store b/doc/Projects/2019/hw2/.DS_Store new file mode 100644 index 000000000..71ee24ec0 Binary files /dev/null and b/doc/Projects/2019/hw2/.DS_Store differ diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs061.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs061.html new file mode 100644 index 000000000..2e05c1953 --- /dev/null +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs061.html @@ -0,0 +1,378 @@ + + + + + + + + +Data Analysis and Machine Learning: From Decision Trees to Forests and all that + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: From Decision Trees to Forests and all that +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Gradient Boosting, Examples of Regression

+

+ + +

import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import train_test_split
+from sklearn.ensemble import GradientBoostingRegressor
+from sklearn.preprocessing import StandardScaler
+import scikitplot as skplt
+from sklearn.metrics import mean_squared_error
+
+n = 100
+maxdegree = 6
+
+# Make data set.
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+
+error = np.zeros(maxdegree)
+bias = np.zeros(maxdegree)
+variance = np.zeros(maxdegree)
+polydegree = np.zeros(maxdegree)
+X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+
+for degree in range(1,maxdegree):
+    model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0)  
+    model.fit(X_train_scaled,y_train)
+    y_pred = model.predict(X_test_scaled)
+    polydegree[degree] = degree
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
+    variance[degree] = np.mean( np.var(y_pred) )
+    print('Max depth:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+
+plt.xlim(1,maxdegree-1)
+plt.plot(polydegree, error, label='Error')
+plt.plot(polydegree, bias, label='bias')
+plt.plot(polydegree, variance, label='Variance')
+plt.legend()
+save_fig("gdregression")
+plt.show()
+
+

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs062.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs062.html new file mode 100644 index 000000000..0268a216d --- /dev/null +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs062.html @@ -0,0 +1,371 @@ + + + + + + + + +Data Analysis and Machine Learning: From Decision Trees to Forests and all that + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: From Decision Trees to Forests and all that +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Gradient Boosting, Classification Example

+

+ + +

import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import  train_test_split 
+from sklearn.datasets import load_breast_cancer
+import scikitplot as skplt
+from sklearn.ensemble import GradientBoostingClassifier
+from sklearn.model_selection import cross_validate
+
+# Load the data
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+#now scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+
+gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0)  
+gd_clf.fit(X_train_scaled, y_train)
+#Cross validation
+accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
+
+import scikitplot as skplt
+y_pred = gd_clf.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+save_fig("gdclassiffierconfusion")
+plt.show()
+y_probas = gd_clf.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test, y_probas)
+save_fig("gdclassiffierroc")
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test, y_probas)
+save_fig("gdclassiffiercgain")
+plt.show()
+
+

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs063.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs063.html new file mode 100644 index 000000000..f9c7f499d --- /dev/null +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs063.html @@ -0,0 +1,343 @@ + + + + + + + + +Data Analysis and Machine Learning: From Decision Trees to Forests and all that + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: From Decision Trees to Forests and all that +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

XGBoost: Extreme Gradient Boosting

+ +

+XGBoost or Extreme Gradient +Boosting, is an optimized distributed gradient boosting library +designed to be highly efficient, flexible and portable. It implements +machine learning algorithms under the Gradient Boosting +framework. XGBoost provides a parallel tree boosting that solve many +data science problems in a fast and accurate way. See the article by Chen and Guestrin. + +

+The authors design and build a highly scalable end-to-end tree +boosting system. It has a theoretically justified weighted quantile +sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning. + +

+It is now the algorithm which wins essentially all ML competitions!!! + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs064.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs064.html new file mode 100644 index 000000000..9e380d4d0 --- /dev/null +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs064.html @@ -0,0 +1,376 @@ + + + + + + + + +Data Analysis and Machine Learning: From Decision Trees to Forests and all that + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: From Decision Trees to Forests and all that +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Regression Case

+ +

+ + +

import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import train_test_split
+import xgboost as xgb
+from sklearn.preprocessing import StandardScaler
+import scikitplot as skplt
+from sklearn.metrics import mean_squared_error
+
+n = 100
+maxdegree = 6
+
+# Make data set.
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+
+error = np.zeros(maxdegree)
+bias = np.zeros(maxdegree)
+variance = np.zeros(maxdegree)
+polydegree = np.zeros(maxdegree)
+X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+
+for degree in range(maxdegree):
+    model =  xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200)
+
+    model.fit(X_train_scaled,y_train)
+    y_pred = model.predict(X_test_scaled)
+    polydegree[degree] = degree
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
+    variance[degree] = np.mean( np.var(y_pred) )
+    print('Max depth:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+
+plt.xlim(1,maxdegree-1)
+plt.plot(polydegree, error, label='Error')
+plt.plot(polydegree, bias, label='bias')
+plt.plot(polydegree, variance, label='Variance')
+plt.legend()
+plt.show()
+
+

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs065.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs065.html new file mode 100644 index 000000000..7ffd1db4e --- /dev/null +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs065.html @@ -0,0 +1,382 @@ + + + + + + + + +Data Analysis and Machine Learning: From Decision Trees to Forests and all that + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: From Decision Trees to Forests and all that +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Xgboost on the Cancer Data

+ +

+As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now. +

+ + +

import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import  train_test_split 
+from sklearn.datasets import load_breast_cancer
+from sklearn.preprocessing import LabelEncoder
+from sklearn.model_selection import cross_validate
+import scikitplot as skplt
+import xgboost as xgb
+# Load the data
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+#now scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+
+xg_clf = xgb.XGBClassifier()
+xg_clf.fit(X_train_scaled,y_train)
+
+y_test = xg_clf.predict(X_test_scaled)
+
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
+
+import scikitplot as skplt
+y_pred = xg_clf.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+save_fig("xdclassiffierconfusion")
+plt.show()
+y_probas = xg_clf.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test, y_probas)
+save_fig("xdclassiffierroc")
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test, y_probas)
+save_fig("gdclassiffiercgain")
+plt.show()
+
+
+xgb.plot_tree(xg_clf,num_trees=0)
+plt.rcParams['figure.figsize'] = [50, 10]
+save_fig("xgtree")
+plt.show()
+
+xgb.plot_importance(xg_clf)
+plt.rcParams['figure.figsize'] = [5, 5]
+save_fig("xgparams")
+plt.show()
+
+

+ +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/baggingboot.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/baggingboot.png new file mode 100644 index 000000000..7933a79b8 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/baggingboot.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/baggingtree.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/baggingtree.png new file mode 100644 index 000000000..cdcb529d0 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/baggingtree.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/boosting_plot.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/boosting_plot.png new file mode 100644 index 000000000..b9cf260dc Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/boosting_plot.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/early_stopping_gbrt_plot.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/early_stopping_gbrt_plot.png new file mode 100644 index 000000000..a1164e265 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/early_stopping_gbrt_plot.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gbrt_learning_rate_plot.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gbrt_learning_rate_plot.png new file mode 100644 index 000000000..864074560 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gbrt_learning_rate_plot.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffiercgain.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffiercgain.png new file mode 100644 index 000000000..28d6cc417 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffiercgain.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffierconfusion.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffierconfusion.png new file mode 100644 index 000000000..ff4188f7b Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffierconfusion.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffierroc.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffierroc.png new file mode 100644 index 000000000..98ae6c098 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdclassiffierroc.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdregression.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdregression.png new file mode 100644 index 000000000..9ff12eab7 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gdregression.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gradient_boosting_plot.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gradient_boosting_plot.png new file mode 100644 index 000000000..cec7811b0 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/gradient_boosting_plot.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/votingsimple.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/votingsimple.png new file mode 100644 index 000000000..b1efa5351 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/votingsimple.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xdclassiffierconfusion.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xdclassiffierconfusion.png new file mode 100644 index 000000000..17bfd0a09 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xdclassiffierconfusion.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xdclassiffierroc.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xdclassiffierroc.png new file mode 100644 index 000000000..72e3c9741 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xdclassiffierroc.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xgparams.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xgparams.png new file mode 100644 index 000000000..4fb84e1d3 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xgparams.png differ diff --git a/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xgtree.png b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xgtree.png new file mode 100644 index 000000000..8407b3fc3 Binary files /dev/null and b/doc/pub/DecisionTrees/ipynb/Results/FigureFiles/xgtree.png differ diff --git a/doc/pub/DimRed/html/._DimRed-bs031.html b/doc/pub/DimRed/html/._DimRed-bs031.html new file mode 100644 index 000000000..3180ae297 --- /dev/null +++ b/doc/pub/DimRed/html/._DimRed-bs031.html @@ -0,0 +1,258 @@ + + + + + + + + +Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

LLE

+ +

+Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction +(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous +algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its +closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where +these local relationships are best preserved (more details shortly). + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/DimRed/html/._DimRed-bs032.html b/doc/pub/DimRed/html/._DimRed-bs032.html new file mode 100644 index 000000000..ed16744de --- /dev/null +++ b/doc/pub/DimRed/html/._DimRed-bs032.html @@ -0,0 +1,262 @@ + + + + + + + + +Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Other techniques

+ +

+There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn. + +

+Here are some of the most popular: + +

    +
  • Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.
    • +
  • Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.
    • +
  • t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).
    • +
  • Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.
    • +
+ + +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/DimRed/html/reveal.js/plugin/leap/leap.js b/doc/pub/DimRed/html/reveal.js/plugin/leap/leap.js new file mode 100644 index 000000000..48084ffb0 --- /dev/null +++ b/doc/pub/DimRed/html/reveal.js/plugin/leap/leap.js @@ -0,0 +1,159 @@ +/* + * Copyright (c) 2013, Leap Motion, Inc. + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: + * + * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. + * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Version 0.2.0 - http://js.leapmotion.com/0.2.0/leap.min.js + * Grab latest versions from http://js.leapmotion.com/ + */ + +!function(e,t,n){function i(n,s){if(!t[n]){if(!e[n]){var o=typeof require=="function"&&require;if(!s&&o)return o(n,!0);if(r)return r(n,!0);throw new Error("Cannot find module '"+n+"'")}var u=t[n]={exports:{}};e[n][0].call(u.exports,function(t){var r=e[n][1][t];return i(r?r:t)},u,u.exports)}return t[n].exports}var r=typeof require=="function"&&require;for(var s=0;s=this.size)return undefined;if(i>=this._buf.length)return undefined;return this._buf[(this.pos-i-1)%this.size]};CircularBuffer.prototype.push=function(o){this._buf[this.pos%this.size]=o;return this.pos++}},{}],3:[function(require,module,exports){var Connection=module.exports=require("./base_connection");Connection.prototype.setupSocket=function(){var connection=this;var socket=new WebSocket(this.getUrl());socket.onopen=function(){connection.handleOpen()};socket.onmessage=function(message){connection.handleData(message.data)};socket.onclose=function(){connection.handleClose()};return socket};Connection.prototype.startHeartbeat=function(){if(!this.protocol.sendHeartbeat||this.heartbeatTimer)return;var connection=this;var propertyName=null;if(typeof document.hidden!=="undefined"){propertyName="hidden"}else if(typeof document.mozHidden!=="undefined"){propertyName="mozHidden"}else if(typeof document.msHidden!=="undefined"){propertyName="msHidden"}else if(typeof document.webkitHidden!=="undefined"){propertyName="webkitHidden"}else{propertyName=undefined}var windowVisible=true;var focusListener=window.addEventListener("focus",function(e){windowVisible=true});var blurListener=window.addEventListener("blur",function(e){windowVisible=false});this.on("disconnect",function(){if(connection.heartbeatTimer){clearTimeout(connection.heartbeatTimer);delete connection.heartbeatTimer}window.removeEventListener(focusListener);window.removeEventListener(blurListener)});this.heartbeatTimer=setInterval(function(){var isVisible=propertyName===undefined?true:document[propertyName]===false;if(isVisible&&windowVisible){connection.sendHeartbeat()}else{connection.setHeartbeatState(false)}},this.opts.heartbeatInterval)}},{"./base_connection":1}],4:[function(require,module,exports){!function(process){var Frame=require("./frame"),CircularBuffer=require("./circular_buffer"),Pipeline=require("./pipeline"),EventEmitter=require("events").EventEmitter,gestureListener=require("./gesture").gestureListener,_=require("underscore");var Controller=module.exports=function(opts){var inNode=typeof process!=="undefined"&&process.title==="node";opts=_.defaults(opts||{},{inNode:inNode});this.inNode=opts.inNode;opts=_.defaults(opts||{},{frameEventName:this.useAnimationLoop()?"animationFrame":"deviceFrame",supressAnimationLoop:false});this.supressAnimationLoop=opts.supressAnimationLoop;this.frameEventName=opts.frameEventName;this.history=new CircularBuffer(200);this.lastFrame=Frame.Invalid;this.lastValidFrame=Frame.Invalid;this.lastConnectionFrame=Frame.Invalid;this.accumulatedGestures=[];if(opts.connectionType===undefined){this.connectionType=this.inBrowser()?require("./connection"):require("./node_connection")}else{this.connectionType=opts.connectionType}this.connection=new this.connectionType(opts);this.setupConnectionEvents()};Controller.prototype.gesture=function(type,cb){var creator=gestureListener(this,type);if(cb!==undefined){creator.stop(cb)}return creator};Controller.prototype.inBrowser=function(){return!this.inNode};Controller.prototype.useAnimationLoop=function(){return this.inBrowser()&&typeof chrome==="undefined"};Controller.prototype.connect=function(){var controller=this;if(this.connection.connect()&&this.inBrowser()&&!controller.supressAnimationLoop){var callback=function(){controller.emit("animationFrame",controller.lastConnectionFrame);window.requestAnimFrame(callback)};window.requestAnimFrame(callback)}};Controller.prototype.disconnect=function(){this.connection.disconnect()};Controller.prototype.frame=function(num){return this.history.get(num)||Frame.Invalid};Controller.prototype.loop=function(callback){switch(callback.length){case 1:this.on(this.frameEventName,callback);break;case 2:var controller=this;var scheduler=null;var immediateRunnerCallback=function(frame){callback(frame,function(){if(controller.lastFrame!=frame){immediateRunnerCallback(controller.lastFrame)}else{controller.once(controller.frameEventName,immediateRunnerCallback)}})};this.once(this.frameEventName,immediateRunnerCallback);break}this.connect()};Controller.prototype.addStep=function(step){if(!this.pipeline)this.pipeline=new Pipeline(this);this.pipeline.addStep(step)};Controller.prototype.processFrame=function(frame){if(frame.gestures){this.accumulatedGestures=this.accumulatedGestures.concat(frame.gestures)}if(this.pipeline){frame=this.pipeline.run(frame);if(!frame)frame=Frame.Invalid}this.lastConnectionFrame=frame;this.emit("deviceFrame",frame)};Controller.prototype.processFinishedFrame=function(frame){this.lastFrame=frame;if(frame.valid){this.lastValidFrame=frame}frame.controller=this;frame.historyIdx=this.history.push(frame);if(frame.gestures){frame.gestures=this.accumulatedGestures;this.accumulatedGestures=[];for(var 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Hand=require("./hand"),Pointable=require("./pointable"),createGesture=require("./gesture").createGesture,glMatrix=require("gl-matrix"),mat3=glMatrix.mat3,vec3=glMatrix.vec3,InteractionBox=require("./interaction_box"),_=require("underscore");var Frame=module.exports=function(data){this.valid=true;this.id=data.id;this.timestamp=data.timestamp;this.hands=[];this.handsMap={};this.pointables=[];this.tools=[];this.fingers=[];if(data.interactionBox){this.interactionBox=new InteractionBox(data.interactionBox)}this.gestures=[];this.pointablesMap={};this._translation=data.t;this._rotation=_.flatten(data.r);this._scaleFactor=data.s;this.data=data;this.type="frame";this.currentFrameRate=data.currentFrameRate;var handMap={};for(var handIdx=0,handCount=data.hands.length;handIdx!=handCount;handIdx++){var hand=new Hand(data.hands[handIdx]);hand.frame=this;this.hands.push(hand);this.handsMap[hand.id]=hand;handMap[hand.id]=handIdx}for(var pointableIdx=0,pointableCount=data.pointables.length;pointableIdx!=pointableCount;pointableIdx++){var pointable=new Pointable(data.pointables[pointableIdx]);pointable.frame=this;this.pointables.push(pointable);this.pointablesMap[pointable.id]=pointable;(pointable.tool?this.tools:this.fingers).push(pointable);if(pointable.handId!==undefined&&handMap.hasOwnProperty(pointable.handId)){var hand=this.hands[handMap[pointable.handId]];hand.pointables.push(pointable);(pointable.tool?hand.tools:hand.fingers).push(pointable)}}if(data.gestures){for(var gestureIdx=0,gestureCount=data.gestures.length;gestureIdx!=gestureCount;gestureIdx++){this.gestures.push(createGesture(data.gestures[gestureIdx]))}}};Frame.prototype.tool=function(id){var pointable=this.pointable(id);return pointable.tool?pointable:Pointable.Invalid};Frame.prototype.pointable=function(id){return this.pointablesMap[id]||Pointable.Invalid};Frame.prototype.finger=function(id){var pointable=this.pointable(id);return!pointable.tool?pointable:Pointable.Invalid};Frame.prototype.hand=function(id){return this.handsMap[id]||Hand.Invalid};Frame.prototype.rotationAngle=function(sinceFrame,axis){if(!this.valid||!sinceFrame.valid)return 0;var rot=this.rotationMatrix(sinceFrame);var cs=(rot[0]+rot[4]+rot[8]-1)*.5;var angle=Math.acos(cs);angle=isNaN(angle)?0:angle;if(axis!==undefined){var rotAxis=this.rotationAxis(sinceFrame);angle*=vec3.dot(rotAxis,vec3.normalize(vec3.create(),axis))}return angle};Frame.prototype.rotationAxis=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();return vec3.normalize(vec3.create(),[this._rotation[7]-sinceFrame._rotation[5],this._rotation[2]-sinceFrame._rotation[6],this._rotation[3]-sinceFrame._rotation[1]])};Frame.prototype.rotationMatrix=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return mat3.create();var transpose=mat3.transpose(mat3.create(),this._rotation);return mat3.multiply(mat3.create(),sinceFrame._rotation,transpose)};Frame.prototype.scaleFactor=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return 1;return Math.exp(this._scaleFactor-sinceFrame._scaleFactor)};Frame.prototype.translation=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();return vec3.subtract(vec3.create(),this._translation,sinceFrame._translation)};Frame.prototype.toString=function(){var str="Frame [ id:"+this.id+" | timestamp:"+this.timestamp+" | Hand count:("+this.hands.length+") | Pointable count:("+this.pointables.length+")";if(this.gestures)str+=" | Gesture count:("+this.gestures.length+")";str+=" ]";return str};Frame.prototype.dump=function(){var out="";out+="Frame Info:
";out+=this.toString();out+="

Hands:
";for(var handIdx=0,handCount=this.hands.length;handIdx!=handCount;handIdx++){out+=" "+this.hands[handIdx].toString()+"
"}out+="

Pointables:
";for(var pointableIdx=0,pointableCount=this.pointables.length;pointableIdx!=pointableCount;pointableIdx++){out+=" "+this.pointables[pointableIdx].toString()+"
"}if(this.gestures){out+="

Gestures:
";for(var gestureIdx=0,gestureCount=this.gestures.length;gestureIdx!=gestureCount;gestureIdx++){out+=" "+this.gestures[gestureIdx].toString()+"
"}}out+="

Raw JSON:
";out+=JSON.stringify(this.data);return out};Frame.Invalid={valid:false,hands:[],fingers:[],tools:[],gestures:[],pointables:[],pointable:function(){return Pointable.Invalid},finger:function(){return Pointable.Invalid},hand:function(){return Hand.Invalid},toString:function(){return"invalid frame"},dump:function(){return this.toString()},rotationAngle:function(){return 0},rotationMatrix:function(){return mat3.create()},rotationAxis:function(){return vec3.create()},scaleFactor:function(){return 1},translation:function(){return vec3.create()}}},{"./gesture":6,"./hand":7,"./interaction_box":9,"./pointable":11,"gl-matrix":19,underscore:20}],6:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3,EventEmitter=require("events").EventEmitter,_=require("underscore");var createGesture=exports.createGesture=function(data){var gesture;switch(data.type){case"circle":gesture=new CircleGesture(data);break;case"swipe":gesture=new SwipeGesture(data);break;case"screenTap":gesture=new ScreenTapGesture(data);break;case"keyTap":gesture=new KeyTapGesture(data);break;default:throw"unkown gesture type"}gesture.id=data.id;gesture.handIds=data.handIds;gesture.pointableIds=data.pointableIds;gesture.duration=data.duration;gesture.state=data.state;gesture.type=data.type;return gesture};var gestureListener=exports.gestureListener=function(controller,type){var handlers={};var gestureMap={};var gestureCreator=function(){var candidateGesture=gestureMap[gesture.id];if(candidateGesture!==undefined)gesture.update(gesture,frame);if(gesture.state=="start"||gesture.state=="stop"){if(type==gesture.type&&gestureMap[gesture.id]===undefined){gestureMap[gesture.id]=new Gesture(gesture,frame);gesture.update(gesture,frame)}if(gesture.state=="stop"){delete gestureMap[gesture.id]}}};controller.on("gesture",function(gesture,frame){if(gesture.type==type){if(gesture.state=="start"||gesture.state=="stop"){if(gestureMap[gesture.id]===undefined){var gestureTracker=new Gesture(gesture,frame);gestureMap[gesture.id]=gestureTracker;_.each(handlers,function(cb,name){gestureTracker.on(name,cb)})}}gestureMap[gesture.id].update(gesture,frame);if(gesture.state=="stop"){delete gestureMap[gesture.id]}}});var builder={start:function(cb){handlers["start"]=cb;return builder},stop:function(cb){handlers["stop"]=cb;return builder},complete:function(cb){handlers["stop"]=cb;return builder},update:function(cb){handlers["update"]=cb;return builder}};return builder};var Gesture=exports.Gesture=function(gesture,frame){this.gestures=[gesture];this.frames=[frame]};Gesture.prototype.update=function(gesture,frame){this.gestures.push(gesture);this.frames.push(frame);this.emit(gesture.state,this)};_.extend(Gesture.prototype,EventEmitter.prototype);var CircleGesture=function(data){this.center=data.center;this.normal=data.normal;this.progress=data.progress;this.radius=data.radius};CircleGesture.prototype.toString=function(){return"CircleGesture ["+JSON.stringify(this)+"]"};var SwipeGesture=function(data){this.startPosition=data.startPosition;this.position=data.position;this.direction=data.direction;this.speed=data.speed};SwipeGesture.prototype.toString=function(){return"SwipeGesture ["+JSON.stringify(this)+"]"};var ScreenTapGesture=function(data){this.position=data.position;this.direction=data.direction;this.progress=data.progress};ScreenTapGesture.prototype.toString=function(){return"ScreenTapGesture ["+JSON.stringify(this)+"]"};var KeyTapGesture=function(data){this.position=data.position;this.direction=data.direction;this.progress=data.progress};KeyTapGesture.prototype.toString=function(){return"KeyTapGesture ["+JSON.stringify(this)+"]"}},{events:17,"gl-matrix":19,underscore:20}],7:[function(require,module,exports){var Pointable=require("./pointable"),glMatrix=require("gl-matrix"),mat3=glMatrix.mat3,vec3=glMatrix.vec3,_=require("underscore");var Hand=module.exports=function(data){this.id=data.id;this.palmPosition=data.palmPosition;this.direction=data.direction;this.palmVelocity=data.palmVelocity;this.palmNormal=data.palmNormal;this.sphereCenter=data.sphereCenter;this.sphereRadius=data.sphereRadius;this.valid=true;this.pointables=[];this.fingers=[];this.tools=[];this._translation=data.t;this._rotation=_.flatten(data.r);this._scaleFactor=data.s;this.timeVisible=data.timeVisible;this.stabilizedPalmPosition=data.stabilizedPalmPosition};Hand.prototype.finger=function(id){var finger=this.frame.finger(id);return finger&&finger.handId==this.id?finger:Pointable.Invalid};Hand.prototype.rotationAngle=function(sinceFrame,axis){if(!this.valid||!sinceFrame.valid)return 0;var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return 0;var rot=this.rotationMatrix(sinceFrame);var cs=(rot[0]+rot[4]+rot[8]-1)*.5;var angle=Math.acos(cs);angle=isNaN(angle)?0:angle;if(axis!==undefined){var rotAxis=this.rotationAxis(sinceFrame);angle*=vec3.dot(rotAxis,vec3.normalize(vec3.create(),axis))}return angle};Hand.prototype.rotationAxis=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return vec3.create();return vec3.normalize(vec3.create(),[this._rotation[7]-sinceHand._rotation[5],this._rotation[2]-sinceHand._rotation[6],this._rotation[3]-sinceHand._rotation[1]])};Hand.prototype.rotationMatrix=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return mat3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return mat3.create();var transpose=mat3.transpose(mat3.create(),this._rotation);var m=mat3.multiply(mat3.create(),sinceHand._rotation,transpose);return m};Hand.prototype.scaleFactor=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return 1;var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return 1;return Math.exp(this._scaleFactor-sinceHand._scaleFactor)};Hand.prototype.translation=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return vec3.create();return[this._translation[0]-sinceHand._translation[0],this._translation[1]-sinceHand._translation[1],this._translation[2]-sinceHand._translation[2]]};Hand.prototype.toString=function(){return"Hand [ id: "+this.id+" | palm velocity:"+this.palmVelocity+" | sphere center:"+this.sphereCenter+" ] "};Hand.Invalid={valid:false,fingers:[],tools:[],pointables:[],pointable:function(){return Pointable.Invalid},finger:function(){return Pointable.Invalid},toString:function(){return"invalid frame"},dump:function(){return this.toString()},rotationAngle:function(){return 0},rotationMatrix:function(){return mat3.create()},rotationAxis:function(){return vec3.create()},scaleFactor:function(){return 1},translation:function(){return vec3.create()}}},{"./pointable":11,"gl-matrix":19,underscore:20}],8:[function(require,module,exports){!function(){module.exports={Controller:require("./controller"),Frame:require("./frame"),Gesture:require("./gesture"),Hand:require("./hand"),Pointable:require("./pointable"),InteractionBox:require("./interaction_box"),Connection:require("./connection"),CircularBuffer:require("./circular_buffer"),UI:require("./ui"),glMatrix:require("gl-matrix"),mat3:require("gl-matrix").mat3,vec3:require("gl-matrix").vec3,loopController:undefined,loop:function(opts,callback){if(callback===undefined){callback=opts;opts={}}if(!this.loopController)this.loopController=new this.Controller(opts);this.loopController.loop(callback)}}}()},{"./circular_buffer":2,"./connection":3,"./controller":4,"./frame":5,"./gesture":6,"./hand":7,"./interaction_box":9,"./pointable":11,"./ui":13,"gl-matrix":19}],9:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3;var InteractionBox=module.exports=function(data){this.valid=true;this.center=data.center;this.size=data.size;this.width=data.size[0];this.height=data.size[1];this.depth=data.size[2]};InteractionBox.prototype.denormalizePoint=function(normalizedPosition){return vec3.fromValues((normalizedPosition[0]-.5)*this.size[0]+this.center[0],(normalizedPosition[1]-.5)*this.size[1]+this.center[1],(normalizedPosition[2]-.5)*this.size[2]+this.center[2])};InteractionBox.prototype.normalizePoint=function(position,clamp){var vec=vec3.fromValues((position[0]-this.center[0])/this.size[0]+.5,(position[1]-this.center[1])/this.size[1]+.5,(position[2]-this.center[2])/this.size[2]+.5);if(clamp){vec[0]=Math.min(Math.max(vec[0],0),1);vec[1]=Math.min(Math.max(vec[1],0),1);vec[2]=Math.min(Math.max(vec[2],0),1)}return vec};InteractionBox.prototype.toString=function(){return"InteractionBox [ width:"+this.width+" | height:"+this.height+" | depth:"+this.depth+" ]"};InteractionBox.Invalid={valid:false}},{"gl-matrix":19}],10:[function(require,module,exports){var Pipeline=module.exports=function(){this.steps=[]};Pipeline.prototype.addStep=function(step){this.steps.push(step)};Pipeline.prototype.run=function(frame){var stepsLength=this.steps.length;for(var i=0;i!=stepsLength;i++){if(!frame)break;frame=this.steps[i](frame)}return frame}},{}],11:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3;var Pointable=module.exports=function(data){this.valid=true;this.id=data.id;this.handId=data.handId;this.length=data.length;this.tool=data.tool;this.width=data.width;this.direction=data.direction;this.stabilizedTipPosition=data.stabilizedTipPosition;this.tipPosition=data.tipPosition;this.tipVelocity=data.tipVelocity;this.touchZone=data.touchZone;this.touchDistance=data.touchDistance;this.timeVisible=data.timeVisible};Pointable.prototype.toString=function(){if(this.tool==true){return"Pointable [ id:"+this.id+" "+this.length+"mmx | with:"+this.width+"mm | direction:"+this.direction+" ]"}else{return"Pointable [ id:"+this.id+" "+this.length+"mmx | direction: "+this.direction+" ]"}};Pointable.Invalid={valid:false}},{"gl-matrix":19}],12:[function(require,module,exports){var Frame=require("./frame");var Event=function(data){this.type=data.type;this.state=data.state};var chooseProtocol=exports.chooseProtocol=function(header){var protocol;switch(header.version){case 1:protocol=JSONProtocol(1,function(data){return new Frame(data)});break;case 2:protocol=JSONProtocol(2,function(data){return new Frame(data)});protocol.sendHeartbeat=function(connection){connection.send(protocol.encode({heartbeat:true}))};break;case 3:protocol=JSONProtocol(3,function(data){return data.event?new Event(data.event):new Frame(data)});protocol.sendHeartbeat=function(connection){connection.send(protocol.encode({heartbeat:true}))};break;default:throw"unrecognized version"}return protocol};var JSONProtocol=function(version,cb){var protocol=cb;protocol.encode=function(message){return JSON.stringify(message)};protocol.version=version;protocol.versionLong="Version "+version;protocol.type="protocol";return protocol}},{"./frame":5}],13:[function(require,module,exports){exports.UI={Region:require("./ui/region"),Cursor:require("./ui/cursor")}},{"./ui/cursor":14,"./ui/region":15}],14:[function(require,module,exports){var Cursor=module.exports=function(){return function(frame){var pointable=frame.pointables.sort(function(a,b){return a.z-b.z})[0];if(pointable&&pointable.valid){frame.cursorPosition=pointable.tipPosition}return frame}}},{}],15:[function(require,module,exports){var EventEmitter=require("events").EventEmitter,_=require("underscore");var Region=module.exports=function(start,end){this.start=new Vector(start);this.end=new Vector(end);this.enteredFrame=null};Region.prototype.hasPointables=function(frame){for(var i=0;i!=frame.pointables.length;i++){var position=frame.pointables[i].tipPosition;if(position.x>=this.start.x&&position.x<=this.end.x&&position.y>=this.start.y&&position.y<=this.end.y&&position.z>=this.start.z&&position.z<=this.end.z){return true}}return false};Region.prototype.listener=function(opts){var region=this;if(opts&&opts.nearThreshold)this.setupNearRegion(opts.nearThreshold);return function(frame){return region.updatePosition(frame)}};Region.prototype.clipper=function(){var region=this;return function(frame){region.updatePosition(frame);return region.enteredFrame?frame:null}};Region.prototype.setupNearRegion=function(distance){var nearRegion=this.nearRegion=new Region([this.start.x-distance,this.start.y-distance,this.start.z-distance],[this.end.x+distance,this.end.y+distance,this.end.z+distance]);var region=this;nearRegion.on("enter",function(frame){region.emit("near",frame)});nearRegion.on("exit",function(frame){region.emit("far",frame)});region.on("exit",function(frame){region.emit("near",frame)})};Region.prototype.updatePosition=function(frame){if(this.nearRegion)this.nearRegion.updatePosition(frame);if(this.hasPointables(frame)&&this.enteredFrame==null){this.enteredFrame=frame;this.emit("enter",this.enteredFrame)}else if(!this.hasPointables(frame)&&this.enteredFrame!=null){this.enteredFrame=null;this.emit("exit",this.enteredFrame)}return frame};Region.prototype.normalize=function(position){return new Vector([(position.x-this.start.x)/(this.end.x-this.start.x),(position.y-this.start.y)/(this.end.y-this.start.y),(position.z-this.start.z)/(this.end.z-this.start.z)])};Region.prototype.mapToXY=function(position,width,height){var normalized=this.normalize(position);var x=normalized.x,y=normalized.y;if(x>1)x=1;else if(x<-1)x=-1;if(y>1)y=1;else if(y<-1)y=-1;return[(x+1)/2*width,(1-y)/2*height,normalized.z]};_.extend(Region.prototype,EventEmitter.prototype)},{events:17,underscore:20}],16:[function(require,module,exports){},{}],17:[function(require,module,exports){!function(process){if(!process.EventEmitter)process.EventEmitter=function(){};var EventEmitter=exports.EventEmitter=process.EventEmitter;var isArray=typeof Array.isArray==="function"?Array.isArray:function(xs){return Object.prototype.toString.call(xs)==="[object Array]"};function indexOf(xs,x){if(xs.indexOf)return xs.indexOf(x);for(var i=0;i0&&this._events[type].length>m){this._events[type].warned=true;console.error("(node) warning: possible EventEmitter memory "+"leak detected. %d listeners added. "+"Use emitter.setMaxListeners() to increase limit.",this._events[type].length);console.trace()}}this._events[type].push(listener)}else{this._events[type]=[this._events[type],listener]}return this};EventEmitter.prototype.on=EventEmitter.prototype.addListener;EventEmitter.prototype.once=function(type,listener){var self=this;self.on(type,function g(){self.removeListener(type,g);listener.apply(this,arguments)});return this};EventEmitter.prototype.removeListener=function(type,listener){if("function"!==typeof listener){throw new Error("removeListener only takes instances of Function")}if(!this._events||!this._events[type])return this;var list=this._events[type];if(isArray(list)){var i=indexOf(list,listener);if(i<0)return this;list.splice(i,1);if(list.length==0)delete this._events[type]}else if(this._events[type]===listener){delete this._events[type]}return this};EventEmitter.prototype.removeAllListeners=function(type){if(arguments.length===0){this._events={};return this}if(type&&this._events&&this._events[type])this._events[type]=null;return this};EventEmitter.prototype.listeners=function(type){if(!this._events)this._events={};if(!this._events[type])this._events[type]=[];if(!isArray(this._events[type])){this._events[type]=[this._events[type]]}return this._events[type]}}(require("__browserify_process"))},{__browserify_process:18}],18:[function(require,module,exports){var process=module.exports={};process.nextTick=function(){var canSetImmediate=typeof window!=="undefined"&&window.setImmediate;var canPost=typeof window!=="undefined"&&window.postMessage&&window.addEventListener;if(canSetImmediate){return function(f){return window.setImmediate(f)}}if(canPost){var queue=[];window.addEventListener("message",function(ev){if(ev.source===window&&ev.data==="process-tick"){ev.stopPropagation();if(queue.length>0){var fn=queue.shift();fn()}}},true);return function nextTick(fn){queue.push(fn);window.postMessage("process-tick","*")}}return function nextTick(fn){setTimeout(fn,0)}}();process.title="browser";process.browser=true;process.env={};process.argv=[];process.binding=function(name){throw new Error("process.binding is not supported")};process.cwd=function(){return"/"};process.chdir=function(dir){throw new Error("process.chdir is not supported")}},{}],19:[function(require,module,exports){!function(){!function(){"use strict";var shim={};if(typeof exports==="undefined"){if(typeof define=="function"&&typeof define.amd=="object"&&define.amd){shim.exports={};define(function(){return shim.exports})}else{shim.exports=window}}else{shim.exports=exports}!function(exports){var vec2={};if(!GLMAT_EPSILON){var GLMAT_EPSILON=1e-6}vec2.create=function(){return new Float32Array(2)};vec2.clone=function(a){var out=new Float32Array(2);out[0]=a[0];out[1]=a[1];return out};vec2.fromValues=function(x,y){var out=new Float32Array(2);out[0]=x;out[1]=y;return out};vec2.copy=function(out,a){out[0]=a[0];out[1]=a[1];return out};vec2.set=function(out,x,y){out[0]=x;out[1]=y;return out};vec2.add=function(out,a,b){out[0]=a[0]+b[0];out[1]=a[1]+b[1];return out};vec2.sub=vec2.subtract=function(out,a,b){out[0]=a[0]-b[0];out[1]=a[1]-b[1];return out};vec2.mul=vec2.multiply=function(out,a,b){out[0]=a[0]*b[0];out[1]=a[1]*b[1];return out};vec2.div=vec2.divide=function(out,a,b){out[0]=a[0]/b[0];out[1]=a[1]/b[1];return out};vec2.min=function(out,a,b){out[0]=Math.min(a[0],b[0]); +out[1]=Math.min(a[1],b[1]);return out};vec2.max=function(out,a,b){out[0]=Math.max(a[0],b[0]);out[1]=Math.max(a[1],b[1]);return out};vec2.scale=function(out,a,b){out[0]=a[0]*b;out[1]=a[1]*b;return out};vec2.dist=vec2.distance=function(a,b){var x=b[0]-a[0],y=b[1]-a[1];return Math.sqrt(x*x+y*y)};vec2.sqrDist=vec2.squaredDistance=function(a,b){var x=b[0]-a[0],y=b[1]-a[1];return x*x+y*y};vec2.len=vec2.length=function(a){var x=a[0],y=a[1];return Math.sqrt(x*x+y*y)};vec2.sqrLen=vec2.squaredLength=function(a){var x=a[0],y=a[1];return x*x+y*y};vec2.negate=function(out,a){out[0]=-a[0];out[1]=-a[1];return out};vec2.normalize=function(out,a){var x=a[0],y=a[1];var len=x*x+y*y;if(len>0){len=1/Math.sqrt(len);out[0]=a[0]*len;out[1]=a[1]*len}return out};vec2.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]};vec2.cross=function(out,a,b){var z=a[0]*b[1]-a[1]*b[0];out[0]=out[1]=0;out[2]=z;return out};vec2.lerp=function(out,a,b,t){var ax=a[0],ay=a[1];out[0]=ax+t*(b[0]-ax);out[1]=ay+t*(b[1]-ay);return out};vec2.transformMat2=function(out,a,m){var x=a[0],y=a[1];out[0]=x*m[0]+y*m[1];out[1]=x*m[2]+y*m[3];return out};vec2.forEach=function(){var vec=new Float32Array(2);return function(a,stride,offset,count,fn,arg){var i,l;if(!stride){stride=2}if(!offset){offset=0}if(count){l=Math.min(count*stride+offset,a.length)}else{l=a.length}for(i=offset;i0){len=1/Math.sqrt(len);out[0]=a[0]*len;out[1]=a[1]*len;out[2]=a[2]*len}return out};vec3.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]+a[2]*b[2]};vec3.cross=function(out,a,b){var ax=a[0],ay=a[1],az=a[2],bx=b[0],by=b[1],bz=b[2];out[0]=ay*bz-az*by;out[1]=az*bx-ax*bz;out[2]=ax*by-ay*bx;return out};vec3.lerp=function(out,a,b,t){var ax=a[0],ay=a[1],az=a[2];out[0]=ax+t*(b[0]-ax);out[1]=ay+t*(b[1]-ay);out[2]=az+t*(b[2]-az);return out};vec3.transformMat4=function(out,a,m){var x=a[0],y=a[1],z=a[2];out[0]=m[0]*x+m[4]*y+m[8]*z+m[12];out[1]=m[1]*x+m[5]*y+m[9]*z+m[13];out[2]=m[2]*x+m[6]*y+m[10]*z+m[14];return out};vec3.transformQuat=function(out,a,q){var x=a[0],y=a[1],z=a[2],qx=q[0],qy=q[1],qz=q[2],qw=q[3],ix=qw*x+qy*z-qz*y,iy=qw*y+qz*x-qx*z,iz=qw*z+qx*y-qy*x,iw=-qx*x-qy*y-qz*z;out[0]=ix*qw+iw*-qx+iy*-qz-iz*-qy;out[1]=iy*qw+iw*-qy+iz*-qx-ix*-qz;out[2]=iz*qw+iw*-qz+ix*-qy-iy*-qx;return out};vec3.forEach=function(){var vec=new Float32Array(3);return function(a,stride,offset,count,fn,arg){var 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Motion integration for Reveal.js. + * James Sun [sun16] + * Rory Hardy [gneatgeek] + */ + +(function () { + var body = document.body, + controller = new Leap.Controller({ enableGestures: true }), + lastGesture = 0, + leapConfig = Reveal.getConfig().leap, + pointer = document.createElement( 'div' ), + config = { + autoCenter : true, // Center pointer around detected position. + gestureDelay : 500, // How long to delay between gestures. + naturalSwipe : true, // Swipe as if it were a touch screen. + pointerColor : '#00aaff', // Default color of the pointer. + pointerOpacity : 0.7, // Default opacity of the pointer. + pointerSize : 15, // Default minimum height/width of the pointer. + pointerTolerance : 120 // Bigger = slower pointer. + }, + entered, enteredPosition, now, size, tipPosition; // Other vars we need later, but don't need to redeclare. + + // Merge user defined settings with defaults + if( leapConfig ) { + for( key in leapConfig ) { + config[key] = leapConfig[key]; + } + } + + 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Reveal.left() : Reveal.right(); + } + else { + config.naturalSwipe ? Reveal.right() : Reveal.left(); + } + } + // Up/down swipe gestures + else { + if( y > 0 ) { + config.naturalSwipe ? Reveal.down() : Reveal.up(); + } + else { + config.naturalSwipe ? Reveal.up() : Reveal.down(); + } + } + + lastGesture = now; + } + } + // Two hand gestures + else if( frame.hands.length === 2 ) { + // Upward two hand swipe gesture + if( gesture.type === 'swipe' && gesture.direction[1] > 0 ) { + Reveal.toggleOverview(); + } + + lastGesture = now; + } + } + }); + + controller.connect(); +})(); diff --git a/doc/pub/DimRed/html/reveal.js/plugin/remotes/remotes.js b/doc/pub/DimRed/html/reveal.js/plugin/remotes/remotes.js new file mode 100644 index 000000000..ba0dbad7b --- /dev/null +++ b/doc/pub/DimRed/html/reveal.js/plugin/remotes/remotes.js @@ -0,0 +1,39 @@ +/** + * Touch-based remote controller for your presentation courtesy + * of the folks at http://remotes.io + */ + +(function(window){ + + /** + * Detects if we are dealing with a touch enabled device (with some false positives) + * Borrowed from modernizr: https://github.com/Modernizr/Modernizr/blob/master/feature-detects/touch.js + */ + var hasTouch = (function(){ + return ('ontouchstart' in window) || window.DocumentTouch && document instanceof DocumentTouch; + })(); + + /** + * Detects if notes are enable and the current page is opened inside an /iframe + * this prevents loading Remotes.io several times + */ + var isNotesAndIframe = (function(){ + return window.RevealNotes && !(self == top); + })(); + + if(!hasTouch && !isNotesAndIframe){ + head.ready( 'remotes.ne.min.js', function() { + new Remotes("preview") + .on("swipe-left", function(e){ Reveal.right(); }) + .on("swipe-right", function(e){ Reveal.left(); }) + .on("swipe-up", function(e){ Reveal.down(); }) + .on("swipe-down", function(e){ Reveal.up(); }) + .on("tap", function(e){ Reveal.next(); }) + .on("zoom-out", function(e){ Reveal.toggleOverview(true); }) + .on("zoom-in", function(e){ Reveal.toggleOverview(false); }) + ; + } ); + + head.js('https://hakim-static.s3.amazonaws.com/reveal-js/remotes.ne.min.js'); + } +})(window); \ No newline at end of file diff --git a/doc/pub/DimRed/ipynb/.ipynb_checkpoints/DimRed-checkpoint.ipynb b/doc/pub/DimRed/ipynb/.ipynb_checkpoints/DimRed-checkpoint.ipynb new file mode 100644 index 000000000..7b001ca63 --- /dev/null +++ b/doc/pub/DimRed/ipynb/.ipynb_checkpoints/DimRed-checkpoint.ipynb @@ -0,0 +1,606 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Oct 14, 2019**\n", + "\n", + "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "## Reducing the number of degrees of freedom, overarching view\n", + "\n", + "Many Machine Learning problems involve thousands or even millions of features for each training\n", + "instance. Not only does this make training extremely slow, it can also make it much harder to find a good\n", + "solution, as we will see. This problem is often referred to as the curse of dimensionality.\n", + "Fortunately, in real-world problems, it is often possible to reduce the number of features considerably,\n", + "turning an intractable problem into a tractable one.\n", + "\n", + "Here we will discuss some of the most popular dimensionality\n", + "reduction techniques: the principal component analysis PCA, Kernel PCA, and Locally Linear Embedding (LLE).\n", + "\n", + "\n", + "\n", + "\n", + "## Preprocessing our data\n", + "\n", + "Before we proceed however, we will discuss how to preprocess our\n", + "data. Till now and in connection with our previous examples we have not met so many cases\n", + "where we are too sensitive to the scaling of our data. Normally the\n", + "data may need a rescaling and/or may be sensitive to extreme\n", + "values. Scaling the data renders our inputs much more suitable for the\n", + "algorithms we want to employ.\n", + "\n", + "**Scikit-Learn** has several functions which allow us to rescale the data, normally resulting in much better results in terms of various accuracy scores. The **StandardScaler** function in **Scikit-Learn** ensures that for each feature/predictor we study the mean value is zero and the variance is zero (every column in the design/feature matrix).\n", + "This scaling has the drawback that it does not ensure that we have a particular maximum or minumum in our data set. Another function included in **Scikit-Learn** is the **MinMaxScaler** which ensures that all features are exactly between $0$ and $1$. The **Normalizer** function scales each column of the design matrix by its Euclidean norm.\n", + "\n", + "\n", + "\n", + "## Simple preprocessing examples, Franke function and regression" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "MSE before scaling: 0.01\n", + "R2 score before scaling 0.94\n", + "Feature min values before scaling:\n", + " [1.00000000e+00 3.65071692e-04 3.95817540e-04 1.33277341e-07\n", + " 1.44501779e-07 1.56671525e-07 4.86557843e-11 5.27535091e-11\n", + " 5.71963387e-11 6.20133374e-11 1.77628495e-14 1.92588128e-14\n", + " 2.08807642e-14 2.26393141e-14 2.45459667e-14 6.48471354e-18\n", + " 7.03084740e-18 7.62297591e-18 8.26497270e-18 8.96103759e-18\n", + " 9.71572413e-18]\n", + "Feature max values before scaling:\n", + " [1. 0.99846102 0.99865465 0.99692442 0.99711774 0.99731111\n", + " 0.99539017 0.9955832 0.99577627 0.99596937 0.99385829 0.99405102\n", + " 0.99424379 0.9944366 0.99462944 0.99232877 0.9925212 0.99271367\n", + " 0.99290618 0.99309873 0.99329132]\n", + "Feature min values after scaling:\n", + " [ 0. -1.68592363 -1.680662 -1.10811004 -1.09082494 -1.07436341\n", + " -0.87879986 -0.86810747 -0.85783478 -0.84797797 -0.74808585 -0.74211553\n", + " -0.73635781 -0.73080352 -0.72544375 -0.6608814 -0.65780307 -0.65482349\n", + " -0.65193637 -0.64913573 -0.64641588]\n", + "Feature max values after scaling:\n", + " [0. 1.72566977 1.72438202 2.24728818 2.21480932 2.18128191\n", + " 2.66918481 2.62627499 2.5831825 2.53999114 3.03235352 2.9851254\n", + " 2.93793971 2.89083203 2.84383422 3.35451279 3.30554819 3.25663591\n", + " 3.20779653 3.15904953 3.11041347]\n", + "MSE after scaling: 0.00\n", + "R2 score for scaled data: 0.97\n" + ] + } + ], + "source": [ + "%matplotlib inline\n", + "\n", + "# Common imports\n", + "import os\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "import sklearn.linear_model as skl\n", + "from sklearn.metrics import mean_squared_error\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer\n", + "from sklearn.svm import SVR\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "\n", + "def FrankeFunction(x,y):\n", + "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", + "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", + "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", + "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", + "\treturn term1 + term2 + term3 + term4\n", + "\n", + "\n", + "def create_X(x, y, n ):\n", + "\tif len(x.shape) > 1:\n", + "\t\tx = np.ravel(x)\n", + "\t\ty = np.ravel(y)\n", + "\n", + "\tN = len(x)\n", + "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", + "\tX = np.ones((N,l))\n", + "\n", + "\tfor i in range(1,n+1):\n", + "\t\tq = int((i)*(i+1)/2)\n", + "\t\tfor k in range(i+1):\n", + "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", + "\n", + "\treturn X\n", + "\n", + "\n", + "# Making meshgrid of datapoints and compute Franke's function\n", + "n = 5\n", + "N = 1000\n", + "x = np.sort(np.random.uniform(0, 1, N))\n", + "y = np.sort(np.random.uniform(0, 1, N))\n", + "z = FrankeFunction(x, y)\n", + "X = create_X(x, y, n=n) \n", + "# split in training and test data\n", + "X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=0.2)\n", + "\n", + "\n", + "svm = SVR(gamma='auto',C=10.0)\n", + "svm.fit(X_train, y_train)\n", + "\n", + "# The mean squared error and R2 score\n", + "print(\"MSE before scaling: {:.2f}\".format(mean_squared_error(svm.predict(X_test), y_test)))\n", + "print(\"R2 score before scaling {:.2f}\".format(svm.score(X_test,y_test)))\n", + "\n", + "scaler = StandardScaler()\n", + "scaler.fit(X_train)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "\n", + "print(\"Feature min values before scaling:\\n {}\".format(X_train.min(axis=0)))\n", + "print(\"Feature max values before scaling:\\n {}\".format(X_train.max(axis=0)))\n", + "\n", + "print(\"Feature min values after scaling:\\n {}\".format(X_train_scaled.min(axis=0)))\n", + "print(\"Feature max values after scaling:\\n {}\".format(X_train_scaled.max(axis=0)))\n", + "\n", + "svm = SVR(gamma='auto',C=10.0)\n", + "svm.fit(X_train_scaled, y_train)\n", + "\n", + "print(\"MSE after scaling: {:.2f}\".format(mean_squared_error(svm.predict(X_test_scaled), y_test)))\n", + "print(\"R2 score for scaled data: {:.2f}\".format(svm.score(X_test_scaled,y_test)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Simple preprocessing examples, breast cancer data and classification\n", + "\n", + "We show here how we can use a simple regression case on the breast cancer data using support vector machine as algorithm for classification" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(426, 30)\n", + "(143, 30)\n", + "Test set accuracy: 0.63\n", + "Feature min values before scaling:\n", + " [6.981e+00 9.710e+00 4.379e+01 1.435e+02 5.263e-02 1.938e-02 0.000e+00\n", + " 0.000e+00 1.060e-01 4.996e-02 1.115e-01 3.628e-01 7.570e-01 7.228e+00\n", + " 1.713e-03 2.252e-03 0.000e+00 0.000e+00 7.882e-03 8.948e-04 7.930e+00\n", + " 1.202e+01 5.041e+01 1.852e+02 7.117e-02 2.729e-02 0.000e+00 0.000e+00\n", + " 1.565e-01 5.504e-02]\n", + "Feature max values before scaling:\n", + " [2.811e+01 3.381e+01 1.885e+02 2.501e+03 1.447e-01 3.114e-01 4.268e-01\n", + " 2.012e-01 3.040e-01 9.744e-02 2.873e+00 4.885e+00 2.198e+01 5.422e+02\n", + " 2.333e-02 1.064e-01 3.960e-01 5.279e-02 6.146e-02 2.984e-02 3.604e+01\n", + " 4.954e+01 2.512e+02 4.254e+03 2.226e-01 1.058e+00 1.252e+00 2.903e-01\n", + " 6.638e-01 2.075e-01]\n", + "Feature min values before scaling:\n", + " [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", + " 0. 0. 0. 0. 0. 0.]\n", + "Feature max values before scaling:\n", + " [1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.\n", + " 1. 1. 1. 1. 1. 1.]\n", + "Test set accuracy scaled data: 0.97\n", + "Test set accuracy scaled data: 0.96\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/usr/local/lib/python3.7/site-packages/sklearn/svm/base.py:196: FutureWarning: The default value of gamma will change from 'auto' to 'scale' in version 0.22 to account better for unscaled features. Set gamma explicitly to 'auto' or 'scale' to avoid this warning.\n", + " \"avoid this warning.\", FutureWarning)\n", + "/usr/local/lib/python3.7/site-packages/sklearn/svm/base.py:196: FutureWarning: The default value of gamma will change from 'auto' to 'scale' in version 0.22 to account better for unscaled features. Set gamma explicitly to 'auto' or 'scale' to avoid this warning.\n", + " \"avoid this warning.\", FutureWarning)\n" + ] + } + ], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.model_selection import train_test_split \n", + "from sklearn.datasets import load_breast_cancer\n", + "from sklearn.svm import SVC\n", + "cancer = load_breast_cancer()\n", + "\n", + "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", + "print(X_train.shape)\n", + "print(X_test.shape)\n", + "\n", + "svm = SVC(C=100)\n", + "svm.fit(X_train, y_train)\n", + "print(\"Test set accuracy: {:.2f}\".format(svm.score(X_test,y_test)))\n", + "\n", + "from sklearn.preprocessing import MinMaxScaler, StandardScaler\n", + "\n", + "scaler = MinMaxScaler()\n", + "scaler.fit(X_train)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "\n", + "print(\"Feature min values before scaling:\\n {}\".format(X_train.min(axis=0)))\n", + "print(\"Feature max values before scaling:\\n {}\".format(X_train.max(axis=0)))\n", + "\n", + "print(\"Feature min values before scaling:\\n {}\".format(X_train_scaled.min(axis=0)))\n", + "print(\"Feature max values before scaling:\\n {}\".format(X_train_scaled.max(axis=0)))\n", + "\n", + "\n", + "svm.fit(X_train_scaled, y_train)\n", + "print(\"Test set accuracy scaled data: {:.2f}\".format(svm.score(X_test_scaled,y_test)))\n", + "\n", + "scaler = StandardScaler()\n", + "scaler.fit(X_train)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "\n", + "svm.fit(X_train_scaled, y_train)\n", + "print(\"Test set accuracy scaled data: {:.2f}\".format(svm.score(X_test_scaled,y_test)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## More on Cancer Data" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + 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+ "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(426, 30)\n", + "(143, 30)\n", + "Test set accuracy: 0.96\n", + "Test set accuracy scaled data: 0.96\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/usr/local/lib/python3.7/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.\n", + " FutureWarning)\n", + "/usr/local/lib/python3.7/site-packages/sklearn/linear_model/logistic.py:432: FutureWarning: Default solver will be changed to 'lbfgs' in 0.22. Specify a solver to silence this warning.\n", + " FutureWarning)\n" + ] + } + ], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.model_selection import train_test_split \n", + "from sklearn.datasets import load_breast_cancer\n", + "from sklearn.svm import SVC\n", + "from sklearn.linear_model import LogisticRegression\n", + "cancer = load_breast_cancer()\n", + "\n", + "fig, axes = plt.subplots(15,2,figsize=(10,20))\n", + "male = cancer.data[cancer.target == 0]\n", + "bene = cancer.data[cancer.target == 1]\n", + "ax = axes.ravel()\n", + "\n", + "for i in range(30):\n", + " _, bins = np.histogram(cancer.data[:,i], bins =50)\n", + " ax[i].hist(male[:,i], bins = bins, alpha = 0.5)\n", + " ax[i].hist(bene[:,i], bins = bins, alpha = 0.5)\n", + " ax[i].set_title(cancer.feature_names[i])\n", + " ax[i].set_yticks(())\n", + "ax[0].set_xlabel(\"Feature magnitude\")\n", + "ax[0].set_ylabel(\"Frequency\")\n", + "ax[0].legend([\"Male\", \"Bene\"], loc =\"best\")\n", + "fig.tight_layout()\n", + "plt.show()\n", + "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", + "print(X_train.shape)\n", + "print(X_test.shape)\n", + "\n", + "logreg = LogisticRegression()\n", + "logreg.fit(X_train, y_train)\n", + "\n", + "print(\"Test set accuracy: {:.2f}\".format(logreg.score(X_test,y_test)))\n", + "\n", + "from sklearn.preprocessing import MinMaxScaler, StandardScaler\n", + "\n", + "\n", + "scaler = StandardScaler()\n", + "scaler.fit(X_train)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "logreg.fit(X_train_scaled, y_train)\n", + "#svm.fit(X_train_scaled, y_train)\n", + "print(\"Test set accuracy scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Principal Component Analysis\n", + "Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.\n", + "First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.\n", + "\n", + "The following Python code uses NumPy’s **svd()** function to obtain all the principal components of the\n", + "training set, then extracts the first two principal components" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "X_centered = X - X.mean(axis=0)\n", + "U, s, V = np.linalg.svd(X_centered)\n", + "c1 = V.T[:, 0]\n", + "c2 = V.T[:, 1]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering\n", + "the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t\n", + "forget to center the data first.\n", + "\n", + "Once you have identified all the principal components, you can reduce the dimensionality of the dataset\n", + "down to $d$ dimensions by projecting it onto the hyperplane defined by the first $d$ principal components.\n", + "Selecting this hyperplane ensures that the projection will preserve as much variance as possible." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "W2 = V.T[:, :2]\n", + "X2D = X_centered.dot(W2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## PCA and scikit-learn\n", + "\n", + "Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The\n", + "following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note\n", + "that it automatically takes care of centering the data):" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.decomposition import PCA\n", + "pca = PCA(n_components = 2)\n", + "X2D = pca.fit_transform(X)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "After fitting the PCA transformer to the dataset, you can access the principal components using the\n", + "components variable (note that it contains the PCs as horizontal vectors, so, for example, the first\n", + "principal component is equal to" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "pca.components_.T[:, 0])." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Another very useful piece of information is the explained variance ratio of each principal component,\n", + "available via the $explained\\_variance\\_ratio$ variable. It indicates the proportion of the dataset’s\n", + "variance that lies along the axis of each principal component. \n", + "More material to come here.\n", + "\n", + "## More on the PCA\n", + "Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to\n", + "choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).\n", + "Unless, of course, you are reducing dimensionality for data visualization — in that case you will\n", + "generally want to reduce the dimensionality down to 2 or 3.\n", + "The following code computes PCA without reducing dimensionality, then computes the minimum number\n", + "of dimensions required to preserve 95% of the training set’s variance:" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "pca = PCA()\n", + "pca.fit(X)\n", + "cumsum = np.cumsum(pca.explained_variance_ratio_)\n", + "d = np.argmax(cumsum >= 0.95) + 1" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "You could then set $n\\_components=d$ and run PCA again. However, there is a much better option: instead\n", + "of specifying the number of principal components you want to preserve, you can set $n\\_components$ to be\n", + "a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "pca = PCA(n_components=0.95)\n", + "X_reduced = pca.fit_transform(X)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Incremental PCA\n", + "One problem with the preceding implementation of PCA is that it requires the whole training set to fit in\n", + "memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have\n", + "been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch\n", + "at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new\n", + "instances arrive).\n", + "\n", + "## Randomized PCA\n", + "\n", + "Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic\n", + "algorithm that quickly finds an approximation of the first d principal components. Its computational\n", + "complexity is $O(m \\times d^2)+O(d^3)$, instead of $O(m \\times n^2) + O(n^3)$, so it is dramatically faster than the\n", + "previous algorithms when $d$ is much smaller than $n$.\n", + "\n", + "\n", + "\n", + "\n", + "## Kernel PCA\n", + "\n", + "The kernel trick is a mathematical technique that implicitly maps instances into a\n", + "very high-dimensional space (called the feature space), enabling nonlinear classification and regression\n", + "with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature\n", + "space corresponds to a complex nonlinear decision boundary in the original space.\n", + "It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear\n", + "projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at\n", + "preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a\n", + "twisted manifold.\n", + "For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.decomposition import KernelPCA\n", + "rbf_pca = KernelPCA(n_components = 2, kernel=\"rbf\", gamma=0.04)\n", + "X_reduced = rbf_pca.fit_transform(X)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## LLE\n", + "\n", + "Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction\n", + "(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous\n", + "algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its\n", + "closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where\n", + "these local relationships are best preserved (more details shortly). \n", + "\n", + "\n", + "\n", + "## Other techniques\n", + "\n", + "\n", + "There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.\n", + "\n", + "Here are some of the most popular:\n", + "* **Multidimensional Scaling (MDS)** reduces dimensionality while trying to preserve the distances between the instances.\n", + "\n", + "* **Isomap** creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.\n", + "\n", + "* **t-Distributed Stochastic Neighbor Embedding** (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).\n", + "\n", + "* Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.4" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/doc/pub/How2ReadData/ipynb/Results/FigureFiles/Masses2016.png b/doc/pub/How2ReadData/ipynb/Results/FigureFiles/Masses2016.png new file mode 100644 index 000000000..780dd5514 Binary files /dev/null and b/doc/pub/How2ReadData/ipynb/Results/FigureFiles/Masses2016.png differ diff --git a/doc/pub/How2ReadData/ipynb/Results/FigureFiles/Masses2016Trees.png b/doc/pub/How2ReadData/ipynb/Results/FigureFiles/Masses2016Trees.png new file mode 100644 index 000000000..ed302c2f7 Binary files /dev/null and b/doc/pub/How2ReadData/ipynb/Results/FigureFiles/Masses2016Trees.png differ diff --git a/doc/pub/LogReg/html/reveal.js/plugin/leap/leap.js b/doc/pub/LogReg/html/reveal.js/plugin/leap/leap.js new file mode 100644 index 000000000..48084ffb0 --- /dev/null +++ b/doc/pub/LogReg/html/reveal.js/plugin/leap/leap.js @@ -0,0 +1,159 @@ +/* + * Copyright (c) 2013, Leap Motion, Inc. + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: + * + * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. + * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Version 0.2.0 - http://js.leapmotion.com/0.2.0/leap.min.js + * Grab latest versions from http://js.leapmotion.com/ + */ + +!function(e,t,n){function i(n,s){if(!t[n]){if(!e[n]){var o=typeof require=="function"&&require;if(!s&&o)return o(n,!0);if(r)return r(n,!0);throw new Error("Cannot find module '"+n+"'")}var u=t[n]={exports:{}};e[n][0].call(u.exports,function(t){var r=e[n][1][t];return i(r?r:t)},u,u.exports)}return t[n].exports}var r=typeof require=="function"&&require;for(var s=0;s=this.size)return undefined;if(i>=this._buf.length)return undefined;return this._buf[(this.pos-i-1)%this.size]};CircularBuffer.prototype.push=function(o){this._buf[this.pos%this.size]=o;return this.pos++}},{}],3:[function(require,module,exports){var Connection=module.exports=require("./base_connection");Connection.prototype.setupSocket=function(){var connection=this;var socket=new WebSocket(this.getUrl());socket.onopen=function(){connection.handleOpen()};socket.onmessage=function(message){connection.handleData(message.data)};socket.onclose=function(){connection.handleClose()};return socket};Connection.prototype.startHeartbeat=function(){if(!this.protocol.sendHeartbeat||this.heartbeatTimer)return;var connection=this;var propertyName=null;if(typeof document.hidden!=="undefined"){propertyName="hidden"}else if(typeof document.mozHidden!=="undefined"){propertyName="mozHidden"}else if(typeof document.msHidden!=="undefined"){propertyName="msHidden"}else if(typeof document.webkitHidden!=="undefined"){propertyName="webkitHidden"}else{propertyName=undefined}var windowVisible=true;var focusListener=window.addEventListener("focus",function(e){windowVisible=true});var blurListener=window.addEventListener("blur",function(e){windowVisible=false});this.on("disconnect",function(){if(connection.heartbeatTimer){clearTimeout(connection.heartbeatTimer);delete connection.heartbeatTimer}window.removeEventListener(focusListener);window.removeEventListener(blurListener)});this.heartbeatTimer=setInterval(function(){var isVisible=propertyName===undefined?true:document[propertyName]===false;if(isVisible&&windowVisible){connection.sendHeartbeat()}else{connection.setHeartbeatState(false)}},this.opts.heartbeatInterval)}},{"./base_connection":1}],4:[function(require,module,exports){!function(process){var Frame=require("./frame"),CircularBuffer=require("./circular_buffer"),Pipeline=require("./pipeline"),EventEmitter=require("events").EventEmitter,gestureListener=require("./gesture").gestureListener,_=require("underscore");var Controller=module.exports=function(opts){var inNode=typeof process!=="undefined"&&process.title==="node";opts=_.defaults(opts||{},{inNode:inNode});this.inNode=opts.inNode;opts=_.defaults(opts||{},{frameEventName:this.useAnimationLoop()?"animationFrame":"deviceFrame",supressAnimationLoop:false});this.supressAnimationLoop=opts.supressAnimationLoop;this.frameEventName=opts.frameEventName;this.history=new CircularBuffer(200);this.lastFrame=Frame.Invalid;this.lastValidFrame=Frame.Invalid;this.lastConnectionFrame=Frame.Invalid;this.accumulatedGestures=[];if(opts.connectionType===undefined){this.connectionType=this.inBrowser()?require("./connection"):require("./node_connection")}else{this.connectionType=opts.connectionType}this.connection=new this.connectionType(opts);this.setupConnectionEvents()};Controller.prototype.gesture=function(type,cb){var creator=gestureListener(this,type);if(cb!==undefined){creator.stop(cb)}return creator};Controller.prototype.inBrowser=function(){return!this.inNode};Controller.prototype.useAnimationLoop=function(){return this.inBrowser()&&typeof chrome==="undefined"};Controller.prototype.connect=function(){var controller=this;if(this.connection.connect()&&this.inBrowser()&&!controller.supressAnimationLoop){var callback=function(){controller.emit("animationFrame",controller.lastConnectionFrame);window.requestAnimFrame(callback)};window.requestAnimFrame(callback)}};Controller.prototype.disconnect=function(){this.connection.disconnect()};Controller.prototype.frame=function(num){return this.history.get(num)||Frame.Invalid};Controller.prototype.loop=function(callback){switch(callback.length){case 1:this.on(this.frameEventName,callback);break;case 2:var controller=this;var scheduler=null;var immediateRunnerCallback=function(frame){callback(frame,function(){if(controller.lastFrame!=frame){immediateRunnerCallback(controller.lastFrame)}else{controller.once(controller.frameEventName,immediateRunnerCallback)}})};this.once(this.frameEventName,immediateRunnerCallback);break}this.connect()};Controller.prototype.addStep=function(step){if(!this.pipeline)this.pipeline=new Pipeline(this);this.pipeline.addStep(step)};Controller.prototype.processFrame=function(frame){if(frame.gestures){this.accumulatedGestures=this.accumulatedGestures.concat(frame.gestures)}if(this.pipeline){frame=this.pipeline.run(frame);if(!frame)frame=Frame.Invalid}this.lastConnectionFrame=frame;this.emit("deviceFrame",frame)};Controller.prototype.processFinishedFrame=function(frame){this.lastFrame=frame;if(frame.valid){this.lastValidFrame=frame}frame.controller=this;frame.historyIdx=this.history.push(frame);if(frame.gestures){frame.gestures=this.accumulatedGestures;this.accumulatedGestures=[];for(var gestureIdx=0;gestureIdx!=frame.gestures.length;gestureIdx++){this.emit("gesture",frame.gestures[gestureIdx],frame)}}this.emit("frame",frame)};Controller.prototype.setupConnectionEvents=function(){var controller=this;this.connection.on("frame",function(frame){controller.processFrame(frame)});this.on(this.frameEventName,function(frame){controller.processFinishedFrame(frame)});this.connection.on("disconnect",function(){controller.emit("disconnect")});this.connection.on("ready",function(){controller.emit("ready")});this.connection.on("connect",function(){controller.emit("connect")});this.connection.on("focus",function(){controller.emit("focus")});this.connection.on("blur",function(){controller.emit("blur")});this.connection.on("protocol",function(protocol){controller.emit("protocol",protocol)});this.connection.on("deviceConnect",function(evt){controller.emit(evt.state?"deviceConnected":"deviceDisconnected")})};_.extend(Controller.prototype,EventEmitter.prototype)}(require("__browserify_process"))},{"./circular_buffer":2,"./connection":3,"./frame":5,"./gesture":6,"./node_connection":16,"./pipeline":10,__browserify_process:18,events:17,underscore:20}],5:[function(require,module,exports){var Hand=require("./hand"),Pointable=require("./pointable"),createGesture=require("./gesture").createGesture,glMatrix=require("gl-matrix"),mat3=glMatrix.mat3,vec3=glMatrix.vec3,InteractionBox=require("./interaction_box"),_=require("underscore");var Frame=module.exports=function(data){this.valid=true;this.id=data.id;this.timestamp=data.timestamp;this.hands=[];this.handsMap={};this.pointables=[];this.tools=[];this.fingers=[];if(data.interactionBox){this.interactionBox=new InteractionBox(data.interactionBox)}this.gestures=[];this.pointablesMap={};this._translation=data.t;this._rotation=_.flatten(data.r);this._scaleFactor=data.s;this.data=data;this.type="frame";this.currentFrameRate=data.currentFrameRate;var handMap={};for(var handIdx=0,handCount=data.hands.length;handIdx!=handCount;handIdx++){var hand=new Hand(data.hands[handIdx]);hand.frame=this;this.hands.push(hand);this.handsMap[hand.id]=hand;handMap[hand.id]=handIdx}for(var pointableIdx=0,pointableCount=data.pointables.length;pointableIdx!=pointableCount;pointableIdx++){var pointable=new Pointable(data.pointables[pointableIdx]);pointable.frame=this;this.pointables.push(pointable);this.pointablesMap[pointable.id]=pointable;(pointable.tool?this.tools:this.fingers).push(pointable);if(pointable.handId!==undefined&&handMap.hasOwnProperty(pointable.handId)){var hand=this.hands[handMap[pointable.handId]];hand.pointables.push(pointable);(pointable.tool?hand.tools:hand.fingers).push(pointable)}}if(data.gestures){for(var gestureIdx=0,gestureCount=data.gestures.length;gestureIdx!=gestureCount;gestureIdx++){this.gestures.push(createGesture(data.gestures[gestureIdx]))}}};Frame.prototype.tool=function(id){var pointable=this.pointable(id);return pointable.tool?pointable:Pointable.Invalid};Frame.prototype.pointable=function(id){return this.pointablesMap[id]||Pointable.Invalid};Frame.prototype.finger=function(id){var pointable=this.pointable(id);return!pointable.tool?pointable:Pointable.Invalid};Frame.prototype.hand=function(id){return this.handsMap[id]||Hand.Invalid};Frame.prototype.rotationAngle=function(sinceFrame,axis){if(!this.valid||!sinceFrame.valid)return 0;var rot=this.rotationMatrix(sinceFrame);var cs=(rot[0]+rot[4]+rot[8]-1)*.5;var angle=Math.acos(cs);angle=isNaN(angle)?0:angle;if(axis!==undefined){var rotAxis=this.rotationAxis(sinceFrame);angle*=vec3.dot(rotAxis,vec3.normalize(vec3.create(),axis))}return angle};Frame.prototype.rotationAxis=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();return vec3.normalize(vec3.create(),[this._rotation[7]-sinceFrame._rotation[5],this._rotation[2]-sinceFrame._rotation[6],this._rotation[3]-sinceFrame._rotation[1]])};Frame.prototype.rotationMatrix=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return mat3.create();var transpose=mat3.transpose(mat3.create(),this._rotation);return mat3.multiply(mat3.create(),sinceFrame._rotation,transpose)};Frame.prototype.scaleFactor=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return 1;return Math.exp(this._scaleFactor-sinceFrame._scaleFactor)};Frame.prototype.translation=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();return vec3.subtract(vec3.create(),this._translation,sinceFrame._translation)};Frame.prototype.toString=function(){var str="Frame [ id:"+this.id+" | timestamp:"+this.timestamp+" | Hand count:("+this.hands.length+") | Pointable count:("+this.pointables.length+")";if(this.gestures)str+=" | Gesture count:("+this.gestures.length+")";str+=" ]";return str};Frame.prototype.dump=function(){var out="";out+="Frame Info:
";out+=this.toString();out+="

Hands:
";for(var handIdx=0,handCount=this.hands.length;handIdx!=handCount;handIdx++){out+=" "+this.hands[handIdx].toString()+"
"}out+="

Pointables:
";for(var pointableIdx=0,pointableCount=this.pointables.length;pointableIdx!=pointableCount;pointableIdx++){out+=" "+this.pointables[pointableIdx].toString()+"
"}if(this.gestures){out+="

Gestures:
";for(var gestureIdx=0,gestureCount=this.gestures.length;gestureIdx!=gestureCount;gestureIdx++){out+=" "+this.gestures[gestureIdx].toString()+"
"}}out+="

Raw JSON:
";out+=JSON.stringify(this.data);return out};Frame.Invalid={valid:false,hands:[],fingers:[],tools:[],gestures:[],pointables:[],pointable:function(){return Pointable.Invalid},finger:function(){return Pointable.Invalid},hand:function(){return Hand.Invalid},toString:function(){return"invalid frame"},dump:function(){return this.toString()},rotationAngle:function(){return 0},rotationMatrix:function(){return mat3.create()},rotationAxis:function(){return vec3.create()},scaleFactor:function(){return 1},translation:function(){return vec3.create()}}},{"./gesture":6,"./hand":7,"./interaction_box":9,"./pointable":11,"gl-matrix":19,underscore:20}],6:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3,EventEmitter=require("events").EventEmitter,_=require("underscore");var createGesture=exports.createGesture=function(data){var gesture;switch(data.type){case"circle":gesture=new CircleGesture(data);break;case"swipe":gesture=new SwipeGesture(data);break;case"screenTap":gesture=new ScreenTapGesture(data);break;case"keyTap":gesture=new KeyTapGesture(data);break;default:throw"unkown gesture type"}gesture.id=data.id;gesture.handIds=data.handIds;gesture.pointableIds=data.pointableIds;gesture.duration=data.duration;gesture.state=data.state;gesture.type=data.type;return gesture};var gestureListener=exports.gestureListener=function(controller,type){var handlers={};var gestureMap={};var gestureCreator=function(){var candidateGesture=gestureMap[gesture.id];if(candidateGesture!==undefined)gesture.update(gesture,frame);if(gesture.state=="start"||gesture.state=="stop"){if(type==gesture.type&&gestureMap[gesture.id]===undefined){gestureMap[gesture.id]=new Gesture(gesture,frame);gesture.update(gesture,frame)}if(gesture.state=="stop"){delete gestureMap[gesture.id]}}};controller.on("gesture",function(gesture,frame){if(gesture.type==type){if(gesture.state=="start"||gesture.state=="stop"){if(gestureMap[gesture.id]===undefined){var gestureTracker=new Gesture(gesture,frame);gestureMap[gesture.id]=gestureTracker;_.each(handlers,function(cb,name){gestureTracker.on(name,cb)})}}gestureMap[gesture.id].update(gesture,frame);if(gesture.state=="stop"){delete gestureMap[gesture.id]}}});var builder={start:function(cb){handlers["start"]=cb;return builder},stop:function(cb){handlers["stop"]=cb;return builder},complete:function(cb){handlers["stop"]=cb;return builder},update:function(cb){handlers["update"]=cb;return builder}};return builder};var Gesture=exports.Gesture=function(gesture,frame){this.gestures=[gesture];this.frames=[frame]};Gesture.prototype.update=function(gesture,frame){this.gestures.push(gesture);this.frames.push(frame);this.emit(gesture.state,this)};_.extend(Gesture.prototype,EventEmitter.prototype);var CircleGesture=function(data){this.center=data.center;this.normal=data.normal;this.progress=data.progress;this.radius=data.radius};CircleGesture.prototype.toString=function(){return"CircleGesture ["+JSON.stringify(this)+"]"};var SwipeGesture=function(data){this.startPosition=data.startPosition;this.position=data.position;this.direction=data.direction;this.speed=data.speed};SwipeGesture.prototype.toString=function(){return"SwipeGesture ["+JSON.stringify(this)+"]"};var ScreenTapGesture=function(data){this.position=data.position;this.direction=data.direction;this.progress=data.progress};ScreenTapGesture.prototype.toString=function(){return"ScreenTapGesture ["+JSON.stringify(this)+"]"};var KeyTapGesture=function(data){this.position=data.position;this.direction=data.direction;this.progress=data.progress};KeyTapGesture.prototype.toString=function(){return"KeyTapGesture ["+JSON.stringify(this)+"]"}},{events:17,"gl-matrix":19,underscore:20}],7:[function(require,module,exports){var Pointable=require("./pointable"),glMatrix=require("gl-matrix"),mat3=glMatrix.mat3,vec3=glMatrix.vec3,_=require("underscore");var Hand=module.exports=function(data){this.id=data.id;this.palmPosition=data.palmPosition;this.direction=data.direction;this.palmVelocity=data.palmVelocity;this.palmNormal=data.palmNormal;this.sphereCenter=data.sphereCenter;this.sphereRadius=data.sphereRadius;this.valid=true;this.pointables=[];this.fingers=[];this.tools=[];this._translation=data.t;this._rotation=_.flatten(data.r);this._scaleFactor=data.s;this.timeVisible=data.timeVisible;this.stabilizedPalmPosition=data.stabilizedPalmPosition};Hand.prototype.finger=function(id){var finger=this.frame.finger(id);return finger&&finger.handId==this.id?finger:Pointable.Invalid};Hand.prototype.rotationAngle=function(sinceFrame,axis){if(!this.valid||!sinceFrame.valid)return 0;var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return 0;var rot=this.rotationMatrix(sinceFrame);var cs=(rot[0]+rot[4]+rot[8]-1)*.5;var angle=Math.acos(cs);angle=isNaN(angle)?0:angle;if(axis!==undefined){var rotAxis=this.rotationAxis(sinceFrame);angle*=vec3.dot(rotAxis,vec3.normalize(vec3.create(),axis))}return angle};Hand.prototype.rotationAxis=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return vec3.create();return vec3.normalize(vec3.create(),[this._rotation[7]-sinceHand._rotation[5],this._rotation[2]-sinceHand._rotation[6],this._rotation[3]-sinceHand._rotation[1]])};Hand.prototype.rotationMatrix=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return mat3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return mat3.create();var transpose=mat3.transpose(mat3.create(),this._rotation);var m=mat3.multiply(mat3.create(),sinceHand._rotation,transpose);return m};Hand.prototype.scaleFactor=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return 1;var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return 1;return Math.exp(this._scaleFactor-sinceHand._scaleFactor)};Hand.prototype.translation=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return vec3.create();return[this._translation[0]-sinceHand._translation[0],this._translation[1]-sinceHand._translation[1],this._translation[2]-sinceHand._translation[2]]};Hand.prototype.toString=function(){return"Hand [ id: "+this.id+" | palm velocity:"+this.palmVelocity+" | sphere center:"+this.sphereCenter+" ] "};Hand.Invalid={valid:false,fingers:[],tools:[],pointables:[],pointable:function(){return Pointable.Invalid},finger:function(){return Pointable.Invalid},toString:function(){return"invalid frame"},dump:function(){return this.toString()},rotationAngle:function(){return 0},rotationMatrix:function(){return mat3.create()},rotationAxis:function(){return vec3.create()},scaleFactor:function(){return 1},translation:function(){return vec3.create()}}},{"./pointable":11,"gl-matrix":19,underscore:20}],8:[function(require,module,exports){!function(){module.exports={Controller:require("./controller"),Frame:require("./frame"),Gesture:require("./gesture"),Hand:require("./hand"),Pointable:require("./pointable"),InteractionBox:require("./interaction_box"),Connection:require("./connection"),CircularBuffer:require("./circular_buffer"),UI:require("./ui"),glMatrix:require("gl-matrix"),mat3:require("gl-matrix").mat3,vec3:require("gl-matrix").vec3,loopController:undefined,loop:function(opts,callback){if(callback===undefined){callback=opts;opts={}}if(!this.loopController)this.loopController=new this.Controller(opts);this.loopController.loop(callback)}}}()},{"./circular_buffer":2,"./connection":3,"./controller":4,"./frame":5,"./gesture":6,"./hand":7,"./interaction_box":9,"./pointable":11,"./ui":13,"gl-matrix":19}],9:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3;var InteractionBox=module.exports=function(data){this.valid=true;this.center=data.center;this.size=data.size;this.width=data.size[0];this.height=data.size[1];this.depth=data.size[2]};InteractionBox.prototype.denormalizePoint=function(normalizedPosition){return vec3.fromValues((normalizedPosition[0]-.5)*this.size[0]+this.center[0],(normalizedPosition[1]-.5)*this.size[1]+this.center[1],(normalizedPosition[2]-.5)*this.size[2]+this.center[2])};InteractionBox.prototype.normalizePoint=function(position,clamp){var vec=vec3.fromValues((position[0]-this.center[0])/this.size[0]+.5,(position[1]-this.center[1])/this.size[1]+.5,(position[2]-this.center[2])/this.size[2]+.5);if(clamp){vec[0]=Math.min(Math.max(vec[0],0),1);vec[1]=Math.min(Math.max(vec[1],0),1);vec[2]=Math.min(Math.max(vec[2],0),1)}return vec};InteractionBox.prototype.toString=function(){return"InteractionBox [ width:"+this.width+" | height:"+this.height+" | depth:"+this.depth+" ]"};InteractionBox.Invalid={valid:false}},{"gl-matrix":19}],10:[function(require,module,exports){var Pipeline=module.exports=function(){this.steps=[]};Pipeline.prototype.addStep=function(step){this.steps.push(step)};Pipeline.prototype.run=function(frame){var stepsLength=this.steps.length;for(var i=0;i!=stepsLength;i++){if(!frame)break;frame=this.steps[i](frame)}return frame}},{}],11:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3;var Pointable=module.exports=function(data){this.valid=true;this.id=data.id;this.handId=data.handId;this.length=data.length;this.tool=data.tool;this.width=data.width;this.direction=data.direction;this.stabilizedTipPosition=data.stabilizedTipPosition;this.tipPosition=data.tipPosition;this.tipVelocity=data.tipVelocity;this.touchZone=data.touchZone;this.touchDistance=data.touchDistance;this.timeVisible=data.timeVisible};Pointable.prototype.toString=function(){if(this.tool==true){return"Pointable [ id:"+this.id+" "+this.length+"mmx | with:"+this.width+"mm | direction:"+this.direction+" ]"}else{return"Pointable [ id:"+this.id+" "+this.length+"mmx | direction: "+this.direction+" ]"}};Pointable.Invalid={valid:false}},{"gl-matrix":19}],12:[function(require,module,exports){var Frame=require("./frame");var Event=function(data){this.type=data.type;this.state=data.state};var chooseProtocol=exports.chooseProtocol=function(header){var protocol;switch(header.version){case 1:protocol=JSONProtocol(1,function(data){return new Frame(data)});break;case 2:protocol=JSONProtocol(2,function(data){return new Frame(data)});protocol.sendHeartbeat=function(connection){connection.send(protocol.encode({heartbeat:true}))};break;case 3:protocol=JSONProtocol(3,function(data){return data.event?new Event(data.event):new Frame(data)});protocol.sendHeartbeat=function(connection){connection.send(protocol.encode({heartbeat:true}))};break;default:throw"unrecognized version"}return protocol};var JSONProtocol=function(version,cb){var protocol=cb;protocol.encode=function(message){return JSON.stringify(message)};protocol.version=version;protocol.versionLong="Version "+version;protocol.type="protocol";return protocol}},{"./frame":5}],13:[function(require,module,exports){exports.UI={Region:require("./ui/region"),Cursor:require("./ui/cursor")}},{"./ui/cursor":14,"./ui/region":15}],14:[function(require,module,exports){var Cursor=module.exports=function(){return function(frame){var pointable=frame.pointables.sort(function(a,b){return a.z-b.z})[0];if(pointable&&pointable.valid){frame.cursorPosition=pointable.tipPosition}return frame}}},{}],15:[function(require,module,exports){var EventEmitter=require("events").EventEmitter,_=require("underscore");var Region=module.exports=function(start,end){this.start=new Vector(start);this.end=new Vector(end);this.enteredFrame=null};Region.prototype.hasPointables=function(frame){for(var i=0;i!=frame.pointables.length;i++){var position=frame.pointables[i].tipPosition;if(position.x>=this.start.x&&position.x<=this.end.x&&position.y>=this.start.y&&position.y<=this.end.y&&position.z>=this.start.z&&position.z<=this.end.z){return true}}return false};Region.prototype.listener=function(opts){var region=this;if(opts&&opts.nearThreshold)this.setupNearRegion(opts.nearThreshold);return function(frame){return region.updatePosition(frame)}};Region.prototype.clipper=function(){var region=this;return function(frame){region.updatePosition(frame);return region.enteredFrame?frame:null}};Region.prototype.setupNearRegion=function(distance){var nearRegion=this.nearRegion=new Region([this.start.x-distance,this.start.y-distance,this.start.z-distance],[this.end.x+distance,this.end.y+distance,this.end.z+distance]);var region=this;nearRegion.on("enter",function(frame){region.emit("near",frame)});nearRegion.on("exit",function(frame){region.emit("far",frame)});region.on("exit",function(frame){region.emit("near",frame)})};Region.prototype.updatePosition=function(frame){if(this.nearRegion)this.nearRegion.updatePosition(frame);if(this.hasPointables(frame)&&this.enteredFrame==null){this.enteredFrame=frame;this.emit("enter",this.enteredFrame)}else if(!this.hasPointables(frame)&&this.enteredFrame!=null){this.enteredFrame=null;this.emit("exit",this.enteredFrame)}return frame};Region.prototype.normalize=function(position){return new Vector([(position.x-this.start.x)/(this.end.x-this.start.x),(position.y-this.start.y)/(this.end.y-this.start.y),(position.z-this.start.z)/(this.end.z-this.start.z)])};Region.prototype.mapToXY=function(position,width,height){var normalized=this.normalize(position);var x=normalized.x,y=normalized.y;if(x>1)x=1;else if(x<-1)x=-1;if(y>1)y=1;else if(y<-1)y=-1;return[(x+1)/2*width,(1-y)/2*height,normalized.z]};_.extend(Region.prototype,EventEmitter.prototype)},{events:17,underscore:20}],16:[function(require,module,exports){},{}],17:[function(require,module,exports){!function(process){if(!process.EventEmitter)process.EventEmitter=function(){};var EventEmitter=exports.EventEmitter=process.EventEmitter;var isArray=typeof Array.isArray==="function"?Array.isArray:function(xs){return Object.prototype.toString.call(xs)==="[object Array]"};function indexOf(xs,x){if(xs.indexOf)return xs.indexOf(x);for(var i=0;i0&&this._events[type].length>m){this._events[type].warned=true;console.error("(node) warning: possible EventEmitter memory "+"leak detected. %d listeners added. "+"Use emitter.setMaxListeners() to increase limit.",this._events[type].length);console.trace()}}this._events[type].push(listener)}else{this._events[type]=[this._events[type],listener]}return this};EventEmitter.prototype.on=EventEmitter.prototype.addListener;EventEmitter.prototype.once=function(type,listener){var self=this;self.on(type,function g(){self.removeListener(type,g);listener.apply(this,arguments)});return this};EventEmitter.prototype.removeListener=function(type,listener){if("function"!==typeof listener){throw new Error("removeListener only takes instances of Function")}if(!this._events||!this._events[type])return this;var list=this._events[type];if(isArray(list)){var i=indexOf(list,listener);if(i<0)return this;list.splice(i,1);if(list.length==0)delete this._events[type]}else if(this._events[type]===listener){delete this._events[type]}return this};EventEmitter.prototype.removeAllListeners=function(type){if(arguments.length===0){this._events={};return this}if(type&&this._events&&this._events[type])this._events[type]=null;return this};EventEmitter.prototype.listeners=function(type){if(!this._events)this._events={};if(!this._events[type])this._events[type]=[];if(!isArray(this._events[type])){this._events[type]=[this._events[type]]}return this._events[type]}}(require("__browserify_process"))},{__browserify_process:18}],18:[function(require,module,exports){var process=module.exports={};process.nextTick=function(){var canSetImmediate=typeof window!=="undefined"&&window.setImmediate;var canPost=typeof window!=="undefined"&&window.postMessage&&window.addEventListener;if(canSetImmediate){return function(f){return window.setImmediate(f)}}if(canPost){var queue=[];window.addEventListener("message",function(ev){if(ev.source===window&&ev.data==="process-tick"){ev.stopPropagation();if(queue.length>0){var fn=queue.shift();fn()}}},true);return function nextTick(fn){queue.push(fn);window.postMessage("process-tick","*")}}return function nextTick(fn){setTimeout(fn,0)}}();process.title="browser";process.browser=true;process.env={};process.argv=[];process.binding=function(name){throw new Error("process.binding is not supported")};process.cwd=function(){return"/"};process.chdir=function(dir){throw new Error("process.chdir is not supported")}},{}],19:[function(require,module,exports){!function(){!function(){"use strict";var shim={};if(typeof exports==="undefined"){if(typeof define=="function"&&typeof define.amd=="object"&&define.amd){shim.exports={};define(function(){return shim.exports})}else{shim.exports=window}}else{shim.exports=exports}!function(exports){var vec2={};if(!GLMAT_EPSILON){var GLMAT_EPSILON=1e-6}vec2.create=function(){return new Float32Array(2)};vec2.clone=function(a){var out=new Float32Array(2);out[0]=a[0];out[1]=a[1];return out};vec2.fromValues=function(x,y){var out=new Float32Array(2);out[0]=x;out[1]=y;return out};vec2.copy=function(out,a){out[0]=a[0];out[1]=a[1];return out};vec2.set=function(out,x,y){out[0]=x;out[1]=y;return out};vec2.add=function(out,a,b){out[0]=a[0]+b[0];out[1]=a[1]+b[1];return out};vec2.sub=vec2.subtract=function(out,a,b){out[0]=a[0]-b[0];out[1]=a[1]-b[1];return out};vec2.mul=vec2.multiply=function(out,a,b){out[0]=a[0]*b[0];out[1]=a[1]*b[1];return out};vec2.div=vec2.divide=function(out,a,b){out[0]=a[0]/b[0];out[1]=a[1]/b[1];return out};vec2.min=function(out,a,b){out[0]=Math.min(a[0],b[0]); +out[1]=Math.min(a[1],b[1]);return out};vec2.max=function(out,a,b){out[0]=Math.max(a[0],b[0]);out[1]=Math.max(a[1],b[1]);return out};vec2.scale=function(out,a,b){out[0]=a[0]*b;out[1]=a[1]*b;return out};vec2.dist=vec2.distance=function(a,b){var x=b[0]-a[0],y=b[1]-a[1];return Math.sqrt(x*x+y*y)};vec2.sqrDist=vec2.squaredDistance=function(a,b){var x=b[0]-a[0],y=b[1]-a[1];return x*x+y*y};vec2.len=vec2.length=function(a){var x=a[0],y=a[1];return Math.sqrt(x*x+y*y)};vec2.sqrLen=vec2.squaredLength=function(a){var x=a[0],y=a[1];return x*x+y*y};vec2.negate=function(out,a){out[0]=-a[0];out[1]=-a[1];return out};vec2.normalize=function(out,a){var x=a[0],y=a[1];var len=x*x+y*y;if(len>0){len=1/Math.sqrt(len);out[0]=a[0]*len;out[1]=a[1]*len}return out};vec2.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]};vec2.cross=function(out,a,b){var z=a[0]*b[1]-a[1]*b[0];out[0]=out[1]=0;out[2]=z;return out};vec2.lerp=function(out,a,b,t){var ax=a[0],ay=a[1];out[0]=ax+t*(b[0]-ax);out[1]=ay+t*(b[1]-ay);return out};vec2.transformMat2=function(out,a,m){var x=a[0],y=a[1];out[0]=x*m[0]+y*m[1];out[1]=x*m[2]+y*m[3];return out};vec2.forEach=function(){var vec=new Float32Array(2);return function(a,stride,offset,count,fn,arg){var i,l;if(!stride){stride=2}if(!offset){offset=0}if(count){l=Math.min(count*stride+offset,a.length)}else{l=a.length}for(i=offset;i0){len=1/Math.sqrt(len);out[0]=a[0]*len;out[1]=a[1]*len;out[2]=a[2]*len}return out};vec3.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]+a[2]*b[2]};vec3.cross=function(out,a,b){var ax=a[0],ay=a[1],az=a[2],bx=b[0],by=b[1],bz=b[2];out[0]=ay*bz-az*by;out[1]=az*bx-ax*bz;out[2]=ax*by-ay*bx;return out};vec3.lerp=function(out,a,b,t){var ax=a[0],ay=a[1],az=a[2];out[0]=ax+t*(b[0]-ax);out[1]=ay+t*(b[1]-ay);out[2]=az+t*(b[2]-az);return out};vec3.transformMat4=function(out,a,m){var x=a[0],y=a[1],z=a[2];out[0]=m[0]*x+m[4]*y+m[8]*z+m[12];out[1]=m[1]*x+m[5]*y+m[9]*z+m[13];out[2]=m[2]*x+m[6]*y+m[10]*z+m[14];return out};vec3.transformQuat=function(out,a,q){var x=a[0],y=a[1],z=a[2],qx=q[0],qy=q[1],qz=q[2],qw=q[3],ix=qw*x+qy*z-qz*y,iy=qw*y+qz*x-qx*z,iz=qw*z+qx*y-qy*x,iw=-qx*x-qy*y-qz*z;out[0]=ix*qw+iw*-qx+iy*-qz-iz*-qy;out[1]=iy*qw+iw*-qy+iz*-qx-ix*-qz;out[2]=iz*qw+iw*-qz+ix*-qy-iy*-qx;return out};vec3.forEach=function(){var vec=new Float32Array(3);return function(a,stride,offset,count,fn,arg){var i,l;if(!stride){stride=3}if(!offset){offset=0}if(count){l=Math.min(count*stride+offset,a.length)}else{l=a.length}for(i=offset;i0){len=1/Math.sqrt(len);out[0]=a[0]*len;out[1]=a[1]*len;out[2]=a[2]*len;out[3]=a[3]*len}return out};vec4.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]+a[2]*b[2]+a[3]*b[3]};vec4.lerp=function(out,a,b,t){var ax=a[0],ay=a[1],az=a[2],aw=a[3];out[0]=ax+t*(b[0]-ax);out[1]=ay+t*(b[1]-ay);out[2]=az+t*(b[2]-az);out[3]=aw+t*(b[3]-aw);return out};vec4.transformMat4=function(out,a,m){var x=a[0],y=a[1],z=a[2],w=a[3];out[0]=m[0]*x+m[4]*y+m[8]*z+m[12]*w;out[1]=m[1]*x+m[5]*y+m[9]*z+m[13]*w;out[2]=m[2]*x+m[6]*y+m[10]*z+m[14]*w;out[3]=m[3]*x+m[7]*y+m[11]*z+m[15]*w;return out};vec4.transformQuat=function(out,a,q){var x=a[0],y=a[1],z=a[2],qx=q[0],qy=q[1],qz=q[2],qw=q[3],ix=qw*x+qy*z-qz*y,iy=qw*y+qz*x-qx*z,iz=qw*z+qx*y-qy*x,iw=-qx*x-qy*y-qz*z;out[0]=ix*qw+iw*-qx+iy*-qz-iz*-qy;out[1]=iy*qw+iw*-qy+iz*-qx-ix*-qz;out[2]=iz*qw+iw*-qz+ix*-qy-iy*-qx;return out};vec4.forEach=function(){var vec=new Float32Array(4);return function(a,stride,offset,count,fn,arg){var i,l;if(!stride){stride=4}if(!offset){offset=0}if(count){l=Math.min(count*stride+offset,a.length)}else{l=a.length}for(i=offset;i=1){if(out!==a){out[0]=ax;out[1]=ay;out[2]=az;out[3]=aw}return out}halfTheta=Math.acos(cosHalfTheta);sinHalfTheta=Math.sqrt(1-cosHalfTheta*cosHalfTheta);if(Math.abs(sinHalfTheta)<.001){out[0]=ax*.5+bx*.5;out[1]=ay*.5+by*.5;out[2]=az*.5+bz*.5;out[3]=aw*.5+bw*.5;return out}ratioA=Math.sin((1-t)*halfTheta)/sinHalfTheta;ratioB=Math.sin(t*halfTheta)/sinHalfTheta;out[0]=ax*ratioA+bx*ratioB;out[1]=ay*ratioA+by*ratioB;out[2]=az*ratioA+bz*ratioB;out[3]=aw*ratioA+bw*ratioB;return out};quat.invert=function(out,a){var a0=a[0],a1=a[1],a2=a[2],a3=a[3],dot=a0*a0+a1*a1+a2*a2+a3*a3,invDot=dot?1/dot:0;out[0]=-a0*invDot;out[1]=-a1*invDot;out[2]=-a2*invDot;out[3]=a3*invDot;return out};quat.conjugate=function(out,a){out[0]=-a[0];out[1]=-a[1];out[2]=-a[2];out[3]=a[3];return 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this._chain?_(obj).chain():obj};_.mixin(_);each(["pop","push","reverse","shift","sort","splice","unshift"],function(name){var method=ArrayProto[name];_.prototype[name]=function(){var obj=this._wrapped;method.apply(obj,arguments);if((name=="shift"||name=="splice")&&obj.length===0)delete obj[0];return result.call(this,obj)}});each(["concat","join","slice"],function(name){var method=ArrayProto[name];_.prototype[name]=function(){return result.call(this,method.apply(this._wrapped,arguments))}});_.extend(_.prototype,{chain:function(){this._chain=true;return this},value:function(){return this._wrapped}})}.call(this)}()},{}],21:[function(require,module,exports){window.requestAnimFrame=function(){return window.requestAnimationFrame||window.webkitRequestAnimationFrame||window.mozRequestAnimationFrame||window.oRequestAnimationFrame||window.msRequestAnimationFrame||function(callback){window.setTimeout(callback,1e3/60)}}();Leap=require("../lib/index")},{"../lib/index":8}]},{},[21]); + +/* + * Leap Motion integration for Reveal.js. + * James Sun [sun16] + * Rory Hardy [gneatgeek] + */ + +(function () { + var body = document.body, + controller = new Leap.Controller({ enableGestures: true }), + lastGesture = 0, + leapConfig = Reveal.getConfig().leap, + pointer = document.createElement( 'div' ), + config = { + autoCenter : true, // Center pointer around detected position. + gestureDelay : 500, // How long to delay between gestures. + naturalSwipe : true, // Swipe as if it were a touch screen. + pointerColor : '#00aaff', // Default color of the pointer. + pointerOpacity : 0.7, // Default opacity of the pointer. + pointerSize : 15, // Default minimum height/width of the pointer. + pointerTolerance : 120 // Bigger = slower pointer. + }, + entered, enteredPosition, now, size, tipPosition; // Other vars we need later, but don't need to redeclare. + + // Merge user defined settings with defaults + if( leapConfig ) { + for( key in leapConfig ) { + config[key] = leapConfig[key]; + } + } + + pointer.id = 'leap'; + + pointer.style.position = 'absolute'; + pointer.style.visibility = 'hidden'; + pointer.style.zIndex = 50; + pointer.style.opacity = config.pointerOpacity; + pointer.style.backgroundColor = config.pointerColor; + + body.appendChild( pointer ); + + // Leap's loop + controller.on( 'frame', function ( frame ) { + // Timing code to rate limit gesture execution + now = new Date().getTime(); + + // Pointer: 1 to 2 fingers. Strictly one finger works but may cause innaccuracies. + // The innaccuracies were observed on a development model and may not be an issue with consumer models. + if( frame.fingers.length > 0 && frame.fingers.length < 3 ) { + // Invert direction and multiply by 3 for greater effect. + size = -3 * frame.fingers[0].tipPosition[2]; + + if( size < config.pointerSize ) { + size = config.pointerSize; + } + + pointer.style.width = size + 'px'; + pointer.style.height = size + 'px'; + pointer.style.borderRadius = size - 5 + 'px'; + pointer.style.visibility = 'visible'; + + tipPosition = frame.fingers[0].tipPosition; + + if( config.autoCenter ) { + + + // Check whether the finger has entered the z range of the Leap Motion. Used for the autoCenter option. + if( !entered ) { + entered = true; + enteredPosition = frame.fingers[0].tipPosition; + } + + pointer.style.top = + (-1 * (( tipPosition[1] - enteredPosition[1] ) * body.offsetHeight / config.pointerTolerance )) + + ( body.offsetHeight / 2 ) + 'px'; + + pointer.style.left = + (( tipPosition[0] - enteredPosition[0] ) * body.offsetWidth / config.pointerTolerance ) + + ( body.offsetWidth / 2 ) + 'px'; + } + else { + pointer.style.top = ( 1 - (( tipPosition[1] - 50) / config.pointerTolerance )) * + body.offsetHeight + 'px'; + + pointer.style.left = ( tipPosition[0] * body.offsetWidth / config.pointerTolerance ) + + ( body.offsetWidth / 2 ) + 'px'; + } + } + else { + // Hide pointer on exit + entered = false; + pointer.style.visibility = 'hidden'; + } + + // Gestures + if( frame.gestures.length > 0 && (now - lastGesture) > config.gestureDelay ) { + var gesture = frame.gestures[0]; + + // One hand gestures + if( frame.hands.length === 1 ) { + // Swipe gestures. 3+ fingers. + if( frame.fingers.length > 2 && gesture.type === 'swipe' ) { + // Define here since some gestures will throw undefined for these. + var x = gesture.direction[0], + y = gesture.direction[1]; + + // Left/right swipe gestures + if( Math.abs( x ) > Math.abs( y )) { + if( x > 0 ) { + config.naturalSwipe ? Reveal.left() : Reveal.right(); + } + else { + config.naturalSwipe ? Reveal.right() : Reveal.left(); + } + } + // Up/down swipe gestures + else { + if( y > 0 ) { + config.naturalSwipe ? Reveal.down() : Reveal.up(); + } + else { + config.naturalSwipe ? Reveal.up() : Reveal.down(); + } + } + + lastGesture = now; + } + } + // Two hand gestures + else if( frame.hands.length === 2 ) { + // Upward two hand swipe gesture + if( gesture.type === 'swipe' && gesture.direction[1] > 0 ) { + Reveal.toggleOverview(); + } + + lastGesture = now; + } + } + }); + + controller.connect(); +})(); diff --git a/doc/pub/LogReg/html/reveal.js/plugin/remotes/remotes.js b/doc/pub/LogReg/html/reveal.js/plugin/remotes/remotes.js new file mode 100644 index 000000000..ba0dbad7b --- /dev/null +++ b/doc/pub/LogReg/html/reveal.js/plugin/remotes/remotes.js @@ -0,0 +1,39 @@ +/** + * Touch-based remote controller for your presentation courtesy + * of the folks at http://remotes.io + */ + +(function(window){ + + /** + * Detects if we are dealing with a touch enabled device (with some false positives) + * Borrowed from modernizr: https://github.com/Modernizr/Modernizr/blob/master/feature-detects/touch.js + */ + var hasTouch = (function(){ + return ('ontouchstart' in window) || window.DocumentTouch && document instanceof DocumentTouch; + })(); + + /** + * Detects if notes are enable and the current page is opened inside an /iframe + * this prevents loading Remotes.io several times + */ + var isNotesAndIframe = (function(){ + return window.RevealNotes && !(self == top); + })(); + + if(!hasTouch && !isNotesAndIframe){ + head.ready( 'remotes.ne.min.js', function() { + new Remotes("preview") + .on("swipe-left", function(e){ Reveal.right(); }) + .on("swipe-right", function(e){ Reveal.left(); }) + .on("swipe-up", function(e){ Reveal.down(); }) + .on("swipe-down", function(e){ Reveal.up(); }) + .on("tap", function(e){ Reveal.next(); }) + .on("zoom-out", function(e){ Reveal.toggleOverview(true); }) + .on("zoom-in", function(e){ Reveal.toggleOverview(false); }) + ; + } ); + + head.js('https://hakim-static.s3.amazonaws.com/reveal-js/remotes.ne.min.js'); + } +})(window); \ No newline at end of file diff --git a/doc/pub/LogReg/ipynb/default of credit card clients.xls b/doc/pub/LogReg/ipynb/default of credit card clients.xls new file mode 100644 index 000000000..91d982f00 Binary files /dev/null and b/doc/pub/LogReg/ipynb/default of credit card clients.xls differ diff --git a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb index 22ab3665e..46a14ec2b 100644 --- a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb +++ b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb @@ -4177,7 +4177,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.7.4" + "version": "3.7.6" } }, "nbformat": 4, diff --git a/doc/pub/Recurrent/html/._Recurrent-bs003.html b/doc/pub/Recurrent/html/._Recurrent-bs003.html new file mode 100644 index 000000000..594657db8 --- /dev/null +++ b/doc/pub/Recurrent/html/._Recurrent-bs003.html @@ -0,0 +1,158 @@ + + + + + + + + +Data Analysis and Machine Learning: Recurrent neural networks + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Recurrent neural networks +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Solving differential equations and eigenvalue problems with RNNs

+ +

+In our discussions of ordinary differential equations and partial +differential equations using neural networks. Here we will discuss how +we can solve say ordinary differential equations and eigenvalue +problems using RNNs. Eigenvalue problems can be solved using RNNs by +rewriting such a problems as a non-linear differential equation. + +

+Instead of starting with a well-known ordinary differential equation, +we start directly with an eigenvaule problem. + +

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/Recurrent/html/._Recurrent-bs004.html b/doc/pub/Recurrent/html/._Recurrent-bs004.html new file mode 100644 index 000000000..23616c340 --- /dev/null +++ b/doc/pub/Recurrent/html/._Recurrent-bs004.html @@ -0,0 +1,148 @@ + + + + + + + + +Data Analysis and Machine Learning: Recurrent neural networks + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Recurrent neural networks +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Long-Short Time Memory

+ +

+Discussions about dynamic unrolling through time. discuss memory cells, input and output + +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/Regression/ipynb/.ipynb_checkpoints/How2ReadData-checkpoint.ipynb b/doc/pub/Regression/ipynb/.ipynb_checkpoints/How2ReadData-checkpoint.ipynb new file mode 100644 index 000000000..2705ffc7b --- /dev/null +++ b/doc/pub/Regression/ipynb/.ipynb_checkpoints/How2ReadData-checkpoint.ipynb @@ -0,0 +1,2015 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Data Analysis and Machine Learning: Getting started, our first data and Machine Learning encounters\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Dec 25, 2019**\n", + "\n", + "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Introduction\n", + "\n", + "Our emphasis throughout this series of lectures \n", + "is on understanding the mathematical aspects of\n", + "different algorithms used in the fields of data analysis and machine learning. \n", + "\n", + "However, where possible we will emphasize the\n", + "importance of using available software. We start thus with a hands-on\n", + "and top-down approach to machine learning. The aim is thus to start with\n", + "relevant data or data we have produced \n", + "and use these to introduce statistical data analysis\n", + "concepts and machine learning algorithms before we delve into the\n", + "algorithms themselves. The examples we will use in the beginning, start with simple\n", + "polynomials with random noise added. We will use the Python\n", + "software package [Scikit-Learn](http://scikit-learn.org/stable/) and\n", + "introduce various machine learning algorithms to make fits of\n", + "the data and predictions. We move thereafter to more interesting\n", + "cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example).\n", + "These are examples where we can easily set up the data and\n", + "then use machine learning algorithms included in for example\n", + "**Scikit-Learn**. \n", + "\n", + "These examples will serve us the purpose of getting\n", + "started. Furthermore, they allow us to catch more than two birds with\n", + "a stone. They will allow us to bring in some programming specific\n", + "topics and tools as well as showing the power of various Python \n", + "libraries for machine learning and statistical data analysis. \n", + "\n", + "Here, we will mainly focus on two\n", + "specific Python packages for Machine Learning, Scikit-Learn and\n", + "Tensorflow (see below for links etc). Moreover, the examples we\n", + "introduce will serve as inputs to many of our discussions later, as\n", + "well as allowing you to set up models and produce your own data and\n", + "get started with programming.\n", + "\n", + "\n", + "\n", + "## What is Machine Learning?\n", + "\n", + "Statistics, data science and machine learning form important fields of\n", + "research in modern science. They describe how to learn and make\n", + "predictions from data, as well as allowing us to extract important\n", + "correlations about physical process and the underlying laws of motion\n", + "in large data sets. The latter, big data sets, appear frequently in\n", + "essentially all disciplines, from the traditional Science, Technology,\n", + "Mathematics and Engineering fields to Life Science, Law, education\n", + "research, the Humanities and the Social Sciences. \n", + "\n", + "It has become more\n", + "and more common to see research projects on big data in for example\n", + "the Social Sciences where extracting patterns from complicated survey\n", + "data is one of many research directions. Having a solid grasp of data\n", + "analysis and machine learning is thus becoming central to scientific\n", + "computing in many fields, and competences and skills within the fields\n", + "of machine learning and scientific computing are nowadays strongly\n", + "requested by many potential employers. The latter cannot be\n", + "overstated, familiarity with machine learning has almost become a\n", + "prerequisite for many of the most exciting employment opportunities,\n", + "whether they are in bioinformatics, life science, physics or finance,\n", + "in the private or the public sector. This author has had several\n", + "students or met students who have been hired recently based on their\n", + "skills and competences in scientific computing and data science, often\n", + "with marginal knowledge of machine learning.\n", + "\n", + "Machine learning is a subfield of computer science, and is closely\n", + "related to computational statistics. It evolved from the study of\n", + "pattern recognition in artificial intelligence (AI) research, and has\n", + "made contributions to AI tasks like computer vision, natural language\n", + "processing and speech recognition. Many of the methods we will study are also \n", + "strongly rooted in basic mathematics and physics research. \n", + "\n", + "Ideally, machine learning represents the science of giving computers\n", + "the ability to learn without being explicitly programmed. The idea is\n", + "that there exist generic algorithms which can be used to find patterns\n", + "in a broad class of data sets without having to write code\n", + "specifically for each problem. The algorithm will build its own logic\n", + "based on the data. You should however always keep in mind that\n", + "machines and algorithms are to a large extent developed by humans. The\n", + "insights and knowledge we have about a specific system, play a central\n", + "role when we develop a specific machine learning algorithm. \n", + "\n", + "Machine learning is an extremely rich field, in spite of its young\n", + "age. The increases we have seen during the last three decades in\n", + "computational capabilities have been followed by developments of\n", + "methods and techniques for analyzing and handling large date sets,\n", + "relying heavily on statistics, computer science and mathematics. The\n", + "field is rather new and developing rapidly. Popular software packages\n", + "written in Python for machine learning like\n", + "[Scikit-learn](http://scikit-learn.org/stable/),\n", + "[Tensorflow](https://www.tensorflow.org/),\n", + "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all\n", + "freely available at their respective GitHub sites, encompass\n", + "communities of developers in the thousands or more. And the number of\n", + "code developers and contributors keeps increasing. Not all the\n", + "algorithms and methods can be given a rigorous mathematical\n", + "justification, opening up thereby large rooms for experimenting and\n", + "trial and error and thereby exciting new developments. However, a\n", + "solid command of linear algebra, multivariate theory, probability\n", + "theory, statistical data analysis, understanding errors and Monte\n", + "Carlo methods are central elements in a proper understanding of many\n", + "of algorithms and methods we will discuss.\n", + "\n", + "\n", + "\n", + "## Types of Machine Learning\n", + "\n", + "\n", + "The approaches to machine learning are many, but are often split into\n", + "two main categories. In *supervised learning* we know the answer to a\n", + "problem, and let the computer deduce the logic behind it. On the other\n", + "hand, *unsupervised learning* is a method for finding patterns and\n", + "relationship in data sets without any prior knowledge of the system.\n", + "Some authours also operate with a third category, namely\n", + "*reinforcement learning*. This is a paradigm of learning inspired by\n", + "behavioral psychology, where learning is achieved by trial-and-error,\n", + "solely from rewards and punishment.\n", + "\n", + "Another way to categorize machine learning tasks is to consider the\n", + "desired output of a system. Some of the most common tasks are:\n", + "\n", + " * Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.\n", + "\n", + " * Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.\n", + "\n", + " * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n", + "\n", + "The methods we cover have three main topics in common, irrespective of\n", + "whether we deal with supervised or unsupervised learning. The first\n", + "ingredient is normally our data set (which can be subdivided into\n", + "training and test data), the second item is a model which is normally a\n", + "function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. \n", + "\n", + "The last ingredient is a so-called **cost**\n", + "function which allows us to present an estimate on how good our model\n", + "is in reproducing the data it is supposed to train. \n", + "At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of **gradient** methods.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Software and needed installations\n", + "\n", + "We will make extensive use of Python as programming language and its\n", + "myriad of available libraries. You will find\n", + "Jupyter notebooks invaluable in your work. You can run **R**\n", + "codes in the Jupyter/IPython notebooks, with the immediate benefit of\n", + "visualizing your data. You can also use compiled languages like C++,\n", + "Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be\n", + "on Python.\n", + "\n", + "\n", + "If you have Python installed (we strongly recommend Python3) and you feel\n", + "pretty familiar with installing different packages, we recommend that\n", + "you install the following Python packages via **pip** as \n", + "\n", + "1. pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow \n", + "\n", + "For Python3, replace **pip** with **pip3**.\n", + "\n", + "For OSX users we recommend, after having installed Xcode, to\n", + "install **brew**. Brew allows for a seamless installation of additional\n", + "software via for example \n", + "\n", + "1. brew install python3\n", + "\n", + "For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,\n", + "you can use **pip** as well and simply install Python as \n", + "\n", + "1. sudo apt-get install python3 (or python for pyhton2.7)\n", + "\n", + "etc etc. \n", + "\n", + "\n", + "\n", + "## Python installers\n", + "\n", + "If you don't want to perform these operations separately and venture\n", + "into the hassle of exploring how to set up dependencies and paths, we\n", + "recommend two widely used distrubutions which set up all relevant\n", + "dependencies for Python, namely \n", + "\n", + "* [Anaconda](https://docs.anaconda.com/), \n", + "\n", + "which is an open source\n", + "distribution of the Python and R programming languages for large-scale\n", + "data processing, predictive analytics, and scientific computing, that\n", + "aims to simplify package management and deployment. Package versions\n", + "are managed by the package management system **conda**. \n", + "\n", + "* [Enthought canopy](https://www.enthought.com/product/canopy/) \n", + "\n", + "is a Python\n", + "distribution for scientific and analytic computing distribution and\n", + "analysis environment, available for free and under a commercial\n", + "license.\n", + "\n", + "Furthermore, [Google's Colab](https://colab.research.google.com/notebooks/welcome.ipynb) is a free Jupyter notebook environment that requires \n", + "no setup and runs entirely in the cloud. Try it out!\n", + "\n", + "## Useful Python libraries\n", + "Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)\n", + "\n", + "* [NumPy](https://www.numpy.org/) is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays\n", + "\n", + "* [The pandas](https://pandas.pydata.org/) library provides high-performance, easy-to-use data structures and data analysis tools \n", + "\n", + "* [Xarray](http://xarray.pydata.org/en/stable/) is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!\n", + "\n", + "* [Scipy](https://www.scipy.org/) (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. \n", + "\n", + "* [Matplotlib](https://matplotlib.org/) is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.\n", + "\n", + "* [Autograd](https://github.com/HIPS/autograd) can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives\n", + "\n", + "* [SymPy](https://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. \n", + "\n", + "* [scikit-learn](https://scikit-learn.org/stable/) has simple and efficient tools for machine learning, data mining and data analysis\n", + "\n", + "* [TensorFlow](https://www.tensorflow.org/) is a Python library for fast numerical computing created and released by Google\n", + "\n", + "* [Keras](https://keras.io/) is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano\n", + "\n", + "* And many more such as [pytorch](https://pytorch.org/), [Theano](https://pypi.org/project/Theano/) etc \n", + "\n", + "## Installing R, C++, cython or Julia\n", + "\n", + "You will also find it convenient to utilize **R**. We will mainly\n", + "use Python during our lectures and in various projects and exercises.\n", + "Those of you\n", + "already familiar with **R** should feel free to continue using **R**, keeping\n", + "however an eye on the parallel Python set ups. Similarly, if you are a\n", + "Python afecionado, feel free to explore **R** as well. Jupyter/Ipython\n", + "notebook allows you to run **R** codes interactively in your\n", + "browser. The software library **R** is really tailored for statistical data analysis\n", + "and allows for an easy usage of the tools and algorithms we will discuss in these\n", + "lectures.\n", + "\n", + "To install **R** with Jupyter notebook \n", + "[follow the link here](https://mpacer.org/maths/r-kernel-for-ipython-notebook)\n", + "\n", + "\n", + "\n", + "\n", + "## Installing R, C++, cython, Numba etc\n", + "\n", + "\n", + "For the C++ aficionados, Jupyter/IPython notebook allows you also to\n", + "install C++ and run codes written in this language interactively in\n", + "the browser. Since we will emphasize writing many of the algorithms\n", + "yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming\n", + "languages.\n", + "\n", + "To add more entropy, **cython** can also be used when running your\n", + "notebooks. It means that Python with the jupyter notebook\n", + "setup allows you to integrate widely popular softwares and tools for\n", + "scientific computing. Similarly, the \n", + "[Numba Python package](https://numba.pydata.org/) delivers increased performance\n", + "capabilities with minimal rewrites of your codes. With its\n", + "versatility, including symbolic operations, Python offers a unique\n", + "computational environment. Your jupyter notebook can easily be\n", + "converted into a nicely rendered **PDF** file or a Latex file for\n", + "further processing. For example, convert to latex as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " pycod jupyter nbconvert filename.ipynb --to latex \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And to add more versatility, the Python package [SymPy](http://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python. \n", + "\n", + "Finally, if you wish to use the light mark-up language \n", + "[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML \n", + "formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using **doconce**.\n", + "\n", + "\n", + "\n", + "## Numpy examples and Important Matrix and vector handling packages\n", + "\n", + "There are several central software libraries for linear algebra and eigenvalue problems. Several of the more\n", + "popular ones have been wrapped into ofter software packages like those from the widely used text **Numerical Recipes**. The original source codes in many of the available packages are often taken from the widely used\n", + "software package LAPACK, which follows two other popular packages\n", + "developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.\n", + "\n", + " * LINPACK: package for linear equations and least square problems.\n", + "\n", + " * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.\n", + "\n", + " * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from .\n", + "\n", + "## Basic Matrix Features\n", + "\n", + "**Matrix properties reminder.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{A} =\n", + " \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", + " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", + " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", + " a_{41} & a_{42} & a_{43} & a_{44}\n", + " \\end{bmatrix}\\qquad\n", + "\\mathbf{I} =\n", + " \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n", + " 0 & 1 & 0 & 0 \\\\\n", + " 0 & 0 & 1 & 0 \\\\\n", + " 0 & 0 & 0 & 1\n", + " \\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The inverse of a matrix is defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \\left (A^{T} \\right )^{-1}$ real orthogonal $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \\left (A^{\\dagger} \\right )^{-1}$ unitary $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$
\n", + "\n", + "\n", + "\n", + "\n", + "### Some famous Matrices\n", + "\n", + " * Diagonal if $a_{ij}=0$ for $i\\ne j$\n", + "\n", + " * Upper triangular if $a_{ij}=0$ for $i > j$\n", + "\n", + " * Lower triangular if $a_{ij}=0$ for $i < j$\n", + "\n", + " * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$\n", + "\n", + " * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$\n", + "\n", + " * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$\n", + "\n", + " * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$\n", + "\n", + " * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$\n", + "\n", + " * Banded, block upper triangular, block lower triangular....\n", + "\n", + "### More Basic Matrix Features\n", + "\n", + "**Some Equivalent Statements.**\n", + "\n", + "For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", + "\n", + " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", + "\n", + " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", + "\n", + " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", + "\n", + " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", + "\n", + " * $\\mathbf{A}$ is a product of elementary matrices.\n", + "\n", + " * $0$ is not eigenvalue of $\\mathbf{A}$.\n", + "\n", + "\n", + "\n", + "\n", + "## Numpy and arrays\n", + "[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution," + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "n = 10\n", + "x = np.random.normal(size=n)\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.\n", + "Another alternative is to declare a vector as follows" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.array([1, 2, 3])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++\n", + "start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \\dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4, 7, 8]))\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the last example we used Numpy's unary function $np.log$. This function is\n", + "highly tuned to compute array elements since the code is vectorized\n", + "and does not require looping. We normaly recommend that you use the\n", + "Numpy intrinsic functions instead of the corresponding **log** function\n", + "from Python's **math** module. The looping is done explicitely by the\n", + "**np.log** function. The alternative, and slower way to compute the\n", + "logarithms of a vector would be to write" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "from math import log\n", + "x = np.array([4, 7, 8])\n", + "for i in range(0, len(x)):\n", + " x[i] = log(x[i])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We note that our code is much longer already and we need to import the **log** function from the **math** module. \n", + "The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4, 7, 8], dtype = np.float64))\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4.0, 7.0, 8.0])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4.0, 7.0, 8.0])\n", + "print(x.itemsize)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Matrices in Python\n", + "\n", + "Having defined vectors, we are now ready to try out matrices. We can\n", + "define a $3 \\times 3 $ real matrix $\\hat{A}$ as (recall that we user\n", + "lowercase letters for vectors and uppercase letters for matrices)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "# print the first column, row-major order and elements start with 0\n", + "print(A[:,0])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can continue this was by printing out other columns or rows. The example here prints out the second column" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "# print the first column, row-major order and elements start with 0\n", + "print(A[1,:])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to zero\n", + "A = np.zeros( (n, n) )\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or initializing all elements to" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to one\n", + "A = np.ones( (n, n) )\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or as unitarily distributed random numbers (see the material on random number generators in the statistics part)" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \\in [0, 1]\n", + "A = np.random.rand(n, n)\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n", + "As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n", + "$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", + " \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n", + " \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where for example" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma_{xy} =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. \n", + "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\hat{W}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n", + " x_1 & y_1 & z_1 \\\\\n", + " x_2 & y_2 & z_2 \\\\\n", + " \\dots & \\dots & \\dots \\\\\n", + " x_{n-2} & y_{n-2} & z_{n-2} \\\\\n", + " x_{n-1} & y_{n-1} & z_{n-1}\n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which in turn is converted into into the $3\\times 3$ covariance matrix\n", + "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", + "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", + "function **np.mean(x)**. We can also extract the eigenvalues of the\n", + "covariance matrix through the **np.linalg.eig()** function." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [], + "source": [ + "# Importing various packages\n", + "import numpy as np\n", + "\n", + "n = 100\n", + "x = np.random.normal(size=n)\n", + "print(np.mean(x))\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "print(np.mean(y))\n", + "z = x**3+np.random.normal(size=n)\n", + "print(np.mean(z))\n", + "W = np.vstack((x, y, z))\n", + "Sigma = np.cov(W)\n", + "print(Sigma)\n", + "Eigvals, Eigvecs = np.linalg.eig(Sigma)\n", + "print(Eigvals)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from scipy import sparse\n", + "eye = np.eye(4)\n", + "print(eye)\n", + "sparse_mtx = sparse.csr_matrix(eye)\n", + "print(sparse_mtx)\n", + "x = np.linspace(-10,10,100)\n", + "y = np.sin(x)\n", + "plt.plot(x,y,marker='x')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Meet the Pandas\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "Another useful Python package is\n", + "[pandas](https://pandas.pydata.org/), which is an open source library\n", + "providing high-performance, easy-to-use data structures and data\n", + "analysis tools for Python. **pandas** stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.\n", + "**pandas** has two major classes, the **DataFrame** class with two-dimensional data objects and tabular data organized in columns and the class **Series** with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. \n", + "**pandas** allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. \n", + "\n", + "The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of **pandas**, in particular in connection with classification of data." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [], + "source": [ + "import pandas as pd\n", + "from IPython.display import display\n", + "data = {'First Name': [\"Frodo\", \"Bilbo\", \"Aragorn II\", \"Samwise\"],\n", + " 'Last Name': [\"Baggins\", \"Baggins\",\"Elessar\",\"Gamgee\"],\n", + " 'Place of birth': [\"Shire\", \"Shire\", \"Eriador\", \"Shire\"],\n", + " 'Date of Birth T.A.': [2968, 2890, 2931, 2980]\n", + " }\n", + "data_pandas = pd.DataFrame(data)\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the above we have imported **pandas** with the shorthand **pd**, the latter has become the standard way we import **pandas**. We make then a list of various variables\n", + "and reorganize the aboves lists into a **DataFrame** and then print out a neat table with specific column labels as *Name*, *place of birth* and *date of birth*.\n", + "Displaying these results, we see that the indices are given by the default numbers from zero to three.\n", + "**pandas** is extremely flexible and we can easily change the above indices by defining a new type of indexing as" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [], + "source": [ + "data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Thereafter we display the content of the row which begins with the index **Aragorn**" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [], + "source": [ + "display(data_pandas.loc['Aragorn'])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can easily append data to this, for example" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "new_hobbit = {'First Name': [\"Peregrin\"],\n", + " 'Last Name': [\"Took\"],\n", + " 'Place of birth': [\"Shire\"],\n", + " 'Date of Birth T.A.': [2990]\n", + " }\n", + "data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here are other examples where we use the **DataFrame** functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix \n", + "of dimensionality $10\\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations." + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "from IPython.display import display\n", + "np.random.seed(100)\n", + "# setting up a 10 x 5 matrix\n", + "rows = 10\n", + "cols = 5\n", + "a = np.random.randn(rows,cols)\n", + "df = pd.DataFrame(a)\n", + "display(df)\n", + "print(df.mean())\n", + "print(df.std())\n", + "display(df**2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Thereafter we can select specific columns only and plot final results" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [], + "source": [ + "df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']\n", + "df.index = np.arange(10)\n", + "\n", + "display(df)\n", + "print(df['Second'].mean() )\n", + "print(df.info())\n", + "print(df.describe())\n", + "\n", + "from pylab import plt, mpl\n", + "plt.style.use('seaborn')\n", + "mpl.rcParams['font.family'] = 'serif'\n", + "\n", + "df.cumsum().plot(lw=2.0, figsize=(10,6))\n", + "plt.show()\n", + "\n", + "\n", + "df.plot.bar(figsize=(10,6), rot=15)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can produce a $4\\times 4$ matrix" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [], + "source": [ + "b = np.arange(16).reshape((4,4))\n", + "print(b)\n", + "df1 = pd.DataFrame(b)\n", + "print(df1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and many other operations. \n", + "\n", + "The **Series** class is another important class included in\n", + "**pandas**. You can view it as a specialization of **DataFrame** but where\n", + "we have just a single column of data. It shares many of the same features as _DataFrame. As with **DataFrame**,\n", + "most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.\n", + "As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.\n", + "For multidimensional arrays, we recommend strongly [xarray](http://xarray.pydata.org/en/stable/). **xarray** has much of the same flexibility as **pandas**, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both **pandas** and **xarray**. \n", + "\n", + "\n", + "\n", + "## Reading Data and fitting\n", + "\n", + "In order to study various Machine Learning algorithms, we need to\n", + "access data. Acccessing data is an essential step in all machine\n", + "learning algorithms. In particular, setting up the so-called **design\n", + "matrix** (to be defined below) is often the first element we need in\n", + "order to perform our calculations. To set up the design matrix means\n", + "reading (and later, when the calculations are done, writing) data\n", + "in various formats, The formats span from reading files from disk,\n", + "loading data from databases and interacting with online sources\n", + "like web application programming interfaces (APIs).\n", + "\n", + "In handling various input formats, as discussed above, we will mainly stay with **pandas**,\n", + "a Python package which allows us, in a seamless and painless way, to\n", + "deal with a multitude of formats, from standard **csv** (comma separated\n", + "values) files, via **excel**, **html** to **hdf5** formats. With **pandas**\n", + "and the **DataFrame** and **Series** functionalities we are able to convert text data\n", + "into the calculational formats we need for a specific algorithm. And our code is going to be \n", + "pretty close the basic mathematical expressions.\n", + "\n", + "Our first data set is going to be a classic from nuclear physics, namely all\n", + "available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set. \n", + "\n", + "We will show some of the\n", + "strengths of packages like **Scikit-Learn** in fitting nuclear binding energies to\n", + "specific functions using linear regression first. Then, as a teaser, we will show you how \n", + "you can easily implement other algorithms like decision trees and random forests and neural networks.\n", + "\n", + "But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,\n", + "(don't be offended) fitting straight lines!\n", + "\n", + "\n", + "### Simple linear regression model using **scikit-learn**\n", + "\n", + "We start with perhaps our simplest possible example, using **Scikit-Learn** to perform linear regression analysis on a data set produced by us. \n", + "\n", + "What follows is a simple Python code where we have defined a function\n", + "$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. \n", + "The numbers in the vector $\\hat{x}$ are given\n", + "by random numbers generated with a uniform distribution with entries\n", + "$x_i \\in [0,1]$ (more about probability distribution functions\n", + "later). These values are then used to define a function $y(x)$\n", + "(tabulated again as a vector) with a linear dependence on $x$ plus a\n", + "random noise added via the normal distribution.\n", + "\n", + "\n", + "The Numpy functions are imported used the **import numpy as np**\n", + "statement and the random number generator for the uniform distribution\n", + "is called using the function **np.random.rand()**, where we specificy\n", + "that we want $100$ random variables. Using Numpy we define\n", + "automatically an array with the specified number of elements, $100$ in\n", + "our case. With the Numpy function **randn()** we can compute random\n", + "numbers with the normal distribution (mean value $\\mu$ equal to zero and\n", + "variance $\\sigma^2$ set to one) and produce the values of $y$ assuming a linear\n", + "dependence as function of $x$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y = 2x+N(0,1),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $N(0,1)$ represents random numbers generated by the normal\n", + "distribution. From **Scikit-Learn** we import then the\n", + "**LinearRegression** functionality and make a prediction $\\tilde{y} =\n", + "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n", + "data $(\\hat{x},\\hat{y})$ for our training data. The Python package\n", + "**scikit-learn** has also a functionality which extracts the above\n", + "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n", + "distinguish between training data and test data.\n", + "\n", + "For plotting we use the Python package\n", + "[matplotlib](https://matplotlib.org/) which produces publication\n", + "quality figures. Feel free to explore the extensive\n", + "[gallery](https://matplotlib.org/gallery/index.html) of examples. In\n", + "this example we plot our original values of $x$ and $y$ as well as the\n", + "prediction **ypredict** ($\\tilde{y}$), which attempts at fitting our\n", + "data with a straight line.\n", + "\n", + "The Python code follows here." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [], + "source": [ + "# Importing various packages\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 2*x+np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "xnew = np.array([[0],[1]])\n", + "ypredict = linreg.predict(xnew)\n", + "\n", + "plt.plot(xnew, ypredict, \"r-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0,1.0,0, 5.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Simple Linear Regression')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This example serves several aims. It allows us to demonstrate several\n", + "aspects of data analysis and later machine learning algorithms. The\n", + "immediate visualization shows that our linear fit is not\n", + "impressive. It goes through the data points, but there are many\n", + "outliers which are not reproduced by our linear regression. We could\n", + "now play around with this small program and change for example the\n", + "factor in front of $x$ and the normal distribution. Try to change the\n", + "function $y$ to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y = 10x+0.01 \\times N(0,1),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $x$ is defined as before. Does the fit look better? Indeed, by\n", + "reducing the role of the noise given by the normal distribution we see immediately that\n", + "our linear prediction seemingly reproduces better the training\n", + "set. However, this testing 'by the eye' is obviouly not satisfactory in the\n", + "long run. Here we have only defined the training data and our model, and \n", + "have not discussed a more rigorous approach to the **cost** function.\n", + "\n", + "We need more rigorous criteria in defining whether we have succeeded or\n", + "not in modeling our training data. You will be surprised to see that\n", + "many scientists seldomly venture beyond this 'by the eye' approach. A\n", + "standard approach for the *cost* function is the so-called $\\chi^2$\n", + "function (a variant of the mean-squared error (MSE))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\chi^2 = \\frac{1}{n}\n", + "\\sum_{i=0}^{n-1}\\frac{(y_i-\\tilde{y}_i)^2}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\sigma_i^2$ is the variance (to be defined later) of the entry\n", + "$y_i$. We may not know the explicit value of $\\sigma_i^2$, it serves\n", + "however the aim of scaling the equations and make the cost function\n", + "dimensionless. \n", + "\n", + "Minimizing the cost function is a central aspect of\n", + "our discussions to come. Finding its minima as function of the model\n", + "parameters ($\\alpha$ and $\\beta$ in our case) will be a recurring\n", + "theme in these series of lectures. Essentially all machine learning\n", + "algorithms we will discuss center around the minimization of the\n", + "chosen cost function. This depends in turn on our specific\n", + "model for describing the data, a typical situation in supervised\n", + "learning. Automatizing the search for the minima of the cost function is a\n", + "central ingredient in all algorithms. Typical methods which are\n", + "employed are various variants of **gradient** methods. These will be\n", + "discussed in more detail later. Again, you'll be surprised to hear that\n", + "many practitioners minimize the above function ''by the eye', popularly dubbed as \n", + "'chi by the eye'. That is, change a parameter and see (visually and numerically) that \n", + "the $\\chi^2$ function becomes smaller. \n", + "\n", + "There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define \n", + "the relative error (why would we prefer the MSE instead of the relative error?) as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\hat{y} -\\hat{\\tilde{y}}\\vert}{\\vert \\hat{y}\\vert}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The squared cost function results in an arithmetic mean-unbiased\n", + "estimator, and the absolute-value cost function results in a\n", + "median-unbiased estimator (in the one-dimensional case, and a\n", + "geometric median-unbiased estimator for the multi-dimensional\n", + "case). The squared cost function has the disadvantage that it has the tendency\n", + "to be dominated by outliers.\n", + "\n", + "We can modify easily the above Python code and plot the relative error instead" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 5*x+0.01*np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "ypredict = linreg.predict(x)\n", + "\n", + "plt.plot(x, np.abs(ypredict-y)/abs(y), \"ro\")\n", + "plt.axis([0,1.0,0.0, 0.5])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$\\epsilon_{\\mathrm{relative}}$')\n", + "plt.title(r'Relative error')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Depending on the parameter in front of the normal distribution, we may\n", + "have a small or larger relative error. Try to play around with\n", + "different training data sets and study (graphically) the value of the\n", + "relative error.\n", + "\n", + "As mentioned above, **Scikit-Learn** has an impressive functionality.\n", + "We can for example extract the values of $\\alpha$ and $\\beta$ and\n", + "their error estimates, or the variance and standard deviation and many\n", + "other properties from the statistical data analysis. \n", + "\n", + "Here we show an\n", + "example of the functionality of **Scikit-Learn**." + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np \n", + "import matplotlib.pyplot as plt \n", + "from sklearn.linear_model import LinearRegression \n", + "from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 2.0+ 5*x+0.5*np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "ypredict = linreg.predict(x)\n", + "print('The intercept alpha: \\n', linreg.intercept_)\n", + "print('Coefficient beta : \\n', linreg.coef_)\n", + "# The mean squared error \n", + "print(\"Mean squared error: %.2f\" % mean_squared_error(y, ypredict))\n", + "# Explained variance score: 1 is perfect prediction \n", + "print('Variance score: %.2f' % r2_score(y, ypredict))\n", + "# Mean squared log error \n", + "print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )\n", + "# Mean absolute error \n", + "print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))\n", + "plt.plot(x, ypredict, \"r-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0.0,1.0,1.5, 7.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Linear Regression fit ')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n", + "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n", + "\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The smaller the value, the better the fit. Ideally we would like to\n", + "have an MSE equal zero. The attentive reader has probably recognized\n", + "this function as being similar to the $\\chi^2$ function defined above.\n", + "\n", + "The **r2score** function computes $R^2$, the coefficient of\n", + "determination. It provides a measure of how well future samples are\n", + "likely to be predicted by the model. Best possible score is 1.0 and it\n", + "can be negative (because the model can be arbitrarily worse). A\n", + "constant model that always predicts the expected value of $\\hat{y}$,\n", + "disregarding the input features, would get a $R^2$ score of $0.0$.\n", + "\n", + "If $\\tilde{\\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have defined the mean value of $\\hat{y}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Another quantity taht we will meet again in our discussions of regression analysis is \n", + " the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error.\n", + "The MAE is defined as follows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\text{MAE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We present the \n", + "squared logarithmic (quadratic) error" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\text{MSLE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\log_e (x)$ stands for the natural logarithm of $x$. This error\n", + "estimate is best to use when targets having exponential growth, such\n", + "as population counts, average sales of a commodity over a span of\n", + "years etc. \n", + "\n", + "\n", + "Finally, another cost function is the Huber cost function used in robust regression.\n", + "\n", + "The rationale behind this possible cost function is its reduced\n", + "sensitivity to outliers in the data set. In our discussions on\n", + "dimensionality reduction and normalization of data we will meet other\n", + "ways of dealing with outliers.\n", + "\n", + "The Huber cost function is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "H_{\\delta}(a)={\\begin{cases}{\\frac {1}{2}}{a^{2}}&{\\text{for }}|a|\\leq \\delta ,\\\\\\delta (|a|-{\\frac {1}{2}}\\delta ),&{\\text{otherwise.}}\\end{cases}}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here $a=\\boldsymbol{y} - \\boldsymbol{\\tilde{y}}$.\n", + "We will discuss in more\n", + "detail these and other functions in the various lectures. We conclude this part with another example. Instead of \n", + "a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "import random\n", + "from sklearn.linear_model import Ridge\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "from sklearn.pipeline import make_pipeline\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x=np.linspace(0.02,0.98,200)\n", + "noise = np.asarray(random.sample((range(200)),200))\n", + "y=x**3*noise\n", + "yn=x**3*100\n", + "poly3 = PolynomialFeatures(degree=3)\n", + "X = poly3.fit_transform(x[:,np.newaxis])\n", + "clf3 = LinearRegression()\n", + "clf3.fit(X,y)\n", + "\n", + "Xplot=poly3.fit_transform(x[:,np.newaxis])\n", + "poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')\n", + "plt.plot(x,yn, color='red', label=\"True Cubic\")\n", + "plt.scatter(x, y, label='Data', color='orange', s=15)\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "def error(a):\n", + " for i in y:\n", + " err=(y-yn)/yn\n", + " return abs(np.sum(err))/len(err)\n", + "\n", + "print (error(y))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### To our real data: nuclear binding energies. Brief reminder on masses and binding energies\n", + "\n", + "Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding\n", + "energies. A basic quantity which can be measured for the ground\n", + "states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with\n", + "atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). \n", + "\n", + "Atomic masses are usually tabulated in terms of the mass excess defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\Delta M(N, Z) = M(N, Z) - uA,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $u$ is the Atomic Mass Unit" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "u = M(^{12}\\mathrm{C})/12 = 931.4940954(57) \\hspace{0.1cm} \\mathrm{MeV}/c^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The nucleon masses are" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "m_p = 1.00727646693(9)u,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "m_n = 939.56536(8)\\hspace{0.1cm} \\mathrm{MeV}/c^2 = 1.0086649156(6)u.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the [2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu](http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf)\n", + "there are data on masses and decays of 3437 nuclei.\n", + "\n", + "The nuclear binding energy is defined as the energy required to break\n", + "up a given nucleus into its constituent parts of $N$ neutrons and $Z$\n", + "protons. In terms of the atomic masses $M(N, Z)$ the binding energy is\n", + "defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.\n", + "In terms of the mass excess the binding energy is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N, Z) = Z\\Delta_H c^2 + N\\Delta_n c^2 -\\Delta(N, Z)c^2 ,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\Delta_H c^2 = 7.2890$ MeV and $\\Delta_n c^2 = 8.0713$ MeV.\n", + "\n", + "\n", + "A popular and physically intuitive model which can be used to parametrize \n", + "the experimental binding energies as function of $A$, is the so-called \n", + "**liquid drop model**. The ansatz is based on the following expression" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N,Z) = a_1A-a_2A^{2/3}-a_3\\frac{Z^2}{A^{1/3}}-a_4\\frac{(N-Z)^2}{A},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit \n", + "to the experimental data. \n", + "\n", + "\n", + "\n", + "\n", + "To arrive at the above expression we have assumed that we can make the following assumptions:\n", + "\n", + " * There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.\n", + "\n", + " * There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.\n", + "\n", + " * There is a Coulomb energy term $a_3\\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding. \n", + "\n", + " * There is an asymmetry term $a_4\\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.\n", + "\n", + "We could also add a so-called pairing term, which is a correction term that\n", + "arises from the tendency of proton pairs and neutron pairs to\n", + "occur. An even number of particles is more stable than an odd number. \n", + "\n", + "\n", + "### Organizing our data\n", + "\n", + "Let us start with reading and organizing our data. \n", + "We start with the compilation of masses and binding energies from 2016.\n", + "After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.\n", + "\n", + "\n", + "We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of **scikit-learn**." + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [], + "source": [ + "# Common imports\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "import sklearn.linear_model as skl\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error\n", + "import os\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "infile = open(data_path(\"MassEval2016.dat\"),'r')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various **matplotlib** commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [], + "source": [ + "from pylab import plt, mpl\n", + "plt.style.use('seaborn')\n", + "mpl.rcParams['font.family'] = 'serif'\n", + "\n", + "def MakePlot(x,y, styles, labels, axlabels):\n", + " plt.figure(figsize=(10,6))\n", + " for i in range(len(x)):\n", + " plt.plot(x[i], y[i], styles[i], label = labels[i])\n", + " plt.xlabel(axlabels[0])\n", + " plt.ylabel(axlabels[1])\n", + " plt.legend(loc=0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Our next step is to read the data on experimental binding energies and\n", + "reorganize them as functions of the mass number $A$, the number of\n", + "protons $Z$ and neutrons $N$ using **pandas**. Before we do this it is\n", + "always useful (unless you have a binary file or other types of compressed\n", + "data) to actually open the file and simply take a look at it!\n", + "\n", + "\n", + "In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with **pandas**. The file begins with some basic format information." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [], + "source": [ + "\"\"\" \n", + "This is taken from the data file of the mass 2016 evaluation. \n", + "All files are 3436 lines long with 124 character per line. \n", + " Headers are 39 lines long. \n", + " col 1 : Fortran character control: 1 = page feed 0 = line feed \n", + " format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 \n", + " These formats are reflected in the pandas widths variable below, see the statement \n", + " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), \n", + " Pandas has also a variable header, with length 39 in this case. \n", + "\"\"\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The data we are interested in are in columns 2, 3, 4 and 11, giving us\n", + "the number of neutrons, protons, mass numbers and binding energies,\n", + "respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will\n", + "covert them into the **pandas** DataFrame structure." + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [], + "source": [ + "# Read the experimental data with Pandas\n", + "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", + " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", + " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", + " header=39,\n", + " index_col=False)\n", + "\n", + "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", + "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", + "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", + "Masses = Masses.dropna()\n", + "# Convert from keV to MeV.\n", + "Masses['Ebinding'] /= 1000\n", + "\n", + "# Group the DataFrame by nucleon number, A.\n", + "Masses = Masses.groupby('A')\n", + "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", + "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We have now read in the data, grouped them according to the variables we are interested in. \n", + "We see how easy it is to reorganize the data using **pandas**. If we\n", + "were to do these operations in C/C++ or Fortran, we would have had to\n", + "write various functions/subroutines which perform the above\n", + "reorganizations for us. Having reorganized the data, we can now start\n", + "to make some simple fits using both the functionalities in **numpy** and\n", + "**Scikit-Learn** afterwards. \n", + "\n", + "Now we define five variables which contain\n", + "the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves." + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [], + "source": [ + "A = Masses['A']\n", + "Z = Masses['Z']\n", + "N = Masses['N']\n", + "Element = Masses['Element']\n", + "Energies = Masses['Ebinding']\n", + "print(Masses)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The next step, and we will define this mathematically later, is to set up the so-called **design matrix**. We will throughout call this matrix $\\boldsymbol{X}$.\n", + "It has dimensionality $p\\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit." + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": {}, + "outputs": [], + "source": [ + "# Now we set up the design matrix X\n", + "X = np.zeros((len(A),5))\n", + "X[:,0] = 1\n", + "X[:,1] = A\n", + "X[:,2] = A**(2.0/3.0)\n", + "X[:,3] = A**(-1.0/3.0)\n", + "X[:,4] = A**(-1.0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "With **scikitlearn** we are now ready to use linear regression and fit our data." + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": {}, + "outputs": [], + "source": [ + "clf = skl.LinearRegression().fit(X, Energies)\n", + "fity = clf.predict(X)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Pretty simple! \n", + "Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data." + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": {}, + "outputs": [], + "source": [ + "# The mean squared error \n", + "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, fity))\n", + "# Explained variance score: 1 is perfect prediction \n", + "print('Variance score: %.2f' % r2_score(Energies, fity))\n", + "# Mean absolute error \n", + "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))\n", + "print(clf.coef_, clf.intercept_)\n", + "\n", + "Masses['Eapprox'] = fity\n", + "# Generate a plot comparing the experimental with the fitted values values.\n", + "fig, ax = plt.subplots()\n", + "ax.set_xlabel(r'$A = N + Z$')\n", + "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", + "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", + " label='Ame2016')\n", + "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", + " label='Fit')\n", + "ax.legend()\n", + "save_fig(\"Masses2016\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Seeing the wood for the trees\n", + "\n", + "As a teaser, let us now see how we can do this with decision trees using **scikit-learn**. Later we will switch to so-called **random forests**!" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "#Decision Tree Regression\n", + "from sklearn.tree import DecisionTreeRegressor\n", + "regr_1=DecisionTreeRegressor(max_depth=5)\n", + "regr_2=DecisionTreeRegressor(max_depth=7)\n", + "regr_3=DecisionTreeRegressor(max_depth=9)\n", + "regr_1.fit(X, Energies)\n", + "regr_2.fit(X, Energies)\n", + "regr_3.fit(X, Energies)\n", + "\n", + "\n", + "y_1 = regr_1.predict(X)\n", + "y_2 = regr_2.predict(X)\n", + "y_3=regr_3.predict(X)\n", + "Masses['Eapprox'] = y_3\n", + "# Plot the results\n", + "plt.figure()\n", + "plt.plot(A, Energies, color=\"blue\", label=\"Data\", linewidth=2)\n", + "plt.plot(A, y_1, color=\"red\", label=\"max_depth=5\", linewidth=2)\n", + "plt.plot(A, y_2, color=\"green\", label=\"max_depth=7\", linewidth=2)\n", + "plt.plot(A, y_3, color=\"m\", label=\"max_depth=9\", linewidth=2)\n", + "\n", + "plt.xlabel(\"$A$\")\n", + "plt.ylabel(\"$E$[MeV]\")\n", + "plt.title(\"Decision Tree Regression\")\n", + "plt.legend()\n", + "save_fig(\"Masses2016Trees\")\n", + "plt.show()\n", + "print(Masses)\n", + "print(np.mean( (Energies-y_1)**2))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### And what about using neural networks?\n", + "\n", + "The **seaborn** package allows us to visualize data in an efficient way. Note that we use **scikit-learn**'s multi-layer perceptron (or feed forward neural network) \n", + "functionality." + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.neural_network import MLPRegressor\n", + "from sklearn.metrics import accuracy_score\n", + "import seaborn as sns\n", + "\n", + "X_train = X\n", + "Y_train = Energies\n", + "n_hidden_neurons = 100\n", + "epochs = 100\n", + "# store models for later use\n", + "eta_vals = np.logspace(-5, 1, 7)\n", + "lmbd_vals = np.logspace(-5, 1, 7)\n", + "# store the models for later use\n", + "DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", + "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "sns.set()\n", + "for i, eta in enumerate(eta_vals):\n", + " for j, lmbd in enumerate(lmbd_vals):\n", + " dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',\n", + " alpha=lmbd, learning_rate_init=eta, max_iter=epochs)\n", + " dnn.fit(X_train, Y_train)\n", + " DNN_scikit[i][j] = dnn\n", + " train_accuracy[i][j] = dnn.score(X_train, Y_train)\n", + "\n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Training Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A first summary\n", + "\n", + "The aim behind these introductory words was to present to you various\n", + "Python libraries and their functionalities, in particular libraries like\n", + "**numpy**, **pandas**, **xarray** and **matplotlib** and other that make our life much easier\n", + "in handling various data sets and visualizing data. \n", + "\n", + "Furthermore,\n", + "**Scikit-Learn** allows us with few lines of code to implement popular\n", + "Machine Learning algorithms for supervised learning. Later we will meet **Tensorflow**, a powerful library for deep learning. \n", + "Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.5" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/doc/pub/Regression/ipynb/How2ReadData.ipynb b/doc/pub/Regression/ipynb/How2ReadData.ipynb new file mode 100644 index 000000000..2705ffc7b --- /dev/null +++ b/doc/pub/Regression/ipynb/How2ReadData.ipynb @@ -0,0 +1,2015 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Data Analysis and Machine Learning: Getting started, our first data and Machine Learning encounters\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Dec 25, 2019**\n", + "\n", + "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Introduction\n", + "\n", + "Our emphasis throughout this series of lectures \n", + "is on understanding the mathematical aspects of\n", + "different algorithms used in the fields of data analysis and machine learning. \n", + "\n", + "However, where possible we will emphasize the\n", + "importance of using available software. We start thus with a hands-on\n", + "and top-down approach to machine learning. The aim is thus to start with\n", + "relevant data or data we have produced \n", + "and use these to introduce statistical data analysis\n", + "concepts and machine learning algorithms before we delve into the\n", + "algorithms themselves. The examples we will use in the beginning, start with simple\n", + "polynomials with random noise added. We will use the Python\n", + "software package [Scikit-Learn](http://scikit-learn.org/stable/) and\n", + "introduce various machine learning algorithms to make fits of\n", + "the data and predictions. We move thereafter to more interesting\n", + "cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example).\n", + "These are examples where we can easily set up the data and\n", + "then use machine learning algorithms included in for example\n", + "**Scikit-Learn**. \n", + "\n", + "These examples will serve us the purpose of getting\n", + "started. Furthermore, they allow us to catch more than two birds with\n", + "a stone. They will allow us to bring in some programming specific\n", + "topics and tools as well as showing the power of various Python \n", + "libraries for machine learning and statistical data analysis. \n", + "\n", + "Here, we will mainly focus on two\n", + "specific Python packages for Machine Learning, Scikit-Learn and\n", + "Tensorflow (see below for links etc). Moreover, the examples we\n", + "introduce will serve as inputs to many of our discussions later, as\n", + "well as allowing you to set up models and produce your own data and\n", + "get started with programming.\n", + "\n", + "\n", + "\n", + "## What is Machine Learning?\n", + "\n", + "Statistics, data science and machine learning form important fields of\n", + "research in modern science. They describe how to learn and make\n", + "predictions from data, as well as allowing us to extract important\n", + "correlations about physical process and the underlying laws of motion\n", + "in large data sets. The latter, big data sets, appear frequently in\n", + "essentially all disciplines, from the traditional Science, Technology,\n", + "Mathematics and Engineering fields to Life Science, Law, education\n", + "research, the Humanities and the Social Sciences. \n", + "\n", + "It has become more\n", + "and more common to see research projects on big data in for example\n", + "the Social Sciences where extracting patterns from complicated survey\n", + "data is one of many research directions. Having a solid grasp of data\n", + "analysis and machine learning is thus becoming central to scientific\n", + "computing in many fields, and competences and skills within the fields\n", + "of machine learning and scientific computing are nowadays strongly\n", + "requested by many potential employers. The latter cannot be\n", + "overstated, familiarity with machine learning has almost become a\n", + "prerequisite for many of the most exciting employment opportunities,\n", + "whether they are in bioinformatics, life science, physics or finance,\n", + "in the private or the public sector. This author has had several\n", + "students or met students who have been hired recently based on their\n", + "skills and competences in scientific computing and data science, often\n", + "with marginal knowledge of machine learning.\n", + "\n", + "Machine learning is a subfield of computer science, and is closely\n", + "related to computational statistics. It evolved from the study of\n", + "pattern recognition in artificial intelligence (AI) research, and has\n", + "made contributions to AI tasks like computer vision, natural language\n", + "processing and speech recognition. Many of the methods we will study are also \n", + "strongly rooted in basic mathematics and physics research. \n", + "\n", + "Ideally, machine learning represents the science of giving computers\n", + "the ability to learn without being explicitly programmed. The idea is\n", + "that there exist generic algorithms which can be used to find patterns\n", + "in a broad class of data sets without having to write code\n", + "specifically for each problem. The algorithm will build its own logic\n", + "based on the data. You should however always keep in mind that\n", + "machines and algorithms are to a large extent developed by humans. The\n", + "insights and knowledge we have about a specific system, play a central\n", + "role when we develop a specific machine learning algorithm. \n", + "\n", + "Machine learning is an extremely rich field, in spite of its young\n", + "age. The increases we have seen during the last three decades in\n", + "computational capabilities have been followed by developments of\n", + "methods and techniques for analyzing and handling large date sets,\n", + "relying heavily on statistics, computer science and mathematics. The\n", + "field is rather new and developing rapidly. Popular software packages\n", + "written in Python for machine learning like\n", + "[Scikit-learn](http://scikit-learn.org/stable/),\n", + "[Tensorflow](https://www.tensorflow.org/),\n", + "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all\n", + "freely available at their respective GitHub sites, encompass\n", + "communities of developers in the thousands or more. And the number of\n", + "code developers and contributors keeps increasing. Not all the\n", + "algorithms and methods can be given a rigorous mathematical\n", + "justification, opening up thereby large rooms for experimenting and\n", + "trial and error and thereby exciting new developments. However, a\n", + "solid command of linear algebra, multivariate theory, probability\n", + "theory, statistical data analysis, understanding errors and Monte\n", + "Carlo methods are central elements in a proper understanding of many\n", + "of algorithms and methods we will discuss.\n", + "\n", + "\n", + "\n", + "## Types of Machine Learning\n", + "\n", + "\n", + "The approaches to machine learning are many, but are often split into\n", + "two main categories. In *supervised learning* we know the answer to a\n", + "problem, and let the computer deduce the logic behind it. On the other\n", + "hand, *unsupervised learning* is a method for finding patterns and\n", + "relationship in data sets without any prior knowledge of the system.\n", + "Some authours also operate with a third category, namely\n", + "*reinforcement learning*. This is a paradigm of learning inspired by\n", + "behavioral psychology, where learning is achieved by trial-and-error,\n", + "solely from rewards and punishment.\n", + "\n", + "Another way to categorize machine learning tasks is to consider the\n", + "desired output of a system. Some of the most common tasks are:\n", + "\n", + " * Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.\n", + "\n", + " * Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.\n", + "\n", + " * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n", + "\n", + "The methods we cover have three main topics in common, irrespective of\n", + "whether we deal with supervised or unsupervised learning. The first\n", + "ingredient is normally our data set (which can be subdivided into\n", + "training and test data), the second item is a model which is normally a\n", + "function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. \n", + "\n", + "The last ingredient is a so-called **cost**\n", + "function which allows us to present an estimate on how good our model\n", + "is in reproducing the data it is supposed to train. \n", + "At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of **gradient** methods.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Software and needed installations\n", + "\n", + "We will make extensive use of Python as programming language and its\n", + "myriad of available libraries. You will find\n", + "Jupyter notebooks invaluable in your work. You can run **R**\n", + "codes in the Jupyter/IPython notebooks, with the immediate benefit of\n", + "visualizing your data. You can also use compiled languages like C++,\n", + "Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be\n", + "on Python.\n", + "\n", + "\n", + "If you have Python installed (we strongly recommend Python3) and you feel\n", + "pretty familiar with installing different packages, we recommend that\n", + "you install the following Python packages via **pip** as \n", + "\n", + "1. pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow \n", + "\n", + "For Python3, replace **pip** with **pip3**.\n", + "\n", + "For OSX users we recommend, after having installed Xcode, to\n", + "install **brew**. Brew allows for a seamless installation of additional\n", + "software via for example \n", + "\n", + "1. brew install python3\n", + "\n", + "For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,\n", + "you can use **pip** as well and simply install Python as \n", + "\n", + "1. sudo apt-get install python3 (or python for pyhton2.7)\n", + "\n", + "etc etc. \n", + "\n", + "\n", + "\n", + "## Python installers\n", + "\n", + "If you don't want to perform these operations separately and venture\n", + "into the hassle of exploring how to set up dependencies and paths, we\n", + "recommend two widely used distrubutions which set up all relevant\n", + "dependencies for Python, namely \n", + "\n", + "* [Anaconda](https://docs.anaconda.com/), \n", + "\n", + "which is an open source\n", + "distribution of the Python and R programming languages for large-scale\n", + "data processing, predictive analytics, and scientific computing, that\n", + "aims to simplify package management and deployment. Package versions\n", + "are managed by the package management system **conda**. \n", + "\n", + "* [Enthought canopy](https://www.enthought.com/product/canopy/) \n", + "\n", + "is a Python\n", + "distribution for scientific and analytic computing distribution and\n", + "analysis environment, available for free and under a commercial\n", + "license.\n", + "\n", + "Furthermore, [Google's Colab](https://colab.research.google.com/notebooks/welcome.ipynb) is a free Jupyter notebook environment that requires \n", + "no setup and runs entirely in the cloud. Try it out!\n", + "\n", + "## Useful Python libraries\n", + "Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)\n", + "\n", + "* [NumPy](https://www.numpy.org/) is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays\n", + "\n", + "* [The pandas](https://pandas.pydata.org/) library provides high-performance, easy-to-use data structures and data analysis tools \n", + "\n", + "* [Xarray](http://xarray.pydata.org/en/stable/) is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!\n", + "\n", + "* [Scipy](https://www.scipy.org/) (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. \n", + "\n", + "* [Matplotlib](https://matplotlib.org/) is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.\n", + "\n", + "* [Autograd](https://github.com/HIPS/autograd) can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives\n", + "\n", + "* [SymPy](https://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. \n", + "\n", + "* [scikit-learn](https://scikit-learn.org/stable/) has simple and efficient tools for machine learning, data mining and data analysis\n", + "\n", + "* [TensorFlow](https://www.tensorflow.org/) is a Python library for fast numerical computing created and released by Google\n", + "\n", + "* [Keras](https://keras.io/) is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano\n", + "\n", + "* And many more such as [pytorch](https://pytorch.org/), [Theano](https://pypi.org/project/Theano/) etc \n", + "\n", + "## Installing R, C++, cython or Julia\n", + "\n", + "You will also find it convenient to utilize **R**. We will mainly\n", + "use Python during our lectures and in various projects and exercises.\n", + "Those of you\n", + "already familiar with **R** should feel free to continue using **R**, keeping\n", + "however an eye on the parallel Python set ups. Similarly, if you are a\n", + "Python afecionado, feel free to explore **R** as well. Jupyter/Ipython\n", + "notebook allows you to run **R** codes interactively in your\n", + "browser. The software library **R** is really tailored for statistical data analysis\n", + "and allows for an easy usage of the tools and algorithms we will discuss in these\n", + "lectures.\n", + "\n", + "To install **R** with Jupyter notebook \n", + "[follow the link here](https://mpacer.org/maths/r-kernel-for-ipython-notebook)\n", + "\n", + "\n", + "\n", + "\n", + "## Installing R, C++, cython, Numba etc\n", + "\n", + "\n", + "For the C++ aficionados, Jupyter/IPython notebook allows you also to\n", + "install C++ and run codes written in this language interactively in\n", + "the browser. Since we will emphasize writing many of the algorithms\n", + "yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming\n", + "languages.\n", + "\n", + "To add more entropy, **cython** can also be used when running your\n", + "notebooks. It means that Python with the jupyter notebook\n", + "setup allows you to integrate widely popular softwares and tools for\n", + "scientific computing. Similarly, the \n", + "[Numba Python package](https://numba.pydata.org/) delivers increased performance\n", + "capabilities with minimal rewrites of your codes. With its\n", + "versatility, including symbolic operations, Python offers a unique\n", + "computational environment. Your jupyter notebook can easily be\n", + "converted into a nicely rendered **PDF** file or a Latex file for\n", + "further processing. For example, convert to latex as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " pycod jupyter nbconvert filename.ipynb --to latex \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And to add more versatility, the Python package [SymPy](http://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python. \n", + "\n", + "Finally, if you wish to use the light mark-up language \n", + "[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML \n", + "formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using **doconce**.\n", + "\n", + "\n", + "\n", + "## Numpy examples and Important Matrix and vector handling packages\n", + "\n", + "There are several central software libraries for linear algebra and eigenvalue problems. Several of the more\n", + "popular ones have been wrapped into ofter software packages like those from the widely used text **Numerical Recipes**. The original source codes in many of the available packages are often taken from the widely used\n", + "software package LAPACK, which follows two other popular packages\n", + "developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.\n", + "\n", + " * LINPACK: package for linear equations and least square problems.\n", + "\n", + " * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.\n", + "\n", + " * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from .\n", + "\n", + "## Basic Matrix Features\n", + "\n", + "**Matrix properties reminder.**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{A} =\n", + " \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", + " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", + " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", + " a_{41} & a_{42} & a_{43} & a_{44}\n", + " \\end{bmatrix}\\qquad\n", + "\\mathbf{I} =\n", + " \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n", + " 0 & 1 & 0 & 0 \\\\\n", + " 0 & 0 & 1 & 0 \\\\\n", + " 0 & 0 & 0 & 1\n", + " \\end{bmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The inverse of a matrix is defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \\left (A^{T} \\right )^{-1}$ real orthogonal $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \\left (A^{\\dagger} \\right )^{-1}$ unitary $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$
\n", + "\n", + "\n", + "\n", + "\n", + "### Some famous Matrices\n", + "\n", + " * Diagonal if $a_{ij}=0$ for $i\\ne j$\n", + "\n", + " * Upper triangular if $a_{ij}=0$ for $i > j$\n", + "\n", + " * Lower triangular if $a_{ij}=0$ for $i < j$\n", + "\n", + " * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$\n", + "\n", + " * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$\n", + "\n", + " * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$\n", + "\n", + " * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$\n", + "\n", + " * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$\n", + "\n", + " * Banded, block upper triangular, block lower triangular....\n", + "\n", + "### More Basic Matrix Features\n", + "\n", + "**Some Equivalent Statements.**\n", + "\n", + "For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", + "\n", + " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", + "\n", + " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", + "\n", + " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", + "\n", + " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", + "\n", + " * $\\mathbf{A}$ is a product of elementary matrices.\n", + "\n", + " * $0$ is not eigenvalue of $\\mathbf{A}$.\n", + "\n", + "\n", + "\n", + "\n", + "## Numpy and arrays\n", + "[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution," + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "n = 10\n", + "x = np.random.normal(size=n)\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.\n", + "Another alternative is to declare a vector as follows" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.array([1, 2, 3])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++\n", + "start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \\dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4, 7, 8]))\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the last example we used Numpy's unary function $np.log$. This function is\n", + "highly tuned to compute array elements since the code is vectorized\n", + "and does not require looping. We normaly recommend that you use the\n", + "Numpy intrinsic functions instead of the corresponding **log** function\n", + "from Python's **math** module. The looping is done explicitely by the\n", + "**np.log** function. The alternative, and slower way to compute the\n", + "logarithms of a vector would be to write" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "from math import log\n", + "x = np.array([4, 7, 8])\n", + "for i in range(0, len(x)):\n", + " x[i] = log(x[i])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We note that our code is much longer already and we need to import the **log** function from the **math** module. \n", + "The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4, 7, 8], dtype = np.float64))\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4.0, 7.0, 8.0])\n", + "print(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "x = np.log(np.array([4.0, 7.0, 8.0])\n", + "print(x.itemsize)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Matrices in Python\n", + "\n", + "Having defined vectors, we are now ready to try out matrices. We can\n", + "define a $3 \\times 3 $ real matrix $\\hat{A}$ as (recall that we user\n", + "lowercase letters for vectors and uppercase letters for matrices)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "# print the first column, row-major order and elements start with 0\n", + "print(A[:,0])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can continue this was by printing out other columns or rows. The example here prints out the second column" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", + "# print the first column, row-major order and elements start with 0\n", + "print(A[1,:])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to zero\n", + "A = np.zeros( (n, n) )\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or initializing all elements to" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to one\n", + "A = np.ones( (n, n) )\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or as unitarily distributed random numbers (see the material on random number generators in the statistics part)" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "n = 10\n", + "# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \\in [0, 1]\n", + "A = np.random.rand(n, n)\n", + "print(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n", + "As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n", + "$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", + " \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n", + " \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where for example" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma_{xy} =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. \n", + "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\hat{W}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n", + " x_1 & y_1 & z_1 \\\\\n", + " x_2 & y_2 & z_2 \\\\\n", + " \\dots & \\dots & \\dots \\\\\n", + " x_{n-2} & y_{n-2} & z_{n-2} \\\\\n", + " x_{n-1} & y_{n-1} & z_{n-1}\n", + " \\end{bmatrix},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which in turn is converted into into the $3\\times 3$ covariance matrix\n", + "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", + "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", + "function **np.mean(x)**. We can also extract the eigenvalues of the\n", + "covariance matrix through the **np.linalg.eig()** function." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [], + "source": [ + "# Importing various packages\n", + "import numpy as np\n", + "\n", + "n = 100\n", + "x = np.random.normal(size=n)\n", + "print(np.mean(x))\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "print(np.mean(y))\n", + "z = x**3+np.random.normal(size=n)\n", + "print(np.mean(z))\n", + "W = np.vstack((x, y, z))\n", + "Sigma = np.cov(W)\n", + "print(Sigma)\n", + "Eigvals, Eigvecs = np.linalg.eig(Sigma)\n", + "print(Eigvals)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from scipy import sparse\n", + "eye = np.eye(4)\n", + "print(eye)\n", + "sparse_mtx = sparse.csr_matrix(eye)\n", + "print(sparse_mtx)\n", + "x = np.linspace(-10,10,100)\n", + "y = np.sin(x)\n", + "plt.plot(x,y,marker='x')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Meet the Pandas\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "Another useful Python package is\n", + "[pandas](https://pandas.pydata.org/), which is an open source library\n", + "providing high-performance, easy-to-use data structures and data\n", + "analysis tools for Python. **pandas** stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.\n", + "**pandas** has two major classes, the **DataFrame** class with two-dimensional data objects and tabular data organized in columns and the class **Series** with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. \n", + "**pandas** allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. \n", + "\n", + "The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of **pandas**, in particular in connection with classification of data." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [], + "source": [ + "import pandas as pd\n", + "from IPython.display import display\n", + "data = {'First Name': [\"Frodo\", \"Bilbo\", \"Aragorn II\", \"Samwise\"],\n", + " 'Last Name': [\"Baggins\", \"Baggins\",\"Elessar\",\"Gamgee\"],\n", + " 'Place of birth': [\"Shire\", \"Shire\", \"Eriador\", \"Shire\"],\n", + " 'Date of Birth T.A.': [2968, 2890, 2931, 2980]\n", + " }\n", + "data_pandas = pd.DataFrame(data)\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the above we have imported **pandas** with the shorthand **pd**, the latter has become the standard way we import **pandas**. We make then a list of various variables\n", + "and reorganize the aboves lists into a **DataFrame** and then print out a neat table with specific column labels as *Name*, *place of birth* and *date of birth*.\n", + "Displaying these results, we see that the indices are given by the default numbers from zero to three.\n", + "**pandas** is extremely flexible and we can easily change the above indices by defining a new type of indexing as" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [], + "source": [ + "data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Thereafter we display the content of the row which begins with the index **Aragorn**" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [], + "source": [ + "display(data_pandas.loc['Aragorn'])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can easily append data to this, for example" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "new_hobbit = {'First Name': [\"Peregrin\"],\n", + " 'Last Name': [\"Took\"],\n", + " 'Place of birth': [\"Shire\"],\n", + " 'Date of Birth T.A.': [2990]\n", + " }\n", + "data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))\n", + "display(data_pandas)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here are other examples where we use the **DataFrame** functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix \n", + "of dimensionality $10\\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations." + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "from IPython.display import display\n", + "np.random.seed(100)\n", + "# setting up a 10 x 5 matrix\n", + "rows = 10\n", + "cols = 5\n", + "a = np.random.randn(rows,cols)\n", + "df = pd.DataFrame(a)\n", + "display(df)\n", + "print(df.mean())\n", + "print(df.std())\n", + "display(df**2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Thereafter we can select specific columns only and plot final results" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [], + "source": [ + "df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']\n", + "df.index = np.arange(10)\n", + "\n", + "display(df)\n", + "print(df['Second'].mean() )\n", + "print(df.info())\n", + "print(df.describe())\n", + "\n", + "from pylab import plt, mpl\n", + "plt.style.use('seaborn')\n", + "mpl.rcParams['font.family'] = 'serif'\n", + "\n", + "df.cumsum().plot(lw=2.0, figsize=(10,6))\n", + "plt.show()\n", + "\n", + "\n", + "df.plot.bar(figsize=(10,6), rot=15)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can produce a $4\\times 4$ matrix" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [], + "source": [ + "b = np.arange(16).reshape((4,4))\n", + "print(b)\n", + "df1 = pd.DataFrame(b)\n", + "print(df1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and many other operations. \n", + "\n", + "The **Series** class is another important class included in\n", + "**pandas**. You can view it as a specialization of **DataFrame** but where\n", + "we have just a single column of data. It shares many of the same features as _DataFrame. As with **DataFrame**,\n", + "most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.\n", + "As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.\n", + "For multidimensional arrays, we recommend strongly [xarray](http://xarray.pydata.org/en/stable/). **xarray** has much of the same flexibility as **pandas**, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both **pandas** and **xarray**. \n", + "\n", + "\n", + "\n", + "## Reading Data and fitting\n", + "\n", + "In order to study various Machine Learning algorithms, we need to\n", + "access data. Acccessing data is an essential step in all machine\n", + "learning algorithms. In particular, setting up the so-called **design\n", + "matrix** (to be defined below) is often the first element we need in\n", + "order to perform our calculations. To set up the design matrix means\n", + "reading (and later, when the calculations are done, writing) data\n", + "in various formats, The formats span from reading files from disk,\n", + "loading data from databases and interacting with online sources\n", + "like web application programming interfaces (APIs).\n", + "\n", + "In handling various input formats, as discussed above, we will mainly stay with **pandas**,\n", + "a Python package which allows us, in a seamless and painless way, to\n", + "deal with a multitude of formats, from standard **csv** (comma separated\n", + "values) files, via **excel**, **html** to **hdf5** formats. With **pandas**\n", + "and the **DataFrame** and **Series** functionalities we are able to convert text data\n", + "into the calculational formats we need for a specific algorithm. And our code is going to be \n", + "pretty close the basic mathematical expressions.\n", + "\n", + "Our first data set is going to be a classic from nuclear physics, namely all\n", + "available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set. \n", + "\n", + "We will show some of the\n", + "strengths of packages like **Scikit-Learn** in fitting nuclear binding energies to\n", + "specific functions using linear regression first. Then, as a teaser, we will show you how \n", + "you can easily implement other algorithms like decision trees and random forests and neural networks.\n", + "\n", + "But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,\n", + "(don't be offended) fitting straight lines!\n", + "\n", + "\n", + "### Simple linear regression model using **scikit-learn**\n", + "\n", + "We start with perhaps our simplest possible example, using **Scikit-Learn** to perform linear regression analysis on a data set produced by us. \n", + "\n", + "What follows is a simple Python code where we have defined a function\n", + "$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. \n", + "The numbers in the vector $\\hat{x}$ are given\n", + "by random numbers generated with a uniform distribution with entries\n", + "$x_i \\in [0,1]$ (more about probability distribution functions\n", + "later). These values are then used to define a function $y(x)$\n", + "(tabulated again as a vector) with a linear dependence on $x$ plus a\n", + "random noise added via the normal distribution.\n", + "\n", + "\n", + "The Numpy functions are imported used the **import numpy as np**\n", + "statement and the random number generator for the uniform distribution\n", + "is called using the function **np.random.rand()**, where we specificy\n", + "that we want $100$ random variables. Using Numpy we define\n", + "automatically an array with the specified number of elements, $100$ in\n", + "our case. With the Numpy function **randn()** we can compute random\n", + "numbers with the normal distribution (mean value $\\mu$ equal to zero and\n", + "variance $\\sigma^2$ set to one) and produce the values of $y$ assuming a linear\n", + "dependence as function of $x$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y = 2x+N(0,1),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $N(0,1)$ represents random numbers generated by the normal\n", + "distribution. From **Scikit-Learn** we import then the\n", + "**LinearRegression** functionality and make a prediction $\\tilde{y} =\n", + "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n", + "data $(\\hat{x},\\hat{y})$ for our training data. The Python package\n", + "**scikit-learn** has also a functionality which extracts the above\n", + "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n", + "distinguish between training data and test data.\n", + "\n", + "For plotting we use the Python package\n", + "[matplotlib](https://matplotlib.org/) which produces publication\n", + "quality figures. Feel free to explore the extensive\n", + "[gallery](https://matplotlib.org/gallery/index.html) of examples. In\n", + "this example we plot our original values of $x$ and $y$ as well as the\n", + "prediction **ypredict** ($\\tilde{y}$), which attempts at fitting our\n", + "data with a straight line.\n", + "\n", + "The Python code follows here." + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [], + "source": [ + "# Importing various packages\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 2*x+np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "xnew = np.array([[0],[1]])\n", + "ypredict = linreg.predict(xnew)\n", + "\n", + "plt.plot(xnew, ypredict, \"r-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0,1.0,0, 5.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Simple Linear Regression')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This example serves several aims. It allows us to demonstrate several\n", + "aspects of data analysis and later machine learning algorithms. The\n", + "immediate visualization shows that our linear fit is not\n", + "impressive. It goes through the data points, but there are many\n", + "outliers which are not reproduced by our linear regression. We could\n", + "now play around with this small program and change for example the\n", + "factor in front of $x$ and the normal distribution. Try to change the\n", + "function $y$ to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y = 10x+0.01 \\times N(0,1),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $x$ is defined as before. Does the fit look better? Indeed, by\n", + "reducing the role of the noise given by the normal distribution we see immediately that\n", + "our linear prediction seemingly reproduces better the training\n", + "set. However, this testing 'by the eye' is obviouly not satisfactory in the\n", + "long run. Here we have only defined the training data and our model, and \n", + "have not discussed a more rigorous approach to the **cost** function.\n", + "\n", + "We need more rigorous criteria in defining whether we have succeeded or\n", + "not in modeling our training data. You will be surprised to see that\n", + "many scientists seldomly venture beyond this 'by the eye' approach. A\n", + "standard approach for the *cost* function is the so-called $\\chi^2$\n", + "function (a variant of the mean-squared error (MSE))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\chi^2 = \\frac{1}{n}\n", + "\\sum_{i=0}^{n-1}\\frac{(y_i-\\tilde{y}_i)^2}{\\sigma_i^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\sigma_i^2$ is the variance (to be defined later) of the entry\n", + "$y_i$. We may not know the explicit value of $\\sigma_i^2$, it serves\n", + "however the aim of scaling the equations and make the cost function\n", + "dimensionless. \n", + "\n", + "Minimizing the cost function is a central aspect of\n", + "our discussions to come. Finding its minima as function of the model\n", + "parameters ($\\alpha$ and $\\beta$ in our case) will be a recurring\n", + "theme in these series of lectures. Essentially all machine learning\n", + "algorithms we will discuss center around the minimization of the\n", + "chosen cost function. This depends in turn on our specific\n", + "model for describing the data, a typical situation in supervised\n", + "learning. Automatizing the search for the minima of the cost function is a\n", + "central ingredient in all algorithms. Typical methods which are\n", + "employed are various variants of **gradient** methods. These will be\n", + "discussed in more detail later. Again, you'll be surprised to hear that\n", + "many practitioners minimize the above function ''by the eye', popularly dubbed as \n", + "'chi by the eye'. That is, change a parameter and see (visually and numerically) that \n", + "the $\\chi^2$ function becomes smaller. \n", + "\n", + "There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define \n", + "the relative error (why would we prefer the MSE instead of the relative error?) as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\hat{y} -\\hat{\\tilde{y}}\\vert}{\\vert \\hat{y}\\vert}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The squared cost function results in an arithmetic mean-unbiased\n", + "estimator, and the absolute-value cost function results in a\n", + "median-unbiased estimator (in the one-dimensional case, and a\n", + "geometric median-unbiased estimator for the multi-dimensional\n", + "case). The squared cost function has the disadvantage that it has the tendency\n", + "to be dominated by outliers.\n", + "\n", + "We can modify easily the above Python code and plot the relative error instead" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 5*x+0.01*np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "ypredict = linreg.predict(x)\n", + "\n", + "plt.plot(x, np.abs(ypredict-y)/abs(y), \"ro\")\n", + "plt.axis([0,1.0,0.0, 0.5])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$\\epsilon_{\\mathrm{relative}}$')\n", + "plt.title(r'Relative error')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Depending on the parameter in front of the normal distribution, we may\n", + "have a small or larger relative error. Try to play around with\n", + "different training data sets and study (graphically) the value of the\n", + "relative error.\n", + "\n", + "As mentioned above, **Scikit-Learn** has an impressive functionality.\n", + "We can for example extract the values of $\\alpha$ and $\\beta$ and\n", + "their error estimates, or the variance and standard deviation and many\n", + "other properties from the statistical data analysis. \n", + "\n", + "Here we show an\n", + "example of the functionality of **Scikit-Learn**." + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np \n", + "import matplotlib.pyplot as plt \n", + "from sklearn.linear_model import LinearRegression \n", + "from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error\n", + "\n", + "x = np.random.rand(100,1)\n", + "y = 2.0+ 5*x+0.5*np.random.randn(100,1)\n", + "linreg = LinearRegression()\n", + "linreg.fit(x,y)\n", + "ypredict = linreg.predict(x)\n", + "print('The intercept alpha: \\n', linreg.intercept_)\n", + "print('Coefficient beta : \\n', linreg.coef_)\n", + "# The mean squared error \n", + "print(\"Mean squared error: %.2f\" % mean_squared_error(y, ypredict))\n", + "# Explained variance score: 1 is perfect prediction \n", + "print('Variance score: %.2f' % r2_score(y, ypredict))\n", + "# Mean squared log error \n", + "print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )\n", + "# Mean absolute error \n", + "print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))\n", + "plt.plot(x, ypredict, \"r-\")\n", + "plt.plot(x, y ,'ro')\n", + "plt.axis([0.0,1.0,1.5, 7.0])\n", + "plt.xlabel(r'$x$')\n", + "plt.ylabel(r'$y$')\n", + "plt.title(r'Linear Regression fit ')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n", + "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n", + "\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The smaller the value, the better the fit. Ideally we would like to\n", + "have an MSE equal zero. The attentive reader has probably recognized\n", + "this function as being similar to the $\\chi^2$ function defined above.\n", + "\n", + "The **r2score** function computes $R^2$, the coefficient of\n", + "determination. It provides a measure of how well future samples are\n", + "likely to be predicted by the model. Best possible score is 1.0 and it\n", + "can be negative (because the model can be arbitrarily worse). A\n", + "constant model that always predicts the expected value of $\\hat{y}$,\n", + "disregarding the input features, would get a $R^2$ score of $0.0$.\n", + "\n", + "If $\\tilde{\\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have defined the mean value of $\\hat{y}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Another quantity taht we will meet again in our discussions of regression analysis is \n", + " the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error.\n", + "The MAE is defined as follows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\text{MAE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We present the \n", + "squared logarithmic (quadratic) error" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\text{MSLE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\log_e (x)$ stands for the natural logarithm of $x$. This error\n", + "estimate is best to use when targets having exponential growth, such\n", + "as population counts, average sales of a commodity over a span of\n", + "years etc. \n", + "\n", + "\n", + "Finally, another cost function is the Huber cost function used in robust regression.\n", + "\n", + "The rationale behind this possible cost function is its reduced\n", + "sensitivity to outliers in the data set. In our discussions on\n", + "dimensionality reduction and normalization of data we will meet other\n", + "ways of dealing with outliers.\n", + "\n", + "The Huber cost function is defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "H_{\\delta}(a)={\\begin{cases}{\\frac {1}{2}}{a^{2}}&{\\text{for }}|a|\\leq \\delta ,\\\\\\delta (|a|-{\\frac {1}{2}}\\delta ),&{\\text{otherwise.}}\\end{cases}}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here $a=\\boldsymbol{y} - \\boldsymbol{\\tilde{y}}$.\n", + "We will discuss in more\n", + "detail these and other functions in the various lectures. We conclude this part with another example. Instead of \n", + "a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn." + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "import random\n", + "from sklearn.linear_model import Ridge\n", + "from sklearn.preprocessing import PolynomialFeatures\n", + "from sklearn.pipeline import make_pipeline\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "x=np.linspace(0.02,0.98,200)\n", + "noise = np.asarray(random.sample((range(200)),200))\n", + "y=x**3*noise\n", + "yn=x**3*100\n", + "poly3 = PolynomialFeatures(degree=3)\n", + "X = poly3.fit_transform(x[:,np.newaxis])\n", + "clf3 = LinearRegression()\n", + "clf3.fit(X,y)\n", + "\n", + "Xplot=poly3.fit_transform(x[:,np.newaxis])\n", + "poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')\n", + "plt.plot(x,yn, color='red', label=\"True Cubic\")\n", + "plt.scatter(x, y, label='Data', color='orange', s=15)\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "def error(a):\n", + " for i in y:\n", + " err=(y-yn)/yn\n", + " return abs(np.sum(err))/len(err)\n", + "\n", + "print (error(y))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### To our real data: nuclear binding energies. Brief reminder on masses and binding energies\n", + "\n", + "Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding\n", + "energies. A basic quantity which can be measured for the ground\n", + "states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with\n", + "atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). \n", + "\n", + "Atomic masses are usually tabulated in terms of the mass excess defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\Delta M(N, Z) = M(N, Z) - uA,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $u$ is the Atomic Mass Unit" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "u = M(^{12}\\mathrm{C})/12 = 931.4940954(57) \\hspace{0.1cm} \\mathrm{MeV}/c^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The nucleon masses are" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "m_p = 1.00727646693(9)u,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "m_n = 939.56536(8)\\hspace{0.1cm} \\mathrm{MeV}/c^2 = 1.0086649156(6)u.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the [2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu](http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf)\n", + "there are data on masses and decays of 3437 nuclei.\n", + "\n", + "The nuclear binding energy is defined as the energy required to break\n", + "up a given nucleus into its constituent parts of $N$ neutrons and $Z$\n", + "protons. In terms of the atomic masses $M(N, Z)$ the binding energy is\n", + "defined by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.\n", + "In terms of the mass excess the binding energy is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N, Z) = Z\\Delta_H c^2 + N\\Delta_n c^2 -\\Delta(N, Z)c^2 ,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\Delta_H c^2 = 7.2890$ MeV and $\\Delta_n c^2 = 8.0713$ MeV.\n", + "\n", + "\n", + "A popular and physically intuitive model which can be used to parametrize \n", + "the experimental binding energies as function of $A$, is the so-called \n", + "**liquid drop model**. The ansatz is based on the following expression" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "BE(N,Z) = a_1A-a_2A^{2/3}-a_3\\frac{Z^2}{A^{1/3}}-a_4\\frac{(N-Z)^2}{A},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit \n", + "to the experimental data. \n", + "\n", + "\n", + "\n", + "\n", + "To arrive at the above expression we have assumed that we can make the following assumptions:\n", + "\n", + " * There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.\n", + "\n", + " * There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.\n", + "\n", + " * There is a Coulomb energy term $a_3\\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding. \n", + "\n", + " * There is an asymmetry term $a_4\\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.\n", + "\n", + "We could also add a so-called pairing term, which is a correction term that\n", + "arises from the tendency of proton pairs and neutron pairs to\n", + "occur. An even number of particles is more stable than an odd number. \n", + "\n", + "\n", + "### Organizing our data\n", + "\n", + "Let us start with reading and organizing our data. \n", + "We start with the compilation of masses and binding energies from 2016.\n", + "After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.\n", + "\n", + "\n", + "We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of **scikit-learn**." + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [], + "source": [ + "# Common imports\n", + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "import sklearn.linear_model as skl\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error\n", + "import os\n", + "\n", + "# Where to save the figures and data files\n", + "PROJECT_ROOT_DIR = \"Results\"\n", + "FIGURE_ID = \"Results/FigureFiles\"\n", + "DATA_ID = \"DataFiles/\"\n", + "\n", + "if not os.path.exists(PROJECT_ROOT_DIR):\n", + " os.mkdir(PROJECT_ROOT_DIR)\n", + "\n", + "if not os.path.exists(FIGURE_ID):\n", + " os.makedirs(FIGURE_ID)\n", + "\n", + "if not os.path.exists(DATA_ID):\n", + " os.makedirs(DATA_ID)\n", + "\n", + "def image_path(fig_id):\n", + " return os.path.join(FIGURE_ID, fig_id)\n", + "\n", + "def data_path(dat_id):\n", + " return os.path.join(DATA_ID, dat_id)\n", + "\n", + "def save_fig(fig_id):\n", + " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", + "\n", + "infile = open(data_path(\"MassEval2016.dat\"),'r')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various **matplotlib** commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [], + "source": [ + "from pylab import plt, mpl\n", + "plt.style.use('seaborn')\n", + "mpl.rcParams['font.family'] = 'serif'\n", + "\n", + "def MakePlot(x,y, styles, labels, axlabels):\n", + " plt.figure(figsize=(10,6))\n", + " for i in range(len(x)):\n", + " plt.plot(x[i], y[i], styles[i], label = labels[i])\n", + " plt.xlabel(axlabels[0])\n", + " plt.ylabel(axlabels[1])\n", + " plt.legend(loc=0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Our next step is to read the data on experimental binding energies and\n", + "reorganize them as functions of the mass number $A$, the number of\n", + "protons $Z$ and neutrons $N$ using **pandas**. Before we do this it is\n", + "always useful (unless you have a binary file or other types of compressed\n", + "data) to actually open the file and simply take a look at it!\n", + "\n", + "\n", + "In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with **pandas**. The file begins with some basic format information." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [], + "source": [ + "\"\"\" \n", + "This is taken from the data file of the mass 2016 evaluation. \n", + "All files are 3436 lines long with 124 character per line. \n", + " Headers are 39 lines long. \n", + " col 1 : Fortran character control: 1 = page feed 0 = line feed \n", + " format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 \n", + " These formats are reflected in the pandas widths variable below, see the statement \n", + " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), \n", + " Pandas has also a variable header, with length 39 in this case. \n", + "\"\"\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The data we are interested in are in columns 2, 3, 4 and 11, giving us\n", + "the number of neutrons, protons, mass numbers and binding energies,\n", + "respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will\n", + "covert them into the **pandas** DataFrame structure." + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [], + "source": [ + "# Read the experimental data with Pandas\n", + "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", + " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", + " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", + " header=39,\n", + " index_col=False)\n", + "\n", + "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", + "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", + "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", + "Masses = Masses.dropna()\n", + "# Convert from keV to MeV.\n", + "Masses['Ebinding'] /= 1000\n", + "\n", + "# Group the DataFrame by nucleon number, A.\n", + "Masses = Masses.groupby('A')\n", + "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", + "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We have now read in the data, grouped them according to the variables we are interested in. \n", + "We see how easy it is to reorganize the data using **pandas**. If we\n", + "were to do these operations in C/C++ or Fortran, we would have had to\n", + "write various functions/subroutines which perform the above\n", + "reorganizations for us. Having reorganized the data, we can now start\n", + "to make some simple fits using both the functionalities in **numpy** and\n", + "**Scikit-Learn** afterwards. \n", + "\n", + "Now we define five variables which contain\n", + "the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves." + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [], + "source": [ + "A = Masses['A']\n", + "Z = Masses['Z']\n", + "N = Masses['N']\n", + "Element = Masses['Element']\n", + "Energies = Masses['Ebinding']\n", + "print(Masses)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The next step, and we will define this mathematically later, is to set up the so-called **design matrix**. We will throughout call this matrix $\\boldsymbol{X}$.\n", + "It has dimensionality $p\\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit." + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": {}, + "outputs": [], + "source": [ + "# Now we set up the design matrix X\n", + "X = np.zeros((len(A),5))\n", + "X[:,0] = 1\n", + "X[:,1] = A\n", + "X[:,2] = A**(2.0/3.0)\n", + "X[:,3] = A**(-1.0/3.0)\n", + "X[:,4] = A**(-1.0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "With **scikitlearn** we are now ready to use linear regression and fit our data." + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": {}, + "outputs": [], + "source": [ + "clf = skl.LinearRegression().fit(X, Energies)\n", + "fity = clf.predict(X)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Pretty simple! \n", + "Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data." + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": {}, + "outputs": [], + "source": [ + "# The mean squared error \n", + "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, fity))\n", + "# Explained variance score: 1 is perfect prediction \n", + "print('Variance score: %.2f' % r2_score(Energies, fity))\n", + "# Mean absolute error \n", + "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))\n", + "print(clf.coef_, clf.intercept_)\n", + "\n", + "Masses['Eapprox'] = fity\n", + "# Generate a plot comparing the experimental with the fitted values values.\n", + "fig, ax = plt.subplots()\n", + "ax.set_xlabel(r'$A = N + Z$')\n", + "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", + "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", + " label='Ame2016')\n", + "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", + " label='Fit')\n", + "ax.legend()\n", + "save_fig(\"Masses2016\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Seeing the wood for the trees\n", + "\n", + "As a teaser, let us now see how we can do this with decision trees using **scikit-learn**. Later we will switch to so-called **random forests**!" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "#Decision Tree Regression\n", + "from sklearn.tree import DecisionTreeRegressor\n", + "regr_1=DecisionTreeRegressor(max_depth=5)\n", + "regr_2=DecisionTreeRegressor(max_depth=7)\n", + "regr_3=DecisionTreeRegressor(max_depth=9)\n", + "regr_1.fit(X, Energies)\n", + "regr_2.fit(X, Energies)\n", + "regr_3.fit(X, Energies)\n", + "\n", + "\n", + "y_1 = regr_1.predict(X)\n", + "y_2 = regr_2.predict(X)\n", + "y_3=regr_3.predict(X)\n", + "Masses['Eapprox'] = y_3\n", + "# Plot the results\n", + "plt.figure()\n", + "plt.plot(A, Energies, color=\"blue\", label=\"Data\", linewidth=2)\n", + "plt.plot(A, y_1, color=\"red\", label=\"max_depth=5\", linewidth=2)\n", + "plt.plot(A, y_2, color=\"green\", label=\"max_depth=7\", linewidth=2)\n", + "plt.plot(A, y_3, color=\"m\", label=\"max_depth=9\", linewidth=2)\n", + "\n", + "plt.xlabel(\"$A$\")\n", + "plt.ylabel(\"$E$[MeV]\")\n", + "plt.title(\"Decision Tree Regression\")\n", + "plt.legend()\n", + "save_fig(\"Masses2016Trees\")\n", + "plt.show()\n", + "print(Masses)\n", + "print(np.mean( (Energies-y_1)**2))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### And what about using neural networks?\n", + "\n", + "The **seaborn** package allows us to visualize data in an efficient way. Note that we use **scikit-learn**'s multi-layer perceptron (or feed forward neural network) \n", + "functionality." + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.neural_network import MLPRegressor\n", + "from sklearn.metrics import accuracy_score\n", + "import seaborn as sns\n", + "\n", + "X_train = X\n", + "Y_train = Energies\n", + "n_hidden_neurons = 100\n", + "epochs = 100\n", + "# store models for later use\n", + "eta_vals = np.logspace(-5, 1, 7)\n", + "lmbd_vals = np.logspace(-5, 1, 7)\n", + "# store the models for later use\n", + "DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", + "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", + "sns.set()\n", + "for i, eta in enumerate(eta_vals):\n", + " for j, lmbd in enumerate(lmbd_vals):\n", + " dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',\n", + " alpha=lmbd, learning_rate_init=eta, max_iter=epochs)\n", + " dnn.fit(X_train, Y_train)\n", + " DNN_scikit[i][j] = dnn\n", + " train_accuracy[i][j] = dnn.score(X_train, Y_train)\n", + "\n", + "fig, ax = plt.subplots(figsize = (10, 10))\n", + "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", + "ax.set_title(\"Training Accuracy\")\n", + "ax.set_ylabel(\"$\\eta$\")\n", + "ax.set_xlabel(\"$\\lambda$\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A first summary\n", + "\n", + "The aim behind these introductory words was to present to you various\n", + "Python libraries and their functionalities, in particular libraries like\n", + "**numpy**, **pandas**, **xarray** and **matplotlib** and other that make our life much easier\n", + "in handling various data sets and visualizing data. \n", + "\n", + "Furthermore,\n", + "**Scikit-Learn** allows us with few lines of code to implement popular\n", + "Machine Learning algorithms for supervised learning. Later we will meet **Tensorflow**, a powerful library for deep learning. \n", + "Now it is time to dive more into the details of various methods. We will start with linear regression and try to take a deeper look at what it entails." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.5" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/doc/pub/Regression/ipynb/Results/FigureFiles/EoSfitting.png b/doc/pub/Regression/ipynb/Results/FigureFiles/EoSfitting.png new file mode 100644 index 000000000..a352c6845 Binary files /dev/null and b/doc/pub/Regression/ipynb/Results/FigureFiles/EoSfitting.png differ diff --git a/doc/pub/Regression/ipynb/Results/FigureFiles/Masses2016OLS.png b/doc/pub/Regression/ipynb/Results/FigureFiles/Masses2016OLS.png new file mode 100644 index 000000000..c1d841a0c Binary files /dev/null and b/doc/pub/Regression/ipynb/Results/FigureFiles/Masses2016OLS.png differ diff --git a/doc/pub/odenn/ipynb/.ipynb_checkpoints/odenn-checkpoint.ipynb b/doc/pub/odenn/ipynb/.ipynb_checkpoints/odenn-checkpoint.ipynb new file mode 100644 index 000000000..a21d0f5a0 --- /dev/null +++ b/doc/pub/odenn/ipynb/.ipynb_checkpoints/odenn-checkpoint.ipynb @@ -0,0 +1,3579 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Data Analysis and Machine Learning: Using Neural networks to solve ODEs and PDEs\n", + "\n", + " \n", + "**Kristine Baluka Hein**, Department of Informatics, University of Oslo, Norway\n", + "\n", + "Date: **Nov 11, 2018**\n", + "\n", + "## Differential equations\n", + "\n", + "The Universal Approximation Theorem states that a neural network can approximate any function at a single hidden layer along with one input and output layer to any given precision.\n", + "Having this in mind, we will look closer at whether a neural network manages to solve for a function in an equation.\n", + "\n", + "A differential equation is an equation where the solution is a function. In the equation, it is given some relations between the function's derivatives subject to some given conditions.\n", + "Typically, a differential equation is solved numerically using approximations of Taylor series.\n", + "These kind of methods usually depends of choosing step sizes along each dimension that are small enough for the approximations to yield accurate results.\n", + "As we will see in the examples, using a neural network manages to outperform some well known methods in selected cases.\n", + "A possible way to solve the equation that we will look into, is to formulate a trial solution involving the result from a neural network.\n", + "The trial solution should hopefully converge into the true solution.\n", + "Be aware though, not all differential equations has an analytical solution.\n", + "We will stick with those that has an analytical solution such that we can check how well the network performs.\n", + "To make the trial solution converge to the true solution, setting up the cost function and updating the weights and biases within the neural network is crucial for the trial solution to converge.\n", + "\n", + "## Description of the equation to solve for\n", + "A differential equation is a equation where the solution is a function.\n", + "The equation describes how the derivatives of the function behaves in a given domain along with some conditions.\n", + "\n", + "Given a differential equation, it is desirable to know how to reformulate it into an equation a neural network can solve.\n", + "Having decided on which activation functions each layer should use, along with the number of hidden layers and neurons within each layer,\n", + "the changeable parameters of a neural network are the weights and biases for each neuron in every layer in the net.\n", + "If a differential equation is reformulated into an equation where minimization of some parameters must be done,\n", + "a neural net could possibly solve this equation.\n", + "\n", + "A trial solution might be tricky to find in general.\n", + "Due to the Universal Approximation Theorem, one could hope that outcome of the deep neural net might solve a given differential equation, even though it is used in a simple trial solution.\n", + "Let us try this idea on some well-known ordinary differential equations and thereafter try to solve for functions defined by two variables, giving partial differential equations.\n", + "\n", + "## Ordinary Differential Equations\n", + "An ordinary differential equation (ODE) is an equation involving functions having one variable.\n", + "\n", + "In general, an ordinary differential equation looks like" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{ode} \\tag{1}\n", + "f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right) = 0\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $g(x)$ is the function to find, and $g^{(n)}(x)$ is the $n$-th derivative of $g(x)$.\n", + "\n", + "The $f\\left(x, g(x), g'(x), g''(x), \\, \\dots \\, , g^{(n)}(x)\\right)$ is just a way to write that there is an expression involving $x$ and $g(x), \\ g'(x), \\ g''(x), \\, \\dots \\, , \\text{ and } g^{(n)}(x)$ on the left side of the equality sign in ([ode](#ode)).\n", + "The highest order of derivative, that is the value of $n$, determines to the order of the equation.\n", + "The equation is referred to as a $n$-th order ODE.\n", + "Along with ([ode](#ode)), some additional conditions of the function $g(x)$ are typically given\n", + "for the solution to be unique.\n", + "\n", + "## The trial solution\n", + "Let the trial solution $g_t(x)$ be" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\tg_t(x) = h_1(x) + h_2(x,N(x,P))\n", + "\\label{_auto1} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $h_1(x)$ is a function that makes $g_t(x)$ satisfy a given set of conditions, $N(x,P)$ a neural network with weights and biases described by $P$ and $h_2(x, N(x,P))$ some expression involving the neural network.\n", + "The role of the function $h_2(x, N(x,P))$, is to ensure that the output from $N(x,P)$ is zero when $g_t(x)$ is evaluated at the values of $x$ where the given conditions must be satisfied.\n", + "The function $h_1(x)$ should alone make $g_t(x)$ satisfy the conditions.\n", + "\n", + "But what about the network $N(x,P)$?\n", + "As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.\n", + "For the minimization to be defined, we need to have a cost function at hand to minimize.\n", + "\n", + "It is given that $f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)$ should be equal to zero in ([ode](#ode)).\n", + "We can choose to consider the mean squared error as the cost function for an input $x$.\n", + "Since we are looking at one input, the cost function is just $f$ squared.\n", + "The cost function $c\\left(x, P \\right)$ can therefore be expressed as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "c\\left(x, P\\right) = \\big(f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)\\big)^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If $N$ inputs are given as a vector $\\vec x$ with elements $x_i$ for $i = 1,\\dots,N$,\n", + "the cost function becomes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{cost} \\tag{3}\n", + "\tc\\left(\\vec x, P\\right) = \\frac{1}{N} \\sum_{i=1}^N \\big(f\\left(x_i, \\, g(x_i), \\, g'(x_i), \\, g''(x_i), \\, \\dots \\, , \\, g^{(n)}(x_i)\\right)\\big)^2\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The neural net should then find some parameters $P$ that minimizes the cost function in\n", + "([cost](#cost)) for a set of $N$ training samples $x_i$.\n", + "\n", + "## Minimizing the cost function using gradient descent and automatic differentiation\n", + "To perform the minimization using gradient descent, the gradient of $c\\left(\\vec x, P\\right)$ is needed.\n", + "It might happen so that finding an analytical expression of the gradient of $c(\\vec x, P)$ from ([cost](#cost)) gets too messy, depending on which cost function one desires to use.\n", + "\n", + "Luckily, there exists libraries that makes the job for us through automatic differentiation.\n", + "Automatic differentiation is a method of finding the derivatives numerically with very high precision.\n", + "\n", + "In the forthcoming examples presenting possible usages of Autograd and TensorFlow,\n", + "it is shown how one could set up a neural network using gradient descent solving a differential\n", + "equation.\n", + "\n", + "## Example: Exponential decay and setting up the network using Autograd\n", + "An exponential decay of a quantity $g(x)$ is described by the equation" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{solve_expdec} \\tag{4}\n", + " g'(x) = -\\gamma g(x)\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $g(0) = g_0$ for some chosen initial value $g_0$.\n", + "\n", + "The analytical solution of ([solve_expdec](#solve_expdec)) is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " g(x) = g_0 \\exp\\left(-\\gamma x\\right)\n", + "\\label{_auto2} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of ([solve_expdec](#solve_expdec)).\n", + "\n", + "In this example, a neural network will be implemented using Autograd in order to perform backpropagation.\n", + "\n", + "## The function to solve for\n", + "\n", + "The program will use a neural network to solve" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{solveode} \\tag{6}\n", + "g'(x) = -\\gamma g(x)\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $g(0) = g_0$ with $\\gamma$ and $g_0$ being some chosen values.\n", + "\n", + "In this example, $\\gamma = 2$ and $g_0 = 10$.\n", + "\n", + "## The trial solution\n", + "To begin with, a trial solution $g_t(t)$ must be chosen. A general trial solution for ordinary differential equations could be" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "g_t(x, P) = h_1(x) + h_2(x, N(x, P))\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and $h_2(x,N(x, P))$ an expression involving $x$ and the output from the neural network $N(x,P)$ with $P $ being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.\n", + "\n", + "In this network, there are no weights and bias at the input layer, so $P = \\{ P_{\\text{hidden}}, P_{\\text{output}} \\}$.\n", + "If there are $N_{\\text{hidden} }$ neurons in the hidden layer, then $P_{\\text{hidden}}$ is a $N_{\\text{hidden} } \\times (1 + N_{\\text{input}})$ matrix, given that there are $N_{\\text{input}}$ neurons in the input layer.\n", + "\n", + "The first column in $P_{\\text{hidden} }$ represents the bias for each neuron in the hidden layer and the second column represents the weights for each neuron in the hidden layer from the input layer.\n", + "If there are $N_{\\text{output} }$ neurons in the output layer, then $P_{\\text{output}} $ is a $N_{\\text{output} } \\times (1 + N_{\\text{hidden} })$ matrix.\n", + "\n", + "Its first column represents the bias of each neuron and the remaining columns represents the weights to each neuron.\n", + "\n", + "It is given that $g(0) = g_0$. The trial solution must fulfill this condition to be a proper solution of ([solveode](#solveode)). A possible way to ensure that $g_t(0, P) = g_0$, is to let $F(N(x,P)) = x \\cdot N(x,P)$ and $A(x) = g_0$. This gives the following trial solution:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{trial} \\tag{7}\n", + "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Reformulating the problem\n", + "We wish that our neural network manages to minimize a given cost function.\n", + "\n", + "A reformulation of out equation, ([solveode](#solveode)), must therefore be done,\n", + "such that it describes the problem a neural network can solve for.\n", + "\n", + "The neural network must find the set of weights and biases $P$ such that the trial solution in ([trial](#trial)) satisfies ([solveode](#solveode)).\n", + "\n", + "The trial solution" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "has been chosen such that it already solves the condition $g(0) = g_0$. What remains, is to find $P$ such that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{nnmin} \\tag{8}\n", + "g_t'(x, P) = - \\gamma g_t(x, P)\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "is fulfilled as *best as possible*.\n", + "\n", + "The left hand side and right hand side of ([nnmin](#nnmin)) must be computed separately, and then the neural network must choose weights and biases, contained in $P$, such that the sides are equal as best as possible.\n", + "This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.\n", + "In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to $P$ of the neural network.\n", + "\n", + "This gives the following cost function our neural network must solve for:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\min_{P}\\Big\\{ \\big(g_t'(x, P) - ( -\\gamma g_t(x, P) \\big)^2 \\Big\\}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "(the notation $\\min_{P}\\{ f(x, P) \\}$ means that we desire to find $P$ that yields the minimum of $f(x, P)$)\n", + "\n", + "or, in terms of weights and biases for the hidden and output layer in our network:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }}\\Big\\{ \\big(g_t'(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) - ( -\\gamma g_t(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) \\big)^2 \\Big\\}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "for an input value $x$.\n", + "\n", + "If the neural network evaluates $g_t(x, P)$ at more values for $x$, say $N$ values $x_i$ for $i = 1, \\dots, N$, then the *total* error to minimize becomes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{min} \\tag{9}\n", + "\\min_{P}\\Big\\{\\frac{1}{N} \\sum_{i=1}^N \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2 \\Big\\}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Letting $\\vec x$ be a vector with elements $x_i$ and $c(\\vec x, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2$ denote the cost function, the minimization problem that our network must solve, becomes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\min_{P} c(\\vec x, P)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In terms of $P_{\\text{hidden} }$ and $P_{\\text{output} }$, this could also be expressed as\n", + "\n", + "$$\n", + "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }} c(\\vec x, \\{P_{\\text{hidden} }, P_{\\text{output} }\\})\n", + "$$\n", + "\n", + "## A possible implementation of a neural network using Autograd\n", + "\n", + "For simplicity, it is assumed that the input is an array $\\vec x = (x_1, \\dots, x_N)$ with $N$ elements. It is at these points the neural network should find $P$ such that it fulfills ([min](#min)).\n", + "\n", + "First, the neural network must feed forward the inputs.\n", + "This means that $\\vec x$ must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further.\n", + "The input layer will consist of $N_{\\text{input} }$ neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be $N_{\\text{hidden} }$.\n", + "\n", + "For the $i$-th in the hidden layer with weight $w_i^{\\text{hidden} }$ and bias $b_i^{\\text{hidden} }$, the weighting from the $j$-th neuron at the input layer is:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "z_{i,j}^{\\text{hidden}} &= b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_j \\\\\n", + "&=\n", + "\\begin{pmatrix}\n", + "b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 \\\\\n", + "x_j\n", + "\\end{pmatrix}\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The result after weighting the inputs at the $i$-th hidden neuron can be written as a vector:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "\\vec{z}_{i}^{\\text{hidden}} &= \\Big( b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_1 , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_2, \\ \\dots \\, , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_N\\Big) \\\\\n", + "&=\n", + "\\begin{pmatrix}\n", + " b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 & 1 & \\dots & 1 \\\\\n", + "x_1 & x_2 & \\dots & x_N\n", + "\\end{pmatrix} \\\\\n", + "&= \\vec{p}_{i, \\text{hidden}}^T X\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The vector $\\vec{p}_{i, \\text{hidden}}^T$ constitutes each row in $P_{\\text{hidden} }$, which contains the weights for the neural network to minimize according to ([min](#min)).\n", + "\n", + "After having found $\\vec{z}_{i}^{\\text{hidden}} $ for every $i$-th neuron within the hidden layer, the vector will be sent to an activation function $a_i(\\vec{z})$.\n", + "\n", + "In this example, the sigmoid function has been chosen to be the activation function for each hidden neuron:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "f(z) = \\frac{1}{1 + \\exp{(-z)}}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is possible to use other activations functions for the hidden layer also.\n", + "\n", + "The output $\\vec{x}_i^{\\text{hidden} }$from each $i$-th hidden neuron is:\n", + "\n", + "$$\n", + "\\vec{x}_i^{\\text{hidden} } = f\\big( \\vec{z}_{i}^{\\text{hidden}} \\big)\n", + "$$\n", + "\n", + "The outputs $\\vec{x}_i^{\\text{hidden} } $ are then sent to the output layer.\n", + "\n", + "The output layer consists of one neuron in this case, and combines the output from each of the neurons in the hidden layers. The output layer combines the results from the hidden layer using some weights $ w_i^{\\text{output}}$ and biases $b_i^{\\text{output}}$. In this case, it is assumes that the number of neurons in the output layer is one.\n", + "\n", + "The procedure of weighting the output neuron $j$ in the hidden layer to the $i$-th neuron in the output layer is similar as for the hidden layer described previously." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "z_{1,j}^{\\text{output}} & =\n", + "\\begin{pmatrix}\n", + "b_1^{\\text{output}} & \\vec{w}_1^{\\text{output}}\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 \\\\\n", + "\\vec{x}_j^{\\text{hidden}}\n", + "\\end{pmatrix}\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Expressing $z_{1,j}^{\\text{output}}$ as a vector gives the following way of weighting the inputs from the hidden layer:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\vec{z}_{1}^{\\text{output}} =\n", + "\\begin{pmatrix}\n", + "b_1^{\\text{output}} & \\vec{w}_1^{\\text{output}}\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 & 1 & \\dots & 1 \\\\\n", + "\\vec{x}_1^{\\text{hidden}} & \\vec{x}_2^{\\text{hidden}} & \\dots & \\vec{x}_N^{\\text{hidden}}\n", + "\\end{pmatrix}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\\vec{z}_{1}^{\\text{output}}$ the neural network has finished its feed forward step, and $\\vec{z}_{1}^{\\text{output}}$ is the final output of the network.\n", + "\n", + "## Backpropagation using Autograd\n", + "The next step is to decide how the parameters should be changed such that they minimize the cost function.\n", + "\n", + "The chosen cost function for this problem is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "c(\\vec x, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In order to minimize the cost function, an optimization method must be chosen.\n", + "\n", + "Here, gradient descent with a constant step size has been chosen.\n", + "\n", + "## Gradient descent\n", + "The idea of the gradient descent algorithm is to update parameters in direction where the cost function decreases goes to a minimum.\n", + "\n", + "In general, the update of some parameters $\\vec \\omega$ given a cost function defined by some weights $\\vec \\omega$, $c(\\vec x, \\vec \\omega)$, goes as follows:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\vec \\omega_{\\text{new} } = \\vec \\omega - \\lambda \\nabla_{\\vec \\omega} c(\\vec x, \\vec \\omega)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "for a number of iterations or until $ \\big|\\big| \\vec \\omega_{\\text{new} } - \\vec \\omega \\big|\\big|$ becomes smaller than some given tolerance.\n", + "\n", + "The value of $\\lambda$ decides how large steps the algorithm must take in the direction of $ \\nabla_{\\vec \\omega} c(\\vec x, \\vec \\omega)$.\n", + "The notation $\\nabla_{\\vec \\omega}$ express the gradient with respect to the elements in $\\vec \\omega$.\n", + "\n", + "In our case, we have to minimize the cost function $c(\\vec x, P)$ with respect to the two sets of weights and biases, that is for the hidden layer $P_{\\text{hidden} }$ and for the output layer $P_{\\text{output} }$ .\n", + "\n", + "This means that $P_{\\text{hidden} }$ and $P_{\\text{output} }$ is updated by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "P_{\\text{hidden},\\text{new}} &= P_{\\text{hidden}} - \\lambda \\nabla_{P_{\\text{hidden}}} c(\\vec x, P) \\\\\n", + "P_{\\text{output},\\text{new}} &= P_{\\text{output}} - \\lambda \\nabla_{P_{\\text{output}}} c(\\vec x, P)\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In general, one could risk using a cost function having gradients that are cumbersome to derive analytically.\n", + "For our case, the cost functions are just the mean squared error.\n", + "One could employ an implementation of the back propagation for this case, but we will emphasis\n", + "on how one could use automatic differentiation in order to train the network.\n", + "\n", + "However, it might be useful to know how automatic differentiation can be used, e.g through Autograd, in order to test an implementation.\n", + "\n", + "## The network with one input, hidden, and output layer" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/usr/local/lib/python3.7/site-packages/autograd/numpy/numpy_vjps.py:444: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.\n", + " return lambda g: g[idxs]\n", + "/usr/local/lib/python3.7/site-packages/autograd/numpy/numpy_boxes.py:13: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.\n", + " def __getitem__(A, idx): return A[idx]\n", + "/usr/local/lib/python3.7/site-packages/autograd/numpy/numpy_vjps.py:597: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.\n", + " onp.add.at(A, idx, x)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Initial cost: 367.01\n" + ] + } + ], + "source": [ + "%matplotlib inline\n", + "\n", + "# Autograd will be used for later, so the numpy wrapper for Autograd must be imported\n", + "import autograd.numpy as np\n", + "from autograd import grad, elementwise_grad\n", + "import autograd.numpy.random as npr\n", + "from matplotlib import pyplot as plt\n", + "\n", + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "# Assuming one input, hidden, and output layer\n", + "def neural_network(params, x):\n", + "\n", + " # Find the weights (including and biases) for the hidden and output layer.\n", + " # Assume that params is a list of parameters for each layer.\n", + " # The biases are the first element for each array in params,\n", + " # and the weights are the remaning elements in each array in params.\n", + "\n", + " w_hidden = params[0]\n", + " w_output = params[1]\n", + "\n", + " # Assumes input x being an one-dimensional array\n", + " num_values = np.size(x)\n", + " x = x.reshape(-1, num_values)\n", + "\n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + "\n", + " ## Hidden layer:\n", + "\n", + " # Add a row of ones to include bias\n", + " x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_input)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " ## Output layer:\n", + "\n", + " # Include bias:\n", + " x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_hidden)\n", + " x_output = z_output\n", + "\n", + " return x_output\n", + "\n", + "# The trial solution using the deep neural network:\n", + "def g_trial(x,params, g0 = 10):\n", + " return g0 + x*neural_network(params,x)\n", + "\n", + "# The right side of the ODE:\n", + "def g(x, g_trial, gamma = 2):\n", + " return -gamma*g_trial\n", + "\n", + "# The cost function:\n", + "def cost_function(P, x):\n", + "\n", + " # Evaluate the trial function with the current parameters P\n", + " g_t = g_trial(x,P)\n", + "\n", + " # Find the derivative w.r.t x of the neural network\n", + " d_net_out = elementwise_grad(neural_network,1)(P,x)\n", + "\n", + " # Find the derivative w.r.t x of the trial function\n", + " d_g_t = elementwise_grad(g_trial,0)(x,P)\n", + "\n", + " # The right side of the ODE\n", + " func = g(x, g_t)\n", + "\n", + " err_sqr = (d_g_t - func)**2\n", + " cost_sum = np.sum(err_sqr)\n", + "\n", + " return cost_sum / np.size(err_sqr)\n", + "\n", + "# Solve the exponential decay ODE using neural network with one input, hidden, and output layer\n", + "def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):\n", + " ## Set up initial weights and biases\n", + "\n", + " # For the hidden layer\n", + " p0 = npr.randn(num_neurons_hidden, 2 )\n", + "\n", + " # For the output layer\n", + " p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included\n", + "\n", + " P = [p0, p1]\n", + "\n", + " print('Initial cost: %g'%cost_function(P, x))\n", + "\n", + " ## Start finding the optimal weights using gradient descent\n", + "\n", + " # Find the Python function that represents the gradient of the cost function\n", + " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", + " cost_function_grad = grad(cost_function,0)\n", + "\n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " # Evaluate the gradient at the current weights and biases in P.\n", + " # The cost_grad consist now of two arrays;\n", + " # one for the gradient w.r.t P_hidden and\n", + " # one for the gradient w.r.t P_output\n", + " cost_grad = cost_function_grad(P, x)\n", + "\n", + " P[0] = P[0] - lmb * cost_grad[0]\n", + " P[1] = P[1] - lmb * cost_grad[1]\n", + "\n", + " print('Final cost: %g'%cost_function(P, x))\n", + "\n", + " return P\n", + "\n", + "def g_analytic(x, gamma = 2, g0 = 10):\n", + " return g0*np.exp(-gamma*x)\n", + "\n", + "# Solve the given problem\n", + "if __name__ == '__main__':\n", + " # Set seed such that the weight are initialized\n", + " # with same weights and biases for every run.\n", + " npr.seed(15)\n", + "\n", + " ## Decide the vales of arguments to the function to solve\n", + " N = 10\n", + " x = np.linspace(0, 1, N)\n", + "\n", + " ## Set up the initial parameters\n", + " num_hidden_neurons = 10\n", + " num_iter = 10000\n", + " lmb = 0.001\n", + "\n", + " # Use the network\n", + " P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", + "\n", + " # Print the deviation from the trial solution and true solution\n", + " res = g_trial(x,P)\n", + " res_analytical = g_analytic(x)\n", + "\n", + " print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))\n", + "\n", + " # Plot the results\n", + " plt.figure(figsize=(10,10))\n", + "\n", + " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", + " plt.plot(x, res_analytical)\n", + " plt.plot(x, res[0,:])\n", + " plt.legend(['analytical','nn'])\n", + " plt.xlabel('x')\n", + " plt.ylabel('g(x)')\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The network with one input layer, specified number of hidden layers, and one output layer output layer\n", + "\n", + "It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.\n", + "\n", + "The number of neurons within each hidden layer are given as a list of integers in the program below." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad, elementwise_grad\n", + "import autograd.numpy.random as npr\n", + "from matplotlib import pyplot as plt\n", + "\n", + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "# The neural network with one input layer and one output layer,\n", + "# but with number of hidden layers specified by the user.\n", + "def deep_neural_network(deep_params, x):\n", + " # N_hidden is the number of hidden layers\n", + "\n", + " N_hidden = np.size(deep_params) - 1 # -1 since params consists of\n", + " # parameters to all the hidden\n", + " # layers AND the output layer.\n", + "\n", + " # Assumes input x being an one-dimensional array\n", + " num_values = np.size(x)\n", + " x = x.reshape(-1, num_values)\n", + "\n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + "\n", + " # Due to multiple hidden layers, define a variable referencing to the\n", + " # output of the previous layer:\n", + " x_prev = x_input\n", + "\n", + " ## Hidden layers:\n", + "\n", + " for l in range(N_hidden):\n", + " # From the list of parameters P; find the correct weigths and bias for this layer\n", + " w_hidden = deep_params[l]\n", + "\n", + " # Add a row of ones to include bias\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_prev)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " # Update x_prev such that next layer can use the output from this layer\n", + " x_prev = x_hidden\n", + "\n", + " ## Output layer:\n", + "\n", + " # Get the weights and bias for this layer\n", + " w_output = deep_params[-1]\n", + "\n", + " # Include bias:\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_prev)\n", + " x_output = z_output\n", + "\n", + " return x_output\n", + "\n", + "# The trial solution using the deep neural network:\n", + "def g_trial_deep(x,params, g0 = 10):\n", + " return g0 + x*deep_neural_network(params, x)\n", + "\n", + "# The right side of the ODE:\n", + "def g(x, g_trial, gamma = 2):\n", + " return -gamma*g_trial\n", + "\n", + "# The same cost function as before, but calls deep_neural_network instead.\n", + "def cost_function_deep(P, x):\n", + "\n", + " # Evaluate the trial function with the current parameters P\n", + " g_t = g_trial_deep(x,P)\n", + "\n", + " # Find the derivative w.r.t x of the neural network\n", + " d_net_out = elementwise_grad(deep_neural_network,1)(P,x)\n", + "\n", + " # Find the derivative w.r.t x of the trial function\n", + " d_g_t = elementwise_grad(g_trial_deep,0)(x,P)\n", + "\n", + " # The right side of the ODE\n", + " func = g(x, g_t)\n", + "\n", + " err_sqr = (d_g_t - func)**2\n", + " cost_sum = np.sum(err_sqr)\n", + "\n", + " return cost_sum / np.size(err_sqr)\n", + "\n", + "# Solve the exponential decay ODE using neural network with one input and one output layer,\n", + "# but with specified number of hidden layers from the user.\n", + "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", + " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", + "\n", + " # The number of elements in the list num_hidden_neurons thus represents\n", + " # the number of hidden layers.\n", + "\n", + " # Find the number of hidden layers:\n", + " N_hidden = np.size(num_neurons)\n", + "\n", + " ## Set up initial weights and biases\n", + "\n", + " # Initialize the list of parameters:\n", + " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", + "\n", + " P[0] = npr.randn(num_neurons[0], 2 )\n", + " for l in range(1,N_hidden):\n", + " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", + "\n", + " # For the output layer\n", + " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", + "\n", + " print('Initial cost: %g'%cost_function_deep(P, x))\n", + "\n", + " ## Start finding the optimal weights using gradient descent\n", + "\n", + " # Find the Python function that represents the gradient of the cost function\n", + " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", + " cost_function_deep_grad = grad(cost_function_deep,0)\n", + "\n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " # Evaluate the gradient at the current weights and biases in P.\n", + " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", + " # in the hidden layers and output layers evaluated at x.\n", + " cost_deep_grad = cost_function_deep_grad(P, x)\n", + "\n", + " for l in range(N_hidden+1):\n", + " P[l] = P[l] - lmb * cost_deep_grad[l]\n", + "\n", + " print('Final cost: %g'%cost_function_deep(P, x))\n", + "\n", + " return P\n", + "\n", + "def g_analytic(x, gamma = 2, g0 = 10):\n", + " return g0*np.exp(-gamma*x)\n", + "\n", + "# Solve the given problem\n", + "if __name__ == '__main__':\n", + " npr.seed(15)\n", + "\n", + " ## Decide the vales of arguments to the function to solve\n", + " N = 10\n", + " x = np.linspace(0, 1, N)\n", + "\n", + " ## Set up the initial parameters\n", + " num_hidden_neurons = np.array([10,10])\n", + " num_iter = 10000\n", + " lmb = 0.001\n", + "\n", + " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", + "\n", + " res = g_trial_deep(x,P)\n", + " res_analytical = g_analytic(x)\n", + "\n", + " plt.figure(figsize=(10,10))\n", + "\n", + " plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')\n", + " plt.plot(x, res_analytical)\n", + " plt.plot(x, res[0,:])\n", + " plt.legend(['analytical','dnn'])\n", + " plt.ylabel('g(x)')\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: Population growth, comparing Autograd, TensorFlow, and Euler's scheme\n", + "\n", + "A logistic model of population growth assumes that a population converges toward an equilibrium.\n", + "The population growth can be modeled by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{log} \\tag{10}\n", + "\tg'(t) = \\alpha g(t)(A - g(t))\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $g(t)$ is the population density at time $t$, $\\alpha > 0$ the growth rate and $A > 0$ is the maximum population number in the environment.\n", + "Also, at $t = 0$ the population has the size $g(0) = g_0$, where $g_0$ is some chosen constant.\n", + "\n", + "In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability\n", + "and high execution time (this might be more apparent in the examples solving PDEs),\n", + "a network has been constructed using TensorFlow also.\n", + "For comparison, the forward Euler method has been implemented in order to see how the networks performs compared to a numerical scheme.\n", + "\n", + "## Setting up the problem\n", + "\n", + "Here, we will model a population $g(t)$ in an environment having carrying capacity $A$.\n", + "The population follows the model" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{solveode_population} \\tag{11}\n", + "g'(t) = \\alpha g(t)(A - g(t))\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $g(0) = g_0$.\n", + "\n", + "In this example, we let $\\alpha = 2$, $A = 1$, and $g_0 = 1.2$.\n", + "\n", + "## The trial solution\n", + "We will get a slightly different trial solution, as the boundary conditions are different\n", + "compared to the case for exponential decay.\n", + "\n", + "A possible trial solution satisfying the condition $g(0) = g_0$ could be\n", + "\n", + "$$\n", + "h_1(t) = g_0 + t \\cdot N(t,P)\n", + "$$\n", + "\n", + "with $N(t,P)$ being the output from the neural network with weights and biases for each layer collected in the set $P$.\n", + "\n", + "The analytical solution is\n", + "\n", + "$$\n", + "g(t) = \\frac{Ag_0}{g_0 + (A - g_0)\\exp(-\\alpha A t)}\n", + "$$\n", + "\n", + "## The program using Autograd\n", + "\n", + "The network will be the similar as for the exponential decay example, but with some small modifications for our problem." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad, elementwise_grad\n", + "import autograd.numpy.random as npr\n", + "from matplotlib import pyplot as plt\n", + "\n", + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "# Function to get the parameters.\n", + "# Done such that one can easily change the paramaters after one's liking.\n", + "def get_parameters():\n", + " alpha = 2\n", + " A = 1\n", + " g0 = 1.2\n", + " return alpha, A, g0\n", + "\n", + "def deep_neural_network(P, x):\n", + " # N_hidden is the number of hidden layers\n", + " N_hidden = np.size(P) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", + "\n", + " # Assumes input x being an one-dimensional array\n", + " num_values = np.size(x)\n", + " x = x.reshape(-1, num_values)\n", + "\n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + "\n", + " # Due to multiple hidden layers, define a variable referencing to the\n", + " # output of the previous layer:\n", + " x_prev = x_input\n", + "\n", + " ## Hidden layers:\n", + "\n", + " for l in range(N_hidden):\n", + " # From the list of parameters P; find the correct weigths and bias for this layer\n", + " w_hidden = P[l]\n", + "\n", + " # Add a row of ones to include bias\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_prev)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " # Update x_prev such that next layer can use the output from this layer\n", + " x_prev = x_hidden\n", + "\n", + " ## Output layer:\n", + "\n", + " # Get the weights and bias for this layer\n", + " w_output = P[-1]\n", + "\n", + " # Include bias:\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_prev)\n", + " x_output = z_output\n", + "\n", + " return x_output\n", + "\n", + "\n", + "def cost_function_deep(P, x):\n", + "\n", + " # Evaluate the trial function with the current parameters P\n", + " g_t = g_trial_deep(x,P)\n", + "\n", + " # Find the derivative w.r.t x of the trial function\n", + " d_g_t = elementwise_grad(g_trial_deep,0)(x,P)\n", + "\n", + " # The right side of the ODE\n", + " func = f(x, g_t)\n", + "\n", + " err_sqr = (d_g_t - func)**2\n", + " cost_sum = np.sum(err_sqr)\n", + "\n", + " return cost_sum / np.size(err_sqr)\n", + "\n", + "# The right side of the ODE:\n", + "def f(x, g_trial):\n", + " alpha,A, g0 = get_parameters()\n", + " return alpha*g_trial*(A - g_trial)\n", + "\n", + "# The trial solution using the deep neural network:\n", + "def g_trial_deep(x, params):\n", + " alpha,A, g0 = get_parameters()\n", + " return g0 + x*deep_neural_network(params,x)\n", + "\n", + "# The analytical solution:\n", + "def g_analytic(t):\n", + " alpha,A, g0 = get_parameters()\n", + " return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))\n", + "\n", + "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", + " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", + "\n", + " # Find the number of hidden layers:\n", + " N_hidden = np.size(num_neurons)\n", + "\n", + " ## Set up initial weigths and biases\n", + "\n", + " # Initialize the list of parameters:\n", + " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", + "\n", + " P[0] = npr.randn(num_neurons[0], 2 )\n", + " for l in range(1,N_hidden):\n", + " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", + "\n", + " # For the output layer\n", + " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", + "\n", + " print('Initial cost: %g'%cost_function_deep(P, x))\n", + "\n", + " ## Start finding the optimal weigths using gradient descent\n", + "\n", + " # Find the Python function that represents the gradient of the cost function\n", + " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", + " cost_function_deep_grad = grad(cost_function_deep,0)\n", + "\n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " # Evaluate the gradient at the current weights and biases in P.\n", + " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", + " # in the hidden layers and output layers evaluated at x.\n", + " cost_deep_grad = cost_function_deep_grad(P, x)\n", + "\n", + " for l in range(N_hidden+1):\n", + " P[l] = P[l] - lmb * cost_deep_grad[l]\n", + "\n", + " print('Final cost: %g'%cost_function_deep(P, x))\n", + "\n", + " return P\n", + "\n", + "if __name__ == '__main__':\n", + " npr.seed(4155)\n", + "\n", + " ## Decide the vales of arguments to the function to solve\n", + " Nt = 10\n", + " T = 1\n", + " t = np.linspace(0,T, Nt)\n", + "\n", + " ## Set up the initial parameters\n", + " num_hidden_neurons = [100, 50, 25]\n", + " num_iter = 1000\n", + " lmb = 1e-3\n", + "\n", + " P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)\n", + "\n", + " g_dnn_ag = g_trial_deep(t,P)\n", + " g_analytical = g_analytic(t)\n", + "\n", + " # Find the maximum absolute difference between the solutons:\n", + " diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))\n", + " print(\"The max absolute difference between the solutions is: %g\"%diff_ag)\n", + "\n", + " plt.figure(figsize=(10,10))\n", + "\n", + " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", + " plt.plot(t, g_analytical)\n", + " plt.plot(t, g_dnn_ag[0,:])\n", + " plt.legend(['analytical','nn'])\n", + " plt.xlabel('t')\n", + " plt.ylabel('g(t)')\n", + "\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using forward Euler to solve the ODE\n", + "\n", + "A straight-forward way of solving an ODE numerically, is to use Euler's method.\n", + "\n", + "Euler's method uses Taylor series to approximate the value at a function $f$ at a step $\\Delta x$ from $x$:\n", + "\n", + "$$\n", + "f(x + \\Delta x) \\approx f(x) + \\Delta x f'(x)\n", + "$$\n", + "\n", + "In our case, using Euler's method to approximate the value of $g$ at a step $\\Delta t$ from $t$ yields" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + " g(t + \\Delta t) &\\approx g(t) + \\Delta t g'(t) \\\\\n", + " &= g(t) + \\Delta t \\big(\\alpha g(t)(A - g(t))\\big)\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "along with the condition that $g(0) = g_0$.\n", + "\n", + "Let $t_i = i \\cdot \\Delta t$ where $\\Delta t = \\frac{T}{N_t-1}$ where $T$ is the final time our solver must solve for and $N_t$ the number of values for $t \\in [0, T]$ for $i = 0, \\dots, N_t-1$.\n", + "\n", + "For $i \\geq 1$, we have that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "t_i &= i\\Delta t \\\\\n", + "&= (i - 1)\\Delta t + \\Delta t \\\\\n", + "&= t_{i-1} + \\Delta t\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now, if $g_i = g(t_i)$ then" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\begin{aligned}\n", + " g_i &= g(t_i) \\\\\n", + " &= g(t_{i-1} + \\Delta t) \\\\\n", + " &\\approx g(t_{i-1}) + \\Delta t \\big(\\alpha g(t_{i-1})(A - g(t_{i-1}))\\big) \\\\\n", + " &= g_{i-1} + \\Delta t \\big(\\alpha g_{i-1}(A - g_{i-1})\\big)\n", + " \\end{aligned}\n", + "\\end{equation} \\label{odenum} \\tag{12}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "for $i \\geq 1$ and $g_0 = g(t_0) = g(0) = g_0$.\n", + "\n", + "Equation ([odenum](#odenum)) could be implemented in the following way,\n", + "extending the program that uses the network using Autograd:" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "# Assume that all function definitions from the example program using Autograd\n", + "# are located here.\n", + "\n", + "if __name__ == '__main__':\n", + " npr.seed(4155)\n", + "\n", + " ## Decide the vales of arguments to the function to solve\n", + " Nt = 10\n", + " T = 1\n", + " t = np.linspace(0,T, Nt)\n", + "\n", + " ## Set up the initial parameters\n", + " num_hidden_neurons = [100,50,25]\n", + " num_iter = 1000\n", + " lmb = 1e-3\n", + "\n", + " P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)\n", + "\n", + " g_dnn_ag = g_trial_deep(t,P)\n", + " g_analytical = g_analytic(t)\n", + "\n", + " # Find the maximum absolute difference between the solutons:\n", + " diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))\n", + " print(\"The max absolute difference between the solutions is: %g\"%diff_ag)\n", + "\n", + " plt.figure(figsize=(10,10))\n", + "\n", + " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", + " plt.plot(t, g_analytical)\n", + " plt.plot(t, g_dnn_ag[0,:])\n", + " plt.legend(['analytical','nn'])\n", + " plt.xlabel('t')\n", + " plt.ylabel('g(t)')\n", + "\n", + " ## Find an approximation to the funtion using forward Euler\n", + "\n", + " alpha, A, g0 = get_parameters()\n", + " dt = T/(Nt - 1)\n", + "\n", + " # Perform forward Euler to solve the ODE\n", + " g_euler = np.zeros(Nt)\n", + " g_euler[0] = g0\n", + "\n", + " for i in range(1,Nt):\n", + " g_euler[i] = g_euler[i-1] + dt*(alpha*g_euler[i-1]*(A - g_euler[i-1]))\n", + "\n", + " # Print the errors done by each method\n", + " diff1 = np.max(np.abs(g_euler - g_analytical))\n", + " diff2 = np.max(np.abs(g_dnn_ag[0,:] - g_analytical))\n", + "\n", + " print('Max absolute difference between Euler method and analytical: %g'%diff1)\n", + " print('Max absolute difference between deep neural network and analytical: %g'%diff2)\n", + "\n", + " # Plot results\n", + " plt.figure(figsize=(10,10))\n", + "\n", + " plt.plot(t,g_euler)\n", + " plt.plot(t,g_analytical)\n", + " plt.plot(t,g_dnn_ag[0,:])\n", + "\n", + " plt.legend(['euler','analytical','dnn'])\n", + " plt.xlabel('Time t')\n", + " plt.ylabel('g(t)')\n", + "\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Running the program gives" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " Max absolute difference between Euler method and analytical: 0.011225\n", + " Max absolute difference between deep neural network and analytical: 0.00424909\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using TensorFlow to model logistic population growth\n", + "\n", + "TensorFlow is a library widely used in the machine learning community.\n", + "A neural network can be set up in a flexible manner, where various optimization algorithms are implemented and different types of networks can be used, making it easier to experiment on solving differential equations using neural networks.\n", + "\n", + "## The general program flow in TensorFlow\n", + "\n", + "Usually, a program in TensorFlow is divided into two parts; the *construction phase* and the *execution phase*.\n", + "In the construction phase, the computational graph that TensorFlow uses to perform its calculations are set up.\n", + "In the execution phase, TensorFlow evaluates any procedure that was defined in the construction phase.\n", + "\n", + "## Program flow in TensorFlow - Construction phase\n", + "\n", + "Here, the architecture for the neural network will be set up, along with the cost function and an optimizer class used during training of the network.\n", + "Note that TensorFlow uses a different convention for the weighting done in each neuron in each layer within the network than in the implementation using Autograd.\n", + "The matrix-vector multiplication between the input from the previous layer and the weighting at the neuron at current layer in the program using Autograd, is the transpose of the convention used in TensorFlow. But it will not affect that much our construction, as TensorFlow takes care of most of the computations. The only thing we have to be aware of, is how the dimensions are for our inputs.\n", + "\n", + "## Program flow in TensorFlow - Execution phase\n", + "\n", + "The computation graph has been defined, and is ready to be evaluated.\n", + "In order to get access to the graph, it has to be initialized and be runned within a Session.\n", + "\n", + "## The full program modeling logistic population growth using TensorFlow" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "import tensorflow as tf\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# Just to reset the graph such that it is possible to rerun this in a\n", + "# Jupyter cell without resetting the whole kernel.\n", + "tf.reset_default_graph()\n", + "\n", + "# Set a seed to ensure getting the same results from every run\n", + "tf.set_random_seed(4155)\n", + "\n", + "Nt = 10\n", + "T = 1\n", + "t = np.linspace(0,T, Nt)\n", + "\n", + "## The construction phase\n", + "\n", + "# Convert the values the trial solution is evaluated at to a tensor.\n", + "t_tf = tf.convert_to_tensor(t.reshape(-1,1),dtype=tf.float64)\n", + "zeros = tf.reshape(tf.convert_to_tensor(np.zeros(t.shape)),shape=(-1,1))\n", + "\n", + "# Define the parameters of the equation\n", + "alpha = tf.constant(2.,dtype=tf.float64)\n", + "A = tf.constant(1.,dtype=tf.float64)\n", + "g0 = tf.constant(1.2,dtype=tf.float64)\n", + "\n", + "num_iter = 100000\n", + "\n", + "# Define the number of neurons at each hidden layer\n", + "num_hidden_neurons = [100,50,25]\n", + "num_hidden_layers = np.size(num_hidden_neurons)\n", + "\n", + "# Construct the network.\n", + "# tf.name_scope is used to group each step in the construction,\n", + "# just for a more organized visualization in TensorBoard\n", + "with tf.name_scope('dnn'):\n", + "\n", + " # Input layer\n", + " previous_layer = t_tf\n", + "\n", + " # Hidden layers\n", + " for l in range(num_hidden_layers):\n", + " current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l], name='hidden%d'%(l+1), activation=tf.nn.sigmoid)\n", + " previous_layer = current_layer\n", + "\n", + " # Output layer\n", + " dnn_output = tf.layers.dense(previous_layer, 1, name='output')\n", + "\n", + "# Define the cost function\n", + "with tf.name_scope('cost'):\n", + " g_trial = g0 + t_tf*dnn_output\n", + " d_g_trial = tf.gradients(g_trial,t_tf)\n", + "\n", + " func = alpha*g_trial*(A - g_trial)\n", + " cost = tf.losses.mean_squared_error(zeros, d_g_trial[0] - func)\n", + "\n", + "\n", + "# Choose the method to minimize the cost function, along with a learning rate\n", + "learning_rate = 1e-2\n", + "with tf.name_scope('train'):\n", + " optimizer = tf.train.GradientDescentOptimizer(learning_rate)\n", + " traning_op = optimizer.minimize(cost)\n", + "\n", + "# Set up a referance to the result from the neural network:\n", + "g_dnn_tf = None\n", + "\n", + "# Define a node that initializes all of the other nodes in the computational graph\n", + "# used by TensorFlow:\n", + "init = tf.global_variables_initializer()\n", + "\n", + "## Execution phase\n", + "\n", + "# Start a session where the graph defined from the construction phase can be evaluated at:\n", + "with tf.Session() as sess:\n", + " # Initialize the whole graph\n", + " init.run()\n", + "\n", + " # Evaluate the initial cost:\n", + " print('Initial cost: %g'%cost.eval())\n", + "\n", + " # The training of the network:\n", + " for i in range(num_iter):\n", + " sess.run(traning_op)\n", + "\n", + " # If one desires to see how the cost function behaves for each iteration:\n", + " #if i % 1000 == 0:\n", + " # print(cost.eval())\n", + "\n", + " # Training is done, and we have an approximate solution to the ODE\n", + " print('Final cost: %g'%cost.eval())\n", + "\n", + " # Store the result\n", + " g_dnn_tf = g_trial.eval()\n", + "\n", + "# Compare with analytical solution\n", + "def get_parameters():\n", + " alpha = 2\n", + " A = 1\n", + " g0 = 1.2\n", + " return alpha, A, g0\n", + "\n", + "def g_analytic(t):\n", + " alpha,A, g0 = get_parameters()\n", + " return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))\n", + "\n", + "g_analytical = g_analytic(t)\n", + "diff_tf = g_dnn_tf - g_analytical.reshape(-1,1)\n", + "\n", + "print('\\nMax absolute difference between the analytical solution and solution from TensorFlow DNN: %g'%np.max(np.abs(diff_tf)))\n", + "\n", + "# Plot the result\n", + "plt.figure(figsize=(10,10))\n", + "\n", + "plt.title('Numerical solutions of the ODE')\n", + "\n", + "plt.plot(t, g_dnn_tf)\n", + "plt.plot(t, g_analytical)\n", + "\n", + "plt.legend(['dnn, tensorflow', 'exact'])\n", + "plt.xlabel('Time t')\n", + "plt.ylabel('g(t)')\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: Solving the one dimensional Poisson equation using Autograd and TensorFlow\n", + "\n", + "The Poisson equation for $g(x)$ in one dimension is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{poisson} \\tag{13}\n", + " -g''(x) = f(x)\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $f(x)$ is a given function for $x \\in (0,1)$.\n", + "\n", + "The conditions that $g(x)$ is chosen to fulfill, are" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + " g(0) &= 0 \\\\\n", + " g(1) &= 0\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used.\n", + "The results from the networks can then be compared to the analytical solution.\n", + "In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.\n", + "\n", + "There exists many different optimization methods implemented in TensorFlow.\n", + "In the examples program using TensorFlow, it could also be of interest to see how\n", + "the choice of an optimization method affects our results.\n", + "In the [TensorFlow documentation about optimizers](https://www.tensorflow.org/versions/r1.2/api_guides/python/train#Optimizers), a list over available optimization methods are shown.\n", + "\n", + "## The specific equation to solve for\n", + "\n", + "Here, the function $g(x)$ to solve for follows the equation" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "-g''(x) = f(x),\\qquad x \\in (0,1)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $f(x)$ is a given function, along with the chosen conditions" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{aligned}\n", + "g(0) = g(1) = 0\n", + "\\end{aligned}\\label{cond} \\tag{14}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this example, we consider the case when $f(x) = (3x + x^2)\\exp(x)$.\n", + "\n", + "For this case, a possible trial solution satisfying the conditions could be" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "g_t(x) = x \\cdot (1-x) \\cdot N(P,x)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The analytical solution for this problem is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "g(x) = x(1 - x)\\exp(x)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Solving the equation using Autograd" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad, elementwise_grad\n", + "import autograd.numpy.random as npr\n", + "from matplotlib import pyplot as plt\n", + "\n", + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "def deep_neural_network(deep_params, x):\n", + " # N_hidden is the number of hidden layers\n", + " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", + "\n", + " # Assumes input x being an one-dimensional array\n", + " num_values = np.size(x)\n", + " x = x.reshape(-1, num_values)\n", + "\n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + "\n", + " # Due to multiple hidden layers, define a variable referencing to the\n", + " # output of the previous layer:\n", + " x_prev = x_input\n", + "\n", + " ## Hidden layers:\n", + "\n", + " for l in range(N_hidden):\n", + " # From the list of parameters P; find the correct weigths and bias for this layer\n", + " w_hidden = deep_params[l]\n", + "\n", + " # Add a row of ones to include bias\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_prev)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " # Update x_prev such that next layer can use the output from this layer\n", + " x_prev = x_hidden\n", + "\n", + " ## Output layer:\n", + "\n", + " # Get the weights and bias for this layer\n", + " w_output = deep_params[-1]\n", + "\n", + " # Include bias:\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_prev)\n", + " x_output = z_output\n", + "\n", + " return x_output\n", + "\n", + "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", + " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", + "\n", + " # Find the number of hidden layers:\n", + " N_hidden = np.size(num_neurons)\n", + "\n", + " ## Set up initial weigths and biases\n", + "\n", + " # Initialize the list of parameters:\n", + " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", + "\n", + " P[0] = npr.randn(num_neurons[0], 2 )\n", + " for l in range(1,N_hidden):\n", + " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", + "\n", + " # For the output layer\n", + " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", + "\n", + " print('Initial cost: %g'%cost_function_deep(P, x))\n", + "\n", + " ## Start finding the optimal weigths using gradient descent\n", + "\n", + " # Find the Python function that represents the gradient of the cost function\n", + " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", + " cost_function_deep_grad = grad(cost_function_deep,0)\n", + "\n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " # Evaluate the gradient at the current weights and biases in P.\n", + " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", + " # in the hidden layers and output layers evaluated at x.\n", + " cost_deep_grad = cost_function_deep_grad(P, x)\n", + "\n", + " for l in range(N_hidden+1):\n", + " P[l] = P[l] - lmb * cost_deep_grad[l]\n", + "\n", + " print('Final cost: %g'%cost_function_deep(P, x))\n", + "\n", + " return P\n", + "\n", + "## Set up the cost function specified for this Poisson equation:\n", + "\n", + "# The right side of the ODE\n", + "def f(x):\n", + " return (3*x + x**2)*np.exp(x)\n", + "\n", + "def cost_function_deep(P, x):\n", + "\n", + " # Evaluate the trial function with the current parameters P\n", + " g_t = g_trial_deep(x,P)\n", + "\n", + " # Find the derivative w.r.t x of the trial function\n", + " d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)\n", + "\n", + " right_side = f(x)\n", + "\n", + " err_sqr = (-d2_g_t - right_side)**2\n", + " cost_sum = np.sum(err_sqr)\n", + "\n", + " return cost_sum/np.size(err_sqr)\n", + "\n", + "# The trial solution:\n", + "def g_trial_deep(x,P):\n", + " return x*(1-x)*deep_neural_network(P,x)\n", + "\n", + "# The analytic solution;\n", + "def g_analytic(x):\n", + " return x*(1-x)*np.exp(x)\n", + "\n", + "if __name__ == '__main__':\n", + " npr.seed(4155)\n", + "\n", + " ## Decide the vales of arguments to the function to solve\n", + " Nx = 10\n", + " x = np.linspace(0,1, Nx)\n", + "\n", + " ## Set up the initial parameters\n", + " num_hidden_neurons = [200,100]\n", + " num_iter = 1000\n", + " lmb = 1e-3\n", + "\n", + " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", + "\n", + " g_dnn_ag = g_trial_deep(x,P)\n", + " g_analytical = g_analytic(x)\n", + "\n", + " # Find the maximum absolute difference between the solutons:\n", + " max_diff = np.max(np.abs(g_dnn_ag - g_analytical))\n", + " print(\"The max absolute difference between the solutions is: %g\"%max_diff)\n", + "\n", + " plt.figure(figsize=(10,10))\n", + "\n", + " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", + " plt.plot(x, g_analytical)\n", + " plt.plot(x, g_dnn_ag[0,:])\n", + " plt.legend(['analytical','nn'])\n", + " plt.xlabel('x')\n", + " plt.ylabel('g(x)')\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Comparing with a numerical scheme\n", + "\n", + "The Poisson equation is possible to solve using Taylor series to approximate the second derivative.\n", + "\n", + "Using Taylor series, the second derivative can be expressed as\n", + "\n", + "$$\n", + "g''(x) = \\frac{g(x + \\Delta x) - 2g(x) + g(x-\\Delta x)}{\\Delta x^2} + E_{\\Delta x}(x)\n", + "$$\n", + "\n", + "where $\\Delta x$ is a small step size and $E_{\\Delta x}(x)$ being the error term.\n", + "\n", + "Looking away from the error terms gives an approximation to the second derivative:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{approx} \\tag{15}\n", + "g''(x) \\approx \\frac{g(x + \\Delta x) - 2g(x) + g(x-\\Delta x)}{\\Delta x^2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If $x_i = i \\Delta x = x_{i-1} + \\Delta x$ and $g_i = g(x_i)$ for $i = 1,\\dots N_x - 2$ with $N_x$ being the number of values for $x$, ([approx](#approx)) becomes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "g''(x_i) &\\approx \\frac{g(x_i + \\Delta x) - 2g(x_i) + g(x_i -\\Delta x)}{\\Delta x^2} \\\\\n", + "&= \\frac{g_{i+1} - 2g_i + g_{i-1}}{\\Delta x^2}\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Since we know from our problem that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "-g''(x) &= f(x) \\\\\n", + "&= (3x + x^2)\\exp(x)\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "along with the conditions $g(0) = g(1) = 0$,\n", + "the following scheme can be used to find an approximate solution for $g(x)$ numerically:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\begin{aligned}\n", + " -\\Big( \\frac{g_{i+1} - 2g_i + g_{i-1}}{\\Delta x^2} \\Big) &= f(x_i) \\\\\n", + " -g_{i+1} + 2g_i - g_{i-1} &= \\Delta x^2 f(x_i)\n", + " \\end{aligned}\n", + "\\end{equation} \\label{odesys} \\tag{16}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "for $i = 1, \\dots, N_x - 2$ where $g_0 = g_{N_x - 1} = 0$ and $f(x_i) = (3x_i + x_i^2)\\exp(x_i)$, which is given for our specific problem.\n", + "\n", + "The equation can be rewritten into a matrix equation:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "\\begin{pmatrix}\n", + "2 & -1 & 0 & \\dots & 0 \\\\\n", + "-1 & 2 & -1 & \\dots & 0 \\\\\n", + "\\vdots & & \\ddots & & \\vdots \\\\\n", + "0 & \\dots & -1 & 2 & -1 \\\\\n", + "0 & \\dots & 0 & -1 & 2\\\\\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "g_1 \\\\\n", + "g_2 \\\\\n", + "\\vdots \\\\\n", + "g_{N_x - 3} \\\\\n", + "g_{N_x - 2}\n", + "\\end{pmatrix}\n", + "&=\n", + "\\Delta x^2\n", + "\\begin{pmatrix}\n", + "f(x_1) \\\\\n", + "f(x_2) \\\\\n", + "\\vdots \\\\\n", + "f(x_{N_x - 3}) \\\\\n", + "f(x_{N_x - 2})\n", + "\\end{pmatrix} \\\\\n", + "A\\vec{g} &= \\vec{f}\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which makes it possible to solve for the vector $\\vec{g}$.\n", + "\n", + "We can then compare the result from this numerical scheme with the output from our network using Autograd:" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad, elementwise_grad\n", + "import autograd.numpy.random as npr\n", + "from matplotlib import pyplot as plt\n", + "\n", + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "def deep_neural_network(deep_params, x):\n", + " # N_hidden is the number of hidden layers\n", + " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", + "\n", + " # Assumes input x being an one-dimensional array\n", + " num_values = np.size(x)\n", + " x = x.reshape(-1, num_values)\n", + "\n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + "\n", + " # Due to multiple hidden layers, define a variable referencing to the\n", + " # output of the previous layer:\n", + " x_prev = x_input\n", + "\n", + " ## Hidden layers:\n", + "\n", + " for l in range(N_hidden):\n", + " # From the list of parameters P; find the correct weigths and bias for this layer\n", + " w_hidden = deep_params[l]\n", + "\n", + " # Add a row of ones to include bias\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_prev)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " # Update x_prev such that next layer can use the output from this layer\n", + " x_prev = x_hidden\n", + "\n", + " ## Output layer:\n", + "\n", + " # Get the weights and bias for this layer\n", + " w_output = deep_params[-1]\n", + "\n", + " # Include bias:\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_prev)\n", + " x_output = z_output\n", + "\n", + " return x_output\n", + "\n", + "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", + " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", + "\n", + " # Find the number of hidden layers:\n", + " N_hidden = np.size(num_neurons)\n", + "\n", + " ## Set up initial weigths and biases\n", + "\n", + " # Initialize the list of parameters:\n", + " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", + "\n", + " P[0] = npr.randn(num_neurons[0], 2 )\n", + " for l in range(1,N_hidden):\n", + " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", + "\n", + " # For the output layer\n", + " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", + "\n", + " print('Initial cost: %g'%cost_function_deep(P, x))\n", + "\n", + " ## Start finding the optimal weigths using gradient descent\n", + "\n", + " # Find the Python function that represents the gradient of the cost function\n", + " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", + " cost_function_deep_grad = grad(cost_function_deep,0)\n", + "\n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " # Evaluate the gradient at the current weights and biases in P.\n", + " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", + " # in the hidden layers and output layers evaluated at x.\n", + " cost_deep_grad = cost_function_deep_grad(P, x)\n", + "\n", + " for l in range(N_hidden+1):\n", + " P[l] = P[l] - lmb * cost_deep_grad[l]\n", + "\n", + " print('Final cost: %g'%cost_function_deep(P, x))\n", + "\n", + " return P\n", + "\n", + "## Set up the cost function specified for this Poisson equation:\n", + "\n", + "# The right side of the ODE\n", + "def f(x):\n", + " return (3*x + x**2)*np.exp(x)\n", + "\n", + "def cost_function_deep(P, x):\n", + "\n", + " # Evaluate the trial function with the current parameters P\n", + " g_t = g_trial_deep(x,P)\n", + "\n", + " # Find the derivative w.r.t x of the trial function\n", + " d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)\n", + "\n", + " right_side = f(x)\n", + "\n", + " err_sqr = (-d2_g_t - right_side)**2\n", + " cost_sum = np.sum(err_sqr)\n", + "\n", + " return cost_sum/np.size(err_sqr)\n", + "\n", + "# The trial solution:\n", + "def g_trial_deep(x,P):\n", + " return x*(1-x)*deep_neural_network(P,x)\n", + "\n", + "# The analytic solution;\n", + "def g_analytic(x):\n", + " return x*(1-x)*np.exp(x)\n", + "\n", + "if __name__ == '__main__':\n", + " npr.seed(4155)\n", + "\n", + " ## Decide the vales of arguments to the function to solve\n", + " Nx = 10\n", + " x = np.linspace(0,1, Nx)\n", + "\n", + " ## Set up the initial parameters\n", + " num_hidden_neurons = [200,100]\n", + " num_iter = 1000\n", + " lmb = 1e-3\n", + "\n", + " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", + "\n", + " g_dnn_ag = g_trial_deep(x,P)\n", + " g_analytical = g_analytic(x)\n", + "\n", + " # Find the maximum absolute difference between the solutons:\n", + "\n", + " plt.figure(figsize=(10,10))\n", + "\n", + " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", + " plt.plot(x, g_analytical)\n", + " plt.plot(x, g_dnn_ag[0,:])\n", + " plt.legend(['analytical','nn'])\n", + " plt.xlabel('x')\n", + " plt.ylabel('g(x)')\n", + "\n", + " ## Perform the computation using the numerical scheme\n", + "\n", + " dx = 1/(Nx - 1)\n", + "\n", + " # Set up the matrix A\n", + " A = np.zeros((Nx-2,Nx-2))\n", + "\n", + " A[0,0] = 2\n", + " A[0,1] = -1\n", + "\n", + " for i in range(1,Nx-3):\n", + " A[i,i-1] = -1\n", + " A[i,i] = 2\n", + " A[i,i+1] = -1\n", + "\n", + " A[Nx - 3, Nx - 4] = -1\n", + " A[Nx - 3, Nx - 3] = 2\n", + "\n", + " # Set up the vector f\n", + " f_vec = dx**2 * f(x[1:-1])\n", + "\n", + " # Solve the equation\n", + " g_res = np.linalg.solve(A,f_vec)\n", + "\n", + " g_vec = np.zeros(Nx)\n", + " g_vec[1:-1] = g_res\n", + "\n", + " # Print the differences between each method\n", + " max_diff1 = np.max(np.abs(g_dnn_ag - g_analytical))\n", + " max_diff2 = np.max(np.abs(g_vec - g_analytical))\n", + " print(\"The max absolute difference between the analytical solution and DNN Autograd: %g\"%max_diff1)\n", + " print(\"The max absolute difference between the analytical solution and numerical scheme: %g\"%max_diff2)\n", + "\n", + " # Plot the results\n", + " plt.figure(figsize=(10,10))\n", + "\n", + " plt.plot(x,g_vec)\n", + " plt.plot(x,g_analytical)\n", + " plt.plot(x,g_dnn_ag[0,:])\n", + "\n", + " plt.legend(['numerical scheme','analytical','dnn'])\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The program prints out:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " The max absolute difference between the analytical solution and DNN Autograd: 0.000464088\n", + " The max absolute difference between the analytical solution and numerical scheme: 0.00266858\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using gradient descent in TensorFlow to solve Poisson equation\n", + "The program follows the similar idea as for the logistic population model.\n", + "\n", + "What has changed, is what the cost function minimizes and the trial solution." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "import tensorflow as tf\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "## Construction phase\n", + "\n", + "# Just to reset the graph such that it is possible to rerun this in a\n", + "# Jupyter cell without resetting the whole kernel.\n", + "tf.reset_default_graph()\n", + "\n", + "tf.set_random_seed(4155)\n", + "\n", + "# Convert the values the trial solution is evaluated at to a tensor.\n", + "Nx = 10\n", + "x = np.linspace(0,1, Nx)\n", + "x_tf = tf.convert_to_tensor(x.reshape(-1,1),dtype=tf.float64)\n", + "\n", + "\n", + "num_iter = 10000\n", + "\n", + "# Define the number of neurons at each hidden layer\n", + "num_hidden_neurons = [20,10]\n", + "num_hidden_layers = np.size(num_hidden_neurons)\n", + "\n", + "# Construct the network.\n", + "# tf.name_scope is used to group each step in the construction,\n", + "# just for a more organized visualization in TensorBoard\n", + "with tf.name_scope('dnn'):\n", + "\n", + " # Input layer\n", + " previous_layer = x_tf\n", + "\n", + " # Hidden layers\n", + " for l in range(num_hidden_layers):\n", + " current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l], name='hidden%d'%(l+1), activation=tf.nn.sigmoid)\n", + " previous_layer = current_layer\n", + "\n", + " # Output layer\n", + " dnn_output = tf.layers.dense(previous_layer, 1, name='output')\n", + "\n", + "# Define the cost function\n", + "with tf.name_scope('cost'):\n", + " g_trial = x_tf*(1-x_tf)*dnn_output\n", + " d_g_trial = tf.gradients(g_trial,x_tf)\n", + " d2_g_trial = tf.gradients(d_g_trial,x_tf)\n", + "\n", + " right_side = (3*x_tf + x_tf**2)*tf.exp(x_tf)\n", + "\n", + " err = tf.square( -d2_g_trial[0] - right_side)\n", + " cost = tf.reduce_sum(err, name = 'cost')\n", + "\n", + "# Choose the method to minimize the cost function, along with a learning rate\n", + "learning_rate = 1e-2\n", + "with tf.name_scope('train'):\n", + " optimizer = tf.train.GradientDescentOptimizer(learning_rate)\n", + " traning_op = optimizer.minimize(cost)\n", + "\n", + "g_dnn_tf = None\n", + "\n", + "# Define a node that initializes all of the other nodes in the computational graph\n", + "# used by TensorFlow:\n", + "init = tf.global_variables_initializer()\n", + "\n", + "\n", + "## Execution phase\n", + "\n", + "# Start a session where the graph defined from the construction phase can be evaluated at:\n", + "\n", + "with tf.Session() as sess:\n", + " # Initialize the whole graph\n", + " init.run()\n", + "\n", + " # Evaluate the initial cost:\n", + " print('Initial cost: %g'%cost.eval())\n", + "\n", + " # The traning of the network:\n", + " for i in range(num_iter):\n", + " sess.run(traning_op)\n", + "\n", + " # Training is done, and we have an approximate solution to the ODE\n", + " print('Final cost: %g'%cost.eval())\n", + "\n", + " # Store the result\n", + " g_dnn_tf = g_trial.eval()\n", + "\n", + " writer = tf.summary.FileWriter(\"./output\", sess.graph)\n", + " writer.close()\n", + "\n", + "# Evaluate the analytical function to compare with\n", + "def g_analytic(x):\n", + " return x*(1-x)*np.exp(x)\n", + "\n", + "g_analytical = g_analytic(x)\n", + "\n", + "diff_tf = g_dnn_tf - g_analytical.reshape(-1,1)\n", + "\n", + "print('\\nMax absolute difference between the analytical solution and solution from TensorFlow DNN: %g'%np.max(np.abs(diff_tf)))\n", + "\n", + "# Plot the result\n", + "plt.figure(figsize=(10,10))\n", + "\n", + "plt.title('Numerical solutions of the ODE')\n", + "\n", + "plt.plot(x, g_dnn_tf)\n", + "plt.plot(x, g_analytical)\n", + "\n", + "plt.legend(['dnn, tensorflow','exact'])\n", + "plt.xlabel('x')\n", + "plt.ylabel('g(x)')\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using a different optimization algorithm implemented in TensorFlow to solve Poisson equation\n", + "\n", + "We can see that the results using GradientDescentOptimizer seems to converge towards the analytical solution.\n", + "But there exists many other methods for optimization also, see [the TensorFlow documentation on Optimizers](https://www.tensorflow.org/versions/r1.2/api_guides/python/train#Optimizers).\n", + "\n", + "Adam is an optimization algorithm that changes its learning rates accordingly to the function it tries to minimize for every iteration.\n", + "The algorithm is described in [this paper](https://arxiv.org/pdf/1412.6980.pdf).\n", + "How much an optimization algorithm has to say for the network to converge, could be interesting to experiment with.\n", + "Using the same TensorFlow program as before, the only change to do, is to replace the variable *optimizer*.\n", + "\n", + "In the program that uses TensorFlow to solve for the Poisson equation, change the line" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " optimizer = tf.train.GradientDescentOptimizer(learning_rate)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " optimizer = tf.train.AdamOptimizer(learning_rate)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The program using the Adam optimizer with a different initial learning rate yields indeed an interesting result:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " Max absolute difference between the analytical solution and solution from TensorFlow DNN: 7.11243e-05\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Partial Differential Equations\n", + "A partial differential equation (PDE) has a solution here the function is defined by multiple variables.\n", + "The equation may involve all kinds of combinations of which variables the function is differentiated with respect to.\n", + "\n", + "In general, a partial differential equation for a function $g(x_1,\\dots,x_N)$ with $N$ variables may be expressed as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{PDE} \\tag{17}\n", + " f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) = 0\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given.\n", + "\n", + "The problem our network must solve for, is similar to the ODE case.\n", + "We must have a trial solution $g_t$ at hand.\n", + "\n", + "For instance, the trial solution could be expressed as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + " g_t(x_1,\\dots,x_N) = h_1(x_1,\\dots,x_N) + h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $h_1(x_1,\\dots,x_N)$ is a function that ensures $g_t(x_1,\\dots,x_N)$ satisfies some given conditions.\n", + "The neural network $N(x_1,\\dots,x_N,P)$ has weights and biases described by $P$ and $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$ is an expression using the output from the neural network in some way.\n", + "\n", + "The role of the function $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$, is to ensure that the output of $N(x_1,\\dots,x_N,P)$ is zero when $g_t(x_1,\\dots,x_N)$ is evaluated at the values of $x_1,\\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\\dots,x_N)$ should alone make $g_t(x_1,\\dots,x_N)$ satisfy the conditions.\n", + "\n", + "The network tries then the minimize the cost function following the same ideas as described for the ODE case, but now with more than one variables to consider.\n", + "The concept still remains the same; find a set of parameters $P$ such that the expression $f$ in ([PDE](#PDE)) is as close to zero as possible.\n", + "\n", + "As for the ODE case, the cost function is the mean squared error that the network must try to minimize. The cost function for the network to minimize is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "c\\left(x_1, \\dots, x_N, P\\right) = \\left( f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) \\right)^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we let $\\vec x = \\big( x_1, \\dots, x_N \\big)$ be an array containing the values for $x_1, \\dots, x_N$ respectively, the cost function can be reformulated into the following:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "c\\left(\\vec{x}, P\\right) = f\\left( \\left( \\vec{x}, \\frac{\\partial g(\\vec x) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\vec x) }{\\partial x_N}, \\frac{\\partial g(\\vec x) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\vec x) }{\\partial x_N^n} \\right) \\right)^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we also have $M$ different sets of values for $x_1, \\dots, x_N$, that is $\\vec{x}_i = \\big(x_1^{(i)}, \\dots, x_N^{(i)}\\big)$ for $i = 1,\\dots,M$ being the rows in matrix $X$, the cost function can be generalized into" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "c\\left(X, P \\right) = \\sum_{i=1}^M f\\left( \\left( \\vec{x}_i, \\frac{\\partial g(\\vec{x}_i) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\vec{x}_i) }{\\partial x_N}, \\frac{\\partial g(\\vec{x}_i) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\vec{x}_i) }{\\partial x_N^n} \\right) \\right)^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: The diffusion equation\n", + "\n", + "In one spatial dimension, the equation reads" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where a possible choice of conditions are" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "g(0,t) &= 0 ,\\qquad t \\geq 0 \\\\\n", + "g(1,t) &= 0, \\qquad t \\geq 0 \\\\\n", + "g(x,0) &= u(x),\\qquad x\\in [0,1]\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $u(x)$ being some given function.\n", + "\n", + "## Defining the problem\n", + "\n", + "For this case, we want to find $g(x,t)$ such that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", + "\\end{equation} \\label{diffonedim} \\tag{18}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "g(0,t) &= 0 ,\\qquad t \\geq 0 \\\\\n", + "g(1,t) &= 0, \\qquad t \\geq 0 \\\\\n", + "g(x,0) &= u(x),\\qquad x\\in [0,1]\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $u(x) = \\sin(\\pi x)$.\n", + "\n", + "First, let us set up the deep neural network.\n", + "The deep neural network will follow the same structure as discussed in the examples solving the ODEs.\n", + "First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions.\n", + "\n", + "\n", + "\n", + "## Setting up the network using Autograd\n", + "\n", + "The only change to do here, is to extend our network such that functions of multiple parameters are correctly handled.\n", + "In this case we have two variables in our function to solve for, that is time $t$ and position $x$.\n", + "The variables will be represented by a one-dimensional array in the program.\n", + "The program will evaluate the network at each possible pair $(x,t)$, given an array for the desired $x$-values and $t$-values to approximate the solution at." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "def deep_neural_network(deep_params, x):\n", + " # x is now a point and a 1D numpy array; make it a column vector\n", + " num_coordinates = np.size(x,0)\n", + " x = x.reshape(num_coordinates,-1)\n", + "\n", + " num_points = np.size(x,1)\n", + "\n", + " # N_hidden is the number of hidden layers\n", + " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", + "\n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + " x_prev = x_input\n", + "\n", + " ## Hidden layers:\n", + "\n", + " for l in range(N_hidden):\n", + " # From the list of parameters P; find the correct weigths and bias for this layer\n", + " w_hidden = deep_params[l]\n", + "\n", + " # Add a row of ones to include bias\n", + " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_prev)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " # Update x_prev such that next layer can use the output from this layer\n", + " x_prev = x_hidden\n", + "\n", + " ## Output layer:\n", + "\n", + " # Get the weights and bias for this layer\n", + " w_output = deep_params[-1]\n", + "\n", + " # Include bias:\n", + " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_prev)\n", + " x_output = z_output\n", + "\n", + " return x_output[0][0]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Setting up the network using Autograd; The trial solution\n", + "The cost function must then iterate through the given arrays containing values for $x$ and $t$, defines a point $(x,t)$ the deep neural network and the trial solution is evaluated at, and then finds the Jacobian of the trial solution.\n", + "\n", + "A possible trial solution for this PDE is\n", + "\n", + "$$\n", + "g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P)\n", + "$$\n", + "\n", + "with $A(x,t)$ being a function ensuring that $g_t(x,t)$ satisfies our given conditions, and $N(x,t,P)$ being the output from the deep neural network using weights and biases for each layer from $P$.\n", + "\n", + "To fulfill the conditions, $A(x,t)$ could be:\n", + "\n", + "$$\n", + "h_1(x,t) = (1-t)\\Big(u(x) - \\big((1-x)u(0) + x u(1)\\big)\\Big) = (1-t)u(x) = (1-t)\\sin(\\pi x)\n", + "$$\n", + "since $(0) = u(1) = 0$ and $u(x) = \\sin(\\pi x)$.\n", + "\n", + "The Jacobian is used because the program must find the derivative of the trial solution with respect to $x$ and $t$.\n", + "\n", + "This gives the necessity of computing the Jacobian matrix, as we want to evaluate the gradient with respect to $x$ and $t$ (note that the Jacobian of a scalar-valued multivariate function is simply its gradient).\n", + "\n", + "In Autograd, the differentiation is by default done with respect to the first input argument of your Python function. Since the points is an array representing $x$ and $t$, the Jacobian is calculated using the values of $x$ and $t$.\n", + "\n", + "To find the second derivative with respect to $x$ and $t$, the Jacobian can be found for the second time. The result is a Hessian matrix, which is the matrix containing all the possible second order mixed derivatives of $g(x,t)$." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "# Set up the trial function:\n", + "def u(x):\n", + " return np.sin(np.pi*x)\n", + "\n", + "def g_trial(point,P):\n", + " x,t = point\n", + " return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)\n", + "\n", + "# The right side of the ODE:\n", + "def f(point):\n", + " return 0.\n", + "\n", + "# The cost function:\n", + "def cost_function(P, x, t):\n", + " cost_sum = 0\n", + "\n", + " g_t_jacobian_func = jacobian(g_trial)\n", + " g_t_hessian_func = hessian(g_trial)\n", + "\n", + " for x_ in x:\n", + " for t_ in t:\n", + " point = np.array([x_,t_])\n", + "\n", + " g_t = g_trial(point,P)\n", + " g_t_jacobian = g_t_jacobian_func(point,P)\n", + " g_t_hessian = g_t_hessian_func(point,P)\n", + "\n", + " g_t_dt = g_t_jacobian[1]\n", + " g_t_d2x = g_t_hessian[0][0]\n", + "\n", + " func = f(point)\n", + "\n", + " err_sqr = ( (g_t_dt - g_t_d2x) - func)**2\n", + " cost_sum += err_sqr\n", + "\n", + " return cost_sum" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Setting up the network using Autograd; The full program\n", + "Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.\n", + "\n", + "The analytical solution of our problem is\n", + "\n", + "$$\n", + "g(x,t) = \\exp(-\\pi^2 t)\\sin(\\pi x)\n", + "$$\n", + "\n", + "A possible way to implement a neural network solving the PDE, is given below.\n", + "Be aware, though, that it is fairly slow for the parameters used.\n", + "A better result is possible, but requires more iterations, and thus longer time to complete.\n", + "\n", + "Using only 20 neurons in one hidden layer, the program managed to make the trial solution have the maximum absolute error of 0.0075. The execution time, however, was approximately one day and 14 hours on a computer having Intel i7-7560U 2.4 GHz CPU.\n", + "\n", + "Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.\n", + "Using TensorFlow in the next example sovling the wave equation, has a much better execution time." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import jacobian,hessian,grad\n", + "import autograd.numpy.random as npr\n", + "from matplotlib import cm\n", + "from matplotlib import pyplot as plt\n", + "from mpl_toolkits.mplot3d import axes3d\n", + "\n", + "## Set up the network\n", + "\n", + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "def deep_neural_network(deep_params, x):\n", + " # x is now a point and a 1D numpy array; make it a column vector\n", + " num_coordinates = np.size(x,0)\n", + " x = x.reshape(num_coordinates,-1)\n", + "\n", + " num_points = np.size(x,1)\n", + "\n", + " # N_hidden is the number of hidden layers\n", + " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", + "\n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + " x_prev = x_input\n", + "\n", + " ## Hidden layers:\n", + "\n", + " for l in range(N_hidden):\n", + " # From the list of parameters P; find the correct weigths and bias for this layer\n", + " w_hidden = deep_params[l]\n", + "\n", + " # Add a row of ones to include bias\n", + " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_prev)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " # Update x_prev such that next layer can use the output from this layer\n", + " x_prev = x_hidden\n", + "\n", + " ## Output layer:\n", + "\n", + " # Get the weights and bias for this layer\n", + " w_output = deep_params[-1]\n", + "\n", + " # Include bias:\n", + " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_prev)\n", + " x_output = z_output\n", + "\n", + " return x_output[0][0]\n", + "\n", + "## Define the trial solution and cost function\n", + "def u(x):\n", + " return np.sin(np.pi*x)\n", + "\n", + "def g_trial(point,P):\n", + " x,t = point\n", + " return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)\n", + "\n", + "# The right side of the ODE:\n", + "def f(point):\n", + " return 0.\n", + "\n", + "# The cost function:\n", + "def cost_function(P, x, t):\n", + " cost_sum = 0\n", + "\n", + " g_t_jacobian_func = jacobian(g_trial)\n", + " g_t_hessian_func = hessian(g_trial)\n", + "\n", + " for x_ in x:\n", + " for t_ in t:\n", + " point = np.array([x_,t_])\n", + "\n", + " g_t = g_trial(point,P)\n", + " g_t_jacobian = g_t_jacobian_func(point,P)\n", + " g_t_hessian = g_t_hessian_func(point,P)\n", + "\n", + " g_t_dt = g_t_jacobian[1]\n", + " g_t_d2x = g_t_hessian[0][0]\n", + "\n", + " func = f(point)\n", + "\n", + " err_sqr = ( (g_t_dt - g_t_d2x) - func)**2\n", + " cost_sum += err_sqr\n", + "\n", + " return cost_sum /( np.size(x)*np.size(t) )\n", + "\n", + "## For comparison, define the analytical solution\n", + "def g_analytic(point):\n", + " x,t = point\n", + " return np.exp(-np.pi**2*t)*np.sin(np.pi*x)\n", + "\n", + "## Set up a function for training the network to solve for the equation\n", + "def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):\n", + " ## Set up initial weigths and biases\n", + " N_hidden = np.size(num_neurons)\n", + "\n", + " ## Set up initial weigths and biases\n", + "\n", + " # Initialize the list of parameters:\n", + " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", + "\n", + " P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias\n", + " for l in range(1,N_hidden):\n", + " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", + "\n", + " # For the output layer\n", + " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", + "\n", + " print('Initial cost: ',cost_function(P, x, t))\n", + "\n", + " cost_function_grad = grad(cost_function,0)\n", + "\n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " cost_grad = cost_function_grad(P, x , t)\n", + "\n", + " for l in range(N_hidden+1):\n", + " P[l] = P[l] - lmb * cost_grad[l]\n", + "\n", + " print('Final cost: ',cost_function(P, x, t))\n", + "\n", + " return P\n", + "\n", + "if __name__ == '__main__':\n", + " ### Use the neural network:\n", + " npr.seed(15)\n", + "\n", + " ## Decide the vales of arguments to the function to solve\n", + " Nx = 10; Nt = 10\n", + " x = np.linspace(0, 1, Nx)\n", + " t = np.linspace(0,1,Nt)\n", + "\n", + " ## Set up the parameters for the network\n", + " num_hidden_neurons = [100, 25]\n", + " num_iter = 250\n", + " lmb = 0.01\n", + "\n", + " P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)\n", + "\n", + " ## Store the results\n", + " g_dnn_ag = np.zeros((Nx, Nt))\n", + " G_analytical = np.zeros((Nx, Nt))\n", + " for i,x_ in enumerate(x):\n", + " for j, t_ in enumerate(t):\n", + " point = np.array([x_, t_])\n", + " g_dnn_ag[i,j] = g_trial(point,P)\n", + "\n", + " G_analytical[i,j] = g_analytic(point)\n", + "\n", + " # Find the map difference between the analytical and the computed solution\n", + " diff_ag = np.abs(g_dnn_ag - G_analytical)\n", + " print('Max absolute difference between the analytical solution and the network: %g'%np.max(diff_ag))\n", + "\n", + " ## Plot the solutions in two dimensions, that being in position and time\n", + "\n", + " T,X = np.meshgrid(t,x)\n", + "\n", + " fig = plt.figure(figsize=(10,10))\n", + " ax = fig.gca(projection='3d')\n", + " ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", + " s = ax.plot_surface(T,X,g_dnn_ag,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + " ax.set_xlabel('Time $t$')\n", + " ax.set_ylabel('Position $x$');\n", + "\n", + "\n", + " fig = plt.figure(figsize=(10,10))\n", + " ax = fig.gca(projection='3d')\n", + " ax.set_title('Analytical solution')\n", + " s = ax.plot_surface(T,X,G_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + " ax.set_xlabel('Time $t$')\n", + " ax.set_ylabel('Position $x$');\n", + "\n", + " fig = plt.figure(figsize=(10,10))\n", + " ax = fig.gca(projection='3d')\n", + " ax.set_title('Difference')\n", + " s = ax.plot_surface(T,X,diff_ag,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + " ax.set_xlabel('Time $t$')\n", + " ax.set_ylabel('Position $x$');\n", + "\n", + " ## Take some slices of the 3D plots just to see the solutions at particular times\n", + " indx1 = 0\n", + " indx2 = int(Nt/2)\n", + " indx3 = Nt-1\n", + "\n", + " t1 = t[indx1]\n", + " t2 = t[indx2]\n", + " t3 = t[indx3]\n", + "\n", + " # Slice the results from the DNN\n", + " res1 = g_dnn_ag[:,indx1]\n", + " res2 = g_dnn_ag[:,indx2]\n", + " res3 = g_dnn_ag[:,indx3]\n", + "\n", + " # Slice the analytical results\n", + " res_analytical1 = G_analytical[:,indx1]\n", + " res_analytical2 = G_analytical[:,indx2]\n", + " res_analytical3 = G_analytical[:,indx3]\n", + "\n", + " # Plot the slices\n", + " plt.figure(figsize=(10,10))\n", + " plt.title(\"Computed solutions at time = %g\"%t1)\n", + " plt.plot(x, res1)\n", + " plt.plot(x,res_analytical1)\n", + " plt.legend(['dnn','analytical'])\n", + "\n", + " plt.figure(figsize=(10,10))\n", + " plt.title(\"Computed solutions at time = %g\"%t2)\n", + " plt.plot(x, res2)\n", + " plt.plot(x,res_analytical2)\n", + " plt.legend(['dnn','analytical'])\n", + "\n", + " plt.figure(figsize=(10,10))\n", + " plt.title(\"Computed solutions at time = %g\"%t3)\n", + " plt.plot(x, res3)\n", + " plt.plot(x,res_analytical3)\n", + " plt.legend(['dnn','analytical'])\n", + "\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: Solving the wave equation using Autograd and TensorFlow\n", + "\n", + "The wave equation is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2\\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $c$ being the specified wave speed.\n", + "\n", + "Here, the chosen conditions are" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "\tg(0,t) &= 0 \\\\\n", + "\tg(1,t) &= 0 \\\\\n", + "\tg(x,0) &= u(x) \\\\\n", + "\t\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0} &= v(x)\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0}$ means the derivative of $g(x,t)$ with respect to $t$ is evaluated at $t = 0$, and $u(x)$ and $v(x)$ being given functions.\n", + "\n", + "## The problem to solve for\n", + "\n", + "The wave equation to solve for, is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \\label{wave} \\tag{19}\n", + "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2 \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $c$ is the given wave speed.\n", + "The chosen conditions for this equation are" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{aligned}\n", + "g(0,t) &= 0, &t \\geq 0 \\\\\n", + "g(1,t) &= 0, &t \\geq 0 \\\\\n", + "g(x,0) &= u(x), &x\\in[0,1] \\\\\n", + "\\frac{\\partial g(x,t)}{\\partial t}\\Big |_{t = 0} &= v(x), &x \\in [0,1]\n", + "\\end{aligned} \\label{condwave} \\tag{20}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this example, let $c = 1$ and $u(x) = \\sin(\\pi x)$ and $v(x) = -\\pi\\sin(\\pi x)$.\n", + "\n", + "\n", + "## The trial solution\n", + "Setting up the network is done in similar matter as for the example of solving the diffusion equation.\n", + "The only things we have to change, is the trial solution such that it satisfies the conditions from ([condwave](#condwave)) and the cost function.\n", + "\n", + "The trial solution becomes slightly different since we have other conditions than in the example of solving the diffusion equation. Here, a possible trial solution $g_t(x,t)$ is\n", + "\n", + "$$\n", + "g_t(x,t) = h_1(x,t) + x(1-x)t^2N(x,t,P)\n", + "$$\n", + "\n", + "where\n", + "\n", + "$$\n", + "h_1(x,t) = (1-t^2)u(x) + tv(x)\n", + "$$\n", + "\n", + "Note that this trial solution satisfies the conditions only if $u(0) = v(0) = u(1) = v(1) = 0$, which is the case in this example.\n", + "\n", + "## The analytical solution\n", + "\n", + "The analytical solution for our specific problem, is\n", + "\n", + "$$\n", + "g(x,t) = \\sin(\\pi x)\\cos(\\pi t) - \\sin(\\pi x)\\sin(\\pi t)\n", + "$$\n", + "\n", + "## Solving the wave equation - the full program using Autograd" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import hessian,grad\n", + "import autograd.numpy.random as npr\n", + "from matplotlib import cm\n", + "from matplotlib import pyplot as plt\n", + "from mpl_toolkits.mplot3d import axes3d\n", + "\n", + "## Set up the trial function:\n", + "def u(x):\n", + " return np.sin(np.pi*x)\n", + "\n", + "def v(x):\n", + " return -np.pi*np.sin(np.pi*x)\n", + "\n", + "def h1(point):\n", + " x,t = point\n", + " return (1 - t**2)*u(x) + t*v(x)\n", + "\n", + "def g_trial(point,P):\n", + " x,t = point\n", + " return h1(point) + x*(1-x)*t**2*deep_neural_network(P,point)\n", + "\n", + "## Define the cost function\n", + "def cost_function(P, x, t):\n", + " cost_sum = 0\n", + "\n", + " g_t_hessian_func = hessian(g_trial)\n", + "\n", + " for x_ in x:\n", + " for t_ in t:\n", + " point = np.array([x_,t_])\n", + "\n", + " g_t_hessian = g_t_hessian_func(point,P)\n", + "\n", + " g_t_d2x = g_t_hessian[0][0]\n", + " g_t_d2t = g_t_hessian[1][1]\n", + "\n", + " err_sqr = ( (g_t_d2t - g_t_d2x) )**2\n", + " cost_sum += err_sqr\n", + "\n", + " return cost_sum / (np.size(t) * np.size(x))\n", + "\n", + "## The neural network\n", + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "def deep_neural_network(deep_params, x):\n", + " # x is now a point and a 1D numpy array; make it a column vector\n", + " num_coordinates = np.size(x,0)\n", + " x = x.reshape(num_coordinates,-1)\n", + "\n", + " num_points = np.size(x,1)\n", + "\n", + " # N_hidden is the number of hidden layers\n", + " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", + "\n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + " x_prev = x_input\n", + "\n", + " ## Hidden layers:\n", + "\n", + " for l in range(N_hidden):\n", + " # From the list of parameters P; find the correct weigths and bias for this layer\n", + " w_hidden = deep_params[l]\n", + "\n", + " # Add a row of ones to include bias\n", + " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_prev)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " # Update x_prev such that next layer can use the output from this layer\n", + " x_prev = x_hidden\n", + "\n", + " ## Output layer:\n", + "\n", + " # Get the weights and bias for this layer\n", + " w_output = deep_params[-1]\n", + "\n", + " # Include bias:\n", + " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_prev)\n", + " x_output = z_output\n", + "\n", + " return x_output[0][0]\n", + "\n", + "## The analytical solution\n", + "def g_analytic(point):\n", + " x,t = point\n", + " return np.sin(np.pi*x)*np.cos(np.pi*t) - np.sin(np.pi*x)*np.sin(np.pi*t)\n", + "\n", + "def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):\n", + " ## Set up initial weigths and biases\n", + " N_hidden = np.size(num_neurons)\n", + "\n", + " ## Set up initial weigths and biases\n", + "\n", + " # Initialize the list of parameters:\n", + " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", + "\n", + " P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias\n", + " for l in range(1,N_hidden):\n", + " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", + "\n", + " # For the output layer\n", + " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", + "\n", + " print('Initial cost: ',cost_function(P, x, t))\n", + "\n", + " cost_function_grad = grad(cost_function,0)\n", + "\n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " cost_grad = cost_function_grad(P, x , t)\n", + "\n", + " for l in range(N_hidden+1):\n", + " P[l] = P[l] - lmb * cost_grad[l]\n", + "\n", + "\n", + " print('Final cost: ',cost_function(P, x, t))\n", + "\n", + " return P\n", + "\n", + "if __name__ == '__main__':\n", + " ### Use the neural network:\n", + " npr.seed(15)\n", + "\n", + " ## Decide the vales of arguments to the function to solve\n", + " Nx = 10; Nt = 10\n", + " x = np.linspace(0, 1, Nx)\n", + " t = np.linspace(0,1,Nt)\n", + "\n", + " ## Set up the parameters for the network\n", + " num_hidden_neurons = [50,20]\n", + " num_iter = 1000\n", + " lmb = 0.01\n", + "\n", + " P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)\n", + "\n", + " ## Store the results\n", + " res = np.zeros((Nx, Nt))\n", + " res_analytical = np.zeros((Nx, Nt))\n", + " for i,x_ in enumerate(x):\n", + " for j, t_ in enumerate(t):\n", + " point = np.array([x_, t_])\n", + " res[i,j] = g_trial(point,P)\n", + "\n", + " res_analytical[i,j] = g_analytic(point)\n", + "\n", + " diff = np.abs(res - res_analytical)\n", + " print(\"Max difference between analytical and solution from nn: %g\"%np.max(diff))\n", + "\n", + " ## Plot the solutions in two dimensions, that being in position and time\n", + "\n", + " T,X = np.meshgrid(t,x)\n", + "\n", + " fig = plt.figure(figsize=(10,10))\n", + " ax = fig.gca(projection='3d')\n", + " ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", + " s = ax.plot_surface(T,X,res,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + " ax.set_xlabel('Time $t$')\n", + " ax.set_ylabel('Position $x$');\n", + "\n", + "\n", + " fig = plt.figure(figsize=(10,10))\n", + " ax = fig.gca(projection='3d')\n", + " ax.set_title('Analytical solution')\n", + " s = ax.plot_surface(T,X,res_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + " ax.set_xlabel('Time $t$')\n", + " ax.set_ylabel('Position $x$');\n", + "\n", + "\n", + " fig = plt.figure(figsize=(10,10))\n", + " ax = fig.gca(projection='3d')\n", + " ax.set_title('Difference')\n", + " s = ax.plot_surface(T,X,diff,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + " ax.set_xlabel('Time $t$')\n", + " ax.set_ylabel('Position $x$');\n", + "\n", + " ## Take some slices of the 3D plots just to see the solutions at particular times\n", + " indx1 = 0\n", + " indx2 = int(Nt/2)\n", + " indx3 = Nt-1\n", + "\n", + " t1 = t[indx1]\n", + " t2 = t[indx2]\n", + " t3 = t[indx3]\n", + "\n", + " # Slice the results from the DNN\n", + " res1 = res[:,indx1]\n", + " res2 = res[:,indx2]\n", + " res3 = res[:,indx3]\n", + "\n", + " # Slice the analytical results\n", + " res_analytical1 = res_analytical[:,indx1]\n", + " res_analytical2 = res_analytical[:,indx2]\n", + " res_analytical3 = res_analytical[:,indx3]\n", + "\n", + " # Plot the slices\n", + " plt.figure(figsize=(10,10))\n", + " plt.title(\"Computed solutions at time = %g\"%t1)\n", + " plt.plot(x, res1)\n", + " plt.plot(x,res_analytical1)\n", + " plt.legend(['dnn','analytical'])\n", + "\n", + " plt.figure(figsize=(10,10))\n", + " plt.title(\"Computed solutions at time = %g\"%t2)\n", + " plt.plot(x, res2)\n", + " plt.plot(x,res_analytical2)\n", + " plt.legend(['dnn','analytical'])\n", + "\n", + " plt.figure(figsize=(10,10))\n", + " plt.title(\"Computed solutions at time = %g\"%t3)\n", + " plt.plot(x, res3)\n", + " plt.plot(x,res_analytical3)\n", + " plt.legend(['dnn','analytical'])\n", + "\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Solving the wave equation - the full program using TensorFlow\n", + "As the program using Autograd is fairly slow, one could hope that using TensorFlow\n", + "could make a naive implementation faster, and more numerically robust.\n", + "\n", + "In addition, having TensorFlow at hand, it could be easier to experiment with different\n", + "optimization algorithms, and other constructions of the network.\n", + "\n", + "The following program solves the given wave equation much faster," + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "import tensorflow as tf\n", + "import numpy as np\n", + "from matplotlib import cm\n", + "from matplotlib import pyplot as plt\n", + "from mpl_toolkits.mplot3d import axes3d\n", + "\n", + "Nx = 10\n", + "x_np = np.linspace(0,1,Nx)\n", + "\n", + "Nt = 10\n", + "t_np = np.linspace(0,1,Nt)\n", + "\n", + "X,T = np.meshgrid(x_np, t_np)\n", + "\n", + "x = X.ravel()\n", + "t = T.ravel()\n", + "\n", + "## The construction phase\n", + "\n", + "zeros = tf.reshape(tf.convert_to_tensor(np.zeros(x.shape)),shape=(-1,1))\n", + "x = tf.reshape(tf.convert_to_tensor(x),shape=(-1,1))\n", + "t = tf.reshape(tf.convert_to_tensor(t),shape=(-1,1))\n", + "\n", + "points = tf.concat([x,t],1)\n", + "\n", + "num_iter = 100000\n", + "num_hidden_neurons = [90]\n", + "\n", + "X = tf.convert_to_tensor(X)\n", + "T = tf.convert_to_tensor(T)\n", + "\n", + "\n", + "with tf.variable_scope('dnn'):\n", + " num_hidden_layers = np.size(num_hidden_neurons)\n", + "\n", + " previous_layer = points\n", + "\n", + " for l in range(num_hidden_layers):\n", + " current_layer = tf.layers.dense(previous_layer, num_hidden_neurons[l],activation=tf.nn.sigmoid)\n", + " previous_layer = current_layer\n", + "\n", + " dnn_output = tf.layers.dense(previous_layer, 1)\n", + "\n", + "\n", + "def u(x):\n", + " return tf.sin(np.pi*x)\n", + "\n", + "def v(x):\n", + " return -np.pi*tf.sin(np.pi*x)\n", + "\n", + "with tf.name_scope('loss'):\n", + " g_trial = (1 - t**2)*u(x) + t*v(x) + x*(1-x)*t**2*dnn_output\n", + "\n", + " g_trial_d2t = tf.gradients(tf.gradients(g_trial,t),t)\n", + " g_trial_d2x = tf.gradients(tf.gradients(g_trial,x),x)\n", + "\n", + " loss = tf.losses.mean_squared_error(zeros, g_trial_d2t[0] - g_trial_d2x[0])\n", + "\n", + "learning_rate = 0.01\n", + "with tf.name_scope('train'):\n", + " optimizer = tf.train.GradientDescentOptimizer(learning_rate)\n", + " traning_op = optimizer.minimize(loss)\n", + "\n", + "init = tf.global_variables_initializer()\n", + "\n", + "g_analytic = tf.sin(np.pi*x)*tf.cos(np.pi*t) - tf.sin(np.pi*x)*tf.sin(np.pi*t)\n", + "g_dnn = None\n", + "\n", + "## The execution phase\n", + "with tf.Session() as sess:\n", + " init.run()\n", + " for i in range(num_iter):\n", + " sess.run(traning_op)\n", + "\n", + " # If one desires to see how the cost function behaves during training\n", + " #if i % 100 == 0:\n", + " # print(loss.eval())\n", + "\n", + " g_analytic = g_analytic.eval()\n", + " g_dnn = g_trial.eval()\n", + "\n", + "\n", + "## Compare with the analutical solution\n", + "diff = np.abs(g_analytic - g_dnn)\n", + "print('Max absolute difference between analytical solution and TensorFlow DNN = ',np.max(diff))\n", + "\n", + "G_analytic = g_analytic.reshape((Nt,Nx))\n", + "G_dnn = g_dnn.reshape((Nt,Nx))\n", + "\n", + "diff = np.abs(G_analytic - G_dnn)\n", + "\n", + "# Plot the results\n", + "\n", + "X,T = np.meshgrid(x_np, t_np)\n", + "\n", + "fig = plt.figure(figsize=(10,10))\n", + "ax = fig.gca(projection='3d')\n", + "ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", + "s = ax.plot_surface(X,T,G_dnn,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + "ax.set_xlabel('Time $t$')\n", + "ax.set_ylabel('Position $x$');\n", + "\n", + "fig = plt.figure(figsize=(10,10))\n", + "ax = fig.gca(projection='3d')\n", + "ax.set_title('Analytical solution')\n", + "s = ax.plot_surface(X,T,G_analytic,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + "ax.set_xlabel('Time $t$')\n", + "ax.set_ylabel('Position $x$');\n", + "\n", + "fig = plt.figure(figsize=(10,10))\n", + "ax = fig.gca(projection='3d')\n", + "ax.set_title('Difference')\n", + "s = ax.plot_surface(X,T,diff,linewidth=0,antialiased=False,cmap=cm.viridis)\n", + "ax.set_xlabel('Time $t$')\n", + "ax.set_ylabel('Position $x$');\n", + "\n", + "## Take some 3D slices\n", + "\n", + "indx1 = 0\n", + "indx2 = int(Nt/2)\n", + "indx3 = Nt-1\n", + "\n", + "t1 = t_np[indx1]\n", + "t2 = t_np[indx2]\n", + "t3 = t_np[indx3]\n", + "\n", + "# Slice the results from the DNN\n", + "res1 = G_dnn[indx1,:]\n", + "res2 = G_dnn[indx2,:]\n", + "res3 = G_dnn[indx3,:]\n", + "\n", + "# Slice the analytical results\n", + "res_analytical1 = G_analytic[indx1,:]\n", + "res_analytical2 = G_analytic[indx2,:]\n", + "res_analytical3 = G_analytic[indx3,:]\n", + "\n", + "# Plot the slices\n", + "plt.figure(figsize=(10,10))\n", + "plt.title(\"Computed solutions at time = %g\"%t1)\n", + "plt.plot(x_np, res1)\n", + "plt.plot(x_np,res_analytical1)\n", + "plt.legend(['dnn','analytical'])\n", + "\n", + "plt.figure(figsize=(10,10))\n", + "plt.title(\"Computed solutions at time = %g\"%t2)\n", + "plt.plot(x_np, res2)\n", + "plt.plot(x_np,res_analytical2)\n", + "plt.legend(['dnn','analytical'])\n", + "\n", + "plt.figure(figsize=(10,10))\n", + "plt.title(\"Computed solutions at time = %g\"%t3)\n", + "plt.plot(x_np, res3)\n", + "plt.plot(x_np,res_analytical3)\n", + "plt.legend(['dnn','analytical'])\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The program manages to find a solution having max absolute difference to the analytical\n", + "at approximately 0.0059, by just using some minutes!\n", + "It was found, by some testing, that one hidden layer with 90 neurons actually performed well.\n", + "\n", + "## Resources\n", + "\n", + "1. [Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al](https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf)\n", + "\n", + "2. [Neural networks for solving differential equations by A. Honchar](https://becominghuman.ai/neural-networks-for-solving-differential-equations-fa230ac5e04c)\n", + "\n", + "3. [Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener](http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf)\n", + "\n", + "4. [Introduction to Partial Differential Equations by A. Tveitio, R. Winther](https://www.springer.com/us/book/9783540225515)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.5" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/doc/pub/summary/html/._summary-bs033.html b/doc/pub/summary/html/._summary-bs033.html new file mode 100644 index 000000000..b9a76312c --- /dev/null +++ b/doc/pub/summary/html/._summary-bs033.html @@ -0,0 +1,227 @@ + + + + + + + + +Summary of course + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Summary of course +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Best wishes to you all and thanks so much for your heroic efforts this semester

+ +

+



+ +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/svm/html/._svm-bs027.html b/doc/pub/svm/html/._svm-bs027.html new file mode 100644 index 000000000..06c88f585 --- /dev/null +++ b/doc/pub/svm/html/._svm-bs027.html @@ -0,0 +1,216 @@ + + + + + + + + +Data Analysis and Machine Learning: Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Support Vector Machines +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Back to the more realistic cases

+ +

+We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \( C \) we have +$$ +\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ +y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ +\dots & \dots & \dots & \dots & \dots \\ +\dots & \dots & \dots & \dots & \dots \\ +y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ +\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, +$$ + +subject to \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). Here we defined the vectors \( \boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n] \) and +\( \boldsymbol{y}=[y_1,y_2,\dots,y_n] \). +With the slack constants this leads to the additional constraint \( 0\leq \lambda_i \leq C \). + +

+code will be added + +

+ +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/svm/html/._svm-bs028.html b/doc/pub/svm/html/._svm-bs028.html new file mode 100644 index 000000000..d13a1be45 --- /dev/null +++ b/doc/pub/svm/html/._svm-bs028.html @@ -0,0 +1,227 @@ + + + + + + + + +Data Analysis and Machine Learning: Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Data Analysis and Machine Learning: Support Vector Machines +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Code Example

+

+ + +

import numpy as np
+from scipy.optimize import minimize
+def rosen(x):
+    return sum(100.0*(x[1:]-x[:-1]**2.0)**2.0 + (1-x[:-1])**2.0)
+def rosen_der(x):
+    xm = x[1:-1]
+    xm_m1 = x[:-2]
+    xm_p1 = x[2:]
+    der = np.zeros_like(x)
+    der[1:-1] = 200*(xm-xm_m1**2) - 400*(xm_p1 - xm**2)*xm - 2*(1-xm)
+    der[0] = -400*x[0]*(x[1]-x[0]**2) - 2*(1-x[0])
+    der[-1] = 200*(x[-1]-x[-2]**2)
+    return der
+
+
+
+x0 = np.array([1.2, 0.1, 0.8, 1.9, 1.2])
+print(x0)
+res = minimize(rosen, x0, method='nelder-mead',options={'xtol': 1e-8, 'disp': True})
+print(x0)
+
+
+res2 = minimize(rosen, x0, method='BFGS', jac=rosen_der, options={'disp': True})
+print(x0)
+
+

+ +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/svm/html/reveal.js/plugin/leap/leap.js b/doc/pub/svm/html/reveal.js/plugin/leap/leap.js new file mode 100644 index 000000000..48084ffb0 --- /dev/null +++ b/doc/pub/svm/html/reveal.js/plugin/leap/leap.js @@ -0,0 +1,159 @@ +/* + * Copyright (c) 2013, Leap Motion, Inc. + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: + * + * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. + * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Version 0.2.0 - http://js.leapmotion.com/0.2.0/leap.min.js + * Grab latest versions from http://js.leapmotion.com/ + */ + +!function(e,t,n){function i(n,s){if(!t[n]){if(!e[n]){var o=typeof require=="function"&&require;if(!s&&o)return o(n,!0);if(r)return r(n,!0);throw new Error("Cannot find module '"+n+"'")}var u=t[n]={exports:{}};e[n][0].call(u.exports,function(t){var r=e[n][1][t];return i(r?r:t)},u,u.exports)}return t[n].exports}var r=typeof require=="function"&&require;for(var s=0;s=this.size)return undefined;if(i>=this._buf.length)return undefined;return this._buf[(this.pos-i-1)%this.size]};CircularBuffer.prototype.push=function(o){this._buf[this.pos%this.size]=o;return this.pos++}},{}],3:[function(require,module,exports){var Connection=module.exports=require("./base_connection");Connection.prototype.setupSocket=function(){var connection=this;var socket=new WebSocket(this.getUrl());socket.onopen=function(){connection.handleOpen()};socket.onmessage=function(message){connection.handleData(message.data)};socket.onclose=function(){connection.handleClose()};return socket};Connection.prototype.startHeartbeat=function(){if(!this.protocol.sendHeartbeat||this.heartbeatTimer)return;var connection=this;var propertyName=null;if(typeof document.hidden!=="undefined"){propertyName="hidden"}else if(typeof document.mozHidden!=="undefined"){propertyName="mozHidden"}else if(typeof document.msHidden!=="undefined"){propertyName="msHidden"}else if(typeof document.webkitHidden!=="undefined"){propertyName="webkitHidden"}else{propertyName=undefined}var windowVisible=true;var focusListener=window.addEventListener("focus",function(e){windowVisible=true});var blurListener=window.addEventListener("blur",function(e){windowVisible=false});this.on("disconnect",function(){if(connection.heartbeatTimer){clearTimeout(connection.heartbeatTimer);delete connection.heartbeatTimer}window.removeEventListener(focusListener);window.removeEventListener(blurListener)});this.heartbeatTimer=setInterval(function(){var isVisible=propertyName===undefined?true:document[propertyName]===false;if(isVisible&&windowVisible){connection.sendHeartbeat()}else{connection.setHeartbeatState(false)}},this.opts.heartbeatInterval)}},{"./base_connection":1}],4:[function(require,module,exports){!function(process){var Frame=require("./frame"),CircularBuffer=require("./circular_buffer"),Pipeline=require("./pipeline"),EventEmitter=require("events").EventEmitter,gestureListener=require("./gesture").gestureListener,_=require("underscore");var Controller=module.exports=function(opts){var inNode=typeof process!=="undefined"&&process.title==="node";opts=_.defaults(opts||{},{inNode:inNode});this.inNode=opts.inNode;opts=_.defaults(opts||{},{frameEventName:this.useAnimationLoop()?"animationFrame":"deviceFrame",supressAnimationLoop:false});this.supressAnimationLoop=opts.supressAnimationLoop;this.frameEventName=opts.frameEventName;this.history=new CircularBuffer(200);this.lastFrame=Frame.Invalid;this.lastValidFrame=Frame.Invalid;this.lastConnectionFrame=Frame.Invalid;this.accumulatedGestures=[];if(opts.connectionType===undefined){this.connectionType=this.inBrowser()?require("./connection"):require("./node_connection")}else{this.connectionType=opts.connectionType}this.connection=new this.connectionType(opts);this.setupConnectionEvents()};Controller.prototype.gesture=function(type,cb){var creator=gestureListener(this,type);if(cb!==undefined){creator.stop(cb)}return creator};Controller.prototype.inBrowser=function(){return!this.inNode};Controller.prototype.useAnimationLoop=function(){return this.inBrowser()&&typeof chrome==="undefined"};Controller.prototype.connect=function(){var controller=this;if(this.connection.connect()&&this.inBrowser()&&!controller.supressAnimationLoop){var callback=function(){controller.emit("animationFrame",controller.lastConnectionFrame);window.requestAnimFrame(callback)};window.requestAnimFrame(callback)}};Controller.prototype.disconnect=function(){this.connection.disconnect()};Controller.prototype.frame=function(num){return this.history.get(num)||Frame.Invalid};Controller.prototype.loop=function(callback){switch(callback.length){case 1:this.on(this.frameEventName,callback);break;case 2:var controller=this;var scheduler=null;var immediateRunnerCallback=function(frame){callback(frame,function(){if(controller.lastFrame!=frame){immediateRunnerCallback(controller.lastFrame)}else{controller.once(controller.frameEventName,immediateRunnerCallback)}})};this.once(this.frameEventName,immediateRunnerCallback);break}this.connect()};Controller.prototype.addStep=function(step){if(!this.pipeline)this.pipeline=new Pipeline(this);this.pipeline.addStep(step)};Controller.prototype.processFrame=function(frame){if(frame.gestures){this.accumulatedGestures=this.accumulatedGestures.concat(frame.gestures)}if(this.pipeline){frame=this.pipeline.run(frame);if(!frame)frame=Frame.Invalid}this.lastConnectionFrame=frame;this.emit("deviceFrame",frame)};Controller.prototype.processFinishedFrame=function(frame){this.lastFrame=frame;if(frame.valid){this.lastValidFrame=frame}frame.controller=this;frame.historyIdx=this.history.push(frame);if(frame.gestures){frame.gestures=this.accumulatedGestures;this.accumulatedGestures=[];for(var gestureIdx=0;gestureIdx!=frame.gestures.length;gestureIdx++){this.emit("gesture",frame.gestures[gestureIdx],frame)}}this.emit("frame",frame)};Controller.prototype.setupConnectionEvents=function(){var controller=this;this.connection.on("frame",function(frame){controller.processFrame(frame)});this.on(this.frameEventName,function(frame){controller.processFinishedFrame(frame)});this.connection.on("disconnect",function(){controller.emit("disconnect")});this.connection.on("ready",function(){controller.emit("ready")});this.connection.on("connect",function(){controller.emit("connect")});this.connection.on("focus",function(){controller.emit("focus")});this.connection.on("blur",function(){controller.emit("blur")});this.connection.on("protocol",function(protocol){controller.emit("protocol",protocol)});this.connection.on("deviceConnect",function(evt){controller.emit(evt.state?"deviceConnected":"deviceDisconnected")})};_.extend(Controller.prototype,EventEmitter.prototype)}(require("__browserify_process"))},{"./circular_buffer":2,"./connection":3,"./frame":5,"./gesture":6,"./node_connection":16,"./pipeline":10,__browserify_process:18,events:17,underscore:20}],5:[function(require,module,exports){var Hand=require("./hand"),Pointable=require("./pointable"),createGesture=require("./gesture").createGesture,glMatrix=require("gl-matrix"),mat3=glMatrix.mat3,vec3=glMatrix.vec3,InteractionBox=require("./interaction_box"),_=require("underscore");var Frame=module.exports=function(data){this.valid=true;this.id=data.id;this.timestamp=data.timestamp;this.hands=[];this.handsMap={};this.pointables=[];this.tools=[];this.fingers=[];if(data.interactionBox){this.interactionBox=new InteractionBox(data.interactionBox)}this.gestures=[];this.pointablesMap={};this._translation=data.t;this._rotation=_.flatten(data.r);this._scaleFactor=data.s;this.data=data;this.type="frame";this.currentFrameRate=data.currentFrameRate;var handMap={};for(var handIdx=0,handCount=data.hands.length;handIdx!=handCount;handIdx++){var hand=new Hand(data.hands[handIdx]);hand.frame=this;this.hands.push(hand);this.handsMap[hand.id]=hand;handMap[hand.id]=handIdx}for(var pointableIdx=0,pointableCount=data.pointables.length;pointableIdx!=pointableCount;pointableIdx++){var pointable=new Pointable(data.pointables[pointableIdx]);pointable.frame=this;this.pointables.push(pointable);this.pointablesMap[pointable.id]=pointable;(pointable.tool?this.tools:this.fingers).push(pointable);if(pointable.handId!==undefined&&handMap.hasOwnProperty(pointable.handId)){var hand=this.hands[handMap[pointable.handId]];hand.pointables.push(pointable);(pointable.tool?hand.tools:hand.fingers).push(pointable)}}if(data.gestures){for(var gestureIdx=0,gestureCount=data.gestures.length;gestureIdx!=gestureCount;gestureIdx++){this.gestures.push(createGesture(data.gestures[gestureIdx]))}}};Frame.prototype.tool=function(id){var pointable=this.pointable(id);return pointable.tool?pointable:Pointable.Invalid};Frame.prototype.pointable=function(id){return this.pointablesMap[id]||Pointable.Invalid};Frame.prototype.finger=function(id){var pointable=this.pointable(id);return!pointable.tool?pointable:Pointable.Invalid};Frame.prototype.hand=function(id){return this.handsMap[id]||Hand.Invalid};Frame.prototype.rotationAngle=function(sinceFrame,axis){if(!this.valid||!sinceFrame.valid)return 0;var rot=this.rotationMatrix(sinceFrame);var cs=(rot[0]+rot[4]+rot[8]-1)*.5;var angle=Math.acos(cs);angle=isNaN(angle)?0:angle;if(axis!==undefined){var rotAxis=this.rotationAxis(sinceFrame);angle*=vec3.dot(rotAxis,vec3.normalize(vec3.create(),axis))}return angle};Frame.prototype.rotationAxis=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();return vec3.normalize(vec3.create(),[this._rotation[7]-sinceFrame._rotation[5],this._rotation[2]-sinceFrame._rotation[6],this._rotation[3]-sinceFrame._rotation[1]])};Frame.prototype.rotationMatrix=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return mat3.create();var transpose=mat3.transpose(mat3.create(),this._rotation);return mat3.multiply(mat3.create(),sinceFrame._rotation,transpose)};Frame.prototype.scaleFactor=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return 1;return Math.exp(this._scaleFactor-sinceFrame._scaleFactor)};Frame.prototype.translation=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();return vec3.subtract(vec3.create(),this._translation,sinceFrame._translation)};Frame.prototype.toString=function(){var str="Frame [ id:"+this.id+" | timestamp:"+this.timestamp+" | Hand count:("+this.hands.length+") | Pointable count:("+this.pointables.length+")";if(this.gestures)str+=" | Gesture count:("+this.gestures.length+")";str+=" ]";return str};Frame.prototype.dump=function(){var out="";out+="Frame Info:
";out+=this.toString();out+="

Hands:
";for(var handIdx=0,handCount=this.hands.length;handIdx!=handCount;handIdx++){out+=" "+this.hands[handIdx].toString()+"
"}out+="

Pointables:
";for(var pointableIdx=0,pointableCount=this.pointables.length;pointableIdx!=pointableCount;pointableIdx++){out+=" "+this.pointables[pointableIdx].toString()+"
"}if(this.gestures){out+="

Gestures:
";for(var gestureIdx=0,gestureCount=this.gestures.length;gestureIdx!=gestureCount;gestureIdx++){out+=" "+this.gestures[gestureIdx].toString()+"
"}}out+="

Raw JSON:
";out+=JSON.stringify(this.data);return out};Frame.Invalid={valid:false,hands:[],fingers:[],tools:[],gestures:[],pointables:[],pointable:function(){return Pointable.Invalid},finger:function(){return Pointable.Invalid},hand:function(){return Hand.Invalid},toString:function(){return"invalid frame"},dump:function(){return this.toString()},rotationAngle:function(){return 0},rotationMatrix:function(){return mat3.create()},rotationAxis:function(){return vec3.create()},scaleFactor:function(){return 1},translation:function(){return vec3.create()}}},{"./gesture":6,"./hand":7,"./interaction_box":9,"./pointable":11,"gl-matrix":19,underscore:20}],6:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3,EventEmitter=require("events").EventEmitter,_=require("underscore");var createGesture=exports.createGesture=function(data){var gesture;switch(data.type){case"circle":gesture=new CircleGesture(data);break;case"swipe":gesture=new SwipeGesture(data);break;case"screenTap":gesture=new ScreenTapGesture(data);break;case"keyTap":gesture=new KeyTapGesture(data);break;default:throw"unkown gesture type"}gesture.id=data.id;gesture.handIds=data.handIds;gesture.pointableIds=data.pointableIds;gesture.duration=data.duration;gesture.state=data.state;gesture.type=data.type;return gesture};var gestureListener=exports.gestureListener=function(controller,type){var handlers={};var gestureMap={};var gestureCreator=function(){var candidateGesture=gestureMap[gesture.id];if(candidateGesture!==undefined)gesture.update(gesture,frame);if(gesture.state=="start"||gesture.state=="stop"){if(type==gesture.type&&gestureMap[gesture.id]===undefined){gestureMap[gesture.id]=new Gesture(gesture,frame);gesture.update(gesture,frame)}if(gesture.state=="stop"){delete gestureMap[gesture.id]}}};controller.on("gesture",function(gesture,frame){if(gesture.type==type){if(gesture.state=="start"||gesture.state=="stop"){if(gestureMap[gesture.id]===undefined){var gestureTracker=new Gesture(gesture,frame);gestureMap[gesture.id]=gestureTracker;_.each(handlers,function(cb,name){gestureTracker.on(name,cb)})}}gestureMap[gesture.id].update(gesture,frame);if(gesture.state=="stop"){delete gestureMap[gesture.id]}}});var builder={start:function(cb){handlers["start"]=cb;return builder},stop:function(cb){handlers["stop"]=cb;return builder},complete:function(cb){handlers["stop"]=cb;return builder},update:function(cb){handlers["update"]=cb;return builder}};return builder};var Gesture=exports.Gesture=function(gesture,frame){this.gestures=[gesture];this.frames=[frame]};Gesture.prototype.update=function(gesture,frame){this.gestures.push(gesture);this.frames.push(frame);this.emit(gesture.state,this)};_.extend(Gesture.prototype,EventEmitter.prototype);var CircleGesture=function(data){this.center=data.center;this.normal=data.normal;this.progress=data.progress;this.radius=data.radius};CircleGesture.prototype.toString=function(){return"CircleGesture ["+JSON.stringify(this)+"]"};var SwipeGesture=function(data){this.startPosition=data.startPosition;this.position=data.position;this.direction=data.direction;this.speed=data.speed};SwipeGesture.prototype.toString=function(){return"SwipeGesture ["+JSON.stringify(this)+"]"};var ScreenTapGesture=function(data){this.position=data.position;this.direction=data.direction;this.progress=data.progress};ScreenTapGesture.prototype.toString=function(){return"ScreenTapGesture ["+JSON.stringify(this)+"]"};var KeyTapGesture=function(data){this.position=data.position;this.direction=data.direction;this.progress=data.progress};KeyTapGesture.prototype.toString=function(){return"KeyTapGesture ["+JSON.stringify(this)+"]"}},{events:17,"gl-matrix":19,underscore:20}],7:[function(require,module,exports){var Pointable=require("./pointable"),glMatrix=require("gl-matrix"),mat3=glMatrix.mat3,vec3=glMatrix.vec3,_=require("underscore");var Hand=module.exports=function(data){this.id=data.id;this.palmPosition=data.palmPosition;this.direction=data.direction;this.palmVelocity=data.palmVelocity;this.palmNormal=data.palmNormal;this.sphereCenter=data.sphereCenter;this.sphereRadius=data.sphereRadius;this.valid=true;this.pointables=[];this.fingers=[];this.tools=[];this._translation=data.t;this._rotation=_.flatten(data.r);this._scaleFactor=data.s;this.timeVisible=data.timeVisible;this.stabilizedPalmPosition=data.stabilizedPalmPosition};Hand.prototype.finger=function(id){var finger=this.frame.finger(id);return finger&&finger.handId==this.id?finger:Pointable.Invalid};Hand.prototype.rotationAngle=function(sinceFrame,axis){if(!this.valid||!sinceFrame.valid)return 0;var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return 0;var rot=this.rotationMatrix(sinceFrame);var cs=(rot[0]+rot[4]+rot[8]-1)*.5;var angle=Math.acos(cs);angle=isNaN(angle)?0:angle;if(axis!==undefined){var rotAxis=this.rotationAxis(sinceFrame);angle*=vec3.dot(rotAxis,vec3.normalize(vec3.create(),axis))}return angle};Hand.prototype.rotationAxis=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return vec3.create();return vec3.normalize(vec3.create(),[this._rotation[7]-sinceHand._rotation[5],this._rotation[2]-sinceHand._rotation[6],this._rotation[3]-sinceHand._rotation[1]])};Hand.prototype.rotationMatrix=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return mat3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return mat3.create();var transpose=mat3.transpose(mat3.create(),this._rotation);var m=mat3.multiply(mat3.create(),sinceHand._rotation,transpose);return m};Hand.prototype.scaleFactor=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return 1;var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return 1;return Math.exp(this._scaleFactor-sinceHand._scaleFactor)};Hand.prototype.translation=function(sinceFrame){if(!this.valid||!sinceFrame.valid)return vec3.create();var sinceHand=sinceFrame.hand(this.id);if(!sinceHand.valid)return vec3.create();return[this._translation[0]-sinceHand._translation[0],this._translation[1]-sinceHand._translation[1],this._translation[2]-sinceHand._translation[2]]};Hand.prototype.toString=function(){return"Hand [ id: "+this.id+" | palm velocity:"+this.palmVelocity+" | sphere center:"+this.sphereCenter+" ] "};Hand.Invalid={valid:false,fingers:[],tools:[],pointables:[],pointable:function(){return Pointable.Invalid},finger:function(){return Pointable.Invalid},toString:function(){return"invalid frame"},dump:function(){return this.toString()},rotationAngle:function(){return 0},rotationMatrix:function(){return mat3.create()},rotationAxis:function(){return vec3.create()},scaleFactor:function(){return 1},translation:function(){return vec3.create()}}},{"./pointable":11,"gl-matrix":19,underscore:20}],8:[function(require,module,exports){!function(){module.exports={Controller:require("./controller"),Frame:require("./frame"),Gesture:require("./gesture"),Hand:require("./hand"),Pointable:require("./pointable"),InteractionBox:require("./interaction_box"),Connection:require("./connection"),CircularBuffer:require("./circular_buffer"),UI:require("./ui"),glMatrix:require("gl-matrix"),mat3:require("gl-matrix").mat3,vec3:require("gl-matrix").vec3,loopController:undefined,loop:function(opts,callback){if(callback===undefined){callback=opts;opts={}}if(!this.loopController)this.loopController=new this.Controller(opts);this.loopController.loop(callback)}}}()},{"./circular_buffer":2,"./connection":3,"./controller":4,"./frame":5,"./gesture":6,"./hand":7,"./interaction_box":9,"./pointable":11,"./ui":13,"gl-matrix":19}],9:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3;var InteractionBox=module.exports=function(data){this.valid=true;this.center=data.center;this.size=data.size;this.width=data.size[0];this.height=data.size[1];this.depth=data.size[2]};InteractionBox.prototype.denormalizePoint=function(normalizedPosition){return vec3.fromValues((normalizedPosition[0]-.5)*this.size[0]+this.center[0],(normalizedPosition[1]-.5)*this.size[1]+this.center[1],(normalizedPosition[2]-.5)*this.size[2]+this.center[2])};InteractionBox.prototype.normalizePoint=function(position,clamp){var vec=vec3.fromValues((position[0]-this.center[0])/this.size[0]+.5,(position[1]-this.center[1])/this.size[1]+.5,(position[2]-this.center[2])/this.size[2]+.5);if(clamp){vec[0]=Math.min(Math.max(vec[0],0),1);vec[1]=Math.min(Math.max(vec[1],0),1);vec[2]=Math.min(Math.max(vec[2],0),1)}return vec};InteractionBox.prototype.toString=function(){return"InteractionBox [ width:"+this.width+" | height:"+this.height+" | depth:"+this.depth+" ]"};InteractionBox.Invalid={valid:false}},{"gl-matrix":19}],10:[function(require,module,exports){var Pipeline=module.exports=function(){this.steps=[]};Pipeline.prototype.addStep=function(step){this.steps.push(step)};Pipeline.prototype.run=function(frame){var stepsLength=this.steps.length;for(var i=0;i!=stepsLength;i++){if(!frame)break;frame=this.steps[i](frame)}return frame}},{}],11:[function(require,module,exports){var glMatrix=require("gl-matrix"),vec3=glMatrix.vec3;var Pointable=module.exports=function(data){this.valid=true;this.id=data.id;this.handId=data.handId;this.length=data.length;this.tool=data.tool;this.width=data.width;this.direction=data.direction;this.stabilizedTipPosition=data.stabilizedTipPosition;this.tipPosition=data.tipPosition;this.tipVelocity=data.tipVelocity;this.touchZone=data.touchZone;this.touchDistance=data.touchDistance;this.timeVisible=data.timeVisible};Pointable.prototype.toString=function(){if(this.tool==true){return"Pointable [ id:"+this.id+" "+this.length+"mmx | with:"+this.width+"mm | direction:"+this.direction+" ]"}else{return"Pointable [ id:"+this.id+" "+this.length+"mmx | direction: "+this.direction+" ]"}};Pointable.Invalid={valid:false}},{"gl-matrix":19}],12:[function(require,module,exports){var Frame=require("./frame");var Event=function(data){this.type=data.type;this.state=data.state};var chooseProtocol=exports.chooseProtocol=function(header){var protocol;switch(header.version){case 1:protocol=JSONProtocol(1,function(data){return new Frame(data)});break;case 2:protocol=JSONProtocol(2,function(data){return new Frame(data)});protocol.sendHeartbeat=function(connection){connection.send(protocol.encode({heartbeat:true}))};break;case 3:protocol=JSONProtocol(3,function(data){return data.event?new Event(data.event):new Frame(data)});protocol.sendHeartbeat=function(connection){connection.send(protocol.encode({heartbeat:true}))};break;default:throw"unrecognized version"}return protocol};var JSONProtocol=function(version,cb){var protocol=cb;protocol.encode=function(message){return JSON.stringify(message)};protocol.version=version;protocol.versionLong="Version "+version;protocol.type="protocol";return protocol}},{"./frame":5}],13:[function(require,module,exports){exports.UI={Region:require("./ui/region"),Cursor:require("./ui/cursor")}},{"./ui/cursor":14,"./ui/region":15}],14:[function(require,module,exports){var Cursor=module.exports=function(){return function(frame){var pointable=frame.pointables.sort(function(a,b){return a.z-b.z})[0];if(pointable&&pointable.valid){frame.cursorPosition=pointable.tipPosition}return frame}}},{}],15:[function(require,module,exports){var EventEmitter=require("events").EventEmitter,_=require("underscore");var Region=module.exports=function(start,end){this.start=new Vector(start);this.end=new Vector(end);this.enteredFrame=null};Region.prototype.hasPointables=function(frame){for(var i=0;i!=frame.pointables.length;i++){var position=frame.pointables[i].tipPosition;if(position.x>=this.start.x&&position.x<=this.end.x&&position.y>=this.start.y&&position.y<=this.end.y&&position.z>=this.start.z&&position.z<=this.end.z){return true}}return false};Region.prototype.listener=function(opts){var region=this;if(opts&&opts.nearThreshold)this.setupNearRegion(opts.nearThreshold);return function(frame){return region.updatePosition(frame)}};Region.prototype.clipper=function(){var region=this;return function(frame){region.updatePosition(frame);return region.enteredFrame?frame:null}};Region.prototype.setupNearRegion=function(distance){var nearRegion=this.nearRegion=new Region([this.start.x-distance,this.start.y-distance,this.start.z-distance],[this.end.x+distance,this.end.y+distance,this.end.z+distance]);var region=this;nearRegion.on("enter",function(frame){region.emit("near",frame)});nearRegion.on("exit",function(frame){region.emit("far",frame)});region.on("exit",function(frame){region.emit("near",frame)})};Region.prototype.updatePosition=function(frame){if(this.nearRegion)this.nearRegion.updatePosition(frame);if(this.hasPointables(frame)&&this.enteredFrame==null){this.enteredFrame=frame;this.emit("enter",this.enteredFrame)}else if(!this.hasPointables(frame)&&this.enteredFrame!=null){this.enteredFrame=null;this.emit("exit",this.enteredFrame)}return frame};Region.prototype.normalize=function(position){return new Vector([(position.x-this.start.x)/(this.end.x-this.start.x),(position.y-this.start.y)/(this.end.y-this.start.y),(position.z-this.start.z)/(this.end.z-this.start.z)])};Region.prototype.mapToXY=function(position,width,height){var normalized=this.normalize(position);var x=normalized.x,y=normalized.y;if(x>1)x=1;else if(x<-1)x=-1;if(y>1)y=1;else if(y<-1)y=-1;return[(x+1)/2*width,(1-y)/2*height,normalized.z]};_.extend(Region.prototype,EventEmitter.prototype)},{events:17,underscore:20}],16:[function(require,module,exports){},{}],17:[function(require,module,exports){!function(process){if(!process.EventEmitter)process.EventEmitter=function(){};var EventEmitter=exports.EventEmitter=process.EventEmitter;var isArray=typeof Array.isArray==="function"?Array.isArray:function(xs){return Object.prototype.toString.call(xs)==="[object Array]"};function indexOf(xs,x){if(xs.indexOf)return xs.indexOf(x);for(var i=0;i0&&this._events[type].length>m){this._events[type].warned=true;console.error("(node) warning: possible EventEmitter memory "+"leak detected. %d listeners added. "+"Use emitter.setMaxListeners() to increase limit.",this._events[type].length);console.trace()}}this._events[type].push(listener)}else{this._events[type]=[this._events[type],listener]}return this};EventEmitter.prototype.on=EventEmitter.prototype.addListener;EventEmitter.prototype.once=function(type,listener){var self=this;self.on(type,function g(){self.removeListener(type,g);listener.apply(this,arguments)});return this};EventEmitter.prototype.removeListener=function(type,listener){if("function"!==typeof listener){throw new Error("removeListener only takes instances of Function")}if(!this._events||!this._events[type])return this;var list=this._events[type];if(isArray(list)){var i=indexOf(list,listener);if(i<0)return this;list.splice(i,1);if(list.length==0)delete this._events[type]}else if(this._events[type]===listener){delete this._events[type]}return this};EventEmitter.prototype.removeAllListeners=function(type){if(arguments.length===0){this._events={};return this}if(type&&this._events&&this._events[type])this._events[type]=null;return this};EventEmitter.prototype.listeners=function(type){if(!this._events)this._events={};if(!this._events[type])this._events[type]=[];if(!isArray(this._events[type])){this._events[type]=[this._events[type]]}return this._events[type]}}(require("__browserify_process"))},{__browserify_process:18}],18:[function(require,module,exports){var process=module.exports={};process.nextTick=function(){var canSetImmediate=typeof window!=="undefined"&&window.setImmediate;var canPost=typeof window!=="undefined"&&window.postMessage&&window.addEventListener;if(canSetImmediate){return function(f){return window.setImmediate(f)}}if(canPost){var queue=[];window.addEventListener("message",function(ev){if(ev.source===window&&ev.data==="process-tick"){ev.stopPropagation();if(queue.length>0){var fn=queue.shift();fn()}}},true);return function nextTick(fn){queue.push(fn);window.postMessage("process-tick","*")}}return function nextTick(fn){setTimeout(fn,0)}}();process.title="browser";process.browser=true;process.env={};process.argv=[];process.binding=function(name){throw new Error("process.binding is not supported")};process.cwd=function(){return"/"};process.chdir=function(dir){throw new Error("process.chdir is not supported")}},{}],19:[function(require,module,exports){!function(){!function(){"use strict";var shim={};if(typeof exports==="undefined"){if(typeof define=="function"&&typeof define.amd=="object"&&define.amd){shim.exports={};define(function(){return shim.exports})}else{shim.exports=window}}else{shim.exports=exports}!function(exports){var vec2={};if(!GLMAT_EPSILON){var GLMAT_EPSILON=1e-6}vec2.create=function(){return new Float32Array(2)};vec2.clone=function(a){var out=new Float32Array(2);out[0]=a[0];out[1]=a[1];return out};vec2.fromValues=function(x,y){var out=new Float32Array(2);out[0]=x;out[1]=y;return out};vec2.copy=function(out,a){out[0]=a[0];out[1]=a[1];return out};vec2.set=function(out,x,y){out[0]=x;out[1]=y;return out};vec2.add=function(out,a,b){out[0]=a[0]+b[0];out[1]=a[1]+b[1];return out};vec2.sub=vec2.subtract=function(out,a,b){out[0]=a[0]-b[0];out[1]=a[1]-b[1];return out};vec2.mul=vec2.multiply=function(out,a,b){out[0]=a[0]*b[0];out[1]=a[1]*b[1];return out};vec2.div=vec2.divide=function(out,a,b){out[0]=a[0]/b[0];out[1]=a[1]/b[1];return out};vec2.min=function(out,a,b){out[0]=Math.min(a[0],b[0]); +out[1]=Math.min(a[1],b[1]);return out};vec2.max=function(out,a,b){out[0]=Math.max(a[0],b[0]);out[1]=Math.max(a[1],b[1]);return out};vec2.scale=function(out,a,b){out[0]=a[0]*b;out[1]=a[1]*b;return out};vec2.dist=vec2.distance=function(a,b){var x=b[0]-a[0],y=b[1]-a[1];return Math.sqrt(x*x+y*y)};vec2.sqrDist=vec2.squaredDistance=function(a,b){var x=b[0]-a[0],y=b[1]-a[1];return x*x+y*y};vec2.len=vec2.length=function(a){var x=a[0],y=a[1];return Math.sqrt(x*x+y*y)};vec2.sqrLen=vec2.squaredLength=function(a){var x=a[0],y=a[1];return x*x+y*y};vec2.negate=function(out,a){out[0]=-a[0];out[1]=-a[1];return out};vec2.normalize=function(out,a){var x=a[0],y=a[1];var len=x*x+y*y;if(len>0){len=1/Math.sqrt(len);out[0]=a[0]*len;out[1]=a[1]*len}return out};vec2.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]};vec2.cross=function(out,a,b){var z=a[0]*b[1]-a[1]*b[0];out[0]=out[1]=0;out[2]=z;return out};vec2.lerp=function(out,a,b,t){var ax=a[0],ay=a[1];out[0]=ax+t*(b[0]-ax);out[1]=ay+t*(b[1]-ay);return out};vec2.transformMat2=function(out,a,m){var x=a[0],y=a[1];out[0]=x*m[0]+y*m[1];out[1]=x*m[2]+y*m[3];return out};vec2.forEach=function(){var vec=new Float32Array(2);return function(a,stride,offset,count,fn,arg){var i,l;if(!stride){stride=2}if(!offset){offset=0}if(count){l=Math.min(count*stride+offset,a.length)}else{l=a.length}for(i=offset;i0){len=1/Math.sqrt(len);out[0]=a[0]*len;out[1]=a[1]*len;out[2]=a[2]*len}return out};vec3.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]+a[2]*b[2]};vec3.cross=function(out,a,b){var ax=a[0],ay=a[1],az=a[2],bx=b[0],by=b[1],bz=b[2];out[0]=ay*bz-az*by;out[1]=az*bx-ax*bz;out[2]=ax*by-ay*bx;return out};vec3.lerp=function(out,a,b,t){var ax=a[0],ay=a[1],az=a[2];out[0]=ax+t*(b[0]-ax);out[1]=ay+t*(b[1]-ay);out[2]=az+t*(b[2]-az);return out};vec3.transformMat4=function(out,a,m){var x=a[0],y=a[1],z=a[2];out[0]=m[0]*x+m[4]*y+m[8]*z+m[12];out[1]=m[1]*x+m[5]*y+m[9]*z+m[13];out[2]=m[2]*x+m[6]*y+m[10]*z+m[14];return out};vec3.transformQuat=function(out,a,q){var x=a[0],y=a[1],z=a[2],qx=q[0],qy=q[1],qz=q[2],qw=q[3],ix=qw*x+qy*z-qz*y,iy=qw*y+qz*x-qx*z,iz=qw*z+qx*y-qy*x,iw=-qx*x-qy*y-qz*z;out[0]=ix*qw+iw*-qx+iy*-qz-iz*-qy;out[1]=iy*qw+iw*-qy+iz*-qx-ix*-qz;out[2]=iz*qw+iw*-qz+ix*-qy-iy*-qx;return out};vec3.forEach=function(){var vec=new Float32Array(3);return function(a,stride,offset,count,fn,arg){var i,l;if(!stride){stride=3}if(!offset){offset=0}if(count){l=Math.min(count*stride+offset,a.length)}else{l=a.length}for(i=offset;i0){len=1/Math.sqrt(len);out[0]=a[0]*len;out[1]=a[1]*len;out[2]=a[2]*len;out[3]=a[3]*len}return out};vec4.dot=function(a,b){return a[0]*b[0]+a[1]*b[1]+a[2]*b[2]+a[3]*b[3]};vec4.lerp=function(out,a,b,t){var ax=a[0],ay=a[1],az=a[2],aw=a[3];out[0]=ax+t*(b[0]-ax);out[1]=ay+t*(b[1]-ay);out[2]=az+t*(b[2]-az);out[3]=aw+t*(b[3]-aw);return out};vec4.transformMat4=function(out,a,m){var x=a[0],y=a[1],z=a[2],w=a[3];out[0]=m[0]*x+m[4]*y+m[8]*z+m[12]*w;out[1]=m[1]*x+m[5]*y+m[9]*z+m[13]*w;out[2]=m[2]*x+m[6]*y+m[10]*z+m[14]*w;out[3]=m[3]*x+m[7]*y+m[11]*z+m[15]*w;return out};vec4.transformQuat=function(out,a,q){var x=a[0],y=a[1],z=a[2],qx=q[0],qy=q[1],qz=q[2],qw=q[3],ix=qw*x+qy*z-qz*y,iy=qw*y+qz*x-qx*z,iz=qw*z+qx*y-qy*x,iw=-qx*x-qy*y-qz*z;out[0]=ix*qw+iw*-qx+iy*-qz-iz*-qy;out[1]=iy*qw+iw*-qy+iz*-qx-ix*-qz;out[2]=iz*qw+iw*-qz+ix*-qy-iy*-qx;return out};vec4.forEach=function(){var vec=new Float32Array(4);return function(a,stride,offset,count,fn,arg){var i,l;if(!stride){stride=4}if(!offset){offset=0}if(count){l=Math.min(count*stride+offset,a.length)}else{l=a.length}for(i=offset;i=1){if(out!==a){out[0]=ax;out[1]=ay;out[2]=az;out[3]=aw}return out}halfTheta=Math.acos(cosHalfTheta);sinHalfTheta=Math.sqrt(1-cosHalfTheta*cosHalfTheta);if(Math.abs(sinHalfTheta)<.001){out[0]=ax*.5+bx*.5;out[1]=ay*.5+by*.5;out[2]=az*.5+bz*.5;out[3]=aw*.5+bw*.5;return out}ratioA=Math.sin((1-t)*halfTheta)/sinHalfTheta;ratioB=Math.sin(t*halfTheta)/sinHalfTheta;out[0]=ax*ratioA+bx*ratioB;out[1]=ay*ratioA+by*ratioB;out[2]=az*ratioA+bz*ratioB;out[3]=aw*ratioA+bw*ratioB;return out};quat.invert=function(out,a){var a0=a[0],a1=a[1],a2=a[2],a3=a[3],dot=a0*a0+a1*a1+a2*a2+a3*a3,invDot=dot?1/dot:0;out[0]=-a0*invDot;out[1]=-a1*invDot;out[2]=-a2*invDot;out[3]=a3*invDot;return out};quat.conjugate=function(out,a){out[0]=-a[0];out[1]=-a[1];out[2]=-a[2];out[3]=a[3];return out};quat.len=quat.length=vec4.length;quat.sqrLen=quat.squaredLength=vec4.squaredLength;quat.normalize=vec4.normalize;quat.str=function(a){return"quat("+a[0]+", "+a[1]+", "+a[2]+", "+a[3]+")"};if(typeof exports!=="undefined"){exports.quat=quat}}(shim.exports)}()}()},{}],20:[function(require,module,exports){!function(){!function(){var root=this;var previousUnderscore=root._;var breaker={};var ArrayProto=Array.prototype,ObjProto=Object.prototype,FuncProto=Function.prototype;var push=ArrayProto.push,slice=ArrayProto.slice,concat=ArrayProto.concat,toString=ObjProto.toString,hasOwnProperty=ObjProto.hasOwnProperty;var nativeForEach=ArrayProto.forEach,nativeMap=ArrayProto.map,nativeReduce=ArrayProto.reduce,nativeReduceRight=ArrayProto.reduceRight,nativeFilter=ArrayProto.filter,nativeEvery=ArrayProto.every,nativeSome=ArrayProto.some,nativeIndexOf=ArrayProto.indexOf,nativeLastIndexOf=ArrayProto.lastIndexOf,nativeIsArray=Array.isArray,nativeKeys=Object.keys,nativeBind=FuncProto.bind;var 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Motion integration for Reveal.js. + * James Sun [sun16] + * Rory Hardy [gneatgeek] + */ + +(function () { + var body = document.body, + controller = new Leap.Controller({ enableGestures: true }), + lastGesture = 0, + leapConfig = Reveal.getConfig().leap, + pointer = document.createElement( 'div' ), + config = { + autoCenter : true, // Center pointer around detected position. + gestureDelay : 500, // How long to delay between gestures. + naturalSwipe : true, // Swipe as if it were a touch screen. + pointerColor : '#00aaff', // Default color of the pointer. + pointerOpacity : 0.7, // Default opacity of the pointer. + pointerSize : 15, // Default minimum height/width of the pointer. + pointerTolerance : 120 // Bigger = slower pointer. + }, + entered, enteredPosition, now, size, tipPosition; // Other vars we need later, but don't need to redeclare. + + // Merge user defined settings with defaults + if( leapConfig ) { + for( key in leapConfig ) { + config[key] = leapConfig[key]; + } + } + + pointer.id = 'leap'; + + pointer.style.position = 'absolute'; + pointer.style.visibility = 'hidden'; + pointer.style.zIndex = 50; + pointer.style.opacity = config.pointerOpacity; + pointer.style.backgroundColor = config.pointerColor; + + body.appendChild( pointer ); + + // Leap's loop + controller.on( 'frame', function ( frame ) { + // Timing code to rate limit gesture execution + now = new Date().getTime(); + + // Pointer: 1 to 2 fingers. Strictly one finger works but may cause innaccuracies. + // The innaccuracies were observed on a development model and may not be an issue with consumer models. + if( frame.fingers.length > 0 && frame.fingers.length < 3 ) { + // Invert direction and multiply by 3 for greater effect. + size = -3 * frame.fingers[0].tipPosition[2]; + + if( size < config.pointerSize ) { + size = config.pointerSize; + } + + pointer.style.width = size + 'px'; + pointer.style.height = size + 'px'; + pointer.style.borderRadius = size - 5 + 'px'; + pointer.style.visibility = 'visible'; + + tipPosition = frame.fingers[0].tipPosition; + + if( config.autoCenter ) { + + + // Check whether the finger has entered the z range of the Leap Motion. Used for the autoCenter option. + if( !entered ) { + entered = true; + enteredPosition = frame.fingers[0].tipPosition; + } + + pointer.style.top = + (-1 * (( tipPosition[1] - enteredPosition[1] ) * body.offsetHeight / config.pointerTolerance )) + + ( body.offsetHeight / 2 ) + 'px'; + + pointer.style.left = + (( tipPosition[0] - enteredPosition[0] ) * body.offsetWidth / config.pointerTolerance ) + + ( body.offsetWidth / 2 ) + 'px'; + } + else { + pointer.style.top = ( 1 - (( tipPosition[1] - 50) / config.pointerTolerance )) * + body.offsetHeight + 'px'; + + pointer.style.left = ( tipPosition[0] * body.offsetWidth / config.pointerTolerance ) + + ( body.offsetWidth / 2 ) + 'px'; + } + } + else { + // Hide pointer on exit + entered = false; + pointer.style.visibility = 'hidden'; + } + + // Gestures + if( frame.gestures.length > 0 && (now - lastGesture) > config.gestureDelay ) { + var gesture = frame.gestures[0]; + + // One hand gestures + if( frame.hands.length === 1 ) { + // Swipe gestures. 3+ fingers. + if( frame.fingers.length > 2 && gesture.type === 'swipe' ) { + // Define here since some gestures will throw undefined for these. + var x = gesture.direction[0], + y = gesture.direction[1]; + + // Left/right swipe gestures + if( Math.abs( x ) > Math.abs( y )) { + if( x > 0 ) { + config.naturalSwipe ? Reveal.left() : Reveal.right(); + } + else { + config.naturalSwipe ? Reveal.right() : Reveal.left(); + } + } + // Up/down swipe gestures + else { + if( y > 0 ) { + config.naturalSwipe ? Reveal.down() : Reveal.up(); + } + else { + config.naturalSwipe ? Reveal.up() : Reveal.down(); + } + } + + lastGesture = now; + } + } + // Two hand gestures + else if( frame.hands.length === 2 ) { + // Upward two hand swipe gesture + if( gesture.type === 'swipe' && gesture.direction[1] > 0 ) { + Reveal.toggleOverview(); + } + + lastGesture = now; + } + } + }); + + controller.connect(); +})(); diff --git a/doc/pub/svm/html/reveal.js/plugin/remotes/remotes.js b/doc/pub/svm/html/reveal.js/plugin/remotes/remotes.js new file mode 100644 index 000000000..ba0dbad7b --- /dev/null +++ b/doc/pub/svm/html/reveal.js/plugin/remotes/remotes.js @@ -0,0 +1,39 @@ +/** + * Touch-based remote controller for your presentation courtesy + * of the folks at http://remotes.io + */ + +(function(window){ + + /** + * Detects if we are dealing with a touch enabled device (with some false positives) + * Borrowed from modernizr: https://github.com/Modernizr/Modernizr/blob/master/feature-detects/touch.js + */ + var hasTouch = (function(){ + return ('ontouchstart' in window) || window.DocumentTouch && document instanceof DocumentTouch; + })(); + + /** + * Detects if notes are enable and the current page is opened inside an /iframe + * this prevents loading Remotes.io several times + */ + var isNotesAndIframe = (function(){ + return window.RevealNotes && !(self == top); + })(); + + if(!hasTouch && !isNotesAndIframe){ + head.ready( 'remotes.ne.min.js', function() { + new Remotes("preview") + .on("swipe-left", function(e){ Reveal.right(); }) + .on("swipe-right", function(e){ Reveal.left(); }) + .on("swipe-up", function(e){ Reveal.down(); }) + .on("swipe-down", function(e){ Reveal.up(); }) + .on("tap", function(e){ Reveal.next(); }) + .on("zoom-out", function(e){ Reveal.toggleOverview(true); }) + .on("zoom-in", function(e){ Reveal.toggleOverview(false); }) + ; + } ); + + head.js('https://hakim-static.s3.amazonaws.com/reveal-js/remotes.ne.min.js'); + } +})(window); \ No newline at end of file diff --git a/doc/src/DimRed/Programs/PCAcancer.py b/doc/src/DimRed/Programs/PCAcancer.py new file mode 100644 index 000000000..10aa6023e --- /dev/null +++ b/doc/src/DimRed/Programs/PCAcancer.py @@ -0,0 +1,43 @@ +import matplotlib.pyplot as plt +import numpy as np +from sklearn.model_selection import train_test_split +from sklearn.datasets import load_breast_cancer +from sklearn.linear_model import LogisticRegression +cancer = load_breast_cancer() +import pandas as pd + +cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names) + + +import seaborn as sns +correlation_matrix = cancerpd.corr().round(1) +# use the heatmap function from seaborn to plot the correlation matrix +# annot = True to print the values inside the square +sns.heatmap(data=correlation_matrix, annot=True) +EigValues, EigVectors = np.linalg.eig(correlation_matrix) +print(EigValues) + +X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) +print(X_train.shape) +print(X_test.shape) + +logreg = LogisticRegression() +logreg.fit(X_train, y_train) +print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train))) + +from sklearn.preprocessing import MinMaxScaler, StandardScaler +scaler = StandardScaler() +scaler.fit(X_train) +X_train_scaled = scaler.transform(X_train) +X_test_scaled = scaler.transform(X_test) + +logreg.fit(X_train_scaled, y_train) +print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train))) + +#thereafter we do a PCA with Scikit-learn +from sklearn.decomposition import PCA +pca = PCA(n_components = 2) +X2D_train = pca.fit_transform(X_train_scaled) + +logreg.fit(X2D_train,y_train) +print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train))) diff --git a/doc/src/DimRed/Programs/PCAexample.py b/doc/src/DimRed/Programs/PCAexample.py new file mode 100644 index 000000000..94cddcdb0 --- /dev/null +++ b/doc/src/DimRed/Programs/PCAexample.py @@ -0,0 +1,31 @@ +import numpy as np +import pandas as pd +from IPython.display import display +np.random.seed(100) +# setting up a 10 x 5 matrix +rows = 10 +cols = 5 +X = np.random.randn(rows,cols) +df = pd.DataFrame(X) +# Pandas does the centering for us +df = df -df.mean() +display(df) + +# we center it ourselves +X_centered = X - X.mean(axis=0) +print(X_centered-df) +#Now we do an SVD +U, s, V = np.linalg.svd(X_centered) +c1 = V.T[:, 0] +c2 = V.T[:, 1] +W2 = V.T[:, :2] +X2D = X_centered.dot(W2) +print(X2D) +#thereafter we do a PCA with Scikit-learn +from sklearn.decomposition import PCA +pca = PCA(n_components = 2) +X2D = pca.fit_transform(X) +print(X2D) + +print(pca.components_.T[:, 0]) + diff --git a/doc/src/DimRed/Programs/PCAsimple.py~ b/doc/src/DimRed/Programs/PCAsimple.py~ new file mode 100644 index 000000000..5ec8a6e82 --- /dev/null +++ b/doc/src/DimRed/Programs/PCAsimple.py~ @@ -0,0 +1,35 @@ +import numpy as np +import pandas as pd +from IPython.display import display + + +n = 10 +mean = (-1, 2) +cov = [[4, 2], [2, 2]] +X = np.random.multivariate_normal(mean, cov, n) +print(X) + +df = pd.DataFrame(X) +# Pandas does the centering for us +display(df) +#df = df -df.mean() +#display(df) + +# we center it ourselves +X_centered = X - X.mean(axis=0) +print(X_centered)#-df) +#Now we do an SVD +U, s, V = np.linalg.svd(X_centered) +c1 = V.T[:, 0] +c2 = V.T[:, 1] +W2 = V.T[:, :2] +X2D = X_centered.dot(W2) +print(X2D) +#thereafter we do a PCA with Scikit-learn +from sklearn.decomposition import PCA +pca = PCA(n_components = 2) +X2Dsl = pca.fit_transform(X) +print(X2D-X2Dsl) + +print(pca.components_.T[:, 0]) + diff --git a/doc/src/DimRed/Programs/cancer.py b/doc/src/DimRed/Programs/cancer.py new file mode 100644 index 000000000..e1c06c60c --- /dev/null +++ b/doc/src/DimRed/Programs/cancer.py @@ -0,0 +1,43 @@ +import matplotlib.pyplot as plt +import numpy as np +from sklearn.model_selection import train_test_split +from sklearn.datasets import load_breast_cancer +from sklearn.svm import SVC +from sklearn.linear_model import LogisticRegression +cancer = load_breast_cancer() + +X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) +print(X_train.shape) +print(X_test.shape) + +logreg = LogisticRegression() +logreg.fit(X_train, y_train) + +#svm = SVC(C=100) +#svm.fit(X_train, y_train) +print("Test set accuracy: {:.2f}".format(logreg.score(X_test,y_test))) + +from sklearn.preprocessing import MinMaxScaler, StandardScaler + +scaler = StandardScaler() +scaler.fit(X_train) +X_train_scaled = scaler.transform(X_train) +X_test_scaled = scaler.transform(X_test) + +print("Feature min values before scaling:\n {}".format(X_train.min(axis=0))) +print("Feature max values before scaling:\n {}".format(X_train.max(axis=0))) + +print("Feature min values before scaling:\n {}".format(X_train_scaled.min(axis=0))) +print("Feature max values before scaling:\n {}".format(X_train_scaled.max(axis=0))) + +logreg.fit(X_train_scaled, y_train) +#svm.fit(X_train_scaled, y_train) +print("Test set accuracy scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) + +scaler = StandardScaler() +scaler.fit(X_train) +X_train_scaled = scaler.transform(X_train) +X_test_scaled = scaler.transform(X_test) +logreg.fit(X_train_scaled, y_train) +#svm.fit(X_train_scaled, y_train) +print("Test set accuracy scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) diff --git a/doc/src/DimRed/Programs/cancerlogreg.py b/doc/src/DimRed/Programs/cancerlogreg.py new file mode 100644 index 000000000..45f03f36a --- /dev/null +++ b/doc/src/DimRed/Programs/cancerlogreg.py @@ -0,0 +1,40 @@ +import matplotlib.pyplot as plt +import numpy as np +from sklearn.model_selection import train_test_split +from sklearn.datasets import load_breast_cancer +from sklearn.linear_model import LogisticRegression +cancer = load_breast_cancer() + +fig, axes = plt.subplots(15,2,figsize=(10,20)) +malignant = cancer.data[cancer.target == 0] +benign = cancer.data[cancer.target == 1] +ax = axes.ravel() + +for i in range(30): + _, bins = np.histogram(cancer.data[:,i], bins =50) + ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5) + ax[i].hist(benign[:,i], bins = bins, alpha = 0.5) + ax[i].set_title(cancer.feature_names[i]) + ax[i].set_yticks(()) +ax[0].set_xlabel("Feature magnitude") +ax[0].set_ylabel("Frequency") +ax[0].legend(["Malignant", "Benign"], loc ="best") +fig.tight_layout() +plt.show() + +X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) +print(X_train.shape) +print(X_test.shape) + +logreg = LogisticRegression() +logreg.fit(X_train, y_train) +print("Test set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test))) + +from sklearn.preprocessing import MinMaxScaler, StandardScaler +scaler = StandardScaler() +scaler.fit(X_train) +X_train_scaled = scaler.transform(X_train) +X_test_scaled = scaler.transform(X_test) + +logreg.fit(X_train_scaled, y_train) +print("Test set accuracy scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) diff --git a/doc/src/DimRed/Programs/cancerownlogreg.py b/doc/src/DimRed/Programs/cancerownlogreg.py new file mode 100644 index 000000000..a87641450 --- /dev/null +++ b/doc/src/DimRed/Programs/cancerownlogreg.py @@ -0,0 +1,44 @@ +import matplotlib.pyplot as plt +import numpy as np +from sklearn.model_selection import train_test_split +from sklearn.datasets import load_breast_cancer +from sklearn.linear_model import LogisticRegression +cancer = load_breast_cancer() + +fig, axes = plt.subplots(15,2,figsize=(10,20)) +malignant = cancer.data[cancer.target == 0] +benign = cancer.data[cancer.target == 1] +ax = axes.ravel() + +for i in range(30): + _, bins = np.histogram(cancer.data[:,i], bins =50) + ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5) + ax[i].hist(benign[:,i], bins = bins, alpha = 0.5) + ax[i].set_title(cancer.feature_names[i]) + ax[i].set_yticks(()) +ax[0].set_xlabel("Feature magnitude") +ax[0].set_ylabel("Frequency") +ax[0].legend(["Malignant", "Benign"], loc ="best") +fig.tight_layout() +plt.show() + +X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) + +logreg = LogisticRegression() +logreg.fit(X_train, y_train) +print("Test set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test))) + + + +beta = np.random.randn(2,1) + +eta = 0.1 +Niterations = 100 + +for iter in range(Niterations): + gradients = 2.0/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta + beta -= eta*gradients + +print(beta) +ypredict = xb @ beta +ypredict2 = xb @ beta_linreg diff --git a/doc/src/DimRed/Programs/covariance.py b/doc/src/DimRed/Programs/covariance.py new file mode 100644 index 000000000..5f47aeac5 --- /dev/null +++ b/doc/src/DimRed/Programs/covariance.py @@ -0,0 +1,24 @@ +# Importing various packages +import numpy as np +n = 100 +# define two vectors +x = np.random.random(size=n) +y = 4+3*x+np.random.normal(size=n) +#scaling the x and y vectors +x = x - np.mean(x) +y = y - np.mean(y) +variance_x = np.sum(x@x)/n +variance_y = np.sum(y@y)/n +print(variance_x) +print(variance_y) +cov_xy = np.sum(x@y)/n +cov_xx = np.sum(x@x)/n +cov_yy = np.sum(y@y)/n +C = np.zeros((2,2)) +C[0,0]= cov_xx/variance_x +C[1,1]= cov_yy/variance_y +C[0,1]= cov_xy/np.sqrt(variance_y*variance_x) +C[1,0]= C[0,1] +print(C) +Eigvals, Eigvecs = np.linalg.eig(C) +print(Eigvals) diff --git a/doc/src/DimRed/Programs/covfrance.py b/doc/src/DimRed/Programs/covfrance.py new file mode 100644 index 000000000..ab57bf2bd --- /dev/null +++ b/doc/src/DimRed/Programs/covfrance.py @@ -0,0 +1,43 @@ +# Common imports +import numpy as np +import pandas as pd + + +def FrankeFunction(x,y): + term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2)) + term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1)) + term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2)) + term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2) + return term1 + term2 + term3 + term4 + + +def create_X(x, y, n ): + if len(x.shape) > 1: + x = np.ravel(x) + y = np.ravel(y) + + N = len(x) + l = int((n+1)*(n+2)/2) # Number of elements in beta + X = np.ones((N,l)) + + for i in range(1,n+1): + q = int((i)*(i+1)/2) + for k in range(i+1): + X[:,q+k] = (x**(i-k))*(y**k) + + return X + + +# Making meshgrid of datapoints and compute Franke's function +n = 4 +N = 1000 +x = np.sort(np.random.uniform(0, 1, N)) +y = np.sort(np.random.uniform(0, 1, N)) +z = FrankeFunction(x, y) +X = create_X(x, y, n=n) + +Xpd = pd.DataFrame(X) +Xpd = Xpd - Xpd.mean() +correlation_matrix = Xpd.cov() +print(correlation_matrix) + diff --git a/doc/src/DimRed/Programs/mlpfranke.py b/doc/src/DimRed/Programs/mlpfranke.py new file mode 100644 index 000000000..639e605ef --- /dev/null +++ b/doc/src/DimRed/Programs/mlpfranke.py @@ -0,0 +1,68 @@ +# Common imports +import numpy as np +from sklearn.neural_network import MLPRegressor +from sklearn.metrics import accuracy_score +import seaborn as sns +import matplotlib.pyplot as plt + +def FrankeFunction(x,y): + term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2)) + term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1)) + term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2)) + term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2) + return term1 + term2 + term3 + term4 + + +def create_X(x, y, n ): + if len(x.shape) > 1: + x = np.ravel(x) + y = np.ravel(y) + + N = len(x) + l = int((n+1)*(n+2)/2) # Number of elements in beta + X = np.ones((N,l)) + + for i in range(1,n+1): + q = int((i)*(i+1)/2) + for k in range(i+1): + X[:,q+k] = (x**(i-k))*(y**k) + + return X + + +# Making meshgrid of datapoints and compute Franke's function +n = 4 +N = 100 +x = np.sort(np.random.uniform(0, 1, N)) +y = np.sort(np.random.uniform(0, 1, N)) +z = FrankeFunction(x, y) +X = create_X(x, y, n=n) + +# only training data, no advanced splitting +X_train = X +Y_train = z +# only one simple layer with 100 neurons +n_hidden_neurons = 100 +epochs = 100 +# store models for later use +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +# store the models for later use +DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +sns.set() +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic', + alpha=lmbd, learning_rate_init=eta, max_iter=epochs) + dnn.fit(X_train, Y_train) + DNN_scikit[i][j] = dnn + train_accuracy[i][j] = dnn.score(X_train, Y_train) + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Training Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + diff --git a/doc/src/DimRed/Programs/newcov.py b/doc/src/DimRed/Programs/newcov.py new file mode 100644 index 000000000..8e6fab13e --- /dev/null +++ b/doc/src/DimRed/Programs/newcov.py @@ -0,0 +1,28 @@ +# Importing various packages +import numpy as np +n = 10 +x = np.random.normal(size=n) +x = x - np.mean(x) +y = 4+3*x+np.random.normal(size=n) +y = y - np.mean(y) +X = (np.vstack((x, y))).T +print(X) +import pandas as pd +Xpd = pd.DataFrame(X) +print(Xpd) +correlation_matrix = Xpd.corr() +print(correlation_matrix) + + + +variance_x = np.sum(x@x)/n +variance_y = np.sum(y@y)/n +cov_xy = np.sum(x@y)/n +cov_xx = np.sum(x@x)/n +cov_yy = np.sum(y@y)/n +C = np.zeros((2,2)) +C[0,0]= cov_xx/variance_x +C[1,1]= cov_yy/variance_y +C[0,1]= cov_xy/np.sqrt(variance_y*variance_x) +C[1,0]= C[0,1] +print(C)