From ce33548cb59f11f6ebc77b4c7c38b9b9a47ad93d Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Tue, 23 Nov 2021 07:35:44 +0100 Subject: [PATCH] added files --- .../.Clustering-reveal_html_file_collection | 2 + doc/src/Clustering/Clustering.dlog | 96 ++ doc/src/How2ReadData/Git.dlog | 4 + doc/src/How2ReadData/Git.do.txt~ | 505 +++++++ doc/src/How2ReadData/add.dlog | 11 + doc/src/How2ReadData/add.do.txt~ | 519 ++++++++ .../GradientMethods-checkpoint.ipynb | 0 .../GradientMethods.ipynb | 0 .../{Splines => Optimization}/Splines.do.txt | 0 .../example_plot-checkpoint.ipynb | 0 ...s_allowed_functions-Copy1-checkpoint.ipynb | 0 ...xamples_allowed_functions-checkpoint.ipynb | 0 .../autodiff/autograd.pdf | Bin .../autodiff/example_plot.ipynb | 0 .../examples_allowed_functions-Copy1.ipynb | 0 .../autodiff/examples_allowed_functions.ipynb | 0 doc/src/Optimization/backup/lecture4.pdf | Bin 0 -> 617258 bytes doc/src/Optimization/backup/lecture5.pdf | Bin 0 -> 297035 bytes 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doc/src/week39/codes/test8.py~ create mode 100644 doc/src/week39/codes/test9.py~ create mode 100644 doc/src/week41/programs/breast.py~ create mode 100644 doc/src/week41/programs/nnown.py~ create mode 100644 doc/src/week41/programs/test.py~ create mode 100644 doc/src/week43/LatexFigures/fig1.aux create mode 100644 doc/src/week43/LatexFigures/fig1.log create mode 100644 doc/src/week43/LatexFigures/fig2.aux create mode 100644 doc/src/week43/LatexFigures/fig2.log create mode 100644 doc/src/week43/LatexFigures/fig2.tex~ create mode 100644 doc/src/week43/LatexFigures/nn.py~ create mode 100644 doc/src/week45/Datafiles/ride.csv~ diff --git a/doc/src/Clustering/.Clustering-reveal_html_file_collection b/doc/src/Clustering/.Clustering-reveal_html_file_collection new file mode 100644 index 000000000..146687edb --- /dev/null +++ b/doc/src/Clustering/.Clustering-reveal_html_file_collection @@ -0,0 +1,2 @@ +Clustering-reveal.html +clustering_example_images/simple_clustering.gif diff --git a/doc/src/Clustering/Clustering.dlog b/doc/src/Clustering/Clustering.dlog new file mode 100644 index 000000000..147687a4f --- /dev/null +++ b/doc/src/Clustering/Clustering.dlog @@ -0,0 +1,96 @@ +Translating doconce text in Clustering.do.txt to html +*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax) +*** error: figure file "clustering_example_images/some_image.jpg" does not exist! +Translating doconce text in Clustering.do.txt to html +*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax) +*** error: syntax error in table! + missing three horizontal rules and heading + in the right places + + lines surrounding the table: + +
    +
  1. For a given cluster assignment \( C \), and \( k \) cluster means + \( \{m_1, \cdots, m_k\} \). We minimize the total cluster variance with respect to + the cluster means \( \{m_k\} \) yielding the means of the currently assigned + clusters. +
  2. Given a current set of \( k \) means \( \{m_k\} \) the total cluster variance is + minimized by assigning each observation to the closest (current) cluster mean. + That is $$C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} + ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2$$ +
  3. Steps 1 and 2 are repeated until the assignments do not change. +
+ +As previously stated the above formulation can be a bit difficult to understand, +at least the first time, due to the dense notation used. But all in all the + + here are the recorded table rows: +NOTE: do not use pipes in horizontal rule of this type: +(write instead |-\boldsymbol{x_i} - \boldsymbol{m_k}-|) +| | \boldsymbol{x_i} - \boldsymbol{m_k} | | + +possible trouble: +1. Not a table, just an opening pipe symbol at the beginning of the line? +2. Something wrong with the syntax in a preceding table? +Translating doconce text in Clustering.do.txt to html +*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax) +*** error: syntax error in table! + missing three horizontal rules and heading + in the right places + + lines surrounding the table: + +
    +
  1. For a given cluster assignment \( C \), and \( k \) cluster means + \( \{m_1, \cdots, m_k\} \). We minimize the total cluster variance with respect to + the cluster means \( \{m_k\} \) yielding the means of the currently assigned + clusters. +
  2. Given a current set of \( k \) means \( \{m_k\} \) the total cluster variance is + minimized by assigning each observation to the closest (current) cluster mean. + That is $$C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} + ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2$$ +
  3. Steps 1 and 2 are repeated until the assignments do not change. +
+ +As previously stated the above formulation can be a bit difficult to understand, +at least the first time, due to the dense notation used. But all in all the + + here are the recorded table rows: +NOTE: do not use pipes in horizontal rule of this type: +(write instead |-\boldsymbol{x_i} - \boldsymbol{m_k}-|) +| | \boldsymbol{x_i} - \boldsymbol{m_k} | | + +possible trouble: +1. Not a table, just an opening pipe symbol at the beginning of the line? +2. Something wrong with the syntax in a preceding table? +Translating doconce text in Clustering.do.txt to html +*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax) +*** error: syntax error in table! + missing three horizontal rules and heading + in the right places + + lines surrounding the table: + +
    +
  1. For a given cluster assignment \( C \), and \( k \) cluster means + \( \left{m_1, \cdots, m_k\right} \). We minimize the total cluster variance with respect to + the cluster means \( \{m_k\} \) yielding the means of the currently assigned + clusters. +
  2. Given a current set of \( k \) means \( \{m_k\} \) the total cluster variance is + minimized by assigning each observation to the closest (current) cluster mean. + That is $$C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} + ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2$$ +
  3. Steps 1 and 2 are repeated until the assignments do not change. +
+ +As previously stated the above formulation can be a bit difficult to understand, +at least the first time, due to the dense notation used. But all in all the + + here are the recorded table rows: +NOTE: do not use pipes in horizontal rule of this type: +(write instead |-\boldsymbol{x_i} - \boldsymbol{m_k}-|) +| | \boldsymbol{x_i} - \boldsymbol{m_k} | | + +possible trouble: +1. Not a table, just an opening pipe symbol at the beginning of the line? +2. Something wrong with the syntax in a preceding table? diff --git a/doc/src/How2ReadData/Git.dlog b/doc/src/How2ReadData/Git.dlog new file mode 100644 index 000000000..ae49b77d9 --- /dev/null +++ b/doc/src/How2ReadData/Git.dlog @@ -0,0 +1,4 @@ +Translating doconce text in Git.do.txt to ipynb +Failed to remove ans_at_end environment +Failed to remove sol_at_end environment +output in Git.ipynb diff --git a/doc/src/How2ReadData/Git.do.txt~ b/doc/src/How2ReadData/Git.do.txt~ new file mode 100644 index 000000000..08f6118ad --- /dev/null +++ b/doc/src/How2ReadData/Git.do.txt~ @@ -0,0 +1,505 @@ + +===== Simple linear regression model using _scikit-learn_ ===== + +We start with perhaps our simplest possible example, using _Scikit-Learn_ to perform linear regression analysis on a data set produced by us. + +What follows is a simple Python code where we have defined a function +$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. +The numbers in the vector $\hat{x}$ are given +by random numbers generated with a uniform distribution with entries +$x_i \in [0,1]$ (more about probability distribution functions +later). These values are then used to define a function $y(x)$ +(tabulated again as a vector) with a linear dependence on $x$ plus a +random noise added via the normal distribution. + + +The Numpy functions are imported used the _import numpy as np_ +statement and the random number generator for the uniform distribution +is called using the function _np.random.rand()_, where we specificy +that we want $100$ random variables. Using Numpy we define +automatically an array with the specified number of elements, $100$ in +our case. With the Numpy function _randn()_ we can compute random +numbers with the normal distribution (mean value $\mu$ equal to zero and +variance $\sigma^2$ set to one) and produce the values of $y$ assuming a linear +dependence as function of $x$ + +!bt +\[ +y = 2x+N(0,1), +\] +!et + +where $N(0,1)$ represents random numbers generated by the normal +distribution. From _Scikit-Learn_ we import then the +_LinearRegression_ functionality and make a prediction $\tilde{y} = +\alpha + \beta x$ using the function _fit(x,y)_. We call the set of +data $(\hat{x},\hat{y})$ for our training data. The Python package +_scikit-learn_ has also a functionality which extracts the above +fitting parameters $\alpha$ and $\beta$ (see below). Later we will +distinguish between training data and test data. + +For plotting we use the Python package +"matplotlib":"https://matplotlib.org/" which produces publication +quality figures. Feel free to explore the extensive +"gallery":"https://matplotlib.org/gallery/index.html" of examples. In +this example we plot our original values of $x$ and $y$ as well as the +prediction _ypredict_ ($\tilde{y}$), which attempts at fitting our +data with a straight line. + +The Python code follows here. +!bc pycod +# Importing various packages +import numpy as np +import matplotlib.pyplot as plt +from sklearn.linear_model import LinearRegression + +x = np.random.rand(100,1) +y = 2*x+np.random.randn(100,1) +linreg = LinearRegression() +linreg.fit(x,y) +xnew = np.array([[0],[1]]) +ypredict = linreg.predict(xnew) + +plt.plot(xnew, ypredict, "r-") +plt.plot(x, y ,'ro') +plt.axis([0,1.0,0, 5.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Simple Linear Regression') +plt.show() +!ec + +This example serves several aims. It allows us to demonstrate several +aspects of data analysis and later machine learning algorithms. The +immediate visualization shows that our linear fit is not +impressive. It goes through the data points, but there are many +outliers which are not reproduced by our linear regression. We could +now play around with this small program and change for example the +factor in front of $x$ and the normal distribution. Try to change the +function $y$ to + +!bt +\[ +y = 10x+0.01 \times N(0,1), +\] +!et + +where $x$ is defined as before. Does the fit look better? Indeed, by +reducing the role of the noise given by the normal distribution we see immediately that +our linear prediction seemingly reproduces better the training +set. However, this testing 'by the eye' is obviouly not satisfactory in the +long run. Here we have only defined the training data and our model, and +have not discussed a more rigorous approach to the _cost_ function. + +We need more rigorous criteria in defining whether we have succeeded or +not in modeling our training data. You will be surprised to see that +many scientists seldomly venture beyond this 'by the eye' approach. A +standard approach for the *cost* function is the so-called $\chi^2$ +function (a variant of the mean-squared error (MSE)) + +!bt +\[ \chi^2 = \frac{1}{n} +\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2}, +\] +!et + +where $\sigma_i^2$ is the variance (to be defined later) of the entry +$y_i$. We may not know the explicit value of $\sigma_i^2$, it serves +however the aim of scaling the equations and make the cost function +dimensionless. + +Minimizing the cost function is a central aspect of +our discussions to come. Finding its minima as function of the model +parameters ($\alpha$ and $\beta$ in our case) will be a recurring +theme in these series of lectures. Essentially all machine learning +algorithms we will discuss center around the minimization of the +chosen cost function. This depends in turn on our specific +model for describing the data, a typical situation in supervised +learning. Automatizing the search for the minima of the cost function is a +central ingredient in all algorithms. Typical methods which are +employed are various variants of _gradient_ methods. These will be +discussed in more detail later. Again, you'll be surprised to hear that +many practitioners minimize the above function ''by the eye', popularly dubbed as +'chi by the eye'. That is, change a parameter and see (visually and numerically) that +the $\chi^2$ function becomes smaller. + +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 +the relative error (why would we prefer the MSE instead of the relative error?) as + +!bt +\[ +\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}. +\] +!et + +The squared cost function results in an arithmetic mean-unbiased +estimator, and the absolute-value cost function results in a +median-unbiased estimator (in the one-dimensional case, and a +geometric median-unbiased estimator for the multi-dimensional +case). The squared cost function has the disadvantage that it has the tendency +to be dominated by outliers. + +We can modify easily the above Python code and plot the relative error instead +!bc pycod +import numpy as np +import matplotlib.pyplot as plt +from sklearn.linear_model import LinearRegression + +x = np.random.rand(100,1) +y = 5*x+0.01*np.random.randn(100,1) +linreg = LinearRegression() +linreg.fit(x,y) +ypredict = linreg.predict(x) + +plt.plot(x, np.abs(ypredict-y)/abs(y), "ro") +plt.axis([0,1.0,0.0, 0.5]) +plt.xlabel(r'$x$') +plt.ylabel(r'$\epsilon_{\mathrm{relative}}$') +plt.title(r'Relative error') +plt.show() +!ec + +Depending on the parameter in front of the normal distribution, we may +have a small or larger relative error. Try to play around with +different training data sets and study (graphically) the value of the +relative error. + +As mentioned above, _Scikit-Learn_ has an impressive functionality. +We can for example extract the values of $\alpha$ and $\beta$ and +their error estimates, or the variance and standard deviation and many +other properties from the statistical data analysis. + +Here we show an +example of the functionality of _Scikit-Learn_. +!bc pycod +import numpy as np +import matplotlib.pyplot as plt +from sklearn.linear_model import LinearRegression +from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error + +x = np.random.rand(100,1) +y = 2.0+ 5*x+0.5*np.random.randn(100,1) +linreg = LinearRegression() +linreg.fit(x,y) +ypredict = linreg.predict(x) +print('The intercept alpha: \n', linreg.intercept_) +print('Coefficient beta : \n', linreg.coef_) +# The mean squared error +print("Mean squared error: %.2f" % mean_squared_error(y, ypredict)) +# Explained variance score: 1 is perfect prediction +print('Variance score: %.2f' % r2_score(y, ypredict)) +# Mean squared log error +print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) ) +# Mean absolute error +print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict)) +plt.plot(x, ypredict, "r-") +plt.plot(x, y ,'ro') +plt.axis([0.0,1.0,1.5, 7.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Linear Regression fit ') +plt.show() + +!ec +The function _coef_ gives us the parameter $\beta$ of our fit while _intercept_ yields +$\alpha$. 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 +!bt +\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +\] +!et + +The smaller the value, the better the fit. Ideally we would like to +have an MSE equal zero. The attentive reader has probably recognized +this function as being similar to the $\chi^2$ function defined above. + +The _r2score_ function computes $R^2$, the coefficient of +determination. It provides a measure of how well future samples are +likely to be predicted by the model. Best possible score is 1.0 and it +can be negative (because the model can be arbitrarily worse). A +constant model that always predicts the expected value of $\hat{y}$, +disregarding the input features, would get a $R^2$ score of $0.0$. + +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 +!bt +\[ +R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, +\] +!et +where we have defined the mean value of $\hat{y}$ as +!bt +\[ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +\] +!et +Another quantity taht we will meet again in our discussions of regression analysis is + 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. +The MAE is defined as follows +!bt +\[ +\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|. +\] +!et +We present the +squared logarithmic (quadratic) error +!bt +\[ +\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2, +\] +!et + +where $\log_e (x)$ stands for the natural logarithm of $x$. This error +estimate is best to use when targets having exponential growth, such +as population counts, average sales of a commodity over a span of +years etc. + + + +We conclude this part with another example. Instead of +a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn. + +!bc pycod +import matplotlib.pyplot as plt +import numpy as np +import random +from sklearn.linear_model import Ridge +from sklearn.preprocessing import PolynomialFeatures +from sklearn.pipeline import make_pipeline +from sklearn.linear_model import LinearRegression + +x=np.linspace(0.02,0.98,200) +noise = np.asarray(random.sample((range(200)),200)) +y=x**3*noise +yn=x**3*100 +poly3 = PolynomialFeatures(degree=3) +X = poly3.fit_transform(x[:,np.newaxis]) +clf3 = LinearRegression() +clf3.fit(X,y) + +Xplot=poly3.fit_transform(x[:,np.newaxis]) +poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit') +plt.plot(x,yn, color='red', label="True Cubic") +plt.scatter(x, y, label='Data', color='orange', s=15) +plt.legend() +plt.show() + +def error(a): + for i in y: + err=(y-yn)/yn + return abs(np.sum(err))/len(err) + +print (error(y)) +!ec + + + +===== The Boston housing data example ===== + +The Boston housing +data set was originally a part of UCI Machine Learning Repository +and has been removed now. The data set is now included in _Scikit-Learn_'s +library. There are 506 samples and 13 feature (predictor) variables +in this data set. The objective is to predict the value of prices of +the house using the features (predictors) listed here. + +The features/predictors are + o CRIM: Per capita crime rate by town + o ZN: Proportion of residential land zoned for lots over 25000 square feet + o INDUS: Proportion of non-retail business acres per town + o CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise) + o NOX: Nitric oxide concentration (parts per 10 million) + o RM: Average number of rooms per dwelling + o AGE: Proportion of owner-occupied units built prior to 1940 + o DIS: Weighted distances to five Boston employment centers + o RAD: Index of accessibility to radial highways + o TAX: Full-value property tax rate per USD10000 + o B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town + o LSTAT: Percentage of lower status of the population + o MEDV: Median value of owner-occupied homes in USD 1000s + + +We start by importing the libraries +!bc pycod +import numpy as np +import matplotlib.pyplot as plt + +import pandas as pd +import seaborn as sns +!ec +and load the Boston Housing DataSet from _Scikit-Learn_ + + +!bc pycod +from sklearn.datasets import load_boston + +boston_dataset = load_boston() + +# boston_dataset is a dictionary +# let's check what it contains +boston_dataset.keys() +!ec +Then we invoke Pandas +!bc pycod +boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names) +boston.head() +boston['MEDV'] = boston_dataset.target +!ec +and preprocess the data +!bc pycod +# check for missing values in all the columns +boston.isnull().sum() +!ec +We can then visualize the data +!bc pycod +# set the size of the figure +sns.set(rc={'figure.figsize':(11.7,8.27)}) + +# plot a histogram showing the distribution of the target values +sns.distplot(boston['MEDV'], bins=30) +plt.show() +!ec + +It is now useful to look at the correlation matrix +!bc pycod +# compute the pair wise correlation for all columns +correlation_matrix = boston.corr().round(2) +# 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) +!ec +From the above coorelation plot we can see that _MEDV_ is strongly correlated to _LSTAT_ and _RM_. We see also that _RAD_ and _TAX_ are stronly correlated, but we don't include this in our features together to avoid multi-colinearity + +!bc pycod +plt.figure(figsize=(20, 5)) + +features = ['LSTAT', 'RM'] +target = boston['MEDV'] + +for i, col in enumerate(features): + plt.subplot(1, len(features) , i+1) + x = boston[col] + y = target + plt.scatter(x, y, marker='o') + plt.title(col) + plt.xlabel(col) + plt.ylabel('MEDV') +!ec +Now we start training our model +!bc pycod +X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM']) +Y = boston['MEDV'] +!ec +We split the data into training and test sets + +!bc pycod +from sklearn.model_selection import train_test_split + +# splits the training and test data set in 80% : 20% +# assign random_state to any value.This ensures consistency. +X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5) +print(X_train.shape) +print(X_test.shape) +print(Y_train.shape) +print(Y_test.shape) +!ec +Then we use the linear regression functionality from _Scikit-Learn_ +!bc pycod +from sklearn.linear_model import LinearRegression +from sklearn.metrics import mean_squared_error, r2_score + +lin_model = LinearRegression() +lin_model.fit(X_train, Y_train) + +# model evaluation for training set + +y_train_predict = lin_model.predict(X_train) +rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict))) +r2 = r2_score(Y_train, y_train_predict) + +print("The model performance for training set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +print("\n") + +# model evaluation for testing set + +y_test_predict = lin_model.predict(X_test) +# root mean square error of the model +rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict))) + +# r-squared score of the model +r2 = r2_score(Y_test, y_test_predict) + +print("The model performance for testing set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +!ec + +!bc pycod +# plotting the y_test vs y_pred +# ideally should have been a straight line +plt.scatter(Y_test, y_test_predict) +plt.show() +!ec + + + + +Many Machine Learning problems involve thousands or even millions of +features for each training instance. Not only does this make training +extremely slow, it can also make it much harder to find a good +solution, as we will see. This problem is often referred to as the +curse of dimensionality. Fortunately, in real-world problems, it is +often possible to reduce the number of features considerably, turning +an intractable problem into a tractable one. + +Later we will discuss some of the most popular dimensionality reduction +techniques: the principal component analysis (PCA), Kernel PCA, and +Locally Linear Embedding (LLE). + + +Principal component analysis and its various variants deal with the +problem of fitting a low-dimensional "affine +subspace":"https://en.wikipedia.org/wiki/Affine_space" to a set of of +data points in a high-dimensional space. With its family of methods it +is one of the most used tools in data modeling, compression and +visualization. + + +Before we proceed however, we will discuss how to preprocess our +data. Till now and in connection with our previous examples we have +not met so many cases where we are too sensitive to the scaling of our +data. Normally the data may need a rescaling and/or may be sensitive +to extreme values. Scaling the data renders our inputs much more +suitable for the algorithms we want to employ. + +_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 one (every column in the design/feature +matrix). This scaling has the drawback that it does not ensure that +we have a particular maximum or minimum 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 + + +The _Normalizer_ scales each data +point such that the feature vector has a euclidean length of one. In other words, it +projects a data point on the circle (or sphere in the case of higher dimensions) with a +radius of 1. This means every data point is scaled by a different number (by the +inverse of it’s length). +This normalization is often used when only the direction (or angle) of the data matters, +not the length of the feature vector. + +The _RobustScaler_ works similarly to the StandardScaler in that it +ensures statistical properties for each feature that guarantee that +they are on the same scale. However, the RobustScaler uses the median +and quartiles, instead of mean and variance. This makes the +RobustScaler ignore data points that are very different from the rest +(like measurement errors). These odd data points are also called +outliers, and might often lead to trouble for other scaling +techniques. + diff --git a/doc/src/How2ReadData/add.dlog b/doc/src/How2ReadData/add.dlog new file mode 100644 index 000000000..8838deedd --- /dev/null +++ b/doc/src/How2ReadData/add.dlog @@ -0,0 +1,11 @@ +Translating doconce text in add.do.txt to ipynb +*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax) + +*** warning: latex envir \begin{cases} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments. +Failed to remove ans_at_end environment +Failed to remove sol_at_end environment +output in add.ipynb +Translating doconce text in add.do.txt to ipynb +Failed to remove ans_at_end environment +Failed to remove sol_at_end environment +output in add.ipynb diff --git a/doc/src/How2ReadData/add.do.txt~ b/doc/src/How2ReadData/add.do.txt~ new file mode 100644 index 000000000..c14a7b2ff --- /dev/null +++ b/doc/src/How2ReadData/add.do.txt~ @@ -0,0 +1,519 @@ + +===== Simple linear regression model using _scikit-learn_ ===== + +We start with perhaps our simplest possible example, using _Scikit-Learn_ to perform linear regression analysis on a data set produced by us. + +What follows is a simple Python code where we have defined a function +$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. +The numbers in the vector $\hat{x}$ are given +by random numbers generated with a uniform distribution with entries +$x_i \in [0,1]$ (more about probability distribution functions +later). These values are then used to define a function $y(x)$ +(tabulated again as a vector) with a linear dependence on $x$ plus a +random noise added via the normal distribution. + + +The Numpy functions are imported used the _import numpy as np_ +statement and the random number generator for the uniform distribution +is called using the function _np.random.rand()_, where we specificy +that we want $100$ random variables. Using Numpy we define +automatically an array with the specified number of elements, $100$ in +our case. With the Numpy function _randn()_ we can compute random +numbers with the normal distribution (mean value $\mu$ equal to zero and +variance $\sigma^2$ set to one) and produce the values of $y$ assuming a linear +dependence as function of $x$ + +!bt +\[ +y = 2x+N(0,1), +\] +!et + +where $N(0,1)$ represents random numbers generated by the normal +distribution. From _Scikit-Learn_ we import then the +_LinearRegression_ functionality and make a prediction $\tilde{y} = +\alpha + \beta x$ using the function _fit(x,y)_. We call the set of +data $(\hat{x},\hat{y})$ for our training data. The Python package +_scikit-learn_ has also a functionality which extracts the above +fitting parameters $\alpha$ and $\beta$ (see below). Later we will +distinguish between training data and test data. + +For plotting we use the Python package +"matplotlib":"https://matplotlib.org/" which produces publication +quality figures. Feel free to explore the extensive +"gallery":"https://matplotlib.org/gallery/index.html" of examples. In +this example we plot our original values of $x$ and $y$ as well as the +prediction _ypredict_ ($\tilde{y}$), which attempts at fitting our +data with a straight line. + +The Python code follows here. +!bc pycod +# Importing various packages +import numpy as np +import matplotlib.pyplot as plt +from sklearn.linear_model import LinearRegression + +x = np.random.rand(100,1) +y = 2*x+np.random.randn(100,1) +linreg = LinearRegression() +linreg.fit(x,y) +xnew = np.array([[0],[1]]) +ypredict = linreg.predict(xnew) + +plt.plot(xnew, ypredict, "r-") +plt.plot(x, y ,'ro') +plt.axis([0,1.0,0, 5.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Simple Linear Regression') +plt.show() +!ec + +This example serves several aims. It allows us to demonstrate several +aspects of data analysis and later machine learning algorithms. The +immediate visualization shows that our linear fit is not +impressive. It goes through the data points, but there are many +outliers which are not reproduced by our linear regression. We could +now play around with this small program and change for example the +factor in front of $x$ and the normal distribution. Try to change the +function $y$ to + +!bt +\[ +y = 10x+0.01 \times N(0,1), +\] +!et + +where $x$ is defined as before. Does the fit look better? Indeed, by +reducing the role of the noise given by the normal distribution we see immediately that +our linear prediction seemingly reproduces better the training +set. However, this testing 'by the eye' is obviouly not satisfactory in the +long run. Here we have only defined the training data and our model, and +have not discussed a more rigorous approach to the _cost_ function. + +We need more rigorous criteria in defining whether we have succeeded or +not in modeling our training data. You will be surprised to see that +many scientists seldomly venture beyond this 'by the eye' approach. A +standard approach for the *cost* function is the so-called $\chi^2$ +function (a variant of the mean-squared error (MSE)) + +!bt +\[ \chi^2 = \frac{1}{n} +\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2}, +\] +!et + +where $\sigma_i^2$ is the variance (to be defined later) of the entry +$y_i$. We may not know the explicit value of $\sigma_i^2$, it serves +however the aim of scaling the equations and make the cost function +dimensionless. + +Minimizing the cost function is a central aspect of +our discussions to come. Finding its minima as function of the model +parameters ($\alpha$ and $\beta$ in our case) will be a recurring +theme in these series of lectures. Essentially all machine learning +algorithms we will discuss center around the minimization of the +chosen cost function. This depends in turn on our specific +model for describing the data, a typical situation in supervised +learning. Automatizing the search for the minima of the cost function is a +central ingredient in all algorithms. Typical methods which are +employed are various variants of _gradient_ methods. These will be +discussed in more detail later. Again, you'll be surprised to hear that +many practitioners minimize the above function ''by the eye', popularly dubbed as +'chi by the eye'. That is, change a parameter and see (visually and numerically) that +the $\chi^2$ function becomes smaller. + +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 +the relative error (why would we prefer the MSE instead of the relative error?) as + +!bt +\[ +\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}. +\] +!et + +The squared cost function results in an arithmetic mean-unbiased +estimator, and the absolute-value cost function results in a +median-unbiased estimator (in the one-dimensional case, and a +geometric median-unbiased estimator for the multi-dimensional +case). The squared cost function has the disadvantage that it has the tendency +to be dominated by outliers. + +We can modify easily the above Python code and plot the relative error instead +!bc pycod +import numpy as np +import matplotlib.pyplot as plt +from sklearn.linear_model import LinearRegression + +x = np.random.rand(100,1) +y = 5*x+0.01*np.random.randn(100,1) +linreg = LinearRegression() +linreg.fit(x,y) +ypredict = linreg.predict(x) + +plt.plot(x, np.abs(ypredict-y)/abs(y), "ro") +plt.axis([0,1.0,0.0, 0.5]) +plt.xlabel(r'$x$') +plt.ylabel(r'$\epsilon_{\mathrm{relative}}$') +plt.title(r'Relative error') +plt.show() +!ec + +Depending on the parameter in front of the normal distribution, we may +have a small or larger relative error. Try to play around with +different training data sets and study (graphically) the value of the +relative error. + +As mentioned above, _Scikit-Learn_ has an impressive functionality. +We can for example extract the values of $\alpha$ and $\beta$ and +their error estimates, or the variance and standard deviation and many +other properties from the statistical data analysis. + +Here we show an +example of the functionality of _Scikit-Learn_. +!bc pycod +import numpy as np +import matplotlib.pyplot as plt +from sklearn.linear_model import LinearRegression +from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error + +x = np.random.rand(100,1) +y = 2.0+ 5*x+0.5*np.random.randn(100,1) +linreg = LinearRegression() +linreg.fit(x,y) +ypredict = linreg.predict(x) +print('The intercept alpha: \n', linreg.intercept_) +print('Coefficient beta : \n', linreg.coef_) +# The mean squared error +print("Mean squared error: %.2f" % mean_squared_error(y, ypredict)) +# Explained variance score: 1 is perfect prediction +print('Variance score: %.2f' % r2_score(y, ypredict)) +# Mean squared log error +print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) ) +# Mean absolute error +print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict)) +plt.plot(x, ypredict, "r-") +plt.plot(x, y ,'ro') +plt.axis([0.0,1.0,1.5, 7.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Linear Regression fit ') +plt.show() + +!ec +The function _coef_ gives us the parameter $\beta$ of our fit while _intercept_ yields +$\alpha$. 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 +!bt +\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} +\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, +\] +!et + +The smaller the value, the better the fit. Ideally we would like to +have an MSE equal zero. The attentive reader has probably recognized +this function as being similar to the $\chi^2$ function defined above. + +The _r2score_ function computes $R^2$, the coefficient of +determination. It provides a measure of how well future samples are +likely to be predicted by the model. Best possible score is 1.0 and it +can be negative (because the model can be arbitrarily worse). A +constant model that always predicts the expected value of $\hat{y}$, +disregarding the input features, would get a $R^2$ score of $0.0$. + +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 +!bt +\[ +R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, +\] +!et +where we have defined the mean value of $\hat{y}$ as +!bt +\[ +\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. +\] +!et +Another quantity taht we will meet again in our discussions of regression analysis is + 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. +The MAE is defined as follows +!bt +\[ +\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|. +\] +!et +We present the +squared logarithmic (quadratic) error +!bt +\[ +\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2, +\] +!et + +where $\log_e (x)$ stands for the natural logarithm of $x$. This error +estimate is best to use when targets having exponential growth, such +as population counts, average sales of a commodity over a span of +years etc. + + +Finally, another cost function is the Huber cost function used in robust regression. + +The rationale behind this possible cost function is its reduced +sensitivity to outliers in the data set. In our discussions on +dimensionality reduction and normalization of data we will meet other +ways of dealing with outliers. + +The Huber cost function is defined as +!bt +\[ +H_{\delta}(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}}. +\] +!et +Here $a=\bm{y} - \bm{\tilde{y}}$. +We will discuss in more +detail these and other functions in the various lectures. We conclude this part with another example. Instead of +a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn. + +!bc pycod +import matplotlib.pyplot as plt +import numpy as np +import random +from sklearn.linear_model import Ridge +from sklearn.preprocessing import PolynomialFeatures +from sklearn.pipeline import make_pipeline +from sklearn.linear_model import LinearRegression + +x=np.linspace(0.02,0.98,200) +noise = np.asarray(random.sample((range(200)),200)) +y=x**3*noise +yn=x**3*100 +poly3 = PolynomialFeatures(degree=3) +X = poly3.fit_transform(x[:,np.newaxis]) +clf3 = LinearRegression() +clf3.fit(X,y) + +Xplot=poly3.fit_transform(x[:,np.newaxis]) +poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit') +plt.plot(x,yn, color='red', label="True Cubic") +plt.scatter(x, y, label='Data', color='orange', s=15) +plt.legend() +plt.show() + +def error(a): + for i in y: + err=(y-yn)/yn + return abs(np.sum(err))/len(err) + +print (error(y)) +!ec + + + +===== The Boston housing data example ===== + +The Boston housing +data set was originally a part of UCI Machine Learning Repository +and has been removed now. The data set is now included in _Scikit-Learn_'s +library. There are 506 samples and 13 feature (predictor) variables +in this data set. The objective is to predict the value of prices of +the house using the features (predictors) listed here. + +The features/predictors are + o CRIM: Per capita crime rate by town + o ZN: Proportion of residential land zoned for lots over 25000 square feet + o INDUS: Proportion of non-retail business acres per town + o CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise) + o NOX: Nitric oxide concentration (parts per 10 million) + o RM: Average number of rooms per dwelling + o AGE: Proportion of owner-occupied units built prior to 1940 + o DIS: Weighted distances to five Boston employment centers + o RAD: Index of accessibility to radial highways + o TAX: Full-value property tax rate per USD10000 + o B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town + o LSTAT: Percentage of lower status of the population + o MEDV: Median value of owner-occupied homes in USD 1000s + + +We start by importing the libraries +!bc pycod +import numpy as np +import matplotlib.pyplot as plt + +import pandas as pd +import seaborn as sns +!ec +and load the Boston Housing DataSet from _Scikit-Learn_ + + +!bc pycod +from sklearn.datasets import load_boston + +boston_dataset = load_boston() + +# boston_dataset is a dictionary +# let's check what it contains +boston_dataset.keys() +!ec +Then we invoke Pandas +!bc pycod +boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names) +boston.head() +boston['MEDV'] = boston_dataset.target +!ec +and preprocess the data +!bc pycod +# check for missing values in all the columns +boston.isnull().sum() +!ec +We can then visualize the data +!bc pycod +# set the size of the figure +sns.set(rc={'figure.figsize':(11.7,8.27)}) + +# plot a histogram showing the distribution of the target values +sns.distplot(boston['MEDV'], bins=30) +plt.show() +!ec + +It is now useful to look at the correlation matrix +!bc pycod +# compute the pair wise correlation for all columns +correlation_matrix = boston.corr().round(2) +# 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) +!ec +From the above coorelation plot we can see that _MEDV_ is strongly correlated to _LSTAT_ and _RM_. We see also that _RAD_ and _TAX_ are stronly correlated, but we don't include this in our features together to avoid multi-colinearity + +!bc pycod +plt.figure(figsize=(20, 5)) + +features = ['LSTAT', 'RM'] +target = boston['MEDV'] + +for i, col in enumerate(features): + plt.subplot(1, len(features) , i+1) + x = boston[col] + y = target + plt.scatter(x, y, marker='o') + plt.title(col) + plt.xlabel(col) + plt.ylabel('MEDV') +!ec +Now we start training our model +!bc pycod +X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM']) +Y = boston['MEDV'] +!ec +We split the data into training and test sets + +!bc pycod +from sklearn.model_selection import train_test_split + +# splits the training and test data set in 80% : 20% +# assign random_state to any value.This ensures consistency. +X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5) +print(X_train.shape) +print(X_test.shape) +print(Y_train.shape) +print(Y_test.shape) +!ec +Then we use the linear regression functionality from _Scikit-Learn_ +!bc pycod +from sklearn.linear_model import LinearRegression +from sklearn.metrics import mean_squared_error, r2_score + +lin_model = LinearRegression() +lin_model.fit(X_train, Y_train) + +# model evaluation for training set + +y_train_predict = lin_model.predict(X_train) +rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict))) +r2 = r2_score(Y_train, y_train_predict) + +print("The model performance for training set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +print("\n") + +# model evaluation for testing set + +y_test_predict = lin_model.predict(X_test) +# root mean square error of the model +rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict))) + +# r-squared score of the model +r2 = r2_score(Y_test, y_test_predict) + +print("The model performance for testing set") +print("--------------------------------------") +print('RMSE is {}'.format(rmse)) +print('R2 score is {}'.format(r2)) +!ec + +!bc pycod +# plotting the y_test vs y_pred +# ideally should have been a straight line +plt.scatter(Y_test, y_test_predict) +plt.show() +!ec + + + + +Many Machine Learning problems involve thousands or even millions of +features for each training instance. Not only does this make training +extremely slow, it can also make it much harder to find a good +solution, as we will see. This problem is often referred to as the +curse of dimensionality. Fortunately, in real-world problems, it is +often possible to reduce the number of features considerably, turning +an intractable problem into a tractable one. + +Later we will discuss some of the most popular dimensionality reduction +techniques: the principal component analysis (PCA), Kernel PCA, and +Locally Linear Embedding (LLE). + + +Principal component analysis and its various variants deal with the +problem of fitting a low-dimensional "affine +subspace":"https://en.wikipedia.org/wiki/Affine_space" to a set of of +data points in a high-dimensional space. With its family of methods it +is one of the most used tools in data modeling, compression and +visualization. + + +Before we proceed however, we will discuss how to preprocess our +data. Till now and in connection with our previous examples we have +not met so many cases where we are too sensitive to the scaling of our +data. Normally the data may need a rescaling and/or may be sensitive +to extreme values. Scaling the data renders our inputs much more +suitable for the algorithms we want to employ. + +_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 one (every column in the design/feature +matrix). This scaling has the drawback that it does not ensure that +we have a particular maximum or minimum 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 + + +The _Normalizer_ scales each data +point such that the feature vector has a euclidean length of one. In other words, it +projects a data point on the circle (or sphere in the case of higher dimensions) with a +radius of 1. This means every data point is scaled by a different number (by the +inverse of it’s length). +This normalization is often used when only the direction (or angle) of the data matters, +not the length of the feature vector. + +The _RobustScaler_ works similarly to the StandardScaler in that it +ensures statistical properties for each feature that guarantee that +they are on the same scale. However, the RobustScaler uses the median +and quartiles, instead of mean and variance. This makes the +RobustScaler ignore data points that are very different from the rest +(like measurement errors). These odd data points are also called +outliers, and might often lead to trouble for other scaling +techniques. + diff --git a/doc/src/Splines/.ipynb_checkpoints/GradientMethods-checkpoint.ipynb b/doc/src/Optimization/.ipynb_checkpoints/GradientMethods-checkpoint.ipynb similarity index 100% rename from doc/src/Splines/.ipynb_checkpoints/GradientMethods-checkpoint.ipynb rename to doc/src/Optimization/.ipynb_checkpoints/GradientMethods-checkpoint.ipynb diff --git a/doc/src/Splines/GradientMethods.ipynb b/doc/src/Optimization/GradientMethods.ipynb similarity index 100% rename from doc/src/Splines/GradientMethods.ipynb rename to doc/src/Optimization/GradientMethods.ipynb diff --git a/doc/src/Splines/Splines.do.txt b/doc/src/Optimization/Splines.do.txt similarity index 100% rename from doc/src/Splines/Splines.do.txt rename to doc/src/Optimization/Splines.do.txt diff --git a/doc/src/Splines/autodiff/.ipynb_checkpoints/example_plot-checkpoint.ipynb b/doc/src/Optimization/autodiff/.ipynb_checkpoints/example_plot-checkpoint.ipynb similarity index 100% rename from doc/src/Splines/autodiff/.ipynb_checkpoints/example_plot-checkpoint.ipynb rename to doc/src/Optimization/autodiff/.ipynb_checkpoints/example_plot-checkpoint.ipynb diff --git a/doc/src/Splines/autodiff/.ipynb_checkpoints/examples_allowed_functions-Copy1-checkpoint.ipynb b/doc/src/Optimization/autodiff/.ipynb_checkpoints/examples_allowed_functions-Copy1-checkpoint.ipynb similarity index 100% rename from doc/src/Splines/autodiff/.ipynb_checkpoints/examples_allowed_functions-Copy1-checkpoint.ipynb rename to doc/src/Optimization/autodiff/.ipynb_checkpoints/examples_allowed_functions-Copy1-checkpoint.ipynb diff --git a/doc/src/Splines/autodiff/.ipynb_checkpoints/examples_allowed_functions-checkpoint.ipynb b/doc/src/Optimization/autodiff/.ipynb_checkpoints/examples_allowed_functions-checkpoint.ipynb similarity index 100% rename from doc/src/Splines/autodiff/.ipynb_checkpoints/examples_allowed_functions-checkpoint.ipynb rename to doc/src/Optimization/autodiff/.ipynb_checkpoints/examples_allowed_functions-checkpoint.ipynb diff --git a/doc/src/Splines/autodiff/autograd.pdf b/doc/src/Optimization/autodiff/autograd.pdf similarity index 100% rename from doc/src/Splines/autodiff/autograd.pdf rename to doc/src/Optimization/autodiff/autograd.pdf diff --git a/doc/src/Splines/autodiff/example_plot.ipynb b/doc/src/Optimization/autodiff/example_plot.ipynb similarity index 100% rename from doc/src/Splines/autodiff/example_plot.ipynb rename to doc/src/Optimization/autodiff/example_plot.ipynb diff --git a/doc/src/Splines/autodiff/examples_allowed_functions-Copy1.ipynb b/doc/src/Optimization/autodiff/examples_allowed_functions-Copy1.ipynb similarity index 100% rename from doc/src/Splines/autodiff/examples_allowed_functions-Copy1.ipynb rename to doc/src/Optimization/autodiff/examples_allowed_functions-Copy1.ipynb diff --git 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a/doc/src/Splines/beamerthemered_shadow.sty b/doc/src/Optimization/beamerthemered_shadow.sty similarity index 100% rename from doc/src/Splines/beamerthemered_shadow.sty rename to doc/src/Optimization/beamerthemered_shadow.sty diff --git a/doc/src/Splines/clean.sh b/doc/src/Optimization/clean.sh similarity index 100% rename from doc/src/Splines/clean.sh rename to doc/src/Optimization/clean.sh diff --git a/doc/src/Splines/logsgd.py b/doc/src/Optimization/logsgd.py similarity index 100% rename from doc/src/Splines/logsgd.py rename to doc/src/Optimization/logsgd.py diff --git a/doc/src/Splines/make.sh b/doc/src/Optimization/make.sh similarity index 100% rename from doc/src/Splines/make.sh rename to doc/src/Optimization/make.sh diff --git a/doc/src/Splines/ridgesgd.py b/doc/src/Optimization/ridgesgd.py similarity index 100% rename from doc/src/Splines/ridgesgd.py rename to doc/src/Optimization/ridgesgd.py diff --git a/doc/src/week37/olscode.py~ b/doc/src/week37/olscode.py~ new file mode 100644 index 000000000..a6094b632 --- /dev/null +++ b/doc/src/week37/olscode.py~ @@ -0,0 +1,47 @@ +import numpy as np +import matplotlib.pyplot as plt +from sklearn.model_selection import KFold +from sklearn.linear_model import Ridge, LinearRegression +from sklearn.model_selection import cross_val_score +from sklearn.preprocessing import PolynomialFeatures + +# A seed just to ensure that the random numbers are the same for every run. +# Useful for eventual debugging. +np.random.seed(3155) + +# Generate the data. +nsamples = 1000 +x = np.random.randn(nsamples) +y = 3*x**2 + np.random.randn(nsamples) + +## Cross-validation on Ridge regression using KFold only + +# Decide degree on polynomial to fit +poly = PolynomialFeatures(degree = 6) + + +# Initialize a KFold instance +k = 10 +kfold = KFold(n_splits = k) + +# Perform the cross-validation to estimate MSE using OLS +scores_KFold = np.zeros((k)) +model = LinearRegression() +j = 0 +for train_inds, test_inds in kfold.split(x): + xtrain = x[train_inds] + ytrain = y[train_inds] + xtest = x[test_inds] + ytest = y[test_inds] + Xtrain = poly.fit_transform(xtrain[:, np.newaxis]) + model.fit(Xtrain, ytrain[:, np.newaxis]) + Xtest = poly.fit_transform(xtest[:, np.newaxis]) + ypred = model.predict(Xtest) + scores_KFold[j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred) + print(f"Score for each fold:{scores_KFold[j]}") + j += 1 + + +estimated_mse_KFold = np.mean(scores_KFold) +print(f"Average OLS score:{estimated_mse_KFold}") + diff --git a/doc/src/week39/codes/test10.py~ b/doc/src/week39/codes/test10.py~ new file mode 100644 index 000000000..fbeb00c02 --- /dev/null +++ b/doc/src/week39/codes/test10.py~ @@ -0,0 +1,44 @@ + +from random import random, seed +import numpy as np +import matplotlib.pyplot as plt +from mpl_toolkits.mplot3d import Axes3D +from matplotlib import cm +from matplotlib.ticker import LinearLocator, FormatStrFormatter +import sys + +# the number of datapoints +n = 100 +x = 2*np.random.rand(n,1) +y = 4+3*x+np.random.randn(n,1) + +X = np.c_[np.ones((n,1)), x] +XT_X = X.T @ X + +#Ridge parameter lambda +lmbda = 0.001 +Id = lmbda* np.eye(XT_X.shape[0]) + +beta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y +print(beta_linreg) +# Start plain gradient descent +beta = np.random.randn(2,1) + +eta = 0.1 +Niterations = 100 + +for iter in range(Niterations): + gradients = 2.0/n*X.T @ (X @ (beta)-y)+2*lmbda*beta + beta -= eta*gradients + +print(beta) +ypredict = X @ beta +ypredict2 = X @ beta_linreg +plt.plot(x, ypredict, "r-") +plt.plot(x, ypredict2, "b-") +plt.plot(x, y ,'ro') +plt.axis([0,2.0,0, 15.0]) +plt.xlabel(r'$x$') +plt.ylabel(r'$y$') +plt.title(r'Gradient descent example for Ridge') +plt.show() diff --git a/doc/src/week39/codes/test12.py~ b/doc/src/week39/codes/test12.py~ new file mode 100644 index 000000000..7abc500c5 --- /dev/null +++ b/doc/src/week39/codes/test12.py~ @@ -0,0 +1,1170 @@ +import os +import sys +import pytest +import numba +from matplotlib import cm +from matplotlib.ticker import LinearLocator, FormatStrFormatter +import matplotlib.pyplot as plt +from mpl_toolkits.mplot3d import Axes3D +import numpy as np +import functools +import time +from numba import jit, njit +from PIL import Image +import pandas as pd +import seaborn as sns +sns.set() +import math + +from sklearn.model_selection import train_test_split, cross_val_score, KFold +from sklearn.preprocessing import StandardScaler +from sklearn.linear_model import SGDRegressor, LinearRegression, LogisticRegression, Lasso, Ridge +from sklearn import datasets +from sklearn.metrics import confusion_matrix, mean_squared_error, r2_score +from sklearn.utils import resample + + +# Bootstrap + +def Bootstrap(x1,x2, y, N_boot=500, method = 'ols', degrees = 5, random_state = 42): + """ + Computes bias^2, variance and the mean squared error using bootstrap resampling method + for the provided data and the method. + + Arguments: + x1: 1D numpy array, covariate + x2: 1D numpy array, covariate + N_boot: integer type, the number of bootstrap samples + method: string type, accepts 'ols', 'ridge' or 'lasso' as arguments + degree: integer type, polynomial degree for generating the design matrix + random_state: integer, ensures the same split when using the train_test_split functionality + + Returns: Bias_vec, Var_vec, MSE_vec, betaVariance_vec + numpy arrays. Bias, Variance, MSE and the variance of beta for the predicted model + """ + ##split x1, x2 and y arrays as a train and test data and generate design matrix + x1_train, x1_test,x2_train, x2_test, y_train, y_test = train_test_split(x1,x2, y, test_size=0.2, random_state = random_state) + y_pred_test = np.zeros((y_test.shape[0], N_boot)) + X_test = designMatrix(x1_test, x2_test, degrees) + + betaMatrix = np.zeros((X_test.shape[1], N_boot)) + + ##resample and fit the corresponding method on the train data + for i in range(N_boot): + x1_,x2_, y_ = resample(x1_train, x2_train, y_train) + X_train = designMatrix(x1_, x2_, degrees) + scaler = StandardScaler() + scaler.fit(X_train) + X_train = scaler.transform(X_train) + X_train[:, 0] = 1 + X_test = designMatrix(x1_test, x2_test, degrees) + X_test = scaler.transform(X_test) + X_test[:, 0] = 1 + + if method == 'ols': + manual_regression = linregOwn(method = 'ols') + beta = manual_regression.fit(X_train, y_) + if method == 'ridge': + manual_regression = linregOwn(method = 'ridge') + beta = manual_regression.fit(X_train, y_, lambda_ = 0.05) + if method == 'lasso': + manual_regression = linregOwn(method = 'lasso') + beta = manual_regression.fit(X_train, y_, lambda_ = 0.05) + + ##predict on the same test data + y_pred_test[:, i] = np.dot(X_test, beta) + betaMatrix[:, i] = beta + y_test = y_test.reshape(len(y_test),1) + + Bias_vec = [] + Var_vec = [] + MSE_vec = [] + betaVariance_vec = [] + R2_score = [] + y_test = y_test.reshape(len(y_test),1) + MSE = np.mean( np.mean((y_test - y_pred_test)**2, axis=1, keepdims=True) ) + bias = np.mean( (y_test - np.mean(y_pred_test, axis=1, keepdims=True))**2 ) + variance = np.mean( np.var(y_pred_test, axis=1, keepdims=True) ) + betaVariance = np.var(betaMatrix, axis=1) + print("-------------------------------------------------------------") + print("Degree: %d" % degrees) + print('MSE:', np.round(MSE, 3)) + print('Bias^2:', np.round(bias, 3)) + print('Var:', np.round(variance,3)) + print('{} >= {} + {} = {}'.format(MSE, bias, variance, bias+variance)) + print("-------------------------------------------------------------") + + Bias_vec.append(bias) + Var_vec.append(variance) + MSE_vec.append(MSE) + betaVariance_vec.append(betaVariance) + return Bias_vec, Var_vec, MSE_vec, betaVariance_vec + + + +class CrossValidation: + """ + A class of cross-validation technique. Performs cross-validation with shuffling. + """ + def __init__(self, LinearRegression, DesignMatrix): + """ + Initialization + + Arguments: + LinearRegression: Instance from the class created by either linregOwn or linregSKl + DesignMatrix: Function that generates design matrix + """ + self.LinearRegression = LinearRegression + self.DesignMatrix = DesignMatrix + + def kFoldCV(self, x1, x2, y, k = 10, lambda_ = 0, degree = 5): + """ + Performs shuffling of the data, holds a split of the data as a test set at each split and evaluates the model + on the rest of the data. + Calculates the MSE , R2_score, variance, bias on the test data and MSE on the train data. + + Arguments: + x1: 1D numpy array + x2: 1D numpy array + y: 1D numpy array + k: integer, the number of splits + lambda_: float type, shrinkage parameter for ridge and lasso methods. + degree: integer type, the number of polynomials, complexity parameter + + """ + self.lambda_ = lambda_ + M = x1.shape[0]//k ## Split input data x in k folds of size M + + + ##save the statistic in the list + MSE_train = [] + MSE_k = [] + R2_k = [] + var_k = [] + bias_k = [] + + ##shuffle the data randomly + shf = np.random.permutation(x1.size) + x1_shuff = x1[shf] + x2_shuff = x2[shf] + y_shuff = y[shf] + + for i in range(k): + # x_k and y_k are the hold out data for fold k + x1_k = x1_shuff[i*M:(i+1)*M] + x2_k = x2_shuff[i*M:(i+1)*M] + y_k = y_shuff[i*M:(i+1)*M] + + ## Generate train data and then scale both train and test + index_true = np.array([True for i in range(x1.shape[0])]) + index_true[i*M:(i+1)*M] = False + X_train = self.DesignMatrix(x1_shuff[index_true], x2_shuff[index_true], degree) + y_train = y_shuff[index_true] + scaler = StandardScaler() + scaler.fit(X_train) + X_train = scaler.transform(X_train) + X_train[:, 0] = 1 + + ### Fit the regression on the train data + beta = self.LinearRegression.fit(X_train, y_train, lambda_) + y_predict_train = np.dot(X_train, beta) + MSE_train.append(np.sum( (y_train-y_predict_train)**2)/len(y_train)) + + ## Predict on the hold out data and calculate statistic of interest + X_k = self.DesignMatrix(x1_k, x2_k, degree) + X_k = scaler.transform(X_k) + X_k[:, 0] = 1 + y_predict = np.dot(X_k,beta) + MSE_k.append(np.sum((y_k-y_predict)**2, axis=0, keepdims=True)/len(y_predict)) + R2_k.append(1.0 - np.sum((y_k - y_predict)**2, axis=0, keepdims=True) / np.sum((y_k - np.mean(y_k))**2, axis=0, keepdims=True) ) + var_k.append(np.var(y_predict,axis=0, keepdims=True)) + bias_k.append((y_k - np.mean(y_predict, axis=0, keepdims=True))**2 ) + + means = [np.mean(MSE_k), np.mean(R2_k), np.mean(var_k), + np.mean(bias_k),np.mean(MSE_train)] + #print('MSE_test: {}' .format(np.round(np.mean(MSE_k),3))) + #print('R2: {}' .format(np.round(np.mean(R2_k),3))) + #print('Variance of the predicted outcome: {}' .format(np.round(np.mean(var_k),3))) + #print('Bias: {}' .format(np.round(np.mean(bias_k),3))) + #print('MSE_train {}' .format(np.round(np.mean(MSE_train),3))) + return means + + +# Franke Function + +def franke(x, y): + """ + Computes Franke function. + Franke's function has two Gaussian peaks of different heights, and a smaller dip. + It is used as a test function in interpolation problems. + + Franke's function is normally defined on the grid [0, 1] for each x, y. + + Arguments of the function: + x : numpy array + y : numpy array + + Output of the function: + f : Franke function values at specific coordinate points of x and y + """ + f = (0.75 * np.exp(-((9*x - 2)**2)/4 - ((9*y - 2)**2)/4 ) + + 0.75 * np.exp(-((9*x + 1)**2)/49 - (9*y + 1) /10) + + 0.5 * np.exp(-((9*x - 7)**2)/4 - ((9*y - 3)**2)/4 ) + - 0.2 * np.exp(-((9*x - 4)**2) - ((9*y - 7)**2) )) + return f + + + + +class linregOwn: + """ + A class of linear regressions. Perform ordinarly least squares (OLS) and Ridge regression manually. Lasso + is performed using scikit-learn functionality. + """ + def __init__(self, method = 'ols'): + """ + Constructor + + Determines the method used in the fitting + + Arguments: + method: string type. Accepts either 'ols', 'ridge' or 'lasso'. + + """ + self.method = method + self.yHat = None + self.X = None + self.y = None + self.beta = None + + self._MSE = None + self._R2 = None + self._betaVariance = None + self.lambda_ = None + + def fit(self, X_train, y_train, lambda_ = 0): + """ + Performs the fit of OLS, Ridge or Lasso, depending on the argument provided initially. + + Arguments: + X_train: Covariate matrix of the train data set, i.e. design matrix of + the shape m x p where m is the number of rows and p is the number of columns + (i.e. p is the complexity parameter). + y_train: Outcome variable, 1D numpy array + lambda_: float type. Shrinkage parameter for ridge and lasso methods. The higher value, higher shrinkage. + lambda_ is set to 0 for the OLS regression + + """ + self.X_train = X_train + self.y_train = y_train + self.lambda_ = lambda_ + if self.method == 'ols': + self._olsFit(X_train, y_train) + if self.method == 'ridge': + self._ridgeFit(X_train, y_train, lambda_) + if self.method == 'lasso': + self._lassoFitSKL(X_train, y_train, lambda_) + return self.beta + + def _olsFit(self, X_train, y_train): + """ + Performs the ordinary least squares (OLS) fit on the provided data using singular value decomposition(SVD). + + + Arguments: + + X_train: Covariate matrix of the train data set, i.e. design matrix of + the shape m x p where m is the number of rows and p is the number of columns + (i.e. p is the complexity parameter). + y_train: Outcome variable, 1D numpy array + + Returns: + beta : numpy.array + The beta parameters from the performed fit + """ + self.X_train = X_train + self.y_test = y_train + U, S, VT = np.linalg.svd(self.X_train, full_matrices=True) + S_inverse = np.zeros(shape=self.X_train.shape) + ##S is a vector, with shape of the number of columns + S_inverse[:S.shape[0], :S.shape[0]] = np.diag(1/S) + self.beta = np.dot(VT.T, np.dot(S_inverse.T, np.dot(U.T, self.y_train))) + #self.beta = np.linalg.inv(np.dot(X.T,X)).dot(X.T, y) + + def _ridgeFit(self, X_train, y_train, lambda_): + """ + Performs the ridge regression fit + + Arguments: + X_train: Covariate matrix of the train data set, design matrix of + the shape m x p (m_train_rows, p_columns). + y_train: Outcome variable, 1D numpy array, dimension m x 1 + lambda_: Integer type. The shrinkage parameter + + Returns: + beta : numpy.array + The beta parameters from the performed fit + """ + self.X_train = X_train + self.y_train = y_train + self.lambda_ = lambda_ + self.beta = np.dot(np.linalg.inv(np.dot(X_train.T,X_train) + self.lambda_ * np.eye(X_train.shape[1])), np.dot(X_train.T,y_train)) + + def _lassoFitSKL(self, X_train, y_train, lambda_): + """ + Performs lasso fit using scikit-learn functionality. + + Arguments: + X_train: Covariate matrix of the train data set, design matrix of + the shape m x p (m_train_datapoints, p_parameters). + y_train: Outcome variable, 1D numpy array, dimension m x 1 + lambda_: Integer type. The shrinkage parameter + + Returns: + self.beta : numpy.array + The beta parameters from the performed fit + """ + self.regression = Lasso(fit_intercept=True, max_iter=1000000, alpha=self.lambda_) + self.regression.fit(X_train,y_train) + self.beta = self.regression.coef_ + self.beta[0] = self.regression.intercept_ + + def predict(self, X_test): + """ + Performs prediction of the fitted model on the provided test data set. + + Arguments: + X_test: Design matrix, covariate matrix, dimension k x p (k_test_rows, p_columns) + + Returns: self.yHat + numpy 1D array, prediction values of dimension k x p + """ + self.X_test = X_test + self._predictOwntest(X_test) + return self.yHat + + def _predictOwntest(self, X_test): + """ + Performs manual prediction of the given model on the train data. + """ + self.X_test = X_test + self.yHat = np.dot(self.X_test, self.beta) + + def MSE(self, y_test): + """ + Calculates the mean squared error (MSE) manually after the fit and prediction have been implemented. + + Arguments: + y_test: Outcome variable, 1D numpy array, dimension k x 1 (k_test_rows, 1_column) + + Returns: self._MSE + The mean squared error of the predicted model + """ + self.y_test = y_test + if self.yHat is None : + self._predictOwntest(X_test) + N = self.yHat.size + self._MSE = (np.sum((self.y_test - self.yHat)**2))/N + return self._MSE + + def R2(self, y_test): + """ + Calculates R2 score manually after the fit and prediction have been implemented. + + Arguments: + y_test: Outcome variable, 1D numpy array, dimension k x 1 (k_test_rows, 1_column) + + Returns: self._R2 + The R2 score of the predicted model + """ + self.y_test = y_test + if self.yHat is None: + self._predictOwntest(X_test) + yMean = (1.0 / self.y_test.size) * np.sum(self.y_test) + self._R2 = 1.0 - np.sum((self.y_test - self.yHat)**2) / np.sum((self.y_test - yMean)**2) + return self._R2 + + def CI(self, y_test): + """ + Calculates confidence intervals manually after the fit and prediction have been implemented. + + Arguments: + y_test: Outcome variable, 1D numpy array, dimension k x 1 (k_test_rows, 1_column) + + Returns: var, Lower, Upper + Variance, Lower and Upper bounds of the confidence intervals for the parameter self.beta + """ + self.y_test = y_test + if self.yHat is None: + self._predictOwntest(X_test) + sigma2 = np.sum(((self.y_test - self.yHat)**2))/(self.y_test.size - self.beta.size) + var = np.diag(np.linalg.inv(np.dot(self.X_test.T, self.X_test))) * sigma2 + Lower = self.beta - 1.96*np.sqrt(var) + Upper = self.beta + 1.96*np.sqrt(var) + return var, Lower, Upper + + + ###Implementation through scikitlearn +class linregSKL: + def __init__(self, method = 'ols'): + """ + A class of linear regressions. Perform ordinarly least squares (OLS) and Ridge and Lasso + using scikit-learn functionality. + + """ + self.method = method + self.yHat = None + self.X = None + self.y = None + self.beta = None + + self._MSE = None + self._R2 = None + self._betaVariance = None + + + def fit(self, X_train, y_train, lambda_ = 0): + self.X_train = X_train + self.y_train = y_train + if self.method == 'ols': + self._olsSKLfit(X_train, y_train) + if self.method == 'ridge': + self._sklRidgeFit(X_train, y_train, lambda_) + if self.method == 'lasso': + self._SKLlassoFit(X_train, y_train, lambda_) + return self.beta + + def _olsSKLfit(self, X_train, y_train): + self.X_train = X_train + self.y_train = y_train + ##We already have standardized data from design matrix + self.ols = LinearRegression().fit(self.X_train, self.y_train) + self.beta = self.ols.coef_ + self.beta[0] = self.ols.intercept_ + + def _SKLlassoFit(self, X_train, y_train, lambda_): + self.regression = Lasso(fit_intercept=True, max_iter=100000, alpha=self.lambda_) + self.regression.fit(X_train,y_train) + self.beta = self.regression.coef_ + self.beta[0] = self.regression.intercept_ + + def _sklRidgeFit(self, X_train, y_train, lambda_): + self.regression = Ridge(fit_intercept=True, alpha=self.lambda_) + self.regression.fit(X,y) + self.beta = self.regression.coef_ + self.beta[0] = self.regression.intercept_ + + def predict(self, X_test): + self.X_test = X_test + if self.method == 'ols': + self._sklPredict(X_test) + return self.yHat + + def _sklPredict(self, X_test): + self.X_test = X_test + ## Since our data contains 1-s, we should subtract intercept, since scikit learn additionally + ##generates the 1-s + self.yHat = self.ols.predict(self.X_test) - self.beta[0] + + def MSE(self, y_test): + self.y_test = y_test + if self.yHat is None : + self._sklPredict(X_test) + self._MSE = mean_squared_error(self.y_test, self.yHat) + return self._MSE + + def R2(self, y_test): + self.y_test = y_test + if self.yHat is None : + self._sklPredict() + self._R2 = r2_score(self.y_test, self.yHat) + return self._R2 + + + +def designMatrix(x, y, k=5): + """ + Generates the design matrix (covariates of polynomial degree k). + Intercept is included in the design matrix. + Scaling does not apply to the intercept term. + if k = 2, generated column vectors: 1, x, y, x^2, xy, y^2 + if k = 3, generated column vectors: 1, x, y, x^2, xy, y^2, x^3, x^2y, xy^2, y^3 + ... + + Arguments: + x: 1D numpy array + y: 1D numpy array + k: integer type. complexity parameter (i.e polynomial degree) + """ + + xb = np.ones((x.size, 1)) + + for i in range(1, k+1): + for j in range(i+1): + xb = np.c_[xb, (x**(i-j))*(y**j)] + + xb[:, 0] = 1 + return xb + + +# # Functions for Project 2 + +# In[7]: + + +# MLP + +class Layer: + """ + Represents a layer (hidden or output) in our neural network. + """ + + def __init__(self, n_input, n_neurons, activation=None, alpha = 0.01, lam = 0.1): + """ + :param int n_input: The input size (coming from the input layer or a previous hidden layer) + :param int n_neurons: The number of neurons in this layer. + :param str activation: The activation function to use (if any). + :param weights: The layer's weights. + :param bias: The layer's bias. + """ + self.activation = activation + self.alpha = alpha + self.lam = lam + self.last_activation = None + self.error = None + self.delta = None + # self.weights = weights if weights is not None else np.random.rand(n_input, n_neurons) + # self.activation = activation + # self.bias = bias if bias is not None else np.random.rand(n_neurons) + + np.random.seed(1921) + # Xavier initializations (http://proceedings.mlr.press/v9/glorot10a/glorot10a.pdf). + if self.activation == 'sigmoid' : + r_inputs = np.sqrt(6.0 / (n_input + n_neurons)) + self.weights = np.random.uniform(-r_inputs, r_inputs, size=(n_input, n_neurons)) + #self.weights = np.random.rand(n_input, n_neurons) + self.bias = np.random.rand(n_neurons) + #self.v_reg = np.random.rand(n_neurons, 1) + self.v_reg = np.random.uniform(-(6/(n_neurons+1)), (6/(n_neurons+1)), size=(n_neurons, 1)) + self.bias_reg = np.random.rand(1) + + elif self.activation == 'tanh' : + r_inputs = 4.0 * np.sqrt(6.0 / (n_input + n_neurons)) + self.weights = np.random.uniform(-r_inputs, r_inputs, size=(n_input, n_neurons)) + #self.weights = np.random.rand(n_input, n_neurons) + self.v_reg = np.random.uniform(-4.0 * np.sqrt(6.0 / (n_neurons+1)), 4.0 * np.sqrt(6.0 / (n_neurons+1)), size=(n_neurons, 1)) + #self.v_reg = np.random.rand(n_neurons, 1) + #self.bias = np.zeros(shape=(n_neurons)) + self.bias = np.random.rand(n_neurons) + self.bias_reg = np.random.rand(1) ##output bias + + # He initializations (https://arxiv.org/pdf/1502.01852.pdf). + elif self.activation == 'relu' or self.activation == 'leaky_relu' or self.activation == 'elu' : + self.weights = np.random.normal(size=(n_input, n_neurons)) * np.sqrt(2.0 / n_input) + #self.weights = np.random.rand(n_input, n_neurons) + #self.v_reg = np.random.rand(n_neurons, 1) + self.v_reg = np.random.normal(size=(n_neurons,1)) * np.sqrt(2.0 / n_neurons) + self.bias = np.random.rand(n_neurons) + self.bias_reg = np.random.rand(1) + + else : + self.weights = np.random.normal(size=(n_input, n_neurons)) + #self.weights = np.random.rand(n_input, n_neurons) + #self.v_reg = np.random.rand(n_neurons, 1) + self.v_reg = np.random.normal(size=(n_neurons, 1)) + self.bias = np.random.rand(n_neurons) + self.bias_reg = np.random.rand(1) + + def activate(self, x): + """ + Calculates the dot product of this layer. + :param x: The input. + :return: The result. + """ + + r = np.dot(x, self.weights) + self.bias + self.last_activation = self._apply_activation(r) + return self.last_activation + + def _apply_activation(self, r): + """ + Applies the chosen activation function (if any). + :param r: The normal value. + :return: The activated value. + """ + + # In case no activation function was chosen + + if self.activation is None: + return r + + if self.activation == 'tanh': + return np.tanh(r) + + if self.activation == 'sigmoid': + return self._sigmoid(r) + + if self.activation == 'relu': + return self._relu(r) + + if self.activation == 'leaky_relu': + return self._leakyrelu(r) + + if self.activation == 'elu': + return self._elu(r) + + if self.activation == 'softmax': + return self._softmax(r) + + if self.activation == 'identity': + return self._identity(r) + + return r + + def apply_activation_derivative(self, r): + """ + Applies the derivative of the activation function (if any). + :param r: The normal value. + :return: The "derived" value. + """ + + # We use 'r' directly here because its already activated, the only values that + + # are used in this function are the last activations that were saved. + + if self.activation is None: + return r + + if self.activation == 'tanh': + return 1 - r ** 2 + + if self.activation == 'sigmoid': + return r * (1 - r) + + if self.activation == 'relu': + + r[r > 0] = self.lam + return r + + if self.activation == 'leaky_relu': + + r[r > 0] = self.lam + r[r <= 0] = self.lam * self.alpha + return r + + if self.activation == 'identity': + return 1 + + if self.activation == 'elu': + + r[r > 0] = 1 + r[r <= 0] = r[r <= 0] + self.alpha + + + return r + + + def _sigmoid(self, x): + return 1.0/ (1.0 + np.exp(-x)) + + + def _tanh(self, x): + + return np.tanh(x) + + + def _relu(self, x): + + x = self.lam * x + x[x <= 0] = 0 + + return x + + + def _leakyrelu(self, x) : + + x = self.lam * x + x[x <= 0] = self.alpha * x[x <= 0] + + return x + + def _identity(self, x) : ##linear activation function + return x + + + def _elu(self, x) : + neg = x<0.0 + x[neg] = self.alpha * (np.exp(x[neg]) - 1.0) + + return x + + def _softmax(self, x) : + exps = np.exp(x - np.max(x)) + return exps / np.sum(exps, axis=0, keepdims=True) ## sum along the column + + + +class NeuralNetwork: + """ + Represents a neural network. + """ + + def __init__(self): + self._layers = [] + + def add_layer(self, layer): + """ + Adds a layer to the neural network. + :param Layer layer: The layer to add. + """ + + self._layers.append(layer) + + + def feed_forward(self, X): + """ + Feed forward the input through the layers. + :param X: The input values. + :return: The result. + """ + + for layer in self._layers: + X = layer.activate(X) + + return X + + """ + N.B: Having a sigmoid activation in the output layer can be interpreted + as expecting probabilities as outputs. + W'll need to choose a winning class, this is usually done by choosing the + index of the biggest probability. + """ + def predict(self, X, net_type = 'regression', n_neurons = 3): + """ + Predicts a class (or classes). + :param X: The input values. + :return: The predictions. + """ + + ff = self.feed_forward(X) + + if net_type == 'classification': + + # One row + + if ff.ndim == 1: + pred = np.argmax(ff) + else: + pred = np.argmax(ff, axis = 1) + + if net_type == 'regression': + + pred = ff + + return pred + + + # # Multiple rows + + # return np.argmax(ff, axis=1) + + + def backpropagation(self, X, y, learning_rate, lmbd, net_type = 'classification'): + """ + Performs the backward propagation algorithm and updates the layers weights. + :param X: The input values. + :param y: The target values. + :param float learning_rate: The learning rate (between 0 and 1). + """ + ntarget = y.size + + # Feed forward for the output + + output = self.feed_forward(X) + + # Loop over the layers backward + + for i in reversed(range(len(self._layers))): + layer = self._layers[i] + + # If this is the output layer + if layer == self._layers[-1]: + layer.error = y - output + + # The output = layer.last_activation in this case + layer.delta = layer.error * layer.apply_activation_derivative(output) + + else: + next_layer = self._layers[i + 1] + layer.error = np.dot(next_layer.weights, next_layer.delta) + layer.delta = layer.error * layer.apply_activation_derivative(layer.last_activation) + + + # Update the weights + + for i in range(len(self._layers)): + layer = self._layers[i] + # The input is either the previous layers output or X itself (for the first hidden layer) + + input_to_use = np.atleast_2d(X if i == 0 else self._layers[i - 1].last_activation) + + layer.weights = layer.weights + layer.delta * input_to_use.T * learning_rate + + if lmbd > 0: ###adding L2 regularization + + layer.weights = layer.weights*(1-lmbd*learning_rate) + layer.delta * input_to_use.T * learning_rate + + layer.bias = layer.bias + layer.delta * learning_rate + + + + def train(self, X, y, learning_rate, max_epochs, net_type = 'regression', lmbd = 0): + """ + Trains the neural network using backpropagation. + :param X: The input values. + :param y: The target values. + :param float learning_rate: The learning rate (between 0 and 1). + :param int max_epochs: The maximum number of epochs (cycles). + :return: The list of calculated MSE errors. + """ + + mses = [] + + for i in range(max_epochs): + for j in range(len(X)): ##len(X) is rows + self.backpropagation(X[j], y[j], learning_rate, lmbd) + + # if i % 10 == 0: #At every 10th epoch, we will print out the Mean Squared Error and save it in mses which we will return at the end. + # nn = NeuralNetwork() + # mse = np.mean(np.square(y - nn.feed_forward(X))) + # mses.append(mse) + # print('Epoch: #%s, MSE: %f' % (i, float(mse))) + # return mses + + + + def MSE(self, y_pred, y_true): + return (1/len(y_true))*np.sum((y_pred - y_true)**2) + + + def accuracy(self, y_pred, y_true): + """ + Calculates the accuracy between the predicted labels and true labels. + :param y_pred: The predicted labels. + :param y_true: The true labels. + :return: The calculated accuracy. + """ + + return (y_pred == y_true).mean() + + def cal_err(self, y_pred, y_true, costf): + + if costf == "squared-error": + err = np.sum((y_pred - y_true)**2) + elif costf == "MSE": + err = (1/len(y_true))*np.sum((y_pred - y_true)**2) + + return err + + def confusion_table(self, y_pred, y_true): + + conf = confusion_matrix(y_true, y_pred, labels=[0, 1]) + + return conf + + def cal_r2(self, y_pred, y_true): + + mu = np.mean(y_true) + SS_tot = np.sum((y_true - mu)**2) + SS_res = np.sum((y_true - y_pred)**2) + + r2 = 1 - (SS_res/SS_tot) + + return r2 + + +# In[8]: + + +# Multiclass Logistig Regression + +class multiclassLogistic: + + def __init__(self, X, y, y_onehot, learning_rate, lambda_): + self.X = X + self.y = y + self.learning_rate = learning_rate + self.lambda_ = lambda_ + self.y_onehot = y_onehot + + + def softmax(self, X) : + exps = np.exp(X - np.max(X)) + return exps / np.sum(exps, axis=1, keepdims=True) + + + def sgd(self, X, y, y_onehot, iterations, lambda_, learning_rate): + w = np.zeros([X.shape[1],len(np.unique(y))]) + losses = [] + + for i in range(0,iterations): + loss,grad = self._getLoss(w,X,y_onehot, lambda_) + losses.append(loss) + w = w - (learning_rate * grad) + return w + + + def _getLoss(self, w, X, y_onehot, lambda_): + m = X.shape[0] #First we get the number of training examples + scores = np.dot(X,w) #Then we compute raw class scores given our input and current weights + prob = self.softmax(scores) #Next we perform a softmax on these scores to get their probabilities + loss = (-1 / m) * np.sum(y_onehot * np.log(prob)) + (lambda_/2)*np.sum(w*w) #We then find the loss of the probabilities + grad = (-1 / m) * np.dot(X.T,(y_onehot - prob)) + lambda_*w #And compute the gradient for that loss + return loss,grad + + + def accuracy(self, someX,someY, w): + prob,prede = self._getProbsAndPreds(someX, w) + accuracy = sum(prede == someY)/(float(len(someY))) + return accuracy + + + def _getProbsAndPreds(self, someX, w): + probs = self.softmax(np.dot(someX,w)) + preds = np.argmax(probs,axis=1) ##returns the highest value index (i.e. class) along the row + return probs,preds + + +# In[9]: + + +# Stochastic Gradient Descent + +from matplotlib.ticker import LinearLocator, FormatStrFormatter + +def compute_square_loss(X, y, theta): + loss = 0 #Initialize the average square loss + + m = len(y) + loss = (1.0/m)*(np.linalg.norm((X.dot(theta) - y)) ** 2) + return loss + + +def gradient_ridge(X, y, beta, lambda_): + return 2*(np.dot(X.T, (X.dot(beta) - y))) + 2*lambda_*beta + +def gradient_ols(X, y, beta): + m = X.shape[0] + + grad = 2/m * X.T.dot(X.dot(beta) - y) + + return grad + +def learning_schedule(t): + t0, t1 = 5, 50 + return t0/(t+t1) + + +def iterate_minibatches(inputs, targets, batchsize, shuffle=True): + assert inputs.shape[0] == targets.shape[0] + if shuffle: + indices = np.random.permutation(inputs.shape[0]) + for start_idx in range(0, inputs.shape[0], batchsize): + end_idx = min(start_idx + batchsize, inputs.shape[0]) + if shuffle: + excerpt = indices[start_idx:end_idx] + else: + excerpt = slice(start_idx, end_idx) + yield inputs[excerpt], targets[excerpt] + + +###sgd +def SGD(X, y, learning_rate = 0.02, n_epochs = 100, lambda_ = 0.01, batch_size = 20, method = 'ols'): + num_instances, num_features = X.shape[0], X.shape[1] + beta = np.random.randn(num_features) ##initialize beta + + for epoch in range(n_epochs+1): + + for batch in iterate_minibatches(X, y, batch_size, shuffle=True): + + X_batch, y_batch = batch + + # for i in range(batch_size): + # learning_rate = learning_schedule(n_epochs*epoch + i) + + if method == 'ols': + gradient = gradient_ols(X_batch, y_batch, beta) + beta = beta - learning_rate*gradient + if method == 'ridge': + gradient = gradient_ridge(X_batch, y_batch, beta, lambda_ = lambda_) + beta = beta - learning_rate*gradient + + mse_ols_train = compute_square_loss(X, y, beta) + mse_ridge_train = compute_square_loss(X, y, beta) + lambda_*np.dot(beta.T, beta) + + return beta + +def compute_test_mse(X_test, y_test, beta, lambda_ = 0.01): + mse_ols_test = compute_square_loss(X_test, y_test, beta) + mse_ridge_test = compute_square_loss(X_test, y_test, beta) + lambda_*np.dot(beta.T, beta) + return mse_ols_test, mse_ridge_test + + +# # Part A + +# In[10]: + + +# a + +##Make synthetic data +n = 1000 +np.random.seed(20) +x1 = np.random.rand(n) +x2 = np.random.rand(n) +X = designMatrix(x1, x2, 4) +y = franke(x1, x2) + +##Train-validation-test samples. +# We choose / play with hyper-parameters on the validation data and then test predictions on the test data + +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=1) +X_train, X_val, y_train, y_val = train_test_split(X_train, y_train, test_size=0.25, random_state=1) # 0.25 x 0.8 = 0.2 + +scaler = StandardScaler() +scaler.fit(X_train) +X_train = scaler.transform(X_train) +X_test = scaler.transform(X_test) +X_val = scaler.transform(X_val) + +X_train[:, 0] = 1 +X_test[:, 0] = 1 +X_val[:, 0] = 1 + + + +linreg = linregOwn(method='ols') +#print('Invert OLS:', linreg.fit(X_train, y_train)) +beta = SGD(X_train, y_train, learning_rate=0.07) +#print('SGD OLS:', beta) + + +linreg = linregOwn(method='ridge') +#print('Invert Ridge:', linreg.fit(X_train, y_train, lambda_= 0.01)) +beta = SGD(X_train, y_train, learning_rate=0.0004, method='ridge') +#print('SGD Ridge:', beta) + + +sgdreg = SGDRegressor(max_iter = 100, penalty=None, eta0=0.1) +sgdreg.fit(X_train[:, 1:],y_train.ravel()) +#print('sklearn:', sgdreg.coef_) +#print('sklearn intercept:', sgdreg.intercept_) + + +def plot_MSE(method = 'ridge', scheme = None): + eta = np.logspace(-5, -3, 10) + lambda_ = np.logspace(-5, -1, 10) + MSE_ols = [] + MSE_ridge = [] + + if scheme == 'joint': + + if method == 'ridge': + + for lmbd in lambda_: + + for i in eta: + + beta = SGD(X_train, y_train, learning_rate=i, lambda_ = lmbd, method = method) + mse_ols_test, mse_ridge_test = compute_test_mse(X_val, y_val, lambda_ = lmbd, beta = beta) + MSE_ridge.append(mse_ridge_test) + + fig = plt.figure() + ax = fig.gca(projection='3d') ##get current axis + lambda_ = np.ravel(lambda_) + eta = np.ravel(eta) + ax.zaxis.set_major_locator(LinearLocator(5)) + ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f')) + ax.xaxis.set_major_formatter(FormatStrFormatter('%.02f')) + ax.yaxis.set_major_formatter(FormatStrFormatter('%.03f')) + ax.plot_trisurf(lambda_, eta, MSE_ridge, cmap='viridis', edgecolor='none') + ax.set_xlabel(r'$\lambda$') + ax.set_ylabel(r'$\eta$') + ax.set_title(r'MSE Ridge') + ax.view_init(30, 60) + plt.show() + + if scheme == 'separate': + + if method == 'ols': + + eta = np.logspace(-5, 0, 10) + + for i in eta: + + beta = SGD(X_train, y_train, learning_rate=i, lambda_ = 0.01, method = method) + mse_ols_test, mse_ridge_test = compute_test_mse(X_val, y_val, beta = beta) + MSE_ols.append(mse_ols_test) + + print('The learning rate {} performs best for the OLS' .format(eta[MSE_ols.index(min(MSE_ols))])) + print('Corresponding minimum MSE for OLS: {}'.format(min(MSE_ols))) + plt.semilogx(eta, MSE_ols) + plt.xlabel(r'Learning rate, $\eta$') + plt.ylabel('MSE OLS') + plt.title('Stochastic Gradient Descent') + plt.show() + if scheme == 'separate': + + if method == 'ridge': + + eta = np.logspace(-5, 0, 10) + + for i in eta: + + beta = SGD(X_train, y_train, learning_rate=i, lambda_ = 0.01, method = method) + mse_ols_test, mse_ridge_test = compute_test_mse(X_val, y_val, beta = beta) + MSE_ols.append(mse_ridge_test) + + print('The learning rate {} performs best for Ridge' .format(eta[MSE_ols.index(min(MSE_ols))])) + print('Corresponding minimum MSE for Ridge: {}'.format(min(MSE_ols))) + + plt.plot(eta, MSE_ols) + plt.xlabel(r'Learning rate, $\eta$') + plt.ylabel('MSE Ridge') + plt.title('Stochastic Gradient Descent') + plt.show() + + + + +####Predict OLS, Ridge on test data after tuning learning rate and lambda on validation data + +def plot_scatter(y_true, method = 'ols'): + if method == 'ols': + beta = SGD(X_train, y_train, learning_rate=0.07, lambda_ = 0, method = method, n_epochs=300) + if method == 'ridge': + beta = SGD(X_train, y_train, learning_rate=0.0001, lambda_ = 0, method = method, n_epochs=300) + y_pred = np.dot(X_test, beta) + mse_ols_test, mse_ridge_test = compute_test_mse(X_test, y_true, beta = beta) + print('Test MSE OLS: {}' .format(mse_ols_test)) + print('Test MSE Ridge: {}' .format(mse_ridge_test)) + a = plt.axes(aspect='equal') + plt.scatter(y_pred, y_pred, color= 'blue', label = "True values") + plt.scatter(y_pred, y_true, color = 'red', label = "Predicted values") + plt.xlabel('True y values') + plt.ylabel('Predicted y') + plt.title(f"Prediction - {method}") + plt.legend() + # if method == 'ols': + # plt.savefig(os.path.join(os.path.dirname(__file__), 'Plots', 'ols_reg_pred.png'), transparent=True, bbox_inches='tight') + # if method == 'ridge': + # plt.savefig(os.path.join(os.path.dirname(__file__), 'Plots', 'ridge_reg_pred.png'), transparent=True, bbox_inches='tight') + + plt.show() + +plot_scatter(y_test, method='ols') + +plot_scatter(y_test, method='ridge') + + + diff --git a/doc/src/week39/codes/test8.py~ b/doc/src/week39/codes/test8.py~ new file mode 100644 index 000000000..7061dcae6 --- /dev/null +++ b/doc/src/week39/codes/test8.py~ @@ -0,0 +1,77 @@ +""" +Code to test Ridge and NNs using Scikit-Learn only +""" + +import numpy as np +import pandas as pd +import matplotlib.pyplot as plt +from sklearn.model_selection import train_test_split +from sklearn import linear_model +from sklearn.neural_network import MLPRegressor +from sklearn.metrics import accuracy_score +import seaborn as sns + + +def MSE(y_data,y_model): + n = np.size(y_model) + return np.sum((y_data-y_model)**2)/n +# A seed just to ensure that the random numbers are the same for every run. +# Useful for eventual debugging. +np.random.seed(315) + +n = 100 +x = np.random.rand(n) +y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + +Maxpolydegree = 5 +X = np.zeros((n,Maxpolydegree-1)) + +for degree in range(1,Maxpolydegree): #No intercept column + X[:,degree-1] = x**(degree) + +# We split the data in test and training data +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2) + +# Decide which values of lambda to use + +nlambdas = 10 +lmbd_vals = np.logspace(-4, 0, nlambdas) +MSERidgePredict = np.zeros(nlambdas) +for i in range(nlambdas): + lmb = lmbd_vals[i] + RegRidge = linear_model.Ridge(lmb) + RegRidge.fit(X_train,y_train) + ypredictRidge = RegRidge.predict(X_test) + MSERidgePredict[i] = MSE(y_test,ypredictRidge) + +plt.figure() +plt.plot(np.log10(lmbd_vals), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test') +plt.xlabel('log10(lambda)') +plt.ylabel('MSE') +plt.legend() +plt.show() + +# Neural Network part + +n_hidden_neurons = 100 +epochs = 100 +# store models for later use +eta_vals = np.logspace(-4, 0, 4) +# store the models for later use +DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) +test_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) + ypredictMLP = dnn.predict(X_test) + test_accuracy[i][j] = MSE(ypredictMLP, y_test) + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_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/week39/codes/test9.py~ b/doc/src/week39/codes/test9.py~ new file mode 100644 index 000000000..e10d5b982 --- /dev/null +++ b/doc/src/week39/codes/test9.py~ @@ -0,0 +1,85 @@ +""" +Code to test Ridge with own gradient descent and SGD +""" + +import numpy as np +import pandas as pd +import matplotlib.pyplot as plt +from sklearn.model_selection import train_test_split +from sklearn import linear_model +from sklearn.neural_network import MLPRegressor +from sklearn.metrics import accuracy_score +import seaborn as sns +import autograd.numpy as np +from autograd import grad + + +def MSE(y_data,y_model): + n = np.size(y_model) + return np.sum((y_data-y_model)**2)/n +# A seed just to ensure that the random numbers are the same for every run. +# Useful for eventual debugging. +np.random.seed(315) + +n = 100 +x = np.random.rand(n) +y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + +Maxpolydegree = 5 +X = np.zeros((n,Maxpolydegree-1)) + +for degree in range(1,Maxpolydegree): #No intercept column + X[:,degree-1] = x**(degree) + +# We split the data in test and training data +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2) + + +nlambdas = 10 +lmbd_vals = np.logspace(-4, 0, nlambdas) +MSERidgePredict = np.zeros(nlambdas) +for i in range(nlambdas): + lmb = lmbd_vals[i] + RegRidge = linear_model.Ridge(lmb,fit_intercept=False) + RegRidge.fit(X_train,y_train) + ypredictRidge = RegRidge.predict(X_test) + MSERidgePredict[i] = MSE(y_test,ypredictRidge) + +beta = np.random.randn(X_train.shape[1],1) +loss = np.mean((y_train.reshape(-1,1) - X_train@beta)**2) +print(loss) +get_grad = grad(loss,argnum=2) +print(get_grad) +#grad_beta = get_grad(X_train,y_train,beta) +#print(grad_beta) + +""" +print(beta) +print( (X_train.T @ y_train).T) +# Make own gradient descent and define precalculated quantities, saves cycles +XT_X = X_train.T @ X_train +XTy = X_train.T @ y_train +MSERidgeGDPredict = np.zeros(nlambdas) +for i in range(nlambdas): + lmb = lmbd_vals[i] + Id = lmb* np.eye(XT_X.shape[0]) + beta = np.random.randn(X_train.shape[1],1) + eta = 0.01 + Niterations = 2 +# beta_linreg = np.linalg.pinv(XT_X+Id) @ X_train.T @ y_train + for iter in range(Niterations): + XX = XT_X @ beta-XTy + gradients = (2.0/n)*XX *lmb*beta + beta -= eta*gradients + ypredictRidgeGD = X_test @ beta + MSERidgeGDPredict[i] = MSE(y_test,ypredictRidgeGD) + +plt.figure() +plt.plot(np.log10(lmbd_vals), MSERidgePredict, 'g--', label = 'MSE Sklearn Ridge Test') +plt.plot(np.log10(lmbd_vals), MSERidgeGDPredict, 'r', label = 'MSE GD Ridge Test') +plt.xlabel('log10(lambda)') +plt.ylabel('MSE') +plt.legend() +plt.show() +""" + diff --git a/doc/src/week41/programs/breast.py~ b/doc/src/week41/programs/breast.py~ new file mode 100644 index 000000000..205822b14 --- /dev/null +++ b/doc/src/week41/programs/breast.py~ @@ -0,0 +1,164 @@ +import tensorflow as tf +from tensorflow.keras.layers import Input +from tensorflow.keras.models import Sequential #This allows appending layers to existing models +from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer +from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop) +from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2) +from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function +import numpy as np +import matplotlib.pyplot as plt +import seaborn as sns +from sklearn.model_selection import train_test_split as splitter +from sklearn.datasets import load_breast_cancer +import pickle +import os + + +"""Load breast cancer dataset""" + +np.random.seed(0) #create same seed for random number every time + +cancer=load_breast_cancer() #Download breast cancer dataset + +inputs=cancer.data #Feature matrix of 569 rows (samples) and 30 columns (parameters) +outputs=cancer.target #Label array of 569 rows (0 for benign and 1 for malignant) +labels=cancer.feature_names[0:30] + +print('The content of the breast cancer dataset is:') #Print information about the datasets +print(labels) +print('-------------------------') +print("inputs = " + str(inputs.shape)) +print("outputs = " + str(outputs.shape)) +print("labels = "+ str(labels.shape)) + +x=inputs #Reassign the Feature and Label matrices to other variables +y=outputs + +# Visualisation of dataset (for correlation analysis) + +plt.figure() +plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral) +plt.xlabel('Mean radius',fontweight='bold') +plt.ylabel('Mean perimeter',fontweight='bold') +plt.show() + +plt.figure() +plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral) +plt.xlabel('Mean compactness',fontweight='bold') +plt.ylabel('Mean concavity',fontweight='bold') +plt.show() + + +plt.figure() +plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral) +plt.xlabel('Mean radius',fontweight='bold') +plt.ylabel('Mean texture',fontweight='bold') +plt.show() + +plt.figure() +plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral) +plt.xlabel('Mean perimeter',fontweight='bold') +plt.ylabel('Mean compactness',fontweight='bold') +plt.show() + + +# Generate training and testing datasets + +#Select features relevant to classification (texture,perimeter,compactness and symmetery) +#and add to input matrix + +temp1=np.reshape(x[:,1],(len(x[:,1]),1)) +temp2=np.reshape(x[:,2],(len(x[:,2]),1)) +X=np.hstack((temp1,temp2)) +temp=np.reshape(x[:,5],(len(x[:,5]),1)) +X=np.hstack((X,temp)) +temp=np.reshape(x[:,8],(len(x[:,8]),1)) +X=np.hstack((X,temp)) + +X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1) #Split datasets into training and testing + +y_train=to_categorical(y_train) #Convert labels to categorical when using categorical cross entropy +y_test=to_categorical(y_test) + +del temp1,temp2,temp + +# Define tunable parameters" + +eta=np.logspace(-3,-1,3) #Define vector of learning rates (parameter to SGD optimiser) +lamda=0.01 #Define hyperparameter +n_layers=2 #Define number of hidden layers in the model +n_neuron=np.logspace(0,3,4,dtype=int) #Define number of neurons per layer +epochs=100 #Number of reiterations over the input data +batch_size=100 #Number of samples per gradient update + +"""Define function to return Deep Neural Network model""" + +def NN_model(inputsize,n_layers,n_neuron,eta,lamda): + model=Sequential() + for i in range(n_layers): #Run loop to add hidden layers to the model + if (i==0): #First layer requires input dimensions + model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize)) + else: #Subsequent layers are capable of automatic shape inferencing + model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda))) + model.add(Dense(2,activation='softmax')) #2 outputs - ordered and disordered (softmax for prob) + sgd=optimizers.SGD(lr=eta) + model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy']) + return model + + +Train_accuracy=np.zeros((len(n_neuron),len(eta))) #Define matrices to store accuracy scores as a function +Test_accuracy=np.zeros((len(n_neuron),len(eta))) #of learning rate and number of hidden neurons for + +for i in range(len(n_neuron)): #run loops over hidden neurons and learning rates to calculate + for j in range(len(eta)): #accuracy scores + DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda) + DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1) + Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1] + Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1] + + +def plot_data(x,y,data,title=None): + + # plot results + fontsize=16 + + + fig = plt.figure() + ax = fig.add_subplot(111) + cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1) + + cbar=fig.colorbar(cax) + cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize) + cbar.set_ticks([0,.2,.4,0.6,0.8,1.0]) + cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%']) + + # put text on matrix elements + for i, x_val in enumerate(np.arange(len(x))): + for j, y_val in enumerate(np.arange(len(y))): + c = "${0:.1f}\\%$".format( 100*data[j,i]) + ax.text(x_val, y_val, c, va='center', ha='center') + + # convert axis vaues to to string labels + x=[str(i) for i in x] + y=[str(i) for i in y] + + + ax.set_xticklabels(['']+x) + ax.set_yticklabels(['']+y) + + ax.set_xlabel('$\\mathrm{learning\\ rate}$',fontsize=fontsize) + ax.set_ylabel('$\\mathrm{hidden\\ neurons}$',fontsize=fontsize) + if title is not None: + ax.set_title(title) + + plt.tight_layout() + + plt.show() + +plot_data(eta,n_neuron,Train_accuracy, 'training') +plot_data(eta,n_neuron,Test_accuracy, 'testing') + + + + + diff --git a/doc/src/week41/programs/nnown.py~ b/doc/src/week41/programs/nnown.py~ new file mode 100644 index 000000000..7875ade5f --- /dev/null +++ b/doc/src/week41/programs/nnown.py~ @@ -0,0 +1,171 @@ + +# import necessary packages +import numpy as np +import matplotlib.pyplot as plt +from sklearn.neural_network import MLPClassifier +from sklearn.metrics import accuracy_score +import seaborn as sns + +# ensure the same random numbers appear every time +np.random.seed(0) + +# Design matrix +X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64) + +# The XOR gate +yXOR = np.array( [ 0, 1 ,1, 0]) +# The OR gate +yOR = np.array( [ 0, 1 ,1, 1]) +# The AND gate +yAND = np.array( [ 0, 0 ,0, 1]) + + +# Defining the neural network +n_inputs, n_features = X.shape +n_hidden_neurons = 2 +n_categories = 2 +n_features = 2 + +def sigmoid(x): + return 1/(1 + np.exp(-x)) + + +class NeuralNetwork: + def __init__( + self, + X_data, + Y_data, + n_hidden_neurons=2, + n_categories=2, + epochs=10, + batch_size=100, + eta=0.1, + lmbd=0.0): + + self.X_data_full = X_data + self.Y_data_full = Y_data + + self.n_inputs = X_data.shape[0] + self.n_features = X_data.shape[1] + self.n_hidden_neurons = n_hidden_neurons + self.n_categories = n_categories + + self.epochs = epochs + self.batch_size = batch_size + self.iterations = self.n_inputs // self.batch_size + self.eta = eta + self.lmbd = lmbd + + self.create_biases_and_weights() + + + def create_biases_and_weights(self): + self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons) + self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01 + + self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories) + self.output_bias = np.zeros(self.n_categories) + 0.01 + + def feed_forward(self): + # feed-forward for training + self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias + self.a_h = sigmoid(self.z_h) + + self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias + + exp_term = np.exp(self.z_o) + self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True) + + def feed_forward_out(self, X): + # feed-forward for output + z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias + a_h = sigmoid(z_h) + + z_o = np.matmul(a_h, self.output_weights) + self.output_bias + + exp_term = np.exp(z_o) + probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True) + return probabilities + + def backpropagation(self): + error_output = self.probabilities - self.Y_data + error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h) + + self.output_weights_gradient = np.matmul(self.a_h.T, error_output) + self.output_bias_gradient = np.sum(error_output, axis=0) + + self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden) + self.hidden_bias_gradient = np.sum(error_hidden, axis=0) + + if self.lmbd > 0.0: + self.output_weights_gradient += self.lmbd * self.output_weights + self.hidden_weights_gradient += self.lmbd * self.hidden_weights + + self.output_weights -= self.eta * self.output_weights_gradient + self.output_bias -= self.eta * self.output_bias_gradient + self.hidden_weights -= self.eta * self.hidden_weights_gradient + self.hidden_bias -= self.eta * self.hidden_bias_gradient + + def predict(self, X): + probabilities = self.feed_forward_out(X) + return np.argmax(probabilities, axis=1) + + def predict_probabilities(self, X): + probabilities = self.feed_forward_out(X) + return probabilities + + def train(self): + data_indices = np.arange(self.n_inputs) + + for i in range(self.epochs): + for j in range(self.iterations): + # pick datapoints with replacement + chosen_datapoints = np.random.choice( + data_indices, size=self.batch_size, replace=False + ) + + # minibatch training data + self.X_data = self.X_data_full[chosen_datapoints] + self.Y_data = self.Y_data_full[chosen_datapoints] + + self.feed_forward() + self.backpropagation() + +epochs = 100 +batch_size = 100 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) +# store the models for later use +DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +# grid search +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + dnn = NeuralNetwork(X, yXOR, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size, + n_hidden_neurons=n_hidden_neurons, n_categories=n_categories) + dnn.train() + DNN_numpy[i][j] = dnn + test_predict = dnn.predict(X) + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Accuracy score on test set: ", accuracy_score(yXOR, test_predict)) + print() + +sns.set() +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + dnn = DNN_numpy[i][j] + test_pred = dnn.predict(X) + test_accuracy[i][j] = accuracy_score(yXOR, test_pred) + +fig, ax = plt.subplots(figsize = (10, 10)) +sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") +ax.set_title("Test Accuracy") +ax.set_ylabel("$\eta$") +ax.set_xlabel("$\lambda$") +plt.show() + + diff --git a/doc/src/week41/programs/test.py~ b/doc/src/week41/programs/test.py~ new file mode 100644 index 000000000..c150131e3 --- /dev/null +++ b/doc/src/week41/programs/test.py~ @@ -0,0 +1,139 @@ +# import necessary packages +import numpy as np +import matplotlib.pyplot as plt +from sklearn import datasets + + +# ensure the same random numbers appear every time +np.random.seed(0) + +# display images in notebook +%matplotlib inline +plt.rcParams['figure.figsize'] = (12,12) + + +# download MNIST dataset +digits = datasets.load_digits() + +# define inputs and labels +inputs = digits.images +labels = digits.target + +# RGB images have a depth of 3 +# our images are grayscale so they should have a depth of 1 +inputs = inputs[:,:,:,np.newaxis] + +print("inputs = (n_inputs, pixel_width, pixel_height, depth) = " + str(inputs.shape)) +print("labels = (n_inputs) = " + str(labels.shape)) + + +# choose some random images to display +n_inputs = len(inputs) +indices = np.arange(n_inputs) +random_indices = np.random.choice(indices, size=5) + +for i, image in enumerate(digits.images[random_indices]): + plt.subplot(1, 5, i+1) + plt.axis('off') + plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest') + plt.title("Label: %d" % digits.target[random_indices[i]]) +plt.show() + +from tensorflow.keras import datasets, layers, models +from tensorflow.keras.layers import Input +from tensorflow.keras.models import Sequential #This allows appending layers to existing models +from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer +from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop) +from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2) +from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function +#from tensorflow.keras import Conv2D +#from tensorflow.keras import MaxPooling2D +#from tensorflow.keras import Flatten + +from sklearn.model_selection import train_test_split + +# representation of labels +labels = to_categorical(labels) + +# split into train and test data +# one-liner from scikit-learn library +train_size = 0.8 +test_size = 1 - train_size +X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size, + test_size=test_size) + +def create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd): + model = Sequential() + model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same', + activation='relu', kernel_regularizer=regularizers.l2(lmbd))) + model.add(layers.MaxPooling2D(pool_size=(2, 2))) + model.add(layers.Flatten()) + model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd))) + model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd))) + + sgd = optimizers.SGD(lr=eta) + model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy']) + + return model + +epochs = 100 +batch_size = 100 +input_shape = X_train.shape[1:4] +receptive_field = 3 +n_filters = 10 +n_neurons_connected = 50 +n_categories = 10 + +eta_vals = np.logspace(-5, 1, 7) +lmbd_vals = np.logspace(-5, 1, 7) + +CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) + +for i, eta in enumerate(eta_vals): + for j, lmbd in enumerate(lmbd_vals): + CNN = create_convolutional_neural_network_keras(input_shape, receptive_field, + n_filters, n_neurons_connected, n_categories, + eta, lmbd) + CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0) + scores = CNN.evaluate(X_test, Y_test) + + CNN_keras[i][j] = CNN + + print("Learning rate = ", eta) + print("Lambda = ", lmbd) + print("Test accuracy: %.3f" % scores[1]) + print() + +# visual representation of grid search +# uses seaborn heatmap, could probably do this in matplotlib +import seaborn as sns + +sns.set() + +train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) +test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) + +for i in range(len(eta_vals)): + for j in range(len(lmbd_vals)): + CNN = CNN_keras[i][j] + + train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1] + test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1] + + +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() + +fig, ax = plt.subplots(figsize 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b/doc/src/week43/LatexFigures/fig2.tex~ @@ -0,0 +1,55 @@ +\documentclass[border=0.125cm]{standalone} +\usepackage{tikz} +\usetikzlibrary{positioning} +\begin{document} + +\tikzset{% + every neuron/.style={ + circle, + draw, + minimum size=1cm + }, + neuron missing/.style={ + draw=none, + scale=4, + text height=0.333cm, + execute at begin node=\color{black}$\vdots$ + }, +} + +\begin{tikzpicture}[x=1.5cm, y=1.5cm, >=stealth] + +\foreach \m/\l [count=\y] in {1,2,3,missing,4} + \node [every neuron/.try, neuron \m/.try] (input-\m) at (0,2.5-\y) {}; + +\foreach \m [count=\y] in {1,missing,2} + \node [every neuron/.try, neuron \m/.try ] (hidden-\m) at (2,2-\y*1.25) {}; + +\foreach \m [count=\y] in {1,missing,2} + \node [every neuron/.try, neuron \m/.try ] (output-\m) at (4,1.5-\y) {}; + +\foreach \l [count=\i] in {1,2,3,n} + \draw [<-] (input-\i) -- ++(-1,0) + node [above, midway] {$I_\l$}; + +\foreach \l [count=\i] in {1,n} + \node [above] at (hidden-\i.north) {$H_\l$}; + +\foreach \l [count=\i] in {1,n} + \draw [->] (output-\i) -- ++(1,0) + node [above, midway] {$O_\l$}; + +\foreach \i in {1,...,4} + \foreach \j in {1,...,2} + \draw [->] (input-\i) -- (hidden-\j); + +\foreach \i in {1,...,2} + \foreach \j in {1,...,2} + \draw [->] (hidden-\i) -- (output-\j); + +\foreach \l [count=\x from 0] in {Input, Hidden, Ouput} + \node [align=center, above] at (\x*2,2) {\l \\ layer}; + +\end{tikzpicture} + +\end{document} diff --git a/doc/src/week43/LatexFigures/nn.py~ b/doc/src/week43/LatexFigures/nn.py~ new file mode 100644 index 000000000..d41673ce2 --- /dev/null +++ b/doc/src/week43/LatexFigures/nn.py~ @@ -0,0 +1,51 @@ +import matplotlib.pyplot as plt + +def draw_neural_net(ax, left, right, bottom, top, layer_sizes): + ''' + Draw a neural network cartoon using matplotilb. + + :usage: + >>> fig = plt.figure(figsize=(12, 12)) + >>> draw_neural_net(fig.gca(), .1, .9, .1, .9, [4, 7, 2]) + + :parameters: + - ax : matplotlib.axes.AxesSubplot + The axes on which to plot the cartoon (get e.g. by plt.gca()) + - left : float + The center of the leftmost node(s) will be placed here + - right : float + The center of the rightmost node(s) will be placed here + - bottom : float + The center of the bottommost node(s) will be placed here + - top : float + The center of the topmost node(s) will be placed here + - layer_sizes : list of int + List of layer sizes, including input and output dimensionality + ''' + n_layers = len(layer_sizes) + v_spacing = (top - bottom)/float(max(layer_sizes)) + h_spacing = (right - left)/float(len(layer_sizes) - 1) + # Nodes + for n, layer_size in enumerate(layer_sizes): + layer_top = v_spacing*(layer_size - 1)/2. + (top + bottom)/2. + for m in range(layer_size): + circle = plt.Circle((n*h_spacing + left, layer_top - m*v_spacing), v_spacing/4., + color='w', ec='k', zorder=4) + ax.add_artist(circle) + # Edges + for n, (layer_size_a, layer_size_b) in enumerate(zip(layer_sizes[:-1], layer_sizes[1:])): + layer_top_a = v_spacing*(layer_size_a - 1)/2. + (top + bottom)/2. + layer_top_b = v_spacing*(layer_size_b - 1)/2. + (top + bottom)/2. + for m in range(layer_size_a): + for o in range(layer_size_b): + line = plt.Line2D([n*h_spacing + left, (n + 1)*h_spacing + left], + [layer_top_a - m*v_spacing, layer_top_b - o*v_spacing], c='k') + ax.add_artist(line) + + +fig = plt.figure(figsize=(12, 12)) +ax = fig.gca() +ax.axis('off') +draw_neural_net(ax, .1, .9, .1, .9, [4, 7, 2]) +fig.savefig('nn.png') +fig.show() diff --git a/doc/src/week45/Datafiles/ride.csv~ b/doc/src/week45/Datafiles/ride.csv~ new file mode 100644 index 000000000..7a908c5f7 --- /dev/null +++ b/doc/src/week45/Datafiles/ride.csv~ @@ -0,0 +1,15 @@ +Outlook,Temperature,Humidity,Wind,Ride +Sunny,Hot,High,Weak,0 +Sunny,Hot,High,Strong,1 +Overcast,Hot,High,Weak,1 +Rain,Mild,High,Weak,1 +Rain,Cool,Normal,Weak,1 +Rain,Cool,Normal,Strong,0 +Overcast,Cool,Normal,Strong,1 +Sunny,Mild,High,Weak,0 +Sunny,Cool,Normal,Weak,1 +Rain,Mild,Normal,Weak,1 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