updating book
This commit is contained in:
Morten Hjorth-Jensen
@@ -0,0 +1,36 @@
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Translating doconce text in chapter1.do.txt to ipynb
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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output in chapter1.ipynb
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Translating doconce text in chapter1.do.txt to ipynb
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*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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output in chapter1.ipynb
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Translating doconce text in chapter1.do.txt to ipynb
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*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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Failed to remove ans_at_end environment
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output in chapter1.ipynb
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Translating doconce text in chapter1.do.txt to ipynb
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*** replacing \bm{...} by \boldsymbol{...} (\bm is not supported by MathJax)
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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*** warning: latex envir \begin{bmatrix} does not work well in Markdown. Stick to \[ ... \], equation, equation*, align, or align* environments in math environments.
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Failed to remove ans_at_end environment
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output in chapter1.ipynb
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@@ -1263,22 +1263,22 @@ matrices as upper case boldfaced letters.
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!bt
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\[
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\frac{\partial (\bm{b}^T\bm{a})}{\partial \bm{a}} = \bm{b},
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\frac{\partial (\bm{b}^T\bm{a})}{\partial\bm{a}}=\bm{b},
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\]
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!et
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!bt
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\[
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\frac{\partial (\bm{a}^T\bm{A}\bm{a})}{\partial \bm{a}} = (\bm{A}+\bm{A}^T)\bm{a},
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\frac{\partial (\bm{a}^T\bm{A}\bm{a})}{\partial\bm{a}}=(\bm{A}+\bm{A}^T)\bm{a},
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\]
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!et
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!bt
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\[
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\frac{\partial tr(\bm{B}\bm{A})}{\partial \bm{A}} = \bm{B}^T,
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\frac{\partial tr(\bm{B}\bm{A})}{\partial\bm{A}}=\bm{B}^T,
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\]
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!et
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!bt
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\[
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\frac{\partial \log{\vert\bm{A}\vert}}{\partial \bm{A}} = (\bm{A}^{-1})^T.
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\frac{\partial\log{\vert\bm{A}\vert}}{\partial \bm{A}}=(\bm{A}^{-1})^T.
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\]
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!et
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@@ -2137,51 +2137,6 @@ where we have defined the mean value of $\hat{y}$ as
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You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions.
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Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.
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!bsol
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The code here is an example of where we define our own design matrix and fit parameters $\beta$.
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!bc pycod
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import os
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import numpy as np
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import pandas as pd
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import matplotlib.pyplot as plt
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from sklearn.model_selection import train_test_split
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def save_fig(fig_id):
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plt.savefig(image_path(fig_id) + ".png", format='png')
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def R2(y_data, y_model):
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return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
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def MSE(y_data,y_model):
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n = np.size(y_model)
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return np.sum((y_data-y_model)**2)/n
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x = np.random.rand(100)
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y = 2.0+5*x*x+0.1*np.random.randn(100)
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# The design matrix now as function of a given polynomial
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X = np.zeros((len(x),3))
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X[:,0] = 1.0
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X[:,1] = x
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X[:,2] = x**2
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# We split the data in test and training data
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X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
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# matrix inversion to find beta
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beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
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print(beta)
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# and then make the prediction
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ytilde = X_train @ beta
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print("Training R2")
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print(R2(y_train,ytilde))
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print("Training MSE")
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print(MSE(y_train,ytilde))
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ypredict = X_test @ beta
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print("Test R2")
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print(R2(y_test,ypredict))
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print("Test MSE")
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print(MSE(y_test,ypredict))
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!ec
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!esol
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@@ -2266,8 +2221,6 @@ Perform an ordinary least squares and compute the means squared error and the $R
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!bsubex
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Add now a model which allows you to make polynomials up to degree $15$. Perform a standard OLS fitting of the training data and compute the MSE and $R2$ for the training and test data and plot both test and training data MSE and $R2$ as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
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!bsol
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Here you simply need to change the degree of the polynomial in the above code to $n=15$.
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!esol
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!esubex
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@@ -0,0 +1,90 @@
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||||
3.3773726001100143E-005, 3.1715032621225665E-002
|
||||
2.7018980800880114E-004, 0.25379346864959029
|
||||
9.1189060202970370E-004, 0.85691856357595775
|
||||
2.1615184640704091E-003, 2.0322700794292126
|
||||
4.2217157501375172E-003, 3.9716946611295496
|
||||
7.2951248162376296E-003, 6.8678138410008760
|
||||
1.1584388018377344E-002, 10.913973030767592
|
||||
1.7292147712563273E-002, 16.304871533080519
|
||||
2.4621046254802003E-002, 23.235226466525493
|
||||
3.3773726001100138E-002, 31.902701432416372
|
||||
4.4952829307464297E-002, 42.504284679798289
|
||||
5.8360998529901037E-002, 55.243642496340065
|
||||
7.4200876024417023E-002, 70.325036648868448
|
||||
9.2675104147018753E-002, 87.967299710326188
|
||||
0.11398632525371297, 108.39264632432710
|
||||
0.13833718170050618 , 131.84282952216495
|
||||
0.16593031584340495 , 158.57124952965228
|
||||
0.19696837003841602 , 188.82820513039681
|
||||
0.203199995458126059, 195.042764094891368
|
||||
0.211199995279312130, 202.937172130123315
|
||||
0.219199995100498229, 210.852580165355278
|
||||
0.227199994921684245, 218.789988200587175
|
||||
0.235199994742870316, 226.748396235819115
|
||||
0.243199994564056388, 234.740804271051104
|
||||
0.251199994385242487, 242.757212306283037
|
||||
0.259199994206428530, 250.797620341515000
|
||||
0.267199994027614574, 258.874028376746878
|
||||
0.275199993848800673, 266.977436411978829
|
||||
0.283199993669986716, 275.092844447210780
|
||||
0.291199993491172815, 283.248252482442751
|
||||
0.299199993312358858, 291.445660517674696
|
||||
0.307199993133544902, 299.655068552906641
|
||||
0.315199992954731001, 307.909476588138546
|
||||
0.323199992775917044, 316.192884623370503
|
||||
0.339199992418289187, 332.846700693834407
|
||||
0.355199992060661329, 349.620516764298316
|
||||
0.371199991703033416, 366.537332834762140
|
||||
0.387199991345405559, 383.604148905225998
|
||||
0.403199990987777701, 400.801964975689941
|
||||
0.419199990630149844, 418.158781046153820
|
||||
0.435199990272521986, 435.624597116617792
|
||||
0.451199989914894073, 453.288413187081574
|
||||
0.467199989557266215, 471.062229257545425
|
||||
0.483199989199638358, 489.048045328009380
|
||||
0.499199988842010500, 507.146861398473277
|
||||
0.515199988484382643, 525.392677468937222
|
||||
0.531199988126754730, 543.829493539401028
|
||||
0.531999988108873390, 544.796634342924222
|
||||
0.550079987704753859, 565.860416502548446
|
||||
0.567999987304210641, 586.865970501467928
|
||||
0.585599986910820047, 607.703068178978242
|
||||
0.603199986517429343, 628.645165856488575
|
||||
0.620639986127614951, 649.578035373294142
|
||||
0.637919985741376872, 670.496676729395176
|
||||
0.655199985355138792, 691.549318085496111
|
||||
0.672479984968900713, 712.766959441597237
|
||||
0.689599984586238834, 733.981372636993456
|
||||
0.706879984200000755, 755.521013993094471
|
||||
0.724159983813762675, 777.228655349195492
|
||||
0.741439983427524596, 799.117296705296553
|
||||
0.758719983041286516, 821.191938061397536
|
||||
0.775999982655048326, 843.460579417498366
|
||||
0.793439982265233934, 866.080448934304059
|
||||
0.810879981875419542, 888.907318451109631
|
||||
0.828479981482028949, 912.100416128620054
|
||||
0.846239981085061932, 935.664741966834868
|
||||
0.864159980684518825, 959.607295965754474
|
||||
0.882079980283975607, 983.787849964674024
|
||||
0.900159979879856187, 1008.36063212429838
|
||||
0.918399979472160344, 1033.33564244462718
|
||||
0.936639979064464612, 1058.56865276495591
|
||||
0.955199978649616255, 1084.36811940669395
|
||||
0.973919978231191585, 1110.59381420913678
|
||||
0.992799977809190715, 1137.25073717228429
|
||||
1.01183997738361353, 1164.35288829613614
|
||||
1.06047997629642499, 1234.41724915034661
|
||||
1.11039997518062594, 1307.72443529019392
|
||||
1.16191997402906422, 1384.74890303708753
|
||||
1.21535997283458719, 1466.00110871243692
|
||||
1.27071997159719463, 1551.73305231624181
|
||||
1.32847997030615828, 1642.68741833061654
|
||||
1.38895996895432461, 1739.56266307697001
|
||||
1.45263996753096580, 1843.32947103741640
|
||||
1.52031996601820008, 1955.19398301547858
|
||||
1.59295996439456933, 2077.14256797538474
|
||||
1.67167996263504026, 2211.44382304206692
|
||||
1.75855996069312082, 2361.74471430468566
|
||||
1.85647995850443825, 2533.64534865592441
|
||||
1.96959995597600934, 2734.93465827410455
|
||||
2.10543995293974895, 2980.03336671234320
|
||||
|
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
@@ -0,0 +1,67 @@
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// This function computes the autocorrelation function for
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// the standard c++ random number generator
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|
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#include <fstream>
|
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#include <iomanip>
|
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#include <iostream>
|
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#include <cmath>
|
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using namespace std;
|
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// output file as global variable
|
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ofstream ofile;
|
||||
|
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// Main function begins here
|
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int main(int argc, char* argv[])
|
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{
|
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int n;
|
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char *outfilename;
|
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|
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cin >> n;
|
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double MCint = 0.; double MCintsqr2=0.;
|
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double invers_period = 1./RAND_MAX; // initialise the random number generator
|
||||
srand(time(NULL)); // This produces the so-called seed in MC jargon
|
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// Compute the variance and the mean value of the uniform distribution
|
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// Compute also the specific values x for each cycle in order to be able to
|
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// the covariance and the correlation function
|
||||
// Read in output file, abort if there are too few command-line arguments
|
||||
if( argc <= 2 ){
|
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cout << "Bad Usage: " << argv[0] <<
|
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" read also output file and number of cycles on same line" << endl;
|
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exit(1);
|
||||
}
|
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else{
|
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outfilename=argv[1];
|
||||
}
|
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ofile.open(outfilename);
|
||||
// Get the number of Monte-Carlo samples
|
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n = atoi(argv[2]);
|
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double *X;
|
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X = new double[n];
|
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for (int i = 0; i < n; i++){
|
||||
double x = double(rand())*invers_period;
|
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X[i] = x;
|
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MCint += x;
|
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MCintsqr2 += x*x;
|
||||
}
|
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double Mean = MCint/((double) n );
|
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MCintsqr2 = MCintsqr2/((double) n );
|
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double STDev = sqrt(MCintsqr2-Mean*Mean);
|
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double Variance = MCintsqr2-Mean*Mean;
|
||||
// Write mean value and standard deviation
|
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cout << " Standard deviation= " << STDev << " Integral = " << Mean << endl;
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|
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// Now we compute the autocorrelation function, setting the distance d between two
|
||||
// to a most 1/4 of the total number of cycles
|
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double *autocor; autocor = new double[n];
|
||||
for (int j = 0; j < n; j++){
|
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double sum = 0.0;
|
||||
for (int k = 0; k < (n-j); k++){
|
||||
sum += (X[k]-Mean)*(X[k+j]-Mean);
|
||||
}
|
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autocor[j] = sum/Variance/((double) n );
|
||||
ofile << setiosflags(ios::showpoint | ios::uppercase);
|
||||
ofile << setw(15) << setprecision(8) << j;
|
||||
ofile << setw(15) << setprecision(8) << autocor[j] << endl;
|
||||
}
|
||||
ofile.close(); // close output file
|
||||
return 0;
|
||||
} // end of main program
|
||||
@@ -0,0 +1,76 @@
|
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// This function computes the autocorrelation function for
|
||||
// the standard c++ random number generator
|
||||
|
||||
#include <fstream>
|
||||
#include <iomanip>
|
||||
#include <iostream>
|
||||
#include <cmath>
|
||||
#include <random>
|
||||
|
||||
using namespace std;
|
||||
// output file as global variable
|
||||
ofstream ofile;
|
||||
|
||||
// Main function begins here
|
||||
int main(int argc, char* argv[])
|
||||
{
|
||||
int n;
|
||||
char *outfilename;
|
||||
|
||||
cin >> n;
|
||||
double MCint = 0.; double MCintsqr2=0.;
|
||||
// Initialize the seed and call the Mersienne algo
|
||||
std::random_device rd;
|
||||
std::mt19937_64 gen(rd());
|
||||
// Set up the uniform distribution for x \in [[0, 1]
|
||||
std::uniform_real_distribution<double> RandomNumberGenerator(0.0,1.0);
|
||||
// Compute the variance and the mean value of the uniform distribution
|
||||
// Compute also the specific values x for each cycle in order to be able to
|
||||
// the covariance and the correlation function
|
||||
// Read in output file, abort if there are too few command-line arguments
|
||||
if( argc <= 2 ){
|
||||
cout << "Bad Usage: " << argv[0] <<
|
||||
" read also output file and number of cycles on same line" << endl;
|
||||
exit(1);
|
||||
}
|
||||
else{
|
||||
outfilename=argv[1];
|
||||
}
|
||||
ofile.open(outfilename);
|
||||
// Get the number of Monte-Carlo samples
|
||||
n = atoi(argv[2]);
|
||||
double *X;
|
||||
X = new double[n];
|
||||
for (int i = 0; i < n; i++){
|
||||
double x = RandomNumberGenerator(gen);
|
||||
X[i] = x;
|
||||
MCint += x;
|
||||
MCintsqr2 += x*x;
|
||||
}
|
||||
double Mean = MCint/((double) n );
|
||||
MCintsqr2 = MCintsqr2/((double) n );
|
||||
double STDev = sqrt(MCintsqr2-Mean*Mean);
|
||||
double Variance = MCintsqr2-Mean*Mean;
|
||||
// Write mean value and standard deviation
|
||||
cout << " Standard deviation= " << STDev << " Integral = " << Mean << endl;
|
||||
|
||||
// Now we compute the autocorrelation function, setting the distance d
|
||||
double *autocor; autocor = new double[n];
|
||||
for (int j = 0; j < n; j++){
|
||||
double sum = 0.0;
|
||||
for (int k = 0; k < (n-j); k++){
|
||||
sum += (X[k]-Mean)*(X[k+j]-Mean);
|
||||
}
|
||||
autocor[j] = sum/Variance/((double) n );
|
||||
ofile << setiosflags(ios::showpoint | ios::uppercase);
|
||||
ofile << setw(15) << setprecision(8) << j;
|
||||
ofile << setw(15) << setprecision(8) << autocor[j] << endl;
|
||||
}
|
||||
ofile.close(); // close output file
|
||||
return 0;
|
||||
} // end of main program
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@@ -0,0 +1,16 @@
|
||||
import numpy as np
|
||||
from matplotlib import pyplot as plt
|
||||
# Load in data file
|
||||
data = np.loadtxt("autocor.dat")
|
||||
data1 = np.loadtxt("automersenne.dat")
|
||||
# Make arrays containing x-axis and binding energies as function of A
|
||||
x = data[:,0]
|
||||
corr = data[:,1]
|
||||
corr2 = data1[:,1]
|
||||
plt.plot(x, corr ,'ro', x, corr2, 'b')
|
||||
plt.axis([0,1000,-0.2, 1.1])
|
||||
plt.xlabel(r'$d$')
|
||||
plt.ylabel(r'$C_d$')
|
||||
plt.title(r'autocorrelation function for RNG')
|
||||
plt.savefig('autocorr.pdf')
|
||||
plt.show()
|
||||
@@ -0,0 +1,90 @@
|
||||
3.3773726001100143E-005, 3.1715032621225665E-002
|
||||
2.7018980800880114E-004, 0.25379346864959029
|
||||
9.1189060202970370E-004, 0.85691856357595775
|
||||
2.1615184640704091E-003, 2.0322700794292126
|
||||
4.2217157501375172E-003, 3.9716946611295496
|
||||
7.2951248162376296E-003, 6.8678138410008760
|
||||
1.1584388018377344E-002, 10.913973030767592
|
||||
1.7292147712563273E-002, 16.304871533080519
|
||||
2.4621046254802003E-002, 23.235226466525493
|
||||
3.3773726001100138E-002, 31.902701432416372
|
||||
4.4952829307464297E-002, 42.504284679798289
|
||||
5.8360998529901037E-002, 55.243642496340065
|
||||
7.4200876024417023E-002, 70.325036648868448
|
||||
9.2675104147018753E-002, 87.967299710326188
|
||||
0.11398632525371297, 108.39264632432710
|
||||
0.13833718170050618 , 131.84282952216495
|
||||
0.16593031584340495 , 158.57124952965228
|
||||
0.19696837003841602 , 188.82820513039681
|
||||
0.203199995458126059, 195.042764094891368
|
||||
0.211199995279312130, 202.937172130123315
|
||||
0.219199995100498229, 210.852580165355278
|
||||
0.227199994921684245, 218.789988200587175
|
||||
0.235199994742870316, 226.748396235819115
|
||||
0.243199994564056388, 234.740804271051104
|
||||
0.251199994385242487, 242.757212306283037
|
||||
0.259199994206428530, 250.797620341515000
|
||||
0.267199994027614574, 258.874028376746878
|
||||
0.275199993848800673, 266.977436411978829
|
||||
0.283199993669986716, 275.092844447210780
|
||||
0.291199993491172815, 283.248252482442751
|
||||
0.299199993312358858, 291.445660517674696
|
||||
0.307199993133544902, 299.655068552906641
|
||||
0.315199992954731001, 307.909476588138546
|
||||
0.323199992775917044, 316.192884623370503
|
||||
0.339199992418289187, 332.846700693834407
|
||||
0.355199992060661329, 349.620516764298316
|
||||
0.371199991703033416, 366.537332834762140
|
||||
0.387199991345405559, 383.604148905225998
|
||||
0.403199990987777701, 400.801964975689941
|
||||
0.419199990630149844, 418.158781046153820
|
||||
0.435199990272521986, 435.624597116617792
|
||||
0.451199989914894073, 453.288413187081574
|
||||
0.467199989557266215, 471.062229257545425
|
||||
0.483199989199638358, 489.048045328009380
|
||||
0.499199988842010500, 507.146861398473277
|
||||
0.515199988484382643, 525.392677468937222
|
||||
0.531199988126754730, 543.829493539401028
|
||||
0.531999988108873390, 544.796634342924222
|
||||
0.550079987704753859, 565.860416502548446
|
||||
0.567999987304210641, 586.865970501467928
|
||||
0.585599986910820047, 607.703068178978242
|
||||
0.603199986517429343, 628.645165856488575
|
||||
0.620639986127614951, 649.578035373294142
|
||||
0.637919985741376872, 670.496676729395176
|
||||
0.655199985355138792, 691.549318085496111
|
||||
0.672479984968900713, 712.766959441597237
|
||||
0.689599984586238834, 733.981372636993456
|
||||
0.706879984200000755, 755.521013993094471
|
||||
0.724159983813762675, 777.228655349195492
|
||||
0.741439983427524596, 799.117296705296553
|
||||
0.758719983041286516, 821.191938061397536
|
||||
0.775999982655048326, 843.460579417498366
|
||||
0.793439982265233934, 866.080448934304059
|
||||
0.810879981875419542, 888.907318451109631
|
||||
0.828479981482028949, 912.100416128620054
|
||||
0.846239981085061932, 935.664741966834868
|
||||
0.864159980684518825, 959.607295965754474
|
||||
0.882079980283975607, 983.787849964674024
|
||||
0.900159979879856187, 1008.36063212429838
|
||||
0.918399979472160344, 1033.33564244462718
|
||||
0.936639979064464612, 1058.56865276495591
|
||||
0.955199978649616255, 1084.36811940669395
|
||||
0.973919978231191585, 1110.59381420913678
|
||||
0.992799977809190715, 1137.25073717228429
|
||||
1.01183997738361353, 1164.35288829613614
|
||||
1.06047997629642499, 1234.41724915034661
|
||||
1.11039997518062594, 1307.72443529019392
|
||||
1.16191997402906422, 1384.74890303708753
|
||||
1.21535997283458719, 1466.00110871243692
|
||||
1.27071997159719463, 1551.73305231624181
|
||||
1.32847997030615828, 1642.68741833061654
|
||||
1.38895996895432461, 1739.56266307697001
|
||||
1.45263996753096580, 1843.32947103741640
|
||||
1.52031996601820008, 1955.19398301547858
|
||||
1.59295996439456933, 2077.14256797538474
|
||||
1.67167996263504026, 2211.44382304206692
|
||||
1.75855996069312082, 2361.74471430468566
|
||||
1.85647995850443825, 2533.64534865592441
|
||||
1.96959995597600934, 2734.93465827410455
|
||||
2.10543995293974895, 2980.03336671234320
|
||||
|
File diff suppressed because it is too large
Load Diff
File diff suppressed because it is too large
Load Diff
@@ -0,0 +1,67 @@
|
||||
// This function computes the autocorrelation function for
|
||||
// the standard c++ random number generator
|
||||
|
||||
#include <fstream>
|
||||
#include <iomanip>
|
||||
#include <iostream>
|
||||
#include <cmath>
|
||||
using namespace std;
|
||||
// output file as global variable
|
||||
ofstream ofile;
|
||||
|
||||
// Main function begins here
|
||||
int main(int argc, char* argv[])
|
||||
{
|
||||
int n;
|
||||
char *outfilename;
|
||||
|
||||
cin >> n;
|
||||
double MCint = 0.; double MCintsqr2=0.;
|
||||
double invers_period = 1./RAND_MAX; // initialise the random number generator
|
||||
srand(time(NULL)); // This produces the so-called seed in MC jargon
|
||||
// Compute the variance and the mean value of the uniform distribution
|
||||
// Compute also the specific values x for each cycle in order to be able to
|
||||
// the covariance and the correlation function
|
||||
// Read in output file, abort if there are too few command-line arguments
|
||||
if( argc <= 2 ){
|
||||
cout << "Bad Usage: " << argv[0] <<
|
||||
" read also output file and number of cycles on same line" << endl;
|
||||
exit(1);
|
||||
}
|
||||
else{
|
||||
outfilename=argv[1];
|
||||
}
|
||||
ofile.open(outfilename);
|
||||
// Get the number of Monte-Carlo samples
|
||||
n = atoi(argv[2]);
|
||||
double *X;
|
||||
X = new double[n];
|
||||
for (int i = 0; i < n; i++){
|
||||
double x = double(rand())*invers_period;
|
||||
X[i] = x;
|
||||
MCint += x;
|
||||
MCintsqr2 += x*x;
|
||||
}
|
||||
double Mean = MCint/((double) n );
|
||||
MCintsqr2 = MCintsqr2/((double) n );
|
||||
double STDev = sqrt(MCintsqr2-Mean*Mean);
|
||||
double Variance = MCintsqr2-Mean*Mean;
|
||||
// Write mean value and standard deviation
|
||||
cout << " Standard deviation= " << STDev << " Integral = " << Mean << endl;
|
||||
|
||||
// Now we compute the autocorrelation function, setting the distance d between two
|
||||
// to a most 1/4 of the total number of cycles
|
||||
double *autocor; autocor = new double[n];
|
||||
for (int j = 0; j < n; j++){
|
||||
double sum = 0.0;
|
||||
for (int k = 0; k < (n-j); k++){
|
||||
sum += (X[k]-Mean)*(X[k+j]-Mean);
|
||||
}
|
||||
autocor[j] = sum/Variance/((double) n );
|
||||
ofile << setiosflags(ios::showpoint | ios::uppercase);
|
||||
ofile << setw(15) << setprecision(8) << j;
|
||||
ofile << setw(15) << setprecision(8) << autocor[j] << endl;
|
||||
}
|
||||
ofile.close(); // close output file
|
||||
return 0;
|
||||
} // end of main program
|
||||
@@ -0,0 +1,76 @@
|
||||
// This function computes the autocorrelation function for
|
||||
// the standard c++ random number generator
|
||||
|
||||
#include <fstream>
|
||||
#include <iomanip>
|
||||
#include <iostream>
|
||||
#include <cmath>
|
||||
#include <random>
|
||||
|
||||
using namespace std;
|
||||
// output file as global variable
|
||||
ofstream ofile;
|
||||
|
||||
// Main function begins here
|
||||
int main(int argc, char* argv[])
|
||||
{
|
||||
int n;
|
||||
char *outfilename;
|
||||
|
||||
cin >> n;
|
||||
double MCint = 0.; double MCintsqr2=0.;
|
||||
// Initialize the seed and call the Mersienne algo
|
||||
std::random_device rd;
|
||||
std::mt19937_64 gen(rd());
|
||||
// Set up the uniform distribution for x \in [[0, 1]
|
||||
std::uniform_real_distribution<double> RandomNumberGenerator(0.0,1.0);
|
||||
// Compute the variance and the mean value of the uniform distribution
|
||||
// Compute also the specific values x for each cycle in order to be able to
|
||||
// the covariance and the correlation function
|
||||
// Read in output file, abort if there are too few command-line arguments
|
||||
if( argc <= 2 ){
|
||||
cout << "Bad Usage: " << argv[0] <<
|
||||
" read also output file and number of cycles on same line" << endl;
|
||||
exit(1);
|
||||
}
|
||||
else{
|
||||
outfilename=argv[1];
|
||||
}
|
||||
ofile.open(outfilename);
|
||||
// Get the number of Monte-Carlo samples
|
||||
n = atoi(argv[2]);
|
||||
double *X;
|
||||
X = new double[n];
|
||||
for (int i = 0; i < n; i++){
|
||||
double x = RandomNumberGenerator(gen);
|
||||
X[i] = x;
|
||||
MCint += x;
|
||||
MCintsqr2 += x*x;
|
||||
}
|
||||
double Mean = MCint/((double) n );
|
||||
MCintsqr2 = MCintsqr2/((double) n );
|
||||
double STDev = sqrt(MCintsqr2-Mean*Mean);
|
||||
double Variance = MCintsqr2-Mean*Mean;
|
||||
// Write mean value and standard deviation
|
||||
cout << " Standard deviation= " << STDev << " Integral = " << Mean << endl;
|
||||
|
||||
// Now we compute the autocorrelation function, setting the distance d
|
||||
double *autocor; autocor = new double[n];
|
||||
for (int j = 0; j < n; j++){
|
||||
double sum = 0.0;
|
||||
for (int k = 0; k < (n-j); k++){
|
||||
sum += (X[k]-Mean)*(X[k+j]-Mean);
|
||||
}
|
||||
autocor[j] = sum/Variance/((double) n );
|
||||
ofile << setiosflags(ios::showpoint | ios::uppercase);
|
||||
ofile << setw(15) << setprecision(8) << j;
|
||||
ofile << setw(15) << setprecision(8) << autocor[j] << endl;
|
||||
}
|
||||
ofile.close(); // close output file
|
||||
return 0;
|
||||
} // end of main program
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@@ -0,0 +1,16 @@
|
||||
import numpy as np
|
||||
from matplotlib import pyplot as plt
|
||||
# Load in data file
|
||||
data = np.loadtxt("autocor.dat")
|
||||
data1 = np.loadtxt("automersenne.dat")
|
||||
# Make arrays containing x-axis and binding energies as function of A
|
||||
x = data[:,0]
|
||||
corr = data[:,1]
|
||||
corr2 = data1[:,1]
|
||||
plt.plot(x, corr ,'ro', x, corr2, 'b')
|
||||
plt.axis([0,1000,-0.2, 1.1])
|
||||
plt.xlabel(r'$d$')
|
||||
plt.ylabel(r'$C_d$')
|
||||
plt.title(r'autocorrelation function for RNG')
|
||||
plt.savefig('autocorr.pdf')
|
||||
plt.show()
|
||||
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|
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|
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@@ -1963,7 +1963,7 @@
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\frac{\\partial \\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}} = (\\boldsymbol{A}^{-1})^T.\n",
|
||||
"\\frac{\\partial\\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}}=(\\boldsymbol{A}^{-1})^T.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
@@ -3359,66 +3359,6 @@
|
||||
"You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. \n",
|
||||
"Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.\n",
|
||||
"\n",
|
||||
"!bsol\n",
|
||||
"The code here is an example of where we define our own design matrix and fit parameters $\\beta$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import os\n",
|
||||
"import numpy as np\n",
|
||||
"import pandas as pd\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"from sklearn.model_selection import train_test_split\n",
|
||||
"\n",
|
||||
"def save_fig(fig_id):\n",
|
||||
" plt.savefig(image_path(fig_id) + \".png\", format='png')\n",
|
||||
"\n",
|
||||
"def R2(y_data, y_model):\n",
|
||||
" return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
|
||||
"def MSE(y_data,y_model):\n",
|
||||
" n = np.size(y_model)\n",
|
||||
" return np.sum((y_data-y_model)**2)/n\n",
|
||||
"\n",
|
||||
"x = np.random.rand(100)\n",
|
||||
"y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"# The design matrix now as function of a given polynomial\n",
|
||||
"X = np.zeros((len(x),3))\n",
|
||||
"X[:,0] = 1.0\n",
|
||||
"X[:,1] = x\n",
|
||||
"X[:,2] = x**2\n",
|
||||
"# We split the data in test and training data\n",
|
||||
"X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
|
||||
"# matrix inversion to find beta\n",
|
||||
"beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train\n",
|
||||
"print(beta)\n",
|
||||
"# and then make the prediction\n",
|
||||
"ytilde = X_train @ beta\n",
|
||||
"print(\"Training R2\")\n",
|
||||
"print(R2(y_train,ytilde))\n",
|
||||
"print(\"Training MSE\")\n",
|
||||
"print(MSE(y_train,ytilde))\n",
|
||||
"ypredict = X_test @ beta\n",
|
||||
"print(\"Test R2\")\n",
|
||||
"print(R2(y_test,ypredict))\n",
|
||||
"print(\"Test MSE\")\n",
|
||||
"print(MSE(y_test,ypredict))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"!esol\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
@@ -3545,9 +3485,7 @@
|
||||
"\n",
|
||||
"!bsubex\n",
|
||||
"Add now a model which allows you to make polynomials up to degree $15$. Perform a standard OLS fitting of the training data and compute the MSE and $R2$ for the training and test data and plot both test and training data MSE and $R2$ as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?\n",
|
||||
"!bsol\n",
|
||||
"Here you simply need to change the degree of the polynomial in the above code to $n=15$.\n",
|
||||
"!esol\n",
|
||||
"\n",
|
||||
"!esubex"
|
||||
]
|
||||
}
|
||||
|
||||
Binary file not shown.
@@ -781,13 +781,13 @@ example of the functionality of <strong>Scikit-Learn</strong>.</p>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The intercept alpha:
|
||||
[2.18954529]
|
||||
[1.98452685]
|
||||
Coefficient beta :
|
||||
[[4.57051369]]
|
||||
Mean squared error: 0.23
|
||||
Variance score: 0.88
|
||||
[[5.00109273]]
|
||||
Mean squared error: 0.19
|
||||
Variance score: 0.92
|
||||
Mean squared log error: 0.01
|
||||
Mean absolute error: 0.39
|
||||
Mean absolute error: 0.35
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter1_13_1.png" src="_images/chapter1_13_1.png" />
|
||||
@@ -888,7 +888,7 @@ a linear <span class="math notranslate nohighlight">\(x\)</span>-dependence we s
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="_images/chapter1_27_0.png" src="_images/chapter1_27_0.png" />
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.004999999999999997
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.004999999999999996
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -997,18 +997,6 @@ After having downloaded this file to our own computer, we are now ready to read
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
|
||||
<span class="ne">FileNotFoundError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||||
<span class="o"><</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">5</span><span class="o">-</span><span class="mi">3</span><span class="n">cd19a0768e1</span><span class="o">></span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">31</span> <span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">32</span>
|
||||
<span class="ne">---> </span><span class="mi">33</span> <span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">"MassEval2016.dat"</span><span class="p">),</span><span class="s1">'r'</span><span class="p">)</span>
|
||||
|
||||
<span class="ne">FileNotFoundError</span>: [Errno 2] No such file or directory: 'DataFiles/MassEval2016.dat'
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various <strong>matplotlib</strong> commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -1049,6 +1037,11 @@ data) to actually open the file and simply take a look at it!</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>' \nThis is taken from the data file of the mass 2016 evaluation. \nAll files are 3436 lines long with 124 character per line. \n Headers are 39 lines long. \n col 1 : Fortran character control: 1 = page feed 0 = line feed \n format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 \n These formats are reflected in the pandas widths variable below, see the statement \n widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), \n Pandas has also a variable header, with length 39 in this case. \n'
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>The data we are interested in are in columns 2, 3, 4 and 11, giving us
|
||||
the number of neutrons, protons, mass numbers and binding energies,
|
||||
@@ -1098,6 +1091,25 @@ the number of nucleons <span class="math notranslate nohighlight">\(A\)</span>,
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> N Z A Element Ebinding
|
||||
A
|
||||
1 0 0 1 1 H 0.000000
|
||||
2 1 1 1 2 H 1.112283
|
||||
3 2 2 1 3 H 2.827265
|
||||
4 6 2 2 4 He 7.073915
|
||||
5 9 3 2 5 He 5.512132
|
||||
... ... ... ... ... ...
|
||||
264 3304 156 108 264 Hs 7.298375
|
||||
265 3310 157 108 265 Hs 7.296247
|
||||
266 3317 158 108 266 Hs 7.298273
|
||||
269 3338 159 110 269 Ds 7.250154
|
||||
270 3344 160 110 270 Ds 7.253775
|
||||
|
||||
[267 rows x 5 columns]
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>The next step, and we will define this mathematically later, is to set up the so-called <strong>design matrix</strong>. We will throughout call this matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.
|
||||
It has dimensionality <span class="math notranslate nohighlight">\(p\times n\)</span>, where <span class="math notranslate nohighlight">\(n\)</span> is the number of data points and <span class="math notranslate nohighlight">\(p\)</span> are the so-called predictors. In our case here they are given by the number of polynomials in <span class="math notranslate nohighlight">\(A\)</span> we wish to include in the fit.</p>
|
||||
@@ -1150,6 +1162,16 @@ Now we can print measures of how our fit is doing, the coefficients from the fit
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Mean squared error: 0.04
|
||||
Variance score: 0.95
|
||||
Mean absolute error: 0.05
|
||||
[ 0.00000000e+00 7.06492086e-03 -1.73091052e-01 -1.66020213e+01
|
||||
1.17385778e+00] 15.212327334149492
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter1_57_1.png" src="_images/chapter1_57_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>As a teaser, let us now see how we can do this with decision trees using <strong>scikit-learn</strong>. Later we will switch to so-called <strong>random forests</strong>!</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -1186,6 +1208,27 @@ Now we can print measures of how our fit is doing, the coefficients from the fit
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="_images/chapter1_59_0.png" src="_images/chapter1_59_0.png" />
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> N Z A Element Ebinding Eapprox
|
||||
A
|
||||
1 0 0 1 1 H 0.000000 0.000000
|
||||
2 1 1 1 2 H 1.112283 1.112283
|
||||
3 2 2 1 3 H 2.827265 2.827265
|
||||
4 6 2 2 4 He 7.073915 7.073915
|
||||
5 9 3 2 5 He 5.512132 5.512132
|
||||
... ... ... ... ... ... ...
|
||||
264 3304 156 108 264 Hs 7.298375 7.298375
|
||||
265 3310 157 108 265 Hs 7.296247 7.297260
|
||||
266 3317 158 108 266 Hs 7.298273 7.297260
|
||||
269 3338 159 110 269 Ds 7.250154 7.250154
|
||||
270 3344 160 110 270 Ds 7.253775 7.253775
|
||||
|
||||
[267 rows x 6 columns]
|
||||
0.009883615646716184
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>The <strong>seaborn</strong> package allows us to visualize data in an efficient way. Note that we use <strong>scikit-learn</strong>’s multi-layer perceptron (or feed forward neural network)
|
||||
functionality.</p>
|
||||
@@ -1223,6 +1266,91 @@ functionality.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
|
||||
warnings.warn(
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter1_61_9.png" src="_images/chapter1_61_9.png" />
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1459,6 +1587,133 @@ our matrix as <span class="math notranslate nohighlight">\(\boldsymbol{X}\in {\m
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_html"><div>
|
||||
<style scoped>
|
||||
.dataframe tbody tr th:only-of-type {
|
||||
vertical-align: middle;
|
||||
}
|
||||
|
||||
.dataframe tbody tr th {
|
||||
vertical-align: top;
|
||||
}
|
||||
|
||||
.dataframe thead th {
|
||||
text-align: right;
|
||||
}
|
||||
</style>
|
||||
<table border="1" class="dataframe">
|
||||
<thead>
|
||||
<tr style="text-align: right;">
|
||||
<th></th>
|
||||
<th>1</th>
|
||||
<th>A</th>
|
||||
<th>A^(2/3)</th>
|
||||
<th>A^(-1/3)</th>
|
||||
<th>1/A</th>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>A</th>
|
||||
<th></th>
|
||||
<th></th>
|
||||
<th></th>
|
||||
<th></th>
|
||||
<th></th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<th>1</th>
|
||||
<td>1.0</td>
|
||||
<td>1.0</td>
|
||||
<td>1.000000</td>
|
||||
<td>1.000000</td>
|
||||
<td>1.000000</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>2</th>
|
||||
<td>1.0</td>
|
||||
<td>2.0</td>
|
||||
<td>1.587401</td>
|
||||
<td>0.793701</td>
|
||||
<td>0.500000</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>3</th>
|
||||
<td>1.0</td>
|
||||
<td>3.0</td>
|
||||
<td>2.080084</td>
|
||||
<td>0.693361</td>
|
||||
<td>0.333333</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>4</th>
|
||||
<td>1.0</td>
|
||||
<td>4.0</td>
|
||||
<td>2.519842</td>
|
||||
<td>0.629961</td>
|
||||
<td>0.250000</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>5</th>
|
||||
<td>1.0</td>
|
||||
<td>5.0</td>
|
||||
<td>2.924018</td>
|
||||
<td>0.584804</td>
|
||||
<td>0.200000</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>...</th>
|
||||
<td>...</td>
|
||||
<td>...</td>
|
||||
<td>...</td>
|
||||
<td>...</td>
|
||||
<td>...</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>264</th>
|
||||
<td>1.0</td>
|
||||
<td>264.0</td>
|
||||
<td>41.153106</td>
|
||||
<td>0.155883</td>
|
||||
<td>0.003788</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>265</th>
|
||||
<td>1.0</td>
|
||||
<td>265.0</td>
|
||||
<td>41.256962</td>
|
||||
<td>0.155687</td>
|
||||
<td>0.003774</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>266</th>
|
||||
<td>1.0</td>
|
||||
<td>266.0</td>
|
||||
<td>41.360688</td>
|
||||
<td>0.155491</td>
|
||||
<td>0.003759</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>269</th>
|
||||
<td>1.0</td>
|
||||
<td>269.0</td>
|
||||
<td>41.671089</td>
|
||||
<td>0.154911</td>
|
||||
<td>0.003717</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>270</th>
|
||||
<td>1.0</td>
|
||||
<td>270.0</td>
|
||||
<td>41.774300</td>
|
||||
<td>0.154720</td>
|
||||
<td>0.003704</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
<p>267 rows × 5 columns</p>
|
||||
</div></div></div>
|
||||
</div>
|
||||
<p>With <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1}\)</span>, it means that we will hereafter write our equations for the approximation as</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
@@ -1610,7 +1865,7 @@ C
|
||||
K</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T.
|
||||
\frac{\partial\log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}}=(\boldsymbol{A}^{-1})^T.
|
||||
\]</div>
|
||||
<p>The residuals <span class="math notranslate nohighlight">\(\boldsymbol{\epsilon}\)</span> are in turn given by</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
@@ -1668,6 +1923,9 @@ write</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="_images/chapter1_132_0.png" src="_images/chapter1_132_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>We can easily test our fit by computing the <span class="math notranslate nohighlight">\(R2\)</span> score that we discussed in connection with the functionality of <strong>Scikit-Learn</strong> in the introductory slides.
|
||||
Since we are not using <strong>Scikit-Learn</strong> here we can define our own <span class="math notranslate nohighlight">\(R2\)</span> function as</p>
|
||||
@@ -1686,6 +1944,11 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9547578478889096
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>We can easily add our <strong>MSE</strong> score as</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -1698,6 +1961,11 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.03787596148305236
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>and finally the relative error as</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -1708,6 +1976,23 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>A
|
||||
1 0 inf
|
||||
2 1 1.123190
|
||||
3 2 0.327631
|
||||
4 6 0.344172
|
||||
5 9 0.044402
|
||||
...
|
||||
264 3304 0.009911
|
||||
265 3310 0.009154
|
||||
266 3317 0.007824
|
||||
269 3338 0.011347
|
||||
270 3344 0.009790
|
||||
Name: Ebinding, Length: 267, dtype: float64
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="the-chi-2-function">
|
||||
<h3><span class="section-number">3.4.1. </span>The <span class="math notranslate nohighlight">\(\chi^2\)</span> function<a class="headerlink" href="#the-chi-2-function" title="Permalink to this headline">¶</a></h3>
|
||||
@@ -1935,6 +2220,19 @@ hyperparameter <span class="math notranslate nohighlight">\(\lambda\)</span>, al
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Mean squared error: 12.36
|
||||
Variance score: 1.00
|
||||
Mean absolute error: 2.83
|
||||
[ 0. 618.32047562 -861.13519106 1404.91549644] -11.057088709963637
|
||||
Mean squared error: 197.93
|
||||
Variance score: 1.00
|
||||
Mean absolute error: 11.69
|
||||
[ 0. 28.18220995 282.79902342 842.30879705] 12.946893955211749
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter1_179_1.png" src="_images/chapter1_179_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>The above simple polynomial in density <span class="math notranslate nohighlight">\(\rho\)</span> gives an excellent fit
|
||||
to the data.</p>
|
||||
@@ -2024,6 +2322,18 @@ but now splitting the data into a training set and a test set.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Training R2
|
||||
0.9999864543345858
|
||||
Training MSE
|
||||
6.180092462880674
|
||||
Test R2
|
||||
0.9999822527140678
|
||||
Test MSE
|
||||
7.205466494327873
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="the-boston-housing-data-example">
|
||||
@@ -2078,6 +2388,11 @@ the house using the features (predictors) listed here.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>dict_keys(['data', 'target', 'feature_names', 'DESCR', 'filename'])
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>Then we invoke Pandas</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -2097,6 +2412,25 @@ the house using the features (predictors) listed here.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>CRIM 0
|
||||
ZN 0
|
||||
INDUS 0
|
||||
CHAS 0
|
||||
NOX 0
|
||||
RM 0
|
||||
AGE 0
|
||||
DIS 0
|
||||
RAD 0
|
||||
TAX 0
|
||||
PTRATIO 0
|
||||
B 0
|
||||
LSTAT 0
|
||||
MEDV 0
|
||||
dtype: int64
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>We can then visualize the data</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -2110,6 +2444,13 @@ the house using the features (predictors) listed here.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/seaborn/distributions.py:2551: FutureWarning: `distplot` is a deprecated function and will be removed in a future version. Please adapt your code to use either `displot` (a figure-level function with similar flexibility) or `histplot` (an axes-level function for histograms).
|
||||
warnings.warn(msg, FutureWarning)
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter1_191_1.png" src="_images/chapter1_191_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>It is now useful to look at the correlation matrix</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -2122,6 +2463,12 @@ the house using the features (predictors) listed here.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><AxesSubplot:>
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter1_193_1.png" src="_images/chapter1_193_1.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>From the above coorelation plot we can see that <strong>MEDV</strong> is strongly correlated to <strong>LSTAT</strong> and <strong>RM</strong>. We see also that <strong>RAD</strong> and <strong>TAX</strong> are stronly correlated, but we don’t include this in our features together to avoid multi-colinearity</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -2142,6 +2489,9 @@ the house using the features (predictors) listed here.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="_images/chapter1_195_0.png" src="_images/chapter1_195_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>Now we start training our model</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -2167,6 +2517,14 @@ the house using the features (predictors) listed here.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(404, 2)
|
||||
(102, 2)
|
||||
(404,)
|
||||
(102,)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>Then we use the linear regression functionality from <strong>Scikit-Learn</strong></p>
|
||||
<div class="cell docutils container">
|
||||
@@ -2205,6 +2563,20 @@ the house using the features (predictors) listed here.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The model performance for training set
|
||||
--------------------------------------
|
||||
RMSE is 5.637129335071195
|
||||
R2 score is 0.6300745149331701
|
||||
|
||||
|
||||
The model performance for testing set
|
||||
--------------------------------------
|
||||
RMSE is 5.137400784702912
|
||||
R2 score is 0.6628996975186952
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
@@ -2215,6 +2587,9 @@ the house using the features (predictors) listed here.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<img alt="_images/chapter1_202_0.png" src="_images/chapter1_202_0.png" />
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="section" id="reducing-the-number-of-degrees-of-freedom-overarching-view">
|
||||
@@ -2364,6 +2739,36 @@ techniques.</p>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>MSE before scaling: 0.00
|
||||
R2 score before scaling 0.99
|
||||
Feature min values before scaling:
|
||||
[1.00000000e+00 1.10094646e-03 9.51523276e-04 1.21208310e-06
|
||||
1.04757618e-06 9.05396545e-07 1.33443859e-09 1.15332528e-09
|
||||
9.96793117e-10 8.61505887e-10 1.46914544e-12 1.26974938e-12
|
||||
1.09741585e-12 9.48471852e-13 8.19742904e-13 1.61745046e-15
|
||||
1.39792608e-15 1.20819609e-15 1.04421672e-15 9.02493044e-16
|
||||
7.80004454e-16]
|
||||
Feature max values before scaling:
|
||||
[1. 0.99825997 0.99883879 0.99652296 0.99710078 0.99767893
|
||||
0.99478898 0.9953658 0.99594294 0.99652042 0.99305802 0.99363383
|
||||
0.99420997 0.99478645 0.99536326 0.99133007 0.99190487 0.99248001
|
||||
0.99305549 0.99363129 0.99420743]
|
||||
Feature min values after scaling:
|
||||
[ 0. -1.61869821 -1.66880047 -1.06209126 -1.08170444 -1.10131725
|
||||
-0.84087101 -0.85396354 -0.86692943 -0.87972591 -0.71351486 -0.7240496
|
||||
-0.73453972 -0.74495014 -0.75524378 -0.62783293 -0.63680118 -0.64580686
|
||||
-0.65482578 -0.66383151 -0.67279536]
|
||||
Feature max values after scaling:
|
||||
[0. 1.78944806 1.68342382 2.34172919 2.25457052 2.16553696
|
||||
2.7899453 2.71281409 2.63374631 2.55280484 3.17117385 3.10307631
|
||||
3.03303359 2.96104648 2.88712946 3.50394742 3.44395541 3.38216436
|
||||
3.31853484 3.25303483 3.1856411 ]
|
||||
MSE after scaling: 0.00
|
||||
R2 score for scaled data: 0.99
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -2460,55 +2865,6 @@ R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)
|
||||
\]</div>
|
||||
<p>You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions.
|
||||
Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.</p>
|
||||
<p>!bsol
|
||||
The code here is an example of where we define our own design matrix and fit parameters <span class="math notranslate nohighlight">\(\beta\)</span>.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
|
||||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">R2</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span> <span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">y_model</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_data</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
|
||||
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">+</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mf">0.1</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
|
||||
|
||||
|
||||
<span class="c1"># The design matrix now as function of a given polynomial</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="mi">3</span><span class="p">))</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
|
||||
<span class="c1"># We split the data in test and training data</span>
|
||||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
|
||||
<span class="c1"># matrix inversion to find beta</span>
|
||||
<span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X_train</span><span class="p">)</span> <span class="o">@</span> <span class="n">X_train</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y_train</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
|
||||
<span class="c1"># and then make the prediction</span>
|
||||
<span class="n">ytilde</span> <span class="o">=</span> <span class="n">X_train</span> <span class="o">@</span> <span class="n">beta</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Training R2"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">R2</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span><span class="n">ytilde</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Training MSE"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span><span class="n">ytilde</span><span class="p">))</span>
|
||||
<span class="n">ypredict</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">@</span> <span class="n">beta</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test R2"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">R2</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredict</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test MSE"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredict</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>!esol</p>
|
||||
</div>
|
||||
<div class="section" id="exercise-normalizing-our-data">
|
||||
<h3><span class="section-number">3.9.3. </span>Exercise: Normalizing our data<a class="headerlink" href="#exercise-normalizing-our-data" title="Permalink to this headline">¶</a></h3>
|
||||
@@ -2549,6 +2905,38 @@ for testing. This can be done as follows with our design matrix <span class="mat
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
|
||||
<span class="ne">ValueError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||||
<span class="o"><</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">38</span><span class="o">-</span><span class="mi">9</span><span class="n">b9cf4fa1a95</span><span class="o">></span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">1</span> <span class="c1"># split in training and test data</span>
|
||||
<span class="ne">----> </span><span class="mi">2</span> <span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
|
||||
|
||||
<span class="nn">~/opt/anaconda3/lib/python3.8/site-packages/sklearn/model_selection/_split.py</span> in <span class="ni">train_test_split</span><span class="nt">(*arrays, **options)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">2125</span> <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s2">"Invalid parameters passed: </span><span class="si">%s</span><span class="s2">"</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">options</span><span class="p">))</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">2126</span>
|
||||
<span class="ne">-> </span><span class="mi">2127</span> <span class="n">arrays</span> <span class="o">=</span> <span class="n">indexable</span><span class="p">(</span><span class="o">*</span><span class="n">arrays</span><span class="p">)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">2128</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">2129</span> <span class="n">n_samples</span> <span class="o">=</span> <span class="n">_num_samples</span><span class="p">(</span><span class="n">arrays</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||||
|
||||
<span class="nn">~/opt/anaconda3/lib/python3.8/site-packages/sklearn/utils/validation.py</span> in <span class="ni">indexable</span><span class="nt">(*iterables)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">290</span> <span class="s2">"""</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">291</span><span class="s2"> result = [_make_indexable(X) for X in iterables]</span>
|
||||
<span class="ne">--> </span><span class="mi">292</span><span class="s2"> check_consistent_length(*result)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">293</span><span class="s2"> return result</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">294</span><span class="s2"> </span>
|
||||
|
||||
<span class="nn">~/opt/anaconda3/lib/python3.8/site-packages/sklearn/utils/validation.py</span> in <span class="ni">check_consistent_length</span><span class="nt">(*arrays)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">253</span><span class="s2"> uniques = np.unique(lengths)</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">254</span><span class="s2"> if len(uniques) > 1:</span>
|
||||
<span class="ne">--> </span><span class="mi">255</span><span class="s2"> raise ValueError("Found input variables with inconsistent numbers of"</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">256</span><span class="s2"> " samples: </span><span class="si">%r</span><span class="s2">" % [int(l) for l in lengths])</span>
|
||||
<span class="g g-Whitespace"> </span><span class="mi">257</span><span class="s2"> </span>
|
||||
|
||||
<span class="ne">ValueError</span>: Found input variables with inconsistent numbers of samples: [1000, 100]
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>Then we can use the standard scaler to scale our data as</p>
|
||||
<div class="cell docutils container">
|
||||
@@ -2587,11 +2975,8 @@ Write a first code which sets up a design matrix <span class="math notranslate n
|
||||
Perform an ordinary least squares and compute the means squared error and the <span class="math notranslate nohighlight">\(R2\)</span> factor for the training data and the test data, with and without scaling.
|
||||
!esubex</p>
|
||||
<p>!bsubex
|
||||
Add now a model which allows you to make polynomials up to degree <span class="math notranslate nohighlight">\(15\)</span>. Perform a standard OLS fitting of the training data and compute the MSE and <span class="math notranslate nohighlight">\(R2\)</span> for the training and test data and plot both test and training data MSE and <span class="math notranslate nohighlight">\(R2\)</span> as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
|
||||
!bsol
|
||||
Here you simply need to change the degree of the polynomial in the above code to <span class="math notranslate nohighlight">\(n=15\)</span>.
|
||||
!esol
|
||||
!esubex</p>
|
||||
Add now a model which allows you to make polynomials up to degree <span class="math notranslate nohighlight">\(15\)</span>. Perform a standard OLS fitting of the training data and compute the MSE and <span class="math notranslate nohighlight">\(R2\)</span> for the training and test data and plot both test and training data MSE and <span class="math notranslate nohighlight">\(R2\)</span> as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?</p>
|
||||
<p>!esubex</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
@@ -0,0 +1,53 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
# split in training and test data
|
||||
X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.2)
|
||||
------------------
|
||||
|
||||
[0;31m---------------------------------------------------------------------------[0m
|
||||
[0;31mValueError[0m Traceback (most recent call last)
|
||||
[0;32m<ipython-input-38-9b9cf4fa1a95>[0m in [0;36m<module>[0;34m[0m
|
||||
[1;32m 1[0m [0;31m# split in training and test data[0m[0;34m[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m----> 2[0;31m [0mX_train[0m[0;34m,[0m [0mX_test[0m[0;34m,[0m [0my_train[0m[0;34m,[0m [0my_test[0m [0;34m=[0m [0mtrain_test_split[0m[0;34m([0m[0mX[0m[0;34m,[0m[0my[0m[0;34m,[0m[0mtest_size[0m[0;34m=[0m[0;36m0.2[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m
|
||||
[0;32m~/opt/anaconda3/lib/python3.8/site-packages/sklearn/model_selection/_split.py[0m in [0;36mtrain_test_split[0;34m(*arrays, **options)[0m
|
||||
[1;32m 2125[0m [0;32mraise[0m [0mTypeError[0m[0;34m([0m[0;34m"Invalid parameters passed: %s"[0m [0;34m%[0m [0mstr[0m[0;34m([0m[0moptions[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 2126[0m [0;34m[0m[0m
|
||||
[0;32m-> 2127[0;31m [0marrays[0m [0;34m=[0m [0mindexable[0m[0;34m([0m[0;34m*[0m[0marrays[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m[1;32m 2128[0m [0;34m[0m[0m
|
||||
[1;32m 2129[0m [0mn_samples[0m [0;34m=[0m [0m_num_samples[0m[0;34m([0m[0marrays[0m[0;34m[[0m[0;36m0[0m[0;34m][0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;32m~/opt/anaconda3/lib/python3.8/site-packages/sklearn/utils/validation.py[0m in [0;36mindexable[0;34m(*iterables)[0m
|
||||
[1;32m 290[0m """
|
||||
[1;32m 291[0m [0mresult[0m [0;34m=[0m [0;34m[[0m[0m_make_indexable[0m[0;34m([0m[0mX[0m[0;34m)[0m [0;32mfor[0m [0mX[0m [0;32min[0m [0miterables[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m--> 292[0;31m [0mcheck_consistent_length[0m[0;34m([0m[0;34m*[0m[0mresult[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m[1;32m 293[0m [0;32mreturn[0m [0mresult[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 294[0m [0;34m[0m[0m
|
||||
|
||||
[0;32m~/opt/anaconda3/lib/python3.8/site-packages/sklearn/utils/validation.py[0m in [0;36mcheck_consistent_length[0;34m(*arrays)[0m
|
||||
[1;32m 253[0m [0muniques[0m [0;34m=[0m [0mnp[0m[0;34m.[0m[0munique[0m[0;34m([0m[0mlengths[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 254[0m [0;32mif[0m [0mlen[0m[0;34m([0m[0muniques[0m[0;34m)[0m [0;34m>[0m [0;36m1[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m--> 255[0;31m raise ValueError("Found input variables with inconsistent numbers of"
|
||||
[0m[1;32m 256[0m " samples: %r" % [int(l) for l in lengths])
|
||||
[1;32m 257[0m [0;34m[0m[0m
|
||||
|
||||
[0;31mValueError[0m: Found input variables with inconsistent numbers of samples: [1000, 100]
|
||||
ValueError: Found input variables with inconsistent numbers of samples: [1000, 100]
|
||||
|
||||
@@ -0,0 +1,29 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
pip3 install tensorflow
|
||||
------------------
|
||||
|
||||
[0;36m File [0;32m"<ipython-input-12-6ea927cc6e88>"[0;36m, line [0;32m1[0m
|
||||
[0;31m pip3 install tensorflow[0m
|
||||
[0m ^[0m
|
||||
[0;31mSyntaxError[0m[0;31m:[0m invalid syntax
|
||||
|
||||
SyntaxError: invalid syntax (<ipython-input-12-6ea927cc6e88>, line 1)
|
||||
|
||||
@@ -0,0 +1,104 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
%matplotlib inline
|
||||
|
||||
from numpy import *
|
||||
from numpy.random import randint, randn
|
||||
from time import time
|
||||
import matplotlib.mlab as mlab
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
# Returns mean of bootstrap samples
|
||||
def stat(data):
|
||||
return mean(data)
|
||||
|
||||
# Bootstrap algorithm
|
||||
def bootstrap(data, statistic, R):
|
||||
t = zeros(R); n = len(data); inds = arange(n); t0 = time()
|
||||
# non-parametric bootstrap
|
||||
for i in range(R):
|
||||
t[i] = statistic(data[randint(0,n,n)])
|
||||
|
||||
# analysis
|
||||
print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
|
||||
print("original bias std. error")
|
||||
print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
|
||||
return t
|
||||
|
||||
|
||||
mu, sigma = 100, 15
|
||||
datapoints = 10000
|
||||
x = mu + sigma*random.randn(datapoints)
|
||||
# bootstrap returns the data sample
|
||||
t = bootstrap(x, stat, datapoints)
|
||||
# the histogram of the bootstrapped data
|
||||
n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
|
||||
|
||||
# add a 'best fit' line
|
||||
y = mlab.normpdf( binsboot, mean(t), std(t))
|
||||
lt = plt.plot(binsboot, y, 'r--', linewidth=1)
|
||||
plt.xlabel('Smarts')
|
||||
plt.ylabel('Probability')
|
||||
plt.axis([99.5, 100.6, 0, 3.0])
|
||||
plt.grid(True)
|
||||
|
||||
plt.show()
|
||||
------------------
|
||||
|
||||
[0;31m---------------------------------------------------------------------------[0m
|
||||
[0;31mAttributeError[0m Traceback (most recent call last)
|
||||
[0;32m<ipython-input-2-772b904ae9cb>[0m in [0;36m<module>[0;34m[0m
|
||||
[1;32m 31[0m [0mt[0m [0;34m=[0m [0mbootstrap[0m[0;34m([0m[0mx[0m[0;34m,[0m [0mstat[0m[0;34m,[0m [0mdatapoints[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 32[0m [0;31m# the histogram of the bootstrapped data[0m[0;34m[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m---> 33[0;31m [0mn[0m[0;34m,[0m [0mbinsboot[0m[0;34m,[0m [0mpatches[0m [0;34m=[0m [0mplt[0m[0;34m.[0m[0mhist[0m[0;34m([0m[0mt[0m[0;34m,[0m [0;36m50[0m[0;34m,[0m [0mnormed[0m[0;34m=[0m[0;36m1[0m[0;34m,[0m [0mfacecolor[0m[0;34m=[0m[0;34m'red'[0m[0;34m,[0m [0malpha[0m[0;34m=[0m[0;36m0.75[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m[1;32m 34[0m [0;34m[0m[0m
|
||||
[1;32m 35[0m [0;31m# add a 'best fit' line[0m[0;34m[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;32m~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/pyplot.py[0m in [0;36mhist[0;34m(x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, data, **kwargs)[0m
|
||||
[1;32m 2683[0m [0morientation[0m[0;34m=[0m[0;34m'vertical'[0m[0;34m,[0m [0mrwidth[0m[0;34m=[0m[0;32mNone[0m[0;34m,[0m [0mlog[0m[0;34m=[0m[0;32mFalse[0m[0;34m,[0m [0mcolor[0m[0;34m=[0m[0;32mNone[0m[0;34m,[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 2684[0m label=None, stacked=False, *, data=None, **kwargs):
|
||||
[0;32m-> 2685[0;31m return gca().hist(
|
||||
[0m[1;32m 2686[0m [0mx[0m[0;34m,[0m [0mbins[0m[0;34m=[0m[0mbins[0m[0;34m,[0m [0mrange[0m[0;34m=[0m[0mrange[0m[0;34m,[0m [0mdensity[0m[0;34m=[0m[0mdensity[0m[0;34m,[0m [0mweights[0m[0;34m=[0m[0mweights[0m[0;34m,[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 2687[0m [0mcumulative[0m[0;34m=[0m[0mcumulative[0m[0;34m,[0m [0mbottom[0m[0;34m=[0m[0mbottom[0m[0;34m,[0m [0mhisttype[0m[0;34m=[0m[0mhisttype[0m[0;34m,[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;32m~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/__init__.py[0m in [0;36minner[0;34m(ax, data, *args, **kwargs)[0m
|
||||
[1;32m 1445[0m [0;32mdef[0m [0minner[0m[0;34m([0m[0max[0m[0;34m,[0m [0;34m*[0m[0margs[0m[0;34m,[0m [0mdata[0m[0;34m=[0m[0;32mNone[0m[0;34m,[0m [0;34m**[0m[0mkwargs[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 1446[0m [0;32mif[0m [0mdata[0m [0;32mis[0m [0;32mNone[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m-> 1447[0;31m [0;32mreturn[0m [0mfunc[0m[0;34m([0m[0max[0m[0;34m,[0m [0;34m*[0m[0mmap[0m[0;34m([0m[0msanitize_sequence[0m[0;34m,[0m [0margs[0m[0;34m)[0m[0;34m,[0m [0;34m**[0m[0mkwargs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m[1;32m 1448[0m [0;34m[0m[0m
|
||||
[1;32m 1449[0m [0mbound[0m [0;34m=[0m [0mnew_sig[0m[0;34m.[0m[0mbind[0m[0;34m([0m[0max[0m[0;34m,[0m [0;34m*[0m[0margs[0m[0;34m,[0m [0;34m**[0m[0mkwargs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;32m~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/axes/_axes.py[0m in [0;36mhist[0;34m(self, x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, **kwargs)[0m
|
||||
[1;32m 6813[0m [0;32mif[0m [0mpatch[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 6814[0m [0mp[0m [0;34m=[0m [0mpatch[0m[0;34m[[0m[0;36m0[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m-> 6815[0;31m [0mp[0m[0;34m.[0m[0mupdate[0m[0;34m([0m[0mkwargs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m[1;32m 6816[0m [0;32mif[0m [0mlbl[0m [0;32mis[0m [0;32mnot[0m [0;32mNone[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 6817[0m [0mp[0m[0;34m.[0m[0mset_label[0m[0;34m([0m[0mlbl[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;32m~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/artist.py[0m in [0;36mupdate[0;34m(self, props)[0m
|
||||
[1;32m 994[0m [0mfunc[0m [0;34m=[0m [0mgetattr[0m[0;34m([0m[0mself[0m[0;34m,[0m [0;34mf"set_{k}"[0m[0;34m,[0m [0;32mNone[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 995[0m [0;32mif[0m [0;32mnot[0m [0mcallable[0m[0;34m([0m[0mfunc[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m--> 996[0;31m raise AttributeError(f"{type(self).__name__!r} object "
|
||||
[0m[1;32m 997[0m f"has no property {k!r}")
|
||||
[1;32m 998[0m [0mret[0m[0;34m.[0m[0mappend[0m[0;34m([0m[0mfunc[0m[0;34m([0m[0mv[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;31mAttributeError[0m: 'Rectangle' object has no property 'normed'
|
||||
AttributeError: 'Rectangle' object has no property 'normed'
|
||||
|
||||
@@ -0,0 +1,89 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
%matplotlib inline
|
||||
|
||||
# Common imports
|
||||
import os
|
||||
import numpy as np
|
||||
import pandas as pd
|
||||
import matplotlib.pyplot as plt
|
||||
from sklearn.linear_model import LinearRegression, Ridge, Lasso
|
||||
from sklearn.model_selection import train_test_split
|
||||
from sklearn.utils import resample
|
||||
from sklearn.metrics import mean_squared_error
|
||||
from IPython.display import display
|
||||
from pylab import plt, mpl
|
||||
plt.style.use('seaborn')
|
||||
mpl.rcParams['font.family'] = 'serif'
|
||||
|
||||
# Where to save the figures and data files
|
||||
PROJECT_ROOT_DIR = "Results"
|
||||
FIGURE_ID = "Results/FigureFiles"
|
||||
DATA_ID = "DataFiles/"
|
||||
|
||||
if not os.path.exists(PROJECT_ROOT_DIR):
|
||||
os.mkdir(PROJECT_ROOT_DIR)
|
||||
|
||||
if not os.path.exists(FIGURE_ID):
|
||||
os.makedirs(FIGURE_ID)
|
||||
|
||||
if not os.path.exists(DATA_ID):
|
||||
os.makedirs(DATA_ID)
|
||||
|
||||
def image_path(fig_id):
|
||||
return os.path.join(FIGURE_ID, fig_id)
|
||||
|
||||
def data_path(dat_id):
|
||||
return os.path.join(DATA_ID, dat_id)
|
||||
|
||||
def save_fig(fig_id):
|
||||
plt.savefig(image_path(fig_id) + ".png", format='png')
|
||||
|
||||
infile = open(data_path("chddata.csv"),'r')
|
||||
|
||||
# Read the chd data as csv file and organize the data into arrays with age group, age, and chd
|
||||
chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
|
||||
chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
|
||||
output = chd['CHD']
|
||||
age = chd['Age']
|
||||
agegroup = chd['Agegroup']
|
||||
numberID = chd['ID']
|
||||
display(chd)
|
||||
|
||||
plt.scatter(age, output, marker='o')
|
||||
plt.axis([18,70.0,-0.1, 1.2])
|
||||
plt.xlabel(r'Age')
|
||||
plt.ylabel(r'CHD')
|
||||
plt.title(r'Age distribution and Coronary heart disease')
|
||||
plt.show()
|
||||
------------------
|
||||
|
||||
[0;31m---------------------------------------------------------------------------[0m
|
||||
[0;31mFileNotFoundError[0m Traceback (most recent call last)
|
||||
[0;32m<ipython-input-1-a77d5ac269b2>[0m in [0;36m<module>[0;34m[0m
|
||||
[1;32m 38[0m [0mplt[0m[0;34m.[0m[0msavefig[0m[0;34m([0m[0mimage_path[0m[0;34m([0m[0mfig_id[0m[0;34m)[0m [0;34m+[0m [0;34m".png"[0m[0;34m,[0m [0mformat[0m[0;34m=[0m[0;34m'png'[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 39[0m [0;34m[0m[0m
|
||||
[0;32m---> 40[0;31m [0minfile[0m [0;34m=[0m [0mopen[0m[0;34m([0m[0mdata_path[0m[0;34m([0m[0;34m"chddata.csv"[0m[0;34m)[0m[0;34m,[0m[0;34m'r'[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m[1;32m 41[0m [0;34m[0m[0m
|
||||
[1;32m 42[0m [0;31m# Read the chd data as csv file and organize the data into arrays with age group, age, and chd[0m[0;34m[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;31mFileNotFoundError[0m: [Errno 2] No such file or directory: 'DataFiles/chddata.csv'
|
||||
FileNotFoundError: [Errno 2] No such file or directory: 'DataFiles/chddata.csv'
|
||||
|
||||
@@ -0,0 +1,41 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
# Import the necessary packages
|
||||
import numpy
|
||||
from cvxopt import matrix
|
||||
from cvxopt import solvers
|
||||
P = matrix(numpy.diag([1,0]), tc=’d’)
|
||||
q = matrix(numpy.array([3,4]), tc=’d’)
|
||||
G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
|
||||
h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
|
||||
# Construct the QP, invoke solver
|
||||
sol = solvers.qp(P,q,G,h)
|
||||
# Extract optimal value and solution
|
||||
sol[’x’]
|
||||
sol[’primal objective’]
|
||||
------------------
|
||||
|
||||
[0;36m File [0;32m"<ipython-input-5-c46dd114b2af>"[0;36m, line [0;32m5[0m
|
||||
[0;31m P = matrix(numpy.diag([1,0]), tc=’d’)[0m
|
||||
[0m ^[0m
|
||||
[0;31mSyntaxError[0m[0;31m:[0m invalid character in identifier
|
||||
|
||||
SyntaxError: invalid character in identifier (<ipython-input-5-c46dd114b2af>, line 5)
|
||||
|
||||
@@ -0,0 +1,66 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
import os
|
||||
from sklearn.datasets import load_breast_cancer
|
||||
from sklearn.tree import DecisionTreeClassifier
|
||||
from sklearn.model_selection import train_test_split
|
||||
from sklearn.metrics import confusion_matrix
|
||||
from sklearn.tree import export_graphviz
|
||||
|
||||
from IPython.display import Image
|
||||
from pydot import graph_from_dot_data
|
||||
import pandas as pd
|
||||
import numpy as np
|
||||
|
||||
|
||||
cancer = load_breast_cancer()
|
||||
X = pd.DataFrame(cancer.data, columns=cancer.feature_names)
|
||||
print(X)
|
||||
y = pd.Categorical.from_codes(cancer.target, cancer.target_names)
|
||||
y = pd.get_dummies(y)
|
||||
print(y)
|
||||
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)
|
||||
tree_clf = DecisionTreeClassifier(max_depth=5)
|
||||
tree_clf.fit(X_train, y_train)
|
||||
|
||||
export_graphviz(
|
||||
tree_clf,
|
||||
out_file="DataFiles/cancer.dot",
|
||||
feature_names=cancer.feature_names,
|
||||
class_names=cancer.target_names,
|
||||
rounded=True,
|
||||
filled=True
|
||||
)
|
||||
cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
|
||||
os.system(cmd)
|
||||
------------------
|
||||
|
||||
[0;31m---------------------------------------------------------------------------[0m
|
||||
[0;31mModuleNotFoundError[0m Traceback (most recent call last)
|
||||
[0;32m<ipython-input-2-7c394b1e8b71>[0m in [0;36m<module>[0;34m[0m
|
||||
[1;32m 7[0m [0;34m[0m[0m
|
||||
[1;32m 8[0m [0;32mfrom[0m [0mIPython[0m[0;34m.[0m[0mdisplay[0m [0;32mimport[0m [0mImage[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m----> 9[0;31m [0;32mfrom[0m [0mpydot[0m [0;32mimport[0m [0mgraph_from_dot_data[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m[1;32m 10[0m [0;32mimport[0m [0mpandas[0m [0;32mas[0m [0mpd[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 11[0m [0;32mimport[0m [0mnumpy[0m [0;32mas[0m [0mnp[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;31mModuleNotFoundError[0m: No module named 'pydot'
|
||||
ModuleNotFoundError: No module named 'pydot'
|
||||
|
||||
@@ -0,0 +1,46 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
heads_proba = 0.51
|
||||
coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
|
||||
cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)
|
||||
plt.figure(figsize=(8,3.5))
|
||||
plt.plot(cumulative_heads_ratio)
|
||||
plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%")
|
||||
plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%")
|
||||
plt.xlabel("Number of coin tosses")
|
||||
plt.ylabel("Heads ratio")
|
||||
plt.legend(loc="lower right")
|
||||
plt.axis([0, 10000, 0.42, 0.58])
|
||||
save_fig("votingsimple")
|
||||
plt.show()
|
||||
------------------
|
||||
|
||||
[0;31m---------------------------------------------------------------------------[0m
|
||||
[0;31mNameError[0m Traceback (most recent call last)
|
||||
[0;32m<ipython-input-1-eface79dac2c>[0m in [0;36m<module>[0;34m[0m
|
||||
[1;32m 1[0m [0mheads_proba[0m [0;34m=[0m [0;36m0.51[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0;32m----> 2[0;31m [0mcoin_tosses[0m [0;34m=[0m [0;34m([0m[0mnp[0m[0;34m.[0m[0mrandom[0m[0;34m.[0m[0mrand[0m[0;34m([0m[0;36m10000[0m[0;34m,[0m [0;36m10[0m[0;34m)[0m [0;34m<[0m [0mheads_proba[0m[0;34m)[0m[0;34m.[0m[0mastype[0m[0;34m([0m[0mnp[0m[0;34m.[0m[0mint32[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[0m[1;32m 3[0m [0mcumulative_heads_ratio[0m [0;34m=[0m [0mnp[0m[0;34m.[0m[0mcumsum[0m[0;34m([0m[0mcoin_tosses[0m[0;34m,[0m [0maxis[0m[0;34m=[0m[0;36m0[0m[0;34m)[0m [0;34m/[0m [0mnp[0m[0;34m.[0m[0marange[0m[0;34m([0m[0;36m1[0m[0;34m,[0m [0;36m10001[0m[0;34m)[0m[0;34m.[0m[0mreshape[0m[0;34m([0m[0;34m-[0m[0;36m1[0m[0;34m,[0m [0;36m1[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 4[0m [0mplt[0m[0;34m.[0m[0mfigure[0m[0;34m([0m[0mfigsize[0m[0;34m=[0m[0;34m([0m[0;36m8[0m[0;34m,[0m[0;36m3.5[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
[1;32m 5[0m [0mplt[0m[0;34m.[0m[0mplot[0m[0;34m([0m[0mcumulative_heads_ratio[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
|
||||
|
||||
[0;31mNameError[0m: name 'np' is not defined
|
||||
NameError: name 'np' is not defined
|
||||
|
||||
@@ -0,0 +1,31 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
import numpy as np
|
||||
x = np.log(np.array([4.0, 7.0, 8.0])
|
||||
print(x)
|
||||
------------------
|
||||
|
||||
[0;36m File [0;32m"<ipython-input-6-f6d7a289d493>"[0;36m, line [0;32m3[0m
|
||||
[0;31m print(x)[0m
|
||||
[0m ^[0m
|
||||
[0;31mSyntaxError[0m[0;31m:[0m invalid syntax
|
||||
|
||||
SyntaxError: invalid syntax (<ipython-input-6-f6d7a289d493>, line 3)
|
||||
|
||||
@@ -0,0 +1,58 @@
|
||||
Traceback (most recent call last):
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution
|
||||
executenb(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute
|
||||
return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped
|
||||
return just_run(coro(*args, **kwargs))
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run
|
||||
return loop.run_until_complete(coro)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete
|
||||
return future.result()
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute
|
||||
await self.async_execute_cell(
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell
|
||||
self._check_raise_for_error(cell, exec_reply)
|
||||
File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error
|
||||
raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content'])
|
||||
nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
|
||||
------------------
|
||||
# Blocking
|
||||
@timeFunction
|
||||
def blocking(self, blockSizeMax = 500):
|
||||
blockSizeMin = 1
|
||||
|
||||
self.blockSizes = []
|
||||
self.meanVec = []
|
||||
self.varVec = []
|
||||
|
||||
for i in range(blockSizeMin, blockSizeMax):
|
||||
if(len(self.data) % i != 0):
|
||||
pass#continue
|
||||
blockSize = i
|
||||
meanTempVec = []
|
||||
varTempVec = []
|
||||
startPoint = 0
|
||||
endPoint = blockSize
|
||||
|
||||
while endPoint <= len(self.data):
|
||||
meanTempVec.append(np.average(self.data[startPoint:endPoint]))
|
||||
startPoint = endPoint
|
||||
endPoint += blockSize
|
||||
mean, var = np.average(meanTempVec), np.var(meanTempVec)/len(meanTempVec)
|
||||
self.meanVec.append(mean)
|
||||
self.varVec.append(var)
|
||||
self.blockSizes.append(blockSize)
|
||||
|
||||
self.blockingAvg = np.average(self.meanVec[-200:])
|
||||
self.blockingVar = (np.average(self.varVec[-200:]))
|
||||
self.blockingStd = np.sqrt(self.blockingVar)
|
||||
------------------
|
||||
|
||||
[0;36m File [0;32m"<ipython-input-6-2ff97f4bf03b>"[0;36m, line [0;32m2[0m
|
||||
[0;31m @timeFunction[0m
|
||||
[0m ^[0m
|
||||
[0;31mIndentationError[0m[0;31m:[0m unexpected indent
|
||||
|
||||
IndentationError: unexpected indent (<ipython-input-6-2ff97f4bf03b>, line 2)
|
||||
|
||||
File diff suppressed because one or more lines are too long
File diff suppressed because one or more lines are too long
@@ -1282,7 +1282,7 @@ C
|
||||
K
|
||||
|
||||
$$
|
||||
\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T.
|
||||
\frac{\partial\log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}}=(\boldsymbol{A}^{-1})^T.
|
||||
$$
|
||||
|
||||
The residuals $\boldsymbol{\epsilon}$ are in turn given by
|
||||
@@ -2103,51 +2103,6 @@ $$
|
||||
You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions.
|
||||
Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.
|
||||
|
||||
!bsol
|
||||
The code here is an example of where we define our own design matrix and fit parameters $\beta$.
|
||||
|
||||
import os
|
||||
import numpy as np
|
||||
import pandas as pd
|
||||
import matplotlib.pyplot as plt
|
||||
from sklearn.model_selection import train_test_split
|
||||
|
||||
def save_fig(fig_id):
|
||||
plt.savefig(image_path(fig_id) + ".png", format='png')
|
||||
|
||||
def R2(y_data, y_model):
|
||||
return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
|
||||
def MSE(y_data,y_model):
|
||||
n = np.size(y_model)
|
||||
return np.sum((y_data-y_model)**2)/n
|
||||
|
||||
x = np.random.rand(100)
|
||||
y = 2.0+5*x*x+0.1*np.random.randn(100)
|
||||
|
||||
|
||||
# The design matrix now as function of a given polynomial
|
||||
X = np.zeros((len(x),3))
|
||||
X[:,0] = 1.0
|
||||
X[:,1] = x
|
||||
X[:,2] = x**2
|
||||
# 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)
|
||||
# matrix inversion to find beta
|
||||
beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train
|
||||
print(beta)
|
||||
# and then make the prediction
|
||||
ytilde = X_train @ beta
|
||||
print("Training R2")
|
||||
print(R2(y_train,ytilde))
|
||||
print("Training MSE")
|
||||
print(MSE(y_train,ytilde))
|
||||
ypredict = X_test @ beta
|
||||
print("Test R2")
|
||||
print(R2(y_test,ypredict))
|
||||
print("Test MSE")
|
||||
print(MSE(y_test,ypredict))
|
||||
|
||||
!esol
|
||||
|
||||
|
||||
|
||||
@@ -2229,7 +2184,5 @@ Perform an ordinary least squares and compute the means squared error and the $R
|
||||
|
||||
!bsubex
|
||||
Add now a model which allows you to make polynomials up to degree $15$. Perform a standard OLS fitting of the training data and compute the MSE and $R2$ for the training and test data and plot both test and training data MSE and $R2$ as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?
|
||||
!bsol
|
||||
Here you simply need to change the degree of the polynomial in the above code to $n=15$.
|
||||
!esol
|
||||
|
||||
!esubex
|
||||
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|
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@@ -1963,7 +1963,7 @@
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\frac{\\partial \\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}} = (\\boldsymbol{A}^{-1})^T.\n",
|
||||
"\\frac{\\partial\\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}}=(\\boldsymbol{A}^{-1})^T.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
@@ -3359,66 +3359,6 @@
|
||||
"You can use the functionality included in scikit-learn. If you feel for it, you can use your own program and define functions which compute the above two functions. \n",
|
||||
"Discuss the meaning of these results. Try also to vary the coefficient in front of the added stochastic noise term and discuss the quality of the fits.\n",
|
||||
"\n",
|
||||
"!bsol\n",
|
||||
"The code here is an example of where we define our own design matrix and fit parameters $\\beta$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import os\n",
|
||||
"import numpy as np\n",
|
||||
"import pandas as pd\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"from sklearn.model_selection import train_test_split\n",
|
||||
"\n",
|
||||
"def save_fig(fig_id):\n",
|
||||
" plt.savefig(image_path(fig_id) + \".png\", format='png')\n",
|
||||
"\n",
|
||||
"def R2(y_data, y_model):\n",
|
||||
" return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
|
||||
"def MSE(y_data,y_model):\n",
|
||||
" n = np.size(y_model)\n",
|
||||
" return np.sum((y_data-y_model)**2)/n\n",
|
||||
"\n",
|
||||
"x = np.random.rand(100)\n",
|
||||
"y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"# The design matrix now as function of a given polynomial\n",
|
||||
"X = np.zeros((len(x),3))\n",
|
||||
"X[:,0] = 1.0\n",
|
||||
"X[:,1] = x\n",
|
||||
"X[:,2] = x**2\n",
|
||||
"# We split the data in test and training data\n",
|
||||
"X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
|
||||
"# matrix inversion to find beta\n",
|
||||
"beta = np.linalg.inv(X_train.T @ X_train) @ X_train.T @ y_train\n",
|
||||
"print(beta)\n",
|
||||
"# and then make the prediction\n",
|
||||
"ytilde = X_train @ beta\n",
|
||||
"print(\"Training R2\")\n",
|
||||
"print(R2(y_train,ytilde))\n",
|
||||
"print(\"Training MSE\")\n",
|
||||
"print(MSE(y_train,ytilde))\n",
|
||||
"ypredict = X_test @ beta\n",
|
||||
"print(\"Test R2\")\n",
|
||||
"print(R2(y_test,ypredict))\n",
|
||||
"print(\"Test MSE\")\n",
|
||||
"print(MSE(y_test,ypredict))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"!esol\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
@@ -3545,9 +3485,7 @@
|
||||
"\n",
|
||||
"!bsubex\n",
|
||||
"Add now a model which allows you to make polynomials up to degree $15$. Perform a standard OLS fitting of the training data and compute the MSE and $R2$ for the training and test data and plot both test and training data MSE and $R2$ as functions of the polynomial degree. Compare what you see with Figure 2.11 of Hastie et al. Comment your results. For which polynomial degree do you find an optimal MSE (smallest value)?\n",
|
||||
"!bsol\n",
|
||||
"Here you simply need to change the degree of the polynomial in the above code to $n=15$.\n",
|
||||
"!esol\n",
|
||||
"\n",
|
||||
"!esubex"
|
||||
]
|
||||
}
|
||||
|
||||
Reference in New Issue
Block a user