# My book ‘Deep Learning from first principles:Second Edition’ now on Amazon

The second edition of my book ‘Deep Learning from first principles:Second Edition- In vectorized Python, R and Octave’, is now available on Amazon, in both paperback ($18.99) and kindle ($9.99/Rs449/-)  versions. Since this book is almost 70% code, all functions, and code snippets have been formatted to use the fixed-width font ‘Lucida Console’. In addition line numbers have been added to all code snippets. This makes the code more organized and much more readable. I have also fixed typos in the book

The book includes the following chapters

Table of Contents
Preface 4
Introduction 6
1. Logistic Regression as a Neural Network 8
2. Implementing a simple Neural Network 23
3. Building a L- Layer Deep Learning Network 48
4. Deep Learning network with the Softmax 85
5. MNIST classification with Softmax 103
6. Initialization, regularization in Deep Learning 121
7. Gradient Descent Optimization techniques 167
8. Gradient Check in Deep Learning 197
1. Appendix A 214
2. Appendix 1 – Logistic Regression as a Neural Network 220
3. Appendix 2 - Implementing a simple Neural Network 227
4. Appendix 3 - Building a L- Layer Deep Learning Network 240
5. Appendix 4 - Deep Learning network with the Softmax 259
6. Appendix 5 - MNIST classification with Softmax 269
7. Appendix 6 - Initialization, regularization in Deep Learning 302
8. Appendix 7 - Gradient Descent Optimization techniques 344
9. Appendix 8 – Gradient Check 405
References 475

To see posts click Index of Posts

# My book ‘Practical Machine Learning in R and Python: Second edition’ on Amazon

Note: The 3rd edition of this book is now available My book ‘Practical Machine Learning in R and Python: Third edition’ on Amazon

The third edition of my book ‘Practical Machine Learning with R and Python – Machine Learning in stereo’ is now available in both paperback ($12.99) and kindle ($9.99/Rs449) versions.  This second edition includes more content,  extensive comments and formatting for better readability.

In this book I implement some of the most common, but important Machine Learning algorithms in R and equivalent Python code.
1. Practical machine with R and Python: Third Edition – Machine Learning in Stereo(Paperback-$12.99) 2. Practical machine with R and Third Edition – Machine Learning in Stereo(Kindle-$9.99/Rs449)

This book is ideal both for beginners and the experts in R and/or Python. Those starting their journey into datascience and ML will find the first 3 chapters useful, as they touch upon the most important programming constructs in R and Python and also deal with equivalent statements in R and Python. Those who are expert in either of the languages, R or Python, will find the equivalent code ideal for brushing up on the other language. And finally,those who are proficient in both languages, can use the R and Python implementations to internalize the ML algorithms better.

Here is a look at the topics covered

Preface …………………………………………………………………………….4
Introduction ………………………………………………………………………6
1. Essential R ………………………………………………………………… 8
2. Essential Python for Datascience ……………………………………………57
3. R vs Python …………………………………………………………………81
4. Regression of a continuous variable ……………………………………….101
5. Classification and Cross Validation ………………………………………..121
6. Regression techniques and regularization ………………………………….146
7. SVMs, Decision Trees and Validation curves ………………………………191
8. Splines, GAMs, Random Forests and Boosting ……………………………222
9. PCA, K-Means and Hierarchical Clustering ………………………………258
References ……………………………………………………………………..269

Hope you have a great time learning as I did while implementing these algorithms!

# My book ‘Practical Machine Learning with R and Python’ on Amazon

Note: The 3rd edition of this book is now available My book ‘Practical Machine Learning in R and Python: Third edition’ on Amazon

My book ‘Practical Machine Learning with R and Python: Second Edition – Machine Learning in stereo’ is now available in both paperback ($10.99) and kindle ($7.99/Rs449) versions. In this book I implement some of the most common, but important Machine Learning algorithms in R and equivalent Python code. This is almost like listening to parallel channels of music in stereo!
1. Practical machine with R and Python: Third Edition – Machine Learning in Stereo(Paperback-$12.99) 2. Practical machine with R and Python Third Edition – Machine Learning in Stereo(Kindle-$8.99/Rs449)
This book is ideal both for beginners and the experts in R and/or Python. Those starting their journey into datascience and ML will find the first 3 chapters useful, as they touch upon the most important programming constructs in R and Python and also deal with equivalent statements in R and Python. Those who are expert in either of the languages, R or Python, will find the equivalent code ideal for brushing up on the other language. And finally,those who are proficient in both languages, can use the R and Python implementations to internalize the ML algorithms better.

Here is a look at the topics covered

Essential R …………………………………….. 7
Essential Python for Datascience ………………..   54
R vs Python ……………………………………. 77
Regression of a continuous variable ………………. 96
Classification and Cross Validation ……………….113
Regression techniques and regularization …………. 134
SVMs, Decision Trees and Validation curves …………175
Splines, GAMs, Random Forests and Boosting …………202
PCA, K-Means and Hierarchical Clustering …………. 234

Hope you have a great time learning as I did while implementing these algorithms!

# Practical Machine Learning with R and Python – Part 4

This is the 4th installment of my ‘Practical Machine Learning with R and Python’ series. In this part I discuss classification with Support Vector Machines (SVMs), using both a Linear and a Radial basis kernel, and Decision Trees. Further, a closer look is taken at some of the metrics associated with binary classification, namely accuracy vs precision and recall. I also touch upon Validation curves, Precision-Recall, ROC curves and AUC with equivalent code in R and Python

This post is a continuation of my 3 earlier posts on Practical Machine Learning in R and Python
1. Practical Machine Learning with R and Python – Part 1
2. Practical Machine Learning with R and Python – Part 2
3. Practical Machine Learning with R and Python – Part 3

The RMarkdown file with the code and the associated data files can be downloaded from Github at MachineLearning-RandPython-Part4

1. Machine Learning in plain English-Part 1
2. Machine Learning in plain English-Part 2
3. Machine Learning in plain English-Part 3

Check out my compact and minimal book  “Practical Machine Learning with R and Python:Third edition- Machine Learning in stereo”  available in Amazon in paperback($12.99) and kindle($8.99) versions. My book includes implementations of key ML algorithms and associated measures and metrics. The book is ideal for anybody who is familiar with the concepts and would like a quick reference to the different ML algorithms that can be applied to problems and how to select the best model. Pick your copy today!!

Support Vector Machines (SVM) are another useful Machine Learning model that can be used for both regression and classification problems. SVMs used in classification, compute the hyperplane, that separates the 2 classes with the maximum margin. To do this the features may be transformed into a larger multi-dimensional feature space. SVMs can be used with different kernels namely linear, polynomial or radial basis to determine the best fitting model for a given classification problem.

In the 2nd part of this series Practical Machine Learning with R and Python – Part 2, I had mentioned the various metrics that are used in classification ML problems namely Accuracy, Precision, Recall and F1 score. Accuracy gives the fraction of data that were correctly classified as belonging to the +ve or -ve class. However ‘accuracy’ in itself is not a good enough measure because it does not take into account the fraction of the data that were incorrectly classified. This issue becomes even more critical in different domains. For e.g a surgeon who would like to detect cancer, would like to err on the side of caution, and classify even a possibly non-cancerous patient as possibly having cancer, rather than mis-classifying a malignancy as benign. Here we would like to increase recall or sensitivity which is  given by Recall= TP/(TP+FN) or we try reduce mis-classification by either increasing the (true positives) TP or reducing (false negatives) FN

On the other hand, search algorithms would like to increase precision which tries to reduce the number of irrelevant results in the search result. Precision= TP/(TP+FP). In other words we do not want ‘false positives’ or irrelevant results to come in the search results and there is a need to reduce the false positives.

When we try to increase ‘precision’, we do so at the cost of ‘recall’, and vice-versa. I found this diagram and explanation in Wikipedia very useful Source: Wikipedia

“Consider a brain surgeon tasked with removing a cancerous tumor from a patient’s brain. The surgeon needs to remove all of the tumor cells since any remaining cancer cells will regenerate the tumor. Conversely, the surgeon must not remove healthy brain cells since that would leave the patient with impaired brain function. The surgeon may be more liberal in the area of the brain she removes to ensure she has extracted all the cancer cells. This decision increases recall but reduces precision. On the other hand, the surgeon may be more conservative in the brain she removes to ensure she extracts only cancer cells. This decision increases precision but reduces recall. That is to say, greater recall increases the chances of removing healthy cells (negative outcome) and increases the chances of removing all cancer cells (positive outcome). Greater precision decreases the chances of removing healthy cells (positive outcome) but also decreases the chances of removing all cancer cells (negative outcome).”

## 1.1a. Linear SVM – R code

In R code below I use SVM with linear kernel

source('RFunctions-1.R')
library(dplyr)
library(e1071)
library(caret)
library(reshape2)
library(ggplot2)
# Read data. Data from SKLearn
cancer$target <- as.factor(cancer$target)

# Split into training and test sets
train_idx <- trainTestSplit(cancer,trainPercent=75,seed=5)
train <- cancer[train_idx, ]
test <- cancer[-train_idx, ]

# Fit a linear basis kernel. DO not scale the data
svmfit=svm(target~., data=train, kernel="linear",scale=FALSE)
ypred=predict(svmfit,test)
#Print a confusion matrix
confusionMatrix(ypred,test$target) ## Confusion Matrix and Statistics ## ## Reference ## Prediction 0 1 ## 0 54 3 ## 1 3 82 ## ## Accuracy : 0.9577 ## 95% CI : (0.9103, 0.9843) ## No Information Rate : 0.5986 ## P-Value [Acc > NIR] : <2e-16 ## ## Kappa : 0.9121 ## Mcnemar's Test P-Value : 1 ## ## Sensitivity : 0.9474 ## Specificity : 0.9647 ## Pos Pred Value : 0.9474 ## Neg Pred Value : 0.9647 ## Prevalence : 0.4014 ## Detection Rate : 0.3803 ## Detection Prevalence : 0.4014 ## Balanced Accuracy : 0.9560 ## ## 'Positive' Class : 0 ##  ## 1.1b Linear SVM – Python code The code below creates a SVM with linear basis in Python and also dumps the corresponding classification metrics import numpy as np import pandas as pd import os import matplotlib.pyplot as plt from sklearn.model_selection import train_test_split from sklearn.svm import LinearSVC from sklearn.datasets import make_classification, make_blobs from sklearn.metrics import confusion_matrix from matplotlib.colors import ListedColormap from sklearn.datasets import load_breast_cancer # Load the cancer data (X_cancer, y_cancer) = load_breast_cancer(return_X_y = True) X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer, random_state = 0) clf = LinearSVC().fit(X_train, y_train) print('Breast cancer dataset') print('Accuracy of Linear SVC classifier on training set: {:.2f}' .format(clf.score(X_train, y_train))) print('Accuracy of Linear SVC classifier on test set: {:.2f}' .format(clf.score(X_test, y_test))) ## Breast cancer dataset ## Accuracy of Linear SVC classifier on training set: 0.92 ## Accuracy of Linear SVC classifier on test set: 0.94 ## 1.2 Dummy classifier Often when we perform classification tasks using any ML model namely logistic regression, SVM, neural networks etc. it is very useful to determine how well the ML model performs agains at dummy classifier. A dummy classifier uses some simple computation like frequency of majority class, instead of fitting and ML model. It is essential that our ML model does much better that the dummy classifier. This problem is even more important in imbalanced classes where we have only about 10% of +ve samples. If any ML model we create has a accuracy of about 0.90 then it is evident that our classifier is not doing any better than a dummy classsfier which can just take a majority count of this imbalanced class and also come up with 0.90. We need to be able to do better than that. In the examples below (1.3a & 1.3b) it can be seen that SVMs with ‘radial basis’ kernel with unnormalized data, for both R and Python, do not perform any better than the dummy classifier. ## 1.2a Dummy classifier – R code R does not seem to have an explicit dummy classifier. I created a simple dummy classifier that predicts the majority class. SKlearn in Python also includes other strategies like uniform, stratified etc. but this should be possible to create in R also. # Create a simple dummy classifier that computes the ratio of the majority class to the totla DummyClassifierAccuracy <- function(train,test,type="majority"){ if(type=="majority"){ count <- sum(train$target==1)/dim(train)[1]
}
count
}

cancer$target <- as.factor(cancer$target)

# Create training and test sets
train_idx <- trainTestSplit(cancer,trainPercent=75,seed=5)
train <- cancer[train_idx, ]
test <- cancer[-train_idx, ]

#Dummy classifier majority class
acc=DummyClassifierAccuracy(train,test)
sprintf("Accuracy is %f",acc)
## [1] "Accuracy is 0.638498"

## 1.2b Dummy classifier – Python code

This dummy classifier uses the majority class.

import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.dummy import DummyClassifier
from sklearn.metrics import confusion_matrix
(X_cancer, y_cancer) = load_breast_cancer(return_X_y = True)
X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer,
random_state = 0)

# Negative class (0) is most frequent
dummy_majority = DummyClassifier(strategy = 'most_frequent').fit(X_train, y_train)
y_dummy_predictions = dummy_majority.predict(X_test)

print('Dummy classifier accuracy on test set: {:.2f}'
.format(dummy_majority.score(X_test, y_test)))

## Dummy classifier accuracy on test set: 0.63

## 1.3a – Radial SVM (un-normalized) – R code

SVMs perform better when the data is normalized or scaled. The 2 examples below show that SVM with radial basis kernel does not perform any better than the dummy classifier

library(dplyr)
library(e1071)
library(caret)
library(reshape2)
library(ggplot2)

train_idx <- trainTestSplit(cancer,trainPercent=75,seed=5)
train <- cancer[train_idx, ]
test <- cancer[-train_idx, ]
# Unnormalized data
ypred=predict(svmfit,test)
confusionMatrix(ypred,test$target) ## Confusion Matrix and Statistics ## ## Reference ## Prediction 0 1 ## 0 0 0 ## 1 57 85 ## ## Accuracy : 0.5986 ## 95% CI : (0.5131, 0.6799) ## No Information Rate : 0.5986 ## P-Value [Acc > NIR] : 0.5363 ## ## Kappa : 0 ## Mcnemar's Test P-Value : 1.195e-13 ## ## Sensitivity : 0.0000 ## Specificity : 1.0000 ## Pos Pred Value : NaN ## Neg Pred Value : 0.5986 ## Prevalence : 0.4014 ## Detection Rate : 0.0000 ## Detection Prevalence : 0.0000 ## Balanced Accuracy : 0.5000 ## ## 'Positive' Class : 0 ##  ## 1.4b – Radial SVM (un-normalized) – Python code import numpy as np import pandas as pd import os import matplotlib.pyplot as plt from sklearn.datasets import load_breast_cancer from sklearn.model_selection import train_test_split from sklearn.svm import SVC # Load the cancer data (X_cancer, y_cancer) = load_breast_cancer(return_X_y = True) X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer, random_state = 0) clf = SVC(C=10).fit(X_train, y_train) print('Breast cancer dataset (unnormalized features)') print('Accuracy of RBF-kernel SVC on training set: {:.2f}' .format(clf.score(X_train, y_train))) print('Accuracy of RBF-kernel SVC on test set: {:.2f}' .format(clf.score(X_test, y_test))) ## Breast cancer dataset (unnormalized features) ## Accuracy of RBF-kernel SVC on training set: 1.00 ## Accuracy of RBF-kernel SVC on test set: 0.63 ## 1.5a – Radial SVM (Normalized) -R Code The data is scaled (normalized ) before using the SVM model. The SVM model has 2 paramaters a) C – Large C (less regularization), more regularization b) gamma – Small gamma has larger decision boundary with more misclassfication, and larger gamma has tighter decision boundary The R code below computes the accuracy as the regularization paramater is changed trainingAccuracy <- NULL testAccuracy <- NULL C1 <- c(.01,.1, 1, 10, 20) for(i in C1){ svmfit=svm(target~., data=train, kernel="radial",cost=i,scale=TRUE) ypredTrain <-predict(svmfit,train) ypredTest=predict(svmfit,test) a <-confusionMatrix(ypredTrain,train$target)
b <-confusionMatrix(ypredTest,test$target) trainingAccuracy <-c(trainingAccuracy,a$overall[1])
testAccuracy <-c(testAccuracy,b$overall[1]) } print(trainingAccuracy) ## Accuracy Accuracy Accuracy Accuracy Accuracy ## 0.6384977 0.9671362 0.9906103 0.9976526 1.0000000 print(testAccuracy) ## Accuracy Accuracy Accuracy Accuracy Accuracy ## 0.5985915 0.9507042 0.9647887 0.9507042 0.9507042 a <-rbind(C1,as.numeric(trainingAccuracy),as.numeric(testAccuracy)) b <- data.frame(t(a)) names(b) <- c("C1","trainingAccuracy","testAccuracy") df <- melt(b,id="C1") ggplot(df) + geom_line(aes(x=C1, y=value, colour=variable),size=2) + xlab("C (SVC regularization)value") + ylab("Accuracy") + ggtitle("Training and test accuracy vs C(regularization)") ## 1.5b – Radial SVM (normalized) – Python The Radial basis kernel is used on normalized data for a range of ‘C’ values and the result is plotted. import numpy as np import pandas as pd import os import matplotlib.pyplot as plt from sklearn.datasets import load_breast_cancer from sklearn.model_selection import train_test_split from sklearn.svm import SVC from sklearn.preprocessing import MinMaxScaler scaler = MinMaxScaler() # Load the cancer data (X_cancer, y_cancer) = load_breast_cancer(return_X_y = True) X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer, random_state = 0) X_train_scaled = scaler.fit_transform(X_train) X_test_scaled = scaler.transform(X_test) print('Breast cancer dataset (normalized with MinMax scaling)') trainingAccuracy=[] testAccuracy=[] for C1 in [.01,.1, 1, 10, 20]: clf = SVC(C=C1).fit(X_train_scaled, y_train) acctrain=clf.score(X_train_scaled, y_train) accTest=clf.score(X_test_scaled, y_test) trainingAccuracy.append(acctrain) testAccuracy.append(accTest) # Create a dataframe C1=[.01,.1, 1, 10, 20] trainingAccuracy=pd.DataFrame(trainingAccuracy,index=C1) testAccuracy=pd.DataFrame(testAccuracy,index=C1) # Plot training and test R squared as a function of alpha df=pd.concat([trainingAccuracy,testAccuracy],axis=1) df.columns=['trainingAccuracy','trainingAccuracy'] fig1=df.plot() fig1=plt.title('Training and test accuracy vs C (SVC)') fig1.figure.savefig('fig1.png', bbox_inches='tight') ## Breast cancer dataset (normalized with MinMax scaling) Output image: ## 1.6a Validation curve – R code Sklearn includes code creating validation curves by varying paramaters and computing and plotting accuracy as gamma or C or changd. I did not find this R but I think this is a useful function and so I have created the R equivalent of this. # The R equivalent of np.logspace seqLogSpace <- function(start,stop,len){ a=seq(log10(10^start),log10(10^stop),length=len) 10^a } # Read the data. This is taken the SKlearn cancer data cancer <- read.csv("cancer.csv") cancer$target <- as.factor(cancer$target) set.seed(6) # Create the range of C1 in log space param_range = seqLogSpace(-3,2,20) # Initialize the overall training and test accuracy to NULL overallTrainAccuracy <- NULL overallTestAccuracy <- NULL # Loop over the parameter range of Gamma for(i in param_range){ # Set no of folds noFolds=5 # Create the rows which fall into different folds from 1..noFolds folds = sample(1:noFolds, nrow(cancer), replace=TRUE) # Initialize the training and test accuracy of folds to 0 trainingAccuracy <- 0 testAccuracy <- 0 # Loop through the folds for(j in 1:noFolds){ # The training is all rows for which the row is != j (k-1 folds -> training) train <- cancer[folds!=j,] # The rows which have j as the index become the test set test <- cancer[folds==j,] # Create a SVM model for this svmfit=svm(target~., data=train, kernel="radial",gamma=i,scale=TRUE) # Add up all the fold accuracy for training and test separately ypredTrain <-predict(svmfit,train) ypredTest=predict(svmfit,test) # Create confusion matrix a <-confusionMatrix(ypredTrain,train$target)
b <-confusionMatrix(ypredTest,test$target) # Get the accuracy trainingAccuracy <-trainingAccuracy + a$overall[1]
testAccuracy <-testAccuracy+b$overall[1] } # Compute the average of accuracy for K folds for number of features 'i' overallTrainAccuracy=c(overallTrainAccuracy,trainingAccuracy/noFolds) overallTestAccuracy=c(overallTestAccuracy,testAccuracy/noFolds) } #Create a dataframe a <- rbind(param_range,as.numeric(overallTrainAccuracy), as.numeric(overallTestAccuracy)) b <- data.frame(t(a)) names(b) <- c("C1","trainingAccuracy","testAccuracy") df <- melt(b,id="C1") #Plot in log axis ggplot(df) + geom_line(aes(x=C1, y=value, colour=variable),size=2) + xlab("C (SVC regularization)value") + ylab("Accuracy") + ggtitle("Training and test accuracy vs C(regularization)") + scale_x_log10() ## 1.6b Validation curve – Python Compute and plot the validation curve as gamma is varied. import numpy as np import pandas as pd import os import matplotlib.pyplot as plt from sklearn.datasets import load_breast_cancer from sklearn.model_selection import train_test_split from sklearn.preprocessing import MinMaxScaler from sklearn.svm import SVC from sklearn.model_selection import validation_curve # Load the cancer data (X_cancer, y_cancer) = load_breast_cancer(return_X_y = True) scaler = MinMaxScaler() X_scaled = scaler.fit_transform(X_cancer) # Create a gamma values from 10^-3 to 10^2 with 20 equally spaced intervals param_range = np.logspace(-3, 2, 20) # Compute the validation curve train_scores, test_scores = validation_curve(SVC(), X_scaled, y_cancer, param_name='gamma', param_range=param_range, cv=10) #Plot the figure fig2=plt.figure() #Compute the mean train_scores_mean = np.mean(train_scores, axis=1) train_scores_std = np.std(train_scores, axis=1) test_scores_mean = np.mean(test_scores, axis=1) test_scores_std = np.std(test_scores, axis=1) fig2=plt.title('Validation Curve with SVM') fig2=plt.xlabel('$\gamma$(gamma)') fig2=plt.ylabel('Score') fig2=plt.ylim(0.0, 1.1) lw = 2 fig2=plt.semilogx(param_range, train_scores_mean, label='Training score', color='darkorange', lw=lw) fig2=plt.fill_between(param_range, train_scores_mean - train_scores_std, train_scores_mean + train_scores_std, alpha=0.2, color='darkorange', lw=lw) fig2=plt.semilogx(param_range, test_scores_mean, label='Cross-validation score', color='navy', lw=lw) fig2=plt.fill_between(param_range, test_scores_mean - test_scores_std, test_scores_mean + test_scores_std, alpha=0.2, color='navy', lw=lw) fig2.figure.savefig('fig2.png', bbox_inches='tight')  Output image: ## 1.7a Validation Curve (Preventing data leakage) – Python code In this course Applied Machine Learning in Python, the Professor states that when we apply the same data transformation to a entire dataset, it will cause a data leakage. “The proper way to do cross-validation when you need to scale the data is not to scale the entire dataset with a single transform, since this will indirectly leak information into the training data about the whole dataset, including the test data (see the lecture on data leakage later in the course). Instead, scaling/normalizing must be computed and applied for each cross-validation fold separately” So I apply separate scaling to the training and testing folds and plot. In the lecture the Prof states that this can be done using pipelines. import numpy as np import pandas as pd import os import matplotlib.pyplot as plt from sklearn.datasets import load_breast_cancer from sklearn.cross_validation import KFold from sklearn.preprocessing import MinMaxScaler from sklearn.svm import SVC # Read the data (X_cancer, y_cancer) = load_breast_cancer(return_X_y = True) # Set the parameter range param_range = np.logspace(-3, 2, 20) # Set number of folds folds=5 #Initialize overallTrainAccuracy=[] overallTestAccuracy=[] # Loop over the paramater range for c in param_range: trainingAccuracy=0 testAccuracy=0 kf = KFold(len(X_cancer),n_folds=folds) # Partition into training and test folds for train_index, test_index in kf: # Partition the data acccording the fold indices generated X_train, X_test = X_cancer[train_index], X_cancer[test_index] y_train, y_test = y_cancer[train_index], y_cancer[test_index] # Scale the X_train and X_test scaler = MinMaxScaler() X_train_scaled = scaler.fit_transform(X_train) X_test_scaled = scaler.transform(X_test) # Fit a SVC model for each C clf = SVC(C=c).fit(X_train_scaled, y_train) #Compute the training and test score acctrain=clf.score(X_train_scaled, y_train) accTest=clf.score(X_test_scaled, y_test) trainingAccuracy += np.sum(acctrain) testAccuracy += np.sum(accTest) # Compute the mean training and testing accuracy overallTrainAccuracy.append(trainingAccuracy/folds) overallTestAccuracy.append(testAccuracy/folds) overallTrainAccuracy=pd.DataFrame(overallTrainAccuracy,index=param_range) overallTestAccuracy=pd.DataFrame(overallTestAccuracy,index=param_range) # Plot training and test R squared as a function of alpha df=pd.concat([overallTrainAccuracy,overallTestAccuracy],axis=1) df.columns=['trainingAccuracy','testAccuracy'] fig3=plt.title('Validation Curve with SVM') fig3=plt.xlabel('$\gamma$(gamma)') fig3=plt.ylabel('Score') fig3=plt.ylim(0.5, 1.1) lw = 2 fig3=plt.semilogx(param_range, overallTrainAccuracy, label='Training score', color='darkorange', lw=lw) fig3=plt.semilogx(param_range, overallTestAccuracy, label='Cross-validation score', color='navy', lw=lw) fig3=plt.legend(loc='best') fig3.figure.savefig('fig3.png', bbox_inches='tight')  Output image: ## 1.8 a Decision trees – R code Decision trees in R can be plotted using RPart package library(rpart) library(rpart.plot) rpart = NULL # Create a decision tree m <-rpart(Species~.,data=iris) #Plot rpart.plot(m,extra=2,main="Decision Tree - IRIS") ## 1.8 b Decision trees – Python code from sklearn.datasets import load_iris from sklearn.tree import DecisionTreeClassifier from sklearn import tree from sklearn.model_selection import train_test_split import graphviz iris = load_iris() X_train, X_test, y_train, y_test = train_test_split(iris.data, iris.target, random_state = 3) clf = DecisionTreeClassifier().fit(X_train, y_train) print('Accuracy of Decision Tree classifier on training set: {:.2f}' .format(clf.score(X_train, y_train))) print('Accuracy of Decision Tree classifier on test set: {:.2f}' .format(clf.score(X_test, y_test))) dot_data = tree.export_graphviz(clf, out_file=None, feature_names=iris.feature_names, class_names=iris.target_names, filled=True, rounded=True, special_characters=True) graph = graphviz.Source(dot_data) graph ## Accuracy of Decision Tree classifier on training set: 1.00 ## Accuracy of Decision Tree classifier on test set: 0.97 ## 1.9a Feature importance – R code I found the following code which had a snippet for feature importance. Sklean has a nice method for this. For some reason the results in R and Python are different. Any thoughts? set.seed(3) # load the library library(mlbench) library(caret) # load the dataset cancer <- read.csv("cancer.csv") cancer$target <- as.factor(cancer$target) # Split as data data <- cancer[,1:31] target <- cancer[,32] # Train the model model <- train(data, target, method="rf", preProcess="scale", trControl=trainControl(method = "cv")) # Compute variable importance importance <- varImp(model) # summarize importance print(importance) # plot importance plot(importance) ## 1.9b Feature importance – Python code import numpy as np import pandas as pd import os import matplotlib.pyplot as plt from sklearn.tree import DecisionTreeClassifier from sklearn.model_selection import train_test_split from sklearn.datasets import load_breast_cancer import numpy as np # Read the data cancer= load_breast_cancer() (X_cancer, y_cancer) = load_breast_cancer(return_X_y = True) X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer, random_state = 0) # Use the DecisionTreClassifier clf = DecisionTreeClassifier(max_depth = 4, min_samples_leaf = 8, random_state = 0).fit(X_train, y_train) c_features=len(cancer.feature_names) print('Breast cancer dataset: decision tree') print('Accuracy of DT classifier on training set: {:.2f}' .format(clf.score(X_train, y_train))) print('Accuracy of DT classifier on test set: {:.2f}' .format(clf.score(X_test, y_test))) # Plot the feature importances fig4=plt.figure(figsize=(10,6),dpi=80) fig4=plt.barh(range(c_features), clf.feature_importances_) fig4=plt.xlabel("Feature importance") fig4=plt.ylabel("Feature name") fig4=plt.yticks(np.arange(c_features), cancer.feature_names) fig4=plt.tight_layout() plt.savefig('fig4.png', bbox_inches='tight')  ## Breast cancer dataset: decision tree ## Accuracy of DT classifier on training set: 0.96 ## Accuracy of DT classifier on test set: 0.94 Output image: ## 1.10a Precision-Recall, ROC curves & AUC- R code I tried several R packages for plotting the Precision and Recall and AUC curve. PRROC seems to work well. The Precision-Recall curves show the tradeoff between precision and recall. The higher the precision, the lower the recall and vice versa.AUC curves that hug the top left corner indicate a high sensitivity,specificity and an excellent accuracy. source("RFunctions-1.R") library(dplyr) library(caret) library(e1071) library(PRROC) # Read the data (this data is from sklearn!) d <- read.csv("digits.csv") digits <- d[2:66] digits$X64 <- as.factor(digits$X64) # Split as training and test sets train_idx <- trainTestSplit(digits,trainPercent=75,seed=5) train <- digits[train_idx, ] test <- digits[-train_idx, ] # Fit a SVM model with linear basis kernel with probabilities svmfit=svm(X64~., data=train, kernel="linear",scale=FALSE,probability=TRUE) ypred=predict(svmfit,test,probability=TRUE) head(attr(ypred,"probabilities")) ## 0 1 ## 6 7.395947e-01 2.604053e-01 ## 8 9.999998e-01 1.842555e-07 ## 12 1.655178e-05 9.999834e-01 ## 13 9.649997e-01 3.500032e-02 ## 15 9.994849e-01 5.150612e-04 ## 16 9.999987e-01 1.280700e-06 # Store the probability of 0s and 1s m0<-attr(ypred,"probabilities")[,1] m1<-attr(ypred,"probabilities")[,2] # Create a dataframe of scores scores <- data.frame(m1,test$X64)

# Class 0 is data points of +ve class (in this case, digit 1) and -ve class (digit 0)
#Compute Precision Recall
pr <- pr.curve(scores.class0=scores[scores$test.X64=="1",]$m1,
scores.class1=scores[scores$test.X64=="0",]$m1,
curve=T)

# Plot precision-recall curve
plot(pr)

#Plot the ROC curve
roc<-roc.curve(m0, m1,curve=TRUE)
plot(roc)

## 1.10b Precision-Recall, ROC curves & AUC- Python code

For Python Logistic Regression is used to plot Precision Recall, ROC curve and compute AUC

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
from sklearn.metrics import precision_recall_curve
from sklearn.metrics import roc_curve, auc
X, y = dataset.data, dataset.target
#Create 2 classes -i) Digit 1 (from digit 1) ii) Digit 0 (from all other digits)
# Make a copy of the target
z= y.copy()
# Replace all non 1's as 0
z[z != 1] = 0

X_train, X_test, y_train, y_test = train_test_split(X, z, random_state=0)
# Fit a LR model
lr = LogisticRegression().fit(X_train, y_train)

#Compute the decision scores
y_scores_lr = lr.fit(X_train, y_train).decision_function(X_test)
y_score_list = list(zip(y_test[0:20], y_scores_lr[0:20]))

#Show the decision_function scores for first 20 instances
y_score_list

precision, recall, thresholds = precision_recall_curve(y_test, y_scores_lr)
closest_zero = np.argmin(np.abs(thresholds))
closest_zero_p = precision[closest_zero]
closest_zero_r = recall[closest_zero]
#Plot
plt.figure()
plt.xlim([0.0, 1.01])
plt.ylim([0.0, 1.01])
plt.plot(precision, recall, label='Precision-Recall Curve')
plt.plot(closest_zero_p, closest_zero_r, 'o', markersize = 12, fillstyle = 'none', c='r', mew=3)
plt.xlabel('Precision', fontsize=16)
plt.ylabel('Recall', fontsize=16)
plt.axes().set_aspect('equal')
plt.savefig('fig5.png', bbox_inches='tight')

#Compute and plot the ROC
y_score_lr = lr.fit(X_train, y_train).decision_function(X_test)
fpr_lr, tpr_lr, _ = roc_curve(y_test, y_score_lr)
roc_auc_lr = auc(fpr_lr, tpr_lr)

plt.figure()
plt.xlim([-0.01, 1.00])
plt.ylim([-0.01, 1.01])
plt.plot(fpr_lr, tpr_lr, lw=3, label='LogRegr ROC curve (area = {:0.2f})'.format(roc_auc_lr))
plt.xlabel('False Positive Rate', fontsize=16)
plt.ylabel('True Positive Rate', fontsize=16)
plt.title('ROC curve (1-of-10 digits classifier)', fontsize=16)
plt.legend(loc='lower right', fontsize=13)
plt.plot([0, 1], [0, 1], color='navy', lw=3, linestyle='--')
plt.axes()
plt.savefig('fig6.png', bbox_inches='tight')


output

## 1.10c Precision-Recall, ROC curves & AUC- Python code

In the code below classification probabilities are used to compute and plot precision-recall, roc and AUC

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.svm import LinearSVC
from sklearn.calibration import CalibratedClassifierCV

X, y = dataset.data, dataset.target
# Make a copy of the target
z= y.copy()
# Replace all non 1's as 0
z[z != 1] = 0

X_train, X_test, y_train, y_test = train_test_split(X, z, random_state=0)
svm = LinearSVC()
# Need to use CalibratedClassifierSVC to redict probabilities for lInearSVC
clf = CalibratedClassifierCV(svm)
clf.fit(X_train, y_train)
y_proba_lr = clf.predict_proba(X_test)
from sklearn.metrics import precision_recall_curve

precision, recall, thresholds = precision_recall_curve(y_test, y_proba_lr[:,1])
closest_zero = np.argmin(np.abs(thresholds))
closest_zero_p = precision[closest_zero]
closest_zero_r = recall[closest_zero]
#plt.figure(figsize=(15,15),dpi=80)
plt.figure()
plt.xlim([0.0, 1.01])
plt.ylim([0.0, 1.01])
plt.plot(precision, recall, label='Precision-Recall Curve')
plt.plot(closest_zero_p, closest_zero_r, 'o', markersize = 12, fillstyle = 'none', c='r', mew=3)
plt.xlabel('Precision', fontsize=16)
plt.ylabel('Recall', fontsize=16)
plt.axes().set_aspect('equal')
plt.savefig('fig7.png', bbox_inches='tight')