My presentations on ‘Elements of Neural Networks & Deep Learning’ -Part1,2,3

I will be uploading a series of presentations on ‘Elements of Neural Networks and Deep Learning’. In these video presentations I discuss the derivations of L -Layer Deep Learning Networks, starting from the basics. The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave‘

1. Elements of Neural Networks and Deep Learning – Part 1
This presentation introduces Neural Networks and Deep Learning. A look at history of Neural Networks, Perceptrons and why Deep Learning networks are required and concluding with a simple toy examples of a Neural Network and how they compute

2. Elements of Neural Networks and Deep Learning – Part 2
This presentation takes logistic regression as an example and creates an equivalent 2 layer Neural network. The presentation also takes a look at forward & backward propagation and how the cost is minimized using gradient descent

The implementation of the discussed 2 layer Neural Network in vectorized R, Python and Octave are available in my post ‘Deep Learning from first principles in Python, R and Octave – Part 1‘

3. Elements of Neural Networks and Deep Learning – Part 3
This 3rd part, discusses a primitive neural network with an input layer, output layer and a hidden layer. The neural network uses tanh activation in the hidden layer and a sigmoid activation in the output layer. The equations for forward and backward propagation are derived.

To see the implementations for the above discussed video see my post ‘Deep Learning from first principles in Python, R and Octave – Part 2‘

Important note: Do check out my later version of these videos at Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8 . These have more content and also include some corrections. Check it out!

To be continued. Watch this space!

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($18.99) and in kindle version($9.99/Rs449).

To see all posts click Index of posts

Practical Machine Learning with R and Python – Part 2

In this 2nd part of the series “Practical Machine Learning with R and Python – Part 2”, I continue where I left off in my first post Practical Machine Learning with R and Python – Part 2. In this post I cover the some classification algorithmns and cross validation. Specifically I touch
-Logistic Regression
-K Nearest Neighbors (KNN) classification
-Leave out one Cross Validation (LOOCV)
-K Fold Cross Validation
in both R and Python.

As in my initial post the algorithms are based on the following courses.

Statistical Learning, Prof Trevor Hastie & Prof Robert Tibesherani, Online Stanford
Applied Machine Learning in Python Prof Kevyn-Collin Thomson, University Of Michigan, Coursera

You can download this R Markdown file along with the data from Github. I hope these posts can be used as a quick reference in R and Python and Machine Learning.I have tried to include the coolest part of either course in this post.

Note: Please listen to my video presentations Machine Learning in youtube
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!!

The following classification problem is based on Logistic Regression. The data is an included data set in Scikit-Learn, which I have saved as csv and use it also for R. The fit of a classification Machine Learning Model depends on how correctly classifies the data. There are several measures of testing a model’s classification performance. They are

–Accuracy = TP + TN / (TP + TN + FP + FN) – Fraction of all classes correctly classified
–Precision = TP / (TP + FP) – Fraction of correctly classified positives among those classified as positive
– Recall = TP / (TP + FN) Also known as sensitivity, or True Positive Rate (True positive) – Fraction of correctly classified as positive among all positives in the data
– F1 = 2 * Precision * Recall / (Precision + Recall)

1a. Logistic Regression – R code

The caret and e1071 package is required for using the confusionMatrix call

source("RFunctions.R")
library(dplyr)
library(caret)
library(e1071)

# Read the data (from sklearn)
cancer <- read.csv("cancer.csv")
# Rename the target variable
names(cancer) <- c(seq(1,30),"output")
# Split as training and test sets
train_idx <- trainTestSplit(cancer,trainPercent=75,seed=5)
train <- cancer[train_idx, ]
test <- cancer[-train_idx, ]

# Fit a generalized linear logistic model, 
fit=glm(output~.,family=binomial,data=train,control = list(maxit = 50))

# Predict the output from the model
a=predict(fit,newdata=train,type="response")
# Set response >0.5 as 1 and <=0.5 as 0
b=ifelse(a>0.5,1,0)
# Compute the confusion matrix for training data
confusionMatrix(b,train$output)

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction   0   1
##          0 154   0
##          1   0 272
##                                      
##                Accuracy : 1          
##                  95% CI : (0.9914, 1)
##     No Information Rate : 0.6385     
##     P-Value [Acc > NIR] : < 2.2e-16  
##                                      
##                   Kappa : 1          
##  Mcnemar's Test P-Value : NA         
##                                      
##             Sensitivity : 1.0000     
##             Specificity : 1.0000     
##          Pos Pred Value : 1.0000     
##          Neg Pred Value : 1.0000     
##              Prevalence : 0.3615     
##          Detection Rate : 0.3615     
##    Detection Prevalence : 0.3615     
##       Balanced Accuracy : 1.0000     
##                                      
##        'Positive' Class : 0          
##

m=predict(fit,newdata=test,type="response")
n=ifelse(m>0.5,1,0)
# Compute the confusion matrix for test output
confusionMatrix(n,test$output)

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction  0  1
##          0 52  4
##          1  5 81
##                                           
##                Accuracy : 0.9366          
##                  95% CI : (0.8831, 0.9706)
##     No Information Rate : 0.5986          
##     P-Value [Acc > NIR] : <2e-16          
##                                           
##                   Kappa : 0.8677          
##  Mcnemar's Test P-Value : 1               
##                                           
##             Sensitivity : 0.9123          
##             Specificity : 0.9529          
##          Pos Pred Value : 0.9286          
##          Neg Pred Value : 0.9419          
##              Prevalence : 0.4014          
##          Detection Rate : 0.3662          
##    Detection Prevalence : 0.3944          
##       Balanced Accuracy : 0.9326          
##                                           
##        'Positive' Class : 0               
##

1b. Logistic Regression – Python code

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.linear_model import LogisticRegression
os.chdir("C:\\Users\\Ganesh\\RandPython")
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)
# Call the Logisitic Regression function
clf = LogisticRegression().fit(X_train, y_train)
fig, subaxes = plt.subplots(1, 1, figsize=(7, 5))
# Fit a model
clf = LogisticRegression().fit(X_train, y_train)

# Compute and print the Accuray scores
print('Accuracy of Logistic regression classifier on training set: {:.2f}'
     .format(clf.score(X_train, y_train)))
print('Accuracy of Logistic regression classifier on test set: {:.2f}'
     .format(clf.score(X_test, y_test)))
y_predicted=clf.predict(X_test)
# Compute and print confusion matrix
confusion = confusion_matrix(y_test, y_predicted)
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
print('Accuracy: {:.2f}'.format(accuracy_score(y_test, y_predicted)))
print('Precision: {:.2f}'.format(precision_score(y_test, y_predicted)))
print('Recall: {:.2f}'.format(recall_score(y_test, y_predicted)))
print('F1: {:.2f}'.format(f1_score(y_test, y_predicted)))

## Accuracy of Logistic regression classifier on training set: 0.96
## Accuracy of Logistic regression classifier on test set: 0.96
## Accuracy: 0.96
## Precision: 0.99
## Recall: 0.94
## F1: 0.97

2. Dummy variables

The following R and Python code show how dummy variables are handled in R and Python. Dummy variables are categorival variables which have to be converted into appropriate values before using them in Machine Learning Model For e.g. if we had currency as ‘dollar’, ‘rupee’ and ‘yen’ then the dummy variable will convert this as
dollar 0 0 0
rupee 0 0 1
yen 0 1 0

2a. Logistic Regression with dummy variables- R code

# Load the dummies library
library(dummies)

df <- read.csv("adult1.csv",stringsAsFactors = FALSE,na.strings = c(""," "," ?"))

# Remove rows which have NA
df1 <- df[complete.cases(df),]
dim(df1)

## [1] 30161    16

# Select specific columns
adult <- df1 %>% dplyr::select(age,occupation,education,educationNum,capitalGain,
                               capital.loss,hours.per.week,native.country,salary)
# Set the dummy data with appropriate values
adult1 <- dummy.data.frame(adult, sep = ".")

#Split as training and test
train_idx <- trainTestSplit(adult1,trainPercent=75,seed=1111)
train <- adult1[train_idx, ]
test <- adult1[-train_idx, ]

# Fit a binomial logistic regression
fit=glm(salary~.,family=binomial,data=train)

# Predict response
a=predict(fit,newdata=train,type="response")

# If response >0.5 then it is a 1 and 0 otherwise
b=ifelse(a>0.5,1,0)
confusionMatrix(b,train$salary)

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction     0     1
##          0 16065  3145
##          1   968  2442
##                                           
##                Accuracy : 0.8182          
##                  95% CI : (0.8131, 0.8232)
##     No Information Rate : 0.753           
##     P-Value [Acc > NIR] : < 2.2e-16       
##                                           
##                   Kappa : 0.4375          
##  Mcnemar's Test P-Value : < 2.2e-16       
##                                           
##             Sensitivity : 0.9432          
##             Specificity : 0.4371          
##          Pos Pred Value : 0.8363          
##          Neg Pred Value : 0.7161          
##              Prevalence : 0.7530          
##          Detection Rate : 0.7102          
##    Detection Prevalence : 0.8492          
##       Balanced Accuracy : 0.6901          
##                                           
##        'Positive' Class : 0               
##

# Compute and display confusion matrix
m=predict(fit,newdata=test,type="response")

## Warning in predict.lm(object, newdata, se.fit, scale = 1, type =
## ifelse(type == : prediction from a rank-deficient fit may be misleading

n=ifelse(m>0.5,1,0)
confusionMatrix(n,test$salary)

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction    0    1
##          0 5263 1099
##          1  357  822
##                                           
##                Accuracy : 0.8069          
##                  95% CI : (0.7978, 0.8158)
##     No Information Rate : 0.7453          
##     P-Value [Acc > NIR] : < 2.2e-16       
##                                           
##                   Kappa : 0.4174          
##  Mcnemar's Test P-Value : < 2.2e-16       
##                                           
##             Sensitivity : 0.9365          
##             Specificity : 0.4279          
##          Pos Pred Value : 0.8273          
##          Neg Pred Value : 0.6972          
##              Prevalence : 0.7453          
##          Detection Rate : 0.6979          
##    Detection Prevalence : 0.8437          
##       Balanced Accuracy : 0.6822          
##                                           
##        'Positive' Class : 0               
##

2b. Logistic Regression with dummy variables- Python code

Pandas has a get_dummies function for handling dummies

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.linear_model import LogisticRegression
from sklearn.metrics import confusion_matrix
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
# Read data
df =pd.read_csv("adult1.csv",encoding="ISO-8859-1",na_values=[""," "," ?"])
# Drop rows with NA
df1=df.dropna()
print(df1.shape)
# Select specific columns
adult = df1[['age','occupation','education','educationNum','capitalGain','capital-loss', 
             'hours-per-week','native-country','salary']]

X=adult[['age','occupation','education','educationNum','capitalGain','capital-loss', 
             'hours-per-week','native-country']]
# Set approporiate values for dummy variables
X_adult=pd.get_dummies(X,columns=['occupation','education','native-country'])
y=adult['salary']

X_adult_train, X_adult_test, y_train, y_test = train_test_split(X_adult, y,
                                                   random_state = 0)
clf = LogisticRegression().fit(X_adult_train, y_train)

# Compute and display Accuracy and Confusion matrix
print('Accuracy of Logistic regression classifier on training set: {:.2f}'
     .format(clf.score(X_adult_train, y_train)))
print('Accuracy of Logistic regression classifier on test set: {:.2f}'
     .format(clf.score(X_adult_test, y_test)))
y_predicted=clf.predict(X_adult_test)
confusion = confusion_matrix(y_test, y_predicted)
print('Accuracy: {:.2f}'.format(accuracy_score(y_test, y_predicted)))
print('Precision: {:.2f}'.format(precision_score(y_test, y_predicted)))
print('Recall: {:.2f}'.format(recall_score(y_test, y_predicted)))
print('F1: {:.2f}'.format(f1_score(y_test, y_predicted)))

## (30161, 16)
## Accuracy of Logistic regression classifier on training set: 0.82
## Accuracy of Logistic regression classifier on test set: 0.81
## Accuracy: 0.81
## Precision: 0.68
## Recall: 0.41
## F1: 0.51

3a – K Nearest Neighbors Classification – R code

The Adult data set is taken from UCI Machine Learning Repository

source("RFunctions.R")
df <- read.csv("adult1.csv",stringsAsFactors = FALSE,na.strings = c(""," "," ?"))
# Remove rows which have NA
df1 <- df[complete.cases(df),]
dim(df1)

## [1] 30161    16

# Select specific columns
adult <- df1 %>% dplyr::select(age,occupation,education,educationNum,capitalGain,
                               capital.loss,hours.per.week,native.country,salary)
# Set dummy variables
adult1 <- dummy.data.frame(adult, sep = ".")

#Split train and test as required by KNN classsification model
train_idx <- trainTestSplit(adult1,trainPercent=75,seed=1111)
train <- adult1[train_idx, ]
test <- adult1[-train_idx, ]
train.X <- train[,1:76]
train.y <- train[,77]
test.X <- test[,1:76]
test.y <- test[,77]

# Fit a model for 1,3,5,10 and 15 neighbors
cMat <- NULL
neighbors <-c(1,3,5,10,15)
for(i in seq_along(neighbors)){
    fit =knn(train.X,test.X,train.y,k=i)
    table(fit,test.y)
    a<-confusionMatrix(fit,test.y)
    cMat[i] <- a$overall[1]
    print(a$overall[1])
}

##  Accuracy 
## 0.7835831 
##  Accuracy 
## 0.8162047 
##  Accuracy 
## 0.8089113 
##  Accuracy 
## 0.8209787 
##  Accuracy 
## 0.8184591

#Plot the Accuracy for each of the KNN models
df <- data.frame(neighbors,Accuracy=cMat)
ggplot(df,aes(x=neighbors,y=Accuracy)) + geom_point() +geom_line(color="blue") +
    xlab("Number of neighbors") + ylab("Accuracy") +
    ggtitle("KNN regression - Accuracy vs Number of Neighors (Unnormalized)")

3b – K Nearest Neighbors Classification – Python code

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.metrics import confusion_matrix
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
from sklearn.neighbors import KNeighborsClassifier
from sklearn.preprocessing import MinMaxScaler

# Read data
df =pd.read_csv("adult1.csv",encoding="ISO-8859-1",na_values=[""," "," ?"])
df1=df.dropna()
print(df1.shape)
# Select specific columns
adult = df1[['age','occupation','education','educationNum','capitalGain','capital-loss', 
             'hours-per-week','native-country','salary']]

X=adult[['age','occupation','education','educationNum','capitalGain','capital-loss', 
             'hours-per-week','native-country']]
             
#Set values for dummy variables
X_adult=pd.get_dummies(X,columns=['occupation','education','native-country'])
y=adult['salary']

X_adult_train, X_adult_test, y_train, y_test = train_test_split(X_adult, y,
                                                   random_state = 0)
                                                   
# KNN classification in Python requires the data to be scaled. 
# Scale the data
scaler = MinMaxScaler()
X_train_scaled = scaler.fit_transform(X_adult_train)
# Apply scaling to test set also
X_test_scaled = scaler.transform(X_adult_test)
# Compute the KNN model for 1,3,5,10 & 15 neighbors
accuracy=[]
neighbors=[1,3,5,10,15]
for i in neighbors:
    knn = KNeighborsClassifier(n_neighbors = i)
    knn.fit(X_train_scaled, y_train)
    accuracy.append(knn.score(X_test_scaled, y_test))
    print('Accuracy test score: {:.3f}'
        .format(knn.score(X_test_scaled, y_test)))

# Plot the models with the Accuracy attained for each of these models    
fig1=plt.plot(neighbors,accuracy)
fig1=plt.title("KNN regression - Accuracy vs Number of neighbors")
fig1=plt.xlabel("Neighbors")
fig1=plt.ylabel("Accuracy")
fig1.figure.savefig('foo1.png', bbox_inches='tight')

## (30161, 16)
## Accuracy test score: 0.749
## Accuracy test score: 0.779
## Accuracy test score: 0.793
## Accuracy test score: 0.804
## Accuracy test score: 0.803

Output image:

4 MPG vs Horsepower

The following scatter plot shows the non-linear relation between mpg and horsepower. This will be used as the data input for computing K Fold Cross Validation Error

4a MPG vs Horsepower scatter plot – R Code

df=read.csv("auto_mpg.csv",stringsAsFactors = FALSE) # Data from UCI
df1 <- as.data.frame(sapply(df,as.numeric))

df2 <- df1 %>% dplyr::select(cylinder,displacement, horsepower,weight, acceleration, year,mpg)
df3 <- df2[complete.cases(df2),]
ggplot(df3,aes(x=horsepower,y=mpg)) + geom_point() + xlab("Horsepower") + 
    ylab("Miles Per gallon") + ggtitle("Miles per Gallon vs Hosrsepower")

4b MPG vs Horsepower scatter plot – Python Code

import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
autoDF =pd.read_csv("auto_mpg.csv",encoding="ISO-8859-1")
autoDF.shape
autoDF.columns
autoDF1=autoDF[['mpg','cylinder','displacement','horsepower','weight','acceleration','year']]
autoDF2 = autoDF1.apply(pd.to_numeric, errors='coerce')
autoDF3=autoDF2.dropna()
autoDF3.shape
#X=autoDF3[['cylinder','displacement','horsepower','weight']]
X=autoDF3[['horsepower']]
y=autoDF3['mpg']

fig11=plt.scatter(X,y)
fig11=plt.title("KNN regression - Accuracy vs Number of neighbors")
fig11=plt.xlabel("Neighbors")
fig11=plt.ylabel("Accuracy")
fig11.figure.savefig('foo11.png', bbox_inches='tight')

5 K Fold Cross Validation

K Fold Cross Validation is a technique in which the data set is divided into K Folds or K partitions. The Machine Learning model is trained on K-1 folds and tested on the Kth fold i.e.
we will have K-1 folds for training data and 1 for testing the ML model. Since we can partition this as $C_{1}^{K}$ or K choose 1, there will be K such partitions. The K Fold Cross
Validation estimates the average validation error that we can expect on a new unseen test data.

The formula for K Fold Cross validation is as follows

$MSE_{K} = \frac{\sum (y-yhat)^{2}}{n_{K}}$
and
$n_{K} = \frac{N}{K}$
and
$CV_{K} = \sum_{K=1}^{K} (\frac{n_{K}}{N}) MSE_{K}$

where $n_{K}$ is the number of elements in partition ‘K’ and N is the total number of elements
$CV_{K} =\sum_{K=1}^{K} MSE_{K}$

$CV_{K} =\frac{\sum_{K=1}^{K} MSE_{K}}{K}$
Leave Out one Cross Validation (LOOCV) is a special case of K Fold Cross Validation where N-1 data points are used to train the model and 1 data point is used to test the model. There are N such paritions of N-1 & 1 that are possible. The mean error is measured The Cross Valifation Error for LOOCV is

$CV_{N} = \frac{1}{n} *\frac{\sum_{1}^{n}(y-yhat)^{2}}{1-h_{i}}$
where $h_{i}$ is the diagonal hat matrix

see [Statistical Learning]

The above formula is also included in this blog post

It took me a day and a half to implement the K Fold Cross Validation formula. I think it is correct. In any case do let me know if you think it is off

5a. Leave out one cross validation (LOOCV) – R Code

R uses the package ‘boot’ for performing Cross Validation error computation

library(boot)
library(reshape2)
# Read data
df=read.csv("auto_mpg.csv",stringsAsFactors = FALSE) # Data from UCI
df1 <- as.data.frame(sapply(df,as.numeric))

# Select complete cases
df2 <- df1 %>% dplyr::select(cylinder,displacement, horsepower,weight, acceleration, year,mpg)
df3 <- df2[complete.cases(df2),]
set.seed(17)
cv.error=rep(0,10)
# For polynomials 1,2,3... 10 fit a LOOCV model
for (i in 1:10){
    glm.fit=glm(mpg~poly(horsepower,i),data=df3)
    cv.error[i]=cv.glm(df3,glm.fit)$delta[1]
    
}
cv.error

##  [1] 24.23151 19.24821 19.33498 19.42443 19.03321 18.97864 18.83305
##  [8] 18.96115 19.06863 19.49093

# Create and display a plot
folds <- seq(1,10)
df <- data.frame(folds,cvError=cv.error)
ggplot(df,aes(x=folds,y=cvError)) + geom_point() +geom_line(color="blue") +
    xlab("Degree of Polynomial") + ylab("Cross Validation Error") +
    ggtitle("Leave one out Cross Validation - Cross Validation Error vs Degree of Polynomial")

5b. Leave out one cross validation (LOOCV) – Python Code

In Python there is no available function to compute Cross Validation error and we have to compute the above formula. I have done this after several hours. I think it is now in reasonable shape. Do let me know if you think otherwise. For LOOCV I use the K Fold Cross Validation with K=N

import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.cross_validation import train_test_split, KFold
from sklearn.preprocessing import PolynomialFeatures
from sklearn.metrics import mean_squared_error
# Read data
autoDF =pd.read_csv("auto_mpg.csv",encoding="ISO-8859-1")
autoDF.shape
autoDF.columns
autoDF1=autoDF[['mpg','cylinder','displacement','horsepower','weight','acceleration','year']]
autoDF2 = autoDF1.apply(pd.to_numeric, errors='coerce')
# Remove rows with NAs
autoDF3=autoDF2.dropna()
autoDF3.shape
X=autoDF3[['horsepower']]
y=autoDF3['mpg']

# For polynomial degree 1,2,3... 10
def computeCVError(X,y,folds):
    deg=[]
    mse=[]
    degree1=[1,2,3,4,5,6,7,8,9,10]
    
    nK=len(X)/float(folds)
    xval_err=0
    # For degree 'j'
    for j in degree1: 
        # Split as 'folds'
        kf = KFold(len(X),n_folds=folds)
        for train_index, test_index in kf:
            # Create the appropriate train and test partitions from the fold index
            X_train, X_test = X.iloc[train_index], X.iloc[test_index]
            y_train, y_test = y.iloc[train_index], y.iloc[test_index]  

            # For the polynomial degree 'j'
            poly = PolynomialFeatures(degree=j)        
            # Transform the X_train and X_test
            X_train_poly = poly.fit_transform(X_train)
            X_test_poly = poly.fit_transform(X_test)
            # Fit a model on the transformed data
            linreg = LinearRegression().fit(X_train_poly, y_train)
            # Compute yhat or ypred
            y_pred = linreg.predict(X_test_poly)   
            # Compute MSE * n_K/N
            test_mse = mean_squared_error(y_test, y_pred)*float(len(X_train))/float(len(X))     
            # Add the test_mse for this partition of the data
            mse.append(test_mse)
        # Compute the mean of all folds for degree 'j'   
        deg.append(np.mean(mse))
        
    return(deg)


df=pd.DataFrame()
print(len(X))
# Call the function once. For LOOCV K=N. hence len(X) is passed as number of folds
cvError=computeCVError(X,y,len(X))

# Create and plot LOOCV
df=pd.DataFrame(cvError)
fig3=df.plot()
fig3=plt.title("Leave one out Cross Validation - Cross Validation Error vs Degree of Polynomial")
fig3=plt.xlabel("Degree of Polynomial")
fig3=plt.ylabel("Cross validation Error")
fig3.figure.savefig('foo3.png', bbox_inches='tight')

6a K Fold Cross Validation – R code

Here K Fold Cross Validation is done for 4, 5 and 10 folds using the R package boot and the glm package

library(boot)
library(reshape2)
set.seed(17)
#Read data
df=read.csv("auto_mpg.csv",stringsAsFactors = FALSE) # Data from UCI
df1 <- as.data.frame(sapply(df,as.numeric))

df2 <- df1 %>% dplyr::select(cylinder,displacement, horsepower,weight, acceleration, year,mpg)
df3 <- df2[complete.cases(df2),]
a=matrix(rep(0,30),nrow=3,ncol=10)
set.seed(17)
# Set the folds as 4,5 and 10
folds<-c(4,5,10)
for(i in seq_along(folds)){
    cv.error.10=rep(0,10)
    for (j in 1:10){
        # Fit a generalized linear model
        glm.fit=glm(mpg~poly(horsepower,j),data=df3)
        # Compute K Fold Validation error
        a[i,j]=cv.glm(df3,glm.fit,K=folds[i])$delta[1]
        
    }
    
}

# Create and display the K Fold Cross Validation Error
b <- t(a)
df <- data.frame(b)
df1 <- cbind(seq(1,10),df)
names(df1) <- c("PolynomialDegree","4-fold","5-fold","10-fold")

df2 <- melt(df1,id="PolynomialDegree")
ggplot(df2) + geom_line(aes(x=PolynomialDegree, y=value, colour=variable),size=2) +
    xlab("Degree of Polynomial") + ylab("Cross Validation Error") +
    ggtitle("K Fold Cross Validation - Cross Validation Error vs Degree of Polynomial")

6b. K Fold Cross Validation – Python code

The implementation of K-Fold Cross Validation Error has to be implemented and I have done this below. There is a small discrepancy in the shapes of the curves with the R plot above. Not sure why!

import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.cross_validation import train_test_split, KFold
from sklearn.preprocessing import PolynomialFeatures
from sklearn.metrics import mean_squared_error
# Read data
autoDF =pd.read_csv("auto_mpg.csv",encoding="ISO-8859-1")
autoDF.shape
autoDF.columns
autoDF1=autoDF[['mpg','cylinder','displacement','horsepower','weight','acceleration','year']]
autoDF2 = autoDF1.apply(pd.to_numeric, errors='coerce')
# Drop NA rows
autoDF3=autoDF2.dropna()
autoDF3.shape
#X=autoDF3[['cylinder','displacement','horsepower','weight']]
X=autoDF3[['horsepower']]
y=autoDF3['mpg']

# Create Cross Validation function
def computeCVError(X,y,folds):
    deg=[]
    mse=[]
    # For degree 1,2,3,..10
    degree1=[1,2,3,4,5,6,7,8,9,10]
    
    nK=len(X)/float(folds)
    xval_err=0
    for j in degree1: 
        # Split the data into 'folds'
        kf = KFold(len(X),n_folds=folds)
        for train_index, test_index in kf:
            # Partition the data acccording the fold indices generated
            X_train, X_test = X.iloc[train_index], X.iloc[test_index]
            y_train, y_test = y.iloc[train_index], y.iloc[test_index]  

            # Scale the X_train and X_test as per the polynomial degree 'j'
            poly = PolynomialFeatures(degree=j)             
            X_train_poly = poly.fit_transform(X_train)
            X_test_poly = poly.fit_transform(X_test)
            # Fit a polynomial regression
            linreg = LinearRegression().fit(X_train_poly, y_train)
            # Compute yhat or ypred
            y_pred = linreg.predict(X_test_poly)  
            # Compute MSE *(nK/N)
            test_mse = mean_squared_error(y_test, y_pred)*float(len(X_train))/float(len(X))  
            # Append to list for different folds
            mse.append(test_mse)
        # Compute the mean for poylnomial 'j' 
        deg.append(np.mean(mse))
        
    return(deg)

# Create and display a plot of K -Folds
df=pd.DataFrame()
for folds in [4,5,10]:
    cvError=computeCVError(X,y,folds)
    #print(cvError)
    df1=pd.DataFrame(cvError)
    df=pd.concat([df,df1],axis=1)
    #print(cvError)
    
df.columns=['4-fold','5-fold','10-fold']
df=df.reindex([1,2,3,4,5,6,7,8,9,10])
df
fig2=df.plot()
fig2=plt.title("K Fold Cross Validation - Cross Validation Error vs Degree of Polynomial")
fig2=plt.xlabel("Degree of Polynomial")
fig2=plt.ylabel("Cross validation Error")
fig2.figure.savefig('foo2.png', bbox_inches='tight')

output

This concludes this 2nd part of this series. I will look into model tuning and model selection in R and Python in the coming parts. Comments, suggestions and corrections are welcome!
To be continued….
Watch this space!

Also see

To see all posts see Index of posts

My travels through the realms of Data Science, Machine Learning, Deep Learning and (AI)

Then felt I like some watcher of the skies

When a new planet swims into his ken;

Or like stout Cortez when with eagle eyes

He star’d at the Pacific—and all his men

Look’d at each other with a wild surmise—

Silent, upon a peak in Darien.

On First Looking into Chapman’s Homer by John Keats

The above excerpt from John Keat’s poem captures the the exhilaration that one experiences, when discovering something for the first time. This also summarizes to some extent my own as enjoyment while pursuing Data Science, Machine Learning and the like.

I decided to write this post, as occasionally youngsters approach me and ask me where they should start their adventure in Data Science & Machine Learning. There are other times, when the ‘not-so-youngsters’ want to know what their next step should be after having done some courses. This post includes my travels through the domains of Data Science, Machine Learning, Deep Learning and (soon to be done AI).

By no means, am I an authority in this field, which is ever-widening and almost bottomless, yet I would like to share some of my experiences in this fascinating field. I include a short review of the courses I have done below. I also include alternative routes through courses which I did not do, but are probably equally good as well. Feel free to pick and choose any course or set of courses. Alternatively, you may prefer to read books or attend bricks-n-mortar classes, In any case, I hope the list below will provide you with some overall direction.

All my learning in the above domains have come from MOOCs and I restrict myself to the top 3 MOOCs, or in my opinion, ‘the original MOOCs’, namely Coursera, edX or Udacity, but may throw in some courses from other online sites if they are only available there. I would recommend these 3 MOOCs over the other numerous online courses and also over face-to-face classroom courses for the following reasons. These MOOCs

Are taken by world class colleges and the lectures are delivered by top class Professors who have a great depth of knowledge and a wealth of experience
The Professors, besides delivering quality content, also point out to important tips, tricks and traps
You can revisit lectures in online courses anytime to refresh your memory
Lectures are usually short between 8 -15 mins (Personally, my attention span is around 15-20 mins at a time!)

Here is a fair warning and something quite obvious. No amount of courses, lectures or books will help if you don’t put it to use through some language like Octave, R or Python.

The journey
My trip through Data Science, Machine Learning started with an off-chance remark,about 3 years ago, from an old friend of mine who spoke to me about having done a few courses at Coursera, and really liked it. He further suggested that I should try. This was the final push which set me sailing into this vast domain.

I have included the list of the courses I have done over the past 5 years (37+ certifications completed and another 9 audited-listened only without doing the assignments). For each of the courses I have included a short review of the course, whether I think the course is mandatory, the language in which the course is based on, and finally whether I have done the course myself etc. I have also included alternative courses, which I may have not done, but which I think are equally good. Finally, I suggest some courses which I have heard of and which are very good and worth taking.

1. Machine Learning, Stanford, Prof Andrew Ng, Coursera
(Requirement: Mandatory, Language:Octave,Status:Completed)
This course provides an excellent foundation to build your Machine Learning citadel on. The course covers the mathematical details of linear, logistic and multivariate regression. There is also a good coverage of topics like Neural Networks, SVMs, Anamoly Detection, underfitting, overfitting, regularization etc. Prof Andrew Ng presents the material in a very lucid manner. It is a great course to start with. It would be a good idea to brush up some basics of linear algebra, matrices and a little bit of calculus, specifically computing the local maxima/minima. You should be able to take this course even if you don’t know Octave as the Prof goes over the key aspects of the language.

2. Statistical Learning, Prof Trevor Hastie & Prof Robert Tibesherani, Online Stanford– (Requirement:Mandatory, Language:R, Status;Completed) –
The course includes linear and polynomial regression, logistic regression. Details also include cross-validation and the bootstrap methods, how to do model selection and regularization (ridge and lasso). It also touches on non-linear models, generalized additive models, boosting and SVMs. Some unsupervised learning methods are also discussed. The 2 Professors take turns in delivering lectures with a slight touch of humor.

3a. Data Science Specialization: Prof Roger Peng, Prof Brian Caffo & Prof Jeff Leek, John Hopkins University (Requirement: Option A, Language: R Status: Completed)
This is a comprehensive 10 module specialization based on R. This Specialization gives a very broad overview of Data Science and Machine Learning. The modules cover R programming, Statistical Inference, Practical Machine Learning, how to build R products and R packages and finally has a very good Capstone project on NLP

3b. Applied Data Science with Python Specialization: University of Michigan (Requirement: Option B, Language: Python, Status: Not done)
In this specialization I only did the Applied Machine Learning in Python (Prof Kevyn-Collin Thomson). This is a very good course that covers a lot of Machine Learning algorithms(linear, logistic, ridge, lasso regression, knn, SVMs etc. Also included are confusion matrices, ROC curves etc. This is based on Python’s Scikit Learn

3c. Machine Learning Specialization, University Of Washington (Requirement:Option C, Language:Python, Status : Not completed). This appears to be a very good Specialization in Python

4. Statistics with R Specialization, Duke University (Requirement: Useful and a must know, Language R, Status:Not Completed)
I audited (listened only) to the following 2 modules from this Specialization.
a.Inferential Statistics
b.Linear Regression and Modeling
Both these courses are taught by Prof Mine Cetikya-Rundel who delivers her lessons with extraordinary clarity. Her lectures are filled with many examples which she walks you through in great detail

5.Bayesian Statistics: From Concept to Data Analysis: Univ of California, Santa Cruz (Requirement: Optional, Language : R, Status:Completed)
This is an interesting course and provides an alternative point of view to frequentist approach

6. Data Science and Engineering with Spark, University of California, Berkeley, Prof Antony Joseph, Prof Ameet Talwalkar, Prof Jon Bates
(Required: Mandatory for Big Data, Status:Completed, Language; pySpark)
This specialization contains 3 modules
a.Introduction to Apache Spark
b.Distributed Machine Learning with Apache Spark
c.Big Data Analysis with Apache Spark

This is an excellent course for those who want to make an entry into Distributed Machine Learning. The exercises are fairly challenging and your code will predominantly be made of map/reduce and lambda operations as you process data that is distributed across Spark RDDs. I really liked the part where the Prof shows how a matrix multiplication on a single machine is of the order of O(nd^2+d^3) (which is the basis of Machine Learning) is reduced to O(nd^2) by taking outer products on data which is distributed.

7. Deep Learning Prof Andrew Ng, Younes Bensouda Mourri, Kian Katanforoosh : Requirement:Mandatory,Language:Python, Tensorflow Status:Completed)

This course had 5 Modules which start from the fundamentals of Neural Networks, their derivation and vectorized Python implementation. The specialization also covers regularization, optimization techniques, mini batch normalization, Convolutional Neural Networks, Recurrent Neural Networks, LSTMs applied to a wide variety of real world problems

The modules are
a. Neural Networks and Deep Learning
In this course Prof Andrew Ng explains differential calculus, linear algebra and vectorized Python implementations of Deep Learning algorithms. The derivation for back-propagation is done and then the Prof shows how to compute a multi-layered DL network
b.Improving Deep Neural Networks: Hyperparameter tuning, Regularization and Optimization
Deep Neural Networks can be very flexible, and come with a lots of knobs (hyper-parameters) to tune with. In this module, Prof Andrew Ng shows a systematic way to tune hyperparameters and by how much should one tune. The course also covers regularization(L1,L2,dropout), gradient descent optimization and batch normalization methods. The visualizations used to explain the momentum method, RMSprop, Adam,LR decay and batch normalization are really powerful and serve to clarify the concepts. As an added bonus,the module also includes a great introduction to Tensorflow.
c.Structuring Machine Learning Projects
A very good module with useful tips, tricks and traps that need to be considered while working on Machine Learning and Deep Learning projects
d. Convolutional Neural Networks
This domain has a lot of really cool ideas, where images represented as 3D volumes, are compressed and stretched longitudinally before applying a multi-layered deep learning neural network to this thin slice for performing classification,detection etc. The Prof provides a glimpse into this fascinating world of image classification, detection andl neural art transfer with frameworks like Keras and Tensorflow.
e. Sequence Models
In this module covers in good detail concepts like RNNs, GRUs, LSTMs, word embeddings, beam search and attention model.

8. Neural Networks for Machine Learning, Prof Geoffrey Hinton,University of Toronto
(Requirement: Mandatory, Language;Octave, Status:Completed)
This is a broad course which starts from the basic of Perceptrons, all the way to Boltzman Machines, RNNs, CNNS, LSTMs etc The course also covers regularisation, learning rate decay, momentum method etc

9.Probabilistic Graphical Models, Stanford Prof Daphne Koller(Language:Octave, Status: Partially completed)
This has 3 courses
a.Probabilistic Graphical Models 1: Representation – Done
b.Probabilistic Graphical Models 2: Inference – To do
c.Probabilistic Graphical Models 3: Learning – To do
This course discusses how a system, which can be represented as a complex interaction
of probability distributions, will behave. This is probably the toughest course I did. I did manage to get through the 1st module, While I felt that grasped a few things, I did not wholly understand the import of this. However I feel this is an important domain and I will definitely revisit this in future

10. Reinforcement Specialization : University of Alberta, Prof Adam White and Prof Martha White
(Requirement: Very important, Language;Python, Status: Partially Completed)
This is a set of 4 courses. I did the first 2 of the 4. Reinforcement Learning appears deceptively simple, but it is anything but simple. Definitely a very critical area to learn.

a.Fundamentals of Reinforcement Learning: This course discusses Markov models, value functions and Bellman equations and dynamic programming.
b.Sample based learning Learning methods: This course touches on Monte Carlo methods, Temporal Difference methods, Q Learning etc.

Reinforcement Learning is a must-have in your AI arsenal.

11. Tensorflow in Practice Specialization – Prof Laurence Moroney – Deep Learning.AI
(Requirement: Important, Language;Python, Status: Completed)
This is a good course but definitely do the Deep Learning Specialization by Prof Andrew Ng
There are 4 courses in this Specialization. I completed all 4 courses. They are fairly straight forward
a. Introduction to TensorFlow – This course introduces you to Tensorflow, image recognition with brute-force method
b. Convolutional Neural Networks in Tensorflow – This course touches on how to build a CNN, image augmentation, transfer learning and multi-class classification
c. Natural Language Processing in Tensorflow – Word embeddings, sentiment analysis, LSTMs, RNNs are discussed.
d. Sequences, time series and prediction – This course discusses using RNNs for time series, auto correlation

12. Natural Language Processing Specialization – Prof Younes Bensouda, Lukasz Kaiser from DeepLearning.AI
(Requirement: Very Important, Language;Python, Status: Partially Completed)
This is the latest specialization from Deep Learning.AI. I have completed the first 2 courses
a.Natural Language Processing with Classification and Vector Spaces -The first course deals with sentiment analysis with Naive Bayes, vector space models, capturing dependencies using PCA etc
b. Natural Language Processing with Probabilistic Models – In this course techniques for auto correction, Markov models and Viterbi algorithm for Parts of Speech tagging, auto completion and word embedding are discussed.

13. Mining Massive Data Sets Prof Jure Leskovec, Prof Anand Rajaraman and ProfJeff Ullman. Online Stanford, Status Partially done.,
I did quickly audit this course, a year back, when it used to be in Coursera. It now seems to have moved to Stanford online. But this is a very good course that discusses key concepts of Mining Big Data of the order a few Petabytes

14. Introduction to Artificial Intelligence, Prof Sebastian Thrun & Prof Peter Norvig, Udacity
This is a really good course. I have started on this course a couple of times and somehow gave up. Will revisit to complete in future. Quite extensive in its coverage.Touches BFS,DFS, A-Star, PGM, Machine Learning etc.

15.Deep Learning (with TensorFlow), Vincent Vanhoucke, Principal Scientist at Google Brain.
Got started on this one and abandoned some time back. In my to do list though

My learning journey is based on Lao Tzu’s dictum of ‘A good traveler has no fixed plans and is not intent on arriving’. You could have a goal and try to plan your courses accordingly.
And so my journey continues…

I hope you find this list useful.
Have a great journey ahead!!!