# 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

# Introducing cricpy:A python package to analyze performances of cricketers

Full many a gem of purest ray serene,
The dark unfathomed caves of ocean bear;
Full many a flower is born to blush unseen,
And waste its sweetness on the desert air.

            Thomas Gray, An Elegy Written In A Country Churchyard


# Introduction

It is finally here! cricpy, the python avatar , of my R package cricketr is now ready to rock-n-roll! My R package cricketr had its genesis about 3 and some years ago and went through a couple of enhancements. During this time I have always thought about creating an equivalent python package like cricketr. Now I have finally done it.

So here it is. My python package ‘cricpy!!!’

This package uses the statistics info available in ESPN Cricinfo Statsguru. The current version of this package supports only Test cricket

You should be able to install the package using pip install cricpy and use the many functions available in the package. Please mindful of the ESPN Cricinfo Terms of Use

Do check out my post on R package cricketr at Re-introducing cricketr! : An R package to analyze performances of cricketers

If you are passionate about cricket, and love analyzing cricket performances, then check out my 2 racy books on cricket! In my books, I perform detailed yet compact analysis of performances of both batsmen, bowlers besides evaluating team & match performances in Tests , ODIs, T20s. You can buy my books on cricket from Amazon at $12.99 for the paperback and$4.99/$6.99 respectively for the kindle versions. The books can be accessed at Cricket analytics with cricketr and Beaten by sheer pace-Cricket analytics with yorkr A must read for any cricket lover! Check it out!! This package uses the statistics info available in ESPN Cricinfo Statsguru. Note: If you would like to do a similar analysis for a different set of batsman and bowlers, you can clone/download my skeleton cricpy-template from Github (which is the R Markdown file I have used for the analysis below). You will only need to make appropriate changes for the players you are interested in. The functions can be executed in RStudio or in a IPython notebook. # The cricpy package The cricpy package has several functions that perform several different analyses on both batsman and bowlers. The package has functions that plot percentage frequency runs or wickets, runs likelihood for a batsman, relative run/strike rates of batsman and relative performance/economy rate for bowlers are available. Other interesting functions include batting performance moving average, forecasting, performance of a player against different oppositions, contribution to wins and losses etc. The data for a particular player can be obtained with the getPlayerData() function. To do this you will need to go to ESPN CricInfo Player and type in the name of the player for e.g Rahul Dravid, Virat Kohli, Alastair Cook etc. This will bring up a page which have the profile number for the player e.g. for Rahul Dravid this would be http://www.espncricinfo.com/india/content/player/28114.html. Hence, Dravid’s profile is 28114. This can be used to get the data for Rahul Dravid as shown below The cricpy package is almost a clone of my R package cricketr. The signature of all the python functions are identical with that of its R avatar namely ‘cricketr’, with only the necessary variations between Python and R. It may be useful to look at my post R vs Python: Different similarities and similar differences. In fact if you are familiar with one of the languages you can look up the package in the other and you will notice the parallel constructs. You can fork/clone the cricpy package at Github cricpy The following 2 examples show the similarity between cricketr and cricpy packages ## 1a.Importing cricketr – R Importing cricketr in R #install.packages("cricketr") library(cricketr) ## 2a. Importing cricpy – Python # Install the package # Do a pip install cricpy # Import cricpy import cricpy # You could either do #1. import cricpy.analytics as ca #ca.batsman4s("../dravid.csv","Rahul Dravid") # Or #2. from cricpy.analytics import * #batsman4s("../dravid.csv","Rahul Dravid")  I would recommend using option 1 namely ca.batsman4s() as I may add an advanced analytics module in the future to cricpy. ## 2 Invoking functions You can seen how the 2 calls are identical for both the R package cricketr and the Python package cricpy ## 2a. Invoking functions with R package ‘cricketr’ library(cricketr) batsman4s("../dravid.csv","Rahul Dravid") ## 2b. Invoking functions with Python package ‘cricpy’ import cricpy.analytics as ca ca.batsman4s("../dravid.csv","Rahul Dravid") # 3a. Getting help from cricketr – R #help("getPlayerData") # 3b. Getting help from cricpy – Python help(ca.getPlayerData) ## Help on function getPlayerData in module cricpy.analytics: ## ## getPlayerData(profile, opposition='', host='', dir='./data', file='player001.csv', type='batting', homeOrAway=[1, 2], result=[1, 2, 4], create=True) ## Get the player data from ESPN Cricinfo based on specific inputs and store in a file in a given directory ## ## Description ## ## Get the player data given the profile of the batsman. The allowed inputs are home,away or both and won,lost or draw of matches. The data is stored in a <player>.csv file in a directory specified. This function also returns a data frame of the player ## ## Usage ## ## getPlayerData(profile,opposition="",host="",dir="./data",file="player001.csv", ## type="batting", homeOrAway=c(1,2),result=c(1,2,4)) ## Arguments ## ## profile ## This is the profile number of the player to get data. This can be obtained from http://www.espncricinfo.com/ci/content/player/index.html. Type the name of the player and click search. This will display the details of the player. Make a note of the profile ID. For e.g For Sachin Tendulkar this turns out to be http://www.espncricinfo.com/india/content/player/35320.html. Hence the profile for Sachin is 35320 ## opposition ## The numerical value of the opposition country e.g.Australia,India, England etc. The values are Australia:2,Bangladesh:25,England:1,India:6,New Zealand:5,Pakistan:7,South Africa:3,Sri Lanka:8, West Indies:4, Zimbabwe:9 ## host ## The numerical value of the host country e.g.Australia,India, England etc. The values are Australia:2,Bangladesh:25,England:1,India:6,New Zealand:5,Pakistan:7,South Africa:3,Sri Lanka:8, West Indies:4, Zimbabwe:9 ## dir ## Name of the directory to store the player data into. If not specified the data is stored in a default directory "./data". Default="./data" ## file ## Name of the file to store the data into for e.g. tendulkar.csv. This can be used for subsequent functions. Default="player001.csv" ## type ## type of data required. This can be "batting" or "bowling" ## homeOrAway ## This is a list with either 1,2 or both. 1 is for home 2 is for away ## result ## This is a list that can take values 1,2,4. 1 - won match 2- lost match 4- draw ## Details ## ## More details can be found in my short video tutorial in Youtube https://www.youtube.com/watch?v=q9uMPFVsXsI ## ## Value ## ## Returns the player's dataframe ## ## Note ## ## Maintainer: Tinniam V Ganesh <tvganesh.85@gmail.com> ## ## Author(s) ## ## Tinniam V Ganesh ## ## References ## ## http://www.espncricinfo.com/ci/content/stats/index.html ## https://gigadom.wordpress.com/ ## ## See Also ## ## getPlayerDataSp ## ## Examples ## ## ## Not run: ## # Both home and away. Result = won,lost and drawn ## tendulkar = getPlayerData(35320,dir=".", file="tendulkar1.csv", ## type="batting", homeOrAway=[1,2],result=[1,2,4]) ## ## # Only away. Get data only for won and lost innings ## tendulkar = getPlayerData(35320,dir=".", file="tendulkar2.csv", ## type="batting",homeOrAway=[2],result=[1,2]) ## ## # Get bowling data and store in file for future ## kumble = getPlayerData(30176,dir=".",file="kumble1.csv", ## type="bowling",homeOrAway=[1],result=[1,2]) ## ## #Get the Tendulkar's Performance against Australia in Australia ## tendulkar = getPlayerData(35320, opposition = 2,host=2,dir=".", ## file="tendulkarVsAusInAus.csv",type="batting") The details below will introduce the different functions that are available in cricpy. ## 3. Get the player data for a player using the function getPlayerData() Important Note This needs to be done only once for a player. This function stores the player’s data in the specified CSV file (for e.g. dravid.csv as above) which can then be reused for all other functions). Once we have the data for the players many analyses can be done. This post will use the stored CSV file obtained with a prior getPlayerData for all subsequent analyses import cricpy.analytics as ca #dravid =ca.getPlayerData(28114,dir="..",file="dravid.csv",type="batting",homeOrAway=[1,2], result=[1,2,4]) #acook =ca.getPlayerData(11728,dir="..",file="acook.csv",type="batting",homeOrAway=[1,2], result=[1,2,4]) import cricpy.analytics as ca #lara =ca.getPlayerData(52337,dir="..",file="lara.csv",type="batting",homeOrAway=[1,2], result=[1,2,4])253802 #kohli =ca.getPlayerData(253802,dir="..",file="kohli.csv",type="batting",homeOrAway=[1,2], result=[1,2,4]) ## 4 Rahul Dravid’s performance – Basic Analyses The 3 plots below provide the following for Rahul Dravid 1. Frequency percentage of runs in each run range over the whole career 2. Mean Strike Rate for runs scored in the given range 3. A histogram of runs frequency percentages in runs ranges import cricpy.analytics as ca import matplotlib.pyplot as plt ca.batsmanRunsFreqPerf("../dravid.csv","Rahul Dravid") ca.batsmanMeanStrikeRate("../dravid.csv","Rahul Dravid") ca.batsmanRunsRanges("../dravid.csv","Rahul Dravid")  ## 5. More analyses import cricpy.analytics as ca ca.batsman4s("../dravid.csv","Rahul Dravid") ca.batsman6s("../dravid.csv","Rahul Dravid")  ca.batsmanDismissals("../dravid.csv","Rahul Dravid") ## 6. 3D scatter plot and prediction plane The plots below show the 3D scatter plot of Dravid Runs versus Balls Faced and Minutes at crease. A linear regression plane is then fitted between Runs and Balls Faced + Minutes at crease import cricpy.analytics as ca ca.battingPerf3d("../dravid.csv","Rahul Dravid") ## 7. Average runs at different venues The plot below gives the average runs scored by Dravid at different grounds. The plot also the number of innings at each ground as a label at x-axis. It can be seen Dravid did great in Rawalpindi, Leeds, Georgetown overseas and , Mohali and Bangalore at home import cricpy.analytics as ca ca.batsmanAvgRunsGround("../dravid.csv","Rahul Dravid") ## 8. Average runs against different opposing teams This plot computes the average runs scored by Dravid against different countries. Dravid has an average of 50+ in England, New Zealand, West Indies and Zimbabwe. import cricpy.analytics as ca ca.batsmanAvgRunsOpposition("../dravid.csv","Rahul Dravid") ## 9 . Highest Runs Likelihood The plot below shows the Runs Likelihood for a batsman. For this the performance of Sachin is plotted as a 3D scatter plot with Runs versus Balls Faced + Minutes at crease. K-Means. The centroids of 3 clusters are computed and plotted. In this plot Dravid’s highest tendencies are computed and plotted using K-Means import cricpy.analytics as ca ca.batsmanRunsLikelihood("../dravid.csv","Rahul Dravid") ## 10. A look at the Top 4 batsman – Rahul Dravid, Alastair Cook, Brian Lara and Virat Kohli The following batsmen have been very prolific in test cricket and will be used for teh analyses 1. Rahul Dravid :Average:52.31,100’s – 36, 50’s – 63 2. Alastair Cook : Average: 45.35, 100’s – 33, 50’s – 57 3. Brian Lara : Average: 52.88, 100’s – 34 , 50’s – 48 4. Virat Kohli: Average: 54.57 ,100’s – 24 , 50’s – 19 The following plots take a closer at their performances. The box plots show the median the 1st and 3rd quartile of the runs ## 11. Box Histogram Plot This plot shows a combined boxplot of the Runs ranges and a histogram of the Runs Frequency import cricpy.analytics as ca ca.batsmanPerfBoxHist("../dravid.csv","Rahul Dravid") ca.batsmanPerfBoxHist("../acook.csv","Alastair Cook") ca.batsmanPerfBoxHist("../lara.csv","Brian Lara") ca.batsmanPerfBoxHist("../kohli.csv","Virat Kohli") ## 12. Contribution to won and lost matches The plot below shows the contribution of Dravid, Cook, Lara and Kohli in matches won and lost. It can be seen that in matches where India has won Dravid and Kohli have scored more and must have been instrumental in the win For the 2 functions below you will have to use the getPlayerDataSp() function as shown below. I have commented this as I already have these files import cricpy.analytics as ca #dravidsp = ca.getPlayerDataSp(28114,tdir=".",tfile="dravidsp.csv",ttype="batting") #acooksp = ca.getPlayerDataSp(11728,tdir=".",tfile="acooksp.csv",ttype="batting") #larasp = ca.getPlayerDataSp(52337,tdir=".",tfile="larasp.csv",ttype="batting") #kohlisp = ca.getPlayerDataSp(253802,tdir=".",tfile="kohlisp.csv",ttype="batting") import cricpy.analytics as ca ca.batsmanContributionWonLost("../dravidsp.csv","Rahul Dravid") ca.batsmanContributionWonLost("../acooksp.csv","Alastair Cook") ca.batsmanContributionWonLost("../larasp.csv","Brian Lara") ca.batsmanContributionWonLost("../kohlisp.csv","Virat Kohli") ## 13. Performance at home and overseas From the plot below it can be seen Dravid has a higher median overseas than at home.Cook, Lara and Kohli have a lower median of runs overseas than at home. This function also requires the use of getPlayerDataSp() as shown above import cricpy.analytics as ca ca.batsmanPerfHomeAway("../dravidsp.csv","Rahul Dravid") ca.batsmanPerfHomeAway("../acooksp.csv","Alastair Cook") ca.batsmanPerfHomeAway("../larasp.csv","Brian Lara") ca.batsmanPerfHomeAway("../kohlisp.csv","Virat Kohli") ## 14 Moving Average of runs in career Take a look at the Moving Average across the career of the Top 4 (ignore the dip at the end of all plots. Need to check why this is so!). Lara’s performance seems to have been quite good before his retirement(wonder why retired so early!). Kohli’s performance has been steadily improving over the years import cricpy.analytics as ca ca.batsmanMovingAverage("../dravid.csv","Rahul Dravid") ca.batsmanMovingAverage("../acook.csv","Alastair Cook") ca.batsmanMovingAverage("../lara.csv","Brian Lara") ca.batsmanMovingAverage("../kohli.csv","Virat Kohli") ## 15 Cumulative Average runs of batsman in career This function provides the cumulative average runs of the batsman over the career. Dravid averages around 48, Cook around 44, Lara around 50 and Kohli shows a steady improvement in his cumulative average. Kohli seems to be getting better with time. import cricpy.analytics as ca ca.batsmanCumulativeAverageRuns("../dravid.csv","Rahul Dravid") ca.batsmanCumulativeAverageRuns("../acook.csv","Alastair Cook") ca.batsmanCumulativeAverageRuns("../lara.csv","Brian Lara") ca.batsmanCumulativeAverageRuns("../kohli.csv","Virat Kohli") ## 16 Cumulative Average strike rate of batsman in career Lara has a terrific strike rate of 52+. Cook has a better strike rate over Dravid. Kohli’s strike rate has improved over the years. import cricpy.analytics as ca ca.batsmanCumulativeStrikeRate("../dravid.csv","Rahul Dravid") ca.batsmanCumulativeStrikeRate("../acook.csv","Alastair Cook") ca.batsmanCumulativeStrikeRate("../lara.csv","Brian Lara") ca.batsmanCumulativeStrikeRate("../kohli.csv","Virat Kohli") ## 17 Future Runs forecast Here are plots that forecast how the batsman will perform in future. Currently ARIMA has been used for the forecast. (To do: Perform Holt-Winters forecast!) import cricpy.analytics as ca ca.batsmanPerfForecast("../dravid.csv","Rahul Dravid") ## ARIMA Model Results ## ============================================================================== ## Dep. Variable: D.runs No. Observations: 284 ## Model: ARIMA(5, 1, 0) Log Likelihood -1522.837 ## Method: css-mle S.D. of innovations 51.488 ## Date: Sun, 28 Oct 2018 AIC 3059.673 ## Time: 09:47:39 BIC 3085.216 ## Sample: 07-04-1996 HQIC 3069.914 ## - 01-24-2012 ## ================================================================================ ## coef std err z P>|z| [0.025 0.975] ## -------------------------------------------------------------------------------- ## const -0.1336 0.884 -0.151 0.880 -1.867 1.599 ## ar.L1.D.runs -0.7729 0.058 -13.322 0.000 -0.887 -0.659 ## ar.L2.D.runs -0.6234 0.071 -8.753 0.000 -0.763 -0.484 ## ar.L3.D.runs -0.5199 0.074 -7.038 0.000 -0.665 -0.375 ## ar.L4.D.runs -0.3490 0.071 -4.927 0.000 -0.488 -0.210 ## ar.L5.D.runs -0.2116 0.058 -3.665 0.000 -0.325 -0.098 ## Roots ## ============================================================================= ## Real Imaginary Modulus Frequency ## ----------------------------------------------------------------------------- ## AR.1 0.5789 -1.1743j 1.3093 -0.1771 ## AR.2 0.5789 +1.1743j 1.3093 0.1771 ## AR.3 -1.3617 -0.0000j 1.3617 -0.5000 ## AR.4 -0.7227 -1.2257j 1.4230 -0.3348 ## AR.5 -0.7227 +1.2257j 1.4230 0.3348 ## ----------------------------------------------------------------------------- ## 0 ## count 284.000000 ## mean -0.306769 ## std 51.632947 ## min -106.653589 ## 25% -33.835148 ## 50% -8.954253 ## 75% 21.024763 ## max 223.152901 ## ## C:\Users\Ganesh\ANACON~1\lib\site-packages\statsmodels\tsa\kalmanf\kalmanfilter.py:646: FutureWarning: Conversion of the second argument of issubdtype from float to np.floating is deprecated. In future, it will be treated as np.float64 == np.dtype(float).type. ## if issubdtype(paramsdtype, float): ## C:\Users\Ganesh\ANACON~1\lib\site-packages\statsmodels\tsa\kalmanf\kalmanfilter.py:650: FutureWarning: Conversion of the second argument of issubdtype from complex to np.complexfloating is deprecated. In future, it will be treated as np.complex128 == np.dtype(complex).type. ## elif issubdtype(paramsdtype, complex): ## C:\Users\Ganesh\ANACON~1\lib\site-packages\statsmodels\tsa\kalmanf\kalmanfilter.py:577: FutureWarning: Conversion of the second argument of issubdtype from float to np.floating is deprecated. In future, it will be treated as np.float64 == np.dtype(float).type. ## if issubdtype(paramsdtype, float): ## 18 Relative Batsman Cumulative Average Runs The plot below compares the Relative cumulative average runs of the batsman for each of the runs ranges of 10 and plots them. The plot indicate the following Range 30 – 100 innings – Lara leads followed by Dravid Range 100+ innings – Kohli races ahead of the rest import cricpy.analytics as ca frames = ["../dravid.csv","../acook.csv","../lara.csv","../kohli.csv"] names = ["Dravid","A Cook","Brian Lara","V Kohli"] ca.relativeBatsmanCumulativeAvgRuns(frames,names) ## 19. Relative Batsman Strike Rate The plot below gives the relative Runs Frequency Percetages for each 10 run bucket. The plot below show Brian Lara towers over the Dravid, Cook and Kohli. However you will notice that Kohli’s strike rate is going up import cricpy.analytics as ca frames = ["../dravid.csv","../acook.csv","../lara.csv","../kohli.csv"] names = ["Dravid","A Cook","Brian Lara","V Kohli"] ca.relativeBatsmanCumulativeStrikeRate(frames,names) ## 20. 3D plot of Runs vs Balls Faced and Minutes at Crease The plot is a scatter plot of Runs vs Balls faced and Minutes at Crease. A prediction plane is fitted import cricpy.analytics as ca ca.battingPerf3d("../dravid.csv","Rahul Dravid") ca.battingPerf3d("../acook.csv","Alastair Cook") ca.battingPerf3d("../lara.csv","Brian Lara") ca.battingPerf3d("../kohli.csv","Virat Kohli") ## 21. Predicting Runs given Balls Faced and Minutes at Crease A multi-variate regression plane is fitted between Runs and Balls faced +Minutes at crease. import cricpy.analytics as ca import numpy as np import pandas as pd BF = np.linspace( 10, 400,15) Mins = np.linspace( 30,600,15) newDF= pd.DataFrame({'BF':BF,'Mins':Mins}) dravid = ca.batsmanRunsPredict("../dravid.csv",newDF,"Dravid") print(dravid) ## BF Mins Runs ## 0 10.000000 30.000000 0.519667 ## 1 37.857143 70.714286 13.821794 ## 2 65.714286 111.428571 27.123920 ## 3 93.571429 152.142857 40.426046 ## 4 121.428571 192.857143 53.728173 ## 5 149.285714 233.571429 67.030299 ## 6 177.142857 274.285714 80.332425 ## 7 205.000000 315.000000 93.634552 ## 8 232.857143 355.714286 106.936678 ## 9 260.714286 396.428571 120.238805 ## 10 288.571429 437.142857 133.540931 ## 11 316.428571 477.857143 146.843057 ## 12 344.285714 518.571429 160.145184 ## 13 372.142857 559.285714 173.447310 ## 14 400.000000 600.000000 186.749436 The fitted model is then used to predict the runs that the batsmen will score for a given Balls faced and Minutes at crease. ## 22 Analysis of Top 3 wicket takers The following 3 bowlers have had an excellent career and will be used for the analysis 1. Glenn McGrath:Wickets: 563, Average = 21.64, Economy Rate – 2.49 2. Kapil Dev : Wickets: 434, Average = 29.64, Economy Rate – 2.78 3. James Anderson: Wickets: 564, Average = 28.64, Economy Rate – 2.88 How do Glenn McGrath, Kapil Dev and James Anderson compare with one another with respect to wickets taken and the Economy Rate. The next set of plots compute and plot precisely these analyses. ## 23. Get the bowler’s data This plot below computes the percentage frequency of number of wickets taken for e.g 1 wicket x%, 2 wickets y% etc and plots them as a continuous line import cricpy.analytics as ca #mcgrath =ca.getPlayerData(6565,dir=".",file="mcgrath.csv",type="bowling",homeOrAway=[1,2], result=[1,2,4]) #kapil =ca.getPlayerData(30028,dir=".",file="kapil.csv",type="bowling",homeOrAway=[1,2], result=[1,2,4]) #anderson =ca.getPlayerData(8608,dir=".",file="anderson.csv",type="bowling",homeOrAway=[1,2], result=[1,2,4]) ## 24. Wicket Frequency Plot This plot below plots the frequency of wickets taken for each of the bowlers import cricpy.analytics as ca ca.bowlerWktsFreqPercent("../mcgrath.csv","Glenn McGrath") ca.bowlerWktsFreqPercent("../kapil.csv","Kapil Dev") ca.bowlerWktsFreqPercent("../anderson.csv","James Anderson") ## 25. Wickets Runs plot The plot below create a box plot showing the 1st and 3rd quartile of runs conceded versus the number of wickets taken import cricpy.analytics as ca ca.bowlerWktsRunsPlot("../mcgrath.csv","Glenn McGrath") ca.bowlerWktsRunsPlot("../kapil.csv","Kapil Dev") ca.bowlerWktsRunsPlot("../anderson.csv","James Anderson") ## 26 Average wickets at different venues The plot gives the average wickets taken by Muralitharan at different venues. McGrath best performances are at Centurion, Lord’s and Port of Spain averaging about 4 wickets. Kapil Dev’s does good at Kingston and Wellington. Anderson averages 4 wickets at Dunedin and Nagpur import cricpy.analytics as ca ca.bowlerAvgWktsGround("../mcgrath.csv","Glenn McGrath") ca.bowlerAvgWktsGround("../kapil.csv","Kapil Dev") ca.bowlerAvgWktsGround("../anderson.csv","James Anderson") ## 27 Average wickets against different opposition The plot gives the average wickets taken by Muralitharan against different countries. The x-axis also includes the number of innings against each team import cricpy.analytics as ca ca.bowlerAvgWktsOpposition("../mcgrath.csv","Glenn McGrath") ca.bowlerAvgWktsOpposition("../kapil.csv","Kapil Dev") ca.bowlerAvgWktsOpposition("../anderson.csv","James Anderson") ## 28 Wickets taken moving average From the plot below it can be see James Anderson has had a solid performance over the years averaging about wickets import cricpy.analytics as ca ca.bowlerMovingAverage("../mcgrath.csv","Glenn McGrath") ca.bowlerMovingAverage("../kapil.csv","Kapil Dev") ca.bowlerMovingAverage("../anderson.csv","James Anderson") ## 29 Cumulative average wickets taken The plots below give the cumulative average wickets taken by the bowlers. mcGrath plateaus around 2.4 wickets, Kapil Dev’s performance deteriorates over the years. Anderson holds on rock steady around 2 wickets import cricpy.analytics as ca ca.bowlerCumulativeAvgWickets("../mcgrath.csv","Glenn McGrath") ca.bowlerCumulativeAvgWickets("../kapil.csv","Kapil Dev") ca.bowlerCumulativeAvgWickets("../anderson.csv","James Anderson") ## 30 Cumulative average economy rate The plots below give the cumulative average economy rate of the bowlers. McGrath’s was very expensive early in his career conceding about 2.8 runs per over which drops to around 2.5 runs towards the end. Kapil Dev’s economy rate drops from 3.6 to 2.8. Anderson is probably more expensive than the other 2. import cricpy.analytics as ca ca.bowlerCumulativeAvgEconRate("../mcgrath.csv","Glenn McGrath") ca.bowlerCumulativeAvgEconRate("../kapil.csv","Kapil Dev") ca.bowlerCumulativeAvgEconRate("../anderson.csv","James Anderson") ## 31 Future Wickets forecast import cricpy.analytics as ca ca.bowlerPerfForecast("../mcgrath.csv","Glenn McGrath") ## ARIMA Model Results ## ============================================================================== ## Dep. Variable: D.Wickets No. Observations: 236 ## Model: ARIMA(5, 1, 0) Log Likelihood -480.815 ## Method: css-mle S.D. of innovations 1.851 ## Date: Sun, 28 Oct 2018 AIC 975.630 ## Time: 09:28:32 BIC 999.877 ## Sample: 11-12-1993 HQIC 985.404 ## - 01-02-2007 ## =================================================================================== ## coef std err z P>|z| [0.025 0.975] ## ----------------------------------------------------------------------------------- ## const 0.0037 0.033 0.113 0.910 -0.061 0.068 ## ar.L1.D.Wickets -0.9432 0.064 -14.708 0.000 -1.069 -0.818 ## ar.L2.D.Wickets -0.7254 0.086 -8.469 0.000 -0.893 -0.558 ## ar.L3.D.Wickets -0.4827 0.093 -5.217 0.000 -0.664 -0.301 ## ar.L4.D.Wickets -0.3690 0.085 -4.324 0.000 -0.536 -0.202 ## ar.L5.D.Wickets -0.1709 0.064 -2.678 0.008 -0.296 -0.046 ## Roots ## ============================================================================= ## Real Imaginary Modulus Frequency ## ----------------------------------------------------------------------------- ## AR.1 0.5630 -1.2761j 1.3948 -0.1839 ## AR.2 0.5630 +1.2761j 1.3948 0.1839 ## AR.3 -0.8433 -1.0820j 1.3718 -0.3554 ## AR.4 -0.8433 +1.0820j 1.3718 0.3554 ## AR.5 -1.5981 -0.0000j 1.5981 -0.5000 ## ----------------------------------------------------------------------------- ## 0 ## count 236.000000 ## mean -0.005142 ## std 1.856961 ## min -3.457002 ## 25% -1.433391 ## 50% -0.080237 ## 75% 1.446149 ## max 5.840050 ## 32 Get player data special As discussed above the next 2 charts require the use of getPlayerDataSp() import cricpy.analytics as ca #mcgrathsp =ca.getPlayerDataSp(6565,tdir=".",tfile="mcgrathsp.csv",ttype="bowling") #kapilsp =ca.getPlayerDataSp(30028,tdir=".",tfile="kapilsp.csv",ttype="bowling") #andersonsp =ca.getPlayerDataSp(8608,tdir=".",tfile="andersonsp.csv",ttype="bowling") ## 33 Contribution to matches won and lost The plot below is extremely interesting Glenn McGrath has been more instrumental in Australia winning than Kapil and Anderson as seems to have taken more wickets when Australia won. import cricpy.analytics as ca ca.bowlerContributionWonLost("../mcgrathsp.csv","Glenn McGrath") ca.bowlerContributionWonLost("../kapilsp.csv","Kapil Dev") ca.bowlerContributionWonLost("../andersonsp.csv","James Anderson") ## 34 Performance home and overseas McGrath and Kapil Dev have performed better overseas than at home. Anderson has performed about the same home and overseas import cricpy.analytics as ca ca.bowlerPerfHomeAway("../mcgrathsp.csv","Glenn McGrath") ca.bowlerPerfHomeAway("../kapilsp.csv","Kapil Dev") ca.bowlerPerfHomeAway("../andersonsp.csv","James Anderson") ## 35 Relative cumulative average economy rate of bowlers The Relative cumulative economy rate shows that McGrath has the best economy rate followed by Kapil Dev and then Anderson. import cricpy.analytics as ca frames = ["../mcgrath.csv","../kapil.csv","../anderson.csv"] names = ["Glenn McGrath","Kapil Dev","James Anderson"] ca.relativeBowlerCumulativeAvgEconRate(frames,names) ## 36 Relative Economy Rate against wickets taken McGrath has been economical regardless of the number of wickets taken. Kapil Dev has been slightly more expensive when he takes more wickets import cricpy.analytics as ca frames = ["../mcgrath.csv","../kapil.csv","../anderson.csv"] names = ["Glenn McGrath","Kapil Dev","James Anderson"] ca.relativeBowlingER(frames,names) ## 37 Relative cumulative average wickets of bowlers in career The plot below shows that McGrath has the best overall cumulative average wickets. Kapil’s leads Anderson till about 150 innings after which Anderson takes over import cricpy.analytics as ca frames = ["../mcgrath.csv","../kapil.csv","../anderson.csv"] names = ["Glenn McGrath","Kapil Dev","James Anderson"] ca.relativeBowlerCumulativeAvgWickets(frames,names) # Key Findings The plots above capture some of the capabilities and features of my cricpy package. Feel free to install the package and try it out. Please do keep in mind ESPN Cricinfo’s Terms of Use. Here are the main findings from the analysis above ## Key insights 1. Brian Lara is head and shoulders above the rest in the overall strike rate 2. Kohli performance has been steadily improving over the years and with the way he is going he will shatter all records. 3. Kohli and Dravid have scored more in matches where India has won than the other two. 4. Dravid has performed very well overseas 5. The cumulative average runs has Kohli just edging out the other 3. Kohli is probably midway in his career but considering that his moving average is improving strongly, we can expect great things of him with the way he is going. 6. McGrath has had some great performances overseas 7. Mcgrath has the best economy rate and has contributed significantly to Australia’s wins. 8.In the cumulative average wickets race McGrath leads the pack. Kapil leads Anderson till about 150 matches after which Anderson takes over. The code for cricpy can be accessed at Github at cricpy Do let me know if you run into issues. ## Conclusion I have long wanted to make a python equivalent of cricketr and I have been able to make it. cricpy is still work in progress. I have add the necessary functions for ODI and Twenty20. Go ahead give ‘cricpy’ a spin!! Stay tuned! Important note: Do check out my other posts using cricpy at cricpy-posts # Big Data-2: Move into the big league:Graduate from R to SparkR This post is a continuation of my earlier post Big Data-1: Move into the big league:Graduate from Python to Pyspark. While the earlier post discussed parallel constructs in Python and Pyspark, this post elaborates similar and key constructs in R and SparkR. While this post just focuses on the programming part of R and SparkR it is essential to understand and fully grasp the concept of Spark, RDD and how data is distributed across the clusters. This post like the earlier post shows how if you already have a good handle of R, you can easily graduate to Big Data with SparkR Note 1: This notebook has also been published at Databricks community site Big Data-2: Move into the big league:Graduate from R to SparkR Note 2: You can download this RMarkdown file from Github at Big Data- Python to Pyspark and R to SparkR 1a. Read CSV- R Note: To upload the CSV to databricks see the video Upload Flat File to Databricks Table # Read CSV file tendulkar= read.csv("/dbfs/FileStore/tables/tendulkar.csv",stringsAsFactors = FALSE,na.strings=c(NA,"-")) #Check the dimensions of the dataframe dim(tendulkar)  [1] 347 12 1b. Read CSV – SparkR # Load the SparkR library library(SparkR) # Initiate a SparkR session sparkR.session() tendulkar1 <- read.df("/FileStore/tables/tendulkar.csv", header = "true", delimiter = ",", source = "csv", inferSchema = "true", na.strings = "") # Check the dimensions of the dataframe dim(tendulkar1)  [1] 347 12 2a. Data frame shape – R # Get the shape of the dataframe in R dim(tendulkar)  [1] 347 12 2b. Dataframe shape – SparkR The same ‘dim’ command works in SparkR too! dim(tendulkar1)  [1] 347 12 3a . Dataframe columns – R # Get the names names(tendulkar) # Also colnames(tendulkar)   [1] "Runs" "Mins" "BF" "X4s" "X6s" [6] "SR" "Pos" "Dismissal" "Inns" "Opposition" [11] "Ground" "Start.Date" 3b. Dataframe columns – SparkR names(tendulkar1)   [1] "Runs" "Mins" "BF" "4s" "6s" [6] "SR" "Pos" "Dismissal" "Inns" "Opposition" [11] "Ground" "Start Date" 4a. Rename columns – R names(tendulkar)=c('Runs','Minutes','BallsFaced','Fours','Sixes','StrikeRate','Position','Dismissal','Innings','Opposition','Ground','StartDate') names(tendulkar)   [1] "Runs" "Minutes" "BallsFaced" "Fours" "Sixes" [6] "StrikeRate" "Position" "Dismissal" "Innings" "Opposition" [11] "Ground" "StartDate" 4b. Rename columns – SparkR names(tendulkar1)=c('Runs','Minutes','BallsFaced','Fours','Sixes','StrikeRate','Position','Dismissal','Innings','Opposition','Ground','StartDate') names(tendulkar1)   [1] "Runs" "Minutes" "BallsFaced" "Fours" "Sixes" [6] "StrikeRate" "Position" "Dismissal" "Innings" "Opposition" [11] "Ground" "StartDate" 5a. Summary – R summary(tendulkar)   Runs Minutes BallsFaced Fours Length:347 Min. : 1.0 Min. : 0.00 Min. : 0.000 Class :character 1st Qu.: 33.0 1st Qu.: 22.00 1st Qu.: 1.000 Mode :character Median : 82.0 Median : 58.50 Median : 4.000 Mean :125.5 Mean : 89.75 Mean : 6.274 3rd Qu.:181.0 3rd Qu.:133.25 3rd Qu.: 9.000 Max. :613.0 Max. :436.00 Max. :35.000 NA's :18 NA's :19 NA's :19 Sixes StrikeRate Position Dismissal Min. :0.0000 Min. : 0.00 Min. :2.00 Length:347 1st Qu.:0.0000 1st Qu.: 38.09 1st Qu.:4.00 Class :character Median :0.0000 Median : 52.25 Median :4.00 Mode :character Mean :0.2097 Mean : 51.79 Mean :4.24 3rd Qu.:0.0000 3rd Qu.: 65.09 3rd Qu.:4.00 Max. :4.0000 Max. :166.66 Max. :7.00 NA's :18 NA's :20 NA's :18 Innings Opposition Ground StartDate Min. :1.000 Length:347 Length:347 Length:347 1st Qu.:1.000 Class :character Class :character Class :character Median :2.000 Mode :character Mode :character Mode :character Mean :2.376 3rd Qu.:3.000 Max. :4.000 NA's :1 5b. Summary – SparkR summary(tendulkar1)  SparkDataFrame[summary:string, Runs:string, Minutes:string, BallsFaced:string, Fours:string, Sixes:string, StrikeRate:string, Position:string, Dismissal:string, Innings:string, Opposition:string, Ground:string, StartDate:string] 6a. Displaying details of dataframe with str() – R str(tendulkar)  'data.frame': 347 obs. of 12 variables:$ Runs      : chr  "15" "DNB" "59" "8" ...
$Minutes : int 28 NA 254 24 124 74 193 1 50 324 ...$ BallsFaced: int  24 NA 172 16 90 51 134 1 44 266 ...
$Fours : int 2 NA 4 1 5 5 6 0 3 5 ...$ Sixes     : int  0 NA 0 0 0 0 0 0 0 0 ...
$StrikeRate: num 62.5 NA 34.3 50 45.5 ...$ Position  : int  6 NA 6 6 7 6 6 6 6 6 ...
$Dismissal : chr "bowled" NA "lbw" "run out" ...$ Innings   : int  2 4 1 3 1 1 3 2 3 1 ...
$Opposition: chr "v Pakistan" "v Pakistan" "v Pakistan" "v Pakistan" ...$ Ground    : chr  "Karachi" "Karachi" "Faisalabad" "Faisalabad" ...
$StartDate : chr "15-Nov-89" "15-Nov-89" "23-Nov-89" "23-Nov-89" ... 6b. Displaying details of dataframe with str() – SparkR str(tendulkar1)  'SparkDataFrame': 12 variables:$ Runs      : chr "15" "DNB" "59" "8" "41" "35"
$Minutes : chr "28" "-" "254" "24" "124" "74"$ BallsFaced: chr "24" "-" "172" "16" "90" "51"
$Fours : chr "2" "-" "4" "1" "5" "5"$ Sixes     : chr "0" "-" "0" "0" "0" "0"
$StrikeRate: chr "62.5" "-" "34.3" "50" "45.55" "68.62"$ Position  : chr "6" "-" "6" "6" "7" "6"
$Dismissal : chr "bowled" "-" "lbw" "run out" "bowled" "lbw"$ Innings   : chr "2" "4" "1" "3" "1" "1"
$Opposition: chr "v Pakistan" "v Pakistan" "v Pakistan" "v Pakistan" "v Pakistan" "v Pakistan"$ Ground    : chr "Karachi" "Karachi" "Faisalabad" "Faisalabad" "Lahore" "Sialkot"
$StartDate : chr "15-Nov-89" "15-Nov-89" "23-Nov-89" "23-Nov-89" "1-Dec-89" "9-Dec-89" 7a. Head & tail -R print(head(tendulkar),3) print(tail(tendulkar),3)   Runs Minutes BallsFaced Fours Sixes StrikeRate Position Dismissal Innings 1 15 28 24 2 0 62.50 6 bowled 2 2 DNB NA NA NA NA NA NA 4 3 59 254 172 4 0 34.30 6 lbw 1 4 8 24 16 1 0 50.00 6 run out 3 5 41 124 90 5 0 45.55 7 bowled 1 6 35 74 51 5 0 68.62 6 lbw 1 Opposition Ground StartDate 1 v Pakistan Karachi 15-Nov-89 2 v Pakistan Karachi 15-Nov-89 3 v Pakistan Faisalabad 23-Nov-89 4 v Pakistan Faisalabad 23-Nov-89 5 v Pakistan Lahore 1-Dec-89 6 v Pakistan Sialkot 9-Dec-89 Runs Minutes BallsFaced Fours Sixes StrikeRate Position Dismissal Innings 342 37 125 81 5 0 45.67 4 caught 2 343 21 71 23 2 0 91.30 4 run out 4 344 32 99 53 5 0 60.37 4 lbw 2 345 1 8 5 0 0 20.00 4 lbw 4 346 10 41 24 2 0 41.66 4 lbw 2 347 74 150 118 12 0 62.71 4 caught 2 Opposition Ground StartDate 342 v Australia Mohali 14-Mar-13 343 v Australia Mohali 14-Mar-13 344 v Australia Delhi 22-Mar-13 345 v Australia Delhi 22-Mar-13 346 v West Indies Kolkata 6-Nov-13 347 v West Indies Mumbai 14-Nov-13 7b. Head – SparkR head(tendulkar1,3)   Runs Minutes BallsFaced Fours Sixes StrikeRate Position Dismissal Innings 1 15 28 24 2 0 62.5 6 bowled 2 2 DNB - - - - - - - 4 3 59 254 172 4 0 34.3 6 lbw 1 Opposition Ground StartDate 1 v Pakistan Karachi 15-Nov-89 2 v Pakistan Karachi 15-Nov-89 3 v Pakistan Faisalabad 23-Nov-89 8a. Determining the column types with sapply -R sapply(tendulkar,class)   Runs Minutes BallsFaced Fours Sixes StrikeRate "character" "integer" "integer" "integer" "integer" "numeric" Position Dismissal Innings Opposition Ground StartDate "integer" "character" "integer" "character" "character" "character" 8b. Determining the column types with printSchema – SparkR printSchema(tendulkar1)  root |-- Runs: string (nullable = true) |-- Minutes: string (nullable = true) |-- BallsFaced: string (nullable = true) |-- Fours: string (nullable = true) |-- Sixes: string (nullable = true) |-- StrikeRate: string (nullable = true) |-- Position: string (nullable = true) |-- Dismissal: string (nullable = true) |-- Innings: string (nullable = true) |-- Opposition: string (nullable = true) |-- Ground: string (nullable = true) |-- StartDate: string (nullable = true) 9a. Selecting columns – R library(dplyr) df=select(tendulkar,Runs,BallsFaced,Minutes) head(df,5)   Runs BallsFaced Minutes 1 15 24 28 2 DNB NA NA 3 59 172 254 4 8 16 24 5 41 90 124 9b. Selecting columns – SparkR library(SparkR) Sys.setenv(SPARK_HOME="/usr/hdp/2.6.0.3-8/spark") .libPaths(c(file.path(Sys.getenv("SPARK_HOME"), "R", "lib"), .libPaths())) # Initiate a SparkR session sparkR.session() tendulkar1 <- read.df("/FileStore/tables/tendulkar.csv", header = "true", delimiter = ",", source = "csv", inferSchema = "true", na.strings = "") df=SparkR::select(tendulkar1, "Runs", "BF","Mins") head(SparkR::collect(df))   Runs BF Mins 1 15 24 28 2 DNB - - 3 59 172 254 4 8 16 24 5 41 90 124 6 35 51 74 10a. Filter rows by criteria – R library(dplyr) df=tendulkar %>% filter(Runs > 50) head(df,5)   Runs Minutes BallsFaced Fours Sixes StrikeRate Position Dismissal Innings 1 DNB NA NA NA NA NA NA 4 2 59 254 172 4 0 34.30 6 lbw 1 3 8 24 16 1 0 50.00 6 run out 3 4 57 193 134 6 0 42.53 6 caught 3 5 88 324 266 5 0 33.08 6 caught 1 Opposition Ground StartDate 1 v Pakistan Karachi 15-Nov-89 2 v Pakistan Faisalabad 23-Nov-89 3 v Pakistan Faisalabad 23-Nov-89 4 v Pakistan Sialkot 9-Dec-89 5 v New Zealand Napier 9-Feb-90 10b. Filter rows by criteria – SparkR df=SparkR::filter(tendulkar1, tendulkar1$Runs > 50)

  Runs Mins  BF 4s 6s    SR Pos Dismissal Inns     Opposition       Ground
1   59  254 172  4  0  34.3   6       lbw    1     v Pakistan   Faisalabad
2   57  193 134  6  0 42.53   6    caught    3     v Pakistan      Sialkot
3   88  324 266  5  0 33.08   6    caught    1  v New Zealand       Napier
4   68  216 136  8  0    50   6    caught    2      v England   Manchester
5  114  228 161 16  0  70.8   4    caught    2    v Australia        Perth
6  111  373 270 19  0 41.11   4    caught    2 v South Africa Johannesburg
Start Date
1  23-Nov-89
2   9-Dec-89
3   9-Feb-90
4   9-Aug-90
5   1-Feb-92
6  26-Nov-92
11a. Unique values -R
unique(tendulkar$Runs)   [1] "15" "DNB" "59" "8" "41" "35" "57" "0" "24" "88" [11] "5" "10" "27" "68" "119*" "21" "11" "16" "7" "40" [21] "148*" "6" "17" "114" "111" "1" "73" "50" "9*" "165" [31] "78" "62" "TDNB" "28" "104*" "71" "142" "96" "43" "11*" [41] "34" "85" "179" "54" "4" "0*" "52*" "2" "122" "31" [51] "177" "74" "42" "18" "61" "36" "169" "9" "15*" "92" [61] "83" "143" "139" "23" "148" "13" "155*" "79" "47" "113" [71] "67" "136" "29" "53" "124*" "126*" "44*" "217" "116" "52" [81] "45" "97" "20" "39" "201*" "76" "65" "126" "36*" "69" [91] "155" "22*" "103" "26" "90" "176" "117" "86" "12" "193" [101] "16*" "51" "32" "55" "37" "44" "241*" "60*" "194*" "3" [111] "32*" "248*" "94" "22" "109" "19" "14" "28*" "63" "64" [121] "101" "122*" "91" "82" "56*" "154*" "153" "49" "10*" "103*" [131] "160" "100*" "105*" "100" "106" "84" "203" "98" "38" "214" [141] "53*" "111*" "146" "14*" "56" "80" "25" "81" "13*" 11b. Unique values – SparkR head(SparkR::distinct(tendulkar1[,"Runs"]),5)   Runs 1 119* 2 7 3 51 4 169 5 32* 12a. Aggregate – Mean, min and max – R library(dplyr) library(magrittr) a <- tendulkar$Runs != "DNB"
tendulkar <- tendulkar[a,]
dim(tendulkar)

# Remove rows with 'TDNB'
c <- tendulkar$Runs != "TDNB" tendulkar <- tendulkar[c,] # Remove rows with absent d <- tendulkar$Runs != "absent"
tendulkar <- tendulkar[d,]
dim(tendulkar)

# Remove the "* indicating not out
tendulkar$Runs <- as.numeric(gsub("\\*","",tendulkar$Runs))
c <- complete.cases(tendulkar)

#Subset the rows which are complete
tendulkar <- tendulkar[c,]
print(dim(tendulkar))
df <-tendulkar %>%  group_by(Ground) %>% summarise(meanRuns= mean(Runs), minRuns=min(Runs), maxRuns=max(Runs))
#names(tendulkar)

[1] 327  12
# A tibble: 6 x 4
Ground       meanRuns minRuns maxRuns

3 Auckland         5.00      5.      5.
4 Bangalore       57.9       4.    214.
5 Birmingham      46.8       1.    122.
6 Bloemfontein    85.0      15.    155.
12b. Aggregate- Mean, Min, Max – SparkR
sparkR.session()

delimiter = ",",
source = "csv",
inferSchema = "true",
na.strings = "")

print(dim(tendulkar1))
tendulkar1 <-SparkR::filter(tendulkar1,tendulkar1$Runs != "DNB") print(dim(tendulkar1)) tendulkar1<-SparkR::filter(tendulkar1,tendulkar1$Runs != "TDNB")
print(dim(tendulkar1))
tendulkar1<-SparkR::filter(tendulkar1,tendulkar1$Runs != "absent") print(dim(tendulkar1)) # Cast the string type Runs to double withColumn(tendulkar1, "Runs", cast(tendulkar1$Runs, "double"))
# Remove the "* indicating not out
tendulkar1$Runs=SparkR::regexp_replace(tendulkar1$Runs, "\\*", "")
df=SparkR::summarize(SparkR::groupBy(tendulkar1, tendulkar1$Ground), mean = mean(tendulkar1$Runs), minRuns=min(tendulkar1$Runs),maxRuns=max(tendulkar1$Runs))

[1] 347  12
[1] 330  12
[1] 329  12
[1] 329  12
Ground       mean minRuns maxRuns
1      Bangalore  54.312500       0      96
3  Colombo (PSS)  37.200000      14      71
4   Christchurch  12.000000       0      24
5       Auckland   5.000000       5       5
6        Chennai  60.625000       0      81
7      Centurion  73.500000     111      36
8       Brisbane   7.666667       0       7
9     Birmingham  46.750000       1      40
11 Colombo (RPS) 143.000000     143     143
12    Chittagong  57.800000     101      36
13     Cape Town  69.857143      14       9
14    Bridgetown  26.000000       0      92
15      Bulawayo  55.000000      36      74
16         Delhi  39.947368       0      76
17    Chandigarh  11.000000      11      11
18  Bloemfontein  85.000000      15     155
19 Colombo (SSC)  77.555556     104       8
20       Cuttack   2.000000       2       2
13a Using SQL with SparkR
sparkR.session()
delimiter = ",",
source = "csv",
inferSchema = "true",
na.strings = "")

# Register this SparkDataFrame as a temporary view.
createOrReplaceTempView(tendulkar1, "tendulkar2")

# SQL statements can be run by using the sql method
df=SparkR::sql("SELECT * FROM tendulkar2 WHERE Ground='Karachi'")


  Runs Mins BF 4s 6s    SR Pos Dismissal Inns Opposition  Ground Start Date
1   15   28 24  2  0  62.5   6    bowled    2 v Pakistan Karachi  15-Nov-89
2  DNB    -  -  -  -     -   -         -    4 v Pakistan Karachi  15-Nov-89
3   23   49 29  5  0 79.31   4    bowled    2 v Pakistan Karachi  29-Jan-06
4   26   74 47  5  0 55.31   4    bowled    4 v Pakistan Karachi  29-Jan-06
Conclusion

This post discusses some of the key constructs in R and SparkR and how one can transition from R to SparkR fairly easily. I will be adding more constructs later. Do check back!

To see all 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 “Deep Learning from first principles” now on Amazon

Note: The 2nd edition of this book is now available on Amazon

My 4th book(self-published), “Deep Learning from first principles – In vectorized Python, R and Octave” (557 pages), is now available on Amazon in both paperback ($18.99) and kindle ($9.99/Rs449). The book starts with the most primitive 2-layer Neural Network and works  its way to a generic L-layer Deep Learning Network, with all the bells and whistles.  The book includes detailed derivations and vectorized implementations in Python, R and Octave.  The code has been extensively  commented and has been included in the Appendix section.

# Deep Learning from first principles in Python, R and Octave – Part 8

## 1. Introduction

You don’t understand anything until you learn it more than one way. Marvin Minsky
No computer has ever been designed that is ever aware of what it’s doing; but most of the time, we aren’t either. Marvin Minsky
A wealth of information creates a poverty of attention. Herbert Simon

This post, Deep Learning from first Principles in Python, R and Octave-Part8, is my final post in my Deep Learning from first principles series. In this post, I discuss and implement a key functionality needed while building Deep Learning networks viz. ‘Gradient Checking’. Gradient Checking is an important method to check the correctness of your implementation, specifically the forward propagation and the backward propagation cycles of an implementation. In addition I also discuss some tips for tuning hyper-parameters of a Deep Learning network based on my experience.

My post in this  ‘Deep Learning Series’ so far were
1. Deep Learning from first principles in Python, R and Octave – Part 1 In part 1, I implement logistic regression as a neural network in vectorized Python, R and Octave
2. Deep Learning from first principles in Python, R and Octave – Part 2 In the second part I implement a simple Neural network with just 1 hidden layer and a sigmoid activation output function
3. Deep Learning from first principles in Python, R and Octave – Part 3 The 3rd part implemented a multi-layer Deep Learning Network with sigmoid activation output in vectorized Python, R and Octave
4. Deep Learning from first principles in Python, R and Octave – Part 4 The 4th part deals with multi-class classification. Specifically, I derive the Jacobian of the Softmax function and enhance my L-Layer DL network to include Softmax output function in addition to Sigmoid activation
5. Deep Learning from first principles in Python, R and Octave – Part 5 This post uses the Softmax classifier implemented to classify MNIST digits using a L-layer Deep Learning network
6. Deep Learning from first principles in Python, R and Octave – Part 6 The 6th part adds more bells and whistles to my L-Layer DL network, by including different initialization types namely He and Xavier. Besides L2 Regularization and random dropout is added.
7. Deep Learning from first principles in Python, R and Octave – Part 7 The 7th part deals with Stochastic Gradient Descent Optimization methods including momentum, RMSProp and Adam
8. Deep Learning from first principles in Python, R and Octave – Part 8 – This post implements a critical function for ensuring the correctness of a L-Layer Deep Learning network implementation using Gradient Checking

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).

You may also like my companion book “Practical Machine Learning with R and Python- Machine Learning in stereo” available in Amazon in paperback($9.99) and Kindle($6.99) versions. This book is ideal for a quick reference of the various ML functions and associated measurements in both R and Python which are essential to delve deep into Deep Learning.

Gradient Checking is based on the following approach. One iteration of Gradient Descent computes and updates the parameters $\theta$ by doing
$\theta := \theta - \frac{d}{d\theta}J(\theta)$.
To minimize the cost we will need to minimize $J(\theta)$
Let $g(\theta)$ be a function that computes the derivative $\frac {d}{d\theta}J(\theta)$. Gradient Checking allows us to numerically evaluate the implementation of the function $g(\theta)$ and verify its correctness.
We know the derivative of a function is given by
$\frac {d}{d\theta}J(\theta) = lim->0 \frac {J(\theta +\epsilon) - J(\theta -\epsilon)} {2*\epsilon}$
Note: The above derivative is based on the 2 sided derivative. The 1-sided derivative  is given by $\frac {d}{d\theta}J(\theta) = lim->0 \frac {J(\theta +\epsilon) - J(\theta)} {\epsilon}$
Gradient Checking is based on the 2-sided derivative because the error is of the order $O(\epsilon^{2})$ as opposed $O(\epsilon)$ for the 1-sided derivative.
Hence Gradient Check uses the 2 sided derivative as follows.
$g(\theta) = lim->0 \frac {J(\theta +\epsilon) - J(\theta -\epsilon)} {2*\epsilon}$

In Gradient Check the following is done
A) Run one normal cycle of your implementation by doing the following
a) Compute the output activation by running 1 cycle of forward propagation
b) Compute the cost using the output activation

B) Perform gradient check steps as below
a) Set $\theta$ . Flatten all ‘weights’ and ‘bias’ matrices and vectors to a column vector.
b) Initialize $\theta+$ by bumping up $\theta$ by adding $\epsilon$ ($\theta + \epsilon$)
c) Perform forward propagation with $\theta+$
d) Compute cost with $\theta+$ i.e. $J(\theta+)$
e) Initialize  $\theta-$ by bumping down $\theta$ by subtracting $\epsilon$ $(\theta - \epsilon)$
f) Perform forward propagation with $\theta-$
g) Compute cost with $\theta-$ i.e.  $J(\theta-)$
h) Compute $\frac {d} {d\theta} J(\theta)$ or ‘gradapprox’ as$\frac {J(\theta+) - J(\theta-) } {2\epsilon}$using the 2 sided derivative.
i) Compute L2norm or the Euclidean distance between ‘grad’ and ‘gradapprox’. If the
diference is of the order of $10^{-5}$ or $10^{-7}$ the implementation is correct. In the Deep Learning Specialization Prof Andrew Ng mentions that if the difference is of the order of $10^{-7}$ then the implementation is correct. A difference of $10^{-5}$ is also ok. Anything more than that is a cause of worry and you should look at your code more closely. To see more details click Gradient checking and advanced optimization

After spending a better part of 3 days, I now realize how critical Gradient Check is for ensuring the correctness of you implementation. Initially I was getting very high difference and did not know how to understand the results or debug my implementation. After many hours of staring at the results, I  was able to finally arrive at a way, to localize issues in the implementation. In fact, I did catch a small bug in my Python code, which did not exist in the R and Octave implementations. I will demonstrate this below

## 1.1a Gradient Check – Sigmoid Activation – Python

import numpy as np
import matplotlib

train_X, train_Y, test_X, test_Y = load_dataset()
#Set layer dimensions
layersDimensions = [2,4,1]
parameters = initializeDeepModel(layersDimensions)
#Perform forward prop
AL, caches, dropoutMat = forwardPropagationDeep(train_X, parameters, keep_prob=1, hiddenActivationFunc="relu",outputActivationFunc="sigmoid")
#Compute cost
cost = computeCost(AL, train_Y, outputActivationFunc="sigmoid")
print("cost=",cost)
gradients = backwardPropagationDeep(AL, train_Y, caches, dropoutMat, lambd=0, keep_prob=1,                                   hiddenActivationFunc="relu",outputActivationFunc="sigmoid")

epsilon = 1e-7
outputActivationFunc="sigmoid"

# Set-up variables
# Flatten parameters to a vector
parameters_values, _ = dictionary_to_vector(parameters)
num_parameters = parameters_values.shape[0]
#Initialize
J_plus = np.zeros((num_parameters, 1))
J_minus = np.zeros((num_parameters, 1))

# Compute gradapprox using 2 sided derivative
for i in range(num_parameters):
# Compute J_plus[i].
thetaplus = np.copy(parameters_values)
thetaplus[i][0] = thetaplus[i][0] + epsilon
AL, caches, dropoutMat = forwardPropagationDeep(train_X, vector_to_dictionary(parameters,thetaplus), keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc)
J_plus[i] = computeCost(AL, train_Y, outputActivationFunc=outputActivationFunc)

# Compute J_minus[i].
thetaminus = np.copy(parameters_values)
thetaminus[i][0] = thetaminus[i][0] - epsilon
AL, caches, dropoutMat  = forwardPropagationDeep(train_X, vector_to_dictionary(parameters,thetaminus), keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc)
J_minus[i] = computeCost(AL, train_Y, outputActivationFunc=outputActivationFunc)

difference =  numerator/denominator

#Check the difference
if difference > 1e-5:
print ("\033[93m" + "There is a mistake in the backward propagation! difference = " + str(difference) + "\033[0m")
else:
print ("\033[92m" + "Your backward propagation works perfectly fine! difference = " + str(difference) + "\033[0m")
print(difference)
print("\n")
# The technique below can be used to identify
# which of the parameters are in error
print(m)
print("\n")
print(n)

## (300, 2)
## (300,)
## cost= 0.6931455556341791
## [92mYour backward propagation works perfectly fine! difference = 1.1604150683743381e-06[0m
## 1.1604150683743381e-06
##
##
## {'dW1': array([[-6.19439955e-06, -2.06438046e-06],
##        [-1.50165447e-05,  7.50401672e-05],
##        [ 1.33435433e-04,  1.74112143e-04],
##        [-3.40909024e-05, -1.38363681e-04]]), 'db1': array([[ 7.31333221e-07],
##        [ 7.98425950e-06],
##        [ 8.15002817e-08],
##        [-5.69821155e-08]]), 'dW2': array([[2.73416304e-04, 2.96061451e-04, 7.51837363e-05, 1.01257729e-04]]), 'db2': array([[-7.22232235e-06]])}
##
##
## {'dW1': array([[-6.19448937e-06, -2.06501483e-06],
##        [-1.50168766e-05,  7.50399742e-05],
##        [ 1.33435485e-04,  1.74112391e-04],
##        [-3.40910633e-05, -1.38363765e-04]]), 'db1': array([[ 7.31081862e-07],
##        [ 7.98472399e-06],
##        [ 8.16013923e-08],
##        [-5.71764858e-08]]), 'dW2': array([[2.73416290e-04, 2.96061509e-04, 7.51831930e-05, 1.01257891e-04]]), 'db2': array([[-7.22255589e-06]])}

## 1.1b Gradient Check – Softmax Activation – Python (Error!!)

In the code below I show, how I managed to spot a bug in your implementation

import numpy as np
N = 100 # number of points per class
D = 2 # dimensionality
K = 3 # number of classes
X = np.zeros((N*K,D)) # data matrix (each row = single example)
y = np.zeros(N*K, dtype='uint8') # class labels
for j in range(K):
ix = range(N*j,N*(j+1))
t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2 # theta
X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
y[ix] = j

# Plot the data
#plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)
layersDimensions = [2,3,3]
y1=y.reshape(-1,1).T
train_X=X.T
train_Y=y1

parameters = initializeDeepModel(layersDimensions)
#Compute forward prop
AL, caches, dropoutMat = forwardPropagationDeep(train_X, parameters, keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc="softmax")
#Compute cost
cost = computeCost(AL, train_Y, outputActivationFunc="softmax")
print("cost=",cost)
gradients = backwardPropagationDeep(AL, train_Y, caches, dropoutMat, lambd=0, keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc="softmax")
# Note the transpose of the gradients for Softmax has to be taken
L= len(parameters)//2
print(L)
gradient_check_n(parameters, gradients, train_X, train_Y, epsilon = 1e-7,outputActivationFunc="softmax")

cost= 1.0986187818144022
2
There is a mistake in the backward propagation! difference = 0.7100295155692544
0.7100295155692544

{'dW1': array([[ 0.00050125,  0.00045194],
[ 0.00096392,  0.00039641],
[-0.00014276, -0.00045639]]), 'db1': array([[ 0.00070082],
[-0.00224399],
[ 0.00052305]]), 'dW2': array([[-8.40953794e-05, -9.52657769e-04, -1.10269379e-04],
[-7.45469382e-04,  9.49795606e-04,  2.29045434e-04],
[ 8.29564761e-04,  2.86216305e-06, -1.18776055e-04]]),
'db2': array([[-0.00253808],
[-0.00505508],
[ 0.00759315]])}

{'dW1': array([[ 0.00050125,  0.00045194],
[ 0.00096392,  0.00039641],
[-0.00014276, -0.00045639]]), 'db1': array([[ 0.00070082],
[-0.00224399],
[ 0.00052305]]), 'dW2': array([[-8.40960634e-05, -9.52657953e-04, -1.10268461e-04],
[-7.45469242e-04,  9.49796908e-04,  2.29045671e-04],
[ 8.29565305e-04,  2.86104473e-06, -1.18776100e-04]]),
'db2': array([[-8.46211989e-06],
[-1.68487446e-05],
[ 2.53108645e-05]])}

Gradient Check gives a high value of the difference of 0.7100295. Inspecting the Gradients and Gradapprox we can see there is a very big discrepancy in db2. After I went over my code I discovered that I my computation in the function layerActivationBackward for Softmax was


# Erroneous code
if activationFunc == 'softmax':
dW = 1/numtraining * np.dot(A_prev,dZ)
db = np.sum(dZ, axis=0, keepdims=True)
dA_prev = np.dot(dZ,W)
# Fixed code
if activationFunc == 'softmax':
dW = 1/numtraining * np.dot(A_prev,dZ)
db = 1/numtraining *  np.sum(dZ, axis=0, keepdims=True)
dA_prev = np.dot(dZ,W)


After fixing this error when I ran Gradient Check I get

## 1.1c Gradient Check – Softmax Activation – Python (Corrected!!)

import numpy as np
N = 100 # number of points per class
D = 2 # dimensionality
K = 3 # number of classes
X = np.zeros((N*K,D)) # data matrix (each row = single example)
y = np.zeros(N*K, dtype='uint8') # class labels
for j in range(K):
ix = range(N*j,N*(j+1))
t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2 # theta
X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
y[ix] = j

# Plot the data
#plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)
layersDimensions = [2,3,3]
y1=y.reshape(-1,1).T
train_X=X.T
train_Y=y1
#Set layer dimensions
parameters = initializeDeepModel(layersDimensions)
#Perform forward prop
AL, caches, dropoutMat = forwardPropagationDeep(train_X, parameters, keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc="softmax")
#Compute cost
cost = computeCost(AL, train_Y, outputActivationFunc="softmax")
print("cost=",cost)
gradients = backwardPropagationDeep(AL, train_Y, caches, dropoutMat, lambd=0, keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc="softmax")
# Note the transpose of the gradients for Softmax has to be taken
L= len(parameters)//2
print(L)
gradient_check_n(parameters, gradients, train_X, train_Y, epsilon = 1e-7,outputActivationFunc="softmax")
## cost= 1.0986193170234435
## 2
## [92mYour backward propagation works perfectly fine! difference = 5.268804859613151e-07[0m
## 5.268804859613151e-07
##
##
## {'dW1': array([[ 0.00053206,  0.00038987],
##        [ 0.00093941,  0.00038077],
##        [-0.00012177, -0.0004692 ]]), 'db1': array([[ 0.00072662],
##        [-0.00210198],
##        [ 0.00046741]]), 'dW2': array([[-7.83441270e-05, -9.70179498e-04, -1.08715815e-04],
##        [-7.70175008e-04,  9.54478237e-04,  2.27690198e-04],
##        [ 8.48519135e-04,  1.57012608e-05, -1.18974383e-04]]), 'db2': array([[-8.52190476e-06],
##        [-1.69954294e-05],
##        [ 2.55173342e-05]])}
##
##
## {'dW1': array([[ 0.00053206,  0.00038987],
##        [ 0.00093941,  0.00038077],
##        [-0.00012177, -0.0004692 ]]), 'db1': array([[ 0.00072662],
##        [-0.00210198],
##        [ 0.00046741]]), 'dW2': array([[-7.83439980e-05, -9.70180603e-04, -1.08716369e-04],
##        [-7.70173925e-04,  9.54478718e-04,  2.27690089e-04],
##        [ 8.48520143e-04,  1.57018842e-05, -1.18973720e-04]]), 'db2': array([[-8.52096171e-06],
##        [-1.69964043e-05],
##        [ 2.55162558e-05]])}

## 1.2a Gradient Check – Sigmoid Activation – R

source("DLfunctions8.R")

x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
#Set layer dimensions
layersDimensions = c(2,5,1)
parameters = initializeDeepModel(layersDimensions)
#Perform forward prop
retvals = forwardPropagationDeep(X, parameters,keep_prob=1, hiddenActivationFunc="relu",
outputActivationFunc="sigmoid")
AL <- retvals[['AL']]
caches <- retvals[['caches']]
dropoutMat <- retvals[['dropoutMat']]
#Compute cost
cost <- computeCost(AL, Y,outputActivationFunc="sigmoid",
numClasses=layersDimensions[length(layersDimensions)])
print(cost)
## [1] 0.6931447
# Backward propagation.
gradients = backwardPropagationDeep(AL, Y, caches, dropoutMat, lambd=0, keep_prob=1, hiddenActivationFunc="relu",
outputActivationFunc="sigmoid",numClasses=layersDimensions[length(layersDimensions)])
epsilon = 1e-07
outputActivationFunc="sigmoid"
#Convert parameter list to vector
parameters_values = list_to_vector(parameters)
num_parameters = dim(parameters_values)[1]
#Initialize
J_plus = matrix(rep(0,num_parameters),
nrow=num_parameters,ncol=1)
J_minus = matrix(rep(0,num_parameters),
nrow=num_parameters,ncol=1)
nrow=num_parameters,ncol=1)

for(i in 1:num_parameters){
# Compute J_plus[i].
thetaplus = parameters_values
thetaplus[i][1] = thetaplus[i][1] + epsilon
retvals = forwardPropagationDeep(X, vector_to_list(parameters,thetaplus), keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc)

AL <- retvals[['AL']]
J_plus[i] = computeCost(AL, Y, outputActivationFunc=outputActivationFunc)

# Compute J_minus[i].
thetaminus = parameters_values
thetaminus[i][1] = thetaminus[i][1] - epsilon
retvals  = forwardPropagationDeep(X, vector_to_list(parameters,thetaminus), keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc)
AL <- retvals[['AL']]
J_minus[i] = computeCost(AL, Y, outputActivationFunc=outputActivationFunc)

}
#Compute L2Norm
difference =  numerator/denominator
if(difference > 1e-5){
cat("There is a mistake, the difference is too high",difference)
} else{
cat("The implementations works perfectly", difference)
}
## The implementations works perfectly 1.279911e-06
# This can be used to check
print("Gradients from backprop")
## [1] "Gradients from backprop"
vector_to_list2(parameters,grad)
## $dW1 ## [,1] [,2] ## [1,] -7.641588e-05 -3.427989e-07 ## [2,] -9.049683e-06 6.906304e-05 ## [3,] 3.401039e-06 -1.503914e-04 ## [4,] 1.535226e-04 -1.686402e-04 ## [5,] -6.029292e-05 -2.715648e-04 ## ##$db1
##               [,1]
## [1,]  6.930318e-06
## [2,] -3.283117e-05
## [3,]  1.310647e-05
## [4,] -3.454308e-05
## [5,] -2.331729e-08
##
## $dW2 ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0.0001612356 0.0001113475 0.0002435824 0.000362149 2.874116e-05 ## ##$db2
##              [,1]
## [1,] -1.16364e-05
print("Grad approx from gradient check")
## [1] "Grad approx from gradient check"
vector_to_list2(parameters,gradapprox)
## $dW1 ## [,1] [,2] ## [1,] -7.641554e-05 -3.430589e-07 ## [2,] -9.049428e-06 6.906253e-05 ## [3,] 3.401168e-06 -1.503919e-04 ## [4,] 1.535228e-04 -1.686401e-04 ## [5,] -6.029288e-05 -2.715650e-04 ## ##$db1
##               [,1]
## [1,]  6.930012e-06
## [2,] -3.283096e-05
## [3,]  1.310618e-05
## [4,] -3.454237e-05
## [5,] -2.275957e-08
##
## $dW2 ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0.0001612355 0.0001113476 0.0002435829 0.0003621486 2.87409e-05 ## ##$db2
##              [,1]
## [1,] -1.16368e-05

## 1.2b Gradient Check – Softmax Activation – R

source("DLfunctions8.R")

# Setup the data
X <- Z[,1:2]
y <- Z[,3]
X <- t(X)
Y <- t(y)
layersDimensions = c(2, 3, 3)
parameters = initializeDeepModel(layersDimensions)
#Perform forward prop
retvals = forwardPropagationDeep(X, parameters,keep_prob=1, hiddenActivationFunc="relu",
outputActivationFunc="softmax")
AL <- retvals[['AL']]
caches <- retvals[['caches']]
dropoutMat <- retvals[['dropoutMat']]
#Compute cost
cost <- computeCost(AL, Y,outputActivationFunc="softmax",
numClasses=layersDimensions[length(layersDimensions)])
print(cost)
## [1] 1.098618
# Backward propagation.
gradients = backwardPropagationDeep(AL, Y, caches, dropoutMat, lambd=0, keep_prob=1, hiddenActivationFunc="relu",
outputActivationFunc="softmax",numClasses=layersDimensions[length(layersDimensions)])
# Need to take transpose of the last layer for Softmax
L=length(parameters)/2
epsilon = 1e-7,outputActivationFunc="softmax")
## The implementations works perfectly 3.903011e-07[1] "Gradients from backprop"
## $dW1 ## [,1] [,2] ## [1,] 0.0007962367 -0.0001907606 ## [2,] 0.0004444254 0.0010354412 ## [3,] 0.0003078611 0.0007591255 ## ##$db1
##               [,1]
## [1,] -0.0017305136
## [2,]  0.0005393734
## [3,]  0.0012484550
##
## $dW2 ## [,1] [,2] [,3] ## [1,] -3.515627e-04 7.487283e-04 -3.971656e-04 ## [2,] -6.381521e-05 -1.257328e-06 6.507254e-05 ## [3,] -1.719479e-04 -4.857264e-04 6.576743e-04 ## ##$db2
##               [,1]
## [1,] -5.536383e-06
## [2,] -1.824656e-05
## [3,]  2.378295e-05
##
## $dW1 ## [,1] [,2] ## [1,] 0.0007962364 -0.0001907607 ## [2,] 0.0004444256 0.0010354406 ## [3,] 0.0003078615 0.0007591250 ## ##$db1
##               [,1]
## [1,] -0.0017305135
## [2,]  0.0005393741
## [3,]  0.0012484547
##
## $dW2 ## [,1] [,2] [,3] ## [1,] -3.515632e-04 7.487277e-04 -3.971656e-04 ## [2,] -6.381451e-05 -1.257883e-06 6.507239e-05 ## [3,] -1.719469e-04 -4.857270e-04 6.576739e-04 ## ##$db2
##               [,1]
## [1,] -5.536682e-06
## [2,] -1.824652e-05
## [3,]  2.378209e-05

## 1.3a Gradient Check – Sigmoid Activation – Octave

source("DL8functions.m")
################## Circles

X=data(:,1:2);
Y=data(:,3);
#Set layer dimensions
layersDimensions = [2 5  1]; #tanh=-0.5(ok), #relu=0.1 best!
[weights biases] = initializeDeepModel(layersDimensions);
#Perform forward prop
[AL forward_caches activation_caches droputMat] = forwardPropagationDeep(X', weights, biases,keep_prob=1,
hiddenActivationFunc="relu", outputActivationFunc="sigmoid");
#Compute cost
cost = computeCost(AL, Y',outputActivationFunc=outputActivationFunc,numClasses=layersDimensions(size(layersDimensions)(2)));
disp(cost);
hiddenActivationFunc="relu", outputActivationFunc="sigmoid",
numClasses=layersDimensions(size(layersDimensions)(2)));
epsilon = 1e-07;
outputActivationFunc="sigmoid";
# Convert paramter cell array to vector
parameters_values = cellArray_to_vector(weights, biases);
#Convert gradient cell array to vector
num_parameters = size(parameters_values)(1);
#Initialize
J_plus = zeros(num_parameters, 1);
J_minus = zeros(num_parameters, 1);
for i = 1:num_parameters
# Compute J_plus[i].
thetaplus = parameters_values;
thetaplus(i,1) = thetaplus(i,1) + epsilon;
[weights1 biases1] =vector_to_cellArray(weights, biases,thetaplus);
[AL forward_caches activation_caches droputMat] = forwardPropagationDeep(X', weights1, biases1, keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc);
J_plus(i) = computeCost(AL, Y', outputActivationFunc=outputActivationFunc);

# Compute J_minus[i].
thetaminus = parameters_values;
thetaminus(i,1) = thetaminus(i,1) - epsilon ;
[weights1 biases1] = vector_to_cellArray(weights, biases,thetaminus);
[AL forward_caches activation_caches droputMat]  = forwardPropagationDeep(X',weights1, biases1, keep_prob=1,
hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc);
J_minus(i) = computeCost(AL, Y', outputActivationFunc=outputActivationFunc);

endfor

#Compute L2Norm
difference =  numerator/denominator;
disp(difference);
#Check difference
if difference > 1e-04
printf("There is a mistake in the implementation ");
disp(difference);
else
printf("The implementation works perfectly");
disp(difference);
endif
disp(weights1);
disp(biases1);
disp(weights2);
disp(biases2);

0.69315
1.4893e-005
The implementation works perfectly 1.4893e-005
{
[1,1] =
5.0349e-005 2.1323e-005
8.8632e-007 1.8231e-006
9.3784e-005 1.0057e-004
1.0875e-004 -1.9529e-007
5.4502e-005 3.2721e-005
[1,2] =
1.0567e-005 6.0615e-005 4.6004e-005 1.3977e-004 1.0405e-004
}
{
[1,1] =
-1.8716e-005
1.1309e-009
4.7686e-005
1.2051e-005
-1.4612e-005
[1,2] = 9.5808e-006
}
{
[1,1] =
5.0348e-005 2.1320e-005
8.8485e-007 1.8219e-006
9.3784e-005 1.0057e-004
1.0875e-004 -1.9762e-007
5.4502e-005 3.2723e-005
[1,2] =
[1,2] =
1.0565e-005 6.0614e-005 4.6007e-005 1.3977e-004 1.0405e-004
}
{
[1,1] =
-1.8713e-005
1.1102e-009
4.7687e-005
1.2048e-005
-1.4609e-005
[1,2] = 9.5790e-006
}


## 1.3b Gradient Check – Softmax Activation – Octave

source("DL8functions.m")

# Setup the data
X=data(:,1:2);
Y=data(:,3);
# Set the layer dimensions
layersDimensions = [2 3  3];
[weights biases] = initializeDeepModel(layersDimensions);
# Run forward prop
[AL forward_caches activation_caches droputMat] = forwardPropagationDeep(X', weights, biases,keep_prob=1,
hiddenActivationFunc="relu", outputActivationFunc="softmax");
# Compute cost
cost = computeCost(AL, Y',outputActivationFunc=outputActivationFunc,numClasses=layersDimensions(size(layersDimensions)(2)));
disp(cost);
# Perform backward prop
hiddenActivationFunc="relu", outputActivationFunc="softmax",
numClasses=layersDimensions(size(layersDimensions)(2)));

#Take transpose of last layer for Softmax
L=size(weights)(2);
outputActivationFunc="softmax",numClasses=layersDimensions(size(layersDimensions)(2)));

 1.0986
The implementation works perfectly  2.0021e-005
{
[1,1] =
-7.1590e-005  4.1375e-005
-1.9494e-004  -5.2014e-005
-1.4554e-004  5.1699e-005
[1,2] =
3.3129e-004  1.9806e-004  -1.5662e-005
-4.9692e-004  -3.7756e-004  -8.2318e-005
1.6562e-004  1.7950e-004  9.7980e-005
}
{
[1,1] =
-3.0856e-005
-3.3321e-004
-3.8197e-004
[1,2] =
1.2046e-006
2.9259e-007
-1.4972e-006
}
{
[1,1] =
-7.1586e-005  4.1377e-005
-1.9494e-004  -5.2013e-005
-1.4554e-004  5.1695e-005
3.3129e-004  1.9806e-004  -1.5664e-005
-4.9692e-004  -3.7756e-004  -8.2316e-005
1.6562e-004  1.7950e-004  9.7979e-005
}
{
[1,1] =
-3.0852e-005
-3.3321e-004
-3.8197e-004
[1,2] =
1.1902e-006
2.8200e-007
-1.4644e-006
}


## 2.1 Tip for tuning hyperparameters

Deep Learning Networks come with a large number of hyper parameters which require tuning. The hyper parameters are

1. $\alpha$ -learning rate
2. Number of layers
3. Number of hidden units
4. Number of iterations
5. Momentum – $\beta$ – 0.9
6. RMSProp – $\beta_{1}$ – 0.9
7. Adam – $\beta_{1}$,$\beta_{2}$ and $\epsilon$
8. learning rate decay
9. mini batch size
10. Initialization method – He, Xavier
11. Regularization

– Among the above the most critical is learning rate $\alpha$ . Rather than just trying out random values, it may help to try out values on a logarithmic scale. So we could try out values -0.01,0.1,1.0,10 etc. If we find that the cost is between 0.01 and 0.1 we could use a technique similar to binary search or bisection, so we can try 0.01, 0.05. If we need to be bigger than 0.01 and 0.05 we could try 0.25  and then keep halving the distance etc.
– The performance of Momentum and RMSProp are very good and work well with values 0.9. Even with this, it is better to try out values of 1-$\beta$ in the logarithmic range. So 1-$\beta$ could 0.001,0.01,0.1 and hence $\beta$ would be 0.999,0.99 or 0.9
– Increasing the number of hidden units or number of hidden layers need to be done gradually. I have noticed that increasing number of hidden layers heavily does not improve performance and sometimes degrades it.
– Sometimes, I tend to increase the number of iterations if I think I see a steady decrease in the cost for a certain learning rate
– It may also help to add learning rate decay if you see there is an oscillation while it decreases.
– Xavier and He initializations also help in a fast convergence and are worth trying out.

## 3.1 Final thoughts

As I come to a close in this Deep Learning Series from first principles in Python, R and Octave, I must admit that I learnt a lot in the process.

* Building a L-layer, vectorized Deep Learning Network in Python, R and Octave was extremely challenging but very rewarding
* One benefit of building vectorized versions in Python, R and Octave was that I was looking at each function that I was implementing thrice, and hence I was able to fix any bugs in any of the languages
* In addition since I built the generic L-Layer DL network with all the bells and whistles, layer by layer I further had an opportunity to look at all the functions in each successive post.
* Each language has its advantages and disadvantages. From the performance perspective I think Python is the best, followed by Octave and then R
* Interesting, I noticed that even if small bugs creep into your implementation, the DL network does learn and does generate a valid set of weights and biases, however this may not be an optimum solution. In one case of an inadvertent bug, I was not updating the weights in the final layer of the DL network. Yet, using all the other layers, the DL network was able to come with a reasonable solution (maybe like random dropout, remaining units can still learn the data!)
* Having said that, the Gradient Check method discussed and implemented in this post can be very useful in ironing out bugs.

## Conclusion

These last couple of months when I was writing the posts and the also churning up the code in Python, R and Octave were  very hectic. There have been times when I found that implementations of some function to be extremely demanding and I almost felt like giving up. Other times, I have spent quite some time on an intractable DL network which would not respond to changes in hyper-parameters. All in all, it was a great learning experience. I would suggest that you start from my first post Deep Learning from first principles in Python, R and Octave-Part 1 and work your way up. Feel free to take the code apart and try out things. That is the only way you will learn.

Hope you had as much fun as I had. Stay tuned. I will be back!!!

To see all post click Index of Posts

# Deep Learning from first principles in Python, R and Octave – Part 7

Artificial Intelligence is the new electricity. – Prof Andrew Ng

Most of human and animal learning is unsupervised learning. If intelligence was a cake, unsupervised learning would be the cake, supervised learning would be the icing on the cake, and reinforcement learning would be the cherry on the cake. We know how to make the icing and the cherry, but we don’t know how to make the cake. We need to solve the unsupervised learning problem before we can even think of getting to true AI.  – Yann LeCun, March 14, 2016 (Facebook)

# Introduction

In this post ‘Deep Learning from first principles with Python, R and Octave-Part 7’, I implement optimization methods used in Stochastic Gradient Descent (SGD) to speed up the convergence. Specifically I discuss and implement the following gradient descent optimization techniques

b.Learning rate decay
c. Momentum method
d. RMSProp

This post, further enhances my generic  L-Layer Deep Learning Network implementations in  vectorized Python, R and Octave to also include the Stochastic Gradient Descent optimization techniques. You can clone/download the code from Github at DeepLearning-Part7

You can view my video  presentation on Gradient Descent Optimization in Neural Networks 7

Incidentally, a good discussion of the various optimizations methods used in Stochastic Gradient Optimization techniques can be seen at Sebastian Ruder’s blog

Note: In the vectorized Python, R and Octave implementations below only a  1024 random training samples were used. This was to reduce the computation time. You are free to use the entire data set (60000 training data) for the computation.

This post is largely based of on Prof Andrew Ng’s Deep Learning Specialization.  All the above optimization techniques for Stochastic Gradient Descent are based on the technique of exponentially weighted average method. So for example if we had some time series data $\theta_{1},\theta_{2},\theta_{3}... \theta_{t}$ then we we can represent the exponentially average value at time ‘t’ as a sequence of the the previous value $v_{t-1}$ and $\theta_{t}$ as shown below
$v_{t} = \beta v_{t-1} + (1-\beta)\theta_{t}$

Here $v_{t}$ represent the average of the data set over $\frac {1}{1-\beta}$  By choosing different values of $\beta$, we can average over a larger or smaller number of the data points.
We can write the equations as follows
$v_{t} = \beta v_{t-1} + (1-\beta)\theta_{t}$
$v_{t-1} = \beta v_{t-2} + (1-\beta)\theta_{t-1}$
$v_{t-2} = \beta v_{t-3} + (1-\beta)\theta_{t-2}$
and
$v_{t-k} = \beta v_{t-(k+1))} + (1-\beta)\theta_{t-k}$
By substitution we have
$v_{t} = (1-\beta)\theta_{t} + \beta v_{t-1}$
$v_{t} = (1-\beta)\theta_{t} + \beta ((1-\beta)\theta_{t-1}) + \beta v_{t-2}$
$v_{t} = (1-\beta)\theta_{t} + \beta ((1-\beta)\theta_{t-1}) + \beta ((1-\beta)\theta_{t-2}+ \beta v_{t-3} )$

Hence it can be seen that the $v_{t}$ is the weighted sum over the previous values $\theta_{k}$, which is an exponentially decaying function.

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).

You may also like my companion book “Practical Machine Learning with R and Python- Machine Learning in stereo” available in Amazon in paperback($9.99) and Kindle($6.99) versions. This book is ideal for a quick reference of the various ML functions and associated measurements in both R and Python which are essential to delve deep into Deep Learning.

## 1.1a. Stochastic Gradient Descent (Vanilla) – Python

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets

lbls=[]
pxls=[]
for i in range(60000):
l,p=training[i]
lbls.append(l)
pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T

# Create  a list of 1024 random numbers.
permutation = list(np.random.permutation(2**10))
# Subset 16384 from the data
X2 = X1[:, permutation]
Y2 = Y1[:, permutation].reshape((1,2**10))
# Set the layer dimensions
layersDimensions=[784, 15,9,10]
# Perform SGD with regular gradient descent
parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu',
outputActivationFunc="softmax",learningRate = 0.01 ,
optimizer="gd",
mini_batch_size =512, num_epochs = 1000, print_cost = True,figure="fig1.png")


## 1.1b. Stochastic Gradient Descent (Vanilla) – R

source("mnist.R")
source("DLfunctions7.R")
x <- t(train$x) X <- x[,1:60000] y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 1024 random samples from MNIST
permutation = c(sample(2^10))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)
# Set layer dimensions
layersDimensions=c(784, 15,9, 10)
# Perform SGD with regular gradient descent
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
hiddenActivationFunc='tanh',
outputActivationFunc="softmax",
learningRate = 0.05,
optimizer="gd",
mini_batch_size = 512,
num_epochs = 5000,
print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvalsSGD$costs df=data.frame(iterations,costs) ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") + ggtitle("Costs vs no of epochs") + xlab("No of epochss") + ylab("Cost") ## 1.1c. Stochastic Gradient Descent (Vanilla) – Octave source("DL7functions.m") #Load and read MNIST load('./mnist/mnist.txt.gz'); #Create a random permutatation from 1024 permutation = randperm(1024); disp(length(permutation)); # Use this 1024 as the batch X=trainX(permutation,:); Y=trainY(permutation,:); # Set layer dimensions layersDimensions=[784, 15, 9, 10]; # Perform SGD with regular gradient descent [weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax", learningRate = 0.005, lrDecay=true, decayRate=1, lambd=0, keep_prob=1, optimizer="gd", beta=0.9, beta1=0.9, beta2=0.999, epsilon=10^-8, mini_batch_size = 512, num_epochs = 5000); plotCostVsEpochs(5000,costs);  ## 2.1. Stochastic Gradient Descent with Learning rate decay Since in Stochastic Gradient Descent,with each epoch, we use slight different samples, the gradient descent algorithm, oscillates across the ravines and wanders around the minima, when a fixed learning rate is used. In this technique of ‘learning rate decay’ the learning rate is slowly decreased with the number of epochs and becomes smaller and smaller, so that gradient descent can take smaller steps towards the minima. There are several techniques employed in learning rate decay a) Exponential decay: $\alpha = decayRate^{epochNum} *\alpha_{0}$ b) 1/t decay : $\alpha = \frac{\alpha_{0}}{1 + decayRate*epochNum}$ c) $\alpha = \frac {decayRate}{\sqrt(epochNum)}*\alpha_{0}$ In my implementation I have used the ‘exponential decay’. The code snippet for Python is shown below if lrDecay == True: learningRate = np.power(decayRate,(num_epochs/1000)) * learningRate  ## 2.1a. Stochastic Gradient Descent with Learning rate decay – Python import numpy as np import matplotlib import matplotlib.pyplot as plt import sklearn.linear_model import pandas as pd import sklearn import sklearn.datasets exec(open("DLfunctions7.py").read()) exec(open("load_mnist.py").read()) # Read the MNIST data training=list(read(dataset='training',path=".\\mnist")) test=list(read(dataset='testing',path=".\\mnist")) lbls=[] pxls=[] for i in range(60000): l,p=training[i] lbls.append(l) pxls.append(p) labels= np.array(lbls) pixels=np.array(pxls) y=labels.reshape(-1,1) X=pixels.reshape(pixels.shape[0],-1) X1=X.T Y1=y.T # Create a list of random numbers of 1024 permutation = list(np.random.permutation(2**10)) # Subset 16384 from the data X2 = X1[:, permutation] Y2 = Y1[:, permutation].reshape((1,2**10)) # Set layer dimensions layersDimensions=[784, 15,9,10] # Perform SGD with learning rate decay parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax", learningRate = 0.01 , lrDecay=True, decayRate=0.9999, optimizer="gd", mini_batch_size =512, num_epochs = 1000, print_cost = True,figure="fig2.png") ## 2.1b. Stochastic Gradient Descent with Learning rate decay – R source("mnist.R") source("DLfunctions7.R") # Read and load MNIST load_mnist() x <- t(train$x)
X <- x[,1:60000]
y <-train$y y1 <- y[1:60000] y2 <- as.matrix(y1) Y=t(y2) # Subset 1024 random samples from MNIST permutation = c(sample(2^10)) # Randomly shuffle the training data X1 = X[, permutation] y1 = Y[1, permutation] y2 <- as.matrix(y1) Y1=t(y2) # Set layer dimensions layersDimensions=c(784, 15,9, 10) # Perform SGD with Learning rate decay retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions, hiddenActivationFunc='tanh', outputActivationFunc="softmax", learningRate = 0.05, lrDecay=TRUE, decayRate=0.9999, optimizer="gd", mini_batch_size = 512, num_epochs = 5000, print_cost = True) #Plot the cost vs iterations iterations <- seq(0,5000,1000) costs=retvalsSGD$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
ggtitle("Costs vs number of epochs") + xlab("No of epochs") + ylab("Cost")

## 2.1c. Stochastic Gradient Descent with Learning rate decay – Octave

source("DL7functions.m")
#Create a random permutatation from 1024
permutation = randperm(1024);
disp(length(permutation));

# Use this 1024 as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);

# Set layer dimensions
layersDimensions=[784, 15, 9, 10];
# Perform SGD with regular Learning rate decay
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.01,
lrDecay=true,
decayRate=0.999,
lambd=0,
keep_prob=1,
optimizer="gd",
beta=0.9,
beta1=0.9,
beta2=0.999,
epsilon=10^-8,
mini_batch_size = 512,
num_epochs = 5000);
plotCostVsEpochs(5000,costs)


## 3.1. Stochastic Gradient Descent with Momentum

Stochastic Gradient Descent with Momentum uses the exponentially weighted average method discusses above and more generally moves faster into the ravine than across it. The equations are
$v_{dW}^l = \beta v_{dW}^l + (1-\beta)dW^{l}$
$v_{db}^l = \beta v_{db}^l + (1-\beta)db^{l}$
$W^{l} = W^{l} - \alpha v_{dW}^l$
$b^{l} = b^{l} - \alpha v_{db}^l$ where
$v_{dW}$ and $v_{db}$ are the momentum terms which are exponentially weighted with the corresponding gradients ‘dW’ and ‘db’ at the corresponding layer ‘l’ The code snippet for Stochastic Gradient Descent with momentum in R is shown below

# Perform Gradient Descent with momentum
# Input : Weights and biases
#       : beta
#       : learning rate
#       : outputActivationFunc - Activation function at hidden layer sigmoid/softmax
#output : Updated weights after 1 iteration

L = length(parameters)/2 # number of layers in the neural network
# Update rule for each parameter. Use a for loop.
for(l in 1:(L-1)){
# Compute velocities
# v['dWk'] = beta *v['dWk'] + (1-beta)*dWk
v[[paste("dW",l, sep="")]] = beta*v[[paste("dW",l, sep="")]] +
v[[paste("db",l, sep="")]] = beta*v[[paste("db",l, sep="")]] +

parameters[[paste("W",l,sep="")]] = parameters[[paste("W",l,sep="")]] -
learningRate* v[[paste("dW",l, sep="")]]
parameters[[paste("b",l,sep="")]] = parameters[[paste("b",l,sep="")]] -
learningRate* v[[paste("db",l, sep="")]]
}
# Compute for the Lth layer
if(outputActivationFunc=="sigmoid"){
v[[paste("dW",L, sep="")]] = beta*v[[paste("dW",L, sep="")]] +
v[[paste("db",L, sep="")]] = beta*v[[paste("db",L, sep="")]] +

parameters[[paste("W",L,sep="")]] = parameters[[paste("W",L,sep="")]] -
learningRate* v[[paste("dW",l, sep="")]]
parameters[[paste("b",L,sep="")]] = parameters[[paste("b",L,sep="")]] -
learningRate* v[[paste("db",l, sep="")]]

}else if (outputActivationFunc=="softmax"){
v[[paste("dW",L, sep="")]] = beta*v[[paste("dW",L, sep="")]] +
v[[paste("db",L, sep="")]] = beta*v[[paste("db",L, sep="")]] +
parameters[[paste("W",L,sep="")]] = parameters[[paste("W",L,sep="")]] -
parameters[[paste("b",L,sep="")]] = parameters[[paste("b",L,sep="")]] -
}
return(parameters)
}

## 3.1a. Stochastic Gradient Descent with Momentum- Python

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
lbls=[]
pxls=[]
for i in range(60000):
l,p=training[i]
lbls.append(l)
pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T

# Create  a list of random numbers of 1024
permutation = list(np.random.permutation(2**10))
# Subset 16384 from the data
X2 = X1[:, permutation]
Y2 = Y1[:, permutation].reshape((1,2**10))
layersDimensions=[784, 15,9,10]
# Perform SGD with momentum
parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu',
outputActivationFunc="softmax",learningRate = 0.01 ,
optimizer="momentum", beta=0.9,
mini_batch_size =512, num_epochs = 1000, print_cost = True,figure="fig3.png")

## 3.1b. Stochastic Gradient Descent with Momentum- R

source("mnist.R")
source("DLfunctions7.R")
x <- t(train$x) X <- x[,1:60000] y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 1024 random samples from MNIST
permutation = c(sample(2^10))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)
layersDimensions=c(784, 15,9, 10)
# Perform SGD with momentum
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
hiddenActivationFunc='tanh',
outputActivationFunc="softmax",
learningRate = 0.05,
optimizer="momentum",
beta=0.9,
mini_batch_size = 512,
num_epochs = 5000,
print_cost = True)


#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvalsSGD$costs df=data.frame(iterations,costs) ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") + ggtitle("Costs vs number of epochs") + xlab("No of epochs") + ylab("Cost") ## 3.1c. Stochastic Gradient Descent with Momentum- Octave source("DL7functions.m") #Load and read MNIST load('./mnist/mnist.txt.gz'); #Create a random permutatation from 60K permutation = randperm(1024); disp(length(permutation)); # Use this 1024 as the batch X=trainX(permutation,:); Y=trainY(permutation,:); # Set layer dimensions layersDimensions=[784, 15, 9, 10]; # Perform SGD with Momentum [weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax", learningRate = 0.01, lrDecay=false, decayRate=1, lambd=0, keep_prob=1, optimizer="momentum", beta=0.9, beta1=0.9, beta2=0.999, epsilon=10^-8, mini_batch_size = 512, num_epochs = 5000); plotCostVsEpochs(5000,costs)  ## 4.1. Stochastic Gradient Descent with RMSProp Stochastic Gradient Descent with RMSProp tries to move faster towards the minima while dampening the oscillations across the ravine. The equations are $s_{dW}^l = \beta_{1} s_{dW}^l + (1-\beta_{1})(dW^{l})^{2}$ $s_{db}^l = \beta_{1} s_{db}^l + (1-\beta_{1})(db^{l})^2$ $W^{l} = W^{l} - \frac {\alpha s_{dW}^l}{\sqrt (s_{dW}^l + \epsilon) }$ $b^{l} = b^{l} - \frac {\alpha s_{db}^l}{\sqrt (s_{db}^l + \epsilon) }$ where $s_{dW}$ and $s_{db}$ are the RMSProp terms which are exponentially weighted with the corresponding gradients ‘dW’ and ‘db’ at the corresponding layer ‘l’ The code snippet in Octave is shown below # Update parameters with RMSProp # Input : parameters # : gradients # : s # : beta # : learningRate # : #output : Updated parameters RMSProp function [weights biases] = gradientDescentWithRMSProp(weights, biases,gradsDW,gradsDB, sdW, sdB, beta1, epsilon, learningRate,outputActivationFunc="sigmoid") L = size(weights)(2); # number of layers in the neural network # Update rule for each parameter. for l=1:(L-1) sdW{l} = beta1*sdW{l} + (1 -beta1) * gradsDW{l} .* gradsDW{l}; sdB{l} = beta1*sdB{l} + (1 -beta1) * gradsDB{l} .* gradsDB{l}; weights{l} = weights{l} - learningRate* gradsDW{l} ./ sqrt(sdW{l} + epsilon); biases{l} = biases{l} - learningRate* gradsDB{l} ./ sqrt(sdB{l} + epsilon); endfor if (strcmp(outputActivationFunc,"sigmoid")) sdW{L} = beta1*sdW{L} + (1 -beta1) * gradsDW{L} .* gradsDW{L}; sdB{L} = beta1*sdB{L} + (1 -beta1) * gradsDB{L} .* gradsDB{L}; weights{L} = weights{L} -learningRate* gradsDW{L} ./ sqrt(sdW{L} +epsilon); biases{L} = biases{L} -learningRate* gradsDB{L} ./ sqrt(sdB{L} + epsilon); elseif (strcmp(outputActivationFunc,"softmax")) sdW{L} = beta1*sdW{L} + (1 -beta1) * gradsDW{L}' .* gradsDW{L}'; sdB{L} = beta1*sdB{L} + (1 -beta1) * gradsDB{L}' .* gradsDB{L}'; weights{L} = weights{L} -learningRate* gradsDW{L}' ./ sqrt(sdW{L} +epsilon); biases{L} = biases{L} -learningRate* gradsDB{L}' ./ sqrt(sdB{L} + epsilon); endif end  ## 4.1a. Stochastic Gradient Descent with RMSProp – Python import numpy as np import matplotlib import matplotlib.pyplot as plt import sklearn.linear_model import pandas as pd import sklearn import sklearn.datasets exec(open("DLfunctions7.py").read()) exec(open("load_mnist.py").read()) # Read and load MNIST training=list(read(dataset='training',path=".\\mnist")) test=list(read(dataset='testing',path=".\\mnist")) lbls=[] pxls=[] for i in range(60000): l,p=training[i] lbls.append(l) pxls.append(p) labels= np.array(lbls) pixels=np.array(pxls) y=labels.reshape(-1,1) X=pixels.reshape(pixels.shape[0],-1) X1=X.T Y1=y.T print("X1=",X1.shape) print("y1=",Y1.shape) # Create a list of random numbers of 1024 permutation = list(np.random.permutation(2**10)) # Subset 16384 from the data X2 = X1[:, permutation] Y2 = Y1[:, permutation].reshape((1,2**10)) layersDimensions=[784, 15,9,10] # Use SGD with RMSProp parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",learningRate = 0.01 , optimizer="rmsprop", beta1=0.7, epsilon=1e-8, mini_batch_size =512, num_epochs = 1000, print_cost = True,figure="fig4.png") ## 4.1b. Stochastic Gradient Descent with RMSProp – R source("mnist.R") source("DLfunctions7.R") load_mnist() x <- t(train$x)
X <- x[,1:60000]
y <-train$y y1 <- y[1:60000] y2 <- as.matrix(y1) Y=t(y2) # Subset 1024 random samples from MNIST permutation = c(sample(2^10)) # Randomly shuffle the training data X1 = X[, permutation] y1 = Y[1, permutation] y2 <- as.matrix(y1) Y1=t(y2) layersDimensions=c(784, 15,9, 10) #Perform SGD with RMSProp retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions, hiddenActivationFunc='tanh', outputActivationFunc="softmax", learningRate = 0.001, optimizer="rmsprop", beta1=0.9, epsilon=10^-8, mini_batch_size = 512, num_epochs = 5000 , print_cost = True) #Plot the cost vs iterations iterations <- seq(0,5000,1000) costs=retvalsSGD$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
ggtitle("Costs vs number of epochs") + xlab("No of epochs") + ylab("Cost")


## 4.1c. Stochastic Gradient Descent with RMSProp – Octave

source("DL7functions.m")
#Create a random permutatation from 1024
permutation = randperm(1024);

# Use this 1024 as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);

# Set layer dimensions
layersDimensions=[784, 15, 9, 10];
#Perform SGD with RMSProp
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.005,
lrDecay=false,
decayRate=1,
lambd=0,
keep_prob=1,
optimizer="rmsprop",
beta=0.9,
beta1=0.9,
beta2=0.999,
epsilon=1,
mini_batch_size = 512,
num_epochs = 5000);
plotCostVsEpochs(5000,costs)


Adaptive Moment Estimate is a combination of the momentum (1st moment) and RMSProp(2nd moment). The equations for Adam are below
$v_{dW}^l = \beta_{1} v_{dW}^l + (1-\beta_{1})dW^{l}$
$v_{db}^l = \beta_{1} v_{db}^l + (1-\beta_{1})db^{l}$
The bias corrections for the 1st moment
$vCorrected_{dW}^l= \frac {v_{dW}^l}{1 - \beta_{1}^{t}}$
$vCorrected_{db}^l= \frac {v_{db}^l}{1 - \beta_{1}^{t}}$

Similarly the moving average for the 2nd moment- RMSProp
$s_{dW}^l = \beta_{2} s_{dW}^l + (1-\beta_{2})(dW^{l})^2$
$s_{db}^l = \beta_{2} s_{db}^l + (1-\beta_{2})(db^{l})^2$
The bias corrections for the 2nd moment
$sCorrected_{dW}^l= \frac {s_{dW}^l}{1 - \beta_{2}^{t}}$
$sCorrected_{db}^l= \frac {s_{db}^l}{1 - \beta_{2}^{t}}$

$W^{l} = W^{l} - \frac {\alpha vCorrected_{dW}^l}{\sqrt (s_{dW}^l + \epsilon) }$
$b^{l} = b^{l} - \frac {\alpha vCorrected_{db}^l}{\sqrt (s_{db}^l + \epsilon) }$
The code snippet of Adam in R is included below

# Perform Gradient Descent with Adam
# Input : Weights and biases
#       : beta1
#       : epsilon
#       : learning rate
#       : outputActivationFunc - Activation function at hidden layer sigmoid/softmax
#output : Updated weights after 1 iteration
beta1=0.9, beta2=0.999, epsilon=10^-8, learningRate=0.1,outputActivationFunc="sigmoid"){

L = length(parameters)/2 # number of layers in the neural network
v_corrected <- list()
s_corrected <- list()
# Update rule for each parameter. Use a for loop.
for(l in 1:(L-1)){
# v['dWk'] = beta *v['dWk'] + (1-beta)*dWk
v[[paste("dW",l, sep="")]] = beta1*v[[paste("dW",l, sep="")]] +
v[[paste("db",l, sep="")]] = beta1*v[[paste("db",l, sep="")]] +

# Compute bias-corrected first moment estimate.
v_corrected[[paste("dW",l, sep="")]] = v[[paste("dW",l, sep="")]]/(1-beta1^t)
v_corrected[[paste("db",l, sep="")]] = v[[paste("db",l, sep="")]]/(1-beta1^t)

# Element wise multiply of gradients
s[[paste("dW",l, sep="")]] = beta2*s[[paste("dW",l, sep="")]] +
s[[paste("db",l, sep="")]] = beta2*s[[paste("db",l, sep="")]] +

# Compute bias-corrected second moment estimate.
s_corrected[[paste("dW",l, sep="")]] = s[[paste("dW",l, sep="")]]/(1-beta2^t)
s_corrected[[paste("db",l, sep="")]] = s[[paste("db",l, sep="")]]/(1-beta2^t)

# Update parameters.
d1=sqrt(s_corrected[[paste("dW",l, sep="")]]+epsilon)
d2=sqrt(s_corrected[[paste("db",l, sep="")]]+epsilon)

parameters[[paste("W",l,sep="")]] = parameters[[paste("W",l,sep="")]] -
learningRate * v_corrected[[paste("dW",l, sep="")]]/d1
parameters[[paste("b",l,sep="")]] = parameters[[paste("b",l,sep="")]] -
learningRate*v_corrected[[paste("db",l, sep="")]]/d2
}
# Compute for the Lth layer
if(outputActivationFunc=="sigmoid"){
v[[paste("dW",L, sep="")]] = beta1*v[[paste("dW",L, sep="")]] +
v[[paste("db",L, sep="")]] = beta1*v[[paste("db",L, sep="")]] +

# Compute bias-corrected first moment estimate.
v_corrected[[paste("dW",L, sep="")]] = v[[paste("dW",L, sep="")]]/(1-beta1^t)
v_corrected[[paste("db",L, sep="")]] = v[[paste("db",L, sep="")]]/(1-beta1^t)

# Element wise multiply of gradients
s[[paste("dW",L, sep="")]] = beta2*s[[paste("dW",L, sep="")]] +
s[[paste("db",L, sep="")]] = beta2*s[[paste("db",L, sep="")]] +

# Compute bias-corrected second moment estimate.
s_corrected[[paste("dW",L, sep="")]] = s[[paste("dW",L, sep="")]]/(1-beta2^t)
s_corrected[[paste("db",L, sep="")]] = s[[paste("db",L, sep="")]]/(1-beta2^t)

# Update parameters.
d1=sqrt(s_corrected[[paste("dW",L, sep="")]]+epsilon)
d2=sqrt(s_corrected[[paste("db",L, sep="")]]+epsilon)

parameters[[paste("W",L,sep="")]] = parameters[[paste("W",L,sep="")]] -
learningRate * v_corrected[[paste("dW",L, sep="")]]/d1
parameters[[paste("b",L,sep="")]] = parameters[[paste("b",L,sep="")]] -
learningRate*v_corrected[[paste("db",L, sep="")]]/d2

}else if (outputActivationFunc=="softmax"){
v[[paste("dW",L, sep="")]] = beta1*v[[paste("dW",L, sep="")]] +
v[[paste("db",L, sep="")]] = beta1*v[[paste("db",L, sep="")]] +

# Compute bias-corrected first moment estimate.
v_corrected[[paste("dW",L, sep="")]] = v[[paste("dW",L, sep="")]]/(1-beta1^t)
v_corrected[[paste("db",L, sep="")]] = v[[paste("db",L, sep="")]]/(1-beta1^t)

# Element wise multiply of gradients
s[[paste("dW",L, sep="")]] = beta2*s[[paste("dW",L, sep="")]] +
s[[paste("db",L, sep="")]] = beta2*s[[paste("db",L, sep="")]] +

# Compute bias-corrected second moment estimate.
s_corrected[[paste("dW",L, sep="")]] = s[[paste("dW",L, sep="")]]/(1-beta2^t)
s_corrected[[paste("db",L, sep="")]] = s[[paste("db",L, sep="")]]/(1-beta2^t)

# Update parameters.
d1=sqrt(s_corrected[[paste("dW",L, sep="")]]+epsilon)
d2=sqrt(s_corrected[[paste("db",L, sep="")]]+epsilon)

parameters[[paste("W",L,sep="")]] = parameters[[paste("W",L,sep="")]] -
learningRate * v_corrected[[paste("dW",L, sep="")]]/d1
parameters[[paste("b",L,sep="")]] = parameters[[paste("b",L,sep="")]] -
learningRate*v_corrected[[paste("db",L, sep="")]]/d2
}
return(parameters)
}


import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
lbls=[]
pxls=[]
print(len(training))
#for i in range(len(training)):
for i in range(60000):
l,p=training[i]
lbls.append(l)
pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T

# Create  a list of random numbers of 1024
permutation = list(np.random.permutation(2**10))
# Subset 16384 from the data
X2 = X1[:, permutation]
Y2 = Y1[:, permutation].reshape((1,2**10))
layersDimensions=[784, 15,9,10]
parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu',
outputActivationFunc="softmax",learningRate = 0.01 ,
optimizer="adam", beta1=0.9, beta2=0.9, epsilon = 1e-8,
mini_batch_size =512, num_epochs = 1000, print_cost = True, figure="fig5.png")

source("mnist.R")
source("DLfunctions7.R")
x <- t(train$x) X <- x[,1:60000] y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 1024 random samples from MNIST
permutation = c(sample(2^10))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)
layersDimensions=c(784, 15,9, 10)
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
hiddenActivationFunc='tanh',
outputActivationFunc="softmax",
learningRate = 0.005,
beta1=0.7,
beta2=0.9,
epsilon=10^-8,
mini_batch_size = 512,
num_epochs = 5000 ,
print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvalsSGD\$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
ggtitle("Costs vs number of epochs") + xlab("No of epochs") + ylab("Cost")

source("DL7functions.m")
#Create a random permutatation from 1024
permutation = randperm(1024);
disp(length(permutation));

# Use this 1024 as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);
# Set layer dimensions
layersDimensions=[784, 15, 9, 10];

# Note the high value for epsilon.
#Otherwise GD with Adam does not seem to converge
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.1,
lrDecay=false,
decayRate=1,
lambd=0,
keep_prob=1,
beta=0.9,
beta1=0.9,
beta2=0.9,
epsilon=100,
mini_batch_size = 512,
num_epochs = 5000);
plotCostVsEpochs(5000,costs)


Conclusion: In this post I discuss and implement several Stochastic Gradient Descent optimization methods. The implementation of these methods enhance my already existing generic L-Layer Deep Learning Network implementation in vectorized Python, R and Octave, which I had discussed in the previous post in this series on Deep Learning from first principles in Python, R and Octave. Check it out, if you haven’t already. As already mentioned the code for this post can be cloned/forked from Github at DeepLearning-Part7

Watch this space! I’ll be back!

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