Pitching yorkpy … short of good length to IPL – Part 1

I fear not the man who has practiced 10,000 kicks once, but I fear the man who has practiced one kick 10,000 times.
Bruce Lee

I’ve missed more than 9000 shots in my career. I’ve lost almost 300 games. 26 times, I’ve been trusted to take the game winning shot and missed. I’ve failed over and over and over again in my life. And that is why I succeed.
Michael Jordan

Man, it doesn’t matter where you come in to bat, the score is still zero
Viv Richards

Introduction

“If cricketr is to cricpy, then yorkr is to _____?”. Yes, you guessed it right, it is yorkpy. In this post, I introduce my 2nd python package, yorkpy, which is a python clone of my R package yorkr. This package is based on data from Cricsheet. yorkpy currently handles IPL T20 matches.

When I created cricpy, the python avatar, of my R package cricketr, see Introducing cricpy:A python package to analyze performances of cricketers, I had decided that I should avoid doing a python avatar of my R package yorkr (see Introducing cricket package yorkr: Part 1- Beaten by sheer pace!) , as it was more involved, and required the parsing of match data available as yaml files.

Just out of curiosity, I tried the python package ‘yaml’ to read the match data, and lo and behold, I was sucked into the developing the package and so, yorkpy was born. Of course, it goes without saying that, usually when I am in the thick of developing something, I occasionally wonder, why I am doing it, for whom and for what purpose? Maybe it is the joy of ideation, the problem-solving,  the programmer’s high, for sharing my ideas etc. Anyway, whatever be the reason, I hope you enjoy this post and also find yorkpy useful.

You can clone/download the code at Github yorkpy
This post has been published to RPubs at yorkpy-Part1
You can download this post as PDF at IPLT20-yorkpy-part1

Note: If you would like to do a similar analysis for a different set of batsman and bowlers, you can clone/download my skeleton yorkpy-template from Github (which is the R Markdown file I have used for the analysis below).

The IPL T20 functions in yorkpy are

2. Install the package using ‘pip install’

import pandas as pd
import yorkpy.analytics as yka
#pip install yorkpy

3. Load a yaml file from Cricsheet

There are 2 functions that can be to convert the IPL Twenty20 yaml files to pandas dataframeare

  1. convertYaml2PandasDataframeT20
  2. convertAllYaml2PandasDataframesT20

Note 1: While I have already converted the IPL T20 files, you will need to use these functions for future IPL matches

4. Convert and save IPL T20 yaml file to pandas dataframe

This function will convert a IPL T20 IPL yaml file, in the format as specified in Cricsheet to pandas dataframe. This will be saved as as CSV file in the target directory. The name of the file wil have the following format team1-team2-date.csv. The IPL T20 zip file can be downloaded from Indian Premier League matches.  An example of how a yaml file can be converted to a dataframe and saved is shown below.

import pandas as pd
import yorkpy.analytics as yka
#convertYaml2PandasDataframe(".\\1082593.yaml","..\ipl", ..\\data")

5. Convert and save all IPL T20 yaml files to dataframes

This function will convert all IPL T20 yaml files from a source directory to dataframes, and save it in the target directory, with the names as mentioned above. Since I have already done this, I will not be executing this again. You can download the zip of all the converted RData files from Github at yorkpyData

import pandas as pd
import yorkpy.analytics as yka
#convertAllYaml2PandasDataframes("..\\ipl", "..\\data")

You can download the the zip of the files and use it directly in the functions as follows.For the analysis below I chosen a set of random IPL matches

The randomly selected IPL T20 matches are

  • Chennai Super Kings vs Kings Xi Punjab, 2014-05-30
  • Deccan Chargers vs Delhi Daredevils, 2012-05-10
  • Gujarat Lions vs Mumbai Indians, 2017-04-29
  • Kolkata Knight Riders vs Rajasthan Royals, 2010-04-17
  • Rising Pune Supergiants vs Royal Challengers Bangalore, 2017-04-29

6. Team batting scorecard

The function below computes the batting score card of a team in an IPL match. The scorecard gives the balls faced, the runs scored, 4s, 6s and strike rate. The example below is based on the CSK KXIP match on 30 May 2014.

You can check against the actual scores in this match Chennai Super Kings-Kings XI Punjab-2014-05-30

import pandas as pd
import yorkpy.analytics as yka
csk_kxip=pd.read_csv(".\\Chennai Super Kings-Kings XI Punjab-2014-05-30.csv")
scorecard,extras=yka.teamBattingScorecardMatch(csk_kxip,"Chennai Super Kings")
print(scorecard)
##         batsman  runs  balls  4s  6s          SR
## 0      DR Smith     7     12   0   0   58.333333
## 1  F du Plessis     0      1   0   0    0.000000
## 2      SK Raina    87     26  12   6  334.615385
## 3   BB McCullum    11     16   0   0   68.750000
## 4     RA Jadeja    27     22   2   1  122.727273
## 5     DJ Hussey     1      3   0   0   33.333333
## 6      MS Dhoni    42     34   3   3  123.529412
## 7      R Ashwin    10     11   0   0   90.909091
## 8     MM Sharma     1      3   0   0   33.333333
print(extras)
##    total  wides  noballs  legbyes  byes  penalty  extras
## 0    428     14        3        5     5        0      27
print("\n\n")
scorecard1,extras1=yka.teamBattingScorecardMatch(csk_kxip,"Kings XI Punjab")
print(scorecard1)
##       batsman  runs  balls  4s  6s          SR
## 0    V Sehwag   122     62  12   8  196.774194
## 1     M Vohra    34     33   1   2  103.030303
## 2  GJ Maxwell    13      8   1   1  162.500000
## 3   DA Miller    38     19   5   1  200.000000
## 4   GJ Bailey     1      2   0   0   50.000000
## 5     WP Saha     6      4   0   1  150.000000
## 6  MG Johnson     1      1   0   0  100.000000
print(extras1)
##    total  wides  noballs  legbyes  byes  penalty  extras
## 0    428     14        3        5     5        0      27

Let’s take another random match between Gujarat Lions and Mumbai Indian on 29 Apr 2017 Gujarat Lions-Mumbai Indians-2017-04-29

import pandas as pd
gl_mi=pd.read_csv(".\\Gujarat Lions-Mumbai Indians-2017-04-29.csv")
import yorkpy.analytics as yka
scorecard,extras=yka.teamBattingScorecardMatch(gl_mi,"Gujarat Lions")
print(scorecard)
##          batsman  runs  balls  4s  6s          SR
## 0   Ishan Kishan    48     38   6   2  126.315789
## 1    BB McCullum     6      4   1   0  150.000000
## 2       SK Raina     1      3   0   0   33.333333
## 3       AJ Finch     0      3   0   0    0.000000
## 4     KD Karthik     2      9   0   0   22.222222
## 5      RA Jadeja    28     22   2   1  127.272727
## 6    JP Faulkner    21     29   2   0   72.413793
## 7      IK Pathan     2      3   0   0   66.666667
## 8         AJ Tye    25     12   2   2  208.333333
## 9   Basil Thampi     2      4   0   0   50.000000
## 10    Ankit Soni     7      2   0   1  350.000000
print(extras)
##    total  wides  noballs  legbyes  byes  penalty  extras
## 0    306      8        3        1     0        0      12
print("\n\n")
scorecard1,extras1=yka.teamBattingScorecardMatch(gl_mi,"Mumbai Indians")
print(scorecard1)
##             batsman  runs  balls  4s  6s          SR
## 0          PA Patel    70     45   9   1  155.555556
## 1        JC Buttler     9      7   2   0  128.571429
## 2            N Rana    19     16   1   1  118.750000
## 3         RG Sharma     5     13   0   0   38.461538
## 4        KA Pollard    15     11   2   0  136.363636
## 5         KH Pandya    29     20   2   1  145.000000
## 6         HH Pandya     4      5   0   0   80.000000
## 7   Harbhajan Singh     0      1   0   0    0.000000
## 8    MJ McClenaghan     1      1   0   0  100.000000
## 9         JJ Bumrah     0      1   0   0    0.000000
## 10       SL Malinga     0      1   0   0    0.000000
print(extras1)
##    total  wides  noballs  legbyes  byes  penalty  extras
## 0    306      8        3        1     0        0      12

7. Plot the team batting partnerships

The functions below plot the team batting partnership in the match. It shows what the partnership were in the mtach

Note: Many of the plots include an additional parameters plot which is either True or False. The default value is plot=True. When plot=True the plot will be displayed. When plot=False the data frame will be returned to the user. The user can use this to create an interactive chart using one of the packages like rcharts, ggvis,googleVis or plotly.

import pandas as pd
import yorkpy.analytics as yka
dc_dd=pd.read_csv(".\\Deccan Chargers-Delhi Daredevils-2012-05-10.csv")
yka.teamBatsmenPartnershipMatch(dc_dd,'Deccan Chargers','Delhi Daredevils')

yka.teamBatsmenPartnershipMatch(dc_dd,'Delhi Daredevils','Deccan Chargers',plot=True)
# Print partnerships as a dataframe

rps_rcb=pd.read_csv(".\\Rising Pune Supergiant-Royal Challengers Bangalore-2017-04-29.csv")
m=yka.teamBatsmenPartnershipMatch(rps_rcb,'Royal Challengers Bangalore','Rising Pune Supergiant',plot=False)
print(m)
##            batsman     non_striker  runs
## 0   AB de Villiers         V Kohli     3
## 1         AF Milne         V Kohli     5
## 2        KM Jadhav         V Kohli     7
## 3           P Negi         V Kohli     3
## 4        S Aravind         V Kohli     0
## 5        S Aravind       YS Chahal     8
## 6         S Badree         V Kohli     2
## 7        STR Binny         V Kohli     1
## 8      Sachin Baby         V Kohli     2
## 9          TM Head         V Kohli     2
## 10         V Kohli  AB de Villiers    17
## 11         V Kohli        AF Milne     5
## 12         V Kohli       KM Jadhav     4
## 13         V Kohli          P Negi     9
## 14         V Kohli       S Aravind     2
## 15         V Kohli        S Badree     8
## 16         V Kohli     Sachin Baby     1
## 17         V Kohli         TM Head     9
## 18       YS Chahal       S Aravind     4

8. Batsmen vs Bowler

The function below computes and plots the performances of the batsmen vs the bowlers. As before the plot parameter can be set to True or False. By default it is plot=True

import pandas as pd
import yorkpy.analytics as yka
gl_mi=pd.read_csv(".\\Gujarat Lions-Mumbai Indians-2017-04-29.csv")
yka.teamBatsmenVsBowlersMatch(gl_mi,"Gujarat Lions","Mumbai Indians", plot=True)
# Print 

csk_kxip=pd.read_csv(".\\Chennai Super Kings-Kings XI Punjab-2014-05-30.csv")
m=yka.teamBatsmenVsBowlersMatch(csk_kxip,'Chennai Super Kings','Kings XI Punjab',plot=False)
print(m)
##          batsman           bowler  runs
## 0    BB McCullum         AR Patel     4
## 1    BB McCullum       GJ Maxwell     1
## 2    BB McCullum  Karanveer Singh     6
## 3      DJ Hussey          P Awana     1
## 4       DR Smith       MG Johnson     7
## 5       DR Smith          P Awana     0
## 6       DR Smith   Sandeep Sharma     0
## 7   F du Plessis       MG Johnson     0
## 8      MM Sharma         AR Patel     0
## 9      MM Sharma       MG Johnson     0
## 10     MM Sharma          P Awana     1
## 11      MS Dhoni         AR Patel    12
## 12      MS Dhoni  Karanveer Singh     2
## 13      MS Dhoni       MG Johnson    11
## 14      MS Dhoni          P Awana    15
## 15      MS Dhoni   Sandeep Sharma     2
## 16      R Ashwin         AR Patel     1
## 17      R Ashwin  Karanveer Singh     4
## 18      R Ashwin       MG Johnson     1
## 19      R Ashwin          P Awana     1
## 20      R Ashwin   Sandeep Sharma     3
## 21     RA Jadeja         AR Patel     5
## 22     RA Jadeja       GJ Maxwell     3
## 23     RA Jadeja  Karanveer Singh    19
## 24     RA Jadeja          P Awana     0
## 25      SK Raina       MG Johnson    21
## 26      SK Raina          P Awana    40
## 27      SK Raina   Sandeep Sharma    26

9. Bowling Scorecard

This function provides the bowling performance, the number of overs bowled, maidens, runs conceded. wickets taken and economy rate for the IPL match

import pandas as pd
import yorkpy.analytics as yka
dc_dd=pd.read_csv(".\\Deccan Chargers-Delhi Daredevils-2012-05-10.csv")
a=yka.teamBowlingScorecardMatch(dc_dd,'Deccan Chargers')
print(a)
##        bowler  overs  runs  maidens  wicket  econrate
## 0  AD Russell      4    39        0       0      9.75
## 1   IK Pathan      4    46        0       1     11.50
## 2    M Morkel      4    32        0       1      8.00
## 3    S Nadeem      4    39        0       0      9.75
## 4    VR Aaron      4    30        0       2      7.50
rps_rcb=pd.read_csv(".\\Rising Pune Supergiant-Royal Challengers Bangalore-2017-04-29.csv")
b=yka.teamBowlingScorecardMatch(rps_rcb,'Royal Challengers Bangalore')
print(b)
##               bowler  overs  runs  maidens  wicket  econrate
## 0          DL Chahar      2    18        0       0      9.00
## 1       DT Christian      4    25        0       1      6.25
## 2        Imran Tahir      4    18        0       3      4.50
## 3         JD Unadkat      4    19        0       1      4.75
## 4        LH Ferguson      4     7        1       3      1.75
## 5  Washington Sundar      2     7        0       1      3.50

10. Wicket Kind

The plots below provide the kind of wicket taken by the bowler (caught, bowled, lbw etc.) for the IPL match

import pandas as pd
import yorkpy.analytics as yka
kkr_rr=pd.read_csv(".\\Kolkata Knight Riders-Rajasthan Royals-2010-04-17.csv")
yka.teamBowlingWicketKindMatch(kkr_rr,'Kolkata Knight Riders','Rajasthan Royals')

csk_kxip=pd.read_csv(".\\Chennai Super Kings-Kings XI Punjab-2014-05-30.csv")
m = yka.teamBowlingWicketKindMatch(csk_kxip,'Chennai Super Kings','Kings-Kings XI Punjab',plot=False)
print(m)
##             bowler     kind  player_out
## 0         AR Patel  run out           1
## 1         AR Patel  stumped           1
## 2  Karanveer Singh  run out           1
## 3       MG Johnson   caught           1
## 4          P Awana   caught           2
## 5   Sandeep Sharma   bowled           1

11. Wicket vs Runs conceded

The plots below provide the wickets taken and the runs conceded by the bowler in the IPL T20 match

import pandas as pd
import yorkpy.analytics as yka
dc_dd=pd.read_csv(".\\Deccan Chargers-Delhi Daredevils-2012-05-10.csv")
yka.teamBowlingWicketMatch(dc_dd,"Deccan Chargers", "Delhi Daredevils",plot=True)

print("\n\n")
rps_rcb=pd.read_csv(".\\Rising Pune Supergiant-Royal Challengers Bangalore-2017-04-29.csv")
a=yka.teamBowlingWicketMatch(rps_rcb,"Royal Challengers Bangalore", "Rising Pune Supergiant",plot=False)
print(a)
##               bowler      player_out  kind
## 0       DT Christian         V Kohli     1
## 1        Imran Tahir        AF Milne     1
## 2        Imran Tahir          P Negi     1
## 3        Imran Tahir        S Badree     1
## 4         JD Unadkat         TM Head     1
## 5        LH Ferguson  AB de Villiers     1
## 6        LH Ferguson       KM Jadhav     1
## 7        LH Ferguson       STR Binny     1
## 8  Washington Sundar     Sachin Baby     1

12. Bowler Vs Batsmen

The functions compute and display how the different bowlers of the IPL team performed against the batting opposition.

import pandas as pd
import yorkpy.analytics as yka
csk_kxip=pd.read_csv(".\\Chennai Super Kings-Kings XI Punjab-2014-05-30.csv")
yka.teamBowlersVsBatsmenMatch(csk_kxip,"Chennai Super Kings","Kings XI Punjab")

print("\n\n")
kkr_rr=pd.read_csv(".\\Kolkata Knight Riders-Rajasthan Royals-2010-04-17.csv")
m =yka.teamBowlersVsBatsmenMatch(kkr_rr,"Rajasthan Royals","Kolkata Knight Riders",plot=False)
print(m)
##        batsman      bowler  runs
## 0     AC Voges    AB Dinda     1
## 1     AC Voges  JD Unadkat     1
## 2     AC Voges   LR Shukla     1
## 3     AC Voges    M Kartik     5
## 4     AJ Finch    AB Dinda     3
## 5     AJ Finch  JD Unadkat     3
## 6     AJ Finch   LR Shukla    13
## 7     AJ Finch    M Kartik     2
## 8     AJ Finch     SE Bond     0
## 9      AS Raut    AB Dinda     1
## 10     AS Raut  JD Unadkat     1
## 11    FY Fazal    AB Dinda     1
## 12    FY Fazal   LR Shukla     3
## 13    FY Fazal    M Kartik     3
## 14    FY Fazal     SE Bond     6
## 15     NV Ojha    AB Dinda    10
## 16     NV Ojha  JD Unadkat     5
## 17     NV Ojha   LR Shukla     0
## 18     NV Ojha    M Kartik     1
## 19     NV Ojha     SE Bond     2
## 20     P Dogra  JD Unadkat     2
## 21     P Dogra   LR Shukla     5
## 22     P Dogra    M Kartik     1
## 23     P Dogra     SE Bond     0
## 24  SK Trivedi    AB Dinda     4
## 25    SK Warne    AB Dinda     2
## 26    SK Warne    M Kartik     1
## 27    SK Warne     SE Bond     0
## 28   SR Watson    AB Dinda     2
## 29   SR Watson  JD Unadkat    13
## 30   SR Watson   LR Shukla     1
## 31   SR Watson    M Kartik    18
## 32   SR Watson     SE Bond    10
## 33   YK Pathan  JD Unadkat     1
## 34   YK Pathan   LR Shukla     7

13. Match worm chart

The plots below provide the match worm graph for the IPL Twenty 20 matches

import pandas as pd
import yorkpy.analytics as yka
dc_dd=pd.read_csv(".\\Deccan Chargers-Delhi Daredevils-2012-05-10.csv")
yka.matchWormChart(dc_dd,"Deccan Chargers", "Delhi Daredevils")

gl_mi=pd.read_csv(".\\Gujarat Lions-Mumbai Indians-2017-04-29.csv")
yka.matchWormChart(gl_mi,"Mumbai Indians","Gujarat Lions")

Feel free to clone/download the code from Github yorkpy

Conclusion

This post included all functions between 2 IPL teams from the package yorkpy for IPL Twenty20 matches. As mentioned above the yaml match files have been already converted to dataframes and are available for download from Github at yorkpyData

After having used Python and R for analytics, Machine Learning and Deep Learning, I have now realized that neither language is superior or inferior. Both have, some good packages and some that are not so well suited.

To be continued. Watch this space!

Important note: Do check out my other posts using yorkpy at yorkpy-posts

You may also like
1.My book ‘Deep Learning from first principles:Second Edition’ now on Amazon
2.My book ‘Practical Machine Learning in R and Python: Second edition’ on Amazon
2. Cricpy takes a swing at the ODIs
3. Introducing cricket package yorkr: Part 1- Beaten by sheer pace!
4. Big Data-1: Move into the big league:Graduate from Python to Pyspark
5. Simulating an Edge Shape in Android

To see all posts click Index of posts

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

Untitled

 

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

Also see
1. My book ‘Practical Machine Learning in R and Python: Second edition’ on Amazon
2. The 3rd paperback & kindle editions of my books on Cricket, now on Amazon
3. De-blurring revisited with Wiener filter using OpenCV
4. TWS-4: Gossip protocol: Epidemics and rumors to the rescue
5. A Cloud medley with IBM Bluemix, Cloudant DB and Node.js
6. Practical Machine Learning with R and Python – Part 6
7. GooglyPlus: yorkr analyzes IPL players, teams, matches with plots and tables
8. Fun simulation of a Chain in Android

To see posts click Index of Posts

Cricpy takes a swing at the ODIs

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 Minksy

“The competent programmer is fully aware of the limited size of his own skull. He therefore approaches his task with full humility, and avoids clever tricks like the plague” Edgser Djikstra

Introduction

In this post, cricpy, the Python avatar of my R package cricketr, learns some new tricks to be able to handle ODI matches. To know more about my R package cricketr see Re-introducing cricketr! : An R package to analyze performances of cricketers

Cricpy 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

To know how to use cricpy see Introducing cricpy:A python package to analyze performances of cricketers. To the original version of cricpy, I have added 3 new functions for ODI. The earlier functions work for Test and ODI.

This post is also hosted on Rpubs at Cricpy takes a swing at the ODIs. You can also down the pdf version of this post at cricpy-odi.pdf

You can fork/clone the package at Github cricpy

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 data for a particular player in ODI can be obtained with the getPlayerDataOD() function. To do you will need to go to ESPN CricInfo Player and type in the name of the player for e.g Virat Kohli, Virendar Sehwag, Chris Gayle etc. This will bring up a page which have the profile number for the player e.g. for Virat Kohli this would be http://www.espncricinfo.com/india/content/player/253802.html. Hence, Kohli’s profile is 253802. This can be used to get the data for Virat Kohlis shown below

The cricpy package is a clone of my R package cricketr. The signature of all the python functions are identical with that of its clone ‘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 familar with one of the lanuguages you can look up the package in the other and you will notice the parallel constructs.

You can fork/clone the package at Github cricpy

Note: The charts are self-explanatory and I have not added much of my owy interpretation to it. Do look at the plots closely and check out the performances for yourself.

1 Importing cricpy – Python

# Install the package
# Do a pip install cricpy
# Import cricpy
import cricpy.analytics as ca 

2. Invoking functions with Python package crlcpy

import cricpy.analytics as ca 
ca.batsman4s("./kohli.csv","Virat Kohli")

3. Getting help from cricpy – Python

import cricpy.analytics as ca 
help(ca.getPlayerDataOD)
## Help on function getPlayerDataOD in module cricpy.analytics:
## 
## getPlayerDataOD(profile, opposition='', host='', dir='./data', file='player001.csv', type='batting', homeOrAway=[1, 2, 3], result=[1, 2, 3, 5], create=True)
##     Get the One day 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
##     
##     getPlayerDataOD(profile, opposition="",host="",dir = "../", file = "player001.csv", 
##     type = "batting", homeOrAway = c(1, 2, 3), result = c(1, 2, 3,5))
##     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 Virender Sehwag this turns out to be http://www.espncricinfo.com/india/content/player/35263.html. Hence the profile for Sehwag is 35263
##     opposition      The numerical value of the opposition country e.g.Australia,India, England etc. The values are Australia:2,Bangladesh:25,Bermuda:12, England:1,Hong Kong:19,India:6,Ireland:29, Netherlands:15,New Zealand:5,Pakistan:7,Scotland:30,South Africa:3,Sri Lanka:8,United Arab Emirates:27, West Indies:4, Zimbabwe:9; Africa XI:405 Note: If no value is entered for opposition then all teams are considered
##     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,Ireland:29,Malaysia:16,New Zealand:5,Pakistan:7, Scotland:30,South Africa:3,Sri Lanka:8,United Arab Emirates:27,West Indies:4, Zimbabwe:9 Note: If no value is entered for host then all host countries are considered
##     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 vector with either or all 1,2, 3. 1 is for home 2 is for away, 3 is for neutral venue
##     result      
##     This is a vector that can take values 1,2,3,5. 1 - won match 2- lost match 3-tied 5- no result
##     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 getPlayerData
##     
##     Examples
##     
##     
##     ## Not run: 
##     # Both home and away. Result = won,lost and drawn
##     sehwag =getPlayerDataOD(35263,dir="../cricketr/data", file="sehwag1.csv",
##     type="batting", homeOrAway=[1,2],result=[1,2,3,4])
##     
##     # Only away. Get data only for won and lost innings
##     sehwag = getPlayerDataOD(35263,dir="../cricketr/data", file="sehwag2.csv",
##     type="batting",homeOrAway=[2],result=[1,2])
##     
##     # Get bowling data and store in file for future
##     malinga = getPlayerData(49758,dir="../cricketr/data",file="malinga1.csv",
##     type="bowling")
##     
##     # Get Dhoni's ODI record in Australia against Australua
##     dhoni = getPlayerDataOD(28081,opposition = 2,host=2,dir=".",
##     file="dhoniVsAusinAusOD",type="batting")
##     
##     ## End(Not run)

The details below will introduce the different functions that are available in cricpy.

4. Get the ODI player data for a player using the function getPlayerDataOD()

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. kohli.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 getPlayerDataOD for all subsequent analyses

import cricpy.analytics as ca
#sehwag=ca.getPlayerDataOD(35263,dir=".",file="sehwag.csv",type="batting")
#kohli=ca.getPlayerDataOD(253802,dir=".",file="kohli.csv",type="batting")
#jayasuriya=ca.getPlayerDataOD(49209,dir=".",file="jayasuriya.csv",type="batting")
#gayle=ca.getPlayerDataOD(51880,dir=".",file="gayle.csv",type="batting")

Included below are some of the functions that can be used for ODI batsmen and bowlers. For this I have chosen, Virat Kohli, ‘the run machine’ who is on-track for breaking many of the Test & ODI records

5 Virat Kohli’s performance – Basic Analyses

The 3 plots below provide the following for Virat Kohli

  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("./kohli.csv","Virat Kohli")

ca.batsmanMeanStrikeRate("./kohli.csv","Virat Kohli")

ca.batsmanRunsRanges("./kohli.csv","Virat Kohli")

6. More analyses

import cricpy.analytics as ca
ca.batsman4s("./kohli.csv","Virat Kohli")

ca.batsman6s("./kohli.csv","Virat Kohli")

ca.batsmanDismissals("./kohli.csv","Virat Kohli")

ca.batsmanScoringRateODTT("./kohli.csv","Virat Kohli")


7. 3D scatter plot and prediction plane

The plots below show the 3D scatter plot of Kohli’s 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("./kohli.csv","Virat Kohli")

Average runs at different venues

The plot below gives the average runs scored by Kohli at different grounds. The plot also the number of innings at each ground as a label at x-axis.

import cricpy.analytics as ca
ca.batsmanAvgRunsGround("./kohli.csv","Virat Kohli")

9. Average runs against different opposing teams

This plot computes the average runs scored by Kohli against different countries.

import cricpy.analytics as ca
ca.batsmanAvgRunsOpposition("./kohli.csv","Virat Kohli")

10 . Highest Runs Likelihood

The plot below shows the Runs Likelihood for a batsman. For this the performance of Kohli 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 Kohli’s highest tendencies are computed and plotted using K-Means

import cricpy.analytics as ca
ca.batsmanRunsLikelihood("./kohli.csv","Virat Kohli")

A look at the Top 4 batsman – Kohli, Jayasuriya, Sehwag and Gayle

The following batsmen have been very prolific in ODI cricket and will be used for the analyses

  1. Virat Kohli: Runs – 10232, Average:59.83 ,Strike rate-92.88
  2. Sanath Jayasuriya : Runs – 13430, Average:32.36 ,Strike rate-91.2
  3. Virendar Sehwag :Runs – 8273, Average:35.05 ,Strike rate-104.33
  4. Chris Gayle : Runs – 9727, Average:37.12 ,Strike rate-85.82

The following plots take a closer at their performances. The box plots show the median the 1st and 3rd quartile of the runs

12. 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("./kohli.csv","Virat Kohli")

ca.batsmanPerfBoxHist("./jayasuriya.csv","Sanath jayasuriya")

ca.batsmanPerfBoxHist("./gayle.csv","Chris Gayle")

ca.batsmanPerfBoxHist("./sehwag.csv","Virendar Sehwag")

13 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!). Kohli’s performance has been steadily improving over the years, so has Sehwag. Gayle seems to be on the way down

import cricpy.analytics as ca
ca.batsmanMovingAverage("./kohli.csv","Virat Kohli")

ca.batsmanMovingAverage("./jayasuriya.csv","Sanath jayasuriya")

ca.batsmanMovingAverage("./gayle.csv","Chris Gayle")

ca.batsmanMovingAverage("./sehwag.csv","Virendar Sehwag")

14 Cumulative Average runs of batsman in career

This function provides the cumulative average runs of the batsman over the career. Kohli seems to be getting better with time and reaches a cumulative average of 45+. Sehwag improves with time and reaches around 35+. Chris Gayle drops from 42 to 35

import cricpy.analytics as ca
ca.batsmanCumulativeAverageRuns("./kohli.csv","Virat Kohli")

ca.batsmanCumulativeAverageRuns("./jayasuriya.csv","Sanath jayasuriya")

ca.batsmanCumulativeAverageRuns("./gayle.csv","Chris Gayle")

ca.batsmanCumulativeAverageRuns("./sehwag.csv","Virendar Sehwag")

15 Cumulative Average strike rate of batsman in career

Sehwag has the best strike rate of almost 90. Kohli and Jayasuriya have a cumulative strike rate of 75.

import cricpy.analytics as ca
ca.batsmanCumulativeStrikeRate("./kohli.csv","Virat Kohli")

ca.batsmanCumulativeStrikeRate("./jayasuriya.csv","Sanath jayasuriya")

ca.batsmanCumulativeStrikeRate("./gayle.csv","Chris Gayle")

ca.batsmanCumulativeStrikeRate("./sehwag.csv","Virendar Sehwag")

16 Relative Batsman Cumulative Average Runs

The plot below compares the Relative cumulative average runs of the batsman . It can be seen that Virat Kohli towers above all others in the runs. He is followed by Chris Gayle and then Sehwag

import cricpy.analytics as ca
frames = ["./sehwag.csv","./gayle.csv","./jayasuriya.csv","./kohli.csv"]
names = ["Sehwag","Gayle","Jayasuriya","Kohli"]
ca.relativeBatsmanCumulativeAvgRuns(frames,names)

Relative Batsman Strike Rate

The plot below gives the relative Runs Frequency Percentages for each 10 run bucket. The plot below show Sehwag has the best strike rate, followed by Jayasuriya

import cricpy.analytics as ca
frames = ["./sehwag.csv","./gayle.csv","./jayasuriya.csv","./kohli.csv"]
names = ["Sehwag","Gayle","Jayasuriya","Kohli"]
ca.relativeBatsmanCumulativeStrikeRate(frames,names)

18. 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 3D prediction plane is fitted

import cricpy.analytics as ca
ca.battingPerf3d("./kohli.csv","Virat Kohli")

ca.battingPerf3d("./jayasuriya.csv","Sanath jayasuriya")

ca.battingPerf3d("./gayle.csv","Chris Gayle")

ca.battingPerf3d("./sehwag.csv","Virendar Sehwag")

3D plot of Runs vs Balls Faced and Minutes at Crease

From the plot below it can be seen that Sehwag has more runs by way of 4s than 1’s,2’s or 3s. Gayle and Jayasuriya have large number of 6s

import cricpy.analytics as ca
frames = ["./sehwag.csv","./kohli.csv","./gayle.csv","./jayasuriya.csv"]
names = ["Sehwag","Kohli","Gayle","Jayasuriya"]
ca.batsman4s6s(frames,names)

20. 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})
kohli= ca.batsmanRunsPredict("./kohli.csv",newDF,"Kohli")
print(kohli)
##             BF        Mins        Runs
## 0    10.000000   30.000000    6.807407
## 1    37.857143   70.714286   36.034833
## 2    65.714286  111.428571   65.262259
## 3    93.571429  152.142857   94.489686
## 4   121.428571  192.857143  123.717112
## 5   149.285714  233.571429  152.944538
## 6   177.142857  274.285714  182.171965
## 7   205.000000  315.000000  211.399391
## 8   232.857143  355.714286  240.626817
## 9   260.714286  396.428571  269.854244
## 10  288.571429  437.142857  299.081670
## 11  316.428571  477.857143  328.309096
## 12  344.285714  518.571429  357.536523
## 13  372.142857  559.285714  386.763949
## 14  400.000000  600.000000  415.991375

The fitted model is then used to predict the runs that the batsmen will score for a given Balls faced and Minutes at crease.

21 Analysis of Top Bowlers

The following 4 bowlers have had an excellent career and will be used for the analysis

  1. Muthiah Muralitharan:Wickets: 534, Average = 23.08, Economy Rate – 3.93
  2. Wasim Akram : Wickets: 502, Average = 23.52, Economy Rate – 3.89
  3. Shaun Pollock: Wickets: 393, Average = 24.50, Economy Rate – 3.67
  4. Javagal Srinath : Wickets:315, Average – 28.08, Economy Rate – 4.44

How do Muralitharan, Akram, Pollock and Srinath compare with one another with respect to wickets taken and the Economy Rate. The next set of plots compute and plot precisely these analyses.

22. 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
#akram=ca.getPlayerDataOD(43547,dir=".",file="akram.csv",type="bowling")
#murali=ca.getPlayerDataOD(49636,dir=".",file="murali.csv",type="bowling")
#pollock=ca.getPlayerDataOD(46774,dir=".",file="pollock.csv",type="bowling")
#srinath=ca.getPlayerDataOD(34105,dir=".",file="srinath.csv",type="bowling")

23. Wicket Frequency Plot

This plot below plots the frequency of wickets taken for each of the bowlers

import cricpy.analytics as ca
ca.bowlerWktsFreqPercent("./murali.csv","M Muralitharan")

ca.bowlerWktsFreqPercent("./akram.csv","Wasim Akram")

ca.bowlerWktsFreqPercent("./pollock.csv","Shaun Pollock")

ca.bowlerWktsFreqPercent("./srinath.csv","J Srinath")

24. 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. Murali’s median runs for wickets ia around 40 while Akram, Pollock and Srinath it is around 32+ runs. The spread around the median is larger for these 3 bowlers in comparison to Murali

import cricpy.analytics as ca
ca.bowlerWktsRunsPlot("./murali.csv","M Muralitharan")

ca.bowlerWktsRunsPlot("./akram.csv","Wasim Akram")

ca.bowlerWktsRunsPlot("./pollock.csv","Shaun Pollock")

ca.bowlerWktsRunsPlot("./srinath.csv","J Srinath")

25 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("./murali.csv","M Muralitharan")

ca.bowlerAvgWktsGround("./akram.csv","Wasim Akram")

ca.bowlerAvgWktsGround("./pollock.csv","Shaun Pollock")

ca.bowlerAvgWktsGround("./srinath.csv","J Srinath")

26 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("./murali.csv","M Muralitharan")

ca.bowlerAvgWktsOpposition("./akram.csv","Wasim Akram")

ca.bowlerAvgWktsOpposition("./pollock.csv","Shaun Pollock")

ca.bowlerAvgWktsOpposition("./srinath.csv","J Srinath")

27 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("./murali.csv","M Muralitharan")

ca.bowlerMovingAverage("./akram.csv","Wasim Akram")

ca.bowlerMovingAverage("./pollock.csv","Shaun Pollock")

ca.bowlerMovingAverage("./srinath.csv","J Srinath")

28 Cumulative average wickets taken

The plots below give the cumulative average wickets taken by the bowlers. Muralitharan has consistently taken wickets at an average of 1.6 wickets per game. Shaun Pollock has an average of 1.5

import cricpy.analytics as ca
ca.bowlerCumulativeAvgWickets("./murali.csv","M Muralitharan")

ca.bowlerCumulativeAvgWickets("./akram.csv","Wasim Akram")

ca.bowlerCumulativeAvgWickets("./pollock.csv","Shaun Pollock")

ca.bowlerCumulativeAvgWickets("./srinath.csv","J Srinath")

29 Cumulative average economy rate

The plots below give the cumulative average economy rate of the bowlers. Pollock is the most economical, followed by Akram and then Murali

import cricpy.analytics as ca
ca.bowlerCumulativeAvgEconRate("./murali.csv","M Muralitharan")

ca.bowlerCumulativeAvgEconRate("./akram.csv","Wasim Akram")

ca.bowlerCumulativeAvgEconRate("./pollock.csv","Shaun Pollock")

ca.bowlerCumulativeAvgEconRate("./srinath.csv","J Srinath")

30 Relative cumulative average economy rate of bowlers

The Relative cumulative economy rate shows that Pollock is the most economical of the 4 bowlers. He is followed by Akram and then Murali

import cricpy.analytics as ca
frames = ["./srinath.csv","./akram.csv","./murali.csv","pollock.csv"]
names = ["J Srinath","Wasim Akram","M Muralitharan", "S Pollock"]
ca.relativeBowlerCumulativeAvgEconRate(frames,names)

31 Relative Economy Rate against wickets taken

Pollock is most economical vs number of wickets taken. Murali has the best figures for 4 wickets taken.

import cricpy.analytics as ca
frames = ["./srinath.csv","./akram.csv","./murali.csv","pollock.csv"]
names = ["J Srinath","Wasim Akram","M Muralitharan", "S Pollock"]
ca.relativeBowlingER(frames,names)

32 Relative cumulative average wickets of bowlers in career

The plot below shows that McGrath has the best overall cumulative average wickets. While the bowlers are neck to neck around 130 innings, you can see Muralitharan is most consistent and leads the pack after 150 innings in the number of wickets taken.

import cricpy.analytics as ca
frames = ["./srinath.csv","./akram.csv","./murali.csv","pollock.csv"]
names = ["J Srinath","Wasim Akram","M Muralitharan", "S Pollock"]
ca.relativeBowlerCumulativeAvgWickets(frames,names)

33. 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

Analysis of Top 4 batsman

The analysis of the Top 4 test batsman Tendulkar, Kallis, Ponting and Sangakkara show the folliwing

  1. Kohli is a mean run machine and has been consistently piling on runs. Clearly records will lay shattered in days to come for Kohli
  2. Virendar Sehwag has the best strike rate of the 4, followed by Jayasuriya and then Kohli
  3. Shaun Pollock is the most economical of the bowlers followed by Wasim Akram
  4. Muralitharan is the most consistent wicket of the lot.

Important note: Do check out my other posts using cricpy at cricpy-posts

Also see
1. Architecting a cloud based IP Multimedia System (IMS)
2. Exploring Quantum Gate operations with QCSimulator
3. Dabbling with Wiener filter using OpenCV
4. Deep Learning from first principles in Python, R and Octave – Part 5
5. Big Data-2: Move into the big league:Graduate from R to SparkR
6. Singularity
7. Practical Machine Learning with R and Python – Part 4
8. Literacy in India – A deepR dive
9. Modeling a Car in Android

To see all posts click Index of 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$Runs > 50)
head(SparkR::collect(df))
  Runs Mins  BF 4s 6s    SR Pos Dismissal Inns     Opposition       Ground
1   59  254 172  4  0  34.3   6       lbw    1     v Pakistan   Faisalabad
2   57  193 134  6  0 42.53   6    caught    3     v Pakistan      Sialkot
3   88  324 266  5  0 33.08   6    caught    1  v New Zealand       Napier
4   68  216 136  8  0    50   6    caught    2      v England   Manchester
5  114  228 161 16  0  70.8   4    caught    2    v Australia        Perth
6  111  373 270 19  0 41.11   4    caught    2 v South Africa Johannesburg
  Start Date
1  23-Nov-89
2   9-Dec-89
3   9-Feb-90
4   9-Aug-90
5   1-Feb-92
6  26-Nov-92
11a. Unique values -R
unique(tendulkar$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)
head(df)
[1] 327  12
# A tibble: 6 x 4
  Ground       meanRuns minRuns maxRuns
                   
1 Adelaide        32.6       0.    153.
2 Ahmedabad       40.1       4.    217.
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()

tendulkar1 <- read.df("/FileStore/tables/tendulkar.csv", 
                header = "true", 
                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$Runs, "double"))
head(SparkR::distinct(tendulkar1[,"Runs"]),20)
# Remove the "* indicating not out
tendulkar1$Runs=SparkR::regexp_replace(tendulkar1$Runs, "\\*", "")
head(SparkR::distinct(tendulkar1[,"Runs"]),20)
df=SparkR::summarize(SparkR::groupBy(tendulkar1, tendulkar1$Ground), mean = mean(tendulkar1$Runs), minRuns=min(tendulkar1$Runs),maxRuns=max(tendulkar1$Runs))
head(df,20)
[1] 347  12
[1] 330  12
[1] 329  12
[1] 329  12
          Ground       mean minRuns maxRuns
1      Bangalore  54.312500       0      96
2       Adelaide  32.600000       0      61
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
10     Ahmedabad  40.125000     100       8
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()
tendulkar1 <- read.df("/FileStore/tables/tendulkar.csv", 
                header = "true", 
                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'")

head(df)

  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!

You may also like
1. Exploring Quantum Gate operations with QCSimulator
2. Deep Learning from first principles in Python, R and Octave – Part 4
3. A Bluemix recipe with MongoDB and Node.js
4. Practical Machine Learning with R and Python – Part 5
5. Introducing cricketr! : An R package to analyze performances of cricketers

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

Table of Contents
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

Pick up your copy today!!
Hope you have a great time learning as I did while implementing these algorithms!

Deep Learning from first principles in Python, R and Octave – Part 6

“Today you are You, that is truer than true. There is no one alive who is Youer than You.”
Dr. Seuss

“Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong.”
H L Mencken

Introduction

In this 6th instalment of ‘Deep Learning from first principles in Python, R and Octave-Part6’, I look at a couple of different initialization techniques used in Deep Learning, L2 regularization and the ‘dropout’ method. Specifically, I implement “He initialization” & “Xavier Initialization”. My earlier posts in this series of Deep Learning included

1. Part 1 – In the 1st part, I implemented logistic regression as a simple 2 layer Neural Network
2. Part 2 – In part 2, implemented the most basic of Neural Networks, with just 1 hidden layer, and any number of activation units in that hidden layer. The implementation was in vectorized Python, R and Octave
3. Part 3 -In part 3, I derive the equations and also implement a L-Layer Deep Learning network with either the relu, tanh or sigmoid activation function in Python, R and Octave. The output activation unit was a sigmoid function for logistic classification
4. Part 4 – This part looks at multi-class classification, and I derive the Jacobian of a Softmax function and implement a simple problem to perform multi-class classification.
5. Part 5 – In the 5th part, I extend the L-Layer Deep Learning network implemented in Part 3, to include the Softmax classification. I also use this L-layer implementation to classify MNIST handwritten digits with Python, R and Octave.

The code in Python, R and Octave are identical, and just take into account some of the minor idiosyncrasies of the individual language. In this post, I implement different initialization techniques (random, He, Xavier), L2 regularization and finally dropout. Hence my generic L-Layer Deep Learning network includes these additional enhancements for enabling/disabling initialization methods, regularization or dropout in the algorithm. It already included sigmoid & softmax output activation for binary and multi-class classification, besides allowing relu, tanh and sigmoid activation for hidden units.

A video presentation of regularization and initialization techniques can be also be viewed in Neural Networks 6

This R Markdown file and the code for Python, R and Octave can be cloned/downloaded from Github at DeepLearning-Part6

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:Second Edition- Machine Learning in stereo” available in Amazon in paperback($10.99) and Kindle($7.99/Rs449) 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. Initialization techniques

The usual initialization technique is to generate Gaussian or uniform random numbers and multiply it by a small value like 0.01. Two techniques which are used to speed up convergence is the He initialization or Xavier. These initialization techniques enable gradient descent to converge faster.

1.1 a Default initialization – Python

This technique just initializes the weights to small random values based on Gaussian or uniform distribution

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("DLfunctions61.py").read())
#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]

# Train a deep learning network with random initialization
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid",learningRate = 0.6, num_iterations = 9000, initType="default", print_cost = True,figure="fig1.png")

# Clear the plot
plt.clf()
plt.close()

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig2.png")

1.1 b He initialization – Python

‘He’ initialization attributed to He et al, multiplies the random weights by
\sqrt{\frac{2}{dimension\ of\ previous\ layer}}

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("DLfunctions61.py").read())

#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]

# Train a deep learning network with He  initialization
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid", learningRate =0.6,    num_iterations = 10000,initType="He",print_cost = True,                           figure="fig3.png")

plt.clf()
plt.close()
# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig4.png")


1.1 c Xavier initialization – Python

Xavier  initialization multiply the random weights by
\sqrt{\frac{1}{dimension\ of\ previous\ layer}}

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("DLfunctions61.py").read())

#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]
 
# Train a L layer Deep Learning network
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid",
                            learningRate = 0.6,num_iterations = 10000, initType="Xavier",print_cost = True,
                            figure="fig5.png")

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig6.png")


1.2a Default initialization – R

source("DLfunctions61.R")
#Load the data
z <- as.matrix(read.csv("circles.csv",header=FALSE)) 
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
#Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="sigmoid",
                            learningRate = 0.5,
                            numIterations = 8000, 
                            initType="default",
                            print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,8000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
 ggtitle("Costs vs iterations") + xlab("No of iterations") + ylab("Cost")

# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",lr=0.5)

1.2b He initialization – R

The code for ‘He’ initilaization in R is included below

# He Initialization model for L layers
# Input : List of units in each layer
# Returns: Initial weights and biases matrices for all layers
# He initilization multiplies the random numbers with sqrt(2/layerDimensions[previouslayer])
HeInitializeDeepModel <- function(layerDimensions){
    set.seed(2)
    
    # Initialize empty list
    layerParams <- list()
    
    # Note the Weight matrix at layer 'l' is a matrix of size (l,l-1)
    # The Bias is a vectors of size (l,1)
    
    # Loop through the layer dimension from 1.. L
    # Indices in R start from 1
    for(l in 2:length(layersDimensions)){
        # Initialize a matrix of small random numbers of size l x l-1
        # Create random numbers of size  l x l-1
        w=rnorm(layersDimensions[l]*layersDimensions[l-1])
        
        # Create a weight matrix of size l x l-1 with this initial weights and
        # Add to list W1,W2... WL
        # He initialization - Divide by sqrt(2/layerDimensions[previous layer])
        layerParams[[paste('W',l-1,sep="")]] = matrix(w,nrow=layersDimensions[l],
                                                      ncol=layersDimensions[l-1])*sqrt(2/layersDimensions[l-1])
        layerParams[[paste('b',l-1,sep="")]] = matrix(rep(0,layersDimensions[l]),
                                                      nrow=layersDimensions[l],ncol=1)
    }
    return(layerParams)
}

The code in R below uses He initialization to learn the data

source("DLfunctions61.R")
# Load the data
z <- as.matrix(read.csv("circles.csv",header=FALSE)) 
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
# Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="sigmoid",
                            learningRate = 0.5,
                            numIterations = 9000, 
                            initType="He",
                            print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,9000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
    ggtitle("Costs vs iterations") + xlab("No of iterations") + ylab("Cost")

# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5,lr=0.5)

1.2c Xavier initialization – R

## Xav initialization 
# Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="sigmoid",
                            learningRate = 0.5,
                            numIterations = 9000, 
                            initType="Xav",
                            print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,9000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
    ggtitle("Costs vs iterations") + xlab("No of iterations") + ylab("Cost")

# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5)

1.3a Default initialization – Octave

source("DL61functions.m")
# Read the data
data=csvread("circles.csv");

X=data(:,1:2);
Y=data(:,3);
# Set the layer dimensions
layersDimensions = [2 11  1]; #tanh=-0.5(ok), #relu=0.1 best!

# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
                               hiddenActivationFunc='relu', 
                               outputActivationFunc="sigmoid",
                               learningRate = 0.5,
                               lambd=0, 
                               keep_prob=1,
                               numIterations = 10000,
                               initType="default");
# Plot cost vs iterations
plotCostVsIterations(10000,costs)  
#Plot decision boundary                            
plotDecisionBoundary(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu")

 

1.3b He initialization – Octave

source("DL61functions.m")
#Load data
data=csvread("circles.csv");
X=data(:,1:2);
Y=data(:,3);
# Set the layer dimensions
layersDimensions = [2 11  1]; #tanh=-0.5(ok), #relu=0.1 best!

# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
                               hiddenActivationFunc='relu', 
                               outputActivationFunc="sigmoid",
                               learningRate = 0.5,
                               lambd=0, 
                               keep_prob=1,
                               numIterations = 8000,
                               initType="He");
plotCostVsIterations(8000,costs)   
#Plot decision boundary                              
plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

1.3c Xavier initialization – Octave

The code snippet for Xavier initialization in Octave is shown below

source("DL61functions.m")
# Xavier Initialization for L layers
# Input : List of units in each layer
# Returns: Initial weights and biases matrices for all layers
function [W b] = XavInitializeDeepModel(layerDimensions)
    rand ("seed", 3);
    # note the Weight matrix at layer 'l' is a matrix of size (l,l-1)
    # The Bias is a vectors of size (l,1)
    
    # Loop through the layer dimension from 1.. L
    # Create cell arrays for Weights and biases

    for l =2:size(layerDimensions)(2)
         W{l-1} = rand(layerDimensions(l),layerDimensions(l-1))* sqrt(1/layerDimensions(l-1)); #  Multiply by .01 
         b{l-1} = zeros(layerDimensions(l),1);       
   
    endfor
end

The Octave code below uses Xavier initialization

source("DL61functions.m")
#Load data
data=csvread("circles.csv");
X=data(:,1:2);
Y=data(:,3);
#Set layer dimensions
layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best!

# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
lambd=0,
keep_prob=1,
numIterations = 8000,
initType="Xav");

plotCostVsIterations(8000,costs)
plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")



 

2.1a Regularization : Circles data – Python

The cross entropy cost for Logistic classification is given as J = \frac{1}{m}\sum_{i=1}^{m}y^{i}log((a^{L})^{(i)}) - (1-y^{i})log((a^{L})^{(i)}) The regularized L2 cost is given by J = \frac{1}{m}\sum_{i=1}^{m}y^{i}log((a^{L})^{(i)}) - (1-y^{i})log((a^{L})^{(i)}) + \frac{\lambda}{2m}\sum \sum \sum W_{kj}^{l}

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("DLfunctions61.py").read())

#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]

# Train a deep learning network
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu',  
                               outputActivationFunc="sigmoid",learningRate = 0.6, lambd=0.1, num_iterations = 9000, 
                               initType="default", print_cost = True,figure="fig7.png")

# Clear the plot
plt.clf()
plt.close()

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig8.png")


plt.clf()
plt.close()
#Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T,keep_prob=0.9), train_X, train_Y,str(2.2),"fig8.png",)

2.1 b Regularization: Spiral data  – 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("DLfunctions61.py").read())
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))
  r = np.linspace(0.0,1,N) # radius
  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)
plt.clf()
plt.close() 
#Set layer dimensions 
layersDimensions = [2,100,3]
y1=y.reshape(-1,1).T
# Train a deep learning network
parameters = L_Layer_DeepModel(X.T, y1, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",
                           learningRate = 1,lambd=1e-3, num_iterations = 5000, print_cost = True,figure="fig9.png")

plt.clf()
plt.close()  
W1=parameters['W1']
b1=parameters['b1']
W2=parameters['W2']
b2=parameters['b2']
plot_decision_boundary1(X, y1,W1,b1,W2,b2,figure2="fig10.png")

 

2.2a Regularization: Circles data  – R

source("DLfunctions61.R")
#Load data
df=read.csv("circles.csv",header=FALSE)
z <- as.matrix(read.csv("circles.csv",header=FALSE)) 
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
#Set layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="sigmoid",
                            learningRate = 0.5,
                            lambd=0.1,
                            numIterations = 9000, 
                            initType="default",
                            print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,9000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
    ggtitle("Costs vs iterations") + xlab("No of iterations") + ylab("Cost")

# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5)

2.2b Regularization:Spiral data – R

# Read the spiral dataset
#Load the data
source("DLfunctions61.R")
Z <- as.matrix(read.csv("spiral.csv",header=FALSE)) 

# Setup the data
X <- Z[,1:2]
y <- Z[,3]
X <- t(X)
Y <- t(y)
layersDimensions = c(2, 100, 3)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.5,
lambd=0.01,
numIterations = 9000,
print_cost = True)
print_cost = True)
parameters<-retvals$parameters
plotDecisionBoundary1(Z,parameters)


2.3a Regularization: Circles data – Octave

source("DL61functions.m")
#Load data
data=csvread("circles.csv");
X=data(:,1:2);
Y=data(:,3);
layersDimensions = [2 11  1]; #tanh=-0.5(ok), #relu=0.1 best!

# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
                               hiddenActivationFunc='relu', 
                               outputActivationFunc="sigmoid",
                               learningRate = 0.5,
                               lambd=0.2,
                               keep_prob=1,
                               numIterations = 8000,
                               initType="default");

plotCostVsIterations(8000,costs)  
#Plot decision boundary                              
plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

2.3b Regularization:Spiral data  2 – Octave

source("DL61functions.m")
data=csvread("spiral.csv");

# Setup the data
X=data(:,1:2);
Y=data(:,3);
layersDimensions = [2 100 3]
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
                               hiddenActivationFunc='relu', 
                               outputActivationFunc="softmax",
                               learningRate = 0.6,
                               lambd=0.2,
                               keep_prob=1,
                               numIterations = 10000);
                              
plotCostVsIterations(10000,costs)
#Plot decision boundary
plotDecisionBoundary1(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")  

3.1 a Dropout: Circles data – Python

The ‘dropout’ regularization technique was used with great effectiveness, to prevent overfitting  by Alex Krizhevsky, Ilya Sutskever and Prof Geoffrey E. Hinton in the Imagenet classification with Deep Convolutional Neural Networks

The technique of dropout works by dropping a random set of activation units in each hidden layer, based on a ‘keep_prob’ criteria in the forward propagation cycle. Here is the code for Octave. A ‘dropoutMat’ is created for each layer which specifies which units to drop Note: The same ‘dropoutMat has to be used which computing the gradients in the backward propagation cycle. Hence the dropout matrices are stored in a cell array.

 for l =1:L-1  
    ...      
    D=rand(size(A)(1),size(A)(2));
    D = (D < keep_prob) ;
    # Zero out some hidden units
    A= A .* D;    
    # Divide by keep_prob to keep the expected value of A the same                                  
    A = A ./ keep_prob; 
    # Store D in a dropoutMat cell array
    dropoutMat{l}=D;
    ...
 endfor

In the backward propagation cycle we have

    for l =(L-1):-1:1
          ...
          D = dropoutMat{l};  
          # Zero out the dAl based on same dropout matrix       
          dAl= dAl .* D;   
          # Divide by keep_prob to maintain the expected value                                       
          dAl = dAl ./ keep_prob;
          ...
    endfor 
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("DLfunctions61.py").read())
#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]

# Train a deep learning network
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu',  
                               outputActivationFunc="sigmoid",learningRate = 0.6, keep_prob=0.7, num_iterations = 9000, 
                               initType="default", print_cost = True,figure="fig11.png")

# Clear the plot
plt.clf()
plt.close()

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T,keep_prob=0.7), train_X, train_Y,str(0.6),figure1="fig12.png")

3.1b Dropout: Spiral data – 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("DLfunctions61.py").read())
# Create an input data set - Taken from CS231n Convolutional Neural networks,
# http://cs231n.github.io/neural-networks-case-study/
               

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))
  r = np.linspace(0.0,1,N) # radius
  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)
plt.clf()
plt.close()  
layersDimensions = [2,100,3]
y1=y.reshape(-1,1).T
# Train a deep learning network
parameters = L_Layer_DeepModel(X.T, y1, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",
                           learningRate = 1,keep_prob=0.9, num_iterations = 5000, print_cost = True,figure="fig13.png")

plt.clf()
plt.close()  
W1=parameters['W1']
b1=parameters['b1']
W2=parameters['W2']
b2=parameters['b2']
#Plot decision boundary
plot_decision_boundary1(X, y1,W1,b1,W2,b2,figure2="fig14.png")

3.2a Dropout: Circles data – R

source("DLfunctions61.R")
#Load data
df=read.csv("circles.csv",header=FALSE)
z <- as.matrix(read.csv("circles.csv",header=FALSE)) 

x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="sigmoid",
                            learningRate = 0.5,
                            keep_prob=0.8,
                            numIterations = 9000, 
                            initType="default",
                            print_cost = True)
# Plot the decision boundary
plotDecisionBoundary(z,retvals,keep_prob=0.6, hiddenActivationFunc="relu",0.5)

3.2b Dropout: Spiral data – R

# Read the spiral dataset
source("DLfunctions61.R")
# Load data
Z <- as.matrix(read.csv("spiral.csv",header=FALSE)) 

# Setup the data
X <- Z[,1:2]
y <- Z[,3]
X <- t(X)
Y <- t(y)

# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="softmax",
                            learningRate = 0.1,
                            keep_prob=0.90,
                            numIterations = 9000, 
                            print_cost = True)

parameters<-retvals$parameters
#Plot decision boundary
plotDecisionBoundary1(Z,parameters)

3.3a Dropout: Circles data – Octave

data=csvread("circles.csv");

X=data(:,1:2);
Y=data(:,3);
layersDimensions = [2 11  1]; #tanh=-0.5(ok), #relu=0.1 best!

# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
                               hiddenActivationFunc='relu', 
                               outputActivationFunc="sigmoid",
                               learningRate = 0.5,
                               lambd=0,
                               keep_prob=0.8,
                               numIterations = 10000,
                               initType="default");
plotCostVsIterations(10000,costs) 
#Plot decision boundary
plotDecisionBoundary1(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu") 

3.3b Dropout  Spiral data – Octave

source("DL61functions.m")
data=csvread("spiral.csv");

# Setup the data
X=data(:,1:2);
Y=data(:,3);

layersDimensions = [numFeats numHidden  numOutput];  
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
                               hiddenActivationFunc='relu', 
                               outputActivationFunc="softmax",
                               learningRate = 0.1,
                               lambd=0,
                               keep_prob=0.8,
                               numIterations = 10000); 

plotCostVsIterations(10000,costs)    
#Plot decision boundary                            
plotDecisionBoundary1(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu")  

Note: The Python, R and Octave code can be cloned/downloaded from Github at DeepLearning-Part6
Conclusion
This post further enhances my earlier L-Layer generic implementation of a Deep Learning network to include options for initialization techniques, L2 regularization or dropout regularization

References
1. Deep Learning Specialization
2. Neural Networks for Machine Learning

Also see
1. Architecting a cloud based IP Multimedia System (IMS)
2. Using Linear Programming (LP) for optimizing bowling change or batting lineup in T20 cricket
3. My book ‘Practical Machine Learning with R and Python’ on Amazon
4. Simulating a Web Joint in Android
5. Inswinger: yorkr swings into International T20s
6. Introducing QCSimulator: A 5-qubit quantum computing simulator in R
7. Computer Vision: Ramblings on derivatives, histograms and contours
8. Bend it like Bluemix, MongoDB using Auto-scale – Part 1!
9. The 3rd paperback & kindle editions of my books on Cricket, now on Amazon

To see all posts click Index of posts

Deep Learning from first principles in Python, R and Octave – Part 4

In this 4th post of my series on Deep Learning from first principles in Python, R and Octave – Part 4, I explore the details of creating a multi-class classifier using the Softmax activation unit in a neural network. The earlier posts in this series were

1. Deep Learning from first principles in Python, R and Octave – Part 1. In this post I implemented logistic regression as a simple Neural Network in vectorized Python, R and Octave
2. Deep Learning from first principles in Python, R and Octave – Part 2. This 2nd part implemented the most elementary neural network with 1 hidden layer and any number of activation units in the hidden layer with sigmoid activation at the output layer
3. Deep Learning from first principles in Python, R and Octave – Part 3. The 3rd implemented a multi-layer Deep Learning network with an arbitrary number if hidden layers and activation units per hidden layer. The output layer was for binary classification which was based on the sigmoid unit. This multi-layer deep network was implemented in vectorized Python, R and Octave.

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).

This 4th part takes a swing at multi-class classification and uses the Softmax as the activation unit in the output layer. Inclusion of the Softmax activation unit in the activation layer requires us to compute the derivative of Softmax, or rather the “Jacobian” of the Softmax function, besides also computing the log loss for this Softmax activation during back propagation. Since the derivation of the Jacobian of a Softmax and the computation of the Cross Entropy/log loss is very involved, I have implemented a basic neural network with just 1 hidden layer with the Softmax activation at the output layer. I also perform multi-class classification based on the ‘spiral’ data set from CS231n Convolutional Neural Networks Stanford course, to test the performance and correctness of the implementations in Python, R and Octave. You can clone download the code for the Python, R and Octave implementations from Github at Deep Learning – Part 4

Note: A detailed discussion of the derivation below can also be seen in my video presentation Neural Networks 5

The Softmax function takes an N dimensional vector as input and generates a N dimensional vector as output.
The Softmax function is given by
S_{j}= \frac{e_{j}}{\sum_{i}^{N}e_{k}}
There is a probabilistic interpretation of the Softmax, since the sum of the Softmax values of a set of vectors will always add up to 1, given that each Softmax value is divided by the total of all values.

As mentioned earlier, the Softmax takes a vector input and returns a vector of outputs.  For e.g. the Softmax of a vector a=[1, 3, 6]  is another vector S=[0.0063,0.0471,0.9464]. Notice that vector output is proportional to the input vector.  Also, taking the derivative of a vector by another vector, is known as the Jacobian. By the way, The Matrix Calculus You Need For Deep Learning by Terence Parr and Jeremy Howard, is very good paper that distills all the main mathematical concepts for Deep Learning in one place.

Let us take a simple 2 layered neural network with just 2 activation units in the hidden layer is shown below

Z_{1}^{1} =W_{11}^{1}x_{1} + W_{21}^{1}x_{2} + b_{1}^{1}
Z_{2}^{1} =W_{12}^{1}x_{1} + W_{22}^{1}x_{2} + b_{2}^{1}
and
A_{1}^{1} = g'(Z_{1}^{1})
A_{2}^{1} = g'(Z_{2}^{1})
where g'() is the activation unit in the hidden layer which can be a relu, sigmoid or a
tanh function

Note: The superscript denotes the layer. The above denotes the equation for layer 1
of the neural network. For layer 2 with the Softmax activation, the equations are
Z_{1}^{2} =W_{11}^{2}x_{1} + W_{21}^{2}x_{2} + b_{1}^{2}
Z_{2}^{2} =W_{12}^{2}x_{1} + W_{22}^{2}x_{2} + b_{2}^{2}
and
A_{1}^{2} = S(Z_{1}^{2})
A_{2}^{2} = S(Z_{2}^{2})
where S() is the Softmax activation function
S=\begin{pmatrix} S(Z_{1}^{2})\\ S(Z_{2}^{2}) \end{pmatrix}
S=\begin{pmatrix} \frac{e^{Z1}}{e^{Z1}+e^{Z2}}\\ \frac{e^{Z2}}{e^{Z1}+e^{Z2}} \end{pmatrix}

The Jacobian of the softmax ‘S’ is given by
\begin{pmatrix} \frac {\partial S_{1}}{\partial Z_{1}} & \frac {\partial S_{1}}{\partial Z_{2}}\\ \frac {\partial S_{2}}{\partial Z_{1}} & \frac {\partial S_{2}}{\partial Z_{2}} \end{pmatrix}
\begin{pmatrix} \frac{\partial}{\partial Z_{1}} \frac {e^{Z1}}{e^{Z1}+ e^{Z2}} & \frac{\partial}{\partial Z_{2}} \frac {e^{Z1}}{e^{Z1}+ e^{Z2}}\\ \frac{\partial}{\partial Z_{1}} \frac {e^{Z2}}{e^{Z1}+ e^{Z2}} & \frac{\partial}{\partial Z_{2}} \frac {e^{Z2}}{e^{Z1}+ e^{Z2}} \end{pmatrix}     – (A)

Now the ‘division-rule’  of derivatives is as follows. If u and v are functions of x, then
\frac{d}{dx} \frac {u}{v} =\frac {vdu -udv}{v^{2}}
Using this to compute each element of the above Jacobian matrix, we see that
when i=j we have
\frac {\partial}{\partial Z1}\frac{e^{Z1}}{e^{Z1}+e^{Z2}} = \frac {\sum e^{Z1} - e^{Z1^{2}}}{\sum ^{2}}
and when i \neq j
\frac {\partial}{\partial Z1}\frac{e^{Z2}}{e^{Z1}+e^{Z2}} = \frac {0 - e^{z1}e^{Z2}}{\sum ^{2}}
This is of the general form
\frac {\partial S_{j}}{\partial z_{i}} = S_{i}( 1-S_{j})  when i=j
and
\frac {\partial S_{j}}{\partial z_{i}} = -S_{i}S_{j}  when i \neq j
Note: Since the Softmax essentially gives the probability the following
notation is also used
\frac {\partial p_{j}}{\partial z_{i}} = p_{i}( 1-p_{j}) when i=j
and
\frac {\partial p_{j}}{\partial z_{i}} = -p_{i}p_{j} when i \neq j
If you throw the “Kronecker delta” into the equation, then the above equations can be expressed even more concisely as
\frac {\partial p_{j}}{\partial z_{i}} = p_{i} (\delta_{ij} - p_{j})
where \delta_{ij} = 1 when i=j and 0 when i \neq j

This reduces the Jacobian of the simple 2 output softmax vectors  equation (A) as
\begin{pmatrix} p_{1}(1-p_{1}) & -p_{1}p_{2} \\ -p_{2}p_{1} & p_{2}(1-p_{2}) \end{pmatrix}
The loss of Softmax is given by
L = -\sum y_{i} log(p_{i})
For the 2 valued Softmax output this is
\frac {dL}{dp1} = -\frac {y_{1}}{p_{1}}
\frac {dL}{dp2} = -\frac {y_{2}}{p_{2}}
Using the chain rule we can write
\frac {\partial L}{\partial w_{pq}} = \sum _{i}\frac {\partial L}{\partial p_{i}} \frac {\partial p_{i}}{\partial w_{pq}} (1)
and
\frac {\partial p_{i}}{\partial w_{pq}} = \sum _{k}\frac {\partial p_{i}}{\partial z_{k}} \frac {\partial z_{k}}{\partial w_{pq}} (2)
In expanded form this is
\frac {\partial L}{\partial w_{pq}} = \sum _{i}\frac {\partial L}{\partial p_{i}} \sum _{k}\frac {\partial p_{i}}{\partial z_{k}} \frac {\partial z_{k}}{\partial w_{pq}}
Also
\frac {\partial L}{\partial Z_{i}} =\sum _{i} \frac {\partial L}{\partial p} \frac {\partial p}{\partial Z_{i}}
Therefore
\frac {\partial L}{\partial Z_{1}} =\frac {\partial L}{\partial p_{1}} \frac {\partial p_{1}}{\partial Z_{1}} +\frac {\partial L}{\partial p_{2}} \frac {\partial p_{2}}{\partial Z_{1}}
\frac {\partial L}{\partial z_{1}}=-\frac {y1}{p1} p1(1-p1) - \frac {y2}{p2}*(-p_{2}p_{1})
Since
\frac {\partial p_{j}}{\partial z_{i}} = p_{i}( 1-p_{j}) when i=j
and
\frac {\partial p_{j}}{\partial z_{i}} = -p_{i}p_{j} when i \neq j
which simplifies to
\frac {\partial L}{\partial Z_{1}} = -y_{1} + y_{1}p_{1} + y_{2}p_{1} =
p_{1}\sum (y_{1} + y_2) - y_{1}
\frac {\partial L}{\partial Z_{1}}= p_{1} - y_{1}
Since
\sum_{i} y_{i} =1
Similarly
\frac {\partial L}{\partial Z_{2}} =\frac {\partial L}{\partial p_{1}} \frac {\partial p_{1}}{\partial Z_{2}} +\frac {\partial L}{\partial p_{2}} \frac {\partial p_{2}}{\partial Z_{2}}
\frac {\partial L}{\partial z_{2}}=-\frac {y1}{p1}*(p_{1}p_{2}) - \frac {y2}{p2}*p_{2}(1-p_{2})
y_{1}p_{2} + y_{2}p_{2} - y_{2}
\frac {\partial L}{\partial Z_{2}} =p_{2}\sum (y_{1} + y_2) - y_{2}\\ = p_{2} - y_{2}
In general this is of the form
\frac {\partial L}{\partial z_{i}} = p_{i} -y_{i}
For e.g if the probabilities computed were p=[0.1, 0.7, 0.2] then this implies that the class with probability 0.7 is the likely class. This would imply that the ‘One hot encoding’ for  yi  would be yi=[0,1,0] therefore the gradient pi-yi = [0.1,-0.3,0.2]

<strong>Note: Further, we could extend this derivation for a Softmax activation output that outputs 3 classes
S=\begin{pmatrix} \frac{e^{z1}}{e^{z1}+e^{z2}+e^{z3}}\\ \frac{e^{z2}}{e^{z1}+e^{z2}+e^{z3}} \\ \frac{e^{z3}}{e^{z1}+e^{z2}+e^{z3}} \end{pmatrix}

We could derive
\frac {\partial L}{\partial z1}= \frac {\partial L}{\partial p_{1}} \frac {\partial p_{1}}{\partial z_{1}} +\frac {\partial L}{\partial p_{2}} \frac {\partial p_{2}}{\partial z_{1}} +\frac {\partial L}{\partial p_{3}} \frac {\partial p_{3}}{\partial z_{1}} which similarly reduces to
\frac {\partial L}{\partial z_{1}}=-\frac {y1}{p1} p1(1-p1) - \frac {y2}{p2}*(-p_{2}p_{1}) - \frac {y3}{p3}*(-p_{3}p_{1})
-y_{1}+ y_{1}p_{1} + y_{2}p_{1} + y_{3}p1 = p_{1}\sum (y_{1} + y_2 + y_3) - y_{1} = p_{1} - y_{1}
Interestingly, despite the lengthy derivations the final result is simple and intuitive!

As seen in my post ‘Deep Learning from first principles with Python, R and Octave – Part 3 the key equations for forward and backward propagation are

Forward propagation equations layer 1
Z_{1} = W_{1}X +b_{1}     and  A_{1} = g(Z_{1})
Forward propagation equations layer 1
Z_{2} = W_{2}A_{1} +b_{2}  and  A_{2} = S(Z_{2})

Using the result (A) in the back propagation equations below we have
Backward propagation equations layer 2
\partial L/\partial W_{2} =\partial L/\partial Z_{2}*A_{1}=(p_{2}-y_{2})*A_{1}
\partial L/\partial b_{2} =\partial L/\partial Z_{2}=p_{2}-y_{2}
\partial L/\partial A_{1} = \partial L/\partial Z_{2} * W_{2}=(p_{2}-y_{2})*W_{2}
Backward propagation equations layer 1
\partial L/\partial W_{1} =\partial L/\partial Z_{1} *A_{0}=(p_{1}-y_{1})*A_{0}
\partial L/\partial b_{1} =\partial L/\partial Z_{1}=(p_{1}-y_{1})

2.0 Spiral data set

As I mentioned earlier, I will be using the ‘spiral’ data from CS231n Convolutional Neural Networks to ensure that my vectorized implementations in Python, R and Octave are correct. Here is the ‘spiral’ data set.

import numpy as np
import matplotlib.pyplot as plt
import os
os.chdir("C:/junk/dl-4/dl-4")
exec(open("././DLfunctions41.py").read())

# Create an input data set - Taken from CS231n Convolutional Neural networks
# http://cs231n.github.io/neural-networks-case-study/
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))
  r = np.linspace(0.0,1,N) # radius
  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)
plt.savefig("fig1.png", bbox_inches='tight')


The implementations of the vectorized Python, R and Octave code are shown diagrammatically below

2.1 Multi-class classification with Softmax – Python code

A simple 2 layer Neural network with a single hidden layer , with 100 Relu activation units in the hidden layer and the Softmax activation unit in the output layer is used for multi-class classification. This Deep Learning Network, plots the non-linear boundary of the 3 classes as shown below

import numpy as np
import matplotlib.pyplot as plt
import os
os.chdir("C:/junk/dl-4/dl-4")
exec(open("././DLfunctions41.py").read())

# Read the input data
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))
  r = np.linspace(0.0,1,N) # radius
  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
  
# Set the number of features, hidden units in hidden layer and number of classess
numHidden=100 # No of hidden units in hidden layer
numFeats= 2 # dimensionality
numOutput = 3 # number of classes

# Initialize the model
parameters=initializeModel(numFeats,numHidden,numOutput)
W1= parameters['W1']
b1= parameters['b1']
W2= parameters['W2']
b2= parameters['b2']

# Set the learning rate
learningRate=0.6 

# Initialize losses
losses=[]
# Perform Gradient descent
for i in range(10000):
    # Forward propagation through hidden layer with Relu units
    A1,cache1= layerActivationForward(X.T,W1,b1,'relu')
    
    # Forward propagation through output layer with Softmax
    A2,cache2 = layerActivationForward(A1,W2,b2,'softmax')
    
    # No of training examples
    numTraining = X.shape[0]
    # Compute log probs. Take the log prob of correct class based on output y
    correct_logprobs = -np.log(A2[range(numTraining),y])
    # Conpute loss
    loss = np.sum(correct_logprobs)/numTraining
    
    # Print the loss
    if i % 1000 == 0:
        print("iteration %d: loss %f" % (i, loss))
        losses.append(loss)

    dA=0

    # Backward  propagation through output layer with Softmax
    dA1,dW2,db2 = layerActivationBackward(dA, cache2, y, activationFunc='softmax')
    # Backward  propagation through hidden layer with Relu unit
    dA0,dW1,db1 = layerActivationBackward(dA1.T, cache1, y, activationFunc='relu')
    
    #Update paramaters with the learning rate
    W1 += -learningRate * dW1
    b1 += -learningRate * db1
    W2 += -learningRate * dW2.T
    b2 += -learningRate * db2.T

#Plot losses vs iterations  
i=np.arange(0,10000,1000)
plt.plot(i,losses)

plt.xlabel('Iterations')
plt.ylabel('Loss')
plt.title('Losses vs Iterations')
plt.savefig("fig2.png", bbox="tight")

#Compute the multi-class Confusion Matrix
from sklearn.metrics import confusion_matrix
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score

# We need to determine the predicted values from the learnt data
# Forward propagation through hidden layer with Relu units
A1,cache1= layerActivationForward(X.T,W1,b1,'relu')
    
# Forward propagation through output layer with Softmax
A2,cache2 = layerActivationForward(A1,W2,b2,'softmax')
#Compute predicted values from weights and biases
yhat=np.argmax(A2, axis=1)

a=confusion_matrix(y.T,yhat.T)
print("Multi-class Confusion Matrix")
print(a)
## iteration 0: loss 1.098507
## iteration 1000: loss 0.214611
## iteration 2000: loss 0.043622
## iteration 3000: loss 0.032525
## iteration 4000: loss 0.025108
## iteration 5000: loss 0.021365
## iteration 6000: loss 0.019046
## iteration 7000: loss 0.017475
## iteration 8000: loss 0.016359
## iteration 9000: loss 0.015703
## Multi-class Confusion Matrix
## [[ 99   1   0]
##  [  0 100   0]
##  [  0   1  99]]

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2.2 Multi-class classification with Softmax – R code

The spiral data set created with Python was saved, and is used as the input with R code. The R Neural Network seems to perform much,much slower than both Python and Octave. Not sure why! Incidentally the computation of loss and the softmax derivative are identical for both R and Octave. yet R is much slower. To compute the softmax derivative I create matrices for the One Hot Encoded yi and then stack them before subtracting pi-yi. I am sure there is a more elegant and more efficient way to do this, much like Python. Any suggestions?

library(ggplot2)
library(dplyr)
library(RColorBrewer)
source("DLfunctions41.R")
# Read the spiral dataset
Z <- as.matrix(read.csv("spiral.csv",header=FALSE)) 
Z1=data.frame(Z)
#Plot the dataset
ggplot(Z1,aes(x=V1,y=V2,col=V3)) +geom_point() + 
  scale_colour_gradientn(colours = brewer.pal(10, "Spectral"))

# Setup the data
X <- Z[,1:2]
y <- Z[,3]
X1 <- t(X)
Y1 <- t(y)

# Initialize number of features, number of hidden units in hidden layer and
# number of classes
numFeats<-2 # No features
numHidden<-100 # No of hidden units
numOutput<-3 # No of classes

# Initialize model
parameters <-initializeModel(numFeats, numHidden,numOutput)

W1 <-parameters[['W1']]
b1 <-parameters[['b1']]
W2 <-parameters[['W2']]
b2 <-parameters[['b2']]

# Set the learning rate
learningRate <- 0.5
# Initialize losses
losses <- NULL
# Perform gradient descent
for(i in 0:9000){

# Forward propagation through hidden layer with Relu units
retvals <- layerActivationForward(X1,W1,b1,'relu')
A1 <- retvals[['A']]
cache1 <- retvals[['cache']]
forward_cache1 <- cache1[['forward_cache1']]
activation_cache <- cache1[['activation_cache']]

# Forward propagation through output layer with Softmax units
retvals = layerActivationForward(A1,W2,b2,'softmax')
A2 <- retvals[['A']]
cache2 <- retvals[['cache']]
forward_cache2 <- cache2[['forward_cache1']]
activation_cache2 <- cache2[['activation_cache']]

# No oftraining examples
numTraining <- dim(X)[1]
dA <-0

# Select the elements where the y values are 0, 1 or 2 and make a vector
a=c(A2[y==0,1],A2[y==1,2],A2[y==2,3])
# Take log
correct_probs = -log(a)
# Compute loss
loss= sum(correct_probs)/numTraining

if(i %% 1000 == 0){
sprintf("iteration %d: loss %f",i, loss)
print(loss)
}
# Backward propagation through output layer with Softmax units
retvals = layerActivationBackward(dA, cache2, y, activationFunc='softmax')
dA1 = retvals[['dA_prev']]
dW2= retvals[['dW']]
db2= retvals[['db']]
# Backward propagation through hidden layer with Relu units
retvals = layerActivationBackward(t(dA1), cache1, y, activationFunc='relu')
dA0 = retvals[['dA_prev']]
dW1= retvals[['dW']]
db1= retvals[['db']]

# Update parameters
W1 <- W1 - learningRate * dW1
b1 <- b1 - learningRate * db1
W2 <- W2 - learningRate * t(dW2)
b2 <- b2 - learningRate * t(db2)
}
## [1] 1.212487
## [1] 0.5740867
## [1] 0.4048824
## [1] 0.3561941
## [1] 0.2509576
## [1] 0.7351063
## [1] 0.2066114
## [1] 0.2065875
## [1] 0.2151943
## [1] 0.1318807

 

#Create iterations
iterations <- seq(0,10)
#df=data.frame(iterations,losses)
ggplot(df,aes(x=iterations,y=losses)) + geom_point() + geom_line(color="blue") +
    ggtitle("Losses vs iterations") + xlab("Iterations") + ylab("Loss")

plotDecisionBoundary(Z,W1,b1,W2,b2)



Multi-class Confusion Matrix

library(caret)
library(e1071)

# Forward propagation through hidden layer with Relu units
retvals <- layerActivationForward(X1,W1,b1,'relu')
A1 <- retvals[['A']]

# Forward propagation through output layer with Softmax units
retvals = layerActivationForward(A1,W2,b2,'softmax')
A2 <- retvals[['A']]
yhat <- apply(A2, 1,which.max) -1
Confusion Matrix and Statistics
          Reference
Prediction  0  1  2
         0 97  0  1
         1  2 96  4
         2  1  4 95

Overall Statistics                                        
               Accuracy : 0.96            
                 95% CI : (0.9312, 0.9792)
    No Information Rate : 0.3333          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.94            
 Mcnemar's Test P-Value : 0.5724          
Statistics by Class:

                     Class: 0 Class: 1 Class: 2
Sensitivity            0.9700   0.9600   0.9500
Specificity            0.9950   0.9700   0.9750
Pos Pred Value         0.9898   0.9412   0.9500
Neg Pred Value         0.9851   0.9798   0.9750
Prevalence             0.3333   0.3333   0.3333
Detection Rate         0.3233   0.3200   0.3167
Detection Prevalence   0.3267   0.3400   0.3333
Balanced Accuracy      0.9825   0.9650   0.9625

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2.3 Multi-class classification with Softmax – Octave code

A 2 layer Neural network with the Softmax activation unit in the output layer is constructed in Octave. The same spiral data set is used for Octave also

source("DL41functions.m")
# Read the spiral data
data=csvread("spiral.csv");
# Setup the data
X=data(:,1:2);
Y=data(:,3);
# Set the number of features, number of hidden units in hidden layer and number of classes
numFeats=2; #No features
numHidden=100; # No of hidden units
numOutput=3; # No of classes
# Initialize model
[W1 b1 W2 b2] = initializeModel(numFeats,numHidden,numOutput);
# Initialize losses
losses=[]
#Initialize learningRate
learningRate=0.5;
for k =1:10000
# Forward propagation through hidden layer with Relu units
[A1,cache1 activation_cache1]= layerActivationForward(X',W1,b1,activationFunc ='relu');
# Forward propagation through output layer with Softmax units
[A2,cache2 activation_cache2] =
layerActivationForward(A1,W2,b2,activationFunc='softmax');
# No of training examples
numTraining = size(X)(1);
# Select rows where Y=0,1,and 2 and concatenate to a long vector
a=[A2(Y==0,1) ;A2(Y==1,2) ;A2(Y==2,3)];
#Select the correct column for log prob
correct_probs = -log(a);
#Compute log loss
loss= sum(correct_probs)/numTraining;
if(mod(k,1000) == 0)
disp(loss);
losses=[losses loss];
endif
dA=0;
# Backward propagation through output layer with Softmax units
[dA1 dW2 db2] = layerActivationBackward(dA, cache2, activation_cache2,Y,activationFunc='softmax');
# Backward propagation through hidden layer with Relu units
[dA0,dW1,db1] = layerActivationBackward(dA1', cache1, activation_cache1, Y, activationFunc='relu');
#Update parameters
W1 += -learningRate * dW1;
b1 += -learningRate * db1;
W2 += -learningRate * dW2';
b2 += -learningRate * db2';
endfor
# Plot Losses vs Iterations
iterations=0:1000:9000
plotCostVsIterations(iterations,losses)
# Plot the decision boundary
plotDecisionBoundary( X,Y,W1,b1,W2,b2)

The code for the Python, R and Octave implementations can be downloaded from Github at Deep Learning – Part 4

Conclusion

In this post I have implemented a 2 layer Neural Network with the Softmax classifier. In Part 3, I implemented a multi-layer Deep Learning Network. I intend to include the Softmax activation unit into the generalized multi-layer Deep Network along with the other activation units of sigmoid,tanh and relu.

Stick around, I’ll be back!!
Watch this space!

References
1. Deep Learning Specialization
2. Neural Networks for Machine Learning
3. CS231 Convolutional Neural Networks for Visual Recognition
4. Eli Bendersky’s Website – The Softmax function and its derivative
5. Cross Validated – Backpropagation with Softmax / Cross Entropy
6. Stackoverflow – CS231n: How to calculate gradient for Softmax loss function?
7. Math Stack Exchange – Derivative of Softmax
8. The Matrix Calculus for Deep Learning

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