R and Datascience

I found this site very interesting datascienceplus

Using R author has categorized

• Data Loading
• Data Management
• Visualization
• Stats

This really helps to align R learning accordingly. I am trying to repeat the pattern for my R learning's

Happy Learning and Happy New Year 2016!!!

Information Retrieval Notes

Token - Sequence of characters, chopping functions and throwing tokens certain characters
Type -  Equivalence class of Tokens
Term - Type in IR Dictionary
Term Frequency - Number of times term t appears in document d
Log Frequency - (1+ log(tf), if tf > 0)
Document Frequency - Number of documents in collection the term appears
Inverse Document Frequency - Log(N/Dft) - (Number of documents in collection / Number of documents term t appears)
IDF = log[Total Docs / Docs contain the term]
Stemming - Crude heuristics chopping end of words. Collapse derivationally related words. Stemming increases recall because morphological variation of words are collapsed into single token enabling higher chances of retrieval
Lemmatization - Return to base word or dictionary form of word. Collapse different inflectional form of words.
Skip Pointers - post of length N, Sqrt(N) evenly placed pointers
Positional Index - Term: DocId <Pos1, Pos2>
Inverted Index - is a dictionary mapping each word token to a set of file names

Boolean Retrieval (AND, OR, NOT)

• Easy to Implement
• Computationally efficient
• Expressiveness and Clarity

Cons of Boolean Retrieval

• No Ranking
• No Weighing

Discounted Cumulative Gain (DCG)

• Highly relevant docs are more useful when they appear earlier in search results list
• Highly relevant docs are more useful than marginally relevant docs

DCG - 2 power (relevance-1) / log2(i+1)
NDCG = DCG / IDCG

HITS - Hyperlink induced Topic Search

• Authorities - Direct answer to information need. Homepage of microsoft.com
• Hub - Good Links to pages answering the information
• Wikipedia good example for both Hub & Authority

Happy Learning!!!

T-Test

T-Test

- Developed in 1908 by William Gosset
- T-test referred as Student's t-test
- Mu, Sigma (Indicate Population parameters)
- X-Dash, S represent mean and standard deviation of sample




Hypothesis Tests in R



One Sample T-Test

Function - t.test example in R

Happy Learning!!!

Hypothesis Testing Basics

After exams I understood my improvement areas in terms of learning. Predominantly these are crucial chapters

- P test using R Programming
- P test using Python Programming
- Hypothesis test using R Programming
- Hypothesis test using Python Programming

I glanced through couple of sites, Bookmarking some of pointers

Normal Distribution Properties




Key Pointers
- Normal distribution unimodal and symmetric
- Mean (Mu)
- Standard Deviation (Sigma)
- 99.7% < 3 Sigma
- 95% < 2 Sigma
- Z > 2 (Unusual)
- pnorm (percentile of observation)
- Qnorm for quantile or cutoff values







Key Pointers 
- Creating Null and Alternate Hypothesis conditions
- Identifying sample space, standard error, population mean, standard deviation from input question
- Computing P value

Happy Learning!!!

t-test and z-test

Problems to workout (Good Compiled List)

References
Link1
Link2

Z - Scores

Z - Scores makes it easy to compare scores from distributions using different scales

Formula #1

Formula #2

Formula #3 for raw score computation is defined by 

Formula #4 for Standard Error

Trying out problems in link 

Problem 2. Suppose X is a normal random variable with a mean of 120 and a standard deviation of 20. Determine the probability that X is greater than 135.

Mean = 120
SD = 135
Z score = (135-120)/20 = 0.75

z score from attached link

Find P(Z < 0.75) = 0.7734
1 - 0.7734 = 0.2266

Problem 4. If the test scores of 400 students are normally distributed with a mean of 100 and a standard deviation of 10, approximately how many students scored between 90 and 110?
Mean = 100
SD = 10

For x = 90, z = (90-100)/10 = -1
For x = 110, z = (110-100)/10 = 1
For Z (< -1), 

= 0.1587
For Z (<1), 

= 0.8413
= 0.8413-0.1587
= 0.6826

Multiply this percentage by 400. After rounding, we get 273 students.

Problem 16. A traffic study shows that the average number of occupants in a car is 1.5 and the standard deviation is .35. In a sample of 45 cars, find the probability that the mean number of occupants is greater than 1.6.

Mean = 1.5
SD = .35

Applying Formula #2

P(mean > 1.6) = 1- P(mean < 1.6)
Z(1.6) = ((1.6-1.5)*sqrt(45)) / 0.35
          = 1.916

P(Z<1.6) =  0.9719
P(Z>1.6) = 1- 0.9719 = 0.0281

Happy Learning!!!

chi-square test for homogeneity

The chi-square test for homogeneity is a test made to determine whether several populations are similar or equal or homogeneous in some characteristics

This link was useful

I tried the problem provided in the link

Problem - Know how to compute the chi-square homegeniety test statistics.

Step 1 

Step 2

Step 3

1-pchisq(19,df=2) - R Command

7.485183e-05

Since it is less than 0.05, you reject the null hypothesis

Happy Learning!!!

Chi Square Test for Independence

• Uses a cross classification table to examine the nature of the relationship between these variables
• Tables are sometimes referred to as contingency tables
• Determine variables are dependent on each other or not

Approach

• H0: chi square test for independence is conducted by assuming that there is no relationship between the two variables
• Ha: alternative hypothesis is that there is some relationship between the variables

The general formula for the degrees of freedom is the number of rows minus one, times the number of columns minus 1.

In terms of independence and dependence these hypotheses could be stated

• H0 : X and Y are independent
• H1 : X and Y are dependent

Expected Frequency = ((row total)*(column total))/Total Population

I liked the example provided in link  

Problem - Test for a Relationship between Sex and Class

X (Sex)

Y (Social Class) Male(M) Female(F) Total

Upper Middle (A) 33 29 62

Middle (B) 153 181             334

Working (C) 103 81 184

Lower (D) 16 14 30

Total 305 305            610

Table 10.12: Social Class Cross Classified by Sex of Respondents

Expected Frequency = ((row total)*(column total))/Total Population

1-pchisq(4.8748,df=3)

 0.1811978

Significance is greater than or equal to 0.05, you don't reject the null hypothesis

Results match with the problem although approach is different. The sum total sum is 610 (Total Sum)

Happy Learning!!!

Stats - Chi-Square Goodness of Fit Test

Purpose -  Test association of variables in two-way tables

The chi-square test is defined for the hypothesis:
H0: The data follow a specified distribution
Ha: The data do not follow the specified distribution
This means that if the significance value is less than 0.05, you reject the null hypothesis; if significance is greater than or equal to 0.05, you don't reject the null hypothesis

Formula is

I liked the example mentioned in notes

Problem - Testing an octadedral die to see if it is biased

Score 1 2 3 4 5 6 7 8

Frequency 7 10 11 9 12 10 14 7 (Observed)

Degree of Freedom = Number of entries - 1. Here is is 8-1 = 7

Test the hypothesis H0 - The Die is Fair

H1: Die is not fair

Significance level alpha = 0.005

Expected frequency is uniform distribution of Ei = Sum of all observed scores / 8(Number of items)

= 80/8 = 10

The expected values will be

Score 1 2 3 4 5 6 7 8

Frequency 10 10 10 10 10 10 10 10 (Expected)

To compute the score we need to find values of (Oi-Ei ), ((Oi-Ei )*(Oi-Ei ))/ Ei

For each element between  both the arrays

Compute chisquare value (R Command)

1-pchisq(4,df=7)

0.7797774

This is above significance level > 0.05. So we cannot reject null hypothesis

Answer - The Die is Fair

Happy Learning!!!

Good Read on Taylor Seris

Two summary points

• A Taylor Series is an expansion of a function into an infinite sum of terms, like these ones
• A derivative gives you the slope of a function at any point

Detailed Notes in link
Taylor series Formula Compilation - link

Happy Learning!!!

Good Formula Lists Compilations

Good Formula Lists Compilations

• Integrals 
• Stats 
• Mathematics - Rotations 
• Partial Derivatives
• Differential Equations
• Laplace Transform
• Elementary Differential and Integral Calculus
• More Reads

Happy Learning!!!

K Means Clustering

I'm slowly moving in Stats with a lot of learning. This post is from my class notes

K-means clustering

• Finding groups of object similar to one another
• Partitioning cluster approach
• Mean moves every time (Within first few iterations it will converge)
• Classify a given data set through a certain number of clusters
• This does not fit well for Sparse / Dense clusters

Great 5 Minute Video



Step 1 - "Figure out centric of region"
Step 2 - "Select K Data points randomly"
Step 3 - "Assign each data point to nearest centre"
Step 4 - "Recalculate the new centroids"
Step 5 - "Repeat Step 3,4"

More Reads - K-Means Clustering

DTW  - Dynamic Time Warping Algorithm. DTW - allowing similar shapes to match even if they are out of phase in the time axis

Ref - Link

Happy Learning!!!

Quick Tip - Python Stemming Module Installation - Windows

Copy the scripts to package folder. Run the command easy_install.py specifying the package containing scripts.

Happy Learning!!!

5 minute quick learning - Naïve Bayes

Good 5 Minute Learning!!!

Conditional Probability - Easy Walkthrough

I take iterations to understand / try out a concept. Going back and learning after sometime it's interesting. This post is a quick explanation on Conditional Probability.

P(A) - Probability of Event A to occur
P(A/B) - Probability of A given that B has already occurred. - This we refer as conditional probability

Problem - Roll a fair die. Let A be event of odd outcomes. B be event where outcome <=3. What is probability of A and Probability of A given B has already occurred

A = Odd Outcomes = {1,3,5} = 3
B = Outcome <=3 = {1,2,3} = 3
Sample space = {1,2,3,4,5,6} = 6

Probability P(A) = |A| / |S| = 3/6 = 1/2

Probability P(A/B) = Probability of A given that B has already occurred

From B outcomes {1,2,3}, Possible A values are = {1,3}

p(A/B) = Events{1,3} / Events of B{1,3,5}
p(A/B)  = 2/3

Bayes Theorem

P(CD) = P(C/D)P(D)

P(CD) = P(D/C)P(C)

Equating both the formulas

P(D/C)P(C) = P(C/D)P(D)

P(D/C) = (P(C/D)P(D)) / P(C)

P(C/D) = ((P(D/C)P(C)) / P(D)

Happy 2 Minute Quick Learning!!!

R Working on Normal Distributions & Binomial Distributions

Cumulative density function - CDF is summation of all the probabilities within a range. The CDF is the integral of the PDF (Probability density function).

Read Jack Lion Heart's answer to What is the difference between a probability density function and a cumulative distribution function? on Quora

• “d” - returns the height of the probability density function
• “p” - returns the cumulative density function
• “q” - returns the inverse cumulative density function (quantiles)
• “r” - returns randomly generated numbers

Reference - Link

R Examples

pnorm 

pnorm(700,500,100)

Mean - 500

Variance - 100

This score 700 is better than 97% of other scores

v <- c(-1,1,2)

dnorm(v)

plot(v,dnorm(v))

plot(v,pnorm(v))

dnorm(0,mean(v),sd(v))

Link Ref

Normal distribution is defined by the following probability density function, where μ is the population mean and σ2 is the variance.

Chi-squared Distribution - If X1,X2,…,Xm are m independent random variables having the standard normal distribution, then the following quantity follows a Chi-Squared distribution with m degrees of freedom

Binormial Distribution - binomial distribution is a discrete probability distribution. It describes the outcome of n independent trials in an experiment. 

Two Extra Parameters - number of trials and the probability of success for a single trial

Distribution function

x <- seq(0,50,by=1)

y <- dbinom(x,50,0.2)

plot(x,y)

50 - Number of Trials

0.2 - Probability of success for each trial

Cumulative Probability Density Function

x <- seq(0,50,by=1)

y <- pbinom(x,50,0.5)

plot(x,y)

50 - Number of Trials

0.5 - Probability of success for each trial

Random Probability Density Function

x <- seq(0,50,by=1)

y <- rbinom(x,50,0.5)

plot(x,y)

Happy Learning, More Learning Needed...It's vast....Lot more efforts needed :)

Maths - Basics

Good Read
Link1

Happy Learning!!!

Class Notes - Information Retrieval

Information Retrieval Notes

• Document Corpus (Collection of links / indexed web structures)

Examples of Information Retrieval

• When user is humming songs, based on it if we identify song then its classified as IR problem
• Multimedia IR (musics, video, analysing music videos)
• Photo Search (Visual IR)

Information Retrieval Application Areas

• Text Information Retrieval
• Web Search
• Social Media Search
• Micro Blogs
• Twitter Blogs

Boolean Information Retrieval

• Simplest model
• Restricted Queries
• queries are boolean expressions

Inverted Index

• For each item we have a list
• Like index of a book (Topic, Pagenumber), Close to glossary
• Document, Tokenize the text
• Inside document order tokens, heuristics to combine multiple tokens for index construction
• Document Frequency - How many times term appears

Challenges

• Ordering (Right to Left)
• Proximity Search Leveraging the context)
• Encoding 
• Normalization and keyword detection based on locale
• Accents, patterns
• Stemming (Chopping end of words to obtain root word)
• Porter algorithm for stemming
• Stopwords, normalization, tokenization, lower casing, stemming, non-latin alphabets, compounds, numbers
• Skip pointers(Find elements common between both lists, increment until they match)

Happy Learning!!!

Central Limit Theorem

Normal Distribution



Standard Normal Distribution - Mean = 0, Variance = 1

Distribution approaches to normal distribution for larger set of variables. 

"As n increases, the distribution of sample mean approaches normal distribution"

Central Limit Theorem - Almost all measurable "random" variables in real world follow some kind of normal distribution.

Good Link

"Sampling distribution of the sampling means approaches a normal distribution as the sample size gets larger"

"Average of your sample means will be the population mean"




Happy Learning!!!

Linear Algebra Playlist

Linear Algebra and Calculus basics Playlist bookmarked based on reference from my colleague

Pauls Online Calculus Notes

MIT Linear Algebra Playlist

Linear Algebra


Calculus

More Reads
Link1
Link2
LA in 4 Pages
LA Machine Learning
Link3
Link4
Link5

Happy Learning!!!

Linear Regression Basics

To Understand Linear Regression basics of Slope, Correlation was useful.

Slope Revision

Finding the slope of a line from its graph: Slope of a line


Slope = Change in Y / Change in X
Slope is constant for a line

Simple Linear Regression

• One Explanatory Variable Simple Regression
• More than one Explanatory Variable multiple Regression
• X - Independant Variable(Explanatory), Y - Dependant Variable (Response)
• Good fitting line (Reasonable for predicting relationship) - Measure of Strength of Relationship (Co-relation)
• Correlation - A & B are observed at Same time
• Methods of Least Squares to estimate B0 and B1
• Residual = Observed - Predicted value (Above Line +ve, Below Line -ve)

Reference Videos

Squared error of regression line: Introduction to the idea that one can find a line that minimizes the squared distances to the points


Happy Learning!!!

Covariance and Correlation - Random Variable - Probability

This video was useful to understand the covariance and correlation relationship

Happy Learning!!!

Working with R - InterQuartile Range

Concept - IQR - InterQuartile Range

IQR = Q3 - Q1 = 3rd Quartile - 1st Quartile

• Median - Arrange data from lowest to highest
• On Even dataset - Average of two most middle numbers
• On Odd dataset - Single Number that is halfway into the set

Dataset - 5,6,12,13,15,18,22,50

Q2 = (13+15)/2 = 14 - Median of Data Value

Q1 = (6+12)/2 = 9 - Median Before Q2

Q3 = (18+22)/2 = 20 - Median After Q2

IQR = Q3-Q1 = 20-9 = 11

BoxPlot is used to identify outliers

For Above Dataset

• Minimum Value - 5
• Q1 - 9
• Q2 - 14
• Q3 - 20
• Maximum Value - 50

This is the mathematical concept. This is used for finding outliers.

Outlier - Much larger or smaller than other values in data set. IQR obtained by subtracting third vs first quartile. 

Finding Outliers

1. Any value < Q1-1.5(IQR) or > Q3+1.5(IQR) is an outlier

2. Any Value < (9-1.5(11)) = -7.5

    Any Value > 20+1.5(11) = 20+16.5 = 36.5

This Video was useful to understand the concept before trying out in R

Computing using R

IQR between 25th percentile and the 75th

dataset <-c( 5,6,12,13,15,18,22,50 )
quantile(x=dataset, probs= c(.25,.75))
IQR(x=dataset)
boxplot(dataset)

Sample Output

Outlier highlighted in circles

Happy Learning!!!

Class 3 - Statistics Notes

This was mostly on probability distribution functions. Couple of one liners from session

Conditional Probability Distribution - Value of one random variable not impacting another then random variables are independent

Variance - How many values < mean and > mean

Covariance - If X and Y are two independent variables Covariance is zero
Correlation Values between -1 and 1
Correlation

Probability "Chebyshev Inequality"
Chebyshev Inequality - For computing mean for specific region (Integrated over smaller region). Probability in tail for any random variable.

Related Reads

• Link1
• Jointly distributed random variables

Happy Learning!!!

R Basic Examples

Listed below are couple of basic examples working from R Console

Example 1 - Set and get working directory

Example 2 - Read from Data Files

Example 3 - Count row and columns in data

Example 4 - Functions

Example 5 - Plotting

Happy Learning!!!

Video Analytics Class 2

My Notes

• Linear Filter - Linear combination of neighbours
• Box filter - All values constant [1's]
• Corr-relation - Masked and Moved across Image
• Gradient  - due to surface normal discontinuity, depth discontinuity, illumination discontinuity
• LOG - Laplace of Gaussian. LOG capable of finding edges
• Salt and Pepper Image - Image has random black and white 

Basics

• Represent Image as a Matrix
• Represent Image as a function
• Point, local operations, histogram equalization, moving average model
• Cross Correlation g = H X F
• Gaussian filter (Removes High frequency, blurring, smoothens image)
• Symmetric Matrix (When you shift rows into columns it would appear the same ( aij = aji, for all indices i and j) example link 

Convolution Basics

Programmatic Walkthru - link

From link 

From link

From Link
FFT

Convolution Applications

• Smoother image
• Gaussian (Point spread function)
• Different Kinds of filter (Box, Gaussian filter)

Cross Correlation - Assess how similar are two different functions. Compares position by position. 

Correlation Walkthru

From Link 

Mathematics concepts to learn

• Vector Product
• Eigen Value decomposition
• First Derivative, Second Derivative

Vertical and horizontal edge detection filters - Sobel, Roberts, Prewitt (Veritical, Horizontal, Diagonal edge detection filters).

Good Read Link
MIT Course Slides link

Related Reads







Happy Learning!!!

OpenCV Python Basics

Basic image loading modules

Example #1

import cv2
import numpy as np
from matplotlib import pyplot as plt
#Load Image
source = cv2.imread('D:\images\Benz.png')

Ref - Link

Example #2

#Printing width and height of image
import cv2
import numpy as np
from matplotlib import pyplot as plt

#Load Image
source = cv2.imread('D:\images\Benz.png')
print source.shape

#Output Number of Rows / Columns
rowCount = source.shape[0]
columnCount = source.shape[1]
print rowCount, columnCount

Ref - Link

Example #3
#Drawing Histogram

import cv2
import numpy as np
from matplotlib import pyplot as plt

#Load Image
source = cv2.imread('D:\images\Benz.png')

# Image, Channel
# Channels - grayscale image - [0], RGB - 0,1,2
# Mask - Supplied None as FULL Region Needed
# histSize - Bin Size
# ranges - 0 to 256
hist = cv2.calcHist([source], [0], None, [256], [0,256])
plt.plot(hist)
plt.show()

Happy Learning!!!

SettingUp OpenCV and Python

Reference Steps - Link

1. Download and Install python from link. Install with Default Settings
2. Download and Install MatPlot Lib (Link in reference steps are fine)
3. Download and Install OpenCV executable. Extract it to C:\OpenCV location

4. Now All Installations are located in C:\

5. Open Python IDLE from program files. In win7 run-it-as admin to open Program Files ->Python IDLE

6. Copy File from below location

      From - C:\OpenCV\opencv\build\python\2.7\x86\cv2.pyd

      To - C:\Python27\Lib\site-packages

7. Got the Error Link

8. Download and install numpy from link (Link provided 1.7 of numpy is incorrect. You need 1.9.2)

9. Validating installation steps

Happy Learning!!!