Day #24 - Python Code Examples

Examples for - for loop, while loop, dictionary, function examples and plotting graphs

	#Example 1 - Working on Float Variables
	print 'Program 1'
	i = 5.44
	j = 3.0
	print j/i
	#Example 2 - For loop
	print 'Program 2'
	for i in range(1,10):
	print i
	i = i+1
	#Example 3 - While Loop with break condition
	print 'Program 3'
	print 'while loop'
	i = 1
	while(i<10):
	print i
	i = i+1
	if(i>5):
	break
	print 'Program 4'
	#Example 4 - Working with Functions and Dictionary
	def computevalues(initval):
	f_ = {}
	i = 0
	while(i<10):
	try:
	f_[i] = i
	i = i+1
	except Exception:
	pass
	return f_
	f_ = {}
	f_ = computevalues(0.01)
	print 'f_'
	for key,value in f_.iteritems():
	print value
	#Example 5 - Plot Graphs
	print 'Program 5'
	import matplotlib.pyplot as plt
	x = [-.001,-0.0005,-0.0001,-0.00005,-0.00001]
	y = [103,178,523,738,880]
	plt.xlabel("X Values")
	plt.ylabel("Y Values")
	plt.plot(x,y,'-o', color = 'g')
	plt.title("Demo X,Y Graph")
	plt.show()
	#Matrix and SVD
	import numpy as np
	from sympy import Matrix
	b = np.arange(1,50,1)
	z = np.eye(3,3)
	print(z)
	print("b")
	print(b)
	a = np.matrix([[2,2,5],[-4,2,3],[0,0,0]])
	print("eig vector")
	print(np.linalg.eig(a))
	print("eig vals")
	print(np.linalg.eigvals(a))
	print("svd")
	print(np.linalg.svd(a))
	print("null space")
	A = Matrix(a)
	print(A*A.nullspace()[0])

view raw pythonbasics.py hosted with ❤ by GitHub

Happy Learning!!

Day #23 - Newton Raphson - Gradient Descent

Newton Raphson

• Optimization Technique
• Newton's method tries to find a point x satisfying f'(x) = 0
• Between two successive approximations
• Stop iteration when difference between x(n+1) and x(n) is close to zero

Formula

• x(n+1) = x(n) - (f(x)/f'(x))
• Choose suitable value for x0

Gradient Descent

• Works for convex function
• x(n+1) = x(n) - af'(x)
• a - learning rate
• Gradient descent tries to find such a minimum x by using information from the first derivative of f
• Both gradient and netwon raphson are similar the update rule is different

More Reads - Link

Optimal Solutions

• Strategy to get bottom of the valley, go-down in steepest slope
• Measures local error function with respect to the parameter vector
• Once gradient zero you have reached the minimum
• Learning rate, steps to converge to a minimum
• How to converge to Global vs Local Minimum
• Gradient descent does guarantee local minimum
• Cost function elongated/circular decides on the convergence

Happy Learning!!!

Day #22 - Data science - Maths Basics

Eigen Vector - Vector along which there is no change in direction

Eigen Value - Amount of Scaling factor defined by Eigen value

Eigen Value Decomposition - Only Square matrix can be performed Eigen Decomposition

Trace - Sum of Eigen Values

Rank of A - Number of Non-Zero Eigen Values

SVD - Singular Value Decomposition

• Swiss Army Knife of Linear Algebra
• SVD - for Stock market Prediction
• SVD - for Data Compression
• SVD - to model sentiments
• SVD is Greatest Gift of Linear Algebra to Data Science
• Square Root of (Eigen Values of AtA) - A Transpose A, becomes Singular Value of

Happy Learning!!! (Revise  - Relearn - Practice)

Day #21 - Data Science - Maths Basics - Vectors and Matrices

Matrix - Combination of rows and columns
Check for Linear Dependence - R2 = R2 - 2R1, When one of the rows is all zeros it is linearly dependent
Span - Linear combination of vectors
Rank - Linearly Independent set

Good Related Read - Span

Vector Space - Space of vectors, collection of many vectors
If V,W belong to space, V+W also belongs to space, multiplied vector will lie in R Square
If the determinant is non-zero, then the vectors are linearly independent. Otherwise, they are linearly dependent

Vector space properties

• Commutative  x+y = y+x
• Associative (x+y)+z = x+(y+z)
• Origin vector - Vector will all zeros, 0+x = x+0 = x
• Additive (Inverse) - For every X there exists -x such that x+(-x) = 0
• Distributivity of scalar sum, r(x+s) = rx+rs
• Distributivity of vector sum, r(x+s) = rx+rs
• Identity multiplication, 1*x = x

Subspace
Vector Space V, Subset W. W is called subspace of V
Properties
W is subspace in following conditions

• Zero vector belongs to W 
• if u and v are vectors, u+v is in W (closure under +)
• if v is any vector in W, and c is any real number, c.v is in W

Subset S belongs to V can be represnted as linear combination
 v = r1v1+ r2v2+... rkvk
v1,v2 distinct vectors from S, r belongs to R

Basis - Linearly Independent spanning set. Vector space is called basis if every vector in the vector space is a linear combination of set. All basis for vector V same cardinality

Null Space, Row Space, Column Space
Let A be m x n matrix

• Null Space - All solutions for Ax = 0, Null space of A, denoted by Null A, is set of all homogenous solution for Ax=0
• Row Space - Subspace of R power N spanned by row vectors is called Row Space
• Column Space -  Subspace of R power N spanned by column vector is called Column Space

Norms - Measure of length and magnitude

• For (1,-1,2), L1 Norm = Absolute value = 1+1+2 = 4
• L1 - Same Angle
• L2 - Plane
• L3 - Sum of vectors in 3D space
• L2 norm (5,2) = 5*5+2*2 = 29
• L infinity - Max of (5,2) = 5

Orthogonal - Dot product equals Zero
Orthogonality - Linearly Independent, perpendicular will be linearly independent
Orthogonal matrix will always have determinant +/-1

Map of Mathematics.

Enlarge the figure to see all the wonderful areas for exploration and imagination. Which topic might you find most fascinating?

By Dominic Walliman, @DominicWalliman, Source: https://t.co/mNu0hWzFGW, Used with permission. pic.twitter.com/kx1azWIhle

— Cliff Pickover (@pickover) August 22, 2022

Differential Equations - Notes - Link

Lectures - Link

Course Notes - Link

Happy Learning!!!

Day #20 - PCA basics

Machine Learning Algorithms adjusts itself based on the input data set. Very different from traditional rules based / logic based systems. The capability to tune itself and work according to changing data set makes it self-learning / self-updating systems. Obviously, the inputs / updated data would be supplied by humans.

Basics

• Line is unidirectional, Square is 2D, Cube is 3D
• Fundamentally shapes are just set of points
• For a N-dimensional space it is represented in N-dimensional hypercube

Feature Extraction

• Converting a feature vector from Higher to lower dimension

PCA (Principal Component Analysis)

• Input is a large number of correlated variables We perform Orthogonal transformation, convert them into uncorrelated variables. We identify principal components based on highest variation
• Orthogonal vector - Dot product equals zero. The components perpendicular to each other
• This is achieved using SVD (Single Value Decomposition)
• SVD internally solves the matrix and identifies the Eigen Vectors
• Eigen vector does not change direction when linear transformation is applied
• PCA is used to explain variations in data. Find principal component with largest variation, Direction with next highest variation (orthogonal for first PCA)
• Rotation or Reflection is referred as Orthogonal Transformation
• PCA - Use components with high variations
• SVD - Express Data as a Matrix

More Reads

PCA Explained

PCA Quora Answer

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Day #19 - Probability Basics

Concepts

• Events - Subset of Sample Space
• Sample Space - Set of all possible outcomes
• Random Variable - Outcome of experiment captured by Random variable
• Permutation - Ordering matters
• Combination - Ordering does not matter
• Binomial - Only two outcomes of trail
• Poisson - Events that take place over and over again. Rate of Event denoted by lambda
• Geometric - Suppose you'd like to figure out how many attempts at something is necessary until the first success occurs, and the probability of success is the same for each trial and the trials are independent of each other, then you'd want to use the geometric distribution
• Conditional Probability - P(A Given B) = P(A) will occur assume B has already occurred
• Normal Distribution - Appears because of central limit theorem (Gaussian and Normal Distribution both are same)

From Quora -  

"Consider a binomial distribution with parameters n and p. The distribution is underlined by only two outcomes in the run of an independent trial- success and failure. A binomial distribution converges to a Poisson distribution when the parameter n tends to infinity and the probability of success p tends to zero. These extreme behaviours of the two parameters make the mean constant i.e. n*p = mean of Poisson distribution "

Read Michael Lamar's answer to Probability (statistics): What is difference between binominal, poisson and normal distribution? on Quora

	#binomial binom discrete
	#Poisson pois discret
	#normal norm continuous
	#chi-squared chisq continuous
	#Student's t t continuous
	#uniform unif continuous
	#dnorm
	#height of the probability density function
	#dnorm() function returns the height of the normal curve at some value along the x-axis
	dnorm(1)
	#you can specify "mean=" and "sd="
	#dnorm(x, mean, sd) - Height of pdf
	dnorm(1,mean=5,sd=2)
	#pnorm
	#cumulative density function
	#It returns the area below the given value of "x"
	pnorm(1)
	#pnorm(q, mean, sd)
	#Left of standard normal curve
	pnorm(1.96, 0, 1)
	#qnorm
	#quantiles or "critical values", you can use the qnorm()
	qnorm(.95)
	#qnorm(p, mean, sd)
	qnorm(0.975, 0, 1)
	#rnorm
	#normally distributed random nos.
	#rnorm(n, mean, sd) will create a vector of length n containing independent,
	x = rnorm(9)
	hist(x)
	#Generate 100 numbers from a normal distribution with mean 5 and sd 0.25
	x = rnorm(100, mean=5, sd=.25)
	hist(x)
	#dbinorm - binomial
	#dbinom(x, s, p)
	#vector of values, x, sample size s and probability of success p
	#report the probability of seeing exactly the number of successes denoted by each value of x
	V = dbinom(0:7, 7, .3)
	V
	barplot(table(V))
	#rbinom(n, s, p) will create a vector of length n containing independent,
	#random draws from a binomial distribution with the size of s and the probability of success p.
	#Bernouli spaces, two outcomes
	rbin(N,n,p)
	#binomial - number of successes in a sequence of n independent yes/no experiments
	#assign 100 independent binomial numbers with parameters n = 7 and p = .3 to a vector object called V.
	#binomial random numbers
	V <- rbinom(100, 7, .3)
	barplot(table(V))
	#poisson
	#Event occurs again and again
	#100 independent Poisson numbers with parameter λ = 0.7 to a vector object called V
	#Poisson distribution - given number of events occurring in a fixed interval of time and/or space
	V <- rpois(100, 0.7)
	barplot(table(V))
	#Geometric
	#something is necessary until the first success
	#Example: Products produced by a machine has a 3% defective rate.
	#What is the probability that the first defective occurs in the fifth item inspected?
	#P(X = 5) = P(1st 4 non-defective )P( 5th defective)
	dgeom (x= 4, prob = .03)

view raw ProbabilityBasics.R hosted with ❤ by GitHub

Happy Learning!!!!

Day #18 - Linear Regression , K Nearest Neighbours

Linear Regression

• Fitting straight line to set of data points
• Create line to predict new values based on previous observations
• Uses OLS (Ordinary Least Squares). Minimize squared error between each point and line
• Maximum likelihood estimation
• R squared - Fraction of total variation in Y
• 0 - R Squared - Terrible
• 1 - R Squared is good
• High R Squared good fit

Linear Regression (Ref - Link )

• ML Model to predict continuous variables based on set of features
• Used where target variable is continuous
• Minimize residuals of points from the line
• Find line of best fit
• y = mx + c
• Residual = sum (y-mx-c)^2
• Reduce residuals
• Assumptions in LR
• Linearity, Residuals Gaussian Distribution, Independence of errors, normal distribution

Updated May 28/ 2020

KNN

• Supervised Machine Learning Technique
• New Data point classify based on distance between existing points
• Choice of K - Small enough to pick neighbours
• Determine value of K based on trial tests
• K nearest neighbours on scatter plot and identify neighbours

Related Read
Recommendation Algo Analysis
Linear Regression
Linear Regression - Concept and Theory
Linear Regression Problem 1
Linear Regression Problem 2
Linear Regression Problem 3

Happy Learning!!!