Stats Lessons

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Naive Bayes Classifier

Naive Based Classifier Notes and Examples

• Work on assumption occurrence of word i is not dependent on occurrence of word i+1
• Usually a sentence will have context only when words occur with appropriate terms and positions
• For example purpose, we have listed below two classes and a testing document to classify the same

Ref - Link

Bayesian data analysis is a fundamental concept in data science. But it took me 2 years to understand its importance. In 2 minutes, I'll share my best findings over the last 2 years exploring Bayesian Modeling. Let's go.

1. Why Bayesian Data Analysis? Bayesian modeling is a… pic.twitter.com/nowKIL4AB4

— 🔥 Matt Dancho (Business Science) 🔥 (@mdancho84) February 6, 2024

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Type I and Type II Error

Type I Error - Rejecting the null hypothesis eventhough it is true
Type II error, also known as a "false negative": the error of not rejecting a null hypothesis when the alternative hypothesis is the true state of nature

I liked below comment from Khan Academy
The easiest way to think about Type 1 and Type 2 errors is in relation to medical tests. A type 1 error is where the person doesn't have the disease, but the test says they do (false positive). A type 2 error is where the person has the disease but the test doesn't pick it up (false negative).

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R + Stats

The Following course material is very useful for R + Stats Combinations. It's a great material for R learning. Captured below are notes from 5,6,7,8 chapters

What is a central limit theorem?

The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal. In practice, some statisticians say that a sample size of 30 is large enough when the population distribution is roughly bell-shaped

Binomial Probability - Only two mutually exclusive events often referred as success, failure. Also called bernouli trial (Link )

R commands - The dbinom and pbinom functions

Discrete Probability Distributions

R command - pnorm

Command Syntax - pnorm(x, mean = , sd = , lower.tail= )

Two-Tailed Tests - Testing for the possibility of the relationship in both directions. This means that .025 is in each tail of the distribution

One-Tailed Tests - one-tailed test allots all of your alpha to testing the statistical significance in the one direction of interest. This means that .05 is in one tail of the distribution of your test statistic.

Alternative hypothesis has the > operator, right-tailed test 

Right-Tailed Tests: P-value = pnorm(zx¯, lower.tail=FALSE)

Alternative hypothesis has the < operator, left-tailed test 

Left-Tailed Tests: P-value = pnorm(zx¯, lower.tail=TRUE)

Alternative hypothesis has the ≠ operator, two-tailed (left and right) test

Two-Tailed Tests: P-value = 2 * pnorm( abs(zx¯), lower.tail=FALSE)

pnorm(x, µ, σ), 

• x is an observation from a normal distribution 
• mean µ 
• standard deviation σ

Computing P value from t value 

pt(abs(t-value), df=degree of freedom)

Reference

Link1

Link2

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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

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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

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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

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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)

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