Correlation and Regression!

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# Correlation and Regression
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## Correlation and Regression
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### Laxmikant Soni [blog](https://laxmikants.github.io) [](https://github.com/laxmiaknts) [](https://twitter.com/laxmikantsoni09)
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# Correlation and Regression

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## Correlation

* **Definition**: Correlation measures the strength and direction of the linear relationship between two variables.
* **Value Range**: Ranges from -1 to +1:
  - **+1**: Perfect positive correlation (variables increase together).
  - **-1**: Perfect negative correlation (one increases as the other decreases).
  - **0**: No linear relationship.
* **Example**: A correlation of 0.8 between height and weight suggests that as height increases, weight tends to increase as well.

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* **Types of Correlation**:
  - **Positive**: Both variables increase together.
  - **Negative**: One variable increases as the other decreases.
  - **No Correlation**: No consistent relationship between variables.

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# Correlation and Regression

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## Regression

* **Definition**: Regression identifies the relationship between a dependent variable (outcome) and one or more independent variables (predictors).
* **Types of Regression**:
  - **Simple Linear Regression**: Uses one predictor for prediction.
  - **Multiple Regression**: Uses multiple predictors.
* **Example**: Predicting house price based on house size. Here, house size is the independent variable, and house price is the dependent variable.

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* **Formula for Simple Linear Regression**:
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# Correlation and Regression

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## Key Differences

- **Correlation**: Quantifies strength and direction of a relationship but does not imply causation.
- **Regression**: Models the relationship and enables prediction.

* **Example**: If study hours and exam scores have a strong correlation, regression can model the relationship to predict a score based on study hours.

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# Statistical Methods for Correlation and Regression

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## Pearson Correlation Coefficient (r)

* **Definition**: Measures the strength and direction of the linear relationship between two continuous variables.

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* **Formula**:
  $$
  r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2} \sqrt{\sum (y_i - \bar{y})^2}}
  $$
  - Values range from -1 (perfect negative correlation) to +1 (perfect positive correlation).

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# Statistical Methods for Correlation and Regression

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## Simple Linear Regression

* **Definition**: Estimates the relationship between a dependent variable and one independent variable.

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* **Formula**:
  $$
  y = \theta_0 + \theta_1 x
  $$
  - This minimizes the cost function to fit a line to the data.

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# Statistical Methods for Correlation and Regression

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## Multiple Linear Regression

* **Definition**: Extends simple linear regression to include multiple predictors.

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* **Formula**:
  $$
  y = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_n x_n
  $$
  - Finds the best-fit line in a multi-dimensional space.

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# Statistical Methods for Correlation and Regression

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## Polynomial Regression

* **Definition**: Models non-linear relationships by including polynomial terms.

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* **Example Formula**:
  $$
  y = \theta_0 + \theta_1 x + \theta_2 x^2 + \ldots + \theta_n x^n
  $$
  - This approach allows fitting curves to data instead of just straight lines.

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# Statistical Methods for Correlation and Regression

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## Logistic Regression

* **Definition**: Used for binary outcomes (classification problems).

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* **Formula**:
  $$
  p(y=1) = \frac{1}{1 + e^{-(\theta_0 + \theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_n x_n)}}
  $$
  - Predicts the probability of a class or event.

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# Statistical Methods for Correlation and Regression

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## Choosing the Right Method

- **Correlation**: Use Pearson for linear relationships, Spearman or Kendall for non-linear or ordinal data.

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- **Regression**: Use simple or multiple regression for linear relationships, polynomial for curves, and logistic regression for classification tasks.

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# Choosing Between Correlation and Regression

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## Nature of Relationship
* Use **correlation** to assess the strength and direction of a linear relationship between two variables. 
* Use **regression** to predict the value of one variable based on another.

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## Variables
* Correlation examines the relationship between two variables without assuming causation. 
* Regression establishes a dependent (outcome) variable and one or more independent (predictor) variables.

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# Choosing Between Correlation and Regression

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## Data Types
* Correlation is suitable for **numerical data**. 
* Regression can handle both **numerical and categorical** independent variables.

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## Analysis Goal
* Choose correlation for **exploratory analysis** to understand relationships. 
* Choose regression for **modeling and predicting** outcomes based on one or more predictors.

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# Examples of Correlation and Regression

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## Suitable for Correlation

* **Example 1**: Examining the relationship between hours studied and test scores to see if more study time is associated with higher scores.
  
* **Example 2**: Analyzing the correlation between temperature and ice cream sales to determine if higher temperatures lead to increased sales.

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## Suitable for Regression

* **Example 1**: Predicting house prices based on features like size, location, and number of bedrooms, where house price is the dependent variable.

* **Example 2**: Estimating a student's future GPA based on their high school GPA and extracurricular activities, using multiple independent variables.

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