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Linear Regression - All About

What is Linear Regression?

Linear regression is a statistical modelling technique used to establish a relationship between a dependent variable and one or more independent variables. It assumes a linear relationship between the variables and aims to find the best-fit line that minimizes the difference between the predicted values and the actual observed values.

In simple linear regression, there is one dependent variable and one independent variable. The relationship between the variables can be represented by a straight-line equation:

Y = mX + c

Where:

Y represents the dependent variable (the variable we want to predict/ Target variable),

X represents the independent variable (the variable used to predict the dependent variable/ Predictor variable),

m represents the slope of the line (the change in y for a unit change in x),

c represents the y-intercept (the value of y when x is 0).

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The Objective of Linear Regression:

The goal of linear regression is to estimate the values of m and c that minimize the sum of squared differences between the predicted values and the actual observed values.

Linear Regression VS Logistics Regression


Where is Linear Regression used?

Linear regression is widely used in various fields, such as economics, finance, social sciences, and machine learning, to analyze and predict relationships between variables. It provides insights into the strength and direction of the relationship and can be used for forecasting and making predictions based on the identified patterns.

Here are some common applications: 

Sales Forecasting: Linear regression can be used to predict future sales based on historical data, identifying patterns and trends that can help businesses make informed decisions about inventory management, production planning, and marketing strategies.


Financial Analysis: Linear regression can be utilized to analyze the relationship between financial variables such as stock prices, interest rates, and economic indicators. It helps in understanding how changes in one variable can impact another, aiding in investment decision-making and risk management.

Demand Analysis: Linear regression can help businesses understand the factors influencing customer demand for their products or services. By analyzing historical sales data and incorporating variables like price, advertising expenditure, and competitor data, businesses can estimate the impact of these factors on demand and optimize pricing and marketing strategies accordingly.

Risk Assessment: Linear regression can be used to assess and predict risk in various domains, such as insurance and credit scoring. By analyzing historical data and identifying patterns, linear regression models can estimate the likelihood of certain events occurring and help in decision-making processes.

Medical Research: Linear regression can be utilized in medical research to examine the relationship between variables such as patient characteristics, lifestyle factors, and disease outcomes. It helps in identifying risk factors, developing predictive models, and understanding the impact of interventions or treatments.

Performance Evaluation: Linear regression can be applied to evaluate the performance of individuals or entities based on various factors. For example, in sports, it can be used to analyze the performance of athletes by considering variables like age, training hours, and past performance records.

Market Research: Linear regression can assist in market research by analyzing data related to consumer preferences, demographics, and purchasing behaviour. It helps in understanding the impact of different factors on consumer decision-making and aids in product development, pricing strategies, and targeted marketing campaigns.

Types of Linear Regression

Simple Linear Regression:

If a single independent variable is used to predict the value of a numerical dependent variable, then such a Linear Regression algorithm is called Simple Linear Regression.

Multiple Linear regression:

If more than one independent variable is used to predict the value of a numerical dependent variable, then such a Linear Regression algorithm is called Multiple Linear Regression.

Multiple linear regression extends this concept to multiple independent variables, allowing for a more complex relationship between the dependent variable and the predictors. 

The equation becomes:

Y = m1*X1 + m2*X2 + ... + mn*Xn + c0

Where:

Y represents the dependent variable,
X1, X2, ..., Xn represent the independent variables,
c0 is the y-intercept,
m1, m2, ..., mn are the coefficients for each independent variable.

The coefficients are estimated using various statistical techniques, such as the least squares method, to find the best-fit line that minimizes the sum of squared differences.

How to find the best-fit regression line using the Least Square Method?

To find the best-fit regression line using the least squares method, follow these steps:
  1. Gather Data: Collect the data for the variables of interest. For simple linear regression, you need a set of paired observations for the independent variable (x) and dependent variable (y).

  2. Plot the Data: Create a scatter plot with the independent variable (x) on the x-axis and the dependent variable (y) on the y-axis. Visualizing the data helps identify the overall trend and any potential outliers.

  3. Define the Regression Line: In simple linear regression, the regression line equation is y = mx + c, where m represents the slope of the line and c is the y-intercept. We need to estimate these values.

  4. Calculate the Mean of x and y: Calculate the mean (average) of the independent variable (x) and the dependent variable (y).

  5. Calculate the Deviations: For each data point, calculate the deviation from the mean for both x and y. The deviation is the difference between the data point and the mean value.

  6. Calculate the Sum of Products of Deviations: Multiply the deviations of x and y for each data point and sum up these products.

  7. Calculate the Sum of Squares of x Deviations: Square the deviations of x for each data point and sum up these squared values.

  8. Calculate the Slope: Calculate the slope (m) of the regression line using the formula: m = Sum of Products of Deviations / Sum of Squares of x Deviations.

  9. Calculate the Y-Intercept: Calculate the y-intercept (b) of the regression line using the formula: b = mean(y) - (m * mean(x)).

  10. Determine the Regression Line Equation: With the calculated values of m and b, you can determine the equation of the regression line, which represents the best fit for the data.

  11. Plot the Regression Line: Add the regression line to the scatter plot created in step 2. The line should pass through the data points as closely as possible.
The least squares method minimizes the sum of squared differences between the observed y-values and the predicted y-values on the regression line. By finding the line that minimizes this sum, you obtain the best-fit regression line that represents the relationship between the variables. 

NOTE: It's worth noting that for multiple linear regression (when there are multiple independent variables), the least squares method is extended to estimate the coefficients for each independent variable.




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