What is linear regression in machine learning for

What is linear regression in machine learning for? Linear regression is a popular machine learning model used for predictive analysis. This is basically a statistical method that is used to establish a relationship between two continuous variables. The aim of linear regression is to predict a dependent variable based on one or more independent variables.

In this blog, we will discuss What is linear regression in machine learning for in detail, including its key concepts, types, and practical applications.

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Mathematical Representation of Linear Regression Equation

Linear regression can be expressed mathematically as: 

y= β0+ β 1x+ ε 

Here, 

Y= Dependent Variable  

X= Independent Variable  

β 0= intercept of the line  

β1 = Linear regression coefficient (slope of the line) 

ε = random error 

Random error ε, is needed to best fit line while values for x and y variables are training datasets for Linear Regression model representation.

What are the 5 assumptions of linear regression?

1. Linearity: 

Linearity is a key assumption of linear regression models. It refers to the relationship between the independent and dependent variables in a linear regression model. In a linear regression model, the relationship between the independent variable(s) and the dependent variable is assumed to be linear, meaning that the change in the dependent variable is proportional to the change in the independent variable(s).

The basic idea behind linearity is that the effect of the independent variable on the dependent variable is constant across the entire range of values for the independent variable. This means that the slope of the regression line is constant, and that the relationship between the independent and dependent variables can be represented by a straight line.

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When there is linearity in a linear regression model, the residuals (the difference between the observed values and the predicted values) should be randomly scattered around the regression line. This indicates that the model is capturing the true relationship between the independent and dependent variables.

However, if there is non-linearity in the relationship between the independent and dependent variables, the residuals will not be randomly scattered around the regression line. Instead, there will be a systematic pattern in the residuals, indicating that the model is not capturing the true relationship between the variables.

To check for linearity, one can use a scatter plot to visually inspect the relationship between the independent and dependent variables. If the relationship appears to be linear, then the linearity assumption is met. However, if there is evidence of non-linearity, it may be necessary to transform the data or use a different type of regression model to capture the relationship between the variables.

2. Normality: 

Normality is another key assumption of linear regression models. It refers to the distribution of the residuals (the difference between the observed values and the predicted values) in a linear regression model. In a linear regression model, the residuals are assumed to follow a normal distribution, with a mean of zero and a constant variance.

The basic idea behind normality is that the residuals should be randomly scattered around the regression line, with no systematic pattern. If the residuals are normally distributed, it means that the errors in the model are random and have a constant variance, indicating that the model is capturing the true relationship between the independent and dependent variables.

However, if the residuals are not normally distributed, it can indicate that there is a problem with the model. For example, if the residuals are skewed, it can indicate that the model is missing an important variable, or that there is a non-linear relationship between the independent and dependent variables.

To check for normality, one can use a histogram or a normal probability plot to visually inspect the distribution of the residuals. If the distribution of the residuals appears to be normal, then the normality assumption is met. However, if there is evidence of non-normality, it may be necessary to transform the data or use a different type of regression model.

In addition to visual inspection, there are statistical tests that can be used to test for normality, such as the Shapiro-Wilk test or the Anderson-Darling test. These tests can provide a quantitative measure of how well the residuals fit a normal distribution.

3. Homoscedasticity:

Homoscedasticity is another key assumption of linear regression models. It refers to the equal variance of the residuals (the difference between the observed values and the predicted values) across the range of values for the independent variable(s). In a linear regression model, the residuals are assumed to have a constant variance (i.e., homoscedastic), meaning that the variability of the residuals is the same for all values of the independent variable(s).

The basic idea behind homoscedasticity is that the scatter of the residuals around the regression line should be constant across the range of values for the independent variable(s). If the residuals have equal variance, it means that the errors in the model are random and do not depend on the value of the independent variable(s), indicating that the model is capturing the true relationship between the independent and dependent variables.

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However, if the residuals have unequal variance, it can indicate that there is a problem with the model. For example, if the variance of the residuals increases as the value of the independent variable(s) increases, it can indicate that the model is missing an important variable, or that there is a non-linear relationship between the independent and dependent variables.

To check for homoscedasticity, one can use a scatter plot of the residuals against the predicted values. If the scatter of the residuals around the regression line appears to be constant across the range of values for the independent variable(s), then the homoscedasticity assumption is met. However, if there is evidence of unequal variance, it may be necessary to transform the data or use a different type of regression model.

In addition to visual inspection, there are statistical tests that can be used to test for homoscedasticity, such as the Breusch-Pagan test or the White test. These tests can provide a quantitative measure of the degree of heteroscedasticity in the residuals.

4. Independence/No Multicollinearity:

Independence and No Multicollinearity are important assumptions in linear regression models.

Independence means that the observations in the data set are independent of each other, i.e., the value of one observation does not influence the value of another observation. In other words, each observation in the data set should be a separate and distinct event. This is important because if the observations are not independent, it can lead to biased estimates and inaccurate predictions.

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No Multicollinearity refers to the absence of high correlation between the independent variables in the model. If two or more independent variables are highly correlated, it can lead to problems in the model. This is because it becomes difficult to distinguish the effect of one independent variable from the effect of the other independent variable(s). As a result, the estimates of the regression coefficients become unstable, and the standard errors become inflated.

To check for independence, it is important to ensure that the data set is collected in such a way that each observation is independent of the others. This can be achieved by ensuring that the sample is random, and that each observation is collected without any bias.

To check for multicollinearity, one can use a correlation matrix to examine the correlations between the independent variables. If there is a high correlation between two or more independent variables, it may be necessary to remove one of the variables or to use a different regression model, such as a ridge regression or a principal component regression.

In addition to the correlation matrix, there are statistical tests that can be used to test for multicollinearity, such as the Variance Inflation Factor (VIF) or the Condition Number. These tests can provide a quantitative measure of the degree of multicollinearity in the model.

5. No Autocorrelation:

No Autocorrelation, also known as the assumption of independence of errors, is another important assumption in linear regression models. It refers to the absence of correlation between the errors in the model, i.e., the residuals from the model should not be correlated with each other.

Autocorrelation occurs when the residuals at one point in time are correlated with the residuals at another point in time. This can happen, for example, in time series data where the observations are collected at regular intervals over time. If there is autocorrelation in the residuals, it can lead to biased estimates and inaccurate predictions.

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To check for autocorrelation, one can use a plot of the residuals against time or a lag plot of the residuals. A plot of the residuals against time can reveal any patterns or trends in the residuals over time, while a lag plot can show any correlation between the residuals at different points in time.

In addition to visual inspection, there are statistical tests that can be used to test for autocorrelation, such as the Durbin-Watson test or the Breusch-Godfrey test. These tests can provide a quantitative measure of the degree of autocorrelation in the residuals.

If autocorrelation is detected, it may be necessary to modify the model or to use a different regression technique that takes into account the time series nature of the data, such as an autoregressive integrated moving average (ARIMA) model.

Key Concepts of Linear Regression:

The linear regression model is based on the following key concepts:

1. Dependent variable: 

The dependent variable, also known as the response variable, is the variable that we are trying to predict or explain. In linear regression, this variable is modeled as a function of one or more independent variables.

2. Independent variables: 

Independent variables, also known as predictor variables or regressors, are the variables that we use to predict or explain the dependent variable. In linear regression, we assume that the relationship between the dependent variable and the independent variables is linear.

3. Linear relationship: 

The relationship between the dependent variable and the independent variables is assumed to be linear, which means that the change in the dependent variable is proportional to the change in the independent variables.

4. Regression line: 

The regression line is the line that best fits the data points in a scatter plot of the dependent variable against the independent variables. The slope and intercept of the regression line are estimated using the least squares method.

5. Coefficient of determination:

The coefficient of determination, also known as R-squared, is a measure of how well the regression line fits the data. It is a value between 0 and 1, with 1 indicating a perfect fit and 0 indicating no fit.

6. Residuals: 

Residuals are the differences between the actual values of the dependent variable and the predicted values of the dependent variable based on the regression line. They are used to check the assumptions of the linear regression model.

7. Assumptions: 

There are several assumptions that must be met for the linear regression model to be valid, including linearity, normality, homoscedasticity, independence, and no multicollinearity. Violations of these assumptions can lead to biased estimates and inaccurate predictions.

8. Model selection: 

There are several types of linear regression models, including simple linear regression, multiple linear regression, polynomial regression, and logistic regression. The choice of model depends on the nature of the data and the research question.

What are the 3 types of linear model?:

1. Simple linear regression: Simple linear regression involves only one independent variable and one dependent variable. The formula for simple linear regression is:

y = mx + c

where y is the dependent variable, x is the independent variable, m is the slope of the line, and c is the intercept.

2. Multiple linear regression: Multiple linear regression involves multiple independent variables and one dependent variable. The formula for multiple linear regression is:

y = b0 + b1x1 + b2x2 + … + bnxn

where y is the dependent variable, x1, x2, …, xn are the independent variables, and b0, b1, b2, …, bn are the coefficients.

3. Non-Linear Regression:

When the best fitting line is not a straight line but a curve, it is referred to as Non-Linear Regression.   

What are some real life examples of regression?:

1. Economics and Finance:

Linear regression is widely used in economics and finance to model and analyze various economic and financial phenomena. For example, it can be used to model the relationship between a company's sales and its advertising spending, or the relationship between a stock's price and its earnings per share. Linear regression can also be used to forecast economic indicators such as GDP, inflation, and unemployment.

2. Healthcare:

Linear regression is used in healthcare to model the relationship between various health outcomes and their predictors. For example, it can be used to model the relationship between a patient's age, gender, and medical history and their risk of developing a certain disease. Linear regression can also be used to analyze clinical trial data and to estimate the effectiveness of a particular treatment or intervention.

3. Marketing:

Linear regression is used in marketing to analyze customer data and to develop marketing strategies. For example, it can be used to model the relationship between a customer's demographics and their purchasing behavior, or the relationship between a customer's satisfaction with a product and their likelihood to recommend it to others. Linear regression can also be used to optimize pricing strategies and to predict customer churn.

4. Social Sciences:

Linear regression is used in social sciences to analyze data and to model the relationship between various social phenomena and their predictors. For example, it can be used to model the relationship between a student's grades and their study habits, or the relationship between a person's income and their education level. Linear regression can also be used to analyze survey data and to estimate the effect of various interventions or policies.

5. Environmental Science:

Linear regression is used in environmental science to model and analyze various environmental phenomena. For example, it can be used to model the relationship between a region's temperature and its precipitation, or the relationship between a river's flow rate and its water quality. Linear regression can also be used to analyze climate data and to forecast future climate patterns.

6. Engineering:

Linear regression is used in engineering to model and analyze various engineering phenomena. For example, it can be used to model the relationship between a bridge's load capacity and its design specifications, or the relationship between a material's strength and its composition. Linear regression can also be used to optimize manufacturing processes and to predict equipment failure rates.

Conclusion:

In conclusion, the linear regression model is a powerful machine learning tool used for predictive analysis. It involves establishing a relationship between one or more independent variables and a dependent variable. The model can be used for a wide range of applications, including sales forecasting, price prediction, customer segmentation, employee performance prediction, and medical diagnosis. Understanding the key concepts of linear regression and its practical applications can help businesses make data-driven decisions and improve their bottom line.