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Linear regression is an supervised learning technique in which it comes under regression technique. It is used when our dependent variable is continuous in nature. In this lesson we will learn about linear regression, it’s cost function, best fit line, etc.
Introduction to linear regression
Linear regression could be a statistical procedure utilized for predicting a reaction or dependent variable from one or more predictor or independent variables. It may be a sort of supervised learning algorithm that produces assumptions around the linear relationship between the input factors (x) and the single output variable (y). The objective of linear regression is to discover the best-fit line for the given information, so that ready to utilize it to anticipate the esteem of the output variable for any given input variable. Linear regression is the foremost commonly utilized predictive modelling procedure and can be utilized for both relapse and classification issues.
In the above dataset Price is the dependent variable(y) i.e. single output variable and Food_Quality and Service_Quality are independent variables that are input variables for our model and we can see the price data is continuous in nature.
What is Best Fit Line?
The best fit line is a line that summarizes the relationship between two variables in a linear regression model.
Types of Linear Regression
There are two types of linear regression:
In simple linear regression, there is only one independent variable. The equation for a simple linear regression model is:
y = mx + b
where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept. The slope (m) represents the change in y for every one-unit change in x, while the y-intercept (b) represents the value of y when x is equal to zero.
In multiple linear regression, there are two or more independent variables. The equation for a multiple linear regression model is:
y = b0 + b1x1 + b2x2 + ... + bnxn
where y is the subordinate variable, x1, x2, ..., xn are the autonomous factors, and b0, b1, b2, ..., bn are the coefficients. The coefficients speak to the alter in y for each one-unit alter within the comparing autonomous variable, holding all other autonomous factors steady.
Hypothesis of Linear Regression
The hypothesis of linear regression is that there's a straight relationship between the autonomous variable(s) and the subordinate variable. In other words, it assumes that the alter within the subordinate variable is specifically relative to the alter within the independent variable(s).
Y = β0 + β1X1 + β2X2 + ... + βpXp + ε
where Y is the subordinate variable, X1, X2, ..., Xp are the autonomous factors, β0 is the intercept or constant term, β1, β2, ..., βp are the coefficients of the independent factors, and ε is the blunder term that speaks to the unexplained assortment inside the subordinate variable.
The speculation assumes that the mistakes are normally distributed, have a mean of zero, and have consistent variance (homoscedasticity). It also assumes that there's no multicollinearity (high correlation) between the independent variables.
Effect of Parameters
The parameters in linear regression are the coefficients (i.e., the weights) that are related with each include of the data set.The coefficients represent the relationship between the highlight and the dependent variable (i.e., the target).The affect of the parameters in linear regression is that they choose the quality and heading of the straight relationship between the highlights and the target. The more prominent the coefficient, the more grounded the relationship between the incorporate and the target. In case the coefficient is positive, it outlines that an increment inside the highlight is related with an increment inside the target; in case it is negative, it outlines that an increment inside the highlight is related with a reduce inside the target.
Error in Simple Linear Regression
Simple linear regression may be a statistical method utilized to foresee the values of one dependent variable based on the values of one autonomous variable. It may be a linear approach to modeling the relationship between two factors, where one variable is considered to be an explanatory variable (x) and the other is considered to be the response variable (y).
For illustration, a company may need to foresee the deals of a item based on the amount of cash went through on publicizing. In this case, the independent variable would be the publicizing budget and the dependent variable would be the deals of the item. The condition for basic linear regression is:
y = P0 + P1x
where P0 is the intercept (the value of y when x = 0), P1 is the slope of the line, and x is the value of the independent variable.
Choose parameters P1,P0 so that the predicted value h(x) is close to Y for our training examples (X,Y).
The sum of the squares of the residual errors are called the Residual Sum of Squares or RSS. The average variation of points around the fitted regression line is called the Residual Standard Error (RSE).
Cost Function in Simple Linear Regression
Choose parameters P1, P0 so that the predicted value h(x) is close to Y for our training examples (X,Y). The cost function helps us to figure out the best possible values for P0 and P1 which would provide the best fit line for the data points. Minimize the error between the predicted value and the actual value.
The Mean Squared Error (MSE) may be a degree of how close a predicted value is to the actual value. It is calculated by taking the contrast between the actual and predicted values, squaring the contrast, and after that taking the average of the squared differences. The smaller the MSE, the closer the anticipated values are to the genuine values.
n = Number of observations
ŷi = Predicted value for observation I
yi = Actual value for observation i
The cost function of multiple linear regression may be a degree of how exact the model is in foreseeing the target variable. It is calculated by taking the sum of the squared errors between the anticipated and real values.The cost function is expressed as:
J(θ) = 1/2m * Σi=1 to m (hθ(x(i)) - y(i))^2
where m is the number of training examples, θ is the vector of model parameters, hθ(x(i)) is the hypothesis function evaluated at x(i), and y(i) is the target value for x(i).
Assumptions of linear regression
It is additionally vital to check for presumptions of linear regression. On the off chance that these assumptions are not met, at that point linear regression might not be suitable, and other regression models or machine learning algorithms ought to be considered.
Before using linear regression algorithm, it is important to check if there is a linear relationship between the independent variable(s) and the dependent variable. One way to check for linearity is to use scatter plots to visually inspect the relationship between the variables.
Multicollinearity is a phenomenon in which two or more predictor variables in a multiple linear regression model are highly correlated. This means that one or more of the predictor variables can be linearly predicted from the others with a substantial degree of accuracy.
Multicollinearity is a problem because it reduces the accuracy of the regression coefficients and makes it difficult to interpret the results. For example, if two predictor variables are highly correlated, then one of them may be redundant and can be removed from the model without significantly affecting the model's accuracy.
What is Homoscedasticity and Heteroscedaticity?
Homoscedasticity and heteroscedasticity are terms used to describe the variance of errors in a regression model.
There are a few strategies for managing with heteroscedasticity, including transforming the data, utilizing weighted least squares, or employing a different regression model altogether, such as a robust regression model.
Residual analysis is an important step in linear regression. It helps to identify potential problems with the linear regression model, such as outliers, non-linearity, and heteroscedasticity.
An equation for residual analysis is as follows:
Residual = Observed Value - Predicted Value
An example of residual analysis would be fitting a linear regression model to predict the price of a house based on its square footage. After the model is fit, we can calculate the residuals for each house by subtracting the predicted price from the observed price. If we find that the residuals are not randomly distributed around zero, then this could indicate that the linear regression model is not the best fit for this dataset.
Normality is an assumption of linear regression that states that the errors or residuals of the model are normally distributed. In other words, the distribution of the residuals should follow a normal or Gaussian distribution with a mean of zero.
Normality is important because it affects the reliability and accuracy of the estimated regression coefficients and predictions. When the residuals are normally distributed, it means that the model is correctly accounting for all the factors that affect the dependent variable, and that there are no systematic errors in the model.
Normality can be assessed by examining a histogram or a Q-Q plot of the residuals. A histogram of the residuals should show a roughly symmetrical distribution around zero, and a Q-Q plot should show the residuals following a straight line. If the histogram or Q-Q plot shows significant deviation from normality, then the model assumptions may not be met.
If the normality assumption is not met, several transformations can be applied to the data to make the residuals more normally distributed. Common transformations include logarithmic, exponential, and Box-Cox transformations. Alternatively, a non-linear regression model or a robust regression model that does not rely on normality assumptions can be used.
Outlier detection in linear regression is a process of identifying and removing or adjusting outliers in a data set that may produce misleading results during linear regression analysis. Outliers are observations that are significantly different from the remaining observations in a dataset. Outliers may be the result of data entry errors, measurement errors, or extreme values that do not reflect the true underlying pattern of the data.
Outliers can have a large influence on the results of linear regression. For example, if we have a data set with 10 observations, and one of the observations is an outlier, it can affect the slope and intercept of the regression line. To detect and remove outliers, we can use the following methods:
Once outliers have been identified, they can be removed or adjusted. For example, if the outlier is due to data entry errors, it can be corrected. If the outlier is due to extreme values, it can be adjusted to a more reasonable value.
When selecting a model for linear regression, it is critical to consider the sort of information that's being utilized, the number of factors, and the reason of the examination.
Eventually, the sort of data, the number of variables, and the reason of the analysis ought to all be taken into thought when selecting a show for linear regression.
Applications of linear regression
Common applications of linear regression incorporate predicting future stock costs, estimating deals, anticipating client churn, understanding the affect of showcasing campaigns, and predicting housing costs. Linear regression can too be utilized to decide the connections between distinctive features in a dataset, and it is commonly utilized in predictive analytics and data mining. It can moreover be utilized to decide the affect of changes in inputs on a model's output, and to distinguish correlations between different variables.
The industry utilized linear regression to foresee the cost of a house given the highlights of the domestic. By preparing a model on a set of known domestic costs, the model was able to accurately predict the cost of a unused home based on its features. This model has been effectively connected within the housing market and has been utilized to assist buyers and venders make educated choices approximately the esteem of a particular home.
Answer: b. The strength of the linear relationship between two or more variables
Answer: c. To predict a continuous variable
Answer: a. It is not suitable for predicting non-linear relationships
Answer: c. To predict a continuous variable