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# Linear Regression Online Courses

Linear regression is a fundamental and extensively used type of predictive analysis. Learn Linear Regression Programs from the World’s Top Universities.

## Linear Regression Course Overview

Linear regression is a fundamental and extensively used type of predictive analysis.

Primarily, the linear regression programs inspect two things:

(i) If a set of predictor variables perform well in predicting a dependent variable (outcome variable)?

(ii) Which variables are significant predictors of the outcome variable, and how do they (specified by the amount and sign of the beta estimates) influence the outcome variable?

The above estimates help illustrate the relationship between a dependent variable and one or multiple independent variables.

The standard form of the linear regression equation comprising a dependent and an independent variable is:

y = b*x + c

here, y is the value of the estimated dependent variable

b is the regression coefficient

x is the value of the independent variable

c is constant

If you want to pursue linear regression training, you need to develop your mindset to correlate linear regression with real-life examples. Most of the linear regression classes and linear regression courses impart training with practical examples.The linear regression online course can help you conveniently access each module.

Let’s take a practical example to understand its meaning.
Suppose we have a dataset with graphics card sizes and the price of these cards. It is assumed that the dataset includes two features, i.e., memory size and price. The more graphics memory we buy for a computer, the more will be the cost.

The ratio of graphics memory to cost may differ between models of graphics cards and manufacturers. The data trends in the linear regression plot begin from the bottom left side and end at the upper right. The bottom left shows graphics cards with smaller capacities and lower prices. The upper right shows those graphics cards with higher capacity and high prices.

Suppose we use X-axis for the graphics card memory and Y-axis for the cost. The line representing a relationship between X and Y variables begins from the bottom left corner and runs up to the upper right.

The regression model shows a linear function between these variables that best explains their relationship. The assumption is a specific combination of the input variables can measure the value of Y. Drawing a line across the points in the graph shows the relationship between the input variables and the target variables.

This line best describes the relationship existing between these variables. For example, they can be related as when the value of X increases by 2, the value of Y increases by 1. The linear regression function aims to plot an optimal regression line that perfectly fits the data.

In each linear regression equation, there will be errors or deviations. The Least-Squares technique mentions the solution for minimising errors or squares of deviations. Usually, this method is implemented in data fitting. The use of linear regression Google Sheets helps you to determine the Least Squares error accurately.

The optimal result intends to decrease the residuals or sum of squared errors showing the differences between the experimental value and the equivalent fitted value stated in the model.

To find least squares, first, we define a linear relationship between the independent variable (X) and dependent variable (Y). It is vital to come up with the formula to determine the sum of errors' squares. Ultimately, this formula helps to find out the variation in observed data.

The linear relationship between these variables is:
Y = c + mX

The aim is to find the values of c and m to determine the minimum error for the specified dataset.

When employing the Least Squares method, we aim to minimise the error. So, now we must proceed with calculating the error. The loss function in machine learning indicates the difference between the actual value and the predicted value.

Let’s use the Quadratic Loss Function to measure the error. Its formula is:

c = y’ –mx’

In the equation of m, x’ shows the mean of all the values in the input X. y’ shows the mean of all the values in output variable Y. Users can retrieve further predictions by implementing corresponding linear regression programs in Python.

Using Python for further analysis of the Least Square method may not yield high accuracy as we simply take a straight line and force it to fit into the specified data optimally. However, it can be helpful to gauge the magnitude of the real value. It serves as an excellent first step for novices in Machine Learning. The Google Sheets linear regression can help measure the least square error.

When we fit a set of points to a regression line, it is assumed that some linear relationship exists between X and Y. The regression line lets you predict the target variable Y for an input value of X.

The following equation corresponds to this:

µY|X = α0 + α1X1

However, for any particular observation, there can be a deviation between the actual value of Y and the predicted value. These deviations are known as errors or residuals. The more efficiently the line fits the data, the smaller the error will be.

But how to find the regression line that best fits these data? Does it help calculate slope and intercept values for the particular regression line?

Finding a line capable of minimising model errors to fit data to the line manually is necessary. However, when you include data to the line, specific errors will be positive, while others will be negative. This means that few actual values would be greater than their predicted values. On the other hand, some of the actual values would also be lower than the predicted values.

When we add all the errors, the sum comes out to zero. So, the challenge is how to determine the overall error? The answer is squaring the errors and finding a line that minimises the sum of the squared errors.

∑e2=∑(Yt−Y’t)2

Here, e = error

Yt - Y’t = deviation between the actual and predicted value of the target variable.

With the above equation, the Least Squares method determines the values of slope and intercept coefficient. These coefficients will minimise the total squared errors. This method makes the sum of the square of the errors as tiny as possible. Hence, the total is the least likely value when all errors are squared and added.

In linear regression, the regression coefficients let you predict an unknown variable's value with a known variable's help. The variables in a regression equation get multiplied by some magnitudes. These magnitudes are regression coefficients. Based on the regression coefficients, the linear regression plots the best-fitted line.

This section helps you thoroughly learn regression coefficients, their formula, and their interpretation.

Regression coefficients are approximations of specific unknown parameters to determine the relationship between a predictor variable and the actual variable. These coefficients help predict the value of an unknown variable with the help of a known variable.

Linear regression analysis measures how a change in an independent variable affects the dependent variable using the best-fitted straight line.

#### Formula to calculate values of Regression Coefficients:

Before finding the values of regression coefficients, you must check whether the variables adhere to a linear relationship or not. You can use the correlation coefficient and interpret the equivalent value to check this.

Linear regression aims to find the straight line equation that establishes the relationship between two or multiple variables. Suppose we have a simple regression equation: y = 5x + 3. Here, 5 is the coefficient, x is the predictor, and 3 is the constant term.

According to the equation of the best-fitted line Y = aX + b, the formulas for the regression coefficients are:

Use this equation to find the coefficient of X:
n is the number of data points in the specified data sets; its formula is:

Now insert the values of regression coefficients in Y= n + mX

#### Interpretation of Regression Coefficients:

Understanding the nature of the regression coefficient assists you in predicting the unknown variable. It gives an idea of the amount the dependent variable changes with a unit change in an independent variable.

If the sign of regression coefficients is positive, there is a direct relationship between these variables. So, if the independent variable increases, the dependent variable increases, and vice versa.

If the sign of regression coefficients is negative, there is an indirect relationship between these variables. So, if the independent variable increases, the dependent variable decreases, and vice versa.

Using regression Google Sheets can provide the exact interpretation of regression coefficients.

Linear regression is a statistical technique to comprehend the relationship between variables x and y. Before conducting linear regression, make sure the below assumptions are met:
If there is a violation of assumptions, linear regression results can be unreliable.

Every assumption discussed below explains how to determine linear regression if it's met and steps to perform if the assumption violates.

#### Assumption-1: Linear Relationship

It assumes the existence of a linear relationship between the dependent variable (y) and the independent variable (x).

The easiest way to determine assumption fulfilment is to prepare a scatter plot of x vs. y. It helps you visually know the linear relationship between these variables. If the plot shows points falling across a straight line, there is some kind of linear relationship between them, and this assumption is fulfiled.

Solutions to try if this assumption is violated:

When you prepare a scatter plot of x and y values, notice no linear relationship exists between them, you have the following options:

i. Implement a non-linear transformation to the independent or dependent variables. You can implement non-linear transformation using log, square root, or reciprocal of the independent or dependent variables.

ii. Add an independent variable to the model.

#### Assumption-2: Independence

In this assumption, the residuals are independent. There is zero correlation between successive residuals in the time series data. It implies residuals do not steadily grow more prominent with time.

Observing a residual time series plot is the easiest way to check this assumption fulfilment, showing a graph of residuals vs. time. Most of the residual autocorrelations must be inside the 95% confidence bands close to zero. These are present at approx. +/- 2 above the square root of n (where n is the sample size). The Durbin-Watson test also helps you check the fulfilment of this assumption.

Solutions to try if this assumption is violated:

Here are a few solutions you can try based on how this assumption is violated:

• If the serial correlation is positive, add lags of the dependent or independent variable to a particular model.
• For the serial correlation to be negative, ensure no variables are over-differenced.
• For periodic correlation, add periodic dummy variables to the model.

#### Assumption-3: Homoscedasticity

In this assumption, the residuals bear constant variance at each level of x. The existence of heteroscedasticity in a regression analysis makes it difficult to rely on the analysis results. Particularly, heteroscedasticity enlarges the difference in the regression coefficient estimates. There are high odds for the regression model to state that a term in the model is statistically substantial, although it’s not.

The easiest way to recognise heteroscedasticity is to create a fitted value vs. residual plot. After fitting a regression line to a data set, you can prepare a scatterplot representing the model’s fitted values vs. residuals of corresponding values. With the increase in the fitted values, the residuals spread out more, and the cone shape shows the existence of heteroscedasticity.

Solutions to try if this assumption is violated:

i. Transformation of the dependent variable:
The common way of transforming the dependent variable is to take its log. For example, suppose we use population size to predict the total number of fruit shops in a town. Here, the population size is the independent variable, and the number of fruit shops is a dependent variable. We can use the log of the dependent variable (population size) instead of the dependent variable itself to predict the number of fruit shops. Following this approach usually eliminates heteroscedasticity.

ii. Weighted regression:

This form of regression allocates weight to every data point depending on the variance of its fitted value. It provides small weights to those data points bearing higher variances, decreasing their squared residuals' value. Overall, proper weights can discard heteroscedasticity.

iii. Redefine the dependent variable:

A typical method to redefine the dependent variable is using a rate instead of the raw value. Let’s consider the example discussed in solution-i. Rather than using the population size to predict the number of fruit shops in a town, use population size to indicate the number of fruit shops per capita.

In many cases, this approach decreases the variability between more significant populations as we measure the number of fruit shops per person instead of the absolute amount of fruit shops

Assumption-4: Normality

The model’s residuals are normally distributed.

How to determine Normality assumption:

i. Visual testing using Q-Q plots.

A Q-Q plot (quantile-quantile plot) is helpful to know whether a model’s residuals obey a normal distribution. The normality assumption is fulfiled when the points on the plot coarsely create a straight diagonal line.

ii. Using formal statistical tests:

This solution checks the normality assumption using formal statistical tests such as Shapiro-Wilk, Jarque-Barre, Kolmogorov-Smirnov, or D’Agostino-Pearson. These tests are sensitive to huge sample sizes. They usually determine that the residuals are not normal when the sample size is big. Therefore, using graphical methods like the Q-Q plot to test this assumption is better.

Solutions to try if this assumption is violated:

Firstly, make sure any outliers don’t lay an immense influence on the distribution. If outliers are present, you need to confirm their real values; no data entry errors are allowed.

You can implement a non-linear transformation to the dependent or independent variable. For example, you can apply the dependent or independent variable's square root, log, or reciprocal.

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