What are the formulas used to calculate the slope and y-intercept in linear regression?
Linear regression is a widely used statistical technique that aims to model the relationship between a dependent variable and one or more independent variables. It is a fundamental tool in the field of machine learning for predicting continuous outcomes. In this context, the slope and y-intercept are essential parameters in linear regression as they capture the relationship between the independent and dependent variables.
To understand how to calculate the slope and y-intercept in linear regression, let's consider a simple case with one independent variable, often referred to as simple linear regression. The goal is to fit a straight line to the data that minimizes the sum of the squared differences between the observed and predicted values.
The slope, often denoted as "m," represents the change in the dependent variable for a unit change in the independent variable. It quantifies the steepness or direction of the line. The formula to calculate the slope in simple linear regression is:
m = Σ((xi – x̄)(yi – ȳ)) / Σ((xi – x̄)²)
where:
– Σ denotes the sum of the values over all data points
– xi represents the value of the independent variable for the ith data point
– yi represents the value of the dependent variable for the ith data point
– x̄ is the mean of the independent variable values
– ȳ is the mean of the dependent variable values
The numerator of the formula calculates the covariance between the independent and dependent variables, while the denominator calculates the variance of the independent variable. By dividing the covariance by the variance, we obtain the slope of the regression line.
Moving on to the y-intercept, often denoted as "b," it represents the value of the dependent variable when the independent variable is zero. In other words, it is the point where the regression line intersects the y-axis. The formula to calculate the y-intercept in simple linear regression is:
b = ȳ – m * x̄
where:
– ȳ is the mean of the dependent variable values
– m is the slope of the regression line
– x̄ is the mean of the independent variable values
By substituting the values into the formula, we can calculate the y-intercept.
To illustrate these concepts, let's consider a simple example. Suppose we have a dataset of housing prices (dependent variable) and the corresponding sizes of the houses (independent variable). We want to fit a regression line to predict the price of a house based on its size.
Using the provided formulas, we can calculate the slope and y-intercept. Let's assume we have the following data:
House Size (x): [1000, 1500, 2000, 2500] Price (y): [300000, 450000, 500000, 550000]
First, we calculate the means of the independent and dependent variables:
x̄ = (1000 + 1500 + 2000 + 2500) / 4 = 1750
ȳ = (300000 + 450000 + 500000 + 550000) / 4 = 450000
Next, we calculate the covariance and variance:
Σ((xi – x̄)(yi – ȳ)) = (1000 – 1750) * (300000 – 450000) + (1500 – 1750) * (450000 – 450000) + (2000 – 1750) * (500000 – 450000) + (2500 – 1750) * (550000 – 450000) = -62500000
Σ((xi – x̄)²) = (1000 – 1750)² + (1500 – 1750)² + (2000 – 1750)² + (2500 – 1750)² = 3500000
Using these values, we can calculate the slope:
m = -62500000 / 3500000 = -17.857
Finally, we calculate the y-intercept:
b = 450000 – (-17.857) * 1750 = 78214.286
Therefore, the regression line for predicting house prices based on size is given by:
Price = -17.857 * Size + 78214.286
The formulas used to calculate the slope and y-intercept in linear regression are:
Slope (m) = Σ((xi – x̄)(yi – ȳ)) / Σ((xi – x̄)²)
Y-intercept (b) = ȳ – m * x̄
These formulas allow us to estimate the relationship between the independent and dependent variables and make predictions based on the fitted regression line.
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