How is the y-intercept of the best fit line calculated in linear regression?
The y-intercept of the best fit line in linear regression is calculated using the formula derived from the ordinary least squares (OLS) method. Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. The best fit line, also known as the regression line, is the line that minimizes the sum of squared residuals between the observed and predicted values.
To calculate the y-intercept, we first need to define the equation of the best fit line. In simple linear regression, where we have one independent variable, the equation takes the form:
y = mx + b
Here, 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 represents the change in y for a unit change in x.
To calculate the y-intercept, we need to estimate the values of m and b. The OLS method provides a way to estimate these values by minimizing the sum of squared residuals. The residual is the difference between the observed value of y and the predicted value of y based on the equation of the line.
Let's consider a simple example to illustrate the calculation of the y-intercept. Suppose we have a dataset with the following values:
x = [1, 2, 3, 4, 5] y = [3, 5, 7, 9, 11]
We can start by calculating the mean of x and y:
mean_x = (1 + 2 + 3 + 4 + 5) / 5 = 3
mean_y = (3 + 5 + 7 + 9 + 11) / 5 = 7
Next, we calculate the deviations from the mean for each data point:
deviation_x = [1 – 3, 2 – 3, 3 – 3, 4 – 3, 5 – 3] = [-2, -1, 0, 1, 2] deviation_y = [3 – 7, 5 – 7, 7 – 7, 9 – 7, 11 – 7] = [-4, -2, 0, 2, 4]
Then, we calculate the sum of the products of the deviations:
sum_product_deviations = (-2 * -4) + (-1 * -2) + (0 * 0) + (1 * 2) + (2 * 4) = 4 + 2 + 0 + 2 + 8 = 16
Next, we calculate the sum of the squared deviations of x:
sum_squared_deviations_x = (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 4 + 1 + 0 + 1 + 4 = 10
Finally, we can calculate the slope and the y-intercept:
m = sum_product_deviations / sum_squared_deviations_x = 16 / 10 = 1.6
b = mean_y – (m * mean_x) = 7 – (1.6 * 3) = 7 – 4.8 = 2.2
Therefore, the equation of the best fit line is y = 1.6x + 2.2, where the y-intercept is 2.2.
The y-intercept of the best fit line in linear regression is calculated using the OLS method. It involves estimating the slope and the y-intercept by minimizing the sum of squared residuals. The y-intercept represents the value of the dependent variable when the independent variable is zero. Understanding how to calculate the y-intercept is fundamental in interpreting and analyzing linear regression models.
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