Question

Show that minimizing the sum of squared vertical distances between the fitted hyperplane and the y values is equivalent to solving min β (y − xβ)t (y − xβ):

a) Derive the formula for the sum of squared vertical distances.

b) Interpret the geometric representation of the hyperplane.

c) Analyze the relationship between the fitted hyperplane and y values.

d) Evaluate the optimization method for minimizing squared distances.

Answer 1

The sum of squared vertical distances can be minimized by solving the equation min β (y − xβ)t (y − xβ). The geometric representation of the hyperplane is a line in a higher-dimensional space that best fits the given data points. The fitted hyperplane represents the line of best fit for the given data points.

Step-by-step explanation:

The formula for the sum of squared vertical distances is:

SSE = (y - Xβ)T(y - Xβ)

Here, y represents the vector of y-values, X represents the matrix of x-values, and β represents the vector of coefficients for the fitted hyperplane.

The geometric representation of the hyperplane is a line in a higher-dimensional space that best fits the given data points. The line minimizes the sum of squared vertical distances between the data points and the line.

The fitted hyperplane represents the line of best fit for the given data points. It is determined by finding the coefficients β that minimize the sum of squared vertical distances between the data points and the hyperplane.

To evaluate and optimize the sum of squared distances, you can use various optimization methods such as solving the normal equations, gradient descent, or using matrix calculus methods.