Problem 85 - Calculus Single Variable

Question

Let \(P_{0}=\left(x_{0}, y_{0}\right)=(4,16),\left(x_{1}, y_{1}\right)=(1,2),\) and \(\left(x_{2}, y_{2}\right)=\) \((2,6) .\) Obtain an expression for the sum of the squares of the errors \(d_{1}^{2}+d_{2}^{2}\) associated with a line through \(P_{0}\) that has slope \(m .\) With \(m\) measured along the horizontal axis and \(d_{1}^{2}+d_{2}^{2}\) measured along the vertical axis, graph \(d_{1}^{2}+d_{2}^{2}\) in the viewing window \([4.5,5] \times[0,1.2] .\) Use your plot to find the slope of the regression line \(\mathcal{L}\) through \(P_{0}\). Now plot the sum of the absolute errors \(\mathrm{d}_{1}+\mathrm{d}_{2}\) in the viewing window \([3.5,6] \times[0,7] .\) Use your plot to find the value of \(m\) that minimizes \(d_{1}+d_{2}\). Finally, on the same coordinate axes, plot the three data points, the line through \(P_{0}\) that minimizes \(d_{1}+d_{2},\) and \(\mathcal{L}\).

Step-by-Step Solution

Answer

The slope of the regression line is the value of \( m \) that minimizes \( d_1^2 + d_2^2 \), as determined from the graph. The slope that minimizes \( d_1 + d_2 \) is found similarly from another graph.

Step 1: Understanding the Problem

We need to find the expression for the sum of the squares of errors for a line passing through point \( P_0 = (4, 16) \) with slope \( m \). The errors \( d_1 \) and \( d_2 \) are the vertical distances from the points \( (1, 2) \) and \( (2, 6) \) to this line.

Step 2: Equation of the Line

The line through point \( P_0 = (4, 16) \) with slope \( m \) is given by the equation \( y - 16 = m(x - 4) \). Rewrite it in slope-intercept form: \( y = mx - 4m + 16 \).

Step 3: Calculate Errors d1 and d2

Calculate the vertical distances (errors) from the points \( (1, 2) \) and \( (2, 6) \) to the line.For point \( (1, 2) \): $$d_1 = |2 - (m(1) - 4m + 16)| = |2 - (m - 4m + 16)| = |14 + 3m|.$$For point \( (2, 6) \): $$d_2 = |6 - (m(2) - 4m + 16)| = |6 - (2m - 4m + 16)| = |10 - 2m|.$$

Step 4: Sum of Squared Errors

The sum of squares of errors is: \( d_1^2 + d_2^2 = (14 + 3m)^2 + (10 - 2m)^2 \). This expands to: \( (14 + 3m)^2 = 196 + 84m + 9m^2 \) and \( (10 - 2m)^2 = 100 - 40m + 4m^2 \). Adding these gives \( d_1^2 + d_2^2 = 296 + 44m + 13m^2 \).

Step 5: Graphing the Error Expression

Graph the function \( d_1^2 + d_2^2 = 296 + 44m + 13m^2 \) using the window \([4.5, 5] \times [0, 1.2]\). Identify the minimum point to find the slope \( m \) of the regression line \( \mathcal{L} \).

Step 6: Sum of Absolute Errors

Express \( d_1 + d_2 = |14 + 3m| + |10 - 2m| \). Graph this function in the window \([3.5, 6] \times [0, 7]\) to find the value of \( m \) that minimizes \( d_1 + d_2 \).

Step 7: Plot Data Points and Lines

Plot the points \((4,16)\), \((1,2)\), and \((2,6)\) along with the two lines: one through \(P_0\) minimizing \( d_1 + d_2 \), and the regression line \( \mathcal{L} \). Adjust and analyze the slopes based on the previous graphs.

Key Concepts

Sum of Squared Errors

When analyzing data and trying to fit a line through a set of points, one standard method used is the sum of squared errors. This approach involves calculating the vertical distances (errors) from each data point to a regression line, squaring these distances, and then summing them up. The idea behind squaring the errors is to ensure that all errors are positive, since negative errors can cancel out positive ones, which doesn't accurately reflect the total error. Moreover, squaring gives more weight to larger errors, penalizing more significant deviations.For instance, in our solution, we calculated the errors for the points \((1, 2)\) and \((2, 6)\) as \(d_1 = |14 + 3m|\) and \(d_2 = |10 - 2m|\), respectively. These were then squared to get \( (14 + 3m)^2 \) and \( (10 - 2m)^2 \). Adding them together gives us the expression for the sum of squared errors: \(d_1^2 + d_2^2 = 296 + 44m + 13m^2\). This quadratic expression can be graphed to find the minimum point, which indicates the best fit slope for the line based on least squared errors.

Slope-Intercept Form

The slope-intercept form is a fundamental concept in linear algebra, used to describe a straight line in the coordinate plane. It's usually expressed as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.In our problem, we're given a point \(P_0 = (4, 16)\) and a slope \(m\), which we use to write the equation of the line passing through \(P_0\). By rearranging the basic line equation \(y - y_0 = m(x - x_0)\), the line can be expressed as: \[y = mx - 4m + 16\]. Here, \(-4m + 16\) acts as our effective y-intercept when considering its context based on point \(P_0\). Understanding how to transition from point-slope to slope-intercept form is vital as it forms the basis for calculating errors in relation to the regression line.

Absolute Errors

Absolute errors give a straightforward measure of discrepancies between the data points and the line. Unlike squared errors, absolute errors are just the raw differences in distances, ignoring whether the deviation is above or below the line.In the exercise, we calculate absolute errors for the points \((1, 2)\) and \((2, 6)\) as \( |14 + 3m| \) and \( |10 - 2m| \) respectively. The sum of these absolute errors is given by: \[d_1 + d_2 = |14 + 3m| + |10 - 2m|\].Plotting the absolute error function helps determine which slope \(m\) would minimize the overall error. This particularly wise method is often used in scenarios where large outliers might skew results if squared errors were used. Using absolute measures tends to provide a more balanced representation of fit.

Error Minimization

Error minimization is, at its core, the objective of regression analysis. It involves modifying the parameters (like the slope \(m\)) to reduce the total error, improving the fit of the line to the data.In the context of our problem, the regression line's effectiveness is verified by obtaining the slope that minimizes the sum of squared errors. Similarly, the line that minimizes the sum of absolute errors is determined by plotting and analyzing the graph of these errors.
By visually assessing these plots, one can identify optimal parameters that result in the smallest total error. In practical terms:

• For squared errors, this often involves finding the low point of a parabola formed by plotting these errors.
• For absolute errors, it means finding the point where changes in slope result in the least increase or decrease in this total absolute sum.

The process highlights the balance between precision and flexibility in regression tasks.