Problem 55 - Calculus Early Transcendentals

Question

Suppose that a scientist has reason to believe that two quan- tities \(x\) and \(y\) are related linearly, that is, \(y=m x+b,\) at least approximately, for some values of \(m\) and \(b\) . The scientist performs an experiment and collects data in the form of points \(\left(X_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) and then plots these points. The points don't lie exactly on a straight line, so the scientist wants $$ m \text { and } b \text { so that the line } y=m x+b$$ points as well as possible. (See the figure.) Let \(d_{i}=y_{i}-\left(m x_{i}+b\right)\) be the vertical deviation of the point \(\left(x_{i}, y_{i}\right)\) from the line. The method of least squares determines \(m\) and \(b\) so as to minimize \(\Sigma_{1-1}^{n} d_{i}^{2}\) , the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when $$\begin{array}{c}{m \sum_{i=1}^{n} x_{i}+b n=\sum_{i=1}^{n} y_{i}} \\ {m \sum_{i=1}^{n} x_{i}^{2}+b \sum_{i=1}^{n} x_{i}=\sum_{i=1}^{n} x_{i} y_{i}}\end{array}$$ Thus the line is found by solving these two equations in the two unknowns \(m\) and \(b\) . See Section 1.2 for a further discus- sion and applications of the method of least squares.)

Step-by-Step Solution

Answer

The best fit line is found using \(m\sum x_i + bn = \sum y_i\) and \(m\sum x_i^2 + b\sum x_i = \sum x_i y_i\).

Step 1: Understand the Least Squares Method

The method of least squares is used to find the line that minimizes the total squared distance from the points to the line. This involves finding values for \(m\) and \(b\) in the equation \(y = mx + b\) that minimize \(\sum_{i=1}^{n} d_i^2\), where \(d_i = y_i - (mx_i + b)\) is the difference between the actual \(y_i\) value and the predicted value \(mx_i + b\).

Step 2: Set Up the Objective Function

The goal is to minimize the function \(S = \sum_{i=1}^{n} (y_i - (mx_i + b))^2\). This can be expanded to \(S = \sum_{i=1}^{n} (y_i^2 - 2y_i(mx_i + b) + (mx_i + b)^2)\).

Step 3: Take Partial Derivatives

To find the minimum of the function \(S\), take the partial derivative of \(S\) with respect to \(m\) and \(b\). The partial derivative with respect to \(m\) is \(\frac{\partial S}{\partial m} = -2\sum x_i(y_i - mx_i - b)\). The partial derivative with respect to \(b\) is \(\frac{\partial S}{\partial b} = -2\sum (y_i - mx_i - b)\).

Step 4: Set Partial Derivatives to Zero

To find the minimum of \(S\), set the partial derivatives equal to zero: \(-2\sum x_i(y_i - mx_i - b) = 0\) and \(-2\sum (y_i - mx_i - b) = 0\).

Step 5: Simplify the Equations

Simplify the equations from Step 4: \(\sum x_i y_i = m\sum x_i^2 + b\sum x_i\) and \(\sum y_i = m\sum x_i + bn\), where \(n\) is the number of data points.

Step 6: Rearrange to Find Equations for m and b

The simplified equations are: 1) \(m \sum x_i + b n = \sum y_i\), and 2) \(m \sum x_i^2 + b \sum x_i = \sum x_i y_i\). These two equations can be solved simultaneously to find \(m\) and \(b\).

Step 7: Conclusion

Having derived the two equations based on the method of least squares, you can solve them to find the values of \(m\) and \(b\) that lead to the best-fitting line according to the least squares criterion.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between two variables. This relationship is established when one variable, typically the dependent variable (often denoted as \(y\)), is assumed to be linearly dependent on another variable, the independent variable (often denoted as \(x\)).
For example, in the context of the exercise, the scientist wants to establish a relationship between \(x\) and \(y\) using the equation: \[y = mx + b\]Here, \(m\) represents the slope of the line, while \(b\) is the intercept on the y-axis. Linear regression helps determine these coefficients \(m\) and \(b\) so that the equation models the data closely. By plotting collected data points on a graph, a line can be fitted through the data points to help make predictions or understand relationships. Some key points to note about linear regression include:- It is one of the simplest forms of regression analysis.- The relationship modeled by linear regression is assumed to be linear.- It uses past data to predict or analyze trends, making future predictions possible.

Best Fit Line

In the context of linear regression, the 'best fit line' is the line that most closely approximates the data points in a scatter plot. This line minimizes the discrepancies—specifically the vertical deviations—between the actual data points and the predicted points on the line. The goal is for the line to have as little total deviation from the actual points as possible, so it "fits" the data well. In the scenario described in the exercise, this is achieved through minimizing the sum of the squares of deviations:- Each point on the graph has a vertical deviation \(d_i = y_i - (mx_i + b)\).- Squaring these deviations, \(d_i^2\), ensures they are non-negative and prioritizes larger deviations more, providing a balanced measurement of accuracy.The least squares method is commonly used to determine this best fit line because:- It mathematically calculates coefficients \(m\) and \(b\) that minimize the sum of \(d_i^2\).- This minimal sum helps in obtaining a line that, on average, offers the best prediction.

Minimization Problem

The minimization problem in the least squares method involves finding the values of \(m\) and \(b\) such that the sum of the squared deviations between actual and predicted values is the smallest possible. The primary function that needs to be minimized is:\[S = \sum_{i=1}^{n} (y_i - (mx_i + b))^2\]Here's how this minimization process generally works:- An objective function \(S\) is formulated, representing the sum of squared differences.- Partial derivatives of \(S\) with respect to \(m\) and \(b\) are calculated to find points of potential minima.- By setting these partial derivatives equal to zero, equations for optimizing \(m\) and \(b\) are derived, as shown in the completed exercise.Solving these equations yields the values for \(m\) and \(b\), which fit the regression line most closely according to the least squares criterion. This process ensures the line is optimized to minimize the prediction errors across all data points and adheres to the minimization principle of the least squares method.