Question

To find the best straight line fit to a set of data points \(\left(x_{n}, y_{n}\right)\) in the "least squares" sense means the following: Assume that the equation of the line is \(y=m x+b\) and verify that the vertical deviation of the line from the point \(\left(x_{n}, y_{n}\right)\) is \(y_{n}-\left(m x_{n}+b\right)\). Write \(S=\) sum of the squares of the deviations, substitute the given values of \(x_{n}, y_{n}\) to give \(S\) as a function of \(m\) and \(b,\) and then find \(m\) and \(b\) to minimize \(S\). Carry through this routine for the set of points: \((-1,-2),(0,0),(1,3) .\) Check your results by computer, and also computer plot (on the same axes) the given points and the approximating line.

Short answer

The best fit line is \(y = x + 1\). Solving the minimization gives \(m = 1\) and \(b = 1\).

Step-by-step solution

Define the line equation and deviations

The equation of the line is given by \(y = mx + b\). The vertical deviation of the line from a point \((x_n, y_n)\) is \(y_n - (m x_n + b)\).

Express the sum of squares of deviations

We need to find the sum of the squares of these deviations, \(S\). This is given by: \[S = \sum_{n=1}^{N} \left(y_n - (m x_n + b)\right)^2\]. For our data points \((-1, -2), (0, 0), (1, 3)\), we substitute \(x_n\) and \(y_n\) into this formula.

Substitute data points into S

Substitute the given data points \((-1, -2), (0, 0), (1, 3)\) into the equation for \(S\): \[S = (-2 - (m (-1) + b))^2 + (0 - (m (0) + b))^2 + (3 - (m (1) + b))^2 \]. Simplify this equation: \[S = (-2 + m - b)^2 + (0 - b)^2 + (3 - m - b)^2 \].

Expand and simplify S

Expand and simplify the equation: \[S = (m - b - 2)^2 + b^2 + (m + b - 3)^2 \]. This becomes: \[S = (m^2 - 2mb + 4m + b^2 - 4b + 4) + b^2 + (m^2 + 2mb - 6m + b^2 + 6b - 9)\]. Combine like terms to get: \[S = 2m^2 + 3b^2 - 2mb - 2m + 2b + 7\].

Take partial derivatives

To minimize \(S\), take the partial derivatives with respect to \(m\) and \(b\) and set them to zero. \[\frac{\partial S}{\partial m} = 4m - 2b - 2 = 0\] \[\frac{\partial S}{\partial b} = 6b - 2m + 2 = 0\].

Solve the equations

Solve the two equations simultaneously: \[4m - 2b - 2 = 0\] \[6b - 2m + 2 = 0\]. From the first equation, \(4m - 2b = 2\) or \(2m - b = 1\). From the second equation, \(6b - 2m = -2\) or \(3b - m = -1\). Solving the system of equations gives \(m = 1\) and \(b = 1\).

Verify results and plot

Check the calculations using a computer and plot the data points \((-1, -2), (0, 0), (1, 3)\) and the line \(y = x + 1\) on the same axes to verify the fit.

Key concepts

Linear Regression

Linear regression is a fundamental statistical method used to model the relationship between two variables by fitting a linear equation to observed data. The equation of the line is typically written as: \( y = mx + b \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) is the slope of the line, and \( b \) is the y-intercept. This method aims to find the best possible line that represents the data by minimizing the distance between the line and the data points. Essentially, in practical applications, linear regression helps us understand how changes in one variable are associated with changes in another variable.
For instance, if we wanted to predict a student's test score based on study hours, linear regression helps us find the best-fit line showing the expected score for various amounts of study time.

Sum of Squares

In linear regression, the sum of squares refers to the sum of the squared differences between the observed values and the values predicted by the regression line. This concept is crucial because it measures how well the model's predictions match the actual data. The formula for the sum of squares of deviations, \( S \), is: \[ S = \sum_{n=1}^{N} \left(y_n - (m x_n + b)\right)^2 \].
Here, \( N \) is the number of data points, \( y_n \) is the observed value, and \( x_n \) is the corresponding value predicted by the model. The goal is to find values of \( m \) and \( b \) that minimize \( S \), ensuring that the line fits the data as closely as possible. Minimizing the sum of squares guarantees that the overall error between the predicted line and the actual data is as small as possible.

Partial Derivatives

In the context of least squares fitting, partial derivatives are used to determine the minimum sum of squares, \( S \). To achieve this, we take the partial derivatives of \( S \) with respect to the slope, \( m \), and the intercept, \( b \), and set them to zero. This process finds the critical points of the function, helping us identify the values of \( m \) and \( b \) that minimize \( S \). The partial derivatives are:
\[\frac{\partial S}{\partial m} = 4m - 2b - 2 = 0\] \[\frac{\partial S}{\partial b} = 6b - 2m + 2 = 0\]
Solving these equations simultaneously gives us the optimal values for \( m \) and \( b \). This technique ensures that the fitted line has the smallest possible error when predicting the observed data, making it the best fit in the least squares sense.

Data Fitting

Data fitting involves finding a mathematical model that best represents a set of data points. In the case of linear regression, data fitting means finding the best straight line that approximates the relationship between the dependent and independent variables. By minimizing the sum of squares of deviations, we ensure this line accurately represents the observed data.
The process involves several steps:

• Formulating the linear equation \( y = mx + b \)
• Calculating the vertical deviations of the data points from the line
• Summing the squares of these deviations to form \( S \)
• Using partial derivatives to minimize \( S \) and find the optimal \( m \) and \( b \)

By performing these steps, we effectively fit the line to the data, capturing the underlying trend and allowing for predictions based on the model. This technique is widely used in various fields, including economics, biology, and engineering, for making informed decisions based on data trends.