Problem 10 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

In the method of least squares we choose \(\alpha\) and \(\beta\) in such a way that the sum of squared residuals \(S(\alpha, \beta)\) is minimal. Since the \(i\) th term in this sum is the squared vertical distance from \(\left(x_{i}, y_{i}\right)\) to the regression line \(y=\alpha+\beta x\), one might also wonder whether it is a good idea to replace this squared distance simply by the distance. So, given a bivariate dataset $$ \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) $$ choose \(\alpha\) and \(\beta\) in such a way that the sum $$ A(\alpha, \beta)=\sum_{i=1}^{n}\left|y_{i}-\alpha-\beta x_{i}\right| $$ is minimal. We will investigate this by a simple example. Consider the following bivariate dataset: $$ (0,2),(1,2),(2,0) $$ \(22.5\) Exercises 339 a. Determine the least squares estimates \(\hat{\alpha}\) and \(\hat{\beta}\), and draw in one figure the scatterplot of the data and the estimated regression line \(y=\hat{\alpha}+\hat{\beta} x\). Finally, determine \(A(\hat{\alpha}, \hat{\beta})\). b. One might wonder whether \(\hat{\alpha}\) and \(\hat{\beta}\) also minimize \(A(\alpha, \beta)\). To investigate this, choose \(\beta=-1\) and find \(\alpha\) 's for which \(A(\alpha,-1)

Step-by-Step Solution

Answer

\(\hat{\alpha} = 2\), \(\hat{\beta} = -1\); minimal \(A(\alpha, \beta)\) occurs at \(\alpha = 2\), \(\beta = -1\).

Step 1: Calculate Least Squares Estimates

For the dataset (0,2), (1,2), and (2,0), we calculate the least squares estimates \(\hat{\alpha}\) and \(\hat{\beta}\). The formulas used are \(\hat{\beta} = \frac{n\sum{x_{i}y_{i}} - \sum{x_{i}}\sum{y_{i}}}{n\sum{x_{i}^{2}} - (\sum{x_{i}})^{2}}\) and \(\hat{\alpha} = \frac{\sum{y_{i}} - \hat{\beta}\sum{x_{i}}}{n}\). Substitute the dataset values into these formulas to find \(\hat{\alpha} = 2\) and \(\hat{\beta} = -1\).

Step 2: Plot Scatterplot and Regression Line

Plot the scatterplot for the points: (0,2), (1,2), and (2,0). Using \(\hat{\alpha} = 2\) and \(\hat{\beta} = -1\), the regression line is \(y = 2 - x\). Draw this line on the scatterplot to visualize how well it fits the data.

Step 3: Calculate the Sum of Absolute Errors

Calculate the sum of absolute errors using \(A(\hat{\alpha}, \hat{\beta}) = \sum_{i=1}^{n}|y_{i} - \hat{\alpha} - \hat{\beta}x_{i}|\). Substitute \(\hat{\alpha} = 2\) and \(\hat{\beta} = -1\) into the formula: \(|2 - 2 - 0(2)| + |2 - (2 - 1)(1)| + |0 - (2 - 2)| = 0 + 1 + 0 = 1\).

Step 4: Investigate New Minimum with β=-1

Now set \(\beta = -1\) and find the value of \(\alpha\) that minimizes \(A(\alpha, -1)\). Plug \(\beta = -1\) into the formula: \(A(\alpha, -1) = |2 - \alpha| + |2 - 1 - \alpha| + |0 + 1 - \alpha|\). Simplifying, this becomes: \(|2 - \alpha| + |1 - \alpha| + |1 - \alpha|\). Test different \(\alpha\) values (e.g., \(\alpha=1\), \(\alpha=2\)) to find that \(\alpha = 2\) gives the minimum \(A(\alpha, -1)\).

Step 5: Minimize A(α, β) Overall

To minimize \(A(\alpha, \beta)\) overall, consider the partial derivatives of \(A(\alpha, \beta)\) with respect to \(\alpha\) and \(\beta\) and apply optimization techniques. By testing sufficient possibilities for \(\alpha\) and \(\beta\), find that for this dataset, \((\alpha, \beta) = (2, -1)\) seems to be the minimal configuration.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In its simplest form, which is simple linear regression, it examines the linear relationship between two variables. The goal is to establish a line, often called the line of best fit, that best predicts the dependent variable from the independent variable.

The equation of this line is typically expressed as:\[ y = \alpha + \beta x \]where:

• \( y \) is the dependent variable (the outcome you are predicting),
• \( x \) is the independent variable (the predictor),
• \( \alpha \) represents the y-intercept (the value of \( y \) when \( x \) is 0), and
• \( \beta \) denotes the slope of the line (how much \( y \) changes for a one-unit increase in \( x \)).

The least squares method is typically used to find the values of \( \alpha \) and \( \beta \) that minimize the sum of the squared differences between the observed values and the values predicted by the model. This is why it's often referred to as the "least squares" method. Essentially, the lesser these differences, the better the line fits the data.

In the exercise, we considered the dataset \((0, 2), (1, 2), (2, 0)\) and calculated \( \hat{\alpha} = 2 \) and \( \hat{\beta} = -1 \) using the least squares method. The line of best fit for this dataset is thus \( y = 2 - x \).

Bivariate Data

Bivariate data involves pairs of linked observations. It is used to analyze the relationship between two variables—each data point contains a value for the first variable and a value for the second variable.
Understanding bivariate data is crucial for many fields, such as economics, biology, and social sciences, as it helps in understanding how changes in one variable are associated with changes in another. The association can be visualized through scatterplots, where one variable is plotted on the x-axis and another on the y-axis.

For the set \((x_1, y_1), (x_2, y_2), ..., (x_n, y_n)\), bivariate data in our exercise, we have:

• The points \((0, 2), (1, 2), (2, 0)\) representing the x and y values,
• These points are plotted on a scatterplot to visually assess potential relationships.

In linear regression, as seen in this exercise, the goal is to determine a linear relationship between these two variables. Drawing a regression line can help reveal trends and forecast future data points. Each point's vertical distance from this line is what the least squares method aims to minimize.

Parameter Estimation

Parameter estimation in linear regression involves determining the coefficients \( \alpha \) and \( \beta \) that define the best-fit line for a given dataset. This process finds the most suitable parameters that minimize the residuals, i.e., the differences between observed and predicted values.

The least squares method is often employed to achieve parameter estimation. This involves calculating the sum of squared residuals, which is minimized by finding the optimal values of \( \alpha \) and \( \beta \). In more advanced scenarios, alternative techniques may be used to find the values that minimize the sum of absolute errors, as explored in this exercise.
For different datasets, parameter estimation might result in different values. Such was the case in the exercise, where further minimization of the sum of absolute errors \( A(\alpha, \beta) \) was explored by altering \( \beta \) and investigating \( \alpha \).-
After experimentation, it was observed that even when \( \beta \) was fixed at -1, varying \( \alpha \) revealed that \( \alpha = 2 \) led to further minimization of errors, confirming the suitability of these parameter choices for the given dataset. Understanding and estimating these parameters accurately allows for better predictions and insights into the relationship captured by the model.