The theoretical regression model:
\[ \underbrace{y_i}_{\text{Dependent Variable}} = \underbrace{\beta_0}_{\text{Intercept}} + \underbrace{\beta_1x_{i1} + \beta_2x_{i2} + \cdots + \beta_jx_{ij}}_{\text{Independent Variables}} + \underbrace{\epsilon_i}_{\text{Random Error}} \]
An estimated regression equation:
\[ \hat{y_i} = \hat{\beta_0} + \hat{\beta_1}x_{i1} + \hat{\beta_2}x_{i2} + \cdots + \hat{\beta_j}x_{ij} \]
The loss function minimized with the OLS procedure:
\[ \sum e_i^2 = \sum (y_i - \hat{y_i})^2 = \sum (y_i - \hat{\beta_0} - \hat{\beta_1}x_{i1} - \hat{\beta_2}x_{i2} - \cdots - \hat{\beta_j}x_{ij})^2 \]
General form:
\[ y_i = f(lat_i,long_i) + \epsilon_i \]
First degree:
\[ y_i = \beta_0 + \beta_1lat_i +\beta_2long_i + \epsilon_i \]
Second degree:
\[ y_i = \beta_0 + \beta_1lat_i^2 +\beta_2lat_i + \\ \beta_3lat_i \cdot long_i +\beta_4long_i + \beta_5long_i^2 + \epsilon_i \]
Expansion Method:
Use the trend surface function in place of a constant coefficient:
\[ y_i = f_{(lat_i, long_i)}^1 + f_{(lat_i, long_i)}^2 x_{i1} + \epsilon_i \]
\[ \begin{array}{l} f_{(lat_i, long_i)}^1 = \beta_{01} +\beta_{02}lat_i + \beta_{03}long_i\\ f_{(lat_i, long_i)}^2 = \beta_{11} +\beta_{12}lat_i + \beta_{13}long_i \end{array} \]
Geographically Weighted Regression:
Instead of fitting a regression model to all the observations, it fits to a subset of them each time based on the bandwidth defined.
It is like a spatial moving average; instead of calculating an average, we are fitting a regression model to each window.
\[ y_i = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + \cdots + \beta_jx_{ij} + \underbrace{\epsilon_i}_{\text{Random Error}} \]
The error term is no longer independent of each other but follows an autoregressive form:
\[ \epsilon_i = \lambda \sum_k w_{ik}\epsilon_k + u_i \]
Noting that the \(u_i\) is the pure random component now.
For lab activity submission, you need to at least conduct descriptive statistics.
Load the data and use the summary method or provide some descriptive plots like histograms.
mpg cyl disp hp
Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
Median :19.20 Median :6.000 Median :196.3 Median :123.0
Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0 2026 Zehui Yin