2. Using the linear model of a researcher calculated the hat-matrix H = X(X'X)-¹X'= Life of [0.2516 0.1101 0.3931 0.1101 0.2044 0.0157 0.3931 0.1572 0.7704 0.0535 0.2421 -0.1352 0.1101 0.2044 0.0157 0.0818 0.2232 -0.0597 10 Kes y = B₁ + B₁x + €, H and the residual e using 6 observations. 0.0534 0.1101 0.0818 0.2421 0.2044 0.2233 -0.1352 0.0157 -0.0597 0.3176 0.2421 0.2233 0.2421 0.2044 0.22327 0.2799 0.2233 0.2516 with 1 at Dund w G vv Ull co uized residua (d) Identify observations with large cook's distance value. e = -0.02] 0.12 -0.56 0.88 -0.01 -0.07
2. Using the linear model of a researcher calculated the hat-matrix H = X(X'X)-¹X'= Life of [0.2516 0.1101 0.3931 0.1101 0.2044 0.0157 0.3931 0.1572 0.7704 0.0535 0.2421 -0.1352 0.1101 0.2044 0.0157 0.0818 0.2232 -0.0597 10 Kes y = B₁ + B₁x + €, H and the residual e using 6 observations. 0.0534 0.1101 0.0818 0.2421 0.2044 0.2233 -0.1352 0.0157 -0.0597 0.3176 0.2421 0.2233 0.2421 0.2044 0.22327 0.2799 0.2233 0.2516 with 1 at Dund w G vv Ull co uized residua (d) Identify observations with large cook's distance value. e = -0.02] 0.12 -0.56 0.88 -0.01 -0.07
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Transcription for Educational Website:
---
**Linear Model Analysis**
**2. Using the Linear Model**
The linear model under consideration is given by:
\[ y = \beta_0 + \beta_1 x + \epsilon, \]
where a researcher calculated the hat matrix \( H \) and the residual vector \( e \) using 6 observations.
The calculation yields:
\[
H = X(X'X)^{-1}X' =
\begin{bmatrix}
0.2516 & 0.1101 & 0.3931 & 0.0534 & 0.1101 & 0.0818 \\
0.1101 & 0.2044 & 0.0157 & 0.2421 & 0.2044 & 0.2233 \\
0.3931 & 0.1572 & 0.7704 & -0.1352 & 0.0157 & -0.0597 \\
0.0535 & 0.2421 & -0.1352 & 0.3176 & 0.2421 & 0.2233 \\
0.1101 & 0.2044 & 0.0157 & 0.2421 & 0.2044 & 0.2233 \\
0.0818 & 0.2232 & -0.0597 & 0.2799 & 0.2233 & 0.2516
\end{bmatrix},
\]
and the residuals are:
\[
e =
\begin{bmatrix}
-0.02 \\
0.12 \\
-0.56 \\
0.88 \\
-0.01 \\
-0.07
\end{bmatrix}
\]
**Tasks:**
(b) & (c) [Unclear: the text was obscured but likely pertains to utilizing the information from \( H \) and \( e \) for further statistical analysis.]
(d) Identify observations with a large Cook's distance value.
**Note:**
The hat matrix \( H \) is crucial in regression diagnostics and influences the leverage of observations. Residuals \( e \) help in assessing the fitting of the model, while Cook’s distance is essential for identifying influential data points. Text in some parts was obscured, highlighting the importance of complete data visibility for thorough analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe168234c-8641-43ba-b22b-28f74f97e6ae%2F88d0eefc-c0c0-401d-a61a-d40f0f03bd37%2Fovf2rpl_processed.png&w=3840&q=75)
Transcribed Image Text:Transcription for Educational Website:
---
**Linear Model Analysis**
**2. Using the Linear Model**
The linear model under consideration is given by:
\[ y = \beta_0 + \beta_1 x + \epsilon, \]
where a researcher calculated the hat matrix \( H \) and the residual vector \( e \) using 6 observations.
The calculation yields:
\[
H = X(X'X)^{-1}X' =
\begin{bmatrix}
0.2516 & 0.1101 & 0.3931 & 0.0534 & 0.1101 & 0.0818 \\
0.1101 & 0.2044 & 0.0157 & 0.2421 & 0.2044 & 0.2233 \\
0.3931 & 0.1572 & 0.7704 & -0.1352 & 0.0157 & -0.0597 \\
0.0535 & 0.2421 & -0.1352 & 0.3176 & 0.2421 & 0.2233 \\
0.1101 & 0.2044 & 0.0157 & 0.2421 & 0.2044 & 0.2233 \\
0.0818 & 0.2232 & -0.0597 & 0.2799 & 0.2233 & 0.2516
\end{bmatrix},
\]
and the residuals are:
\[
e =
\begin{bmatrix}
-0.02 \\
0.12 \\
-0.56 \\
0.88 \\
-0.01 \\
-0.07
\end{bmatrix}
\]
**Tasks:**
(b) & (c) [Unclear: the text was obscured but likely pertains to utilizing the information from \( H \) and \( e \) for further statistical analysis.]
(d) Identify observations with a large Cook's distance value.
**Note:**
The hat matrix \( H \) is crucial in regression diagnostics and influences the leverage of observations. Residuals \( e \) help in assessing the fitting of the model, while Cook’s distance is essential for identifying influential data points. Text in some parts was obscured, highlighting the importance of complete data visibility for thorough analysis.
![**Cook's Distance Measure**
- **Purpose**: Detects influential observations that are outliers from either \( x \) (hat value) or \( y \) (studentized residual).
- **Label in R**: Known as "cooks.distance".
**Description**:
- \( r_i \) represents the \( i^{th} \) standardized residual.
- \( h_{ii} \) represents the \( i^{th} \) hat value.
- The \( i^{th} \) Cook’s Distance (\( D_i \)) is defined by the formula:
\[
D_i = \frac{{r_i^2}}{{k + 1}} \times \frac{{h_{ii}}}{{1 - h_{ii}}}
\]
- \( r_i^2 \) is the squared \( i^{th} \) standardized residual.
- \( h_{ii} \) is the \( i^{th} \) diagonal entry of the hat matrix.
- **Interpretation**:
- The formula indicates that \( D_i \) is large if either \( r_i \) or \( h_{ii} \) is large.
- There is no significant test for \( D_i \), but an observation is considered generally influential if:
\[
D_i > \frac{4}{n - (k + 1)}
\]
This measure is essential in regression analysis to identify points that can disproportionately influence the estimated parameters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe168234c-8641-43ba-b22b-28f74f97e6ae%2F88d0eefc-c0c0-401d-a61a-d40f0f03bd37%2F37w8luj_processed.png&w=3840&q=75)
Transcribed Image Text:**Cook's Distance Measure**
- **Purpose**: Detects influential observations that are outliers from either \( x \) (hat value) or \( y \) (studentized residual).
- **Label in R**: Known as "cooks.distance".
**Description**:
- \( r_i \) represents the \( i^{th} \) standardized residual.
- \( h_{ii} \) represents the \( i^{th} \) hat value.
- The \( i^{th} \) Cook’s Distance (\( D_i \)) is defined by the formula:
\[
D_i = \frac{{r_i^2}}{{k + 1}} \times \frac{{h_{ii}}}{{1 - h_{ii}}}
\]
- \( r_i^2 \) is the squared \( i^{th} \) standardized residual.
- \( h_{ii} \) is the \( i^{th} \) diagonal entry of the hat matrix.
- **Interpretation**:
- The formula indicates that \( D_i \) is large if either \( r_i \) or \( h_{ii} \) is large.
- There is no significant test for \( D_i \), but an observation is considered generally influential if:
\[
D_i > \frac{4}{n - (k + 1)}
\]
This measure is essential in regression analysis to identify points that can disproportionately influence the estimated parameters.
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