PCR_FITTED - PCR Fitting Values

Returns an array of cells for the fitted values of the conditional mean, residuals or leverage measures.

 

Syntax

PCR_FITTED(X, Mask, Y, Intercept, Return_type)

X is the independent variables data matrix, such that each column represents one variable.

Mask is the boolean array to choose the explanatory variables in the model. If missing, all variables in X are included.

Y is the response or the dependent variable data array (one dimensional array of cells (e.g. rows or columns)).

Intercept is the constant or the intercept value to fix (e.g. zero). If missing, an intercept will not be fixed and is computed normally.

Return_type is a switch to select the return output (1 = fitted values (default), 2 = residuals, 3 = standardized residuals, 4 = leverage, 5 = Cook's distance).

Method Description
1 Fitted/conditional mean
2 Residuals
3 Standardized (aka. Studentized) residuals
4 Leverage (H)
5 Cook's distance (D)
 

Remarks

  1. The underlying model is described here.
  2. The regression fitted (aka estimated) conditional mean is calculated as follows:

    $$\hat y_i = E \left[ Y| x_i1\cdots x_ip \right] = \alpha + \hat \beta_1 \times x_i1 + \cdots + \beta_p \times x_ip$$

    Residuals are defined as follows:

    $$ e_i = y_i - \hat y_i $$

    The standardized (aka studentized) residuals are calculated as follows:

    $$\bar e_i = \frac{e_i}{\hat \sigma_i} $$

    Where:
    • $\hat y $ is the estimated regression value.
    • $e $ is the error term in the regression.
    • $\hat e $ is the standardized error term.
    • $\hat \sigma_i $ is the standard error for the i-th observation.
  3. For the influential data analysis, PCR_FITTED computes two values: leverage statistics and Cook's distance for observations in our sample data.
  4. Leverage statistics describe the influence that each observed value has on the fitted value for that same observation. By definition, the diagonal elements of the hat matrix are the leverages.

    $$H = X \left(X^\top X \right)^{-1} X^\top$$

    $$L_i = h_{ii}$$ Where:
    • $H$ is the Hat matrix for uncorrelated error terms.
    • $\mathbf{X}$ is a (N x p+1) matrix of explanatory variable where the first column is all ones.
    • $L_i$ is the leverage statistics for the i-th observation.
    • $h_{ii}$ is the i-th diagonal element in the hat matrix.
  5. Cook's distance measures the effect of deleting a given observation. Data points with large residuals (outliers) and/or high leverage may distort the outcome and accuracy of a regression. Points with a large Cook's distance are considered to merit closer examination in the analysis.

    $$D_i = \frac{e_i^2}{p \ \mathrm{MSE}}\left[\frac{h_{ii}}{(1-h_{ii})^2}\right]$$
    Where:
    • $D_i$ is the Cook's distance for the i-th observation.
    • $h_{ii}$ is the leverage statistics (or the i-th diagonal element in the hat matrix).
    • $\mathrm{MSE}$ is the mean square error of the regression model.
    • $p$ is the number of explanatory variables.
    • $e_i$ is the error term (residual) for the i-th observation.
  6. The sample data may include missing values.
  7. Each column in the input matrix corresponds to a separate variable.
  8. Each row in the input matrix corresponds to an observation.
  9. Observations (i.e. rows) with missing values in X or Y are removed.
  10. The number of rows of the response variable (Y) must be equal to the number of rows of the explanatory variable (X).
  11. The MLR_FITTED function is available starting with version 1.60 APACHE.

Files Examples

References

  • Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
  • Kenney, J. F. and Keeping, E. S. (1962) "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285
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