| int __stdcall NDK_PCR_FITTED | ( | double ** | X, |
| size_t | nXSize, | ||
| size_t | nXVars, | ||
| LPBYTE | mask, | ||
| size_t | nMaskLen, | ||
| double * | Y, | ||
| size_t | nYSize, | ||
| double | intercept, | ||
| WORD | nRetType | ||
| ) |
Returns an array of cells for the i-th principal component (or residuals).
- Returns
- status code of the operation
- Return values
-
NDK_SUCCESS Operation successful NDK_FAILED Operation unsuccessful. See Macros for full list.
- Parameters
-
[in] X is the independent variables data matrix, such that each column represents one variable [in] nXSize is the number of observations (i.e. rows) in X [in] nXVars is the number of variables (i.e. columns) in X [in] mask is the boolean array to select a subset of the input variables in X. If missing (i.e. NULL), all variables in X are included. [in] nMaskLen is the number of elements in mask [in,out] Y is the response or the dependent variable data array (one dimensional array) [in] nYSize is the number of elements in Y [in] intercept is the constant or the intercept value to fix (e.g. zero). If missing (NaN), an intercept will not be fixed and is computed normally [in] nRetType is a switch to select the return output - fitted values (default),
- residuals,
- standardized residuals,
- leverage (H),
- Cook's distance.
- Remarks
-
- li>The underlying model is described here.
- 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.
- For the influential data analysis, PCR_FITTED computes two values: leverage statistics and Cook's distance for observations in our sample data.
- 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.
- 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.
- The sample data may include missing values.
- Each column in the input matrix corresponds to a separate variable.
- Each row in the input matrix corresponds to an observation.
- Observations (i.e. row) with missing values in X or Y are removed.
- The number of rows of the response variable (Y) must be equal to the number of rows of the explanatory variables (X).
- The MLR_FITTED function is available starting with version 1.60 APACHE.
- Requirements
-
Header SFSDK.H Library SFSDK.LIB DLL SFSDK.DLL
