# NDK_MLR_FITTED

 int __stdcall NDK_MLR_FITTED ( double ** X, size_t nXSize, size_t nXVars, LPBYTE mask, size_t nMaskLen, double * Y, size_t nYSize, double intercept, WORD nRetType )

Returns the fitted values of the conditional mean, residuals or leverage measures.

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 (explanatory) variables data matrix, such that each column represents one variable. [in] nXSize is the number of observations (rows) in X. [in] nXVars is the number of independent (explanatory) variables (columns) in X. [in] mask is the boolean array to choose the explanatory variables in the model. If missing, all variables in X are included. [in] nMaskLen is the number of elements in the "mask." [in] Y is the response or dependent variable data array (one dimensional array of cells). [in] nYSize is the number of observations in Y. [in] intercept is the constant or intercept value to fix (e.g. zero). If missing (i.e. NaN), an intercept will not be fixed and is computed normally. [in] nRetType is a switch to select the return output (1=fitted values (default), 2=residuals, 3=standardized residuals, 4=leverage, 5=Cook's distance). Fitted/conditional mean Residuals Standardized residuals Leverage factor (H) 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 residuals are calculated as follow: $\bar e_i = \frac{e_i}{\hat \sigma_i}$ 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, SLR_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 variables 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. row) with missing values in X or Y are removed.
10. The number of rows of the response variable (Y) must be equal to number of rows of the explanatory variables (X).
11. The MLR_FITTED function is available starting with version 1.60 APACHE.
Requirements
 Namespace: NumXLAPI Class: SFSDK Scope: Public Lifetime: Static
 int __stdcall NDK_MLR_FITTED ( double ** X, size_t nXSize, size_t nXVars, LPBYTE mask, size_t nMaskLen, double * Y, size_t nYSize, double intercept, WORD nRetType )

Returns the fitted values of the conditional mean, residuals or leverage measures.

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 (explanatory) variables data matrix, such that each column represents one variable. [in] nXSize is the number of observations (rows) in X. [in] nXVars is the number of independent (explanatory) variables (columns) in X. [in] mask is the boolean array to choose the explanatory variables in the model. If missing, all variables in X are included. [in] nMaskLen is the number of elements in the "mask." [in] Y is the response or dependent variable data array (one dimensional array of cells). [in] nYSize is the number of observations in Y. [in] intercept is the constant or intercept value to fix (e.g. zero). If missing (i.e. NaN), an intercept will not be fixed and is computed normally. [in] nRetType is a switch to select the return output (1=fitted values (default), 2=residuals, 3=standardized residuals, 4=leverage, 5=Cook's distance). Fitted/conditional mean Residuals Standardized residuals Leverage factor (H) 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 residuals are calculated as follow: $\bar e_i = \frac{e_i}{\hat \sigma_i}$ 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, SLR_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 variables 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. row) with missing values in X or Y are removed.
10. The number of rows of the response variable (Y) must be equal to number of rows of the explanatory variables (X).
11. The MLR_FITTED function is available starting with version 1.60 APACHE.
Requirements
References
* Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
* Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740
* D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
* Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848