# LinearFitEx

Advanced Analysis Library Only

AnalysisLibErrType LinearFitEx (double arrayX[], double arrayY[], double weight[], int numberOfElements, int fitMethod, double tolerance, double fittedData[], double *slope, double *intercept, double *residue);

## Purpose

Fits the data set (x, y) to the linear model using the Least Square, Least Absolute Residual, or Bisquare method. The following equation represents the linear model:

where | f represents the fittedData
| a is the slope
| x represents the input sequence arrayX
| b is the intercept

If fitMethod is LEAST_SQUARE, the function finds the slope and intercept of the linear model by minimizing the residue as follows:

where | n is numberOfElements
| wi is the i-th element of weight
| fi is the i-th element of fittedData
| yi is the i-th element of arrayY

If fitMethod is LEAST_ABSOLUTE_RESIDUAL, the function finds the slope and intercept of the linear model by minimizing the residue as follows:

If fitMethod is BISQUARE, the function finds slope and intercept of the linear model by using reweighted least square fitting iteratively, as shown in the following flowchart:

## Parameters

Input
Name | Type | Description
arrayX | double-precision array | The x value of the data set (x, y).
arrayY | double-precision array | The y value of the data set (x, y). If fittedData is NULL, arrayY is overwritten by the best fitted array.
weight | double-precision array | The weight of each data point. If weight is NULL, the function sets all the elements of weight to 1.
numberOfElements | integer | The length of arrayX, arrayY, and weight.
fitMethod | integer | The fitting method. fitMethod must be one of the following values: LEAST_SQUARE (0) LEAST_ABSOLUTE_RESIDUAL (1) BISQUARE (2) The Least Square method is preferable if the noise in arrayY is Gaussian distributed. The Least Absolute Residual and Bisquare method are robust fitting methods. Therefore, they are preferable if there are outliers in the observations. In most cases, the Bisquare method is less sensitive to outliers than the Least Absolute Residual method.
tolerance | double-precision | The stop criteria used in the Least Absolute Residual method or Bisquare method. In these two methods, the function adjusts the slope and intercept iteratively. If the relative difference of residue in two successive iterations is less than tolerance, the function returns the resulting slope and intercept. If tolerance is less than or equal to 0, the function sets tolerance to 0.0001.
Output
Name | Type | Description
fittedData | double-precision array | The y values calculated using the fitted linear model. If fittedData is NULL, the best fit array overwrites arrayY.
slope | double-precision | The slope of the fitted linear model.
intercept | double-precision | The intercept of the fitted linear model.
residue | double-precision | The weighted mean error of the linear fit. If fitMethod is LEAST_ABSOLUTE_RESIDUAL, residue is the weighted mean absolute error, as follows: If fitMethod is LEAST_SQUARE or BISQUARE, residue is the weighted mean square error, as follows:

## Return Value

Name | Type | Description
status | AnalysisLibErrType | A value that specifies the type of error that occurred. Refer to analysis.h for definitions of these constants.