CDPL.Math.FMLRModel Class Reference

Performs Multiple Linear Regression [WLIREG] on a set of data points \( (y_i, \vec{X}_i) \). More...

Inheritance diagram for CDPL.Math.FMLRModel:

Public Member Functions
None	__init__ ()
	Constructs and initializes a regression model with an empty data set.
None	__init__ (FMLRModel model)
	Initializes a copy of the FMLRModel instance model. More...
int	getObjectID ()
	Returns the numeric identifier (ID) of the wrapped C++ class instance. More...
FMLRModel	assign (FMLRModel model)
	Replaces the current state of self with a copy of the state of the FMLRModel instance model. More...
None	resizeDataSet (int num_points, int num_vars)
	Resizes the data set to hold num_points data points with num_vars independent variables. More...
None	clearDataSet ()
	Clears the data set. More...
None	setXYData (int i, ConstFVectorExpression x_vars, float y)
	Sets the i-th data point \( (y_i, \vec{X}_i) \) of the data set. More...
None	setXYData (int i, ConstDVectorExpression x_vars, float y)
	Sets the i-th data point \( (y_i, \vec{X}_i) \) of the data set. More...
None	setXYData (int i, ConstLVectorExpression x_vars, float y)
	Sets the i-th data point \( (y_i, \vec{X}_i) \) of the data set. More...
None	setXYData (int i, ConstULVectorExpression x_vars, float y)
	Sets the i-th data point \( (y_i, \vec{X}_i) \) of the data set. More...
None	addXYData (ConstFVectorExpression x_vars, float y)
	Adds a new data point \( (y, \vec{X}) \) to the current data set. More...
None	addXYData (ConstDVectorExpression x_vars, float y)
	Adds a new data point \( (y, \vec{X}) \) to the current data set. More...
None	addXYData (ConstLVectorExpression x_vars, float y)
	Adds a new data point \( (y, \vec{X}) \) to the current data set. More...
None	addXYData (ConstULVectorExpression x_vars, float y)
	Adds a new data point \( (y, \vec{X}) \) to the current data set. More...
FMatrix	getXMatrix ()
	Returns a read-only matrix where each row represents the vector \( \vec{X}_i \) with independent variables of the currently stored data points \( (y_i, \vec{X}_i), \, i = 1, 2, \ldots, N \). More...
FVector	getYValues ()
	Returns a read-only vector containing the dependent variables \( y_i \) of the currently stored data points \( (y_i, \vec{X}_i), \, i = 1, 2, \ldots, N \). More...
None	buildModel ()
	Performs linear least squares regression modeling of the set of currently stored data points \( (y_i, \vec{X}_i), \, i = 1, 2, \ldots, N \). More...
None	calcStatistics ()
	Calculates various statistical parameters describing the built regression model. More...
float	calcYValue (ConstFVectorExpression x_vars)
	Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars. More...
float	calcYValue (ConstDVectorExpression x_vars)
	Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars. More...
float	calcYValue (ConstLVectorExpression x_vars)
	Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars. More...
float	calcYValue (ConstULVectorExpression x_vars)
	Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars. More...
FVector	getCoefficients ()
	Returns a read-only vector containing the estimated regression coefficients \( \beta_i \) which were calculated by buildModel(). More...
float	getChiSquare ()
	Returns the sum of squared residuals \( \chi^2 \). More...
float	getGoodnessOfFit ()
	Returns the goodness of fit \( Q \). More...
float	getCorrelationCoefficient ()
	Returns the correlation coefficient \( r \). More...
float	getStandardDeviation ()
	Returns the standard deviation of the residuals \( s_r \). More...
float	__call__ (ConstFVectorExpression x_vars)
	Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars. More...
float	__call__ (ConstDVectorExpression x_vars)
	Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars. More...
float	__call__ (ConstLVectorExpression x_vars)
	Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars. More...
float	__call__ (ConstULVectorExpression x_vars)
	Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars. More...

Properties
	objectID = property(getObjectID)
	xMatrix = property(getXMatrix)
	yValues = property(getYValues)
	coefficients = property(getCoefficients)
	chiSquare = property(getChiSquare)
	goodnessOfFit = property(getGoodnessOfFit)
	correlationCoefficient = property(getCorrelationCoefficient)
	standardDeviation = property(getStandardDeviation)

Detailed Description

Performs Multiple Linear Regression [WLIREG] on a set of data points \( (y_i, \vec{X}_i) \).

For each data point, \( y_i \) is the dependent (response) variable and \( \vec{X}_i \) is a \( M \)-dimensional vector containing the independent (explanatory) variables of the modeled function \( y = f(\vec{X}) \). It is assumed that the relationship between the dependent variables \( y_i \) and the independent variables \( \vec{X}_i \) can be modeled by a linear function of \( M \) parameters \( \beta_i, \, i = 1, 2, \ldots, M \) (regression coefficients) plus an error term \( \epsilon_i \):

[ y_i = \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_M x_{iM} + \epsilon_i ]

The parameters \( \beta_i \) are estimated by Least Squares Analysis [WLSQRS] which minimizes the sum of squared residuals \( \chi^2 \)

[ \chi^2 = \sum_{i=1}^{N} (y_i - f(\vec{X}_i, \vec{\beta}))^2 ]

of the given set of \( N \) data points with respect to the adjustable parameters \( \vec{\beta} \). The parameters \( \beta_i \) are computed using Singular Value Decomposition [WSVD] as implemented in [NRIC]. This method is computationally intensive, but is particularly useful if the \( X \) matrix is ill-conditioned.

Constructor & Destructor Documentation

__init__()

None CDPL.Math.FMLRModel.__init__

(

FMLRModel

model

)

Initializes a copy of the FMLRModel instance model.

Parameters

model

The FMLRModel instance to copy.

Member Function Documentation

getObjectID()

int CDPL.Math.FMLRModel.getObjectID

(

)

Returns the numeric identifier (ID) of the wrapped C++ class instance.

Different Python FMLRModel instances may reference the same underlying C++ class instance. The commonly used Python expression a is not b thus cannot tell reliably whether the two FMLRModel instances a and b reference different C++ objects. The numeric identifier returned by this method allows to correctly implement such an identity test via the simple expression a.getObjectID() != b.getObjectID().

Returns

The numeric ID of the internally referenced C++ class instance.

assign()

FMLRModel CDPL.Math.FMLRModel.assign

(

FMLRModel

model

)

Replaces the current state of self with a copy of the state of the FMLRModel instance model.

Parameters

model

The FMLRModel instance to copy.

Returns

self

resizeDataSet()

None CDPL.Math.FMLRModel.resizeDataSet	(	int	num_points,
		int	num_vars
	)

Resizes the data set to hold num_points data points with num_vars independent variables.

Parameters

num_points	The number of data points.
num_vars	The number of independent variables per data point.

clearDataSet()

None CDPL.Math.FMLRModel.clearDataSet

(

)

Clears the data set.

Equivalent to calling resizeDataSet() with both arguments beeing zero.

setXYData() [1/4]

None CDPL.Math.FMLRModel.setXYData	(	int	i,
		ConstFVectorExpression	x_vars,
		float	y
	)

Sets the i-th data point \( (y_i, \vec{X}_i) \) of the data set.

If i is larger or equal to the number of currently stored data points or if the number of independent variables provided by x_vars is larger than the maximum number so far, the data set will be resized accordingly. Emerging space between the data points so far and the new data point is filled up with zeros. If the number of independent variables provided by x_vars is smaller than the number of variables in the current data set, the missing independent variables are assumed to be zero.

Parameters

i	The zero-based index of the data point in the data set.
x_vars	The vector \( \vec{X}_i \) with independent variables.
y	The dependent variable \( y_i \).

setXYData() [2/4]

None CDPL.Math.FMLRModel.setXYData	(	int	i,
		ConstDVectorExpression	x_vars,
		float	y
	)

Sets the i-th data point \( (y_i, \vec{X}_i) \) of the data set.

If i is larger or equal to the number of currently stored data points or if the number of independent variables provided by x_vars is larger than the maximum number so far, the data set will be resized accordingly. Emerging space between the data points so far and the new data point is filled up with zeros. If the number of independent variables provided by x_vars is smaller than the number of variables in the current data set, the missing independent variables are assumed to be zero.

Parameters

i	The zero-based index of the data point in the data set.
x_vars	The vector \( \vec{X}_i \) with independent variables.
y	The dependent variable \( y_i \).

setXYData() [3/4]

None CDPL.Math.FMLRModel.setXYData	(	int	i,
		ConstLVectorExpression	x_vars,
		float	y
	)

Sets the i-th data point \( (y_i, \vec{X}_i) \) of the data set.

If i is larger or equal to the number of currently stored data points or if the number of independent variables provided by x_vars is larger than the maximum number so far, the data set will be resized accordingly. Emerging space between the data points so far and the new data point is filled up with zeros. If the number of independent variables provided by x_vars is smaller than the number of variables in the current data set, the missing independent variables are assumed to be zero.

Parameters

i	The zero-based index of the data point in the data set.
x_vars	The vector \( \vec{X}_i \) with independent variables.
y	The dependent variable \( y_i \).

setXYData() [4/4]

None CDPL.Math.FMLRModel.setXYData	(	int	i,
		ConstULVectorExpression	x_vars,
		float	y
	)

Sets the i-th data point \( (y_i, \vec{X}_i) \) of the data set.

If i is larger or equal to the number of currently stored data points or if the number of independent variables provided by x_vars is larger than the maximum number so far, the data set will be resized accordingly. Emerging space between the data points so far and the new data point is filled up with zeros. If the number of independent variables provided by x_vars is smaller than the number of variables in the current data set, the missing independent variables are assumed to be zero.

Parameters

i	The zero-based index of the data point in the data set.
x_vars	The vector \( \vec{X}_i \) with independent variables.
y	The dependent variable \( y_i \).

addXYData() [1/4]

None CDPL.Math.FMLRModel.addXYData	(	ConstFVectorExpression	x_vars,
		float	y
	)

Adds a new data point \( (y, \vec{X}) \) to the current data set.

If the number of independent variables provided by x_vars is larger than the number in the current data set, the data set is resized accordingly and any emerging space will be filled with zeros. If the number of independent variables provided by x_vars is smaller than the number in the current data set, the missing independent variables are assumed to be zero.

Parameters

x_vars	The vector \( \vec{X} \) with independent variables.
y	The dependent variable \( y \).

Note

If the final size of the data set is known in advance, a call to resizeDataSet() followed by calls to setXYData() for each data point is more efficient than a build-up of the data set by repeatedly calling addXYData().

addXYData() [2/4]

None CDPL.Math.FMLRModel.addXYData	(	ConstDVectorExpression	x_vars,
		float	y
	)

Adds a new data point \( (y, \vec{X}) \) to the current data set.

If the number of independent variables provided by x_vars is larger than the number in the current data set, the data set is resized accordingly and any emerging space will be filled with zeros. If the number of independent variables provided by x_vars is smaller than the number in the current data set, the missing independent variables are assumed to be zero.

Parameters

x_vars	The vector \( \vec{X} \) with independent variables.
y	The dependent variable \( y \).

Note

If the final size of the data set is known in advance, a call to resizeDataSet() followed by calls to setXYData() for each data point is more efficient than a build-up of the data set by repeatedly calling addXYData().

addXYData() [3/4]

None CDPL.Math.FMLRModel.addXYData	(	ConstLVectorExpression	x_vars,
		float	y
	)

Adds a new data point \( (y, \vec{X}) \) to the current data set.

If the number of independent variables provided by x_vars is larger than the number in the current data set, the data set is resized accordingly and any emerging space will be filled with zeros. If the number of independent variables provided by x_vars is smaller than the number in the current data set, the missing independent variables are assumed to be zero.

Parameters

x_vars	The vector \( \vec{X} \) with independent variables.
y	The dependent variable \( y \).

Note

If the final size of the data set is known in advance, a call to resizeDataSet() followed by calls to setXYData() for each data point is more efficient than a build-up of the data set by repeatedly calling addXYData().

addXYData() [4/4]

None CDPL.Math.FMLRModel.addXYData	(	ConstULVectorExpression	x_vars,
		float	y
	)

Adds a new data point \( (y, \vec{X}) \) to the current data set.

If the number of independent variables provided by x_vars is larger than the number in the current data set, the data set is resized accordingly and any emerging space will be filled with zeros. If the number of independent variables provided by x_vars is smaller than the number in the current data set, the missing independent variables are assumed to be zero.

Parameters

x_vars	The vector \( \vec{X} \) with independent variables.
y	The dependent variable \( y \).

Note

If the final size of the data set is known in advance, a call to resizeDataSet() followed by calls to setXYData() for each data point is more efficient than a build-up of the data set by repeatedly calling addXYData().

getXMatrix()

FMatrix CDPL.Math.FMLRModel.getXMatrix

(

)

Returns a read-only matrix where each row represents the vector \( \vec{X}_i \) with independent variables of the currently stored data points \( (y_i, \vec{X}_i), \, i = 1, 2, \ldots, N \).

Returns

A reference to the matrix with the independent variables \( \vec{X}_i \).

getYValues()

FVector CDPL.Math.FMLRModel.getYValues

(

)

Returns a read-only vector containing the dependent variables \( y_i \) of the currently stored data points \( (y_i, \vec{X}_i), \, i = 1, 2, \ldots, N \).

Returns

A reference to the vector with the dependent variables \( y_i \).

buildModel()

None CDPL.Math.FMLRModel.buildModel

(

)

Performs linear least squares regression modeling of the set of currently stored data points \( (y_i, \vec{X}_i), \, i = 1, 2, \ldots, N \).

Exceptions

Base.CalculationFailed

if the data set is empty or the singular value decomposition of the \( X \) matrix failed.

calcStatistics()

None CDPL.Math.FMLRModel.calcStatistics

(

)

Calculates various statistical parameters describing the built regression model.

Exceptions

Base.CalculationFailed

if the data set is in an inconsistent state (e.g. the number of estimated regression coefficients does not match the number of independent variables that make up the data points).

See also

buildModel(), getChiSquare(), getGoodnessOfFit(), getCorrelationCoefficient(), getStandardDeviation()

calcYValue() [1/4]

float CDPL.Math.FMLRModel.calcYValue

(

ConstFVectorExpression

x_vars

)

Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars.

Parameters

x_vars

The vector \( \vec{X} \) of independent variables.

Returns

The predicted value for \( y \).

Exceptions

Base.CalculationFailed

if the number of regression coefficients \( \beta_i \) does not match the size of x_vars.

calcYValue() [2/4]

float CDPL.Math.FMLRModel.calcYValue

(

ConstDVectorExpression

x_vars

)

Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars.

Parameters

x_vars

The vector \( \vec{X} \) of independent variables.

Returns

The predicted value for \( y \).

Exceptions

Base.CalculationFailed

if the number of regression coefficients \( \beta_i \) does not match the size of x_vars.

calcYValue() [3/4]

float CDPL.Math.FMLRModel.calcYValue

(

ConstLVectorExpression

x_vars

)

Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars.

Parameters

x_vars

The vector \( \vec{X} \) of independent variables.

Returns

The predicted value for \( y \).

Exceptions

Base.CalculationFailed

if the number of regression coefficients \( \beta_i \) does not match the size of x_vars.

calcYValue() [4/4]

float CDPL.Math.FMLRModel.calcYValue

(

ConstULVectorExpression

x_vars

)

Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars.

Parameters

x_vars

The vector \( \vec{X} \) of independent variables.

Returns

The predicted value for \( y \).

Exceptions

Base.CalculationFailed

if the number of regression coefficients \( \beta_i \) does not match the size of x_vars.

getCoefficients()

FVector CDPL.Math.FMLRModel.getCoefficients

(

)

Returns a read-only vector containing the estimated regression coefficients \( \beta_i \) which were calculated by buildModel().

Returns

A reference to the vector with the estimated regression coefficients \( \beta_i \).

getChiSquare()

float CDPL.Math.FMLRModel.getChiSquare

(

)

Returns the sum of squared residuals \( \chi^2 \).

\( \chi^2 \) is calculated by:

[ \chi^2 = \sum_{i=1}^{N} (y_i - \sum_{j=1}^{M}(x_{ij} \beta_j))^2 ]

Returns

The sum of squared residuals \( \chi^2 \).

Note

The returned value is only valid if calcStatistics() has been called before.

getGoodnessOfFit()

float CDPL.Math.FMLRModel.getGoodnessOfFit

(

)

Returns the goodness of fit \( Q \).

The goodness of fit \( Q \) is given by:

[ Q = gammaq(\frac{N - 2}{2}, \frac{\chi^2}{2}) ]

where \( gammaq \) is the incomplete gamma function (see [NRIC] for details).

Returns

The goodness of fit \( Q \).

Note

The returned value is only valid if calcStatistics() has been called before.

See also

Math.gammaQ()

getCorrelationCoefficient()

float CDPL.Math.FMLRModel.getCorrelationCoefficient

(

)

Returns the correlation coefficient \( r \).

The correlation coefficient \( r \) is calculated by:

[ r = \frac{\sum_{i=1}^{N} (\hat{y}_i - \bar{\hat{y}})(y_i - \bar{y})} {\sqrt{\sum_{i=1}^{N} (\hat{y}_i - \bar{\hat{y}})^2 \sum_{i=1}^{N} (y_i - \bar{y})^2 }} ]

where

\begin{eqnarray*} \hat{y}_i &=& \sum_{j=1}^{M}(x_{ij} \beta_j) \ \bar{\hat{y}} &=& \frac{\sum_{i=1}^{N} \hat{y}_i}{N} \ \bar{y} &=& \frac{\sum_{i=1}^{N} y_i}{N} \end{eqnarray*}

Returns

The correlation coefficient \( r \).

Note

The returned value is only valid if calcStatistics() has been called before.

getStandardDeviation()

float CDPL.Math.FMLRModel.getStandardDeviation

(

)

Returns the standard deviation of the residuals \( s_r \).

The standard deviation \( s_r \) is calculated by:

[ s_r = \sqrt{\frac{\sum_{i=1}^{N} (y_i - \sum_{j=1}^{M} (x_{ij} \beta_j))^2} {N - M}} ]

Returns

The standard deviation of the residuals \( s_r \).

Note

\( s_r \) is only definded if \( N > M \) and calcStatistics() has been called before.

__call__() [1/4]

float CDPL.Math.FMLRModel.__call__

(

ConstFVectorExpression

x_vars

)

Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars.

Equivalent to calling calcYValue().

Parameters

x_vars

The vector \( \vec{X} \) of independent variables.

Returns

The predicted value for \( y \).

Exceptions

Base.CalculationFailed

if the number of regression coefficients \( \beta_i \) does not match the size of x_vars.

__call__() [2/4]

float CDPL.Math.FMLRModel.__call__

(

ConstDVectorExpression

x_vars

)

Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars.

Equivalent to calling calcYValue().

Parameters

x_vars

The vector \( \vec{X} \) of independent variables.

Returns

The predicted value for \( y \).

Exceptions

Base.CalculationFailed

if the number of regression coefficients \( \beta_i \) does not match the size of x_vars.

__call__() [3/4]

float CDPL.Math.FMLRModel.__call__

(

ConstLVectorExpression

x_vars

)

Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars.

Equivalent to calling calcYValue().

Parameters

x_vars

The vector \( \vec{X} \) of independent variables.

Returns

The predicted value for \( y \).

Exceptions

Base.CalculationFailed

if the number of regression coefficients \( \beta_i \) does not match the size of x_vars.

__call__() [4/4]

float CDPL.Math.FMLRModel.__call__

(

ConstULVectorExpression

x_vars

)

Predicts the value of the dependent variable \( y \) for a vector \( \vec{X} \) of independent variables given by x_vars.

Equivalent to calling calcYValue().

Parameters

x_vars

The vector \( \vec{X} \) of independent variables.

Returns

The predicted value for \( y \).

Exceptions

Base.CalculationFailed

if the number of regression coefficients \( \beta_i \) does not match the size of x_vars.