LinearModelResult¶

class LinearModelResult(*args)¶

Result of a LinearModelAlgorithm.

Parameters:

inputSample2-d sequence of float: The input sample of a model.
basisBasis: Functional basis to estimate the trend.
designMatrix: The design matrix $\mat{\Psi}$ .
outputSample2-d sequence of float: The output sample $(Y_i)_{1 \leq i \leq \sampleSize}$ .
metaModelFunction: The meta model.
coefficientssequence of float: The estimated coefficients $\vect{\hat{a}}$ .
formulastr: The formula description.
coefficientsNamessequence of str: The coefficients names of the basis.
sampleResiduals2-d sequence of float: The residual errors $(\varepsilon_i)_{1 \leq i \leq \sampleSize}$ .
standardizedSampleResiduals2-d sequence of float: The standardized residuals defined in (9).
diagonalGramInversesequence of float: The diagonal of the Gram inverse matrix.
leveragessequence of float: The leverages $(\ell_i)_{1 \leq i \leq \sampleSize}$ defined in (7).
cookDistancessequence of float: Cook’s distances defined in (2).
residualsVariancefloat: The unbiased variance estimator of the residuals defined in (8).

Methods

`buildMethod`()	Accessor to the least squares method.
`getAdjustedRSquared`()	Accessor to the Adjusted R-squared test.
`getBasis`()	Accessor to the basis.
`getClassName`()	Accessor to the object's name.
`getCoefficients`()	Accessor to the coefficients of the linear model.
`getCoefficientsNames`()	Accessor to the coefficients names.
`getCoefficientsStandardErrors`()	Accessor to the coefficients of standard error.
`getCookDistances`()	Accessor to the cook's distances.
`getDegreesOfFreedom`()	Accessor to the degrees of freedom.
`getDesign`()	Accessor to the design matrix.
`getDiagonalGramInverse`()	Accessor to the diagonal gram inverse matrix.
`getFittedSample`()	Accessor to the fitted sample.
`getFormula`()	Accessor to the formula.
`getInputSample`()	Accessor to the input sample.
`getLeverages`()	Accessor to the leverages.
`getMetaModel`()	Accessor to the metamodel.
`getName`()	Accessor to the object's name.
`getNoiseDistribution`()	Accessor to the normal distribution of the residuals.
`getOutputSample`()	Accessor to the output sample.
`getRSquared`()	Accessor to the R-squared test.
`getResidualsVariance`()	Accessor to the unbiased sample variance of the residuals.
`getSampleResiduals`()	Accessor to the residuals.
`getStandardizedResiduals`()	Accessor to the standardized residuals.
`hasIntercept`()	Returns if intercept is provided in the basis or not.
`hasName`()	Test if the object is named.
`involvesModelSelection`()	Get the model selection flag.
`setInputSample`(sampleX)	Accessor to the input sample.
`setInvolvesModelSelection`(involvesModelSelection)	Set the model selection flag.
`setMetaModel`(metaModel)	Accessor to the metamodel.
`setName`(name)	Accessor to the object's name.
`setOutputSample`(sampleY)	Accessor to the output sample.

getRelativeErrors
getResiduals
setRelativeErrors
setResiduals

See also

LinearModelAlgorithm

__init__(*args)¶

buildMethod()¶

Accessor to the least squares method.

Returns:

leastSquaresMethod: LeastSquaresMethod: The least squares method.

Notes

The least squares method used to estimate the coefficients is precised in the ResourceMap class, entry LinearModelAlgorithm-DecompositionMethod.

getAdjustedRSquared()¶

Accessor to the Adjusted R-squared test.

Returns:

adjustedRSquaredfloat: The $R_{ad}^2$ indicator.

Notes

The $R_{ad}^2$ value quantifies the quality of the linear approximation. With respect to $R^2$ , $R_{ad}^2$ takes into account the data set size and the number of hyperparameters.

If the model is defined by (3) such that the basis does not contain any intercept (constant function), then $R_{ad}^2$ is defined by:

$R_{ad}^2 = 1 - \left(\dfrac{\sampleSize}{dof}\right)(1 - R^2)$

where $dof$ is the degrees of freedom of the model defined in (4) and $\sampleSize$ the number of experiences.

Otherwise, when the model is defined by (1) or by (3) with an intercept, $R_{ad}^2$ is defined by:

$R_{ad}^2 = 1 - \left(\dfrac{\sampleSize - 1}{dof}\right)(1 - R^2)$

where $dof$ is defined in (3) or (4).

If the degree of freedom $dof$ is null, $R_{ad}^2$ is not defined.

getBasis()¶

Accessor to the basis.

Returns:

basisBasis: The basis of the regression model.

Notes

If a functional basis has been provided in the constructor, then we get it back: $(\phi_j)_{1 \leq j \leq p'}$ . Its size is $p'$ .

Otherwise, the functional basis is composed of the projections $\phi_k : \Rset^p \rightarrow \Rset$ such that $\phi_k(\vect{x}) = x_k$ for $1 \leq k \leq p$ , completed with the constant function: $\phi_0 : \vect{x} \rightarrow 1$ . Its size is $p+1$ .

getClassName()¶

Accessor to the object’s name.

Returns:

class_namestr: The object class name (object.__class__.__name__).

getCoefficients()¶

Accessor to the coefficients of the linear model.

Returns:

coefficientsPoint: The estimated of the coefficients $\hat{\vect{a}}$ .

getCoefficientsNames()¶

Accessor to the coefficients names.

Returns:

coefficientsNamesDescription

Notes

The name of the coefficient $a_k$ is the name of the regressor $X_k$ .

getCoefficientsStandardErrors()¶

Accessor to the coefficients of standard error.

Returns:

standardErrorsPoint

Notes

The standard deviation $\sigma(a_k)$ of the estimator $\hat{a}_k$ is defined by:

(1)¶ $\sigma(a_k)^2 = \sigma^2 \left(\Tr{\mat{\Psi}}\mat{\Psi}\right)^{-1}_{k+1, k+1}$

where:

the variance $\sigma^2$ of the residual $\varepsilon$ is approximated by its unbiaised estimator $\hat{\sigma}^2$ defined in (8),
the matrix $\mat{\Psi}$ is the design matrix defined in (5) or (6).

getCookDistances()¶

Accessor to the cook’s distances.

Returns:

cookDistancesPoint: The Cook’s distance of each experience $(CookD_i)_{1 \leq i \leq \sampleSize}$ .

Notes

The Cook’s distance measures the impact of every experience on the linear regression. See [rawlings2001] (section 11.2.1, Cook’s D page 362) for more details.

The Cook distance of experience $i$ is defined by:

(2)¶ $CookD_i = \left(\dfrac{1}{\sampleSize-dof}\right) \left( \dfrac{\ell_i}{1 - \ell_i} \right) (\varepsilon_i^{st})^2$

where $\varepsilon_i^{st}$ is the standardized residual defined in (9) and $dof$ is the degress of freedom defined in (3) or (4).

getDegreesOfFreedom()¶

Accessor to the degrees of freedom.

Returns:

dofint, $\geq 1$: Number of degrees of freedom.

Notes

If the linear model is defined by (1), the degrees of freedom $dof$ is:

(3)¶ $dof = \sampleSize - (p + 1)$

where $p$ is the number of regressors.

Otherwise, the linear model is defined by (3) and its $dof$ is:

(4)¶ $dof = \sampleSize - p'$

where $p'$ is the number of functions in the provided basis.

getDesign()¶

Accessor to the design matrix.

Returns:

design: Matrix: The design matrix $\mat{\Psi}$ .

Notes

If the linear model is defined by (1), the design matrix is:

(5)¶ $\mat{\Psi} = (\vect{1}, \vect{X}_1, \dots, \vect{X}_{p})$

where $\vect{1} = \Tr{(1, \dots, 1)}$ and $\vect{X}_k = \Tr{(X_k^1, \dots, X_k^\sampleSize)}$ is the values of the regressor $k$ in the $\sampleSize$ experiences. Thus, $\mat{\Psi}$ has $\sampleSize$ rows and $p+1$ columns.

If the linear model is defined by (3), the design matrix is:

(6)¶ $\mat{\Psi} = (\vect{\phi}_1, \dots, \vect{\phi}_{p'})$

where $\vect{\phi_j} = \Tr{(\phi_j(\vect{X}^1), \dots, \phi_j(\vect{X}^\sampleSize))}$ is the values of the function $\phi_j$ at the $\sampleSize$ experiences. Thus, $\mat{\Psi}$ has $\sampleSize$ rows and $p'$ columns.

getDiagonalGramInverse()¶

Accessor to the diagonal gram inverse matrix.

Returns:

diagonalGramInversePoint: The diagonal of the Gram inverse matrix.

Notes

The Gram matrix is $\Tr{\mat{\Psi}}\mat{\Psi}$ where $\mat{\Psi}$ is the design matrix defined in (5) or (6).

getFittedSample()¶

Accessor to the fitted sample.

Returns:

outputSampleSample

Notes

The fitted sample is $(\hat{Y}_1, \dots, \hat{Y}_\sampleSize)$ where $\hat{Y}_i$ is defined in (2) or (4).

getFormula()¶

Accessor to the formula.

Returns:

condensedFormulastr

Notes

This formula gives access to the linear model.

getInputSample()¶

Accessor to the input sample.

Returns:

inputSampleSample: The input sample.

getLeverages()¶

Accessor to the leverages.

Returns:

leveragesPoint: The leverage of all the experiences $(\ell_i)_{1 \leq i \leq \sampleSize}$ .

Notes

We denote by $\hat{\vect{Y}} = (\hat{Y}_1, \dots, \hat{Y}_n)$ the fitted values of the $n$ experiences. Then we have:

$\hat{\vect{Y}} = \mat{\Psi} \hat{\vect{a}}$

where $\mat{\Psi}$ is the design matrix defined in (5). It leads to:

$\Var{\hat{\vect{Y}}} & = \mat{\Psi} \Var{\hat{\vect{a}}}\Tr{\mat{\Psi}} \\ & = \sigma^2 \mat{H}$

where:

$\mat{H} = \mat{\Psi} (\Tr{\mat{\Psi}}\mat{\Psi})^{-1} \Tr{\mat{\Psi}}$

Thus, for the experience $i$ , we get:

(7)¶ $\Var{\hat{Y}_i} = \sigma^2 \ell_{ii}$

where $\ell_{ii}$ is the $i$ -th element of the diagonal of $\mat{H}$ : $\ell_{ii}$ is the leverage $\ell_i$ of experience $i$ .

getMetaModel()¶

Accessor to the metamodel.

Returns:

metaModelFunction: Metamodel.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getNoiseDistribution()¶

Accessor to the normal distribution of the residuals.

Returns:

noiseDistributionNormal: The normal distribution estimated from the residuals.

Notes

The noise distribution is the distribution of the residuals. It is assumed to be Gaussian. The normal distribution has zero mean and its variance is estimated from the residuals sample $(\varepsilon_i)_{1 \leq i \leq \sampleSize}$ defined in (5), using the unbiaised estimator defined in (8).

If the residuals are not Gaussian, this distribution is not appropriate and should not be used.

getOutputSample()¶

Accessor to the output sample.

Returns:

outputSampleSample: The output sample.

getRSquared()¶

Accessor to the R-squared test.

Returns:

rSquaredfloat: The indicator $R^2$ .

Notes

The $R^2$ value quantifies the quality of the linear approximation.

If the model is defined by (3) such that the basis does not contain any intercept (constant function), then $R^2$ is defined by:

$R^2 = 1- \dfrac{\sum_{i=1}^\sampleSize \varepsilon_i^2}{\sum_{i=1}^\sampleSize Y_i^2}$

where the $\varepsilon_i$ are the residuals defined in (5) and $Y_i$ the output sample values.

Otherwise, when the model is defined by (1) or by (3) with an intercept, $R^2$ is defined by:

$R^2 = 1- \dfrac{\sum_{i=1}^\sampleSize \varepsilon_i^2}{\sum_{i=1}^\sampleSize (Y_i-\bar{Y})^2}$

where $\bar{Y} = \dfrac{1}{\sampleSize} \sum_{i=1}^\sampleSize Y_i$ .

getResidualsVariance()¶

Accessor to the unbiased sample variance of the residuals.

Returns:

residualsVariancefloat: The residuals variance estimator.

Notes

The residual variance estimator is the unbiaised empirical variance of the residuals:

(8)¶ $\hat{\sigma}^2 = \dfrac{1}{dof} \sum_{i=1}^\sampleSize \varepsilon_i^2$

where $dof$ is the degrees of freedom of the model defined in (3) or (4).

getSampleResiduals()¶

Accessor to the residuals.

Returns:

sampleResidualsSample: The sample of the residuals.

Notes

The residuals sample is $(\varepsilon_i)_{1 \leq i \leq \sampleSize}$ defined in (5).

getStandardizedResiduals()¶

Accessor to the standardized residuals.

Returns:

standardizedResidualsSample: The standarduzed residuals $(\varepsilon_i^{st})_{1 \leq i \leq \sampleSize}$ .

Notes

The standardized residuals are defined by:

(9)¶ $\varepsilon_i^{st} = \dfrac{\varepsilon_i}{\sqrt{\hat{\sigma}^2(1 - \ell_i)}}$

where $\hat{\sigma}^2$ is the unbiaised residuals variance defined in (8) and $\ell_i$ is the leverage of experience $i$ defined in (7).

hasIntercept()¶

Returns if intercept is provided in the basis or not.

Returns:

interceptBool: Tells if the model has a constant regressor.

Notes

The intercept is True when:

the model is defined in (1),
the model is defined in (3) such that the basis contains a constant function.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

involvesModelSelection()¶

Get the model selection flag.

A model selection method can be used to select the coefficients to best predict the output. Model selection can lead to a sparse model.

Returns:

involvesModelSelectionbool: True if the method involves a model selection method.

setInputSample(sampleX)¶

Accessor to the input sample.

Parameters:

inputSampleSample: The input sample.

setInvolvesModelSelection(involvesModelSelection)¶

Set the model selection flag.

A model selection method can be used to select the coefficients to best predict the output. Model selection can lead to a sparse model.

Parameters:

involvesModelSelectionbool: True if the method involves a model selection method.

setMetaModel(metaModel)¶

Accessor to the metamodel.

Parameters:

metaModelFunction: Metamodel.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setOutputSample(sampleY)¶

Accessor to the output sample.

Parameters:

outputSampleSample: The output sample.

Examples using the class¶

Build and validate a linear model

Create a linear model

Perform stepwise regression

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LinearModelResult¶

Examples using the class¶