naginterfaces.library.correg.quantile_​linreg_​easy

naginterfaces.library.correg.quantile_linreg_easy(x, y, tau, io_manager=None)

quantile_linreg_easy performs a multiple linear quantile regression, returning the parameter estimates and associated confidence limits based on an assumption of Normal, independent, identically distributed errors. quantile_linreg_easy is a simplified version of quantile_linreg().

For full information please refer to the NAG Library document for g02qf

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02qff.html

Parameters

xfloat, array-like, shape $(n,m)$

$X$, the design matrix, with the $\text{i}$th value for the $\text{j}$th variate supplied in $x[\text{i}-1,\text{j}-1]$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

yfloat, array-like, shape $\left(n\right)$

$y$, the observations on the dependent variable.

taufloat, array-like, shape $\left(\text{ntau}\right)$

The vector of quantiles of interest. A separate model is fitted to each quantile.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

dffloat

The degrees of freedom given by $n-k$, where $n$ is the number of observations and $k$ is the rank of the cross-product matrix $X^{T}X$.

bfloat, ndarray, shape $(m,\text{ntau})$

$\hat{\beta }$, the estimates of the parameters of the regression model, with $b[j-1,l-1]$ containing the coefficient for the variable in column $j$ of $x$, estimated for $\tau =tau[l-1]$.

blfloat, ndarray, shape $(m,\text{ntau})$

${\hat{\beta }}_L$, the lower limit of a $95\%$ confidence interval for $\hat{\beta }$, with $bl[j-1,l-1]$ holding the lower limit associated with $b[j-1,l-1]$.

bufloat, ndarray, shape $(m,\text{ntau})$

${\hat{\beta }}_U$, the upper limit of a $95\%$ confidence interval for $\hat{\beta }$, with $bu[j-1,l-1]$ holding the upper limit associated with $b[j-1,l-1]$.

infoint, ndarray, shape $\left(\text{ntau}\right)$

$info\left[l\right]$ holds additional information concerning the model fitting and confidence limit calculations when $\tau =tau\left[l\right]$.

Code	Warning
$0$	Model fitted and confidence limits calculated successfully.
$1$	The function did not converge whilst calculating the parameter estimates. The returned values are based on the estimate at the last iteration.
$2$	A singular matrix was encountered during the optimization. The model was not fitted for this value of $\tau$.
$8$	The function did not converge whilst calculating the confidence limits. The returned limits are based on the estimate at the last iteration.
$16$	Confidence limits for this value of $\tau$ could not be calculated. The returned upper and lower limits are set to a large positive and large negative value respectively.

It is possible for multiple warnings to be applicable to a single model.

In these cases the value returned in $info$ is the sum of the corresponding individual nonzero warning codes.

Raises

NagValueError

(errno $11$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $21$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $1\le m<n$.

(errno $51$)

On entry, $\text{ntau}=⟨value⟩$.

Constraint: $\text{ntau}\ge 1$.

(errno $61$)

On entry, $tau[⟨value⟩]=⟨value⟩$.

Constraint: $\sqrt{ϵ}<tau[\text{l}-1]<1-\sqrt{ϵ}$ where $ϵ$ is the machine precision returned by machine.precision, for all $\text{ntau}$.

Warns

NagAlgorithmicWarning

(errno $111$)

A potential problem occurred whilst fitting the model(s).

Additional information has been returned in $info$.

Notes

Given a vector of $n$ observed values, $y=\{y_i:i=1,2,\dots ,n\}$, an $n\times p$ design matrix $X$, a column vector, $x$, of length $p$ holding the $i$th row of $X$ and a quantile $\tau \in (0,1)$, quantile_linreg_easy estimates the $p$-element vector $\beta$ as the solution to

$${\text{minimize}}_{\beta \in R^p}\sum _1^n{\rho }_{\tau }(y_i-x_i^T\beta )$$

where ${\rho }_{\tau }$ is the piecewise linear loss function ${\rho }_{\tau }\left(z\right)=z(\tau -I(z<0))$, and $I(z<0)$ is an indicator function taking the value $1$ if $z<0$ and $0$ otherwise.

quantile_linreg_easy assumes Normal, independent, identically distributed (IID) errors and calculates the asymptotic covariance matrix from

$$\Sigma =\frac{\tau (1-\tau )}{n}{\left(s\left(\tau \right)\right)}^2{\left(X^{T}X\right)}^{-1}$$

where $s$ is the sparsity function, which is estimated from the residuals, $r_i=y_i-x_i^T\hat{\beta }$ (see Koenker (2005)).

Given an estimate of the covariance matrix, $\hat{\Sigma }$, lower, ${\hat{\beta }}_L$, and upper, ${\hat{\beta }}_U$, limits for a $95\%$ confidence interval are calculated for each of the $p$ parameters, via

$${\hat{\beta }}_{Li}={\hat{\beta }}_i-t_{n-p,0.975}\sqrt{{\hat{\Sigma }}_{ii}},{\hat{\beta }}_{Ui}={\hat{\beta }}_i+t_{n-p,0.975}\sqrt{{\hat{\Sigma }}_{ii}}$$

where $t_{n-p,0.975}$ is the $97.5$ percentile of the Student’s $t$ distribution with $n-k$ degrees of freedom, where $k$ is the rank of the cross-product matrix $X^{T}X$.

Further details of the algorithms used by quantile_linreg_easy can be found in the documentation for quantile_linreg().

References

Koenker, R, 2005, Quantile Regression, Econometric Society Monographs, Cambridge University Press, New York