naginterfaces.library.correg.lars_​param

naginterfaces.library.correg.lars_param(b, fitsum, ktype, nk)

lars_param calculates additional parameter estimates following Least Angle Regression (LARS), forward stagewise linear regression or Least Absolute Shrinkage and Selection Operator (LASSO) as performed by lars() and lars_xtx().

For full information please refer to the NAG Library document for g02mc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02mcf.html

Parameters

bfloat, array-like, shape $(\text{ip},\text{nstep}+1)$

$\beta$ the parameter estimates, as returned by lars() and lars_xtx(), with $b[\text{j}-1,k-1]={\beta }_{k\text{j}}$, the parameter estimate for the $\text{j}$th variable, for $\text{j}=1,2,\dots ,p$, at the $k$th step of the model fitting process.

fitsumfloat, array-like, shape $(6,\text{nstep}+1)$

Summaries of the model fitting process, as returned by lars() and lars_xtx().

ktypeint

Indicates what target values are held in $nk$.

$ktype=1$

$nk$ holds (fractional) LARS step numbers.

$ktype=2$

$nk$ holds values for $L_1$ norm of the (scaled) parameters.

$ktype=3$

$nk$ holds ratios with respect to the largest (scaled) $L_1$ norm.

$ktype=4$

$nk$ holds values for the $L_1$ norm of the (unscaled) parameters.

$ktype=5$

$nk$ holds ratios with respect to the largest (unscaled) $L_1$ norm.

If lars() was called with $\text{pred}=0$ or $1$ or lars_xtx() was called with $\text{pred}=0$ then the model fitting routine did not rescale the independent variables, $X$, prior to fitting the model and, therefore, there is no difference between $ktype=2$ or $3$ and $ktype=4$ or $5$.

nkfloat, array-like, shape $\left(\text{lnk}\right)$

Target values used for predicting the new set of parameter estimates.

Returns

nbfloat, ndarray, shape $(\text{ip},\text{lnk})$

$\tilde{\beta }$ the predicted parameter estimates, with $b[j-1,i-1]={\tilde{\beta }}_{ij}$, the parameter estimate for variable $j$, $j=1,2,\dots ,p$ at the point in the fitting process associated with $nk[i-1]$, $i=1,2,\dots ,\text{lnk}$.

Raises

NagValueError

(errno $11$)

On entry, $\text{nstep}=⟨value⟩$.

Constraint: $\text{nstep}\ge 0$.

(errno $21$)

On entry, $\text{ip}=⟨value⟩$.

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

(errno $31$)

$b$ has been corrupted since the last call to lars() or lars_xtx().

(errno $51$)

$fitsum$ has been corrupted since the last call to lars() or lars_xtx().

(errno $61$)

On entry, $ktype=⟨value⟩$.

Constraint: $ktype=1$, $2$, $3$, $4$ or $5$.

(errno $71$)

On entry, $ktype=1$, $nk[⟨value⟩]=⟨value⟩$ and $\text{nstep}=⟨value⟩$

Constraint: $0\le nk\left[i\right]\le \text{nstep}$, for all $i$.

(errno $72$)

On entry, $ktype=2$, $nk[⟨value⟩]=⟨value⟩$, $\text{nstep}=⟨value⟩$ and $fitsum[0,\text{nstep}-1]=⟨value⟩$.

Constraint: $0\le nk\left[i\right]\le fitsum[0,\text{nstep}-1]$, for all $i$.

(errno $73$)

On entry, $ktype=3$ or $5$, $nk[⟨value⟩]=⟨value⟩$.

Constraint: $0\le nk\left[i\right]\le 1$, for all $i$.

(errno $74$)

On entry, $ktype=4$, $nk[⟨value⟩]=⟨value⟩$ and ${\parallel {\beta }_K\parallel }_1=⟨value⟩$

Constraint: $0\le nk\left[i\right]\le {\parallel {\beta }_K\parallel }_1$, for all $i$.

(errno $81$)

On entry, $\text{lnk}=⟨value⟩$.

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

Notes

lars() and lars_xtx() fit either a LARS, forward stagewise linear regression, LASSO or positive LASSO model to a vector of $n$ observed values, $y=\{y_i:i=1,2,\dots ,n\}$ and an $n\times p$ design matrix $X$, where the $j$th column of $X$ is given by the $j$th independent variable $x_j$. The models are fit using the LARS algorithm of Efron et al. (2004).

[figure omitted]

The full solution path for all four of these models follow a similar pattern where the parameter estimate for a given variable is piecewise linear. One such path, for a LARS model with six variables $(p=6)$ can be seen in Figure [label omitted]. Both lars() and lars_xtx() return the vector of $p$ parameter estimates, ${\beta }_k$, at $K$ points along this path (so $k=1,2,\dots ,K$). Each point corresponds to a step of the LARS algorithm. The number of steps taken depends on the model being fitted. In the case of a LARS model, $K=p$ and each step corresponds to a new variable being included in the model. In the case of the LASSO models, each step corresponds to either a new variable being included in the model or an existing variable being removed from the model; the value of $K$ is, therefore, no longer bound by the number of parameters. For forward stagewise linear regression, each step no longer corresponds to the addition or removal of a variable;, therefore, the number of possible steps is often markedly greater than for a corresponding LASSO model.

lars_param uses the piecewise linear nature of the solution path to predict the parameter estimates, $\tilde{\beta }$, at a different point on this path. The location of the solution can either be defined in terms of a (fractional) step number or a function of the $L_1$ norm of the parameter estimates.

References

Efron, B, Hastie, T, Johnstone, I and Tibshirani, R, 2004, Least Angle Regression, The Annals of Statistics (Volume 32) (2), 407–499

Hastie, T, Tibshirani, R and Friedman, J, 2001, The Elements of Statistical Learning: Data Mining, Inference and Prediction, Springer (New York)

Tibshirani, R, 1996, Regression Shrinkage and Selection via the Lasso, Journal of the Royal Statistics Society, Series B (Methodological) (Volume 58) (1), 267–288

Weisberg, S, 1985, Applied Linear Regression, Wiley

See also

naginterfaces.library.examples.correg.lars_param_ex.main()