naginterfaces.library.correg.lars

naginterfaces.library.correg.lars(mtype, d, y, pred=3, prey=1, isx=None, mnstep=None, ropt=None, io_manager=None)

lars performs Least Angle Regression (LARS), forward stagewise linear regression or Least Absolute Shrinkage and Selection Operator (LASSO).

For full information please refer to the NAG Library document for g02ma

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02maf.html

Parameters

mtypeint

Indicates the type of model to fit.

$mtype=1$

LARS is performed.

$mtype=2$

Forward linear stagewise regression is performed.

$mtype=3$

LASSO model is fit.

$mtype=4$

A positive LASSO model is fit.

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

$D$, the data, which along with $pred$ and $isx$, defines the design matrix $X$. The $\text{i}$th observation for the $\text{j}$th variable must be supplied in $d[\text{i}-1,\text{j}-1]$, 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.

predint, optional

Indicates the type of data preprocessing to perform on the independent variables supplied in $d$ to comply with the standardized form of the design matrix.

$pred=0$

No preprocessing is performed.

$pred=1$

Each of the independent variables, $x_{\text{j}}$, for $\text{j}=1,2,\dots ,p$, are mean centred prior to fitting the model. The means of the independent variables, $\overline{x}$, are returned in $b$, with ${\overline{x}}_{\text{j}}=b[\text{j}-1,nstep+1]$, for $\text{j}=1,2,\dots ,p$.

$pred=2$

Each independent variable is normalized, with the $j$th variable scaled by $1/\sqrt{x_j^T{x}_j}$. The scaling factor used by variable $j$ is returned in $b[\text{j}-1,nstep]$.

$pred=3$

As $pred=1$ or $2$, all of the independent variables are mean centred prior to being normalized.

preyint, optional

Indicates the type of data preprocessing to perform on the dependent variable supplied in $y$.

$prey=0$

No preprocessing is performed, this is equivalent to setting $\alpha =0$.

$prey=1$

The dependent variable, $y$, is mean centred prior to fitting the model, so $\alpha =\overline{y}$. Which is equivalent to fitting a non-penalized intercept to the model and the degrees of freedom etc. are adjusted accordingly.

The value of $\alpha$ used is returned in $fitsum[0,nstep]$.

isxNone or int, array-like, shape $\left(\text{lisx}\right)$, optional

Indicates which independent variables from $d$ will be included in the design matrix, $X$.

If $isx$ is None, all variables are included in the design matrix.

Otherwise $isx[\text{j}-1]$ must be set as follows, for $\text{j}=1,2,\dots ,m$:

$isx[j-1]=1$

To indicate that the $j$th variable, as supplied in $d$, is included in the design matrix;

$isx[j-1]=0$

To indicated that the $j$th variable, as supplied in $d$, is not included in the design matrix;

and $p={\sum }_1^{m}isx[\text{j}-1]$.

mnstepNone or int, optional

Note: if this argument is None then a default value will be used, determined as follows: if $mtype=1$: $m$; otherwise: $200\times m$.

The maximum number of steps to carry out in the model fitting process.

If $mtype=1$, i.e., a LARS is being performed, the maximum number of steps the algorithm will take is $min(p,n)$ if $prey=0$, otherwise $min(p,n-1)$.

If $mtype=2$, i.e., a forward linear stagewise regression is being performed, the maximum number of steps the algorithm will take is likely to be several orders of magnitude more and is no longer bound by $p$ or $n$.

If $mtype=3$ or $4$, i.e., a LASSO or positive LASSO model is being fit, the maximum number of steps the algorithm will take lies somewhere between that of the LARS and forward linear stagewise regression, again it is no longer bound by $p$ or $n$.

roptNone or float, array-like, shape $\left(\text{lropt}\right)$, optional

Options to control various aspects of the LARS algorithm.

The default value will be used for $ropt[i-1]$ if the length of $ropt$ is less than $i$, therefore, to use the default values for all options $ropt$ need not be set and may be None.

The default value will also be used if an invalid value is supplied for a particular argument, for example, setting $ropt[i-1]=-1$ will use the default value for argument $i$.

$ropt\left[0\right]$

The minimum step size that will be taken.

Default is $100\times \text{eps}$, where $\text{eps}$ is the machine precision returned by machine.precision.

$ropt\left[1\right]$

General tolerance, used amongst other things, for comparing correlations.

Default is $ropt\left[0\right]$.

$ropt\left[2\right]$

If set to $1$, parameter estimates are rescaled before being returned.

If set to $0$, no rescaling is performed.

This argument has no effect when $pred=0$ or $1$.

Default is for the parameter estimates to be rescaled.

$ropt\left[3\right]$

If set to $1$, it is assumed that the model contains an intercept during the model fitting process and when calculating the degrees of freedom.

If set to $0$, no intercept is assumed.

This has no effect on the amount of preprocessing performed on $y$.

Default is to treat the model as having an intercept when $prey=1$ and as not having an intercept when $prey=0$.

$ropt\left[4\right]$

As implemented, the LARS algorithm can either work directly with $y$ and $X$, or it can work with the cross-product matrices, $X^{T}y$ and $X^{T}X$. In most cases it is more efficient to work with the cross-product matrices. This flag allows you direct control over which method is used, however, the default value will usually be the best choice.

If $ropt\left[4\right]=1$, $y$ and $X$ are worked with directly.

If $ropt\left[4\right]=0$, the cross-product matrices are used.

Default is $1$ when $p\ge 500$ and $n<p$ and $0$ otherwise.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

ipint

$p$, number of parameter estimates.

If $isx$ is None, $p=m$, i.e., the number of variables in $d$.

Otherwise $p$ is the number of nonzero values in $isx$.

nstepint

$K$, the actual number of steps carried out in the model fitting process.

bfloat, ndarray, shape $(ip,nstep+2)$

$\beta$ the parameter estimates, with $b[j-1,k-1]={\beta }_{kj}$, the parameter estimate for the $j$th variable, $j=1,2,\dots ,p$ at the $k$th step of the model fitting process, $k=1,2,\dots ,nstep$.

By default, when $pred=2$ or $3$ the parameter estimates are rescaled prior to being returned.

If the parameter estimates are required on the normalized scale, then this can be overridden via $ropt$.

The values held in the remaining part of $b$ depend on the type of preprocessing performed.

If $pred=0$,

$\begin{array}{ccc}b[j-1,nstep]& =& 1\\ b[j-1,nstep+1]& =& 0\end{array}$

If $pred=1$,

$\begin{array}{ccc}b[j-1,nstep]& =& 1\\ b[j-1,nstep+1]& =& {\overline{x}}_j\end{array}$

If $pred=2$,

$\begin{array}{ccc}b[j-1,nstep]& =& 1/\sqrt{x_j^T{x}_j}\\ b[j-1,nstep+1]& =& 0\end{array}$

If $pred=3$,

$\begin{array}{ccc}b[j-1,nstep]& =& 1/\sqrt{{(x_j-{\overline{x}}_j)}^T(x_j-{\overline{x}}_j)}\\ b[j-1,nstep+1]& =& {\overline{x}}_j\end{array}$

for $j=1,2,\dots ,p$.

fitsumfloat, ndarray, shape $(6,mnstep+1)$

Summaries of the model fitting process. When $k=1,2,\dots ,nstep$,

$fitsum[0,k-1]$

${\parallel {\beta }_k\parallel }_1$, the sum of the absolute values of the parameter estimates for the $k$th step of the modelling fitting process. If $pred=2$ or $3$, the scaled parameter estimates are used in the summation.

$fitsum[1,k-1]$

${\text{RSS}}_k$, the residual sums of squares for the $k$th step, where ${\text{RSS}}_k={\Vert y-X^T{\beta }_k\Vert }^2$.

$fitsum[2,k-1]$

${\nu }_k$, approximate degrees of freedom for the $k$th step.

$fitsum[3,k-1]$

$C_p^{\left(k\right)}$, a $C_p$-type statistic for the $k$th step, where $C_p^{\left(k\right)}=\frac{{\text{RSS}}_k}{{\sigma }^2}-n+2{\nu }_k$.

$fitsum[4,k-1]$

${\hat{C}}_k$, correlation between the residual at step $k-1$ and the most correlated variable not yet in the active set $A$, where the residual at step $0$ is $y$.

$fitsum[5,k-1]$

${\hat{\gamma }}_k$, the step size used at step $k$.

In addition

$fitsum[0,nstep]$

$\alpha$, with $\alpha =\overline{y}$ if $prey=1$ and $0$ otherwise.

$fitsum[1,nstep]$

${\text{RSS}}_0$, the residual sums of squares for the null model, where ${\text{RSS}}_0=y^{T}y$ when $prey=0$ and ${\text{RSS}}_0={(y-\overline{y})}^T(y-\overline{y})$ otherwise.

$fitsum[2,nstep]$

${\nu }_0$, the degrees of freedom for the null model, where ${\nu }_0=0$ if $prey=0$ and ${\nu }_0=1$ otherwise.

$fitsum[3,nstep]$

$C_p^{\left(0\right)}$, a $C_p$-type statistic for the null model, where $C_p^{\left(0\right)}=\frac{{\text{RSS}}_0}{{\sigma }^2}-n+2{\nu }_0$.

$fitsum[4,nstep]$

${\sigma }^2$, where ${\sigma }^2=\frac{n-{\text{RSS}}_K}{{\nu }_K}$ and $K=nstep$.

Although the $C_p$ statistics described above are returned when $errno$ = 112 they may not be meaningful due to the estimate ${\sigma }^2$ not being based on the saturated model.

Raises

NagValueError

(errno $11$)

On entry, $mtype=⟨value⟩$.

Constraint: $mtype=1$, $2$, $3$ or $4$.

(errno $21$)

On entry, $pred=⟨value⟩$.

Constraint: $pred=0$, $1$, $2$ or $3$.

(errno $31$)

On entry, $prey=⟨value⟩$.

Constraint: $prey=0$ or $1$.

(errno $41$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $51$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $81$)

On entry, $isx[⟨value⟩]=⟨value⟩$.

Constraint: $isx\left[i\right]=0$ or $1$, for all $i$.

(errno $82$)

On entry, all values of $isx$ are zero.

Constraint: at least one value of $isx$ must be nonzero.

(errno $91$)

On entry, $\text{lisx}=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $\text{lisx}=0$ or $m$.

(errno $111$)

On entry, $mnstep=⟨value⟩$.

Constraint: $mnstep\ge 1$.

(errno $151$)

On entry, $\text{ldb}=⟨value⟩$ and $m=⟨value⟩$.

Constraint: if $\text{lisx}=0$ then $\text{ldb}\ge m$.

(errno $152$)

On entry, $\text{ldb}=⟨value⟩$ and $p=⟨value⟩$.

Constraint: if $\text{lisx}=m$ then $\text{ldb}\ge p$.

(errno $181$)

On entry, $\text{lropt}=⟨value⟩$.

Constraint: $0\le \text{lropt}\le 5$.

Warns

NagAlgorithmicWarning

(errno $112$)

Fitting process did not finish in $mnstep$ steps. Try increasing the size of $mnstep$ and supplying larger output arrays.

All output is returned as documented, up to step $mnstep$, however, $\sigma$ and the $C_p$ statistics may not be meaningful.

(errno $161$)

${\sigma }^2$ is approximately zero and hence the $C_p$-type criterion cannot be calculated. All other output is returned as documented.

(errno $162$)

${\nu }_K=n$, therefore, $\sigma$ has been set to a large value. Output is returned as documented.

(errno $163$)

Degenerate model, no variables added and $nstep=0$. Output is returned as documented.

Notes

lars implements the LARS algorithm of Efron et al. (2004) as well as the modifications needed to perform forward stagewise linear regression and fit LASSO and positive LASSO models.

Given 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 $X$, denoted $x_j$, is a vector of length $n$ representing the $j$th independent variable $x_j$, standardized such that ${\sum }_1^n{x}_{ij}=0$, and ${\sum }_1^n{x}_{ij}^2=1$ and a set of model parameters $\beta$ to be estimated from the observed values, the LARS algorithm can be summarised as:

1. Set $k=1$ and all coefficients to zero, that is $\beta =0$.

2. Find the variable most correlated with $y$, say $x_{j_1}$. Add $x_{j_1}$ to the ‘most correlated’ set $A$. If $p=1$ go to (8).

3. Take the largest possible step in the direction of $x_{j_1}$ (i.e., increase the magnitude of ${\beta }_{j_1}$) until some other variable, say $x_{j_2}$, has the same correlation with the current residual, $y-x_{j_1}{\beta }_{j_1}$.

4. Increment $k$ and add $x_{j_k}$ to $A$.

5. If $\left|A\right|=p$ go to (8).

6. Proceed in the ‘least angle direction’, that is, the direction which is equiangular between all variables in $A$, altering the magnitude of the parameter estimates of those variables in $A$, until the $k$th variable, $x_{j_k}$, has the same correlation with the current residual.

7. Go to (4).

8. Let $K=k$.

As well as being a model selection process in its own right, with a small number of modifications the LARS algorithm can be used to fit the LASSO model of Tibshirani (1996), a positive LASSO model, where the independent variables enter the model in their defined direction (i.e., ${\beta }_{kj}\ge 0$), forward stagewise linear regression (Hastie et al. (2001)) and forward selection (Weisberg (1985)). Details of the required modifications in each of these cases are given in Efron et al. (2004).

The LASSO model of Tibshirani (1996) is given by

$${\text{minimize}}_{\alpha ,{\beta }_k\in R^p}{\Vert y-\alpha -X^T{\beta }_k\Vert }^2\text{subject to}{\parallel {\beta }_k\parallel }_1\le t_k$$

for all values of $t_k$, where $\alpha =\overline{y}=n^{-1}{\sum }_1^n{y}_i$. The positive LASSO model is the same as the standard LASSO model, given above, with the added constraint that

$${\beta }_{kj}\ge 0\text{,}j=1,2,\dots ,p\text{.}$$

Unlike the standard LARS algorithm, when fitting either of the LASSO models, variables can be dropped as well as added to the set $A$. Therefore, the total number of steps $K$ is no longer bounded by $p$.

Forward stagewise linear regression is an iterative procedure of the form:

1. Initialize $k=1$ and the vector of residuals $r_0=y-\alpha$.

2. For each $j=1,2,\dots ,p$ calculate $c_j=x_j^T{r}_{k-1}$. The value $c_j$ is, therefore, proportional to the correlation between the $j$th independent variable and the vector of previous residual values, $r_k$.

3. Calculate $j_k={\text{argmax}}_j\left|c_j\right|$, the value of $j$ with the largest absolute value of $c_j$.

4. If $\left|c_{j_k}\right|<ϵ$ then go to (7).

5. Update the residual values, with

   $$r_k=r_{k-1}+\delta sign\left(c_{j_k}\right)x_{j_k}$$

   where $\delta$ is a small constant and $sign\left(c_{j_k}\right)=-1$ when $c_{j_k}<0$ and $1$ otherwise.

6. Increment $k$ and go to (2).

7. Set $K=k$.

If the largest possible step were to be taken, that is $\delta =\left|c_{j_k}\right|$ then forward stagewise linear regression reverts to the standard forward selection method as implemented in linregm_fit_onestep().

The LARS procedure results in $K$ models, one for each step of the fitting process. In order to aid in choosing which is the most suitable Efron et al. (2004) introduced a $C_p$-type statistic given by

$$C_p^{\left(k\right)}=\frac{{\Vert y-X^T{\beta }_k\Vert }^2}{{\sigma }^2}-n+2{\nu }_k\text{,}$$

where ${\nu }_k$ is the approximate degrees of freedom for the $k$th step and

$${\sigma }^2=\frac{n-y^{T}y}{{\nu }_K}\text{.}$$

One way of choosing a model is, therefore, to take the one with the smallest value of $C_p^{\left(k\right)}$.

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_ex.main()