The function may be called by the names: g02mac, nag_correg_lars or nag_lars.
g02mac 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 observed values,
and an design matrix , where the th column of , denoted , is a vector of length representing the th independent variable , standardized such that
and a set of model parameters to be estimated from the observed values, the LARS algorithm can be summarised as:
1.Set and all coefficients to zero, that is .
2.Find the variable most correlated with , say . Add to the ‘most correlated’ set . If go to 8.
3.Take the largest possible step in the direction of (i.e., increase the magnitude of ) until some other variable, say , has the same correlation with the current residual, .
6.Proceed in the ‘least angle direction’, that is, the direction which is equiangular between all variables in , altering the magnitude of the parameter estimates of those variables in , until the th variable, , has the same correlation with the current residual.
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., ), 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).
If the largest possible step were to be taken, that is then forward stagewise linear regression reverts to the standard forward selection method as implemented in g02eec.
The LARS procedure results in 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 -type statistic given by
where is the approximate degrees of freedom for the th step and
One way of choosing a model is, therefore, to take the one with the smallest value of .
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
1: – Nag_LARSModelTypeInput
On entry: indicates the type of model to fit.
LARS is performed.
Forward linear stagewise regression is performed.
LASSO model is fit.
A positive LASSO model is fit.
, , or .
2: – Nag_LARSPreProcessInput
On entry: 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.
No preprocessing is performed.
Each of the independent variables,
, for , are mean centred prior to fitting the model. The means of the independent variables, , are returned in b, with
, for .
Each independent variable is normalized, with the th variable scaled by . The scaling factor used by variable is returned in .
As and , all of the independent variables are mean centred prior to being normalized.
, , or .
3: – Nag_LARSPreProcessInput
On entry: indicates the type of data preprocessing to perform on the dependent variable supplied in y.
No preprocessing is performed, this is equivalent to setting .
The dependent variable, , is mean centred prior to fitting the model, so . Which is equivalent to fitting a non-penalized intercept to the model and the degrees of freedom etc. are adjusted accordingly.
The value of used is returned in .
4: – IntegerInput
On entry: , the number of observations.
5: – IntegerInput
On entry: , the total number of independent variables.
6: – const doubleInput
Note: the dimension, dim, of the array d
must be at least
On entry: , the data, which along with pred and isx, defines the design matrix . The th observation for the th variable must be supplied in
, for and .
7: – IntegerInput
On entry: the stride separating row elements in the two-dimensional data stored in the array d.
8: – const IntegerInput
Note: the dimension, dim, of the array isx
must be at least
, when .
On entry: indicates which independent variables from d will be included in the design matrix, .
If isx is NULL, all variables are included in the design matrix.
Otherwise must be set as follows, for :
To indicate that the th variable, as supplied in d, is included in the design matrix;
To indicated that the th variable, as supplied in d, is not included in the design matrix;
or and at least one value of , for .
9: – const doubleInput
On entry: , the observations on the dependent variable.
10: – IntegerInput
On entry: the maximum number of steps to carry out in the model fitting process.
If , the maximum number of steps the algorithm will take is if , otherwise .
If , the maximum number of steps the algorithm will take is likely to be several orders of magnitude more and is no longer bound by or .
If or , 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 or .
11: – Integer *Output
On exit: , number of parameter estimates.
If isx is NULL, , i.e., the number of variables in d.
On exit: , the actual number of steps carried out in the model fitting process.
13: – doubleOutput
Note: the dimension, dim, of the array b
must be at least
On exit: the parameter estimates, with , the parameter estimate for the th variable, at the th step of the model fitting process, .
By default, when or 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.
14: – IntegerInput
On entry: the stride separating row elements in the two-dimensional data stored in the array b.
, where is the number of parameter estimates as described in ip.
15: – doubleOutput
On exit: summaries of the model fitting process. When ,
, the sum of the absolute values of the parameter estimates for the th step of the modelling fitting process. If or , the scaled parameter estimates are used in the summation.
, the residual sums of squares for the th step, where .
, approximate degrees of freedom for the th step.
, a -type statistic for the th step, where .
, correlation between the residual at step and the most correlated variable not yet in the active set , where the residual at step is .
, the step size used at step .
, with if and otherwise.
, the residual sums of squares for the null model, where when and otherwise.
, the degrees of freedom for the null model, where if and otherwise.
, a -type statistic for the null model, where .
, where and .
Although the statistics described above are returned when NW_LIMIT_REACHED they may not be meaningful due to the estimate not being based on the saturated model.
16: – const doubleInput
On entry: optional parameters to control various aspects of the LARS algorithm.
The default value will be used for if , therefore, setting will use the default values for all optional parameters and ropt need not be set and may be NULL. The default value will also be used if an invalid value is supplied for a particular argument, for example, setting will use the default value for argument .
The minimum step size that will be taken.
Default is , where is the machine precision returned by X02AJC.
General tolerance, used amongst other things, for comparing correlations.
Default is .
If set to , parameter estimates are rescaled before being returned.
If set to , no rescaling is performed.
This argument has no effect when or .
Default is for the parameter estimates to be rescaled.
If set to , it is assumed that the model contains an intercept during the model fitting process and when calculating the degrees of freedom.
If set to , 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 and as not having an intercept when .
As implemented, the LARS algorithm can either work directly with and , or it can work with the cross-product matrices, and . 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.
On entry, and .
Constraint: if isx is not NULL then .
On entry, and .
On entry, argument had an illegal value.
On entry, .
On entry, .
On entry, all values of isx are zero.
Constraint: at least one value of isx must be nonzero.
On entry, .
Constraint: or , for all .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
On entry, .
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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, and the statistics may not be meaningful.
, therefore, has been set to a large value. Output is returned as documented.
is approximately zero and hence the -type criterion cannot be calculated. All other output is returned as documented.
Degenerate model, no variables added and . Output is returned as documented.
8Parallelism and Performance
g02mac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02mac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
g02mac returns the parameter estimates at various points along the solution path of a LARS, LASSO or stagewise regression analysis. If the solution is required at a different set of points, for example when performing cross-validation, then g02mcc can be used.
For datasets with a large number of observations, , it may be impractical to store the full matrix in memory in one go. In such instances the cross-product matrices and can be calculated, using for example, multiple calls to g02buc and g02bzc, and g02mbc called to perform the analysis.
The amount of workspace used by g02mac depends on whether the cross-product matrices are being used internally (as controlled by ropt). If the cross-product matrices are being used then g02mac internally allocates approximately elements of real storage compared to elements when and are used directly. In both cases approximately elements of integer storage are also used. If a forward linear stagewise analysis is performed than an additional elements of real storage are required.
This example performs a LARS on a simulated dataset with observations and independent variables.