Computes residual sums of squares for all possible linear regressions for a set of independent variables

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NAG[g02eac] NAG[nag_all_regsn] - Computes residual sums of squares for all possible linear regressions for a set of independent variables

Calling Sequence

g02eac(mean, x, var_names, sx, y, nmod, model, rss, nterms, mrank, 'n'=n, 'm'=m, 'wt'=wt, 'fail'=fail)

nag_all_regsn(. . .)

Parameters

mean - String;

On entry: indicates if a mean term is to be included.

A mean term, intercept, will be included in the model.

The model will pass through the origin, zero-point.

Constraint: "Nag_MeanInclude" or "Nag_MeanZero". .

x - Matrix(1..dim1, 1..m, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: must contain the th observation for the th independent variable, for ; .

var_names - Vector(1..m, datatype=REF(string[512]));

On entry: must contain the name of the independent variable in row of x, for .

sx - Vector(1..m, datatype=integer[kernelopts('wordsize')/8]);

On entry: indicates which independent variables are to be considered in the model.

The variable contained in the th column of x is included in all regression models, i.e., is a forced variable.

The variable contained in the th column of x is included in the set from which the regression models are chosen, i.e., is a free variable.

The variable contained in the th column of x is not included in the models.

We denote the total number of free variables a , where is the number of free variables in the model, see nmod to mrank.

Constraint: , for and at least one value of . .

y - Vector(1..n, datatype=float[8]);

On entry: must contain the th observation on the dependent variable, , for .

nmod - assignable;

Note: On exit the variable nmod will have a value of type integer.

On exit: the total number of models for which residual sums of squares have been calculated. If there are independent variables then .

model - Vector(1..dim, datatype=REF(string[512]));

Note: the dimension, dim, of the array model must be at least big enough to hold the names of all the free independent variables which appear in all the models. This will never exceed , where is the number of free variables in the model.

On exit: the names of the independent variables in each model, represented as pointers to the names provided by you in var_names. The model names are stored as follows:

if the first model has three names, i.e., ; then , and will contain these three names;

if the second model has two names, i.e., ; then , will contain these two names.

rss - Vector(1.., datatype=float[8]);

Note: the dimension, dim, of the array rss must be at least .

On exit: contains the residual sum of squares for the th model, for .

nterms - Vector(1.., datatype=integer[kernelopts('wordsize')/8]);

Note: the dimension, dim, of the array nterms must be at least .

On exit: contains the number of independent variables in the th model, not including the mean if one is fitted, for .

mrank - Vector(1.., datatype=integer[kernelopts('wordsize')/8]);

Note: the dimension, dim, of the array mrank must be at least .

On exit: contains the rank of the residual sum of squares for the th model, i.e., model with smallest sum of squares has rank 1.

'n'=n - integer; (optional)

On entry: the number of observations.

Constraint: . .

'm'=m - integer; (optional)

On entry: the maximum number of variables contained in x.

Constraint: . .

'wt'=wt - Vector(1..dim, datatype=float[8]); (optional)

Note: the dimension, dim, of the array wt must be at least .

On entry: optionally, the weights to be used in the weighted regression.

If , then the th observation is not included in the model, in which case the effective number of observations is the number of observations with non-zero weights.

If weights are not provided then wt must be set to the null pointer, i.e., (double *)0, and the effective number of observations is n.

Constraint: if , for . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_all_regsn (g02eac) calculates the residual sums of squares for all possible linear regressions for a given set of independent variables.

Description

For a set of possible independent variables there are linear regression models with from zero to independent variables in each model. For example if and the variables are , and then the possible models are:

null model

and

, and .

nag_all_regsn (g02eac) calculates the residual sums of squares from each of the possible models. The method used involves a decomposition of the matrix of possible independent variables. Independent variables are then moved into and out of the model by a series of Givens rotations and the residual sums of squares computed for each model; see Clark (1981) and Smith and Bremner (1989).

The computed residual sums of squares are then ordered first by increasing number of terms in the model, then by decreasing size of residual sums of squares. So the first model will always have the largest residual sum of squares and the th will always have the smallest. This aids you in selecting the best possible model from the given set of independent variables.

nag_all_regsn (g02eac) allows you to specify some independent variables that must be in the model, the forced variables. The other independent variables from which the possible models are to be formed are the free variables.

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_FREE_VARS"

There are no free x variables.

"NE_FULL_RANK"

Full model is not of full rank.

"NE_INDEP_VARS_OBS"

Number of requested -variables number of observations.

"NE_INT"

On entry, . Constraint: .

"NE_INT_ARRAY_ELEM_CONS"

On entry, .

"NE_INTERNAL_ERROR"

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.

"NE_REAL_ARRAY_ELEM_CONS"

On entry, .

	Accuracy
	For a discussion of the improved accuracy obtained by using a method based on the decomposition see Smith and Bremner (1989).

	Further Comments
	g02ecc (nag_cp_stat) may be used to compute and -values from the results of nag_all_regsn (g02eac). If a mean has been included in the model and no variables are forced in then contains the total sum of squares and in many situations a reasonable estimate of the variance of the errors is given by .

Examples

mean := "Nag_MeanInclude":
n := 20:
m := 6:
x := Matrix([[0, 1125, 232, 7160, 85.90000000000001, 8905], [7, 920, 268, 8804, 86.5, 7388], [15, 835, 271, 8108, 85.2, 5348], [22, 1000, 237, 6370, 83.8, 8056], [29, 1150, 192, 6441, 82.09999999999999, 6960], [37, 990, 202, 5154, 79.2, 5690], [44, 840, 184, 5896, 81.2, 6932], [58, 650, 200, 5336, 80.59999999999999, 5400], [65, 640, 180, 5041, 78.40000000000001, 3177], [72, 583, 165, 5012, 79.3, 4461], [80, 570, 151, 4825, 78.7, 3901], [86, 570, 171, 4391, 78, 5002], [93, 510, 243, 4320, 72.3, 4665], [100, 555, 147, 3709, 74.90000000000001, 4642], [107, 460, 286, 3969, 74.40000000000001, 4840], [122, 275, 198, 3558, 72.5, 4479], [129, 510, 196, 4361, 57.7, 4200], [151, 165, 210, 3301, 71.8, 3410], [171, 244, 327, 2964, 72.5, 3360], [220, 79, 334, 2777, 71.90000000000001, 2599]], datatype=float[8]):
var_names := Vector(["DAY", "BOD", "TKN", "TS", "TVS", "COD"], datatype=string):
sx := Vector([0, 1, 1, 1, 1, 1], datatype=integer[kernelopts('wordsize')/8]):
free_vars := 1:
for j from 1 to m do
  if sx[j] = 1 then
     free_vars := free_vars*2:
  end if:
end do:
y := Vector([1.5563, 0.8976, 0.7482, 0.716, 0.301, 0.3617, 0.1139, 0.1139, -0.2218, -0.1549, 0, 0, -0.0969, -0.2218, -0.3979, -0.1549, -0.2218, -0.3979, -0.5229, -0.0458], datatype=float[8]):
wt := Vector([], datatype=float[8]):
model := Vector(free_vars*m, datatype=string, fill=""):
rss := Vector(free_vars, datatype=float[8]):
nterms := Vector(free_vars, datatype=integer[kernelopts('wordsize')/8]):
mrank := Vector(free_vars, datatype=integer[kernelopts('wordsize')/8]):
NAG:-g02eac(mean, x, var_names, sx, y, nmod, model, rss, nterms, mrank, 'n' = n, 'm' = m, 'wt' = wt):

nmod;

rss;

nterms;

mrank;

	See Also
	Clark M R B (1981) A Givens algorithm for moving from one linear model to another without going back to the data Appl. Statist. 30 198–203 Smith D M and Bremner J M (1989) All possible subset regressions using the decomposition Comput. Statist. Data Anal. 7 217–236 Weisberg S (1985) Applied Linear Regression Wiley g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.