Estimates of parameters of a general linear regression model for given constraints

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NAG[g02dkc] NAG[nag_regsn_mult_linear_tran_model] - Estimates of parameters of a general linear regression model for given constraints

Calling Sequence

g02dkc(p, c, b, rss, df, se, cov, 'ip'=ip, 'iconst'=iconst, 'tdc'=tdc, 'fail'=fail)

nag_regsn_mult_linear_tran_model(. . .)

Parameters

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

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

On entry: p as returned by g02dac (nag_regsn_mult_linear) and g02ddc (nag_regsn_mult_linear_upd_model).

c - Matrix(1..ip, 1..tdc, datatype=float[8], order=C_order);

On entry: the iconst constraints stored by column, i.e., the th constraint is stored in the th column of c.

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

On entry: the argument estimates computed by using the singular value decomposition, .

On exit: the argument estimates of the arguments with the constraints imposed, .

rss - float;

On entry: the residual sum of squares as returned by g02dac (nag_regsn_mult_linear) or g02ddc (nag_regsn_mult_linear_upd_model).

Constraint: . .

df - float;

On entry: the degrees of freedom associated with the residual sum of squares as returned by g02dac (nag_regsn_mult_linear) or g02ddc (nag_regsn_mult_linear_upd_model).

Constraint: . .

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

On exit: the standard error of the argument estimates in b.

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

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

On exit: the upper triangular part of the variance-covariance matrix of the ip argument estimates given in b. They are stored packed by column, i.e., the covariance between the argument estimate given in and the argument estimate given in , , is stored in , for and .

'ip'=ip - integer; (optional)

Default value: the first dimension of the arrays c, b, se.

On entry: the number of terms in the linear model, .

Constraint: . .

'iconst'=iconst - integer; (optional)

Default value: the second dimension of the array c.

On entry: the number of constraints to be imposed on the arguments, .

Constraint: . .

'tdc'=tdc - integer; (optional)

On entry: the second dimension of the array c as declared in the function from which nag_regsn_mult_linear_tran_model (g02dkc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_regsn_mult_linear_tran_model (g02dkc) calculates the estimates of the arguments of a general linear regression model for given constraints from the singular value decomposition results.

Description

nag_regsn_mult_linear_tran_model (g02dkc) computes the estimates given a set of linear constraints for a general linear regression model which is not of full rank. It is intended for use after a call to g02dac (nag_regsn_mult_linear) or g02ddc (nag_regsn_mult_linear_upd_model).

In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the argument estimates, , and their variance-covariance matrix. Details of the SVD are made available, in the form of the matrix :

P^(*)=([[D^(-1)P[1]^T],[P[0]^T]])

as described by g02dac (nag_regsn_mult_linear) and g02ddc (nag_regsn_mult_linear_upd_model).

Alternative solutions can be formed by imposing constraints on the arguments. If there are arguments and the rank of the model is , then constraints will have to be imposed to obtain a unique solution.

Let be a by matrix of constraints, such that

then the new argument estimates are given by:

[[(beta)[c],=,A(beta)[svd]],[,=,((I-P)[0],(C^T,P[0])^(-1))(beta)[svd]]],

where is the identity matrix, and the variance-covariance matrix is given by:

provided exists.

	Error Indicators and Warnings
	"NE_2_INT_ARG_GE" On entry, while . These arguments must satisfy . "NE_ALLOC_FAIL" Dynamic memory allocation failed. "NE_INT_ARG_LE" On entry, iconst must not be less than or equal to 0: . "NE_INT_ARG_LT" On entry, ip must not be less than 1: . "NE_MAT_NOT_FULL_RANK" Matrix c does not give a model of full rank. "NE_REAL_ARG_LE" On entry, rss must not be less than or equal to 0.0: .

	Accuracy
	It should be noted that due to rounding errors an argument that should be zero when the constraints have been imposed may be returned as a value of order machine precision.

	Further Comments
	nag_regsn_mult_linear_tran_model (g02dkc) is intended for use in situations in which dummy (0-1) variables have been used such as in the analysis of designed experiments when the user does not wish to change the arguments of the model to give a full rank model. The function is not intended for situations in which the relationships between the independent variables are only approximate.

Examples

ip := 5:
iconst := 1:
tdc := 5:
rss := 22.22679999999998:
df := 8:
p := Vector([1.135198279858991, 1.13078348304906, 1.234006904805161, 1.228003039942231, 1.148419798367432, 3.872983346207417, 1.732050807568878, 1.732050807568877, 1.732050807568877, 3.166264259090703e-16, 0.2309401076758502, 0.05773502691896256, 0.05773502691896257, 0.0577350269189626, 0.05773502691896259, -8.986850838355147e-17, -0.2180297316413536, -0.3448196554213736, 0.3466016006011757, 0.2162477864615517, 3.472144456476812e-18, 0.4492089773025322, -0.3565253366776346, -0.03799964639604183, -0.05468399422885582, 4.280983223691876e-18, -0.02596788076090121, -0.06315923956148034, -0.3583620478433494, 0.4474891681657309, -0.447213595499958, 0.4472135954999579, 0.447213595499958, 0.4472135954999578, 0.4472135954999579], datatype=float[8]):
c := Matrix([[0, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0]], datatype=float[8], order='C_order'):
b := Vector([30.55666666666667, 5.446666666666667, 6.743333333333339, 11.04666666666667, 7.320000000000003], datatype=float[8]):
se := Vector(5, datatype=float[8]):
cov := Vector(15, datatype=float[8]):
NAG:-g02dkc(p, c, b, rss, df, se, cov, 'ip' = ip, 'iconst' = iconst, 'tdc' = tdc):

se;

cov;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20 (3) 2–25 Searle S R (1971) Linear Models Wiley g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.