Add/delete an observation to/from a general linear regression model

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NAG[g02dcc] NAG[nag_regsn_mult_linear_addrem_obs] - Add/delete an observation to/from a general linear regression model

Calling Sequence

g02dcc(update, mean, sx, q, x, ix, y, rss, 'm'=m, 'tdq'=tdq, 'ip'=ip, 'nr'=nr, 'tdx'=tdx, 'wt'=wt, 'fail'=fail)

nag_regsn_mult_linear_addrem_obs(. . .)

Parameters

update - String;

On entry: indicates if an observation is to be added or deleted.

The observation is added.

The observation is deleted.

Constraint: "Nag_ObservAdd" or "Nag_ObservDel". .

mean - String;

On entry: indicates if a mean has been used in the model.

A mean term or intercept will have been included in the model by g02dac (nag_regsn_mult_linear).

A model with no mean term or intercept will have been fitted by g02dac (nag_regsn_mult_linear).

Constraint: "Nag_MeanInclude" or "Nag_MeanZero". .

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

On entry: if is greater than 0, then the value contained in is to be included as a value of , an observation on an independent variable, for .

Constraint: if , then exactly elements of sx must be and if , then exactly ip elements of sx must be . .

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

On entry: q must be array q as output by g02dac (nag_regsn_mult_linear), g02dec (nag_regsn_mult_linear_add_var), g02dfc (nag_regsn_mult_linear_delete_var), or a previous call to nag_regsn_mult_linear_addrem_obs (g02dcc).

On exit: the first ip elements of the first column of q will contain , the upper triangular part of columns 2 to will contain , the remainder is unchanged.

x - Matrix(1..nr, 1..tdx, datatype=float[8], order=C_order);

On entry: the ip values for the dependent variables of the observation to be added or deleted, . The positions of the values x extracted depends on ix and tdx.

ix - integer;

On entry: the row of the notional two dimensional array x that contains the values for the dependent variables of the observation to be added or deleted.

Constraint: . .

y - float;

On entry: the value of the dependent variable for the observation to be added or deleted, .

rss - assignable;

Note: On exit the variable rss will have a value of type float.

On entry: the value of the residual sums of squares for the original set of observations.

Constraint: . .

On exit: the updated values of the residual sums of squares.

Note: this will only be valid if the model is of full rank.

'm'=m - integer; (optional)

Default value: the first dimension of the array sx and the second dimension of the array sxthe array x.

On entry: the total number of independent variables in the data set.

Constraint: . .

'tdq'=tdq - integer; (optional)

On entry: the second dimension of the array q as declared in the function from which nag_regsn_mult_linear_addrem_obs (g02dcc) is called.

Constraint: . .

'ip'=ip - integer; (optional)

Default value: the dimension of the array q.

On entry: the number of linear terms in general linear regression model (including mean if there is one).

Constraint: . .

'nr'=nr - integer; (optional)

Default value: the first dimension of the array x.

On entry: the number of rows of the notional two dimensional array x.

Constraint: . .

'tdx'=tdx - integer; (optional)

On entry: the second dimension of the array x as declared in the function from which nag_regsn_mult_linear_addrem_obs (g02dcc) is called.x

Constraint: . .

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

On entry: if the new observation is to be weighted, then wt must contain the weight to be used with the new observation. If , then the observation is not included in the model. If the new observation is to be unweighted, then a null pointer, (double *)0, must be passed.

Constraint: if the new observation is to be weighted .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_regsn_mult_linear_addrem_obs (g02dcc) adds or deletes an observation from a general regression model fitted by g02dac (nag_regsn_mult_linear).

Description

g02dac (nag_regsn_mult_linear) fits a general linear regression model to a data set. The user may wish to change the model by either adding or deleting an observation from the data set. nag_regsn_mult_linear_addrem_obs (g02dcc) takes the results from g02dac (nag_regsn_mult_linear) and makes the required changes to the vector and the upper triangular matrix produced by g02dac (nag_regsn_mult_linear). The regression coefficients, standard errors and the variance-covariance matrix of the regression coefficients can be obtained from g02ddc (nag_regsn_mult_linear_upd_model) after all required changes to the data set have been made.

g02dac (nag_regsn_mult_linear) performs a decomposition on the (weighted) matrix of independent variables. To add a new observation to a model with arguments the upper triangular matrix and vector , the first elements of , are augmented by the new observation on independent variables in and dependent variable . Givens rotations are then used to restore the upper triangular form.

([[R],[ ],[c[1]],[x,y]]) ⟶ ([[R^(*),c[1]^(*)],[y^(*),0]])

To delete an observation Givens rotations are applied to give:

([[R,c[1]]]) ⟶ ([[R^(*),c[1]^(*)],[x,y]])

Note: only the and upper part of the are updated, the remainder of the matrix is unchanged.

Error Indicators and Warnings

"NE_2_INT_ARG_GT"

On entry, while . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, update had an illegal value.

"NE_INT_ARG_LT"

On entry, ip must not be less than 1: .

"NE_IP_INCOMP_WITH_SX"

On entry, for , number of non-zero values of sx must be equal to : number of non-zero values of , .

"NE_MAT_NOT_UPD"

The matrix could not be updated: to, either, delete non-existent observation, or, add an observation to matrix with zero diagonal element.

"NE_REAL_ARG_LT"

On entry, wt must not be less than 0.0: .

"NE_RSS_NOT_UPD"

The rss could not be updated because the input rss was less than the calculated decrease in rss when the new observation was deleted.

	Accuracy
	Higher accuracy is achieved by updating the matrix rather than the traditional methods of updating X'X.

Further Comments

Care should be taken with the use of this function.

a. It is possible to delete observations which were not included in the original model.

b. If several additions/deletions have been performed the user is advised to recompute the regression using g02dac (nag_regsn_mult_linear).

c. Adding or deleting observations can alter the rank of the model. Such changes will only be detected when a call to g02ddc (nag_regsn_mult_linear_upd_model) has been made. g02ddc (nag_regsn_mult_linear_upd_model) should also be used to compute the new residual sum of squares when the model is not of full rank.

nag_regsn_mult_linear_addrem_obs (g02dcc) may also be used after g02dec (nag_regsn_mult_linear_add_var) and g02dfc (nag_regsn_mult_linear_delete_var).

Examples

update := "Nag_ObservDel":
mean := "Nag_MeanZero":
m := 4:
tdq := 5:
ip := 4:
nr := 12:
tdx := 4:
ix := 12:
y := 37.89:
rss := 5274.815851111111:
sx := Vector([1, 1, 1, 1], datatype=integer[kernelopts('wordsize')/8]):
q := Matrix([[-72.94999999999999, -2, -0.5, -0.5, -0.5], [45.48298632955564, 0, 1.6583123951777, 0.4522670168666454, 0.4522670168666454], [56.11154475279979, 0, -0.6030226891555273, 1.882937743382544, 0.2896827297511605], [-42.04198673812751, 0, -0, -0.4963538360840667, -1.860521018838127]], datatype=float[8], order='C_order'):
x := Matrix([[1, 0, 0, 0], [0, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 1, 0, 0], [0, 0, 0, 1], [1, 0, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [0, 0, 1, 0], [1, 1, 1, 1]], datatype=float[8], order='C_order'):
wt := Vector([], datatype=float[8]):
NAG:-g02dcc(update, mean, sx, q, x, ix, y, rss, 'm' = m, 'tdq' = tdq, 'ip' = ip, 'nr' = nr, 'tdx' = tdx, 'wt' = wt):

rss;

	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 g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.