Estimates and standard errors of parameters of a general linear model for given constraints

はじめる前に
新機能一覧
Maple ワークシートの作成
Mapleワークシートを共有
Maple ウィンドウのカスタマイズ
Maple システムのカスタマイズ
コネクティビティ
Mathematics
数学
物理
プログラミング
グラフィックス
Student パッケージ
Science and Engineering
科学・エンジニアリング
アプリケーションと例題
リファレンス
システム
MapleSim
MapleSim ツールボックス
MapleSim ヘルプ
Tasks
Toolboxes
- BlockImporter
- Global Optimization
- Grid Computing
- Maple-NAG Connector
  - General information
  - Computational Routines
    - a02 (Complex Arithmetic)
    - c02 (Zeros of Polynomials)
    - c05 (Roots of One or More Transcendental Equations)
    - c06 (Fourier Transforms)
    - d01 (Quadrature)
    - d02 (Ordinary Differential Equations)
    - d03 (Partial Differential Equations)
    - d06 (Mesh Generation)
    - e01 (Interpolation)
    - e02 (Curve and Surface Fitting)
    - e04 (Minimizing or Maximizing a Function)
    - f01 (Matrix Factorizations)
    - f02 (Eigenvalues and Eigenvectors)
    - f03 (Determinants)
    - f04 (Simultaneous Linear Equations)
    - f06 (Linear Algebra Support Functions)
    - f07 (Linear Equations (LAPACK))
    - f08 (Least-squares and Eigenvalue Problems (LAPACK))
    - f11 (Large Scale Linear Systems)
    - f16 (NAG Interface to BLAS)
    - g01 (Simple Calculations on Statistical Data)
    - g02 (Correlation and Regression Analysis)
      - g02 Introduction
      - g02brc (nag_ken_spe_corr_coeff)
      - g02btc (nag_sum_sqs_update)
      - g02buc (nag_sum_sqs)
      - g02bwc (nag_cov_to_corr)
      - g02bxc (nag_corr_cov)
      - g02byc (nag_partial_corr)
      - g02cac (nag_simple_linear_regression)
      - g02cbc (nag_regress_confid_interval)
      - g02dac (nag_regsn_mult_linear)
      - g02dcc (nag_regsn_mult_linear_addrem_obs)
      - g02ddc (nag_regsn_mult_linear_upd_model)
      - g02dec (nag_regsn_mult_linear_add_var)
      - g02dfc (nag_regsn_mult_linear_delete_var)
      - g02dgc (nag_regsn_mult_linear_newyvar)
      - g02dkc (nag_regsn_mult_linear_tran_model)
      - g02dnc (nag_regsn_mult_linear_est_func)
      - g02eac (nag_all_regsn)
      - g02ecc (nag_cp_stat)
      - g02eec (nag_step_regsn)
      - g02fac (nag_regsn_std_resid_influence)
      - g02fcc (nag_durbin_watson_stat)
      - g02gac (nag_glm_normal)
      - g02gbc (nag_glm_binomial)
      - g02gcc (nag_glm_poisson)
      - g02gdc (nag_glm_gamma)
      - g02gkc (nag_glm_tran_model)
      - g02gnc (nag_glm_est_func)
      - g02hac (nag_robust_m_regsn_estim)
      - g02hbc (nag_robust_m_regsn_wts)
      - g02hdc (nag_robust_m_regsn_user_fn)
      - g02hfc (nag_robust_m_regsn_param_var)
      - g02hkc (nag_robust_corr_estim)
      - g02hlc (nag_robust_m_corr_user_fn)
      - g02hmc (nag_robust_m_corr_user_fn_no_derr)
    - g03 (Multivariate Methods)
    - g04 (Analysis of Variance)
    - g05 (Random Number Generators)
    - g07 (Univariate Estimation)
    - g08 (Nonparametric Statistics)
    - g10 (Smoothing in Statistics)
    - g11 (Contingency Table Analysis)
    - g12 (Survival Analysis)
    - g13 (Time Series Analysis)
    - h (Operations Research)
    - m01 (Sorting)
    - s (Approximations of Special Functions)
    - x01 (Mathematical Constants)
    - x02 (Machine Constants)
    - x04 (Input-Output Utilities)
  - Getting Started
  - Example Worksheet
エラーメッセージガイド
マニュアル
数学アプリ

NAG[g02gkc] NAG[nag_glm_tran_model] - Estimates and standard errors of parameters of a general linear model for given constraints

Calling Sequence

g02gkc(v, c, b, scale, se, cov, 'ip'=ip, 'nclin'=nclin, 'tdv'=tdv, 'tdc'=tdc, 'fail'=fail)

nag_glm_tran_model(. . .)

Parameters

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

On entry: v as returned by g02gac (nag_glm_normal), g02gbc (nag_glm_binomial), g02gcc (nag_glm_poisson) or g02gdc (nag_glm_gamma).

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

On entry: the nclin 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, .

scale - float;

On entry: the estimate of the scale argument.

For results from g02gac (nag_glm_normal) and g02gdc (nag_glm_gamma) then scale is the scale argument, for the model and respectively.

For results from g02gbc (nag_glm_binomial) and g02gcc (nag_glm_poisson) then scale should be set to 1.0.

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 v, c, b, se and the second dimension of the arrays v, c, b, sethe array v.

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

Constraint: . .

'nclin'=nclin - integer; (optional)

Default value: the second dimension of the array c.

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

Constraint: . .

'tdv'=tdv - integer; (optional)

On entry: the second dimension of the array v as declared in the function from which nag_glm_tran_model (g02gkc) is called.

Constraint: ..

tdv should be as supplied to g02gac (nag_glm_normal), g02gbc (nag_glm_binomial), g02gcc (nag_glm_poisson) or g02gdc (nag_glm_gamma).

'tdc'=tdc - integer; (optional)

On entry: the second dimension of the array c as declared in the function from which nag_glm_tran_model (g02gkc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_glm_tran_model (g02gkc) calculates the estimates of the arguments of a generalized linear model for given constraints from the singular value decomposition results.

Description

nag_glm_tran_model (g02gkc) computes the estimates given a set of linear constraints for a generalized linear model which is not of full rank. It is intended for use after a call to g02gac (nag_glm_normal), g02gbc (nag_glm_binomial), g02gcc (nag_glm_poisson) or g02gdc (nag_glm_gamma).

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 g02gac (nag_glm_normal), g02gbc (nag_glm_binomial), g02gcc (nag_glm_poisson) and g02gdc (nag_glm_gamma).

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, nclin 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, scale 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_glm_tran_model (g02gkc) 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 := 9:
nclin := 2:
tdv := 15:
tdc := 2:
scale := 1:
v := Matrix([[4.890297476493483, 132.9931304975524, 0.08671323636775562, 11.53226476012203, 0.6875039713237752, 0.6035396163882445, 0.01901067844788449, 0.008405335397184032, 0.008587821997748728, 0.002017521052951722, 0.006018698150011753, 0.002569439508216105, 0.00571026811894922, 0.003195800397035, 0.001516472273672411], [4.150630280659896, 63.47399412403458, 0.1255168658332746, 7.96705680436851, 0.4385677120269487, 0.5137644803624517, -0.0002104052850739223, -0.0336264460190696, 0.03352834001769895, -0.0001122992837032632, -8.410523978668439e-05, -1.834202533241134e-05, -7.350180559028091e-05, -2.503731813921335e-05, -9.418896225330081e-06], [4.847173050021615, 127.379784101069, 0.08860326907783024, 11.28626528578294, -1.207211262022789, 0.5962906923855552, -0.0005932662634519351, -0.0003445349204235177, -0.0003331755811170331, 8.444423808861337e-05, 0.04164067761137812, -0.0008346027101391711, -0.03974432490123236, -0.001281738894244943, -0.0003732773692135901], [4.347583499449089, 77.29146221158084, 0.1137455037136346, 8.791556302019616, 0.1936290267369067, 0.5316079856446034, -0.005604448606553031, -0.004730762234353659, -0.00466040874579493, 0.003786722373595564, 0.03079195529372065, -0.01961182893501798, 0.03434336417204357, -0.04455401443460469, -0.006573924702694607], [3.660007365784039, 38.86162911664574, 0.160412976990412, 6.23390961729842, 0.02218333268101244, 0.4819807366090338, -0.01246501079735565, 0.02562719062269762, 0.02538801446619449, -0.06348021588624773, -0.003268009116713832, 0.01218311975746821, -0.003443738617825775, -0.01974708083814366, 0.00181069801785938], [4.906081344202833, 135.1089303020152, 0.08603159451608586, 11.62363670724508, -0.3553126814370607, 0.6083327470988833, -0.007494797532342898, 0.006087596749368361, 0.006034528387724049, -0.01961692266943529, 0.009789429893167638, -0.06230084141169705, 0.01026746382279116, 0.04102802603588974, -0.006278875872494371], [4.166414148369245, 64.48380766743846, 0.1245301935277972, 8.030181048235367, 0.1880789664600944, 0.5196429754104228, -0.007856295692343652, -0.001520639263699584, -0.001513684000176263, -0.004821972428467786, 0.01450100243772307, 0.03053649776467426, 0.01481400539841175, 0.02215701246445585, -0.08986481375760853], [4.862956917730965, 129.4062806673598, 0.08790676991659417, 11.37568814038781, 1.174924303092439, 0.6011714612100961, 0.4548904250162034, -0.05928158200771749, -0.05928158200771735, -0.05928158200771735, -0.3956088430084861, -0.395608843008486, -0.3956088430084862, -0.395608843008486, -0.3956088430084861], [4.363367367158438, 78.52109911103648, 0.1128513646061788, 8.861213185057478, -0.7464706890225739, 0.5372707561039685, 0.3753675374684118, -0.5073019409109862, -0.507301940910986, -0.507301940910986, 0.1319344034425741, 0.1319344034425741, 0.1319344034425743, 0.1319344034425742, 0.1319344034425742]], datatype=float[8], order='C_order'):
c := Matrix([[0, 0], [1, 0], [1, 0], [1, 0], [0, 1], [0, 1], [0, 1], [0, 1], [0, 1]], datatype=float[8], order='C_order'):
b := Vector([2.597657842414577, 1.261948923584132, 1.277732791293482, 0.05797612753696271, 1.030690710494774, 0.2910235146611871, 0.987566284022906, 0.4879767334503798, -0.1995994002146697], datatype=float[8]):
se := Vector(9, datatype=float[8]):
cov := Vector(45, datatype=float[8]):
NAG:-g02gkc(v, c, b, scale, se, cov, 'ip' = ip, 'nclin' = nclin, 'tdv' = tdv, 'tdc' = tdc):

se;

cov;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore McCullagh P and Nelder J A (1983) Generalized Linear Models Chapman and Hall Searle S R (1971) Linear Models Wiley g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.