Mahalanobis squared distances, following g03dac (nag_mv_discrim)

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)
    - g03 (Multivariate Methods)
      - g03 Introduction
      - g03aac (nag_mv_prin_comp)
      - g03acc (nag_mv_canon_var)
      - g03adc (nag_mv_canon_corr)
      - g03bac (nag_mv_orthomax)
      - g03bcc (nag_mv_procustes)
      - g03cac (nag_mv_factor)
      - g03ccc (nag_mv_fac_score)
      - g03dac (nag_mv_discrim)
      - g03dbc (nag_mv_discrim_mahaldist)
      - g03dcc (nag_mv_discrim_group)
      - g03eac (nag_mv_distance_mat)
      - g03ecc (nag_mv_hierar_cluster_analysis)
      - g03efc (nag_mv_kmeans_cluster_analysis)
      - g03ehc (nag_mv_dendrogram)
      - g03ejc (nag_mv_cluster_indicator)
      - g03fac (nag_mv_prin_coord_analysis)
      - g03fcc (nag_mv_ordinal_multidimscale)
      - g03zac (nag_mv_z_scores)
    - 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[g03dbc] NAG[nag_mv_discrim_mahaldist] - Mahalanobis squared distances, following g03dac (nag_mv_discrim)

Calling Sequence

g03dbc(equal, mode, gmean, gcov, isx, x, d, 'nvar'=nvar, 'ng'=ng, 'tdg'=tdg, 'nobs'=nobs, 'm'=m, 'tdx'=tdx, 'tdd'=tdd, 'fail'=fail)

nag_mv_discrim_mahaldist(. . .)

Parameters

equal - String;

On entry: indicates whether or not the within-group variance-covariance matrices are assumed to be equal and the pooled variance-covariance matrix used.

The within-group variance-covariance matrices are assumed equal and the matrix stored in the first elements of gcov is used.

The within-group variance-covariance matrices are assumed to be unequal and the matrices , for , stored in the remainder of gcov are used.

Constraint: "Nag_EqualCovar" or "Nag_NotEqualCovar". .

mode - String;

On entry: indicates whether distances from sample points are to be calculated or distances between the group means.

The distances between the sample points given in x and the group means are calculated.

The distances between the group means will be calculated.

Constraint: "Nag_SamplePoints" or "Nag_GroupMeans". .

gmean - Matrix(1..ng, 1..tdg, datatype=float[8], order=C_order);

On entry: the th row of gmean contains the means of the selected variables for the th group, for . These are returned by g03dac (nag_mv_discrim).

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

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

On entry: the first elements of gcov should contain the upper triangular matrix and the next blocks of elements should contain the upper triangular matrices . All matrices must be stored packed by column. These matrices are returned by g03dac (nag_mv_discrim).

If only the first elements are referenced.

If only the elements to are referenced.

Constraints:

if , the diagonal elements of ;

if , the diagonal elements of the , for .

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

On entry: if , indicates if the th variable in x is to be included in the distance calculations. If , the th variable is included, for ; otherwise the th variable is not referenced.

If , then isx is not referenced and may be set to the NULL pointer (Integer *)0.

Constraint: if , for nvar values of . .

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

On entry: if , the th row of x must contain . That is, must contain the th sample value for the th variable for ; . Otherwise x is not referenced and may be set to the NULL pointer (double *)0.

d - Matrix(1.., 1..tdd, datatype=float[8], order=C_order);

On exit: the squared distances.

If , contains the squared distance of the th sample point from the th group mean, , for ; .

If and , contains the squared distance between the th mean and the th mean, , for ; . The elements are not referenced for .

If and , contains the squared distance between the th mean and the th mean, , for ; . Since the elements are not referenced, for ; .

'nvar'=nvar - integer; (optional)

Default value: the second dimension of the array gmean.

On entry: the number of variables, , in the variance-covariance matrices as specified to g03dac (nag_mv_discrim).

Constraint: . .

'ng'=ng - integer; (optional)

Default value: the first dimension of the array gmean and the second dimension of the array gmeanthe array d.

On entry: the number of groups, .

Constraint: . .

'tdg'=tdg - integer; (optional)

On entry: the second dimension of the array gmean as declared in the function from which nag_mv_discrim_mahaldist (g03dbc) is called.

Constraint: . .

'nobs'=nobs - integer; (optional)

Default value: the first dimension of the array x.

On entry: if the number of sample points in x for which distances are to be calculated.

If , nobs is not referenced.

Constraint: if , . .

'm'=m - integer; (optional)

Default value: the first dimension of the array isx and the second dimension of the array isxthe array x.

On entry: if the number of variables in the data array x.

If , then m is not referenced.

Constraint: if , . .

'tdx'=tdx - integer; (optional)

On entry: the second dimension of the array x as declared in the function from which nag_mv_discrim_mahaldist (g03dbc) is called.

Constraint: . .

'tdd'=tdd - integer; (optional)

On entry: the second dimension of the array d as declared in the function from which nag_mv_discrim_mahaldist (g03dbc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_mv_discrim_mahaldist (g03dbc) computes Mahalanobis squared distances for group or pooled variance-covariance matrices. It is intended for use after g03dac (nag_mv_discrim).

Description

Consider variables observed on populations or groups. Let be the sample mean and the within-group variance-covariance matrix for the th group and let be the th sample point in a data set. A measure of the distance of the point from the th population or group is given by the Mahalanobis distance, :

If the pooled estimated of the variance-covariance matrix is used rather than the within-group variance-covariance matrices, then the distance is:

Instead of using the variance-covariance matrices and , nag_mv_discrim_mahaldist (g03dbc) uses the upper triangular matrices and supplied by g03dac (nag_mv_discrim) such that and . can then be calculated as where or as appropriate.

A particular case is when the distance between the group or population means is to be estimated. The Mahalanobis distance between the th and th groups is:

Note: and that in the case when the pooled variance-covariance matrix is used so in this case only the lower triangular values of , , are computed. .

Error Indicators and Warnings

"NE_2_INT_ARG_ENUM_CONS"

On entry, while and . These arguments must satisfy when .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument equal had an illegal value.

"NE_DIAG_0_COND"

A diagonal element of is zero when .

"NE_DIAG_0_J_COND"

A diagonal element of is zero for some , when .

"NE_INT_ARG_ENUM_CONS"

On entry, while . These arguments must satisfy when .

"NE_INT_ARG_LT"

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

"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_VAR_INCL_COND"

The number of variables, nvar in the analysis , while number of variables included in the analysis via array . Constraint: These two numbers must be the same when .

	Accuracy
	The accuracy will depend upon the accuracy of the input or matrices.

	Further Comments
	If the distances are to be used for discrimination, see also g03dcc (nag_mv_discrim_group).

Examples

equal := "Nag_NotEqualCovar":
mode := "Nag_SamplePoints":
nvar := 2:
ng := 3:
tdg := 2:
nobs := 6:
m := 2:
tdx := 2:
tdd := 3:
gmean := Matrix([[1.0433, -0.6034166666666667], [2.00727, -0.20604], [2.70974, 1.5998]], datatype=float[8], order='C_order'):
gcov := Vector([-0.5099642881287538, -0.2797054723861329, -1.217327847040481, -0.3326727521153483, -0.3723518779712079, -1.987589395382754, -0.4603014906920608, -0.7041634974247671, 0.4737334252803499, 0.7451327720614629, -0.3251057349548681, -0.4275545007358186], datatype=float[8]):
isx := Vector([1, 1], datatype=integer[kernelopts('wordsize')/8]):
x := Matrix([[1.6292, -0.9163], [2.5572, 1.6094], [2.5649, -0.2231], [0.9555, -2.3026], [3.4012, -2.3026], [3.0204, -0.2231]], datatype=float[8], order='C_order'):
d := Matrix(6, 3, datatype=float[8], order='C_order'):
NAG:-g03dbc(equal, mode, gmean, gcov, isx, x, d, 'nvar' = nvar, 'ng' = ng, 'tdg' = tdg, 'nobs' = nobs, 'm' = m, 'tdx' = tdx, 'tdd' = tdd):

	See Also
	Aitchison J and Dunsmore I R (1975) Statistical Prediction Analysis Cambridge Kendall M G and Stuart A (1976) The Advanced Theory of Statistics (Volume 3) (3rd Edition) Griffin Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press g03 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.