g02bef computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data omitting completely any cases with a missing observation for any variable.
where ${x}_{ij}$ is the $i$th observation on the $j$th variable. In addition, each of the $m$ variables may optionally have associated with it a value which is to be considered as representing a missing observation for that variable; the missing value for the $j$th variable is denoted by ${\mathit{xm}}_{j}$. Missing values need not be specified for all variables.
Let ${w}_{i}=0$ if observation $i$ contains a missing value for any of those variables for which missing values have been declared, i.e., if ${x}_{ij}={\mathit{xm}}_{j}$ for any $j$ for which an ${\mathit{xm}}_{j}$ has been assigned (see also Section 7); and ${w}_{i}=1$ otherwise, for $\mathit{i}=1,2,\dots ,n$.
If ${\stackrel{~}{S}}_{jj}$ or ${\stackrel{~}{S}}_{kk}$ is zero, ${\stackrel{~}{R}}_{jk}$ is set to zero.
4References
None.
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations or cases.
Constraint:
${\mathbf{n}}\ge 2$.
2: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of variables.
Constraint:
${\mathbf{m}}\ge 2$.
3: $\mathbf{x}({\mathbf{ldx}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}(\mathit{i},\mathit{j})$ must be set to ${x}_{\mathit{i}\mathit{j}}$, the value of the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
4: $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02bef is called.
On entry: ${\mathbf{miss}}\left(j\right)$ must be set equal to $1$ if a missing value, $x{m}_{j}$, is to be specified for the $j$th variable in the array x, or set equal to $0$ otherwise. Values of miss must be given for all $m$ variables in the array x.
On exit: the array miss is overwritten by the routine, and the information it contained on entry is lost.
6: $\mathbf{xmiss}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{xmiss}}\left(j\right)$ must be set to the missing value, $x{m}_{j}$, to be associated with the $j$th variable in the array x, for those variables for which missing values are specified by means of the array miss (see Section 7).
On exit: the array xmiss is overwritten by the routine, and the information it contained on entry is lost.
7: $\mathbf{xbar}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the mean value,
${\overline{x}}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
8: $\mathbf{std}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the standard deviation,
${s}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
9: $\mathbf{sspz}({\mathbf{ldsspz}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{sspz}}(\mathit{j},\mathit{k})$ is the cross-product about zero, ${\stackrel{~}{S}}_{\mathit{j}\mathit{k}}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
10: $\mathbf{ldsspz}$ – IntegerInput
On entry: the first dimension of the array sspz as declared in the (sub)program from which g02bef is called.
Constraint:
${\mathbf{ldsspz}}\ge {\mathbf{m}}$.
11: $\mathbf{rz}({\mathbf{ldrz}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{rz}}(\mathit{j},\mathit{k})$ is the correlation-like coefficient, ${\stackrel{~}{R}}_{\mathit{j}\mathit{k}}$, between the $\mathit{j}$th and $\mathit{k}$th variables, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
12: $\mathbf{ldrz}$ – IntegerInput
On entry: the first dimension of the array rz as declared in the (sub)program from which g02bef is called.
Constraint:
${\mathbf{ldrz}}\ge {\mathbf{m}}$.
13: $\mathbf{ncases}$ – IntegerOutput
On exit: the number of cases actually used in the calculations (when cases involving missing values have been eliminated).
14: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 2$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldrz}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldrz}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldsspz}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldsspz}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=4$
After observations with missing values were omitted, no cases remained.
${\mathbf{ifail}}=5$
After observations with missing values were omitted, only one case remained.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
g02bef does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large $n$.
You are warned of the need to exercise extreme care in your selection of missing values. g02bef treats all values in the inclusive range $(1\pm {0.1}^{({\mathbf{x02bef}}-2)})\times {xm}_{j}$, where ${\mathit{xm}}_{j}$ is the missing value for variable $j$ specified in xmiss.
You must, therefore, ensure that the missing value chosen for each variable is sufficiently different from all valid values for that variable so that none of the valid values fall within the range indicated above.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g02bef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02bef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The time taken by g02bef depends on $n$ and $m$, and the occurrence of missing values.
The routine uses a two-pass algorithm.
9.1Internal Changes
Internal changes have been made to this routine as follows:
At Mark 27: The algorithm underlying this routine was altered to improve efficiency for large problem sizes on a multi-threaded system.
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.
10Example
This example reads in a set of data consisting of five observations on each of three variables. Missing values of $0.0$ are declared for the first and third variables; no missing value is specified for the second variable. The means, standard deviations, sums of squares and cross-products about zero, and correlation-like coefficients for all three variables are then calculated and printed, omitting completely all cases containing missing values; cases $3$ and $4$ are, therefore, eliminated, leaving only three cases in the calculations.