# NAG CL Interfaceg03fcc (multidimscal_​ordinal)

## 1Purpose

g03fcc performs non-metric (ordinal) multidimensional scaling.

## 2Specification

 #include
 void g03fcc (Nag_ScaleCriterion type, Integer n, Integer ndim, const double d[], double x[], Integer tdx, double *stress, double dfit[], Nag_E04_Opt *options, NagError *fail)
The function may be called by the names: g03fcc, nag_mv_multidimscal_ordinal or nag_mv_ordinal_multidimscale.

## 3Description

For a set of $n$ objects, a distance or dissimilarity matrix $D$ can be calculated such that ${d}_{ij}$ is a measure of how ‘far apart’ objects $i$ and $j$ are. If $p$ variables ${x}_{k}$ have been recorded for each observation this measure may be based on Euclidean distance, ${d}_{ij}={\sum }_{k=1}^{p}{\left({x}_{ki}-{x}_{kj}\right)}^{2}$, or some other calculation such as the number of variables for which ${x}_{kj}\ne {x}_{ki}$. Alternatively, the distances may be the result of a subjective assessment. For a given distance matrix, multidimensional scaling produces a configuration of $n$ points in a chosen number of dimensions, $m$, such that the distance between the points in some way best matches the distance matrix. For some distance measures, such as Euclidean distance, the size of distance is meaningful, for other measures of distance all that can be said is that one distance is greater or smaller than another. For the former, metric scaling can be used, see g03fac, for the latter, a non-metric scaling is more appropriate.
For non-metric multidimensional scaling, the criterion used to measure the closeness of the fitted distance matrix to the observed distance matrix is known as $\mathit{stress}$. $\mathit{stress}$ is given by,
 $∑ i=1 n ∑ j=1 i-1 d ^ ij - d ~ ij 2 ∑ i=1 n ∑ j=1 i-1 d ^ ij 2$
where ${\stackrel{^}{d}}_{ij}^{2}$ is the Euclidean squared distance between points $i$ and $j$ and ${\stackrel{~}{d}}_{ij}$ is the fitted distance obtained when ${\stackrel{^}{d}}_{ij}$ is monotonically regressed on ${d}_{ij}$, that is, ${\stackrel{~}{d}}_{ij}$ is monotonic relative to ${d}_{ij}$ and is obtained from ${\stackrel{^}{d}}_{ij}$ with the smallest number of changes. So $\mathit{stress}$ is a measure of by how much the set of points preserve the order of the distances in the original distance matrix. Non-metric multidimensional scaling seeks to find the set of points that minimize the $\mathit{stress}$.
An alternate measure is squared $\mathit{stress}$, $SSTRESS$,
 $∑ i=1 n ∑ j=1 i-1 d ^ ij 2 - d ~ ij 2 2 ∑ i=1 n ∑ j=1 i-1 d ^ ij 4$
in which the distances in $\mathit{stress}$ are replaced by squared distances.
In order to perform a non-metric scaling, an initial configuration of points is required. This can be obtained from principal coordinate analysis, see g03fac. Given an initial configuration, g03fcc uses the optimization function e04dgc to find the configuration of points that minimizes $\mathit{stress}$ or $SSTRESS$. The function e04dgc uses a conjugate gradient algorithm. g03fcc will find an optimum that may only be a local optimum, to be more sure of finding a global optimum several different initial configurations should be used; these can be obtained by randomly perturbing the original initial configuration using functions from the G05 Chapter Introduction.

## 4References

Chatfield C and Collins A J (1980) Introduction to Multivariate Analysis Chapman and Hall
Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

## 5Arguments

1: $\mathbf{type}$Nag_ScaleCriterion Input
On entry: indicates whether $\mathit{stress}$ or $SSTRESS$ is to be used as the criterion.
${\mathbf{type}}=\mathrm{Nag_Stress}$
$\mathit{stress}$ is used.
${\mathbf{type}}=\mathrm{Nag_SStress}$
$SSTRESS$ is used.
Constraint: ${\mathbf{type}}=\mathrm{Nag_Stress}$ or $\mathrm{Nag_SStress}$.
2: $\mathbf{n}$Integer Input
On entry: the number of objects in the distance matrix, $n$.
Constraint: ${\mathbf{n}}>{\mathbf{ndim}}$.
3: $\mathbf{ndim}$Integer Input
On entry: the number of dimensions used to represent the data, $m$.
Constraint: ${\mathbf{ndim}}\ge 1$.
4: $\mathbf{d}\left[{\mathbf{n}}×\left({\mathbf{n}}-1\right)/2\right]$const double Input
On entry: the lower triangle of the distance matrix $D$ stored packed by rows. That is ${\mathbf{d}}\left[\left(\mathit{i}-1\right)×\left(i-2\right)/2+\mathit{j}-1\right]$ must contain ${d}_{\mathit{i}\mathit{j}}$, for $\mathit{i}=2,3,\dots ,n$ and $\mathit{j}=1,2,\dots ,i-1$. If ${d}_{ij}$ is missing then set ${d}_{ij}<0$; For further comments on missing values see Section 9.
5: $\mathbf{x}\left[{\mathbf{n}}×{\mathbf{tdx}}\right]$double Input/Output
Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{tdx}}+j-1\right]$.
On entry: the $i$th row must contain an initial estimate of the coordinates for the $i$th point, $i=1,2,\dots ,n$. One method of computing these is to use g03fac.
On exit: the $i$th row contains $m$ coordinates for the $i$th point, $i=1,2,\dots ,n$.
6: $\mathbf{tdx}$Integer Input
On entry: the stride separating matrix column elements in the array x.
Constraint: ${\mathbf{tdx}}\ge {\mathbf{ndim}}$.
7: $\mathbf{stress}$double * Output
On exit: the value of $\mathit{stress}$ or $SSTRESS$ at the final iteration.
8: $\mathbf{dfit}\left[2×{\mathbf{n}}×\left({\mathbf{n}}-1\right)\right]$double Output
On exit: auxiliary outputs. If ${\mathbf{type}}=\mathrm{Nag_Stress}$, the first $n\left(n-1\right)/2$ elements contain the distances, ${\stackrel{^}{d}}_{ij}$, for the points returned in x, the second set of $n\left(n-1\right)/2$ contains the distances ${\stackrel{^}{d}}_{ij}$ ordered by the input distances, ${d}_{ij}$, the third set of $n\left(n-1\right)/2$ elements contains the monotonic distances, ${\stackrel{~}{d}}_{ij}$, ordered by the input distances, ${d}_{ij}$ and the final set of $n\left(n-1\right)/2$ elements contains fitted monotonic distances, ${\stackrel{~}{d}}_{ij}$, for the points in x. The ${\stackrel{~}{d}}_{ij}$ corresponding to distances which are input as missing are set to zero. If ${\mathbf{type}}=\mathrm{Nag_SStress}$, the results are as above except that the squared distances are returned.
Each distance matrix is stored in lower triangular packed form in the same way as the input matrix $D$.
9: $\mathbf{options}$Nag_E04_Opt * Input/Output
On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for e04dgc. These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. You are referred to the e04dgc document for further details.
The default values used by g03fcc when the options argument is set to the NAG defined null pointer, E04_DEFAULT, are as follows:
• ${\mathbf{optim_tol}}=0.00001$;
• ${\mathbf{print_level}}=\mathrm{Nag_NoPrint}$;
• ${\mathbf{list}}=\mathrm{Nag_FALSE}$;
• ${\mathbf{verify_grad}}=\mathrm{Nag_FALSE}$;
• ${\mathbf{max_iter}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{n}}×{\mathbf{ndim}}\right)$.
If a different value is required for any of these four structure members or if other options available in e04dgc are to be used, then the structure options should be declared and initialized by a call to e04xxc and supplied as an argument to g03fcc. In this case, the structure members listed above except for ${\mathbf{list}}$ will have the default values as specified above; ${\mathbf{list}}=\mathrm{Nag_TRUE}$ in this case.
10: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_2_INT_ARG_LE
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ while ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{n}}>{\mathbf{ndim}}$.
NE_2_INT_ARG_LT
On entry, ${\mathbf{tdx}}=〈\mathit{\text{value}}〉$ while ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tdx}}\ge {\mathbf{ndim}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument type had an illegal value.
NE_INT_ARG_LT
On entry, ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ndim}}\ge 1$.
NE_INTERNAL_ERROR
Additional error messages are output if the optimization fails to converge or if the options are set incorrectly, Details of these can be found in the e04dgc document.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_NEG_OR_ZERO_ARRAY
All elements of array ${\mathbf{d}}\le 0.0$.
Constraint: At least one element of d must be positive.

## 7Accuracy

After a successful optimization, the relative accuracy of $\mathit{stress}$ should be approximately $\epsilon$, as specified by ${\mathbf{optim_tol}}$.

## 8Parallelism and Performance

g03fcc is not threaded in any implementation.

Missing values in the input distance matrix can be specified by a negative value and providing there are not more than about two thirds of the values missing, the algorithm may still work. However, the function g03fac does not allow for missing values so an alternative method of obtaining an initial set of coordinates is required. It may be possible to estimate the missing values with some form of average and then use g03fac to give an initial set of coordinates.

## 10Example

The data, given by Krzanowski (1990), are dissimilarities between water vole populations in Europe. Initial estimates are provided by the first two principal coordinates computed.

### 10.1Program Text

Program Text (g03fcce.c)

### 10.2Program Data

Program Data (g03fcce.d)

### 10.3Program Results

Program Results (g03fcce.r)