naginterfaces.library.correg.coeffs_​kspearman_​overwrite

naginterfaces.library.correg.coeffs_kspearman_overwrite(x, itype)

coeffs_kspearman_overwrite computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data; the data array is overwritten with the ranks of the observations.

For full information please refer to the NAG Library document for g02bn

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g02/g02bnf.html

Parameters

xfloat, ndarray, shape $(n,m)$, modified in place

On entry: $x[\text{i}-1,\text{j}-1]$ must be set to $x_{\text{i}\text{j}}$, the value of the $\text{i}$th observation on the $\text{j}$th variable, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

On exit: $x[\text{i}-1,\text{j}-1]$ contains the rank $y_{\text{i}\text{j}}$ of the observation $x_{\text{i}\text{j}}$, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

itypeint

The type of correlation coefficients which are to be calculated.

$itype=-1$

Only Kendall’s tau coefficients are calculated.

$itype=0$

Both Kendall’s tau and Spearman’s coefficients are calculated.

$itype=1$

Only Spearman’s coefficients are calculated.

Returns

rrfloat, ndarray, shape $(m,m)$

The requested correlation coefficients.

If only Kendall’s tau coefficients are requested ($itype=-1$), $rr[j-1,k-1]$ contains Kendall’s tau for the $j$th and $k$th variables.

If only Spearman’s coefficients are requested ($itype=1$), $rr[j-1,k-1]$ contains Spearman’s rank correlation coefficient for the $j$th and $k$th variables.

If both Kendall’s tau and Spearman’s coefficients are requested ($itype=0$), the upper triangle of $rr$ contains the Spearman coefficients and the lower triangle the Kendall coefficients.

That is, for the $\text{j}$th and $\text{k}$th variables, where $\text{j}$ is less than $\text{k}$, $rr[\text{j}-1,\text{k}-1]$ contains the Spearman rank correlation coefficient, and $rr[\text{k}-1,\text{j}-1]$ contains Kendall’s tau, for $\text{k}=1,2,\dots ,m$, for $\text{j}=1,2,\dots ,m$.

(Diagonal terms, $rr[j-1,j-1]$, are unity for all three values of $itype$.)

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $2$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 2$.

(errno $4$)

On entry, $itype=⟨value⟩$.

Constraint: $itype=-1$ or $1$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The input data consists of $n$ observations for each of $m$ variables, given as an array

$$\left[x_{ij}\right]\text{,}i=1,2,\dots ,n(n\ge 2),j=1,2,\dots ,m(m\ge 2)\text{,}$$

where $x_{ij}$ is the $i$th observation of the $j$th variable.

The quantities calculated are:

1. Ranks

   For a given variable, $j$ say, each of the $n$ observations, $x_{1j},x_{2j},\dots ,x_{nj}$, has associated with it an additional number, the ‘rank’ of the observation, which indicates the magnitude of that observation relative to the magnitudes of the other $n-1$ observations on that same variable.

   The smallest observation for variable $j$ is assigned the rank $1$, the second smallest observation for variable $j$ the rank $2$, the third smallest the rank $3$, and so on until the largest observation for variable $j$ is given the rank $n$.

   If a number of cases all have the same value for the given variable, $j$, then they are each given an ‘average’ rank, e.g., if in attempting to assign the rank $h+1$, $k$ observations were found to have the same value, then instead of giving them the ranks

   $$h+1,h+2,\dots ,h+k\text{,}$$

   all $k$ observations would be assigned the rank

   $$\frac{2h+k+1}{2}$$

   and the next value in ascending order would be assigned the rank

   $$h+k+1\text{.}$$

   The process is repeated for each of the $m$ variables.

   Let $y_{ij}$ be the rank assigned to the observation $x_{ij}$ when the $j$th variable is being ranked. The actual observations $x_{ij}$ are replaced by the ranks $y_{ij}$.

2. Nonparametric rank correlation coefficients

   1. Kendall’s tau:

      $$R_{jk}=\frac{{\sum }_{h=1}^n{\sum }_{i=1}^{n}sign(y_{hj}-y_{ij})sign(y_{hk}-y_{ik})}{\sqrt{[n(n-1)-T_j][n(n-1)-T_k]}}\text{,}j,k=1,2,\dots ,m\text{,}$$

      where	$sign\left(u\right)=1$ if $u>0$,
      	$sign\left(u\right)=0$ if $u=0$,
      	$sign\left(u\right)=-1$ if $u<0$,

      and $T_j=\sum t_j(t_j-1)$, where $t_j$ is the number of ties of a particular value of variable $j$, and the summation is over all tied values of variable $j$.

   2. Spearman’s:

      $$R_{jk}^{\ast }=\frac{n(n^2-1)-6{\sum }_{i=1}^n{(y_{ij}-y_{ik})}^2-\frac{1}{2}(T_j^{\ast }+T_k^{\ast })}{\sqrt{[n(n^2-1)-T_j^{\ast }][n(n^2-1)-T_k^{\ast }]}}\text{,}j,k=1,2,\dots ,m\text{,}$$

      where $T_j^{\ast }=\sum t_j(t_j^2-1)$, $t_j$ being the number of ties of a particular value of variable $j$, and the summation being over all tied values of variable $j$.

References

Siegel, S, 1956, Non-parametric Statistics for the Behavioral Sciences, McGraw–Hill