naginterfaces.library.correg.corrmat

naginterfaces.library.correg.corrmat(x, nonzwt='W', wt=None)

corrmat calculates the sample means, the standard deviations, the variance-covariance matrix, and the matrix of Pearson product-moment correlation coefficients for a set of data. Weights may be used.

For full information please refer to the NAG Library document for g02bx

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g02/g02bxf.html

Parameters

xfloat, array-like, shape $(n,m)$

$x[\text{i}-1,\text{j}-1]$ must contain the $\text{i}$th observation for the $\text{j}$th variable, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

nonzwtstr, length 1, optional

The variance calculation uses a divisor which is either the number of weights or the number of nonzero weights.

wtNone or float, array-like, shape $\left(n\right)$, optional

$w$, the optional frequency weighting for each observation, with $wt[i-1]=w_i$. Usually $w_i$ will be an integral value corresponding to the number of observations associated with the $i$th data value, or zero if the $i$th data value is to be ignored. If $wt\text{is}None$, $w_i$ is set to $1$ for all $i$.

Returns

xbarfloat, ndarray, shape $\left(m\right)$

The sample means. $xbar[j-1]$ contains the mean of the $j$th variable.

stdfloat, ndarray, shape $\left(m\right)$

The standard deviations. $std[j-1]$ contains the standard deviation for the $j$th variable.

vfloat, ndarray, shape $(m,m)$

The variance-covariance matrix. $v[\text{j}-1,\text{k}-1]$ contains the covariance between variables $\text{j}$ and $\text{k}$, for $\text{k}=1,2,\dots ,m$, for $\text{j}=1,2,\dots ,m$.

rfloat, ndarray, shape $(m,m)$

The matrix of Pearson product-moment correlation coefficients. $r[j-1,k-1]$ contains the correlation coefficient between variables $j$ and $k$.

Raises

NagValueError

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

(errno $2$)

On entry, $\text{weight}=⟨value⟩$.

Constraint: $\text{weight}=\text{'U'}$, $\text{'V'}$ or $\text{'W'}$

(errno $3$)

On entry, at least one value of $wt$ is negative.

Constraint: $wt[i-1]\ge 0$, for $i=1,2,\dots ,n$.

(errno $4$)

On entry, $⟨value⟩$ observations have nonzero weight.

Constraint: at least two observations must have a nonzero weight.

(errno $4$)

On entry, Sum of the weights is $⟨value⟩$.

Constraint: Sum of the weights must be greater than $1$.

Warns

NagAlgorithmicWarning

(errno $5$)

A variable has a zero variance. In this case $v$ and $std$ are returned as calculated but $r$ will contain zero for any correlation involving a variable with zero variance.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

For $n$ observations on $m$ variables the one-pass algorithm of West (1979) as implemented in ssqmat() is used to compute the means, the standard deviations, the variance-covariance matrix, and the Pearson product-moment correlation matrix for $p$ selected variables. Suitables weights may be used to indicate multiple observations and to remove missing values. The quantities are defined by:

1. The means

$${\overline{x}}_j=\frac{{\sum }_{i=1}^n{w}_i{x}_{ij}}{{\sum }_{i=1}^n{w}_i}j=1,\dots ,p$$

2. The variance-covariance matrix

$$C_{jk}=\frac{{\sum }_{i=1}^n{w}_i(x_{ij}-{\overline{x}}_j)(x_{ik}-{\overline{x}}_k)}{{\sum }_{i=1}^n{w}_i-1}j,k=1,\dots ,p$$

3. The standard deviations

$$s_j=\sqrt{C_{jj}}j=1,\dots ,p$$

4. The Pearson product-moment correlation coefficients

$$R_{jk}=\frac{C_{jk}}{\sqrt{C_{jj}{C}_{kk}}}j,k=1,\dots ,p$$

where $x_{ij}$ is the value of the $i$th observation on the $j$th variable and $w_i$ is the weight for the $i$th observation which will be $1$ in the unweighted case.

Note that the denominator for the variance-covariance is ${\sum }_{i=1}^n{w}_i-1$, so the weights should be scaled so that the sum of weights reflects the true sample size.

References

Chan, T F, Golub, G H and Leveque, R J, 1982, Updating Formulae and a Pairwise Algorithm for Computing Sample Variances, Compstat, Physica-Verlag

West, D H D, 1979, Updating mean and variance estimates: An improved method, Comm. ACM (22), 532–555