naginterfaces.library.correg.ssqmat_​update

naginterfaces.library.correg.ssqmat_update(wt, x, sw, xbar, c, mean='M')

ssqmat_update updates the sample means and sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean, for a new observation. The data may be weighted.

For full information please refer to the NAG Library document for g02bt

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g02/g02btf.html

Parameters

wtfloat

The weight to use for the current observation, $w_i$.

For unweighted means and cross-products set $wt=1.0$.

The use of a suitable negative value of $wt$, e.g., $-w_i$ will have the effect of deleting the observation.

xfloat, array-like, shape $(m\times \text{incx})$

$x[(j-1)\times \text{incx}]$ must contain the value of the $j$th variable for the current observation, $j=1,2,\dots ,m$.

swfloat

The sum of weights for the previous observations, $W_{i-1}$.

$sw=0.0$

The update procedure is initialized.

$sw+wt=0.0$

All elements of $xbar$ and $c$ are set to zero.

xbarfloat, array-like, shape $\left(m\right)$

If $sw=0.0$, $xbar$ is initialized, otherwise $xbar[\text{j}-1]$ must contain the weighted mean of the $\text{j}$th variable for the previous $(\text{i}-1)$ observations, ${\overline{x}}_{\text{j}}(\text{i}-1)$, for $\text{j}=1,2,\dots ,m$.

cfloat, array-like, shape $\left((m\times m+m)/2\right)$

If $sw\ne 0.0$, $c$ must contain the upper triangular part of the matrix of weighted sums of squares and cross-products or weighted sums of squares and cross-products of deviations about the mean. It is stored packed form by column, i.e., the cross-product between the $j$th and $k$th variable, $k\ge j$, is stored in $c[k\times (k-1)/2+j-1]$.

meanstr, length 1, optional

Indicates whether ssqmat_update is to calculate sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean.

$mean=\text{'M'}$

The sums of squares and cross-products of deviations about the mean are calculated.

$mean=\text{'Z'}$

The sums of squares and cross-products are calculated.

Returns

swfloat

Contains the updated sum of weights, $W_i$.

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

$xbar[\text{j}-1]$ contains the weighted mean of the $\text{j}$th variable, ${\overline{x}}_{\text{j}}\left(\text{i}\right)$, for $\text{j}=1,2,\dots ,m$.

cfloat, ndarray, shape $\left((m\times m+m)/2\right)$

The update sums of squares and cross-products stored as on input.

Raises

NagValueError

(errno $1$)

On entry, $\text{incx}=⟨value⟩$.

Constraint: $\text{incx}\ge 1$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $2$)

On entry, $sw=⟨value⟩$.

Constraint: $sw\ge 0.0$.

(errno $3$)

On entry, $(sw+wt)=⟨value⟩$.

Constraint: $(sw+wt)\ge 0.0$.

(errno $4$)

On entry, $mean=⟨value⟩$.

Constraint: $mean=\text{'M'}$ or $\text{'Z'}$.

Notes

ssqmat_update is an adaptation of West’s WV2 algorithm; see West (1979). This function updates the weighted means of variables and weighted sums of squares and cross-products or weighted sums of squares and cross-products of deviations about the mean for observations on $m$ variables $X_j$, for $j=1,2,\dots ,m$. For the first $i-1$ observations let the mean of the $j$th variable be ${\overline{x}}_j(i-1)$, the cross-product about the mean for the $j$th and $k$th variables be $c_{jk}(i-1)$ and the sum of weights be $W_{i-1}$. These are updated by the $i$th observation, $x_{ij}$, for $\text{j}=1,2,\dots ,m$, with weight $w_i$ as follows:

$$W_i=W_{i-1}+w_i\text{,}{\overline{x}}_j\left(i\right)={\overline{x}}_j(i-1)+\frac{w_i}{W_i}(x_j-{\overline{x}}_j(i-1))\text{,}j=1,2,\dots ,m$$

and

$$c_{jk}\left(i\right)=c_{jk}(i-1)+\frac{w_i}{W_i}(x_j-{\overline{x}}_j(i-1))(x_k-{\overline{x}}_k(i-1))W_{i-1}\text{,}j=1,2,\dots ,m\text{;}k=j,j+1,2,\dots ,m\text{.}$$

The algorithm is initialized by taking ${\overline{x}}_j\left(1\right)=x_{1j}$, the first observation and $c_{ij}\left(1\right)=0.0$.

For the unweighted case $w_i=1$ and $W_i=i$ for all $i$.

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