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\RequirePackage{url}
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\providecommand{\email}[1]{\href{mailto:#1}{\normalfont\texttt{#1}}}

\providecommand{\ndf}[1]{\mathSUB{n}{#1}}
\providecommand{\twt}[1]{\mathSUB{W}{#1}}
\providecommand{\wmupow}[2]{\mathUL{\mu}{#2}{#1}}
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\providecommand{\wmu}[1]{\wmupow{#1}{}}
\providecommand{\mompow}[3]{\mathUL{\mathcal{S}}{#3}{#1,#2}}
\providecommand{\mom}[2]{\mathSUB{\mathcal{S}}{#1,#2}}
\providecommand{\cmom}[2]{\mathSUB{\mathcal{M}}{#1,#2}}
\providecommand{\scmom}[2]{\mathSUB{\mathcal{Y}}{#1,#2}}

\providecommand{\ccum}[2]{\mathSUB{\mathcal{K}}{#1,#2}}
\providecommand{\sccum}[2]{\mathSUB{\mathcal{G}}{#1,#2}}

\providecommand{\tiltwt}[1]{\mathSUB{\tilde{W}}{#1}}
\providecommand{\tilwmu}[1]{\mathSUB{\tilde{\mu}}{#1}}
\providecommand{\tilmom}[2]{\mathSUB{\tilde{\mathcal{S}}}{#1,#2}}

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% cf https://math.stackexchange.com/a/1169347/610
\providecommand{\kron}{\otimes}
\providecommand{\bigkron}{\bigotimes}
\providecommand{\kpow}[2]{\mathSUP{#1}{\kron {#2}}}
\providecommand{\kommUL}[2]{\mathUL{\mathcal{K}}{#1}{#2}}
\providecommand{\komm}[1][{}]{\kommUL{}{{#1}}}
\providecommand{\meye}[1][{}]{\mathSUB{\mathbf{I}}{#1}}
\providecommand{\mcoef}[1]{\mathSUB{\mathbf{C}}{#1}}

\providecommand{\vwmupow}[2]{\mathUL{\boldsymbol{\mu}}{\mtP{#2}{}{\kron {#2}}}{#1}}
\providecommand{\vwmu}[1]{\vwmupow{#1}{}}

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<<'preamble', include=FALSE, warning=FALSE, message=FALSE>>=
library(knitr)

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compile.time <- Sys.time()

# from the environment

# only recompute if FORCE_RECOMPUTE=True w/out case match.
FORCE_RECOMPUTE <- 
	(toupper(Sys.getenv('FORCE_RECOMPUTE',unset='False')) == "TRUE")

# compiler flags!

# not used yet
LONG.FORM <- FALSE
@
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% document incantations%FOLDUP
\begin{document}

\title{Computing Moments with fromo}
\author{Steven E. Pav %
\thanks{\email{shabbychef@gmail.com}}}
%\date{\today, \currenttime}

\maketitle
%UNFOLD

\begin{abstract}%FOLDUP
The \fromo  package provides fast robust summation using the Welford-Terriberry method.
The update formula used therein is described, as well as the output produced
by various functions.
\end{abstract}%UNFOLD

\section{The update formula}%FOLDUP
\label{sec:the_update_formula}

Let \sta be a set of indices over the data \data{i} and corresponding weights
$\wt{i} > 0$. The weights are \emph{replication weights}, and are intended
to simulate having observed multiple identical, though independent, values of
\data{i}. In the standard setting the weights are identically 1.

Define the total elements, sum of weights, and (weighted) mean over \sta via
\begin{align*}
\ndf{\sta} &= \abs{\sta},\\
\twt{\sta} &= \sum_{i \in \sta} \wt{i},\\
\wmu{\sta} &= \frac{\sum_{i \in \sta} \wt{i} \data{i}}{\twt{\sta}}.
\end{align*}
Then go on to define the \kth{k} centered weighted sum via
$$
\mom{\sta}{k} = \sum_{i \in \sta} \wt{i} \wrapParens{\data{i} - \wmu{\sta}}^k.
$$
Note that we have
$$
\mom{\sta}{0} = \twt{\sta},\qquad\mbox{and}\qquad\mom{\sta}{1} = 0.
$$
When \sta consists of a single observation, that is when $\ndf{\sta}=1$, we have
$\wmu{\sta} = \data{a}$ for the unique $a \in \sta$, and 
$\mom{\sta}{k} = 0$ for all $k\ge 1$.

Let \stb and \stc be sets of indices with the
restriction that 
$\sta \cap \stb = \emptyset$, 
$\stc \subseteq \sta$, and define
$$
\std = \sta \cup \stb \setminus \stc.
$$
Thus \std is \sta `plus' \stb `minus' \stc. 
Consider how to compute \twt{\std}, \wmu{\std}, and \mom{\std}{k} from
the sums of weights, weighted means, and centered sums over the
sets \sta, \stb, and \stc. \cite{bennett,cook,terriberry}
We have:
\begin{align*}
	\ndf{\std} &= \ndf{\sta} + \ndf{\stb} - \ndf{\stc},\\
	\twt{\std} &= \twt{\sta} + \twt{\stb} - \twt{\stc},\\
	\wmu{\std} &= \frac{\twt{\sta}\wmu{\sta} + \twt{\stb}\wmu{\stb} - \twt{\stc}\wmu{\stc}}{\twt{\std}},\\
	&= \wmu{\sta} + \frac{\twt{\stb}\wrapParens{\wmu{\stb} - \wmu{\sta}} - \twt{\stc}\wrapParens{\wmu{\stc}-\wmu{\sta}}}{\twt{\std}},
\end{align*}
and
\begin{align*}
	\mom{\std}{k}&=\sum_{i \in \std} \wt{i} \wrapParens{\data{i} - \wmu{\std}}^k,\\
	&=\sum_{i \in \sta} \wt{i} \wrapParens{\data{i} - \wmu{\std}}^k +
	\sum_{i \in \stb} \wt{i} \wrapParens{\data{i} - \wmu{\std}}^k -
	\sum_{i \in \stc} \wt{i} \wrapParens{\data{i} - \wmu{\std}}^k,\\
	&=\sum_{i \in \sta} \wt{i} \wrapParens{\data{i} - \wmu{\sta} + \wmu{\sta} - \wmu{\std}}^k +
	\sum_{i \in \stb} \wt{i} \wrapParens{\data{i} - \wmu{\stb} + \wmu{\stb} - \wmu{\std}}^k -
	\sum_{i \in \stc} \wt{i} \wrapParens{\data{i} - \wmu{\stc} + \wmu{\stc} - \wmu{\std}}^k,\\
	&= \sum_{i \in \sta} \sum_{0 \le j \le k}{k \choose j}
		\wt{i} \wrapParens{\data{i} - \wmu{\sta}}^j \wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j}\\
	&\phantom{=}\,+
	\sum_{i \in \stb} \sum_{0 \le j \le k}{k \choose j}
		\wt{i} \wrapParens{\data{i} - \wmu{\stb}}^j \wrapParens{\wmu{\stb} - \wmu{\std}}^{k-j}\\
	&\phantom{=}\,-
	\sum_{i \in \stc} \sum_{0 \le j \le k}{k \choose j}
		\wt{i} \wrapParens{\data{i} - \wmu{\stc}}^j \wrapParens{\wmu{\stc} - \wmu{\std}}^{k-j},\\
	&=\sum_{0 \le j \le k}{k \choose j}\wrapBraces{
		\mom{\sta}{j} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j} +
		\mom{\stb}{j} \wrapParens{\wmu{\stb} - \wmu{\std}}^{k-j} -
		\mom{\stc}{j} \wrapParens{\wmu{\stc} - \wmu{\std}}^{k-j}},\\
	&= \mom{\sta}{k} + \mom{\stb}{k} - \mom{\stc}{k} +
		\twt{\sta} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k} +
		\twt{\stb} \wrapParens{\wmu{\stb} - \wmu{\std}}^{k} -
		\twt{\stc} \wrapParens{\wmu{\stc} - \wmu{\std}}^{k}\\
  &\phantom{=}\,+ \sum_{2 \le j < k}{k \choose j}\wrapBraces{
		\mom{\sta}{j} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j} +
		\mom{\stb}{j} \wrapParens{\wmu{\stb} - \wmu{\std}}^{k-j} -
		\mom{\stc}{j} \wrapParens{\wmu{\stc} - \wmu{\std}}^{k-j}}.
\end{align*}

Note that if the centered sums are to be computed \emph{in place}, that is,
overwriting the vector of \mom{\sta}{j} with values of \mom{\std}{j}, then
they should be computed in decreasing order, as updates to higher order
sums require the old values of lower order sums.

It should also be noted that we did not use the restriction that
$\wt{i} > 0$. True, negative values of $\wt{i}$ can cause 
\twt{\std} to be zero, and therefore \wmu{\std} is not defined, nor
is \mom{\std}{k}. Negative weights also make no sense for most 
statistical uses. However, notice what happens when we subsitute
every $\wt{c}$ with $-\wt{c}$ for $c \in \stc$. If we compute 
the total weight, \twt{\stc}, its sign has flipped, as has that
of \mom{\stc}{k}, but not of \wmu{\stc}. In this case, using the
negative weights we have
\begin{align*}
	\mom{\std}{k} 
	&= \mom{\sta}{k} + \mom{\stb}{k} + \tilmom{\stc}{k} +
		\twt{\sta} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k} +
		\twt{\stb} \wrapParens{\wmu{\stb} - \wmu{\std}}^{k} +
		\tiltwt{\stc} \wrapParens{\tilwmu{\stc} - \wmu{\std}}^{k}\\
  &\phantom{=}\,+ \sum_{2 \le j < k}{k \choose j}\wrapBraces{
		\mom{\sta}{j} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j} +
		\mom{\stb}{j} \wrapParens{\wmu{\stb} - \wmu{\std}}^{k-j} +
		\tilmom{\stc}{j} \wrapParens{\tilwmu{\stc} - \wmu{\std}}^{k-j}},
\end{align*}
where the \tiltwt{\stc}, \tilwmu{\stc}, and \tilmom{\stc}{j} denote that
they are computed with negative weights.
Now the update formula for removing \stc from \sta looks just like that
for adding \stb, except negative weights are used.



%Now consider the difference in means. We have
%\begin{align*}
	%\wmu{\sta} - \wmu{\std} &= 
		%\frac{\sum_{i \in \sta} \wt{i} \data{i}}{\twt{\sta}} -
		%\frac{\sum_{i \in \sta} \wt{i} \data{i} +
		%\sum_{i \in \stb} \wt{i} \data{i} -
		%\sum_{i \in \stc} \wt{i} \data{i}}{\twt{\std}},\\
		%&= \frac{\twt{\std} - \twt{\sta}}{\twt{\std}\twt{\sta}}
		%\sum_{i \in \sta} \wt{i} \data{i}
		%- \frac{\sum_{i \in \stb} \wt{i} \data{i} - \sum_{i \in \stc} \wt{i} \data{i}}{\twt{\std}}.
%\end{align*}
%Similar formul{\ae} 
%exist for 
%$\wmu{\stb} - \wmu{\std}$ and
%$\wmu{\stc} - \wmu{\std}$, though these are only useful for simple cases.

%We recover Welford's method in the special case of $k=2$, with
%\begin{align*}
	%\mom{\std}{2}&= \mom{\sta}{2} + \mom{\stb}{2} - \mom{\stc}{2} +
		%\twt{\sta} \wrapParens{\wmu{\sta} - \wmu{\std}}^{2} +
		%\twt{\stb} \wrapParens{\wmu{\stb} - \wmu{\std}}^{2} -
		%\twt{\stc} \wrapParens{\wmu{\stc} - \wmu{\std}}^{2},\\
	%&= \mom{\sta}{2} + \mom{\stb}{2} - \mom{\stc}{2} 
		%+ \wrapParens{\twt{\sta} + \twt{\stb} - \twt{\stc}} \wmupow{\std}{2}\\
	%&\phantom{=}\,\, 
		%+ \twt{\sta} \wmupow{\sta}{2}
		%+ \twt{\stb} \wmupow{\stb}{2}
		%- \twt{\stc} \wmupow{\stc}{2}
		%- 2 \wmu{\std} \wrapParens{
			%\twt{\sta} \wmu{\sta}
			%+ \twt{\stb} \wmu{\stb}
			%- \twt{\stc} \wmu{\stc}},\\
	%&= \mom{\sta}{2} + \mom{\stb}{2} - \mom{\stc}{2} 
		%- \twt{\std} \wmupow{\std}{2}
		%+ \twt{\sta} \wmupow{\sta}{2}
		%+ \twt{\stb} \wmupow{\stb}{2}
		%- \twt{\stc} \wmupow{\stc}{2}.
%\end{align*}
%Again, further simplifications are possible when \stb or \stc consist of a single index.

%There is temptation to store the uncentered second sums, as one notices they are additive:
%$$
%\mom{\std}{2} + \twt{\std}\wmupow{\std}{2} = 
%\mom{\sta}{2} + \twt{\sta}\wmupow{\sta}{2} 
%+ \mom{\stb}{2} + \twt{\stb}\wmupow{\stb}{2} 
%- \mom{\stc}{2} + \twt{\stc}\wmupow{\stc}{2}.
%$$
%However, recovering the centered sums from the uncentered sums is not numerically robust, and
%can result in significant roundoff errors.

\subsection{Adding a single observation}%FOLDUP

We now consider the special case where \stc is empty, and \stb consists of the single index
$b$. We then have $\twt{\std} = \twt{\sta} + \wt{b},$
\begin{align*}
\wmu{\std} &= \frac{\twt{\sta}\wmu{\sta} + \wt{b}\data{b}}{\twt{\sta} + \wt{b}},\\
	&= \wmu{\sta} + \wt{b}\frac{\data{b} - \wmu{\sta}}{\twt{\sta} + \wt{b}},\\
	&= \wmu{\sta} + \wt{b}\frac{\data{b} - \wmu{\sta}}{\twt{\std}}.
\end{align*}
So then
$$
\wmu{\sta} - \wmu{\std} = -\wt{b}\frac{\data{b} - \wmu{\sta}}{\twt{\std}}.
$$
Also
\begin{align*}
\data{b} - \wmu{\std} &= \data{b} - \wmu{\sta} + \wmu{\sta} - \wmu{\std},\\
	&= \data{b} - \wmu{\sta} -\wt{b}\frac{\data{b} - \wmu{\sta}}{\twt{\std}},\\
	&= \twt{\sta}\frac{\data{b} - \wmu{\sta}}{\twt{\std}}.
\end{align*}
Since \stb is a single index, $\mom{\stb}{k} = 0$ for $k > 0$, then we have
\begin{align*}
	\mom{\std}{k} &= \mom{\sta}{k} 
	+	\twt{\sta} \wrapParens{-\wt{b}\frac{\data{b} - \wmu{\sta}}{\twt{\std}}}^{k} 
	+ \wt{b} \wrapParens{\twt{\sta}\frac{\data{b} - \wmu{\sta}}{\twt{\std}}}^{k}\\
	&\phantom{=}\,
	+ \sum_{2 \le j < k}{k \choose j}{\mom{\sta}{j}\wrapParens{-\wt{b}\frac{\data{b} - \wmu{\sta}}{\twt{\std}}}^{k-j}},\\
	&= \mom{\sta}{k} 
	+ \wt{b}\wrapParens{\data{b} - \wmu{\std}}^{k} 
	\wrapBracks{1 + \wrapParens{\frac{-\wt{b}}{\twt{\sta}}}^{k-1}}\\
	&\phantom{=}\,
	+ \sum_{2 \le j < k}{k \choose j}{\mom{\sta}{j}\wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j}},\\
	&= \mom{\sta}{k} 
	+ \twt{\sta}\wrapParens{\wmu{\sta} - \wmu{\std}}^{k}
	\wrapBracks{1 - \wrapParens{\frac{\twt{\sta}}{-\wt{b}}}^{k-1}}\\
	&\phantom{=}\,
	+ \sum_{2 \le j < k}{k \choose j}{\mom{\sta}{j}\wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j}}.
\end{align*}
For the $k=2$ case this further simplifies to
\begin{align*}
	\mom{\std}{2} &= \mom{\sta}{2} +	\frac{\twt{\sta} \wt{b}}{\twt{\std}} \wrapParens{\data{b} - \wmu{\sta}}^{2},\\
	&= \mom{\sta}{2} +	\frac{\twt{\std} \wt{b}}{\twt{\sta}} \wrapParens{\data{b} - \wmu{\std}}^{2},\\
	&= \mom{\sta}{2} + \twt{\std} \wrapParens{\data{b} - \wmu{\std}} \wrapParens{\wmu{\std} - \wmu{\sta}}.
\end{align*}
Note that the quantity $\twt{\std}\wrapParens{\wmu{\std} - \wmu{\sta}} = 
\wt{b}\wrapParens{\data{b} - \wmu{\sta}}$ is computed as an intermediary in updating
the weighted mean, resulting in some computational savings.
%UNFOLD
\subsection{Removing a single observation}%FOLDUP

As noted above, removing a single observation should look just like adding a single
observation, except signs on the weights and weighted sums are flipped. Let $c$ be the
single index in \stc. 
We then have $\twt{\std} = \twt{\sta} - \wt{c},$ and
\begin{align*}
\wmu{\std} &= \wmu{\sta} - \wt{c}\frac{\data{c} - \wmu{\sta}}{\twt{\std}}.
\end{align*}
The centered sum update formula is
%&= \mom{\sta}{k} 
%+	\twt{\sta} \wrapParens{\wt{c}\frac{\data{c} - \wmu{\sta}}{\twt{\std}}}^{k} 
%- \wt{c} \wrapParens{\twt{\sta}\frac{\data{c} - \wmu{\sta}}{\twt{\std}}}^{k}\\
%&\phantom{=}\,
%+ \sum_{2 \le j < k}{k \choose j}{\mom{\sta}{j}\wrapParens{\wt{c}\frac{\data{c} - \wmu{\sta}}{\twt{\std}}}^{k-j}},\\
\begin{align*}
	\mom{\std}{k} 
	&= \mom{\sta}{k} 
	- \wt{c}\wrapParens{\data{c} - \wmu{\std}}^{k} 
	\wrapBracks{1 + \wrapParens{\frac{\wt{c}}{\twt{\sta}}}^{k-1}}\\
	&\phantom{=}\,
	+ \sum_{2 \le j < k}{k \choose j}{\mom{\sta}{j}\wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j}},\\
	&= \mom{\sta}{k} 
	+ \twt{\sta}\wrapParens{\wmu{\sta} - \wmu{\std}}^{k}
	\wrapBracks{1 - \wrapParens{\frac{\twt{\sta}}{\wt{c}}}^{k-1}}\\
	&\phantom{=}\,
	+ \sum_{2 \le j < k}{k \choose j}{\mom{\sta}{j}\wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j}}.
\end{align*}
%For the $k=2$ case this further simplifies to
%\begin{align*}
	%\mom{\std}{2} &= \mom{\sta}{2} +	\frac{\twt{\sta} \twt{\stb}}{\twt{\std}} \wrapParens{\wmu{\stb} - \wmu{\sta}}^{2},\\
	%&= \mom{\sta}{2} + \twt{\std} \wrapParens{\wmu{\stb} - \wmu{\std}} \wrapParens{\wmu{\std} - \wmu{\sta}}.
%\end{align*}
For the $k=2$ case this further simplifies to
\begin{align*}
	\mom{\std}{2} 
	&= \mom{\sta}{2} -\frac{\twt{\sta} \wt{c}}{\twt{\std}} \wrapParens{\data{c} - \wmu{\sta}}^{2},\\
	&= \mom{\sta}{2} -\frac{\twt{\std} \wt{c}}{\twt{\sta}} \wrapParens{\data{c} - \wmu{\std}}^{2},\\
	&= \mom{\sta}{2} + \twt{\std} \wrapParens{\data{c} - \wmu{\std}} \wrapParens{\wmu{\std} - \wmu{\sta}}.
\end{align*}
%UNFOLD
\subsection{Adding and removing a single observation}%FOLDUP

Consider the case of adding a single $b \in\stb$ and removing a single
$c\in\stc$.  The total number elements, \ndf{\sta}, is unchanged, of course. 
We have
\begin{align*}
\twt{\std}&=\twt{\sta} + \wt{b} - \wt{c},\\
	\wmu{\std}&=\wmu{\sta} + \frac{\wt{b}\wrapParens{\data{b} - \wmu{\sta}} - \wt{c}\wrapParens{\data{c}-\wmu{\sta}}}{\twt{\std}}.
\end{align*}
The update formula for \mom{\std}{k} is
\begin{align*}
	\mom{\std}{k}
	&= \mom{\sta}{k} + \mom{\stb}{k} - \mom{\stc}{k} +
		\twt{\sta} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k} +
		\wt{b} \wrapParens{\data{b} - \wmu{\std}}^{k} -
		\wt{c} \wrapParens{\data{c} - \wmu{\std}}^{k}\\
  &\phantom{=}\,+ \sum_{2 \le j < k}{k \choose j}\wrapBraces{
		\mom{\sta}{j} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j} +
		\mom{\stb}{j} \wrapParens{\data{b} - \wmu{\std}}^{k-j} -
		\mom{\stc}{j} \wrapParens{\data{c} - \wmu{\std}}^{k-j}}.
\end{align*}
In the $k=2$ case this simplifies to 
\begin{align*}
	\mom{\std}{2}
	&= \mom{\sta}{2} + 
		\twt{\sta} \wrapParens{\wmu{\sta} - \wmu{\std}}^{2} +
		\wt{b} \wrapParens{\data{b} - \wmu{\std}}^{2} -
		\wt{c} \wrapParens{\data{c} - \wmu{\std}}^{2},\\
	&= \mom{\sta}{2} 
	+ \wrapParens{\wmu{\sta}-\wmu{\std}}^2 \wrapParens{\twt{\sta} - \twt{\std}}
	+ \wt{b} \wrapParens{\data{b} - \wmu{\std}}\wrapParens{\data{b} - \wmu{\sta}}
	- \wt{c} \wrapParens{\data{c} - \wmu{\std}}\wrapParens{\data{c} - \wmu{\sta}},\\
	&= \mom{\sta}{2} 
	+ \frac{\twt{\sta}\wt{b} \wrapParens{\data{b}-\wmu{\sta}}^2 - \twt{\std}\wt{c}\wrapParens{\data{c}-\wmu{\std}}^2}{\twt{\sta} + \wt{b}}.
\end{align*}
Further simplifications are possible when $\wt{b}=\wt{c}$ (as occurs in the vanilla
case of equal weighted moments), and we arrive at
\begin{align*}
\wmu{\std}&=\wmu{\sta} + \frac{\wt{b}\wrapParens{\data{b} - \data{c}}}{\twt{\std}},\\
\mom{\std}{2} &= \mom{\sta}{2} 
	+ \wt{b} \wrapParens{\data{b}+\data{c}-\wmu{\sta}-\wmu{\std}} \wrapParens{\data{b}-\data{c}}.
\end{align*}

% 2FIX: almost done. start from here...
%2FIX: continue from here...
	%&= \mom{\sta}{2} 
	%+ \wt{b}\wrapParens{\data{b}-\wmu{\sta}}\wrapParens{\data{b}-\wmu{\std}}
	%- \wt{c}\wrapParens{\data{c}-\wmu{\sta}}\wrapParens{\data{c}-\wmu{\std}},
%UNFOLD

%UNFOLD

\section{Vector Moments}%FOLDUP

Consider now the case where the data are a $m$ dimensional vector,
that is we observe $\vdata{1}, \vdata{2},$ and so on.
Moments will be expressed in terms of the \emph{Kronecker product}.
While defined for general matrices, we use the Kronecker product only on vectors.
Given vectors $\vec{a}$ and $\vec{b}$, we write
$$\vec{a}\kron\vec{b} = \left[\begin{array}{c} a_1 \vec{b}\\a_2\vec{b}\\a_3\vec{b}\\\vdots\\a_k\vec{b}\end{array}\right].
$$
If $\vec{a}, \vec{b}$ are both $m$-dimensional then $\vec{a}\kron\vec{b}$ is an $m^2$-dimensional vector.

Note that the Kronecker product is not commutative.  
However, there is a special matrix, the \emph{Commutation matrix}, \komm[m,m], that performs commutation \cite{magnus1979commutation}:
$$
\vec{a}\kron\vec{b} = \komm[m,m]\left(\vec{b}\kron\vec{a}\right).
$$
The subscript refers to the number of columns of the vectors in the Kronecker product, here $m$ and $m$,
but the matrix $\komm[m,m]$ is $m^2 \times m^2$.
The Commutation matrix is involutary, that is $\kommUL{2}{m,m} = \meye$.

We use special notation to denote the repeated application of Kronecker product to the same vector:
$$
\kpow{\vec{a}}{n} = \bigkron_{i=1}^{n} \vec{a} = \underbrace{\vec{a}\kron\vec{a}\kron \ldots \vec{a}}_{n\,\,\text{times}}.
$$
If $\vec{a}$ is $m$-dimensional, then $\kpow{\vec{a}}{n}$ is $m^n$-dimensional.

We can now quote a binomial expansion for Kronecker products.
$$
\kpow{\left(\vec{a} + \vec{b}\right)}{k} = \sum_j^k \mcoef{j,k} \kpow{\vec{a}}{j} \kron \kpow{\vec{b}}{k-j},
$$
for some binomial coefficient matrices, $\mcoef{j,k}$. 
These matrices are $m^k \times m^k$ matrices when $\vec{a}, \vec{b}$ are $m$-dimensional vectors.
We have the following base and recurrence relationships between the coefficient matrices:
\begin{align*}
\mcoef{0,k} &= \mcoef{k,k} = \meye[m^k],\\
	\mcoef{j,k} &= \meye[m] \kron \mcoef{j-1,k-1} + \komm[m^{k-1},m]\left(\mcoef{j,k-1}\kron\meye[m]\right),\quad\text{for}\,0 < j < k.
\end{align*}

Thus in particular,
\begin{align*}
	\kpow{\left(\vec{a} + \vec{b}\right)}{2} &= \meye[m^2] \kpow{\vec{a}}{2} + \left(\meye[m^2] + \komm[m,m]\right) \vec{a}\kron\vec{b} + \meye[m^2]\kpow{\vec{b}}{2},\\
	\kpow{\left(\vec{a} + \vec{b}\right)}{3} &= \meye[m^3]\kpow{\vec{a}}{3} 
	+ \left(\meye[m^3] + \komm[m^2,m] + \komm[m^2,m]\left(\komm[m,m]\kron\meye[m]\right)\right) \kpow{\vec{a}}{2}\kron\vec{b}\\
	&\phantom{=}\,
	+ \left(\meye[m^3] + \meye[m]\kron\komm[m,m] + \komm[m^2,m] \right)\vec{a}\kron\kpow{\vec{b}}{2} 
	+ \meye[m^3]\kpow{\vec{b}}{3},
\end{align*}
and so on.
Notice how the coefficent matrices generalize the scalar binomial coefficients. For example,
\begin{equation*}
	\mcoef{1,3} \vec{a}\kron\kpow{\vec{b}}{2} = 
	\vec{a}\kron\vec{b}\kron\vec{b} + 
	\vec{b}\kron\vec{a}\kron\vec{b} + 
	\vec{b}\kron\vec{b}\kron\vec{a}.
\end{equation*}

\subsection{Update Formula for Vector Moments}

We can now proceed as in Section~\ref{sec:the_update_formula}.
Let \sta be a set of indices over the data \vdata{i} and corresponding weights
$\wt{i} > 0$
Define the total elements, sum of weights, and (weighted) mean over \sta via
\begin{align*}
\ndf{\sta} &= \abs{\sta},\\
\twt{\sta} &= \sum_{i \in \sta} \wt{i},\\
\vwmu{\sta} &= \frac{\sum_{i \in \sta} \wt{i} \vdata{i}}{\twt{\sta}}.
\end{align*}
Then go on to define the \kth{k} centered weighted sum via
$$
\mom{\sta}{k} = \sum_{i \in \sta} \wt{i} \kpow{\wrapParens{\vdata{i} - \vwmu{\sta}}}{k}.
$$
Again, $\mom{\sta}{0} = \twt{\sta},$ and $\mom{\sta}{1} = 0$.

Let \stb and \stc be sets of indices with the
restriction that 
$\sta \cap \stb = \emptyset$, 
$\stc \subseteq \sta$, and define
$$
\std = \sta \cup \stb \setminus \stc.
$$
% Thus \std is \sta `plus' \stb `minus' \stc. 
We have:
\begin{align*}
	\ndf{\std} &= \ndf{\sta} + \ndf{\stb} - \ndf{\stc},\\
	\twt{\std} &= \twt{\sta} + \twt{\stb} - \twt{\stc},\\
	\vwmu{\std} &= \frac{\twt{\sta}\vwmu{\sta} + \twt{\stb}\vwmu{\stb} - \twt{\stc}\vwmu{\stc}}{\twt{\std}},\\
	&= \vwmu{\sta} + \frac{\twt{\stb}\wrapParens{\vwmu{\stb} - \vwmu{\sta}} - \twt{\stc}\wrapParens{\vwmu{\stc}-\vwmu{\sta}}}{\twt{\std}},
\end{align*}
and
\begin{align*}
	\mom{\std}{k}&=\sum_{i \in \std} \wt{i} \kpow{\wrapParens{\vdata{i} - \vwmu{\std}}}{k},\\
	%&=\sum_{i \in \sta} \wt{i} \wrapParens{\data{i} - \vwmu{\std}}^k +
	%\sum_{i \in \stb} \wt{i} \wrapParens{\data{i} - \vwmu{\std}}^k -
	%\sum_{i \in \stc} \wt{i} \wrapParens{\data{i} - \vwmu{\std}}^k,\\
	%&=\sum_{i \in \sta} \wt{i} \wrapParens{\data{i} - \vwmu{\sta} + \vwmu{\sta} - \vwmu{\std}}^k +
	%\sum_{i \in \stb} \wt{i} \wrapParens{\data{i} - \vwmu{\stb} + \vwmu{\stb} - \vwmu{\std}}^k -
	%\sum_{i \in \stc} \wt{i} \wrapParens{\data{i} - \vwmu{\stc} + \vwmu{\stc} - \vwmu{\std}}^k,\\
	%&= \sum_{i \in \sta} \sum_{0 \le j \le k}{k \choose j}
		%\wt{i} \wrapParens{\data{i} - \vwmu{\sta}}^j \wrapParens{\vwmu{\sta} - \vwmu{\std}}^{k-j}\\
	%&\phantom{=}\,+
	%\sum_{i \in \stb} \sum_{0 \le j \le k}{k \choose j}
		%\wt{i} \wrapParens{\data{i} - \vwmu{\stb}}^j \wrapParens{\vwmu{\stb} - \vwmu{\std}}^{k-j}\\
	%&\phantom{=}\,-
	%\sum_{i \in \stc} \sum_{0 \le j \le k}{k \choose j}
		%\wt{i} \wrapParens{\data{i} - \vwmu{\stc}}^j \wrapParens{\vwmu{\stc} - \vwmu{\std}}^{k-j},\\
	%&=\sum_{0 \le j \le k}{k \choose j}\wrapBraces{
		%\mom{\sta}{j} \wrapParens{\vwmu{\sta} - \vwmu{\std}}^{k-j} +
		%\mom{\stb}{j} \wrapParens{\vwmu{\stb} - \vwmu{\std}}^{k-j} -
		%\mom{\stc}{j} \wrapParens{\vwmu{\stc} - \vwmu{\std}}^{k-j}},\\
	&= \mom{\sta}{k} + \mom{\stb}{k} - \mom{\stc}{k}\\
  &\phantom{=}\, +
		\twt{\sta} \kpow{\wrapParens{\vwmu{\sta} - \vwmu{\std}}}{k} +
		\twt{\stb} \kpow{\wrapParens{\vwmu{\stb} - \vwmu{\std}}}{k} -
		\twt{\stc} \kpow{\wrapParens{\vwmu{\stc} - \vwmu{\std}}}{k}\\
	&\phantom{=}\,+ \sum_{2 \le j < k}\mcoef{j,k}\wrapBraces{
		\mom{\sta}{j} \kron \kpow{\wrapParens{\vwmu{\sta} - \vwmu{\std}}}{k-j} +
		\mom{\stb}{j} \kron \kpow{\wrapParens{\vwmu{\stb} - \vwmu{\std}}}{k-j}}\\
	&\phantom{=}\,- \sum_{2 \le j < k}\mcoef{j,k}\wrapBraces{
		\mom{\stc}{j} \kron \kpow{\wrapParens{\vwmu{\stc} - \vwmu{\std}}}{k-j}}.
\end{align*}

This formula is a bit unweildy for the general case.
Moreover, the number of elements in $\mom{\std}{k}$ quickly grows in $k$,
even when accounting for symmetry.

The case $k=2$ is of the highest interest.
The matrix binomial coefficients do not enter into the computation, and we merely have:

\begin{align*}
	\mom{\std}{2}
	&= \mom{\sta}{2} + \mom{\stb}{2} - \mom{\stc}{2}\\
  &\phantom{=}\, +
		\twt{\sta} \kpow{\wrapParens{\vwmu{\sta} - \vwmu{\std}}}{2} +
		\twt{\stb} \kpow{\wrapParens{\vwmu{\stb} - \vwmu{\std}}}{2} -
		\twt{\stc} \kpow{\wrapParens{\vwmu{\stc} - \vwmu{\std}}}{2}.
\end{align*}


\subsection{Adding a single observation}%FOLDUP

We now consider the special case where \stc is empty, and \stb consists of the single index $b$. 
We then have 
\begin{align*}
\twt{\std} &= \twt{\sta} + \wt{b},\\
\vwmu{\std} 
%&= \frac{\twt{\sta}\wmu{\sta} + \wt{b}\data{b}}{\twt{\sta} + \wt{b}},\\
	%&= \wmu{\sta} + \wt{b}\frac{\data{b} - \wmu{\sta}}{\twt{\sta} + \wt{b}},\\
	&= \vwmu{\sta} + \wt{b}\frac{\vdata{b} - \vwmu{\sta}}{\twt{\std}}.
\end{align*}
For the $k=2$ case the centered moment is updated as
\begin{align*}
	\mom{\std}{2} &= \mom{\sta}{2} +	\frac{\twt{\sta} \wt{b}}{\twt{\std}} \kpow{\wrapParens{\vdata{b} - \vwmu{\sta}}}{2},\\
	%&= \mom{\sta}{2} +	\frac{\twt{\std} \wt{b}}{\twt{\sta}} \wrapParens{\data{b} - \wmu{\std}}^{2},\\
	&= \mom{\sta}{2} + \twt{\std} \wrapParens{\vdata{b} - \vwmu{\std}} \kron\wrapParens{\vwmu{\std} - \vwmu{\sta}}.
\end{align*}
Note that the quantity $\twt{\std}\wrapParens{\vwmu{\std} - \vwmu{\sta}} = 
\wt{b}\wrapParens{\vdata{b} - \vwmu{\sta}}$ is computed as an intermediary in updating
the weighted mean, resulting in some computational savings.
%UNFOLD
\subsection{Removing a single observation}%FOLDUP

As noted above, removing a single observation should look just like adding a single
observation, except signs on the weights and weighted sums are flipped. 
Let $c$ be the single index in \stc. 
We then have $\twt{\std} = \twt{\sta} - \wt{c},$ and
\begin{align*}
\wmu{\std} &= \vwmu{\sta} - \wt{c}\frac{\vdata{c} - \vwmu{\sta}}{\twt{\std}}.
\end{align*}
For the $k=2$ case the centered sum computation simplifies to
\begin{align*}
	\mom{\std}{2} 
	%&= \mom{\sta}{2} -\frac{\twt{\sta} \wt{c}}{\twt{\std}} \wrapParens{\data{c} - \wmu{\sta}}^{2},\\
	%&= \mom{\sta}{2} -\frac{\twt{\std} \wt{c}}{\twt{\sta}} \wrapParens{\data{c} - \wmu{\std}}^{2},\\
	&= \mom{\sta}{2} + \twt{\std} \wrapParens{\vdata{c} - \vwmu{\std}} \kron\wrapParens{\vwmu{\std} - \vwmu{\sta}}.
\end{align*}
%UNFOLD
\subsection{Adding and removing a single observation}%FOLDUP

Consider the case of adding a single $b \in\stb$ and removing a single
$c\in\stc$.  
%The total number elements, \ndf{\sta}, is unchanged, of course. 
We have
\begin{align*}
\twt{\std}&=\twt{\sta} + \wt{b} - \wt{c},\\
	\vwmu{\std}&=\vwmu{\sta} + \frac{\wt{b}\wrapParens{\vdata{b} - \vwmu{\sta}} - \wt{c}\wrapParens{\vdata{c}-\vwmu{\sta}}}{\twt{\std}}.
\end{align*}
%The update formula for \mom{\std}{k} is
%\begin{align*}
	%\mom{\std}{k}
	%&= \mom{\sta}{k} + \mom{\stb}{k} - \mom{\stc}{k} +
		%\twt{\sta} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k} +
		%\wt{b} \wrapParens{\data{b} - \wmu{\std}}^{k} -
		%\wt{c} \wrapParens{\data{c} - \wmu{\std}}^{k}\\
  %&\phantom{=}\,+ \sum_{2 \le j < k}{k \choose j}\wrapBraces{
		%\mom{\sta}{j} \wrapParens{\wmu{\sta} - \wmu{\std}}^{k-j} +
		%\mom{\stb}{j} \wrapParens{\data{b} - \wmu{\std}}^{k-j} -
		%\mom{\stc}{j} \wrapParens{\data{c} - \wmu{\std}}^{k-j}}.
%\end{align*}
%In the $k=2$ case this simplifies to 

The update formula for \mom{\std}{2} is
\begin{align*}
	\mom{\std}{2}
	%&= \mom{\sta}{2} + 
		%\twt{\sta} \wrapParens{\wmu{\sta} - \wmu{\std}}^{2} +
		%\wt{b} \wrapParens{\data{b} - \wmu{\std}}^{2} -
		%\wt{c} \wrapParens{\data{c} - \wmu{\std}}^{2},\\
	%&= \mom{\sta}{2} 
	%+ \wrapParens{\wmu{\sta}-\wmu{\std}}^2 \wrapParens{\twt{\sta} - \twt{\std}}
	%+ \wt{b} \wrapParens{\data{b} - \wmu{\std}}\wrapParens{\data{b} - \wmu{\sta}}
	%- \wt{c} \wrapParens{\data{c} - \wmu{\std}}\wrapParens{\data{c} - \wmu{\sta}},\\
	&= \mom{\sta}{2} 
	+ \frac{\twt{\sta}\wt{b} \kpow{\wrapParens{\vdata{b}-\vwmu{\sta}}}{2} - \twt{\std}\wt{c}\kpow{\wrapParens{\vdata{c}-\vwmu{\std}}}{2}}{\twt{\sta} + \wt{b}}.
\end{align*}
Because Kronecer products are not commutative, the trick to simplify this expression
in the case where $\wt{b}=\wt{c}$, involving completing the square, cannot be easily applied.

%Further simplifications are possible when $\wt{b}=\wt{c}$ (as occurs in the vanilla
%case of equal weighted moments), and we arrive at
%\begin{align*}
%\vwmu{\std}&=\vwmu{\sta} + \frac{\wt{b}\wrapParens{\vdata{b} - \vdata{c}}}{\twt{\std}},\\
%\mom{\std}{2} &= \mom{\sta}{2} 
	%+ \wt{b} \wrapParens{\vdata{b}+\vdata{c}-\vwmu{\sta}-\vwmu{\std}} \kron\wrapParens{\vdata{b}-\vdata{c}}.
%\end{align*}
%UNFOLD



%UNFOLD

\section{The output}%FOLDUP

Several kinds of output are produced by the \fromo  package. We describe them in more detail here.

\begin{itemize}
	\item[mean] The (weighted) mean over the index set \sta is simply \wmu{\sta}.
	\item[standard deviation] The standard deviation over the index set \sta is 
$$
		\sdev{\sta} = \sqrt{\frac{\mom{\sta}{2}}{\twt{\sta} - \nu}},
$$
where $\nu$ are the `consumed degrees of freedom', and is typically set to $1$. 
When the normalization flag is set to true, however, it is intended that the weights
be normalized to have mean value $1$. In this case the standard deviation over 
\sta is defined as
$$
		\sdev{\sta} = \sqrt{\frac{\mom{\sta}{2}}{\twt{\sta}}\frac{\ndf{\sta}}{\ndf{\sta} - \nu}}.
$$
When $\nu=0$, these are identical.
	\item[centered moment] The \kth{k} centered moment is defined as
$$
\cmom{\sta}{k} = \frac{\mom{\sta}{k}}{\twt{\sta}}.
$$
Note that for $k>2$ we do not support consumed degrees of freedom, and so
normalization of weights has no effect.
		The first centered moment is zero: $\cmom{\sta}{1} = 0$.
	\item[standardized moment] The \kth{k} standardized (and centered) moment is defined as
$$
\scmom{\sta}{k} = \frac{\cmom{\sta}{k}}{\sdevpow{\sta}{k}}.
$$
Normalization and consumed degrees of freedom affect the computation of \sdevpow{\sta}{k},
		and so affect the standardized moments.
	\item[centered value] For a given index $i$ and some fixed set \sta, the centered version of
		\data{i} is merely $\data{i} - \wmu{\sta}$.
	\item[standardized value] For a given index $i$ and some fixed set \sta, the standardized version of
		\data{i} is merely $\data{i}/\sdev{\sta}$. 
		It is affected by normalization and consumed degrees of freedom.
	\item[z-scored value] For a given index $i$ and some fixed set \sta, the z-scored version of
		\data{i} is $\wrapParens{\data{i} - \wmu{\sta}}/\sdev{\sta}$. 
		It is affected by normalization and consumed degrees of freedom.

	\item[Cumulants] Sometimes called ``centered cumulants'' in the package, though this is 
		redundant and not common usage. Cumulants are defined from the centered moments by
		the recursive formula: \cite{willink2003}
\begin{equation}
	\ccum{\sta}{r} = \cmom{\sta}{r} - \sum_{j=1}^{r - 2} {r-1 \choose j} \cmom{\sta}{j} \ccum{\sta}{r-j}.
\end{equation}
	%\ccum{\sta}{r} = \cmom{\sta}{r} - \sum_{j=0}^{r - 2} \binom{r-1}{j} \cmom{\sta}{j} \ccum{\sta}{r-j}.
		Note that the sum is over an empty index for $r=2$, and that $\cmom{\sta}{1} = 0$, so we have
\begin{align*}
	\ccum{\sta}{2} &= \cmom{\sta}{2},\\
	\ccum{\sta}{3} &= \cmom{\sta}{3},\\
	\ccum{\sta}{4} &= \cmom{\sta}{4} - 3\cmom{\sta}{2}^2,\\
	\ccum{\sta}{5} &= \cmom{\sta}{5} - 10\cmom{\sta}{3}\cmom{\sta}{2},
\end{align*}
and so on.

	\item[Standardized Cumulants] These are the regular cumulants normalized by the
		computed standard deviation:
\begin{equation}
	\sccum{\sta}{r} = \frac{\ccum{\sta}{r}}{\sdevpow{\sta}{r}}.
\end{equation}


\end{itemize}

\subsection{Output for Bivariate Input}

Consider now the case where the user provides input \xdata{i} and \ydata{i}.
We denote the (weighted) mean over the index set \sta of the \xdata{i} as \wmu{\sta,x},
and similarly \wmu{\sta,y} is the (weighted) mean of the \ydata{i}.
Write $\mom{\sta}{x,x}$, 
$\mom{\sta}{x,y}$, and
$\mom{\sta}{y,y}$ for the weighted sums of centered second power terms:
\begin{align*}
\mom{\sta}{x,x} &= \sum_{i \in \sta} \wt{i} \wrapParens{\xdata{i} - \wmu{\sta,x}}^2,\\
\mom{\sta}{x,y} &= \sum_{i \in \sta} \wt{i} \wrapParens{\xdata{i} - \wmu{\sta,x}}\wrapParens{\ydata{i} - \wmu{\sta,y}},\\
\mom{\sta}{y,y} &= \sum_{i \in \sta} \wt{i} \wrapParens{\ydata{i} - \wmu{\sta,y}}^2.
\end{align*}
We then have the following output:

\begin{itemize}
	\item[correlation] The correlation is computed as
$$
\frac{\mom{\sta}{x,y}}{\sqrt{\mom{\sta}{x,x} \mom{\sta}{y,y}}}.
$$
	\item[regression slope] We consider the ordinary least squares regression of \ydata{i} against \xdata{i}. 
		The data are supposed to follow $\ydata{i} = \beta_0 + \beta_1 \xdata{i}$, and this computation estimates $\beta_1$.
		This has the value
$$
\frac{\mom{\sta}{x,y}}{\mom{\sta}{x,x}}.
$$
	\item[regression intercept] We again consider the OLS regression of \ydata{i} against \xdata{i} and return an estimate of $\beta_0$
		taking value
$$
\wmu{\sta,y} - \wmu{\sta,x} \frac{\mom{\sta}{x,y}}{\mom{\sta}{x,x}}.
$$
	\item[regression standard error] We again consider the OLS regression of \ydata{i} against \xdata{i}. 
		We write $\sigma^2$ to denote the variance of $\ydata{i} - \wrapParens{\beta_0 + \beta_1 \ydata{i}}$.
		The ``regression standard error'' is an estimate of $\sigma$.
It is computed as 
$$
	\sqrt{\frac{\mom{\sta}{y,y} - {\mompow{\sta}{x,y}{2}}/{\mom{\sta}{x,x}}}{\ndf{\sta} - \nu}},
$$
		where $\nu$ are the `consumed degrees of freedom'. When $\nu=0$ this quantity is often denoted as $\hat{\sigma}^2$;
		when $\nu=2$ it is often written as $s^2$.
	\item[regression slope standard error] This estimates the standard error of the estimate of $\beta_1$ in the regression.
It is computed as
$$
\sqrt{\frac{s^2}{\mom{\sta}{x,x}}},
$$
where $s^2$ is the regression standard error computed with $\nu=2$.
	\item[regression intercept std. err.] This estimates the standard error of the estimate of $\beta_0$ in the regression.
It is computed as
$$
\sqrt{\frac{s^2\wrapParens{\mom{\sta}{x,x}/\ndf{\sta} + \wmu{\sta,x}^2}}{\mom{\sta}{x,x}}},
$$
where $s^2$ is the regression standard error computed with $\nu=2$.
	\item[covariance] We estimate the covariance of \ydata{i} and \xdata{i} by computing
$$
\frac{\mom{\sta}{x,y}}{\ndf{\sta} - \nu},
$$
where $\nu$ are the `consumed degrees of freedom', typically set to $2$. 
	\item[covariance 3] We estimate the full covariance matrix of \ydata{i} and \xdata{i} by computing the lower triangle of the matrix
$$
\frac{1}{\ndf{\sta} - \nu}\left[
\begin{array}{cc}
\mom{\sta}{x,x} & \mom{\sta}{x,y}\\
\mom{\sta}{x,y} & \mom{\sta}{y,y}
\end{array}
\right],
$$
where $\nu$ are the `consumed degrees of freedom', typically set to $2$. 



\end{itemize}


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\section{Running Moments}%FOLDUP

The \fromo  package can also perform 
running\footnote{Variously known also as \emph{rolling}, \emph{boxcar}, or \emph{finite impulse response} computations.} computations
of moments, centered moments, centered values, standardized values, z-scored values and so on. Given an integral window, $W$,
and data and weights vectors $\data{i},$ $\wt{i}$, we compute output \outp{i} as follows: let \std be the set of $j$ with
$i - W < j \le i$. Then compute the desired moment over \std, and return it as \outp{i}.  The `comparison' operations 
(computed centered, standardized, or z-scored versions of a variable) admit a \emph{lookahead} parameter, $l$. In this case,
we let \std be the set of $j$ with $i - W + l < j \le i + l$, then compute the moments over \std and normalize \data{i} with
respect to those moments. When $l > 0$, we are using future information to compute \outp{i}. That is, \outp{i} will depend
on \data{j} with $j > i$. This is not the case for the typical moments computations or for the case where the lookahead is
non-positive.

We also support a time (or other counter) based running computation. Here the input are the data, and weights vectors, \data{i}, \wt{i},
but also a vector of time indices, say \tim{i} which are non-decreasing: $\tim{1} \le \tim{2} \le \ldots$. It is assumed that
$\tim{0}$ is essentially $-\infty$. The window, $W$ is now a time-based window. The output \outp{i} is defined in terms of the
moments computations over the set \std which are all $j$ such that $\tim{i} - W < \tim{j} \le \tim{i}$. For comparison functions,
we again admit a lookahead so that \std are all $j$ such that $\tim{i} - W + l < \tim{j} \le \tim{i} + l$. Obviously if we 
let $\tim{i} = i$, we recover the `vanilla' running moments computations. We also support supplying the \tim{i} implicitly as the
sum of positive deltas: let vector \dltim{i} of strictly positive elements be given. Then we assume
$$
\tim{i} = \sum_{1 \le j \le i} \dltim{j},
$$
and then perform the time-based running computation. We also support the case where the \dltim{i} are actually just the \wt{i}.

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