A Review of Welford's Paper¶

In 1962, B.P. Welford published a paper titled Note on a Method for Calculating Corrected Sums of Squares and Products. Let's review some important identities/derivations from this paper as they will come in handy later. Note that we use zero-based indexing when referencing elements of our time series.

Deriving Equation (1)¶

\begin{align} \mu_{Q_{i,n}} ={}& \sum \limits _{0 \leq j \lt n} \frac{q_{i+j}}{n} \\ ={}& \frac{q_{i}}{n} + \frac{q_{i+1}}{n} + \ldots + \frac{q_{i+n-2}}{n} + \frac{q_{i+n-1}}{n} \\ ={}& 1 \cdot \frac{q_{i}}{n} + 1 \cdot \frac{q_{i+1}}{n} + \ldots + 1 \cdot \frac{q_{i+n-2}}{n} + \frac{q_{i+n-1}}{n} \\ ={}& \frac{n-1}{n-1} \frac{q_{i}}{n} + \frac{n-1}{n-1} \frac{q_{i+1}}{n} + \ldots + \frac{n-1}{n-1} \frac{q_{i+n-2}}{n} + \frac{q_{i+n-1}}{n} \\ ={}& \frac{n-1}{n} \frac{q_{i}}{n-1} + \frac{n-1}{n} \frac{q_{i+1}}{n-1} + \ldots + \frac{n-1}{n} \frac{q_{i+n-2}}{n} + \frac{q_{i+n-1}}{n} \\ ={}& \frac{n-1}{n} \left( \frac{q_{i}}{n-1} + \frac{q_{i+1}}{n-1} + \ldots + \frac{q_{i+n-2}}{n-1} \right) + \frac{q_{i+n-1}}{n} \\ ={}& \frac{n-1}{n} \sum \limits _{0 \leq j \lt n-1} \frac{q_{i+j}}{n-1} + \frac{q_{i+n-1}}{n} \\ ={}& \frac{n-1}{n} \mu_{Q_{i,n-1}} + \frac{q_{i+n-1}}{n} \\ \mu_{Q_{i,m}} ={}& \frac{m-1}{m} \mu_{Q_{i,m-1}} + \frac{q_{i+m-1}}{m} \\ \end{align}

Deriving Equation (2)¶

\begin{align} j < n-1, \;\;\;\;\; q_{i+j} - \mu_{Q_{i,n}} ={}& q_{i+j} - \frac{n-1}{n} \mu_{Q_{i,n-1}} - \frac{q_{i+n-1}}{n} \\ ={}& q_{i+j} - \frac{n}{n} \mu_{Q_{i,n-1}} + \frac{\mu_{Q_{i,n-1}}}{n} - \frac{q_{i+n-1}}{n} \\ ={}& q_{i+j} - \mu_{Q_{i,n-1}} + \frac{1}{n} \left( -q_{i+n-1} + \mu_{Q_{i,n-1}} \right) \\ ={}& q_{i+j} - \mu_{Q_{i,n-1}} - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \\ q_{i+j} - \mu_{Q_{i,m}} ={}& q_{i+j} - \mu_{Q_{i,m-1}} - \frac{1}{m} \left( q_{i+m-1} - \mu_{Q_{i,m-1}} \right) \\ \end{align}

Deriving Equation (3)¶

\begin{align} j = n-1, \;\;\;\;\; q_{i+n-1} - \mu_{Q_{i,n}} &= q_{i+n-1} - \frac{n-1}{n} \mu_{Q_{i,n-1}} - \frac{q_{i+n-1}}{n} \\ &= - \frac{n-1}{n} \mu_{Q_{i,n-1}} - \frac{q_{i+n-1}}{n} + q_{i+n-1} \\ &= - \frac{n-1}{n} \mu_{Q_{i,n-1}} + q_{i+n-1} \left( -\frac{1}{n} + 1 \right) \\ &= - \frac{n-1}{n} \mu_{Q_{i,n-1}} + q_{i+n-1} \left( -\frac{1}{n} + \frac{n}{n} \right) \\ &= - \frac{n-1}{n} \mu_{Q_{i,n-1}} + q_{i+n-1} \left( -\frac{1}{n} + \frac{n}{n} \right) \\ &= - \frac{n-1}{n} \mu_{Q_{i,n-1}} + q_{i+n-1} \left( \frac{n-1}{n} \right) \\ &= \frac{n-1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \\ q_{i+m-1} - \mu_{Q_{i,m}} &= \frac{m-1}{m} \left( q_{i+m-1} - \mu_{Q_{i,m-1}} \right) \end{align}

Deriving Equation (I) - The Corrected Sum of Squares¶

for $n = 1, 2, \ldots, m$

\begin{align} S(Q_{i,n}) ={}& \sum \limits _{0 \leq j \lt n} \left( q_{i+j} - \mu_{Q_{i,n}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} (q_{i+j} - \mu_{Q_{i,n}})^2 + \left( q_{i+n-1} - \mu_{Q_{i,n}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left[ q_{i+j} - \mu_{Q_{i,n-1}} - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \right] ^2 + \left[ \frac{n-1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \right] ^2 \\ ={}& \sum \limits _{0 \leq j \lt m-1} \left[ \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \right] ^2 + \left( \frac{n-1}{n} \right) ^2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left[ \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) ^2 - \frac{2}{n} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) + \frac{1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \right] + \frac{(n-1)^2}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) ^2 - \sum \limits _{0 \leq j \lt n-1} \frac{2}{n} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) + \sum \limits _{0 \leq j \lt n-1} \frac{1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 + \frac{(n-1)^2}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) ^2 - 2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \frac{1}{n} \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) + \sum \limits _{0 \leq j \lt n-1} \frac{1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 + \frac{(n-1)^2}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) ^2 - 2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \cdot 0 + \sum \limits _{0 \leq j \lt n-1} \frac{1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 + \frac{(n-1)^2}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) ^2 + \frac{n-1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 + \frac{(n-1)^2}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) ^2 + \left[ \frac{n-1}{n^2} + \frac{(n-1)^2}{n^2} \right] \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) ^2 + \left[ \frac{n-1+n^2-2n+1}{n^2} \right] \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) ^2 + \left[ \frac{n^2-n}{n^2} \right] \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ ={}& S(Q_{i,n-1}) + \left( \frac{n-1}{n} \right) \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) ^2 \\ S(Q_{i,m}) ={}& S({Q_{i,m-1}}) + \left( \frac{m-1}{m} \right) \left( q_{i+m-1} - \mu_{Q_{i,m-1}} \right) ^2 \\ \end{align}

Deriving Equation (II) - The Corrected Sum of Products¶

\begin{align} S(Q_{i,n}, T_{i,n}) ={}& \sum \limits _{0 \leq j \lt n} \left( q_{i+j} - \mu_{Q_{i,n}} \right) \left( t_{i+j} - M_{T_{i,n}} \right) \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n}} \right) \left( t_{i+j} - M_{T_{i,n}} \right) + \left( q_{i+n-1} - \mu_{Q_{i,n}} \right) \left( t_{i+n-1} - M_{T_{i,n}} \right) \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left[ q_{i+j} - \mu_{Q_{i,n-1}} - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \right] \left[ t_{i+j} - M_{T_{i,n-1}} - \frac{1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \right] + \left[ \frac{n-1}{n} \left( q_{i+n-1} - \mu_{Q_{i+n-1}} \right) \right] \left[ \frac{n-1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \right] \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left[ \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \right] \left[ \left( t_{i+j} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \right] + \left( \frac{n-1}{n} \right)^2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left[ \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) - \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \frac{1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) + \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \frac{1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \right] \\ {}& + \left( \frac{n-1}{n} \right)^2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) -\frac{1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \sum \limits _{0 \leq j \lt n-1} \left( t_{i+j} - M_{T_{i,n-1}} \right) +\frac{1}{n^2} \sum \limits _{0 \leq j \lt n-1} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ {}& + \left( \frac{n-1}{n} \right)^2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) -\frac{1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \sum \limits _{0 \leq j \lt n-1} \left( t_{i+j} - M_{T_{i,n-1}} \right) +\frac{n-1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ {}& + \left( \frac{n-1}{n} \right)^2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1}, T_{i,n-1}) -\frac{1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \sum \limits _{0 \leq j \lt n-1} \left( t_{i+j} - M_{T_{i,n-1}} \right) +\frac{n-1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ {}& + \left( \frac{n-1}{n} \right)^2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1}, T_{i,n-1}) -\frac{1}{n} \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \cdot 0 - \frac{1}{n} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \cdot 0 +\frac{n-1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) + \left( \frac{n-1}{n} \right)^2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1}, T_{i,n-1}) +\frac{n-1}{n^2} \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) + \left( \frac{n-1}{n} \right)^2 \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1}, T_{i,n-1}) + \left[ \frac{n-1}{n^2} + \left( \frac{n-1}{n} \right)^2 \right] \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1}, T_{i,n-1}) + \left[ \frac{n-1}{n^2} + \frac{n^2-2n+1}{n^2} \right] \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i+n-1}, T_{i,n-1}) + \left( \frac{n-1+n^2-2n+1}{n^2} \right) \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1}, T_{i,n-1}) + \left( \frac{n^2-n}{n^2} \right) \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1}, T_{i,n-1}) + \left( \frac{n-1}{n} \right) \left( q_{i+n-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i+n-1} - M_{T_{i,n-1}} \right) \\ \end{align}

\begin{align} S(Q_{i,m}, T_{i,m}) ={}& S(Q_{i,m-1}, T_{i,m-1}) + \left( \frac{m-1}{m} \right) \left( q_{i+m-1} - \mu_{Q_{i,m-1}} \right) \left( t_{i+m-1} - M_{T_{i,m-1}} \right) \\ S(Q_{i-1,m}, T_{i-1,m}) ={}& S(Q_{i-1,m-1}, T_{i-1,m-1}) + \left( \frac{m-1}{m} \right) \left( q_{i+m-2} - \mu_{Q_{i-1,m-1}} \right) \left( t_{i+m-2} - M_{T_{i-1,m-1}} \right) \\ \end{align}

Building on Equation (II) and also the fact that:

\begin{align} \mu_{Q_{i-1,n}} ={}& \frac{n-1}{n} \mu_{Q_{i-1,n-1}} + \frac{q_{i+n-2}}{n} \\ ={}& \mu_{Q_{i-1,n-1}} - \frac{1}{n} \mu_{Q_{i-1,n-1}} + \frac{q_{i+n-2}}{n} \\ ={}& \mu_{Q_{i-1,n-1}} + \frac{q_{i+n-2}}{n} - \frac{\mu_{Q_{i-1,n-1}}}{n} \\ ={}& \frac{1}{n} \sum \limits _{0 \leq j \lt n-1} q_{i+j-1} + \frac{q_{i+n-2}}{n} - \frac{\mu_{Q_{i-1,n-1}}}{n} \\ ={}& \frac{q_{i-1}}{n} + \frac{1}{n} \sum \limits _{1 \leq j \lt n-1} q_{i+j-1} + \frac{q_{i+n-2}}{n} - \frac{\mu_{Q_{i-1,n-1}}}{n} \\ ={}& \frac{q_{i-1}}{n} + \frac{1}{n} \sum \limits _{0 \leq j \lt n-1} q_{i+j} - \frac{\mu_{Q_{i-1,n-1}}}{n} \\ ={}& \frac{q_{i-1}}{n} + \mu_{Q_{i,n-1}} - \frac{\mu_{Q_{i-1,n-1}}}{n} \\ ={}& \mu_{Q_{i,n-1}} + \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \\ \end{align}

we can actually express $S(Q_{i-1,n},T_{i-1,n})$ with respect to $S(Q_{i,n-1},T_{i,n-1}) = \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right)$ and $\left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right)$:

\begin{align} S(Q_{i-1,n}, T_{i-1,n}) ={}& \sum \limits _{0 \leq j \lt n} \left( q_{i+j-1} - \mu_{Q_{i-1,n}} \right) \left( t_{i+j-1} - M_{T_{i-1,n}} \right) \\ ={}& \left( q_{i-1} - \mu_{Q_{i-1,n}} \right) \left( t_{i-1} - M_{T_{i-1,n}} \right) + \sum \limits _{1 \leq j \lt n} \left( q_{i+j-1} - \mu_{Q_{i-1,n}} \right) \left( t_{i+j-1} - M_{T_{i-1,n}} \right) \\ ={}& \left( q_{i-1} - \mu_{Q_{i-1,n}} \right) \left( t_{i-1} - M_{T_{i-1,n}} \right) + \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i-1,n}} \right) \left( t_{i+j} - M_{T_{i-1,n}} \right) \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i-1,n}} \right) \left( t_{i+j} - M_{T_{i-1,n}} \right) + \left( q_{i-1} - \mu_{Q_{i-1,n}} \right) \left( t_{i-1} - M_{T_{i-1,n}} \right) \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left[ \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \right] \left[ \left( t_{i+j} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \right] + \left[ \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \right] \left[ \left( t_{i-1} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \right] \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left[ \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) - \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) + \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \right] \\ &+ \left[ \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \right] \left[ \left( t_{i-1} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \right] \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) - \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) - \sum \limits _{0 \leq j \lt n-1} \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) + \sum \limits _{0 \leq j \lt n-1} \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ &+ \left[ \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \right] \left[ \left( t_{i-1} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \right] \\ ={}& S(Q_{i,n-1},T_{i,n-1}) - \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \sum \limits _{0 \leq j \lt n-1} \left( t_{i+j} - M_{T_{i,n-1}} \right) + \frac{n-1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ &+ \left[ \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \right] \left[ \left( t_{i-1} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \right] \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \cdot 0 - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \cdot 0 + \frac{n-1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ &+ \left[ \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \right] \left[ \left( t_{i-1} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \right] \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( q_{i+j} - \mu_{Q_{i,n-1}} \right) \left( t_{i+j} - M_{T_{i,n-1}} \right) + \frac{n-1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ &+ \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) - \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \frac{1}{n} \left( t_{i-1} - M_{T_{i-1,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) + \frac{1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ ={}& S(Q_{i,n-1},T_{i,n-1}) + \frac{n-1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ &+ \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) - \frac{n}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) + \frac{1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ ={}& S(Q_{i,n-1},T_{i,n-1}) + \frac{n-1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) - \frac{n}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) + \frac{1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ &+ \frac{n}{n} \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) - \frac{1}{n} \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) \\ ={}& S(Q_{i,n-1},T_{i,n-1}) + \frac{n-1-n+1}{n^2} \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) + \frac{n-1}{n} \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1},T_{i,n-1}) + 0 \cdot \left( q_{i-1} - \mu_{Q_{i-1,n-1}} \right) \left( t_{i-1} - M_{T_{i-1,n-1}} \right) + \frac{n-1}{n} \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1},T_{i,n-1}) + \frac{n-1}{n} \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) \\ ={}& S(Q_{i,n-1},T_{i,n-1}) + \frac{n-1}{n} \left( q_{i-1} - \mu_{Q_{i,n-1}} \right) \left( t_{i-1} - M_{T_{i,n-1}} \right) \\ S(Q_{i-1,m}, T_{i-1,m}) ={}& S(Q_{i,m-1},T_{i,m-1}) + \frac{m-1}{m} \left( q_{i-1} - \mu_{Q_{i,m-1}} \right) \left( t_{i-1} - M_{T_{i,m-1}} \right) \\ \end{align}

Deriving Rolling Window Variance (or Standard Deviation) with Welford's Method¶

For any sequence $T$, we can compute the mean, $\mu_{T_{i,n}}$, for any subsequence starting at position $i$ in $T$ and with subsequence length $n$ according to the following definition:

\begin{align} \mu_{T_{i,n}} ={}& \sum \limits _{0 \leq j \lt n} \frac{t_{i+j}}{n} \\ ={}& \frac{t_{i}}{n} + \frac{t_{i+1}}{n} + \ldots + \frac{t_{i+n-2}}{n} + \frac{t_{i+n-1}}{n} \\ ={}& 0 + \frac{t_{i}}{n} + \frac{t_{i+1}}{n} + \ldots + \frac{t_{i+n-2}}{n} + \frac{t_{i+n-1}}{n} \\ ={}& \left(\frac{t_{i-1}}{n} - \frac{t_{i-1}}{n}\right) + \frac{t_{i}}{n} + \frac{t_{i+1}}{n} + \ldots + \frac{t_{i+n-2}}{n} + \frac{t_{i+n-1}}{n} \\ ={}& \frac{t_{i-1}}{n} + \frac{t_{i}}{n} + \frac{t_{i+1}}{n} + \ldots + \frac{t_{i+n-2}}{n} + \frac{t_{i+n-1}}{n} - \frac{t_{i-1}}{n} \\ ={}& \sum \limits _{-1 \leq j \lt n-1} \frac{t_{i+j}}{n} + \frac{t_{i+n-1}}{n} - \frac{t_{i-1}}{n} \\ ={}& \mu_{T_{i-1,n}} + \frac{t_{i+n-1} - t_{i-1}}{n} \\ ={}& \mu_{T_{i-1,n}} + \frac{t_{i+n-1} - t_{i-1}}{n} \\ \end{align}

Then, it follows that:

\begin{align} t_{i+j} - \mu_{T_{i,n}} ={}& t_{i+j} - \mu_{T_{i-1,n}} - \frac{t_{i+n-1} - t_{i-1}}{n} \\ \end{align}

Finally, we can derive the rolling window variance, $S$, by first calculating the correctect sum-of-squares, $CSS$:

\begin{align} CSS(T_{i,n}) ={}& \sum \limits _{0 \leq j \lt n} \left( t_{i+j} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i,n}} \right) ^2 + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i,n}} \right) ^2 + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 + 0 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i,n}} \right) ^2 + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 + \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{0 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i,n}} \right) ^2 + \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{-1 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i,n}} \right) ^2 + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{-1 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i-1,n}} - \frac{t_{i+n-1} - t_{i-1}}{n} \right) ^2 + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{-1 \leq j \lt n-1} \left[ \left( t_{i+j} - \mu_{T_{i-1,n}} \right) - \frac{t_{i+n-1} - t_{i-1}}{n} \right] ^2 + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{-1 \leq j \lt n-1} \left[ \left( t_{i+j} - \mu_{T_{i-1,n}} \right)^2 - 2 \left(t_{i+n-1} - t_{i-1} \right)\frac{1}{n} \left( t_{i+j} - \mu_{T_{i-1,n}} \right) + \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n^2} \right] + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{-1 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i-1,n}} \right)^2 - \sum \limits _{-1 \leq j \lt n-1} 2 \left(t_{i+n-1} - t_{i-1} \right)\frac{1}{n} \left( t_{i+j} - \mu_{T_{i-1,n}} \right) + \sum \limits _{-1 \leq j \lt n-1} \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n^2} + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{-1 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i-1,n}} \right)^2 - 2 \left(t_{i+n-1} - t_{i-1} \right)\frac{1}{n} \sum \limits _{-1 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i-1,n}} \right) + \sum \limits _{-1 \leq j \lt n-1} \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n^2} + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& \sum \limits _{-1 \leq j \lt n-1} \left( t_{i+j} - \mu_{T_{i-1,n}} \right)^2 - 2 \left(t_{i+n-1} - t_{i-1} \right) \cdot 0 + \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n} + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& CSS(T_{i-1,n}) + \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n} + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) ^2 - \left( t_{i-1} - \mu_{T_{i,n}} \right) ^2 \\ ={}& CSS(T_{i-1,n}) + \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n} + \left( t_{i+n-1}^2 - 2t_{i+n-1}\mu_{T_{i,n}} + \mu_{T_{i,n}}^2 \right) - \left( t_{i-1}^2 - 2t_{i-1}\mu_{T_{i,n}} + \mu_{T_{i,n}}^2 \right) \\ ={}& CSS(T_{i-1,n}) + \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n} + t_{i+n-1}^2 - 2t_{i+n-1}\mu_{T_{i,n}} + \mu_{T_{i,n}}^2 - t_{i-1}^2 + 2t_{i-1}\mu_{T_{i,n}} - \mu_{T_{i,n}}^2 \\ ={}& CSS(T_{i-1,n}) + \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n} + t_{i+n-1}^2 - 2t_{i+n-1}\mu_{T_{i,n}} - t_{i-1}^2 + 2t_{i-1}\mu_{T_{i,n}} \\ ={}& CSS(T_{i-1,n}) + \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n} + \left( t_{i+n-1}^2 - t_{i-1}^2 \right) - \left( 2t_{i+n-1}\mu_{T_{i,n}} - 2t_{i-1}\mu_{T_{i,n}} \right) \\ ={}& CSS(T_{i-1,n}) + \frac{\left(t_{i+n-1} - t_{i-1}\right)^2}{n} + \left( t_{i+n-1} - t_{i-1} \right) \left( t_{i+n-1} + t_{i-1} \right) - 2\mu_{T_{i,n}} \left( t_{i+n-1} - t_{i-1} \right) \\ ={}& CSS(T_{i-1,n}) + \left(t_{i+n-1} - t_{i-1}\right) \left[ \frac{t_{i+n-1} - t_{i-1}}{n} + \left( t_{i+n-1} + t_{i-1} \right) - 2\mu_{T_{i,n}} \right] \\ ={}& CSS(T_{i-1,n}) + \left(t_{i+n-1} - t_{i-1}\right) \left[ \frac{t_{i+n-1} - t_{i-1}}{n} + \left( t_{i+n-1} + t_{i-1} \right) - \mu_{T_{i,n}} - \mu_{T_{i,n}} \right] \\ ={}& CSS(T_{i-1,n}) + \left(t_{i+n-1} - t_{i-1}\right) \left[ \frac{t_{i+n-1} - t_{i-1}}{n} + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) + \left( t_{i-1} - \mu_{T_{i,n}} \right) \right] \\ ={}& CSS(T_{i-1,n}) + \left(t_{i+n-1} - t_{i-1}\right) \left[ \frac{t_{i+n-1} - t_{i-1}}{n} + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) + \left( t_{i-1} - \mu_{T_{i-1,n}} - \frac{t_{i+n-1} - t_{i-1}}{n} \right) \right] \\ ={}& CSS(T_{i-1,n}) + \left(t_{i+n-1} - t_{i-1}\right) \left[ \frac{t_{i+n-1} - t_{i-1}}{n} - \frac{t_{i+n-1} - t_{i-1}}{n} + \left( t_{i+n-1} - \mu_{T_{i,n}} \right) + \left( t_{i-1} - \mu_{T_{i-1,n}} \right) \right] \\ ={}& CSS(T_{i-1,n}) + \left(t_{i+n-1} - t_{i-1}\right) \left[ t_{i+n-1} - \mu_{T_{i,n}} + t_{i-1} - \mu_{T_{i-1,n}} \right] \\ S^2(T_{i,n}) ={}& \frac{\sum \limits _{0 \leq j \lt n} \left( t_{i+j} - \mu_{T_{i,n}} \right) ^2 }{n} \\ ={}& \frac{CSS(T_{i,n})}{n} \\ ={}& \frac{CSS(T_{i-1,n})}{n} + \frac{\left(t_{i+n-1} - t_{i-1}\right) \left[ t_{i+n-1} - \mu_{T_{i,n}} + t_{i-1} - \mu_{T_{i-1,n}} \right]}{n} \\ ={}& S^2(T_{i-1,n}) + \frac{\left(t_{i+n-1} - t_{i-1}\right) \left[ t_{i+n-1} - \mu_{T_{i,n}} + t_{i-1} - \mu_{T_{i-1,n}} \right]}{n} \\ \end{align}