Computing A Signal Variance in Real Time

28 Jul 2017

The standard way to compute the variance of a data set is using the sqaure of the standard deviation. When we have scalar values $x_1, x_2, x_3, …, x_N$, the mean $\mu_N$ standard deviation $\sigma_N$ for all the N values is computed as follows:

\[\mu_N = \frac{\sum_{k=1}^N x_k}{N} ... (1)\] \[\sigma_N = \frac{\sqrt{\sum_{k=1}^N (x_k - \mu_N) }}{N} ... (1)\]

I already described in my previous blog post how to compute the mean in real time using the following algorithm at runtime:

\[\begin{align} & \mu_0 = 0; \nonumber \\\ & \mathrm{for} \hspace{2mm} k = 1:1:N \nonumber \\\ & \hspace{4mm} \mu_k = \left( \frac{k-1}{k} \right) \mu_{k-1} + \frac{x_k}{k}; \nonumber \\\ & \mathrm{endfor} \nonumber \end{align}\]

This saves memory and is more elegent.

Similarly, we can compute the variance of a data set using another recursive algorithm:

\[\begin{align} & \mu_0 = 0; \nonumber \\\ & S_0 = 0; \nonumber \\\ & \mathrm{for} \hspace{2mm} k = 1:1:N \nonumber \\\ & \hspace{4mm} \mu_k = \left( \frac{k-1}{k} \right) \mu_{k-1} + \frac{x_k}{k}; \nonumber \\\ & \hspace{4mm} S_k = S_{k-1} + \left( (x_k - \mu_k)(x_k - \mu_{k-1}) \right); \nonumber \\\ & \hspace{4mm} \sigma_k = \sqrt\frac{S_k}{k} ; \nonumber \\\ & \hspace{4mm} Var_k = \sigma _k^2 = \frac{S_k}{k}; \nonumber \\\ & \mathrm{endfor} \nonumber \end{align}\]

Note that, to compute the variance, we also require to compute the mean.

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