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 $x_1,x_2,x_3,\dots ,x_N$, 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:

$$\begin{array}{rl}& {\mu }_0=0;\\ & \mathrm{for}k=1:1:N\\ & {\mu }_k=\left(\frac{k-1}{k}\right){\mu }_{k-1}+\frac{x_k}{k};\\ & \mathrm{endfor}\end{array}$$

This saves memory and is more elegent.

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

$$\begin{array}{rl}& {\mu }_0=0;\\ & S_0=0;\\ & \mathrm{for}k=1:1:N\\ & {\mu }_k=\left(\frac{k-1}{k}\right){\mu }_{k-1}+\frac{x_k}{k};\\ & S_k=S_{k-1}+((x_k-{\mu }_k)(x_k-{\mu }_{k-1}));\\ & {\sigma }_k=\sqrt{\frac{S_k}{k}};\\ & Va{r}_k={\sigma }_k^2=\frac{S_k}{k};\\ & \mathrm{endfor}\end{array}$$

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

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