# Downsampling with CUSUM Filter

Filtering out the noise and keeping only the informative parts of your data.

Last updated

Filtering out the noise and keeping only the informative parts of your data.

Last updated

Typically financial time series suffer from a low signal-to-noise ratio. When the entire financial dataset is used the model will focus too much on noisy samples and not enough on highly informative samples. A way to improve the signal-to-noise ratio is to downsample the dataset, but randomly downsampling is not effective as the ratio of noisy to informative sample will persist. Instead one could apply a CUSUM filter which only creates a sample when the next values deviate sufficiently from the previous value.

Consider a locally stationary process generating IID observations $\{y_t\}_{t=1,..,T}$. The cumulative sums can then be defined as

$S_t = \max(0, S_{t-1} +y_t - E_{t-1}[y_t])$

with boundary condition $S_{0} = 0.$ A sample is only created when $S_{t} \ge h,$ for some threshold $h.$This can be further extended to a symmetric CUSUM filter to include run-ups and run-downs such that

$S^+_t = \max(0, S^+_{t-1} + y_t - E_{t-1}[y_t]), S^+_0 = 0 \\
S^-_t = \min(0, S^-_{t-1} + y_t - E_{t-1}[y_t]), S^-_0 = 0 \\
S_t = \max(S^+_t, -S^-_t)$