Downsampling with CUSUM Filter

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

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

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