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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
. The cumulative sums can then be defined as
St=max(0,St1+ytEt1[yt])S_t = \max(0, S_{t-1} +y_t - E_{t-1}[y_t])
with boundary condition
S0=0.S_{0} = 0.
A sample is only created when
Sth,S_{t} \ge h,
for some threshold
This can be further extended to a symmetric CUSUM filter to include run-ups and run-downs such that
St+=max(0,St1++ytEt1[yt]),S0+=0St=min(0,St1+ytEt1[yt]),S0=0St=max(St+,St)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)