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