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