CUSUM charts in Python (and why pandas.cumsum is not one)

The task: detect small sustained shifts, around one sigma, as early as possible. The CUSUM chart is the classic answer, and it is frequently confused with a plain cumulative sum. They are not the same thing.

The three-line version

import shewhart as sw

r = sw.cusum(df, value="torque")          # k=0.5, h=4 by default
r = sw.cusum(df, value="torque", center=10.0, sigma=1.0)   # known parameters

cusum is not cumsum

pandas.Series.cumsum() adds up raw values. A CUSUM chart accumulates deviations beyond a reference value and resets at zero:

C+_i = max(0, x_i - (center + K) + C+_{i-1})
C-_i = max(0, (center - K) - x_i + C-_{i-1})

The reference value K = k * sigma (k = 0.5 by convention) makes the sums ignore noise smaller than half a sigma; the reset at zero means in-control periods do not build up; and the decision interval H = h * sigma turns the plot into a test. A raw cumulative sum has none of these, drifts with any nonzero mean, and cannot serve as a control chart.

Worked example

For values [10, 10, 10, 11, 12, 13] with center 10, sigma 1, k = 0.5: the upper sums are [0, 0, 0, 0.5, 2.0, 4.5], and with h = 4 the last point crosses the decision interval, so the chart signals at observation 5. This example runs verbatim in the test suite.

CUSUM or EWMA?

Both detect small shifts well; both are autocorrelated by construction, which is why shewhart offers no run rules on either (zone tests would be statistically invalid). Practical guidance: CUSUM is slightly sharper for a known shift size (tune k to half the shift you care about); EWMA is easier to read as a smoothed level. Use sw.imr() alongside either to keep sensitivity to large isolated spikes.