Page-Hinkley sequential test for a shift in the mean

Formula: with \(\bar x_T\) the running mean of the first \(T\) observations, \(m_T = \sum_{t\le T}(x_t - \bar x_t - \delta)\), \(M_T = \min_{t \le T} m_t\), \(PH_T = m_T - M_T\), and a change is flagged at the first \(T\) with \(PH_T > \lambda\). The mirrored form detects a decrease. \(\delta\) is the magnitude below which a deviation is treated as noise; threshold trades detection delay against false alarms.

Usage

Pagehink(x, threshold, delta = 0.005, direction = "increase")

Arguments

x

Stream in arrival order.

threshold

Alarm level lambda.

delta

Tolerated magnitude of drift.

direction

"increase" or "decrease".

Value

List with statistic, detected, changepoint (1-based, 0 if none), ph, n, method.

References

Page (1954), Biometrika 41:100-115; Hinkley (1971), Biometrika 58:509-523. Both paywalled at JSTOR (HTTP 403); the running-mean form used here is the one standardised in Gama, Zliobaite, Bifet, Pechenizkiy and Bouchachia (2014), A survey on concept drift adaptation, ACM Computing Surveys 46(4):44, sec. 4.2.

Examples

Pagehink(x = c(1, 2, 3, 4, 5, 6, 7, 8), threshold = 0.5)
#> $statistic
#> [1] 13.965
#> 
#> $detected
#> [1] TRUE
#> 
#> $changepoint
#> [1] 3
#> 
#> $ph
#> [1]  0.000  0.495  1.490  2.985  4.980  7.475 10.470 13.965
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Page-Hinkley change detector"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"