bu.test: Buishand U Test for Change-Point Detection

In trend: Non-Parametric Trend Tests and Change-Point Detection

Buishand U Test for Change-Point Detection

Description

Performes the Buishand U test for change-point detection of a normal variate.

Usage

bu.test(x, m = 20000)

Arguments

x	a vector of class "numeric" or a time series object of class "ts"
m	numeric, number of Monte-Carlo replicates, defaults to 20000

Details

Let X denote a normal random variate, then the following model with a single shift (change-point) can be proposed:

x_i = \left\{ \begin{array}{lcl} \mu + \epsilon_i, & \qquad & i = 1, \ldots, m \\ \mu + \Delta + \epsilon_i & \qquad & i = m + 1, \ldots, n \\ \end{array} \right.

with \epsilon \approx N(0,\sigma). The null hypothesis \Delta = 0 is tested against the alternative \Delta \ne 0.

In the Buishand U test, the rescaled adjusted partial sums are calculated as

S_k = \sum_{i=1}^k \left(x_i - \bar{x}\right) \qquad (1 \le i \le n)

The sample standard deviation is

D_x = \sqrt{n^{-1} \sum_{i=1}^n \left(x_i - \bar{x}\right)}

The test statistic is calculated as:

U = \left[n \left(n + 1 \right) \right]^{-1} \sum_{k=1}^{n-1} \left(S_k / D_x \right)^2

.

The p.value is estimated with a Monte Carlo simulation using m replicates.

Critical values based on m = 19999 Monte Carlo simulations are tabulated for U by Buishand (1982, 1984).

Value

A list with class "htest" and "cptest"

data.name	character string that denotes the input data
p.value	the p-value
statistic	the test statistic
null.value	the null hypothesis
estimates	the time of the probable change point
alternative	the alternative hypothesis
method	character string that denotes the test
data	numeric vector of Sk for plotting

Note

The current function is for complete observations only.

References

T. A. Buishand (1982), Some Methods for Testing the Homogeneity of Rainfall Records, Journal of Hydrology 58, 11–27.

T. A. Buishand (1984), Tests for Detecting a Shift in the Mean of Hydrological Time Series, Journal of Hydrology 73, 51–69.

See Also

efp sctest.efp

Examples

data(Nile)
(out <- bu.test(Nile))
plot(out)

data(PagesData)
bu.test(PagesData)
 

trend documentation built on Oct. 10, 2023, 9:06 a.m.