trend (version 1.1.6)

snh.test: Standard Normal Homogeinity Test (SNHT) for Change-Point Detection

Description

Performes the Standard Normal Homogeinity Test (SNHT) for change-point detection of a normal variate.

Usage

snh.test(x, m = 20000)

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 Tk for plotting

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 \(\Delta \ne 0\).

The test statistic for the SNHT test is calculated as follows:

$$T_k = k z_1^2 + \left(n - k\right) z_2^2 \qquad (1 \le k < n)$$

where

$$ \begin{array}{l l} z_1 = \frac{1}{k} \sum_{i=1}^k \frac{x_i - \bar{x}}{\sigma} & z_2 = \frac{1}{n-k} \sum_{i=k+1}^n \frac{x_i - \bar{x}}{\sigma}. \\ \end{array}$$

The critical value is: $$T = \max T_k.$$

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

Critical values based on \(m = 1,000,000\) Monte Carlo simulations are tabulated for \(T\) by Khaliq and Ouarda (2007).

References

H. Alexandersson (1986), A homogeneity test applied to precipitation data, Journal of Climatology 6, 661--675.

M. N. Khaliq, T. B. M. J. Ouarda (2007), On the critical values of the standard normal homogeneity test (SNHT), International Journal of Climatology 27, 681--687.

G. Verstraeten, J. Poesen, G. Demaree, C. Salles (2006), Long-term (105 years) variability in rain erosivity as derived from 10-min rainfall depth data for Ukkel (Brussels, Belgium): Implications for assessing soil erosion rates. Journal of Geophysical Research 111, D22109.

See Also

efp sctest.efp

Examples

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

data(PagesData) ; snh.test(PagesData)