getBatchThreshold {cpm}

Returns the Threshold Associated with a Type I Error Probability .

Description

When performing Phase I analysis within the CPM framework for a sequence of length n, the null hypothesis of no change is rejected if D_n > h_n for some threshold h_n. Typically this threshold is chosen to be the upper alpha quantile of the distribution of D_n under the null hypothesis of no change. Given a particular choice of alpha and n, this function returns the associated h_n threshold. Because these thresholds are laborious to compute, the package contains pre-computed values of h_n for alpha = 0.05, 0.01, 0.005 and 0.001, and for n < 10000.

For a fuller overview of this function including a description of the CPM framework and examples of how to use the various functions, please consult the package manual "Parametric and Nonparametric Sequential Change Detection in R: The cpm Package" available from www.gordonjross.co.uk

Usage

	getBatchThreshold(cpmType, alpha, n, lambda=0.3)
    

Arguments

cpmType

The type of CPM which is used. Possible arguments are:

• Student: Student-t test statistic, as in [Hawkins et al, 2003]. Use to detect mean changes in a Gaussian sequence.

• Bartlett: Bartlett test statistic, as in [Hawkins and Zamba, 2005]. Use to detect variance changes in a Gaussian sequence.

• GLR: Generalized Likelihood Ratio test statistic, as in [Hawkins and Zamba, 2005b]. Use to detect both mean and variance changes in a Gaussian sequence.

• Exponential: Generalized Likelihood Ratio test statistic for the Exponential distribution, as in [Ross, 2013]. Used to detect changes in the parameter of an Exponentially distributed sequence.

• GLRAdjusted and ExponentialAdjusted: Identical to the GLR and Exponential statistics, except with the finite-sample correction discussed in [Ross, 2013] which can lead to more powerful change detection.

• FET: Fishers Exact Test statistic, as in [Ross and Adams, 2012b]. Use to detect parameter changes in a Bernoulli sequence.

• Mann-Whitney: Mann-Whitney test statistic, as in [Ross et al, 2011]. Use to detect location shifts in a stream with a (possibly unknown) non-Gaussian distribution.

• Mood: Mood test statistic, as in [Ross et al, 2011]. Use to detect scale shifts in a stream with a (possibly unknown) non-Gaussian distribution.

• Lepage: Lepage test statistics in [Ross et al, 2011]. Use to detect location and/or shifts in a stream with a (possibly unknown) non-Gaussian distribution.

• Kolmogorov-Smirnov: Kolmogorov-Smirnov test statistic, as in [Ross et al 2012]. Use to detect arbitrary changes in a stream with a (possibly unknown) non-Gaussian distribution.

• Cramer-von-Mises: Cramer-von-Mises test statistic, as in [Ross et al 2012]. Use to detect arbitrary changes in a stream with a (possibly unknown) non-Gaussian distribution.

alpha

the null hypothesis of no change is rejected if D_n > h_n where n is the length of the sequence and h_n is the upper alpha percentile of the test statistic distribution.

n

the sequence length the value should be calculated for, i.e. the value of n in D_n.

lambda

A smoothing parameter which is used to reduce the discreteness of the test statistic when using the FET CPM. See [Ross and Adams, 2012b] in the References section for more details on how this parameter is used. Currently the package only contains sequences of ARL0 thresholds corresponding to lambda=0.1 and lambda=0.3, so using other values will result in an error. If no value is specified, the default value will be 0.1.

Author(s)

Gordon J. Ross [e-mail]

References

Hawkins, D. , Zamba, K. (2005) – A Change-Point Model for a Shift in Variance, Journal of Quality Technology, 37, 21-31

Hawkins, D. , Zamba, K. (2005b) – Statistical Process Control for Shifts in Mean or Variance Using a Changepoint Formulation, Technometrics, 47(2), 164-173

Hawkins, D., Qiu, P., Kang, C. (2003) – The Changepoint Model for Statistical Process Control, Journal of Quality Technology, 35, 355-366.

Ross, G. J., Tasoulis, D. K., Adams, N. M. (2011) – A Nonparametric Change-Point Model for Streaming Data, Technometrics, 53(4)

Ross, G. J., Adams, N. M. (2012) – Two Nonparametric Control Charts for Detecting Arbitary Distribution Changes, Journal of Quality Technology, 44:102-116

Ross, G. J., Adams, N. M. (2013) – Sequential Monitoring of a Proportion, Computational Statistics, 28(2)

Ross, G. J., (2014) – Sequential Change Detection in the Presence of Unknown Parameters, Statistics and Computing 24:1017-1030

Ross, G. J., (2015) – Parametric and Nonparametric Sequential Change Detection in R: The cpm Package, Journal of Statistical Software, forthcoming

See Also

detectChangePointBatch.

Examples

     ## Returns the threshold for n=1000, alpha=0.05 and the Mann-Whitney CPM
     h <- getBatchThreshold("Mann-Whitney", 0.05, 1000)