LRCUSUM.runlength {surveillance} | R Documentation |
Compute run length for a count data or categorical CUSUM. The computations are based on a Markov representation of the likelihood ratio based CUSUM.
LRCUSUM.runlength(mu,mu0,mu1,h,dfun, n, g=5,outcomeFun=NULL,...)
mu |
k-1 times T matrix with true proportions, i.e. equal to mu0 or mu1 if one wants to compute e.g. ARL_0 or ARL_1. |
mu0 |
k-1 times T matrix with in-control proportions |
mu1 |
k-1 times T matrix with out-of-control proportion |
h |
The threshold h which is used for the CUSUM. |
dfun |
The probability mass function or density used to compute
the likelihood ratios of the CUSUM. In a negative binomial CUSUM
this is dnbinom , in a binomial CUSUM dbinom and in a
multinomial CUSUM dmultinom . |
n |
Vector of length T containing the total number of experiments for each time point. |
g |
The number of levels to cut the state space into when performing the Markoc chain approximation. Sometimes also denoted M. |
outcomeFun |
A hook function to compute all possible outcome
states to compute the likelihood ratio for. If NULL then the
default function outcomeFunStandard(k,n) is used. This
function uses the Cartesian product of 0:n for k components. |
... |
Additional arguments to send to dfun . |
Brook and Evans (1972) formulated an approximate approach based on Markov chains to determine the PMF of the run length of a time-constant CUSUM detector. They describe the dynamics of the CUSUM statistic by a Markov chain with a discretized state space of size g+2. This is adopted to the time varying case in Höhle (2010) and implemented in R using the ... notation such that it works for a very large class of distributions.
A list with five components
P |
An array of g+2 times g+2 transition matrices of the approximation Markov chain. |
pmf |
Probability mass function (up to length T) of the run length variable. |
cdf |
Cumulative density function (up to length T) of the run length variable. |
arl |
If the model is time homogenous (i.e. if T==1) then the ARL is computed based on the stationary distribution of the Markov chain. See the eqns in the reference for details. |
M. Höhle
Höhle, M. (2010), Changepoint detection in categorical time series, Book chapter to appear in T. Kneib and G. Tutz (Eds.), Statistical Modelling and Regression Structures, Springer.
Höhle, M. and Mazick, A. (2009), Aberration detection in R illustrated by Danish mortality monitoring, Book chapter to appear in T. Kass-Hout and X. Zhang (Eds.) Biosurveillance: A Health Protection Priority, CRCPress.
Brook, D. and Evans, D. A. (1972), An approach to the probability distribution of Cusum run length, Biometrika, 59:3, pp. 539–549.
###################################################### #Run length of a time constant negative binomial CUSUM ###################################################### #In-control and out of control parameters mu0 <- 10 alpha <- 1/2 kappa <- 2 #Density for comparison in the negative binomial distribution dY <- function(y,mu,log=FALSE, alpha, ...) { dnbinom(y, mu=mu, size=1/alpha, log=log) } #In this case "n" is the maximum value to investigate the LLR for #It is assumed that beyond n the LLR is too unlikely to be worth #computing. LRCUSUM.runlength( mu=t(mu0), mu0=t(mu0), mu1=kappa*t(mu0), h=5, dfun = dY, n=rep(100,length(mu0)), alpha=alpha) h.grid <- seq(3,6,by=0.1) arls <- sapply(h.grid, function(h) { LRCUSUM.runlength( mu=t(mu0), mu0=t(mu0), mu1=kappa*t(mu0), h=h, dfun = dY, n=rep(100,length(mu0)), alpha=alpha,g=20)$arl }) plot(h.grid, arls,type="l",xlab="threshold h",ylab=expression(ARL[0])) ###################################################### #Run length of a time varying negative binomial CUSUM ###################################################### mu0 <- matrix(5*sin(2*pi/52 * 1:200) + 10,ncol=1) rl <- LRCUSUM.runlength( mu=t(mu0), mu0=t(mu0), mu1=kappa*t(mu0), h=2, dfun = dY, n=rep(100,length(mu0)), alpha=alpha,g=20) plot(1:length(mu0),rl$pmf,type="l",xlab="t",ylab="PMF") plot(1:length(mu0),rl$cdf,type="l",xlab="t",ylab="CDF") ######################################################## # Further examples contain the binomial, beta-binomial # and multinomial CUSUMs. Hopefully, these will be added # in the future. ########################################################