Count data regression charts for the monitoring of surveillance time series as proposed by Höhle and Paul (2008). The implementation is described in Salmon et al. (2016).

algo.glrnb(disProgObj, control = list(range=range, c.ARL=5,
mu0=NULL, alpha=0, Mtilde=1, M=-1, change="intercept",
theta=NULL, dir=c("inc","dec"),
ret=c("cases","value"), xMax=1e4))

algo.glrpois(disProgObj, control = list(range=range, c.ARL=5,
mu0=NULL, Mtilde=1, M=-1, change="intercept",
theta=NULL, dir=c("inc","dec"),
ret=c("cases","value"), xMax=1e4))

## Arguments

disProgObj

object of class disProg to do surveillance for

control

A list controlling the behaviour of the algorithm

range

vector of indices in the observed vector to monitor (should be consecutive)

mu0

A vector of in-control values of the mean of the Poisson / negative binomial distribution with the same length as range. If NULL the observed values in 1:(min(range)-1) are used to estimate the beta vector through a generalized linear model. To fine-tune the model one can instead specify mu0 as a list with two components:

S

integer number of harmonics to include (typically 1 or 2)

trend

A Boolean indicating whether to include a term t in the GLM model

The fitting is controlled by the estimateGLRNbHook function. The in-control mean model is re-fitted after every alarm. The fitted models can be found as a list mod in the control slot after the call. Note: If a value for alpha is given, then the inverse of this value is used as fixed theta in a negative.binomial glm. If is.null(alpha) then the parameter is estimated as well (using glm.nb) -- see the description of this parameter for details.
alpha

The (known) dispersion parameter of the negative binomial distribution, i.e. the parametrization of the negative binomial is such that the variance is $$mean + alpha*mean^2$$. Note: This parametrization is the inverse of the shape parametrization used in R -- for example in dnbinom and glr.nb. Hence, if alpha=0 then the negative binomial distribution boils down to the Poisson distribution and a call of algo.glrnb is equivalent to a call to algo.glrpois. If alpha=NULL the parameter is calculated as part of the in-control estimation. However, the parameter is estimated only once from the first fit. Subsequent fittings are only for the parameters of the linear predictor with alpha fixed.

c.ARL

threshold in the GLR test, i.e. $$c_{\gamma}$$

Mtilde

number of observations needed before we have a full rank the typical setup for the "intercept" and "epi" charts is Mtilde=1

M

number of time instances back in time in the window-limited approach, i.e. the last value considered is $$\max{1,n-M}$$. To always look back until the first observation use M=-1.

change

a string specifying the type of the alternative. Currently the two choices are intercept and epi. See the SFB Discussion Paper 500 for details.

theta

if NULL then the GLR scheme is used. If not NULL the prespecified value for $$\kappa$$ or $$\lambda$$ is used in a recursive LR scheme, which is faster.

dir

a string specifying the direction of testing in GLR scheme. With "inc" only increases in $$x$$ are considered in the GLR-statistic, with "dec" decreases are regarded.

ret

a string specifying the type of upperbound-statistic that is returned. With "cases" the number of cases that would have been necessary to produce an alarm or with "value" the GLR-statistic is computed (see below).

xMax

Maximum value to try for x to see if this is the upperbound number of cases before sounding an alarm (Default: 1e4). This only applies for the GLR using the NegBin when ret="cases" -- see details.

## Value

algo.glrpois simply calls algo.glrnb with control\$alpha set to 0.

algo.glrnb returns a list of class survRes (surveillance result), which includes the alarm value for recognizing an outbreak (1 for alarm, 0 for no alarm), the threshold value for recognizing the alarm and the input object of class disProg. The upperbound slot of the object are filled with the current $$GLR(n)$$ value or with the number of cases that are necessary to produce an alarm at any time point $$<=n$$. Both lead to the same alarm timepoints, but "cases" has an obvious interpretation.

## Details

This function implements the seasonal count data chart based on generalized likelihood ratio (GLR) as described in the Höhle and Paul (2008) paper. A moving-window generalized likelihood ratio detector is used, i.e. the detector has the form $$N = \inf\left\{ n : \max_{1\leq k \leq n} \left[ \sum_{t=k}^n \log \left\{ \frac{f_{\theta_1}(x_t|z_t)}{f_{\theta_0}(x_t|z_t)} \right\} \right] \geq c_\gamma \right\}$$ where instead of $$1\leq k \leq n$$ the GLR statistic is computed for all $$k \in \{n-M, \ldots, n-\tilde{M}+1\}$$. To achieve the typical behaviour from $$1\leq k\leq n$$ use Mtilde=1 and M=-1.

So $$N$$ is the time point where the GLR statistic is above the threshold the first time: An alarm is given and the surveillance is reset starting from time $$N+1$$. Note that the same c.ARL as before is used, but if mu0 is different at $$N+1,N+2,\ldots$$ compared to time $$1,2,\ldots$$ the run length properties differ. Because c.ARL to obtain a specific ARL can only be obtained my Monte Carlo simulation there is no good way to update c.ARL automatically at the moment. Also, FIR GLR-detectors might be worth considering.

In case is.null(theta) and alpha>0 as well as ret="cases" then a brute-force search is conducted for each time point in range in order to determine the number of cases necessary before an alarm is sounded. In case no alarm was sounded so far by time $$t$$, the function increases $$x[t]$$ until an alarm is sounded any time before time point $$t$$. If no alarm is sounded by xMax, a return value of 1e99 is given. Similarly, if an alarm was sounded by time $$t$$ the function counts down instead. Note: This is slow experimental code!

At the moment, window limited intercept'' charts have not been extensively tested and are at the moment not supported. As speed is not an issue here this doesn't bother too much. Therefore, a value of M=-1 is always used in the intercept charts.

## Author

M. Höhle with contributions by V. Wimmer

## References

Höhle, M. and Paul, M. (2008): Count data regression charts for the monitoring of surveillance time series. Computational Statistics and Data Analysis, 52 (9), 4357-4368.

Salmon, M., Schumacher, D. and Höhle, M. (2016): Monitoring count time series in R: Aberration detection in public health surveillance. Journal of Statistical Software, 70 (10), 1-35. doi: 10.18637/jss.v070.i10

## Examples

##Simulate data and apply the algorithm
S <- 1 ; t <- 1:120 ; m <- length(t)
beta <- c(1.5,0.6,0.6)
omega <- 2*pi/52
#log mu_{0,t}
base <- beta[1] + beta[2] * cos(omega*t) + beta[3] * sin(omega*t)
#Generate example data with changepoint and tau=tau
tau <- 100
kappa <- 0.4
mu0 <- exp(base)
mu1 <- exp(base  + kappa)

## Poisson example
#Generate data
set.seed(42)
x <- rpois(length(t),mu0*(exp(kappa)^(t>=tau)))
s.ts <- create.disProg(week=1:length(t),observed=x,state=(t>=tau))
#Plot the data
plot(s.ts,legend=NULL,xaxis.years=FALSE)
#Run
cntrl = list(range=t,c.ARL=5, Mtilde=1, mu0=mu0,
change="intercept",ret="value",dir="inc")
glr.ts <- algo.glrpois(s.ts,control=cntrl)
plot(glr.ts,xaxis.years=FALSE)
lr.ts  <- algo.glrpois(s.ts,control=c(cntrl,theta=0.4))
plot(lr.ts,xaxis.years=FALSE)

## NegBin example
#Generate data
set.seed(42)
alpha <- 0.2
x <- rnbinom(length(t),mu=mu0*(exp(kappa)^(t>=tau)),size=1/alpha)
s.ts <- create.disProg(week=1:length(t),observed=x,state=(t>=tau))

#Plot the data
plot(s.ts,legend=NULL,xaxis.years=FALSE)

#Run GLR based detection
cntrl = list(range=t,c.ARL=5, Mtilde=1, mu0=mu0, alpha=alpha,
change="intercept",ret="value",dir="inc")
glr.ts <- algo.glrnb(s.ts,control=c(cntrl))
plot(glr.ts,xaxis.years=FALSE)

#CUSUM LR detection with backcalculated number of cases
cntrl2 = list(range=t,c.ARL=5, Mtilde=1, mu0=mu0, alpha=alpha,
change="intercept",ret="cases",dir="inc",theta=1.2)
glr.ts2 <- algo.glrnb(s.ts,control=c(cntrl2))
plot(glr.ts2,xaxis.years=FALSE)