# CUSUM method

`algo.cusum.Rd`

Approximate one-side CUSUM method for a Poisson variate based on the cumulative sum of the deviation between a reference value k and the transformed observed values. An alarm is raised if the cumulative sum equals or exceeds a prespecified decision boundary h. The function can handle time varying expectations.

## Usage

```
algo.cusum(disProgObj, control = list(range = range, k = 1.04, h = 2.26,
m = NULL, trans = "standard", alpha = NULL))
```

## Arguments

- disProgObj
object of class disProg (including the observed and the state chain)

- control
control object:

`range`

determines the desired time points which should be evaluated

`k`

is the reference value

`h`

the decision boundary

`m`

how to determine the expected number of cases – the following arguments are possible

`numeric`

a vector of values having the same length as

`range`

. If a single numeric value is specified then this value is replicated`length(range)`

times.`NULL`

A single value is estimated by taking the mean of all observations previous to the first

`range`

value.`"glm"`

A GLM of the form $$\log(m_t) = \alpha + \beta t + \sum_{s=1}^S (\gamma_s \sin(\omega_s t) + \delta_s \cos(\omega_s t)),$$ where \(\omega_s = \frac{2\pi}{52}s\) are the Fourier frequencies is fitted. Then this model is used to predict the

`range`

values.

`trans`

one of the following transformations (warning: Anscombe and NegBin transformations are experimental)

`rossi`

standardized variables z3 as proposed by Rossi

`standard`

standardized variables z1 (based on asymptotic normality) - This is the default.

`anscombe`

anscombe residuals – experimental

`anscombe2nd`

anscombe residuals as in Pierce and Schafer (1986) based on 2nd order approximation of E(X) – experimental

`pearsonNegBin`

compute Pearson residuals for NegBin – experimental

`anscombeNegBin`

anscombe residuals for NegBin – experimental

`none`

no transformation

`alpha`

parameter of the negative binomial distribution, s.t. the variance is \(m+\alpha *m^2\)

## Value

`algo.cusum`

gives a list of class `"survRes"`

which includes the
vector of alarm values for every timepoint in `range`

and the vector
of cumulative sums for every timepoint in `range`

for the system
specified by `k`

and `h`

, the range and the input object of
class `"disProg"`

.

The `upperbound`

entry shows for each time instance the number of diseased individuals
it would have taken the cusum to signal. Once the CUSUM signals no resetting is applied, i.e.
signals occurs until the CUSUM statistic again returns below the threshold.

In case `control$m="glm"`

was used, the returned
`control$m.glm`

entry contains the fitted `"glm"`

object.

## References

G. Rossi, L. Lampugnani and M. Marchi (1999), An approximate CUSUM procedure for surveillance of health events, Statistics in Medicine, 18, 2111–2122

D. A. Pierce and D. W. Schafer (1986), Residuals in Generalized Linear Models, Journal of the American Statistical Association, 81, 977–986

## Examples

```
# Xi ~ Po(5), i=1,...,500
set.seed(321)
stsObj <- sts(observed = rpois(500,lambda=5))
# there should be no alarms as mean doesn't change
res <- cusum(stsObj, control = list(range = 100:500, trans = "anscombe"))
plot(res, xaxis.labelFormat = NULL)
# simulated data
disProgObj <- sim.pointSource(p = 1, r = 1, length = 250,
A = 0, alpha = log(5), beta = 0, phi = 10,
frequency = 10, state = NULL, K = 0)
plot(disProgObj)
# Test weeks 200 to 250 for outbreaks
surv0 <- algo.cusum(disProgObj, control = list(range = 200:250))
plot(surv0, xaxis.years = FALSE)
# alternatively, using the newer "sts" interface
stsObj <- disProg2sts(disProgObj)
surv <- cusum(stsObj, control = list(range = 200:250))
plot(surv)
stopifnot(upperbound(surv) == surv0$upperbound)
```