The function takes range values of the surveillance time
series sts and for each time point computes a threshold for the number of counts
based on values from the recent past.
This is then compared to the observed
number of counts. If the observation is above a specific quantile of
the prediction interval, then an alarm is raised. This method is especially useful
for data without many historic values, since it only needs counts from the recent past.
earsC(sts, control = list(range = NULL, method = "C1",
baseline = 7, minSigma = 0,
alpha = 0.001))An object of class sts with the slots upperbound and alarm filled
by the chosen method.
object of class sts (including the observed and the state time series) , which is to be monitored.
Control object
rangeSpecifies the index in the sts object of
all the timepoints which
should be monitored. If range is NULL the maximum number
of possible timepoints is used (this number depends on the method chosen):
all timepoints from the observation with index baseline + 1
can be monitored,
timepoints from index baseline + 3 can be
monitored,
timepoints starting from the index baseline + 5 can be monitored.
methodString indicating which method to use:
"C1"for EARS C1-MILD method (Default),
"C2"for EARS C2-MEDIUM method,
"C3"for EARS C3-HIGH method.
See Details for further information about the methods.
baselinehow many time points to use for calculating the baseline, see details
minSigmaBy default 0. If minSigma is higher than 0, for C1 and
C2, the quantity zAlpha * minSigma is then the alerting threshold if the
baseline is zero. Howard Burkom suggests using a value of 0.5 or 1 for sparse data.
alphaAn approximate (two-sided) \((1-\alpha)\cdot 100\%\) prediction
interval is calculated. By default if alpha is NULL
the value 0.001 is assumed
for C1 and C2 whereas 0.025 is assumed for C3. These different choices are the one made at the CDC.
%
M. Salmon, H. Burkom
The three methods are different in terms of baseline used for calculation of the expected value and in terms of method for calculating the expected value:
in C1 and C2 the expected value is the moving average of counts over the sliding window of the baseline and the prediction interval depends on the standard derivation of the observed counts in this window. They can be considered as Shewhart control charts with a small sample used for calculations.
in C3 the expected value is based on the sum over 3 timepoints (assessed timepoints and the two previous timepoints) of the discrepancy between observations and predictions, predictions being calculated with the C2 method. This method has similarities with a CUSUM method due to it adding discrepancies between predictions and observations over several timepoints, but is not a CUSUM (sum over 3 timepoints, not accumulation over a whole range), even if it sometimes is presented as such.
Here is what the function does for each method, see the literature sources for further details:
For C1 the baseline are the baseline (default 7) timepoints before the assessed timepoint t,
t-baseline to t-1. The expected value is the mean of the baseline. An approximate
(two-sided) \((1-\alpha)\cdot 100\%\) prediction interval is calculated based on the
assumption that the difference between the expected value and the observed
value divided by the standard derivation of counts over the sliding window,
called \(C_1(t)\), follows a standard normal distribution in the absence
of outbreaks:
$$C_1(t)= \frac{Y(t)-\bar{Y}_1(t)}{S_1(t)},$$
where
$$\bar{Y}_1(t)= \frac{1}{\code{baseline}} \sum_{i=t-1}^{t-\code{baseline}} Y(i)$$
and
$$ S^2_1(t)= \frac{1}{6} \sum_{i=t-1}^{t-\code{baseline}} [Y(i) - \bar{Y}_1(i)]^2.$$
Then under the null hypothesis of no outbreak,
$$C_1(t) \mathcal \> \sim \> {N}(0,1)$$
An alarm is raised if $$C_1(t)\ge z_{1-\alpha}$$
with \(z_{1-\alpha}\) the \((1-\alpha)^{th}\) quantile of the standard normal distribution.
The upperbound \(U_1(t)\) is then defined by: $$U_1(t)= \bar{Y}_1(t) + z_{1-\alpha}S_1(t).$$
C2 is very similar to C1 apart from a 2-day lag in the baseline definition.
In other words the baseline for C2 is baseline (Default: 7) timepoints with a 2-day lag before the monitored
timepoint t, i.e. \((t-\code{baseline}-2)\) to \(t-3\). The expected value is the mean of the baseline. An approximate
(two-sided) \((1-\alpha)\cdot 100\%\) prediction interval is calculated based on the
assumption that the difference between the expected value and the observed
value divided by the standard derivation of counts over the sliding window,
called \(C_2(t)\), follows a standard normal distribution in the absence
of outbreaks:
$$C_2(t)= \frac{Y(t)-\bar{Y}_2(t)}{S_2(t)},$$
where
$$\bar{Y}_2(t)= \frac{1}{\code{baseline}} \sum_{i=t-3}^{t-\code{baseline}-2} Y(i)$$
and
$$ S^2_2(t)= \frac{1}{\code{baseline}-1} \sum_{i=t-3}^{t-\code{baseline}-2} [Y(i) - \bar{Y}_2(i)]^2.$$
Then under the null hypothesis of no outbreak,
$$C_2(t) \mathcal \sim {N}(0,1)$$
An alarm is raised if $$C_2(t)\ge z_{1-\alpha},$$
with \(z_{1-\alpha}\) the \((1-\alpha)^{th}\) quantile of the standard normal distribution.
The upperbound \(U_{2}(t)\) is then defined by:
$$U_{2}(t)= \bar{Y}_{2}(t) + z_{1-\alpha}S_{2}(t).$$
C3 is quite different from the two other methods, but it is based on C2.
Indeed it uses \(C_2(t)\) from timepoint t and the two previous timepoints.
This means the baseline consists of the timepoints \(t-(\code{baseline}+4)\) to \(t-3\).
The statistic \(C_3(t)\) is the sum of discrepancies between observations and
predictions.
$$C_3(t)= \sum_{i=t}^{t-2} \max(0,C_2(i)-1)$$
Then under the null hypothesis of no outbreak,
$$C_3(t) \mathcal \sim {N}(0,1)$$
An alarm is raised if $$C_3(t)\ge z_{1-\alpha},$$
with \(z_{1-\alpha}\) the \((1-\alpha)^{th}\) quantile of the standard normal distribution.
The upperbound \(U_3(t)\) is then defined by: $$U_3(t)= \bar{Y}_2(t) + S_2(t)\left(z_{1-\alpha}-\sum_{i=t-1}^{t-2} \max(0,C_2(i)-1)\right).$$
#Sim data and convert to sts object
disProgObj <- sim.pointSource(p = 0.99, r = 0.5, length = 208, A = 1,
alpha = 1, beta = 0, phi = 0,
frequency = 1, state = NULL, K = 1.7)
stsObj <- disProg2sts( disProgObj)
# Call earsC function and show result
res1 <- earsC(stsObj, control = list(range = 20:208, method="C1"))
plot(res1, legend.opts=list(horiz=TRUE, x="topright"))
# Compare C3 upperbounds depending on alpha
res3 <- earsC(stsObj, control = list(range = 20:208,method="C3",alpha = 0.001))
plot(upperbound(res3), type='l')
res3 <- earsC(stsObj, control = list(range = 20:208,method="C3"))
lines(upperbound(res3), col='red')
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