earsC  R Documentation 
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))
sts 
object of class sts (including the 
control 
Control object

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,
tbaseline
to t1. The expected value is the mean of the baseline. An approximate
(twosided) (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=t1}^{t\code{baseline}} Y(i)
and
S^2_1(t)= \frac{1}{6} \sum_{i=t1}^{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 2day lag in the baseline definition.
In other words the baseline for C2 is baseline
(Default: 7) timepoints with a 2day lag before the monitored
timepoint t, i.e. (t\code{baseline}2)
to t3
. The expected value is the mean of the baseline. An approximate
(twosided) (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=t3}^{t\code{baseline}2} Y(i)
and
S^2_2(t)= \frac{1}{\code{baseline}1} \sum_{i=t3}^{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 t3
.
The statistic C_3(t)
is the sum of discrepancies between observations and
predictions.
C_3(t)= \sum_{i=t}^{t2} \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=t1}^{t2} \max(0,C_2(i)1)\right).
An object of class sts
with the slots upperbound
and alarm
filled
by the chosen method.
M. Salmon, H. Burkom
Fricker, R.D., Hegler, B.L, and Dunfee, D.A. (2008). Comparing syndromic surveillance detection methods: EARS versus a CUSUMbased methodology, 27:34073429, Statistics in medicine.
Salmon, M., Schumacher, D. and HÃ¶hle, M. (2016): Monitoring count time series in R: Aberration detection in public health surveillance. Journal of Statistical Software, 70 (10), 135. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v070.i10")}
#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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