EWMA: Exponentially Weighted Moving Average

In WarnEpi: A Comprehensive Tool for Early Warning in Infectious Disease

Exponentially Weighted Moving Average

Description

Detects anomalies in infectious disease surveillance data using an Exponentially Weighted Moving Average (EWMA) algorithm. Designed for time series data, it flags potential outbreaks by smoothing past observations with decayed weights and comparing against control thresholds.

Usage

EWMA(data, column, lambda = 0.5, k = 3, move_t, ignore_t = 2)

Arguments

data	A data frame containing the warning indicator columns, arranged in time-based order.
column	A column name or column number, used to specify the warning indicator.
lambda	The weight factor \lambda, ranging from 0 to 1(higher values prioritize recent observations).
k	The standard deviation coefficient k.
move_t	The moving period t_{move}.
ignore_t	The number of nearest time units to be ignored by the model, t_{ignore}.

Details

Let \mathbf{X} = (X_1,\ldots,X_T)^\top be an observed time series of disease case counts, where X_t represents the aggregated counts at time t (e.g., daily, weekly, or monthly observations). We assume X_t \sim N(\mu, \sigma^2) for the underlying distribution.

The EWMA (Exponentially Weighted Moving Average) model is defined as:

Z_1 = X_1

Z_t = \lambda X_t + (1-\lambda)Z_{t-1}

UCL_t = \hat{\mu}_t + k\hat{\sigma}_t\sqrt{\frac{\lambda}{2-\lambda}}

where:

• Z_t: The EWMA statistic at time t, representing an exponentially weighted average of current and past observations.

• \lambda: Weight factor (0 < \lambda < 1), higher values prioritize recent observations

• k: Standard deviation coefficient (typically 2-3)

• UCL_t: Upper Control Limit at time t, forming a dynamic threshold for anomaly detection.

• \hat{\mu}_t, \hat{\sigma}_t: Estimated from moving window (X_{t-t_{move}-t_{ignore}},\ldots,X_{t-1-t_{ignore}})

An alarm is triggered when Z_t > UCL_t, with the alarm set defined as:

\mathcal{T} = \{t: Z_t > UCL_t\}

Value

A data frame containing warning results. The value of the warning column is 1 for warning and 0 for no warning.

References

Wang X, Zeng D, Seale H, et al. Comparing early outbreak detection algorithms based on their optimized parameter values. J Biomed Inform, 2010,43(1):97-103.

Examples

## simulate reported cases
set.seed(123)
cases <- c(round(rnorm(10, 10, 1)), seq(12,21,3), seq(15,5,-5))
dates <- seq(as.Date("2025-01-01"), by = "7 days", length.out = length(cases))
data_frame <- data.frame(date = dates, case = cases)

## modeling
output <- EWMA(data_frame,'case',lambda = 0.5, k = 3, move_t = 4, ignore_t = 2)
output

## visualize alerts
plot(output$date, output$case, type = "l")
points(output$date[output$warning == 1],
       output$case[output$warning == 1], col = "red")

WarnEpi documentation built on Aug. 27, 2025, 9:10 a.m.