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

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")

Adjusted Serfling

Description

Adjusted Serfling regression for periodic disease surveillance, automating epidemic baseline estimation through iterative threshold optimization. Enhances traditional Serfling models by objectively determining epidemic periods and improving peak detection accuracy.

Usage

aSerfling(data, col_name, cycles)

Arguments

Details

Implements an iterative periodic regression for time series with at least 2 full cycles. Key features:

1. Dynamic Epidemic Filtering:

   • Automatically excludes outbreak points via iterative prediction-CI comparison

   • Terminates when adjusted R-squared stabilizes (maximized model fit)

2. Flexible Seasonality Modeling:

   Y = \beta_0 + \beta_1 t + \beta_2 t^2 + \sum_{k=1}^K \left[\gamma_k \sin\left(\frac{2\pi t}{C_k}\right) + \delta_k \cos\left(\frac{2\pi t}{C_k}\right)\right] + \epsilon

   • Supports multiple cycles via cycles parameter (e.g., c(52,26) for weekly annual + semi-annual patterns)

   • Self-adapts to pathogen seasonality shifts

3. Peak-Centric Alerting:

   • Flags peaks via optimized threshold (final model's 95% CI upper bound)

   • Avoids subjective epidemic-onset definitions

Value

A list containing:

• output: Full dataset with warning flags (1=alert, 0=normal)

• best_fit: Final lm model object

• fit_times: Iteration count for convergence

• cycles: Input cycle parameters

References

Wang X, Wu S, MacIntyre CR, et al. Using an adjusted Serfling regression model to improve the early warning at the arrival of peak timing of influenza in Beijing. PLoS One, 2015,10(3):e0119923.

Examples

## modeling
data(sample_ili)
sf <- aSerfling(data = sample_ili, 'case', cycles = c(52, 26))
sf

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

Apply Adjusted Serfling Model to Subsequent Time Periods

Description

Projects an existing Serfling model onto new temporally contiguous data to detect epidemic signals. Requires test data to immediately follow training data chronologically to maintain periodicity.

Usage

aSerfling_predict(sf, df_test)

Arguments

Details

This function extends the surveillance capability of an established aSerfling model by:

• Automatically generating time indices continuing from the training set

• Preserving all terms from the original model fit

• Calculating prediction intervals using the trained coefficients

• Flagging values exceeding the 95% upper prediction bound as warnings

Critical requirements:

1. Test data must maintain the same time resolution (weekly/monthly) as training data

2. The first test observation must be the immediate next time point after the last training observation

3. Column names and cycle parameters must match the original model specification

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, Wu S, MacIntyre CR, et al. Using an adjusted Serfling regression model to improve the early warning at the arrival of peak timing of influenza in Beijing. PLoS One, 2015,10(3):e0119923.

Examples

data(sample_ili)

## Split into sequential training/test sets
df_train <- sample_ili[1:150,]
df_test <- sample_ili[151:200,]

## modeling
sf <- aSerfling(df_train, 'case', cycles = c(52, 26))

## apply the model to test set
pre <- aSerfling_predict(sf, df_test)

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

Modified Cumulative Sum

Description

Modified CUSUM method for outbreak detection in infectious disease surveillance data. Implements three variants (C1', C2', C3') with dynamic thresholds for time series analysis.

Usage

mCUSUM(data, column, k = 1, h = 2, move_t)

Arguments

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 modified CUSUM models accumulate excess cases beyond control limits:

C1'_0 = C2'_0 = 0

C1'_t = \max\left(0, X_t - (\hat{\mu}_t + k\hat{\sigma}_t) + C1'_{t-1}\right)

C2'_t = \max\left(0, X_t - (\hat{\mu}_t + k\hat{\sigma}_t) + C2'_{t-1}\right)

C3'_t = C2'_t + C2'_{t-1} + C2'_{t-2}

H_t = h\hat{\sigma}_t

where:

• k: Standard deviation coefficient (typical range 0.5–1.5), adjusts sensitivity to deviations

• h: Threshold coefficient (typical range 2–5), controls alarm stringency

• H: Threshold

Model specifications:

• C1': Baseline \hat{\mu}_t, \hat{\sigma}_t estimated from (X_{t-t_{move}},...,X_{t-1})

• C2': Baseline \hat{\mu}_t, \hat{\sigma}_t estimated from (X_{t-2-t_{move}},...,X_{t-3}) to avoid recent outbreaks

• C3': 3-day cumulative sum of C2' values

• Alarms trigger when Cx'_t > H_t for each model (x = 1,2,3)

Value

A data frame containing C1', C2' and C3' 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 <- mCUSUM(data_frame, 'case', k = 1, h = 2.5, move_t = 4)
output

## visualize alerts
### C1'
plot(output$date, output$case, type = "l")
points(output$date[output$C1_prime_warning == 1],
       output$case[output$C1_prime_warning == 1], col = "red")

### C2'
plot(output$date, output$case, type = "l")
points(output$date[output$C2_prime_warning == 1],
       output$case[output$C2_prime_warning == 1], col = "red")

### C3'
plot(output$date, output$case, type = "l")
points(output$date[output$C3_prime_warning == 1],
       output$case[output$C3_prime_warning == 1], col = "red")

Simulated ILI Surveillance Data

Description

A dataset containing 200 weeks of simulated influenza-like illness case counts.

Usage

data(sample_ili)

Format

A data frame with 200 rows and 2 variables:

• date: Date of observation (weekly)

• case: Integer count of reported cases