Detecting Trends in the AMS Variability

This section describes the methods used to detect trends or changes in the variability (e.g., variance or standard deviation) of annual maximum series (AMS) data.

Moving Window Mann-Kendall (MW-MK) Test

The MW-MK test detects statistically significant monotonic trends in the standard deviation of AMS data.

  • Null hypothesis: No significant trend in the standard deviation.
  • Alternative hypothesis: Significant monotonic trend in the standard deviation.

Steps

To compute the standard deviations of the AMS data, we use a moving window algorithm. Let \(w\) be the length of the moving window and \(s\) be the step size. Then,

  1. Initialize the moving window at indices \([1, w]\).
  2. Compute the sample standard deviation within the window.
  3. Move the window forward by \(s\) steps.
  4. Repeat steps 2 and 3 until the window reaches the end of the data.

This produces a time series of moving-window standard deviations. Then, the Mann-Kendall Test is used to test for a monotonic trend in the standard deviation.

Sen's Trend Estimator

Used to estimate the slope of a trend in the standard deviations (see here).

Runs Test

Used to the check residuals of trend fitted to the standard deviations for randomness (see here).

White Test

The White Test detects changes in variability (heteroskedasticity) in a time series.

  • Null hypothesis: Constant variability (homoskedasticity).
  • Alternative hypothesis: Time-dependent variability (heteroskedasticity).

Steps

  1. Fit a simple linear regression model using ordinary least squares:

    \[y_{i} = \beta_{0} + \beta_{1} x_{i} + \epsilon_{i}\]

  2. Compute the squared residuals:

    \[ \hat{r}_i^2 = \left(y_i - \hat{y}_i\right)^2 \]

  3. Fit an auxiliary regression model to the squared residuals. This model includes each regressor, the square of each regressor, and the cross products between all regressors. Since \(x\) is the only regressor, the regression model is simply:

    \[ \hat{r}_i^2 = \alpha_0 + \alpha_1 x_i + \alpha_2 x_i^2 + u_i \]

  4. Compute the coefficient of determination \(R^2\) for the auxillary model.

  5. Compute the test statistic \(nR^2 \sim \chi_{d}^2\) where \(n\) is the number of observations and \(d = 2\) is the number of regressors, excluding the intercept.

  6. If \(nR^2 > \chi^2_{1-\alpha, d}\), we reject the null hypothesis and conclude that the time series exhibits heteroskedasticity.