Model Selection

This module selects a statistical model for S-FFA or NS-FFA based on the annual maximum series.

• S-FFA: A time-invariant probability distribution is selected from the candidate distributions.
• NS-FFA: A distribution is chosen along with a nonstationary structure to capture its evolution over time. In piecewise NS-FFA, the series is segmented into subperiods, each modelled with either time-invariant or time-varying distributions.

The framework uses the L-moment ratio method to identify the best-fit distribution family by comparing sample L-moments with the L-moments of various distribution families.

For NS-FFA, the series is decomposed to isolate its stationary component following Vidrio-Sahagún and He (2022). This decomposed sample is then used for distribution selection, as in S-FFA.

An Introduction to L-Moments

Definition 1: The \(k\)-th Order Statistic of a statistical sample is its \(k\)-th smallest value.

Definition 2: The \(r\)-th Population L-moment \(\lambda_{r}\) is a linear combination of the expectation of the order statistics. Let \(X_{k:n}\) be the \(k\)-th order statistic from a sample of size \(n\). Then,

Definition 3: A Probability Weighted Moment (PWM) encodes information about a value's position on the cumulative distribution function. The \(r\)-th PWM, denoted \(\beta_{r}\), is:

For an ordered sample \(x_{1:n} \leq \dots \leq x_{n:n}\), the sample PWM is often estimated as:

Sample L-Moments (from PWMs) and L-Moment Ratios

The first four sample L-moments can be computed as linear combinations of the PWMs:

The L-moments are used to compute the Sample L-variance \(t_{2}\), Sample L-skewness \(t_{3}\) and the Sample L-kurtosis \(t_{4}\) using the following formulas:

Then, we compare these statistics, specifically the L-skewness and L-kurtosis to their theoretical values (given here) using one of three different metrics to select a distribution.

Note: Probability distributions with two parameters have constant L-skewness \(\tau_{3}\) and L-kurtosis \(\tau_{4}\) regardless of their parameters. The L-skewness and L-kurtosis of probability distributions with three parameters is a function of the shape parameter \(\kappa\). The notation \(\tau_{3}(\kappa)\) and \(\tau_{4}(\kappa)\) refers to the L-skewness and L-kurtosis curves for three parameter distributions.

Example Plot

Shown below are the L-moment curves of the GEV, GLO, GNO, PE3/LP3, and WEI distributions as well as the L-moment ratios of the two parameter distributions GUM and NOR/LNO. This L-moment diagram depicts the "L-distance" selection metric, which compares the euclidian distance between the sample and theoretical L-moment ratios. The inset shows that the GEV distribution (yellow line) has the closest L-moments to the data.

Selection Metrics

1. L-Distance

The Euclidean distance between the sample \((t_3, t_4)\) and theoretical \((\tau_3, \tau_4)\) for each candidate distribution. For 3-parameter distributions, this is the minimum distance along their L-moment ratio curve.

2. L-Kurtosis

The L-kurtosis method is only used for three-parameter probability distributions. First, the shape parameter \(\kappa^{*}\) such that \(t_{3} = \tau _{3}(\kappa ^{*})\) is identified. Then, the difference between the sample L-kurtosis and the theoretical L-kurtosis is computed using the metric \(|\tau_{4}(\kappa ^{*}) - t_{4} |\).

3. Z-statistic

The Z-statistic selection metric is calculated as follows (for three-parameter distributions):

1. Fit the four-parameter Kappa (K4D) distribution to the sample.
2. Generate \(N_{\text{sim}}\) bootstrap samples from the fitted K4D distribution.
3. Calculate the sample L-kurtosis \(t_{4}^{[i]}\) of each synthetic dataset.

4. Calculate the bias and standard deviation of the bootstrap distribution:

   \[ B_{4} = N_{\text{sim} }^{-1} \sum_{i = 1}^{N_{\text{sim} }} \left(t_{4}^{[i]} - t_{4}^{s}\right) \]
   \[ \sigma _{4} = \left[(N_{\text{sim} } - 1)^{-1} \left\{\sum_{i - 1}^{N_{\text{sim} }} \left(t_{4}^{[i]} - t_{4}^{s}\right)^2 - N_{\text{sim} } B_{4}^2\right\} \right] ^{\frac{1}{2}} \]

5. Identify the shape parameter \(\kappa^{*}\) such that \(t_{3} = \tau _{3}(\kappa ^{*})\).

6. Use the bootstrap distribution to compute the Z-statistic for each distribution:

   \[ z = \frac{\tau_{4} (\kappa ^{*}) - t_{4} + B_{4} }{ \sigma _{4}} \]

7. Choose the distribution with the smallest Z-statistic.

Handling Nonstationarity

When nonstationarity is detected, the annual maximum series is decomposed before model selection. We consider three nonstationary scenarios that can be identified in EDA:

1. Trend in mean only.
2. Trend in standard deviation only.
3. Trend in both mean and standard deviation.

Decomposition Steps

Scenario 1: Trend in mean

1. Use Sen's Trend Estimator to approximate the slope \(b_1\) and intercept \(b_0\).
2. Detrend: subtract the linear function \((b_{1} \cdot \text{Covariate})\) from the time series, where the covariate is a time index calculated using the formula \((\text{Years} - 1900) / 100\).
3. Ensure positivity: if necessary, shift series by adding a constant such that \(\min(\text{data}) = 1\).

Scenario 2: Trend in standard deviation

1. Generate a time series of standard deviations using the moving windows method.
2. Use Sen's Trend Estimator to identify the slope \(c_{1}\) and intercept \(c_{0}\) of the trend in the standard deviations.

3. Normalize the data to have mean \(0\), then divide out the scale factor \(g_{t}\).

   \[ g_{t} = \frac{(c_{1} \cdot \text{Covariate} ) + c_{0}}{c_{0}} \]

4. Add back the long-term mean \(\mu\), and then ensure positivity as in Scenario 1.

Scenario 3: Trend in both mean and standard deviation

1. Remove the linear trend in mean exactly as in Scenario 1.
2. On that detrended series, generate a rolling‐window STD series and fit its trend.
3. Divide the detrended data by the time-varying scale factor \(g_{t}\) (as in Scenario 2).
4. Shift to preserve the series mean and ensure positivity.