Model Selection

Our framework uses the method of L-moment ratios to choose a suitable probability model for frequency analysis. This technique involves comparing the L-moments of the data with the known L-moments of various probability distributions. The CRAN package lmom is used extensively in this portion of the framework.

An Introduction to L-Moments

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

Definition: 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: 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:

Remark: 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 to their theoretical values to select a distribution.

List of Candidate Distributions

The four-parameter kappa distribution (K4D), generalizes all ten of the distributions above.

Note: Probability distributions with less than three 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 \(\kappa\).

Selection Metrics

L-Distance

Compare the euclidean distance between the sample L-skewness and sample L-kurtosis \((t_{3}, t_{4})\) and the known L-moment ratios \((\tau_{3}, \tau_{4})\) for each candidate distribution. For probability distributions with three parameters, we use the minimum distance between the L-moment ratio curve \((\tau _{3}(\kappa ), \tau _{4}(\kappa ))\)and the L-moment ratios of the sample \((t_{3}, t_{4})\).

L-Kurtosis

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

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 data using \(t_{2}\), \(t_{3}\), and \(t_{4}\).
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 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 Non-Stationarity

There are three non-stationary scenarios that can be identified during EDA:

1. Significant trend in the mean only.
2. Significant trend in the STD only.
3. Significant trend in both the mean and STD.

To determine the best probability distribution for non-stationary data, we decompose the data into stationary and non-stationary components and then use one of the methods described above.

Decomposition (Scenario 1)

1. Use Sen's Trend Estimator to identify the slope \(b_{1}\) and intercept \(b_{0}\) of the trend.
2. Detrend the data by subtracting the linear function \((b_{1} \cdot \text{Covariate})\) from the data, where the covariate is a value between \([0, 1]\) derived from the index.
3. If necessary, enforce positivity by adding a constant such that \(\min(\text{data}) = 1\) .

Decomposition (Scenario 2)

1. Use a moving-window algorithm to compute the variance of the data.
2. Use Sen's Trend Estimator to identify the slope \(c_{1}\) and intercept \(c_{0}\) of the trend in the variance.

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.

Decomposition (Scenario 3)

1. Remove the linear (additive) trend exactly as in Scenario 1.
2. On that detrended series, compute 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.