# Dispersion Index

An alternative way of checking that the sampled excesses over a threshold approximate a Poisson process is the Dispersion Index ($DI$) [^cunnane]. Remember that we wanted the number of excesses per year (or any other time block) to follow a Poisson distribution, $X \sim Poisson(\lambda)$. The $DI$ is defined as the intensity or rate of the Poisson process, $\lambda$ over the mean of the number of excesses per year, $\mu = E(X)$. By definition in a Poisson process [^poisson], the rate is equal to the variance of the process, $\lambda = \sigma^2$. Thus, the $DI$ can be calculated as

$
DI=\sigma^2/\mu
$

where $\sigma^2$ and $\mu$ are the variance and mean of $X$.

Also, by definition, the rate in a Poisson process is equal to the mean, $\lambda = \sigma^2 = \mu$. Thus, if $X$ is Poisson distributed, $\sigma^2/\mu \approx 1$, indicating that the sampled excesses are *iid*.

The $DI$ plot presents in the x-axis a range of values of the threshold and in the y-axis the corresponding value of $DI$. We can identify the range of thresholds which are valid for a given declustering time as those where $DI$ approximates to 1. Moreover, confidence interval for $DI$ can be calculated by testing against a $\chi^2$ distribution with $M−1$ degrees of freedom, where $M$ is the number of years in the sample. The assumption that the sampled excesses follow the Poisson distribution is not rejected if the estimated $DI$ is within the range

$
(\frac{\chi^2_{\alpha/2, M-1}}{(M/1)}, \frac{\chi^2_{1-\alpha/2, M-1}}{(M/1)})
$

where $\alpha$ is the significance level.

We have computed the $DI$ plot for our $H_s$ data obtaining the figure below and calculated the confidence intervals with $\alpha=0.05$ and a number of years $M=21$ as

$
(\frac{\chi^2_{0.025, 20}}{20}, \frac{\chi^2_{0.975, 20}}{20}) = (34.17/20, 9.591/20) = (1.71, 0.48)
$

Therefore, the null hypothesis that the exceedances come from a Poisson distribution cannot be rejected for values of $DI \in (1.71, 0.48)$.

```{figure} ./figures/Threshold_DI.png

---
DI por wave data.
```

We can see that we can assume that our process is Poisson distributed with $th \in [1.3, 2.6]$ for our selected $dl = 48h$. Note that the plot would change if we modify the value of $dl$. Therefore, our parameters of $th = 2.5m \ and \ dl=48h$ are reasonable according to $DI$ plot.

## Let's code it!

Pseudo code is presented. Note that here the first element in a vector corresponds to index 1.

    read observations

    #define parameters
    dl = 48 #in hours
    th = linspace(min_threshold, max_threshold, step) #range_thresholds

    for i in length(th):
        excesses = find_peaks(observations, threshold = th[i], distance = dl) - th[i]
        for j in length(years):
            n_excesses[j] = count(excesses[j])
        e_mean[i] = mean(n_excesses)
        var_mean[i] = var(n_excesses)
    
    plot(x = th, y = e_mean/var_mean)


[^cunnane]: Cunnane (1979). A note on the Poisson assumption in partial duration series models. *Water Resources Research*, 15 (2), 489-494.

[^poisson]: Don't you remember it? Go back to our section in Poisson processes!