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Measures, Metrics

Risks

This page outlines tools under investigation for real-time areas at risk identification.

Rates of Change of River Level

Let

\[\mathcal{l}_{t}, \: \mathcal{l}_{t - \tau}, \: \mathcal{l}_{t - 2\tau}, \: \mathcal{l}_{t - 3\tau}, \: \ldots\]

represent the time series of a gauge's river level measures, separated by \(\tau\) hours, and unit of measure ↠ millimetres. Then the weighted-rate-of-change, w.r.t. each \(\tau\) hours interval, is

\[r_{t - i\tau} = \frac{1}{\tau} \bigl(\mathcal{l}_{t - i\tau} - \mathcal{l}_{t - (i + 1)\tau}\bigr) \times \frac{1}{\mathcal{l}_{t - (i + 1)\tau}} \bigl(\mathcal{l}_{t - i\tau} - \mathcal{l}_{t - (i + 1)\tau}\bigr) \]

wherein

  • \(i = 0, 1, 2, 3, \ldots\)
  • \(t\:\) is a/the current time point.
  • \(r_{t - i\tau}\:\) is the rate of change at time point \(t - i\tau\), and its unit of measure is millimetres/hour.

Noting that

\[\frac{1}{\mathcal{l}_{t - (i + 1)\tau}} \bigl(\mathcal{l}_{t - i\tau} - \mathcal{l}_{t - (i + 1)\tau}\bigr)\]

determines the relative river level change w.r.t. consecutive river level values, i.e., values that are \(\tau\) hours apart.



Drift

For time series drift calculations, this hub depends on

  • Jensen-Shannon Distance \(J_{dist}\)
  • Wasserstein Distance \(\mathcal{W}\)


wherein

\[J_{dist} = \sqrt{J_{div}}\]

\(J_{div}\) is the Jensen-Shannon Divergence, a method for determining the similarity of a pair of distributions1. The similarity between a pair of distributions increases as \(J_{dist} \rightarrow 0\). Note, for a pair of distributions

\[J_{div} \in [0 \quad 1]\]

therefore

\[J_{dist} \in [0 \quad 1]\]

The Wasserstein Distance is a distance measure2.

Important, do not consider scores in isolation, also consider the pattern of the scores over time.











  1. Study Divergence Measures Based on Shannon Entropy for an in-depth understanding of Jensen-Shannon Divergence. ↩

  2. Part 6 of From GAN to WGAN has an in-depth discussion of the Wasserstein Distance. GAN: Generative Adversarial Network, WGAN: Wasserstein Generative Adversarial Network ↩