Statistical Hazard Modelling

A very brief introduction

Zak Varty

Why do we care about hazard modelling?


Climate

  • Heat-waves and cold-snaps,
  • Drought and floods,
  • Earthquakes and wildfires.

Finance

  • Market Crashes,
  • Portfolio Optimisation,
  • Insurance and Reinsurance.

Industry

  • Quality assurance,
  • Reliability modelling,
  • Asset protection.

Risk vs Hazard


Hazard \(\approx\) probability: the chance of an event at least as severe as some value happening within in a given space-time window.


\[\Pr(X > x) = 1 - F_X(x).\]

Risk \(\approx\) cost: the potential economic, social and environmental consequences of perilous events that may occur in a specified period of time or space.

\[ \text{VaR}_\alpha (X) = F_X^{-1}(\alpha) \quad or \quad \text{ES}_\alpha(X) = \mathbb{E}[X | X < \text{VaR}_\alpha(X)].\]

  • Subjectivity in cost function
  • Convolution of hazard and geographic / demographic information

We can focus on modelling large values

Depending on the peril we are considering, the definition of a “bad” outcome differs:

  • the largest negative returns in finance,
  • the largest positive amounts of rain,
  • the smallest failure stress/time in engineering.


Without loss of generality, we can focus on modelling large positive values, by transforming our data and results as appropriate.

\[g(X_i) \quad \text{e.g.} \quad -X \quad \text{or} \quad X^{-1}.\]

What is wrong with OLS?



An issue with risk / hazard modelling is that we are by definition interested in the rare events, which make up only a very small proportion of our data.

  • Standard modelling techniques describe the expected outcome.
  • Each point is weighted equally and typical values are most common and so dominate measures of model fit.



Robust Regression and Quantile Regression

Robust regression:

  • Models the conditional median \(Y_{0.5} \ |\ X\).
  • Reduces sensitivity to “outliers”
  • The opposite of what we want to do!

Generalises to quantile regression:

  • Model for conditional quantile \(Y_p \ | \ X\)
  • Okay in sample but sample not always large enough.



Extreme Value Theory

What if we need to do beyond the historical record?

  • e.g. estimate a 1 in 1000-year flood from 50 years of data.

  • Extreme Value Theory allows principled extrapolation beyond the range of the observed data.

    • Return values and return periods.

  • Focuses on the most extreme observations.

    • bulk vs tail of distribution.


If we care about


\[M_n = \max \{X_1, \ldots, X_n\}\]


How can we model


\[\begin{align*} F_{M_n}(x) &= \Pr(X_1 \leq x,\ \ldots, \ X_n \leq x) \\ &= \Pr(X \leq x) ^n \\ &= F_X(x)^n? \end{align*}\]

Extremal Types Theorem

Analogue of CLT for Sample Maxima. Let’s revisit the CLT:


Suppose \(X_1, X_2, X_3, \ldots,\) is a sequence of i.i.d. random variables with \(\mathbb{E}[X_i] = \mu\) and \(\text{Var}[X_i] = \sigma^2 < \infty\).

As \(n \rightarrow \infty\), the random variables \(\frac{\sqrt{n} (\bar{X}_n - \mu)}{\sigma}\) converge in distribution to a standard normal distribution.


\[ \frac{\sqrt{n} (\bar{X}_n - \mu)}{\sigma} \overset{d}{\longrightarrow} \mathcal{N}(0,1).\]


Rephrasing this as the partial sums rather than partial means:

\[\frac{S_n}{\sigma\sqrt{n}} - \frac{\mu}{\sigma / \sqrt{n}} \overset{d}{\longrightarrow} \mathcal{N}(0,1).\]

Extremal Types Theorem

Analogue of CLT for Sample Maxima. Let’s revisit the CLT:


Under weak conditions on \(F_X\) and where appropriate sequences of constants \(\{a_n\}\) and \(\{b_n\}\) exist:

\[a_n S_n - b_n \overset{d}{\longrightarrow} \mathcal{N}(0,1).\]


  • Does not depend on \(F_X(x)\), subject to weak conditions (mean exists and finite variance).
  • Scaling and shifting to avoid degeneracy.
  • Motivates Gaussian errors as sum of many non-Gaussian errors.
  • Generalises to non-iid sequences.

Extremal Types Theorem


If suitable sequences of normalising constants exist, then as \(n \rightarrow \infty\):

\[\begin{equation} \label{eqn:lit_extremes_normalising} \Pr\left(\frac{M_n - b_n}{a_n} \leq x \right) \rightarrow G(x), \end{equation}\]

where \(G\) is distribution function of a Fréchet, Gumbel or negative Weibull random variable.


This links to the concept of Maximal Domain of Attraction: if we know \(F_X(x)\) then we can identify \(G(x)\).


But we don’t know \(F\)!

Unified Extremal Types Theorem

These distributional forms are united in a single parameterisation by the Unified Extremal Types Theorem.

The resulting generalised extreme value (GEV) family of distribution functions has the form

\[\begin{equation} \label{eqn:lit_extremes_GEV} G(x) = \exp\left\{ -\left[ 1 + \xi \frac{x - \mu}{\sigma} \right]_{+}^{-1/\xi}\right\}, \end{equation}\]

where \(x_+ = \max(x,0)\), \(\sigma \in \mathbb{R}^+\) and \(\mu , \xi \in \mathbb{R}\). The parameters \(\mu, \sigma\) and \(\xi\) have respective interpretations as location, scale and shape parameters.

  • \(\xi > 0\) correspond to a Fr'echet distribution and a heavy upper tail.
  • \(\xi = 0\) the GEV is equivalent to a Gumbel distribution and has an exponential upper tail.
  • \(\xi < 0\) correspond to a negative Weibull distribution, which is light tailed and has a finite upper end point.

GEV in Action

  • CLT and UETT are asymptotic results, we use them as approximations for finite \(n\).

  • Split the data into \(m\) blocks of length \(k\) and model the \(M_k\).

  • How to pick the block size? Trade-off between bias and variance.

  • Annual blocks often uses as a pragmatic choice to handle seasonal trends.