Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extremal properties of max-autoregressive moving average processes for modelling extreme river flows

Eleanor D'Arcy, Jonathan A Tawn

Abstract

Max-autogressive moving average (Max-ARMA) processes are powerful tools for modelling time series data with heavy-tailed behaviour; these are a non-linear version of the popular autoregressive moving average models. River flow data typically have features of heavy tails and non-linearity, as large precipitation events cause sudden spikes in the data that then exponentially decay. Therefore, stationary Max-ARMA models are a suitable candidate for capturing the unique temporal dependence structure exhibited by river flows. This paper contributes to advancing our understanding of the extremal properties of stationary Max-ARMA processes. We detail the first approach for deriving the extremal index, the lagged asymptotic dependence coefficient, and an efficient simulation for a general Max-ARMA process. We use the extremal properties, coupled with the belief that Max-ARMA processes provide only an approximation to extreme river flow, to fit such a model which can broadly capture river flow behaviour over a high threshold. We make our inference under a reparametrisation which gives a simpler parameter space that excludes cases where any parameter is non-identifiable. We illustrate results for river flow data from the UK River Thames.

Extremal properties of max-autoregressive moving average processes for modelling extreme river flows

Abstract

Max-autogressive moving average (Max-ARMA) processes are powerful tools for modelling time series data with heavy-tailed behaviour; these are a non-linear version of the popular autoregressive moving average models. River flow data typically have features of heavy tails and non-linearity, as large precipitation events cause sudden spikes in the data that then exponentially decay. Therefore, stationary Max-ARMA models are a suitable candidate for capturing the unique temporal dependence structure exhibited by river flows. This paper contributes to advancing our understanding of the extremal properties of stationary Max-ARMA processes. We detail the first approach for deriving the extremal index, the lagged asymptotic dependence coefficient, and an efficient simulation for a general Max-ARMA process. We use the extremal properties, coupled with the belief that Max-ARMA processes provide only an approximation to extreme river flow, to fit such a model which can broadly capture river flow behaviour over a high threshold. We make our inference under a reparametrisation which gives a simpler parameter space that excludes cases where any parameter is non-identifiable. We illustrate results for river flow data from the UK River Thames.
Paper Structure (18 sections, 3 theorems, 51 equations, 6 figures, 1 table)

This paper contains 18 sections, 3 theorems, 51 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model definition and parameter constraints
  3. Estimating extremal measures
  4. Extremal index
  5. Extremal dependence measure
  6. Simulation of Max-ARMA$(p,q)$ processes
  7. Inference
  8. Model parameterisation
  9. Inference strategy
  10. Marginal inference
  11. Illustrative analysis of River Thames extreme flows
  12. Proofs
  13. Proof of Proposition \ref{['Max-ARMA:prop:identifiability']}
  14. Proof of Proposition \ref{['Max-ARMA:prop::extremalindex']}
  15. Case $n=3$:
  16. ...and 3 more sections

Key Result

Proposition 1

For a stationary Max-ARMA($p,q$) process with $p\geq 2$, if $\alpha_i$ for all $i=1,\ldots,(k-1)<p$ are identifiable, then $\alpha_k$ is identifiable if where $\lfloor\cdot\rfloor$ denotes the floor function. Therefore, all $\alpha_i$ for $i=1,\ldots,p$ are identifiable if the above holds for all $i$.

Figures (6)

  • Figure 1: Left: River flow time series of the River Thames for the winter season (October - March) in 1894/95 (black), 1927/28 (dark blue), 1973/74 (light blue) and 2019/2020 (green) EA_river_data. Right: Pearson's correlation coefficient (dashed line) and empirical estimates of $\chi_{\kappa}(u)$ (solid lines; see Section \ref{['Max-ARMA:subsec::chi']}) for the Thames data over different lags, $\kappa$ (in days). Quantiles of $u=0.9$ (black), 0.95 (dark blue) and 0.975 (light blue) are used for estimating $\chi_\kappa(u)$.
  • Figure 2: Simulations from stationary Max-ARMA$(p,q)$ processes $\{X_t\}$, presented on Gumbel margins, i.e., for $\log X_t$, with sample sizes $n=1000$: $(p,q)=(3,0)$ (top row) and $(p,q)=(3,3)$ (bottom row) with parameters $\bm\alpha=(0.85,0.77,0.7)$ (top left), $\bm\alpha=(0.3,0,0.1)$ (top right), $\bm\alpha=(0.85,0.77,0.7)$ and $\bm\beta=(2,1,0.9)$ (bottom left), and $\bm\alpha=(0.85,0.77,0.7)$ and $\bm\beta=(50,10,5)$ (bottom right).
  • Figure 3: Left: River flow trace plot of the River Thames on Gumbel margins for the winter season (October - March) in 1894/95 (black), 1927/28 (dark blue), 1973/74 (light blue) and 2019/2020 (green). Right: QQ plot of the marginal Pareto tail model fitted to the River Thames exceedances of $u_M$ on Gumbel margins with 95% tolerance bounds.
  • Figure 4: The minimised objective function value $\mathcal{M}(\hat{\bm\delta}_{p,q},\hat{\bm\epsilon}_{p,q};p,q)$ for our moments-based inference of expression \ref{['Max-ARMA:eqn:obj_function']} for Max-ARMA fits of different orders $p=1,2,3$ ($x$-axis) and $q=0,\ldots,4$ ($y$-axis) to the River Thames data using a threshold $u$ of the 0.9 (left) and 0.95 (right) quantiles. Darker (lighter) red indicates a higher (lower) objective function value.
  • Figure 5: Estimates of $\theta(u)$ (left) and $\chi_\kappa(u)$ (right) for $k=1,7$ and $14$ (black, blue and green, respectively) using a threshold $u$ of the 0.95 quantile for the River Thames with empirical estimates (solid lines) and estimates using fitted Max-ARMA models (points) with varying orders $(p,q)$ ($x$-axis). Horizontal dashed lines show 95% confidence intervals for the empirical estimates of $\theta(u)$ and $\chi_\kappa(u)$ for the data.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Remark 3
  • Remark 4
  • Proposition 2
  • Proposition 3
  • Remark 5
  • Remark 6
  • proof
  • ...and 2 more