Extreme events by definition are rare events that occur infrequently but their impacts on both physical and socioeconomic resources are very enormous. Extreme climate events such as heavy precipitation, drought, tropical cyclones, hurricanes and heat waves are known to have tremendous impact on the society. Over the last few decades, our understanding of the mean behavior of the climate and its normal variability has improved to a large extend but the same cannot be said of climate extremes. Climate extremes represent nonlinear systems that are very hard to study and even harder to make predictions on them. The objective of this paper is to assess how these extreme events relate to modes of climatic variability such as El Nino-Southern Oscillation, the Pacific Decadal Oscillation and the North Atlantic Oscillation by utilizing the familiar distributions that arise out of the extreme value theory such as the generalized extreme value distribution and the generalized Pareto distribution. Nonstationarity is ensured by expressing the parameters of the distribution as functions of the covariates.
Extreme events are rare events that occur infrequently
but have enormous impact on both physical and
socioeconomic resources. Extreme climate events such as heavy
precipitation, temperature, drought, tropical cyclones,
hurricanes and heat waves have had tremendous impact
on the society, costing lives and property. There is
therefore no surprise that considerable attention has been given
to climate studies in the past decades. However, in
analyzing precipitation events, the majority of the existing
methods are based on the assumption that precipitation
time series are stationary, implying that the distribution of
precipitation events is not significantly affected by
climatic trends, long-term cycles or modes of climate variability
Although relaxing the assumption of stationarity can
lead to accurate models, the results can be potentially
misleading. Hence, there is the need to modify the
assumption of a series of independently and identically
distributed data with constant properties through time
(stationarity) to reflect the effect of long-term climate change
on the variable of interest. For instance, the maximum
time series of climatic variables such as temperature and
precipitation could show trends over time
ENSO events, in particular have influence on the
occurrence of precipitation events. It has also been shown
that there is a well-established connection between the
two phases of ENSO and the North American
precipitation
The objective of this research is to assess how these
extreme events relate to modes of climatic variability such
as the ENSO, NAO and PDO. Similar work has been
carried out on non-stationary extreme events where they
considered only trend in their analysis
The foundation of Extreme Value Theory (EVT) is
the Generalized Extreme Value (GEV) distribution
Let Mn=max{X1, X2,…, Xn} suppose there exist
normalizing constants an>0 and bn>0 such that
prMn bn / an x G(x) as n then the
cumulative distribution function for the GEV distribution
is defined as shown in Equation (1)
Model 1: o (2)
(t) o 1t ( y) o 1 y (t, y) o 1t 2 y (3) (4) (5)
The combined effect of both time and the El Nino Southern Oscillation is shown in Equation (5) where t (time) is the year in which the maxima is taken and y the covariate representing the Nino 3.4. The above GEV distribution models are fitted to both the annual monthly maxima of precipitation data at Pasquotank, North Carolina and the mean sea level data at Pensacola, Florida.
All the model parameters are obtained using the
maximum likelihood estimation (MLE) procedure.
Although other methods such as the Method of Moments
(MOM), Probability Weighted Moments (PWM) can be
used, we exclusively used the MLE because of its easy
adaptability to non-stationary conditions
If G(x(t);µ(t),σ(t),ξ(t)) is the probability density
function of a random variable x with µ(t),σ(t) and ξ(t) as
parameters, the log likelihood of the GEV distribution is
simply given by
n L( , , ) g(x(t); (t), (t), (t)) (6) t1
Both the stationary and nonstationary models of the
GEV distribution can be fitted to the time series of the
random variable x by maximizing the log-likelihood of
the function
n x (t)
1 (t) i
i1 (t)
which is obtained by taking the log of Equation (6).The
maximum likelihood estimates are the values of the
parameters µ, σ and ξ that maximize the log likelihood
function in (7). Time is used as an explanatory variable
(covariate). The parameters µ, σ and ξ can also be expressed
as functions of other explanatory variables and similar
procedure is followed to estimate them. Instead of
maximizing the log likelihood, we can rather minimize the
negative log likelihood of Equation (7). Numerical
methods such as the Newton-Raphson iteration algorithm can
be used to solve Equation (7)
Model choice is usually necessary when you have
more than one model to choose from. For instance, to
compare two nested models (usually between a simpler
model and a complex model), we can easily apply the
Likelihood Ratio Test (LRT) to select the best model by
computing the test statistic and determining whether it is
significant or not. However, the use of the likelihood
ratio test becomes cumbersome when there are more than
two models to choose from. Model selection techniques
such as the Akaike Information Criterion (AIC) and the
Bayesian Information Criterion (BIC) can be used to
select the best model among a collection of nested models.
The BIC selection criterion is applied here to select the
best model among a collection of nested models
Monthly precipitation data in North Carolina for the time period 1950-2008 was obtained from the National Climatic Data Center (NCDC). We selected the station near the Atlantic Ocean. Figure 1 shows a time series plot of the annual maxima of monthly precipitation data at Pasquotank as well as a scatter plot showing how the precipitation vary with the El Nino Southern Oscillation Index (Nino3.4).
Also, we have analyzed the annual mean sea level at Pensacola, Florida during the same time period of 19502008. The sea level data was obtained from the University of Hawaii Sea Level Center. Figure 2 shows a time series plot of the mean sea level data as well as a scatter plot of the sea level versus the El Nino Southern Oscillation Index (Nino3.4). Most climate indices such as ENSO, NAO and PDO do not contain values beyond 1950. Hence, in order to analyze the effects of these indices on climate extremes, we chose a time period 1950-2008 so as to have the same data length. (b) Figure. 1. (a) Time series plot of annual maxima of precipitation and (b) Scatter plot of precipitation and El Nino 3.4 at Pasquotank, North Carolina. Trend is indicated by the solid line.
We investigated the use of the GEV distribution to model both extreme precipitation and sea level in North Carolina and Florida respectively. We modeled these events using both stationary and non-stationary models for the time period 1950-2008. (b) Figure. 2. (a) Time series plot of mean Sea level and (b) Scatter plot of Sea level and Nino 3.4 at Pensacola, Florida. Trend indicated by the solid line.
The effects of both time and El Nino Southern Oscillation index (Nino 3.4) were taken into account. The results are summarized in Tables 1 and 2. The values in parenthesis are the standard errors of the estimates for the parameters. The minimized negative log-likelihood as well as the BIC values is shown in the tables. From the results, it can be seen from Table 1 that when the ENSO was used as a covariate, the negative log-likelihood was minimum as observed in model 3. However, there was an increase in the BIC value. This increase in the BIC value implies that though its introduction has an effect, the change is not significant as compared to the computation complexity involved. In Table 2, model 2 for the Florida sea level has the most minimized negative log likelihood as well the least BIC value, and hence is selected by the BIC as the best model. This means that the model with the linear trend in time is the most appropriate model to be considered for the sea level data at Pensacola.
The effects of both time and ENSO have been analyzed in this research unlike previous work where only linear trend in time was analyzed. From the results, it is realized from the log likelihood values that the sea level at Pensacola is affected by both time and the ENSO as seen in Model 4. Model 4, which is a combination of both time and the ENSO, has the lowest negative log likelihood as compared to the stationary model in Model 1. Comparing the results of models 2 and 4, it is observed that the combined effect of both time and the ENSO is greater than that of time alone but due to computational complexity the BIC results favor the time dependent model in Model 2. Similar observations can be made of from the precipitation data at Pasquotank. However, for this station, the impact of ENSO seems to be greater than that of time as can be seen from both the negative log likelihood and BIC values. This is also evident from the time series plots shown in Fig.1. Again, due to computational complexity, the BIC results favors the stationary model. This suggests that the effect is not significant enough as compared to the computation complexity involved. Hence, the stationary model may be suitable for the precipitation data according to the BIC values. The work presented here only considered simple forms of nonstationarity, where we only relied on linear models of the covariates. The linear models used might have influenced the less significance of the effects of the covariates. As such, as future work, we will consider nonlinear forms of nonstationarity such as vector generalized additive models (VGAM) or generalized additive models for location, scale and shape (GAMLSS) parameters of the distribution. Acknowledgments This work is partially supported by the Expeditions in Computing by the National Science Foundation under Award CCF-1029731.