Below is my notes and thoughts.
On my desk was a copy of
Coles, S. (2001). An introduction to statistical modeling of extreme values / Stuart Coles, London ; New York : Springer, [2001].
Fitting the Generalized Extreme Value distribution function to the Port Jervis, New York annual maximum winter temperature data using "extRemes" R package
library(extremes)
data("PORTw")
plot(PORTw$TMX1, type = "l", xlab = "Year", ylab = "Maximum winter temperature", col = "darkblue")
Default Run
fit1 <- fevd(TMX1, PORTw, units = "deg C")
fit1
fevd(x = TMX1, data = PORTw, units = "deg C")
[1] "Estimation Method used: MLE"
Negative Log-Likelihood Value: 172.7426
Estimated parameters: location scale shape 15.1406132 2.9724952 -0.2171486
Standard Error Estimates: location scale shape 0.39745119 0.27521741 0.07438302
Estimated parameter covariance matrix. location scale shapelocation 0.15796745 0.01028664 -0.010869596scale 0.01028664 0.07574462 -0.010234809shape -0.01086960 -0.01023481 0.005532834
AIC = 351.4853
BIC = 358.1438
GEV model has 3 parameters
- locatation
- scale
- shape
When the Gumbel df. is a good fit, the estimate of the shape parameter should be approx. 0
Here shape is not close to 0. Gumbel probably not good.
3 measures of the goodness of fit provided.
- Negative Log-Likelihood Value
- AIC
- BIC
Gumbel Run
fit0 <- fevd(TMX1, PORTw, type = "Gumbel", units = "deg C")
fit0
fevd(x = TMX1, data = PORTw, type = "Gumbel", units = "deg C")
[1] "Estimation Method used: MLE"
Negative Log-Likelihood Value: 175.7782
Estimated parameters:
location scale
14.799982 2.886128
Standard Error Estimates:
location scale
0.3709054 0.2585040
Estimated parameter covariance matrix.
location scale
location 0.13757080 0.03173887
scale 0.03173887 0.06682429
AIC = 355.5563
BIC = 359.9954All the goodness of fit measure are all larger as expected.
lr.test Likelihood-Ratio Test
Use the likelihood ratio test to confirm this
lr.test(fit1,fit0)
Likelihood-ratio Test
data: TMX1TMX1
Likelihood-ratio = 6.0711, chi-square critical value = 3.841, alpha = 0.050,
Degrees of Freedom = 1.000, p-value = 0.01374 alternative hypothesis: greater
I look at the p-value, because it makes sense to me.
Bayesian estimation
> fB <- fevd(TMX1, PORTw, method = "Bayesian")
> fB
fevd(x = TMX1, data = PORTw, method = "Bayesian")
[1] "Estimation Method used: Bayesian"
Acceptance Rates:
log.scale shape
0.5379076 0.4971994
fevd(x = TMX1, data = PORTw, method = "Bayesian")
[1] "Quantiles of MCMC Sample from Posterior Distribution"
2.5% Posterior Mean 97.5%
location 14.2452034 15.1496558 16.0945492194
scale 2.5276179 3.0935399 3.8803429287
shape -0.3654228 -0.1975376 -0.0004970114
Estimated parameter covariance matrix.
location log.scale shape
location 0.240496184 0.009813165 -0.015884315
log.scale 0.009813165 0.124166098 -0.013129196
shape -0.015884315 -0.013129196 0.008814744
DIC = 1045.476
L-moments
fitLM <- fevd(TMX1, PORTw, method = "Lmoments")
fitLM
fevd(x = TMX1, data = PORTw, method = "Lmoments")
[1] "GEV Fitted to TMX1 of PORTw data frame, using L-moments estimation."
location scale shape
15.1775146 3.0286294 -0.2480594
Significance of Co-variance Information
plot(PORTw$TMX1, PORTw$AOindex)
> fit2 <- fevd(TMX1, PORTw, location.fun = ~AOindex, units = "deg C")
> fit2
fevd(x = TMX1, data = PORTw, location.fun = ~AOindex, units = "deg C")
[1] "Estimation Method used: MLE"
Negative Log-Likelihood Value: 166.7992
Estimated parameters:
mu0 mu1 scale shape
15.2538412 1.1518782 2.6809613 -0.1812824
Standard Error Estimates:
mu0 mu1 scale shape
0.35592663 0.31800904 0.24186870 0.06725912
Estimated parameter covariance matrix.
mu0 mu1 scale shape
mu0 0.126683767 0.002230374 0.010009100 -0.008065698
mu1 0.002230374 0.101129752 -0.002538585 0.002075487
scale 0.010009100 -0.002538585 0.058500466 -0.007020374
shape -0.008065698 0.002075487 -0.007020374 0.004523789
AIC = 341.5984
BIC = 350.4764What is going on here?
location parameterisation now has 2 parts mu0 and mu1
mu(x) = mu0 + mu1 * x, where x = AO Indexlr-test
All the goodness of fit measure are lower when AO Index is consider
Expect utilises the AO index information improves the fit of the model
lr.test(fit1,fit2)
Likelihood-ratio Test
data: TMX1TMX1
Likelihood-ratio = 11.8869, chi-square critical value = 3.841, alpha = 0.050,
Degrees of Freedom = 1.000, p-value = 0.0005653
p-value very small, result as expected
Is the covariance information significant to the scale parameter?
fit3 <- fevd(TMX1, PORTw, location.fun= ~AOindex, scale.fun = ~AOindex, units = "deg C")
> fit3
fevd(x = TMX1, data = PORTw, location.fun = ~AOindex, scale.fun = ~AOindex,
units = "deg C")
[1] "Estimation Method used: MLE"
Negative Log-Likelihood Value: 166.6593
Estimated parameters:
mu0 mu1 sigma0 sigma1 shape
15.2616373 1.1802783 2.6782395 -0.1464682 -0.1860603
Standard Error Estimates:
mu0 mu1 sigma0 sigma1 shape
0.35610192 0.31724472 0.24242220 0.27629143 0.06862845
Estimated parameter covariance matrix.
mu0 mu1 sigma0 sigma1 shape
mu0 0.126808580 -0.0085301401 0.0098703920 -0.001720545 -0.0084519671
mu1 -0.008530140 0.1006442141 -0.0001225007 -0.014111326 0.0004891586
sigma0 0.009870392 -0.0001225007 0.0587685234 -0.005405442 -0.0073208829
sigma1 -0.001720545 -0.0141113262 -0.0054054420 0.076336953 0.0015286996
shape -0.008451967 0.0004891586 -0.0073208829 0.001528700 0.0047098637
AIC = 343.3185
BIC = 354.4161
lr-test
All the goodness of fit measure are HIGHER when AO Index is consider for location and scale
than only location
lr.test(fit2,fit3)
Likelihood-ratio Test
data: TMX1TMX1
Likelihood-ratio = 0.2798, chi-square critical value = 3.841, alpha = 0.050,
Degrees of Freedom = 1.000, p-value = 0.5968
alternative hypothesis: greater
p-value is large,