The following Lab exercise uses Rstudio with extRemes and in2extRemes with the following references:

Coles (2001), Gilleland and Katz (2016a) and Gilleland and Katz (2016b)

1 Maxiumn Sea-level Date

1.1 Q1

1.2 Q2

1.2.1 Normalization of Year Unit

  Year SeaLevel   SOI       time       timeSq
1 1897     1.58 -0.67 0.00000000 0.0000000000
2 1898     1.71  0.57 0.01086957 0.0001181474
3 1899     1.40  0.16 0.02173913 0.0004725898
4 1900     1.34 -0.65 0.03260870 0.0010633270
5 1901     1.43  0.06 0.04347826 0.0018903592
7 1903     1.19  0.47 0.06521739 0.0042533081

1.2.2 fremantle_df_M1

fremantle_df_M01 = fevd(x = fremantle_df$SeaLevel)
fremantle_df_M01


fevd(x = fremantle_df$SeaLevel)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  -43.56663 


 Estimated parameters:
  location      scale      shape 
 1.4823417  0.1412723 -0.2174282 

 Standard Error Estimates:
  location      scale      shape 
0.01672527 0.01149706 0.06378114 

 Estimated parameter covariance matrix.
              location         scale         shape
location  2.797348e-04  1.555836e-05 -0.0003818987
scale     1.555836e-05  1.321823e-04 -0.0003554911
shape    -3.818987e-04 -3.554911e-04  0.0040680341

 AIC = -81.13326 

 BIC = -73.77022

1.2.3 fremantle_df_M2

cov_year_df = data.frame(fremantle_df$time)
colnames(cov_year_df) = "time"
fremantle_df_M02 = fevd(x = fremantle_df$SeaLevel, data = cov_year_df, location.fun = ~cov_year_df$time, 
    scale.fun = ~cov_year_df$time, use.phi = TRUE)
fremantle_df_M02


fevd(x = fremantle_df$SeaLevel, data = cov_year_df, location.fun = ~cov_year_df$time, 
    scale.fun = ~cov_year_df$time, use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  -50.75242 


 Estimated parameters:
       mu0        mu1       phi0       phi1      shape 
 1.3918440  0.1707783 -1.9200454 -0.3270439 -0.1362358 

 Standard Error Estimates:
       mu0        mu1       phi0       phi1      shape 
0.03158419 0.04790219 0.15667422 0.25214080 0.07496857 

 Estimated parameter covariance matrix.
                mu0          mu1         phi0         phi1         shape
mu0    0.0009975612 -0.001338059  0.001201300 -0.001693688 -0.0004948027
mu1   -0.0013380589  0.002294620 -0.001608736  0.002950810  0.0001382900
phi0   0.0012012995 -0.001608736  0.024546811 -0.033190832 -0.0029743290
phi1  -0.0016936880  0.002950810 -0.033190832  0.063574985  0.0003215570
shape -0.0004948027  0.000138290 -0.002974329  0.000321557  0.0056202868

 AIC = -91.50484 

 BIC = -79.2331

1.2.4 fremantle_df_M3

cov_year_yearSq_df = fremantle_df[, c("time", "timeSq")]
fremantle_df_M03 = fevd(x = fremantle_df$SeaLevel, data = cov_year_yearSq_df, location.fun = ~cov_year_yearSq_df$time + 
    cov_year_yearSq_df$timeSq, scale.fun = ~cov_year_yearSq_df$time, use.phi = TRUE)
fremantle_df_M03


fevd(x = fremantle_df$SeaLevel, data = cov_year_yearSq_df, location.fun = ~cov_year_yearSq_df$time + 
    cov_year_yearSq_df$timeSq, scale.fun = ~cov_year_yearSq_df$time, 
    use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  -51.67405 


 Estimated parameters:
       mu0        mu1        mu2       phi0       phi1      shape 
 1.3388206  0.4421351 -0.2530953 -1.9238939 -0.3657534 -0.1136636 

 Standard Error Estimates:
       mu0        mu1        mu2       phi0       phi1      shape 
0.05059924 0.20530340 0.18521625 0.15879905 0.25579564 0.07839770 

 Estimated parameter covariance matrix.
                mu0           mu1          mu2          phi0          phi1
mu0    0.0025602835 -0.0092523325  0.007355854  0.0005034633  0.0003423894
mu1   -0.0092523325  0.0421494880 -0.036999929  0.0006870302 -0.0044998660
mu2    0.0073558545 -0.0369999288  0.034305061 -0.0014353542  0.0055372055
phi0   0.0005034633  0.0006870302 -0.001435354  0.0252171389 -0.0341095659
phi1   0.0003423894 -0.0044998660  0.005537206 -0.0341095659  0.0654314093
shape -0.0010280608  0.0027689734 -0.002429921 -0.0030523062  0.0002270933
              shape
mu0   -0.0010280608
mu1    0.0027689734
mu2   -0.0024299210
phi0  -0.0030523062
phi1   0.0002270933
shape  0.0061461992

 AIC = -91.34809 

 BIC = -76.62201

1.2.5 fremantle_df_M4

cov_year_SOI_df = fremantle_df[, c("time", "SOI")]
fremantle_df_M04 = fevd(x = fremantle_df$SeaLevel, data = cov_year_SOI_df, location.fun = ~cov_year_SOI_df$time + 
    cov_year_SOI_df$SOI, scale.fun = ~cov_year_SOI_df$time + cov_year_SOI_df$SOI, 
    use.phi = TRUE)
fremantle_df_M04


fevd(x = fremantle_df$SeaLevel, data = cov_year_SOI_df, location.fun = ~cov_year_SOI_df$time + 
    cov_year_SOI_df$SOI, scale.fun = ~cov_year_SOI_df$time + 
    cov_year_SOI_df$SOI, use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  -57.90522 


 Estimated parameters:
        mu0         mu1         mu2        phi0        phi1        phi2 
 1.40186193  0.17541020  0.06591733 -1.89868529 -0.41994918  0.26517960 
      shape 
-0.22291118 

 Standard Error Estimates:
       mu0        mu1        mu2       phi0       phi1       phi2      shape 
0.02982290 0.04320254 0.01755776 0.15115112 0.23001018 0.11521237 0.07573829 

 Estimated parameter covariance matrix.
                mu0           mu1           mu2          phi0          phi1
mu0    8.894052e-04 -1.117092e-03  6.439313e-05  2.633663e-04 -0.0001435531
mu1   -1.117092e-03  1.866460e-03  7.364112e-05  4.241739e-05 -0.0001542734
mu2    6.439313e-05  7.364112e-05  3.082748e-04  1.424826e-04 -0.0001186999
phi0   2.633663e-04  4.241739e-05  1.424826e-04  2.284666e-02 -0.0286385740
phi1  -1.435531e-04 -1.542734e-04 -1.186999e-04 -2.863857e-02  0.0529046850
phi2   2.306938e-04 -1.052208e-04  2.532632e-04 -2.556215e-04  0.0022543772
shape -5.620074e-04  1.916362e-04 -1.786980e-04 -4.986159e-03  0.0027837765
               phi2         shape
mu0    0.0002306938 -0.0005620074
mu1   -0.0001052208  0.0001916362
mu2    0.0002532632 -0.0001786980
phi0  -0.0002556215 -0.0049861589
phi1   0.0022543772  0.0027837765
phi2   0.0132738902 -0.0009703609
shape -0.0009703609  0.0057362888

 AIC = -101.8104 

 BIC = -84.63002

1.2.6 fremantle_df_M5

cov_year_SOI_df = fremantle_df[, c("time", "SOI")]
fremantle_df_M05 = fevd(x = fremantle_df$SeaLevel, data = cov_year_SOI_df, location.fun = ~cov_year_SOI_df$time + 
    cov_year_SOI_df$SOI, scale.fun = ~cov_year_SOI_df$SOI, use.phi = TRUE)
fremantle_df_M05


fevd(x = fremantle_df$SeaLevel, data = cov_year_SOI_df, location.fun = ~cov_year_SOI_df$time + 
    cov_year_SOI_df$SOI, scale.fun = ~cov_year_SOI_df$SOI, use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  -56.32075 


 Estimated parameters:
       mu0        mu1        mu2       phi0       phi1      shape 
 1.3958835  0.1808483  0.0642604 -2.1125622  0.2726217 -0.1879142 

 Standard Error Estimates:
       mu0        mu1        mu2       phi0       phi1      shape 
0.02968694 0.04597623 0.01804519 0.08394512 0.11946431 0.06623236 

 Estimated parameter covariance matrix.
                mu0           mu1           mu2          phi0          phi1
mu0    8.813146e-04 -1.179281e-03  9.829679e-05  5.463821e-04  0.0005687221
mu1   -1.179281e-03  2.113814e-03  2.362338e-05 -6.761357e-04 -0.0006122714
mu2    9.829679e-05  2.362338e-05  3.256289e-04  8.202556e-05  0.0004030277
phi0   5.463821e-04 -6.761357e-04  8.202556e-05  7.046783e-03  0.0014227545
phi1   5.687221e-04 -6.122714e-04  4.030277e-04  1.422755e-03  0.0142717206
shape -9.512628e-04  1.101412e-03 -1.648930e-04 -2.814421e-03 -0.0018697343
              shape
mu0   -0.0009512628
mu1    0.0011014117
mu2   -0.0001648930
phi0  -0.0028144207
phi1  -0.0018697343
shape  0.0043867260

 AIC = -100.6415 

 BIC = -85.91541

1.2.7 fremantle_df_M6

cov_year_SOI_df = fremantle_df[, c("time", "SOI")]
fremantle_df_M06 = fevd(x = fremantle_df$SeaLevel, data = cov_year_SOI_df, location.fun = ~cov_year_SOI_df$time + 
    cov_year_SOI_df$SOI)
fremantle_df_M06


fevd(x = fremantle_df$SeaLevel, data = cov_year_SOI_df, location.fun = ~cov_year_SOI_df$time + 
    cov_year_SOI_df$SOI)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  -53.89875 


 Estimated parameters:
        mu0         mu1         mu2       scale       shape 
 1.38436092  0.19443976  0.05452387  0.12073531 -0.15005059 

 Standard Error Estimates:
       mu0        mu1        mu2      scale      shape 
0.03001756 0.04774502 0.01963362 0.01012689 0.06664585 

 Estimated parameter covariance matrix.
                mu0           mu1           mu2         scale         shape
mu0    9.010542e-04 -1.257547e-03  3.651179e-05  6.650006e-05 -0.0008624140
mu1   -1.257547e-03  2.279587e-03 -1.554483e-06 -7.169983e-05  0.0009745994
mu2    3.651179e-05 -1.554483e-06  3.854791e-04  2.889890e-05 -0.0003553078
scale  6.650006e-05 -7.169983e-05  2.889890e-05  1.025538e-04 -0.0003100925
shape -8.624140e-04  9.745994e-04 -3.553078e-04 -3.100925e-04  0.0044416698

 AIC = -97.7975 

 BIC = -85.52576

1.3 Q3

1.3.1 Answer

lr.test(fremantle_df_M01, fremantle_df_M04)


    Likelihood-ratio Test

data:  fremantle_df$SeaLevelfremantle_df$SeaLevel
Likelihood-ratio = 28.677, chi-square critical value = 9.4877, alpha =
0.0500, Degrees of Freedom = 4.0000, p-value = 9.091e-06
alternative hypothesis: greater

lr.test(fremantle_df_M02, fremantle_df_M04)


    Likelihood-ratio Test

data:  fremantle_df$SeaLevelfremantle_df$SeaLevel
Likelihood-ratio = 14.306, chi-square critical value = 5.9915, alpha =
0.0500, Degrees of Freedom = 2.0000, p-value = 0.0007827
alternative hypothesis: greater

lr.test(fremantle_df_M03, fremantle_df_M04)


    Likelihood-ratio Test

data:  fremantle_df$SeaLevelfremantle_df$SeaLevel
Likelihood-ratio = 12.462, chi-square critical value = 3.8415, alpha =
0.0500, Degrees of Freedom = 1.0000, p-value = 0.0004152
alternative hypothesis: greater

lr.test(fremantle_df_M05, fremantle_df_M04)


    Likelihood-ratio Test

data:  fremantle_df$SeaLevelfremantle_df$SeaLevel
Likelihood-ratio = 3.169, chi-square critical value = 3.8415, alpha =
0.0500, Degrees of Freedom = 1.0000, p-value = 0.07505
alternative hypothesis: greater

lr.test(fremantle_df_M06, fremantle_df_M04)


    Likelihood-ratio Test

data:  fremantle_df$SeaLevelfremantle_df$SeaLevel
Likelihood-ratio = 8.0129, chi-square critical value = 5.9915, alpha =
0.0500, Degrees of Freedom = 2.0000, p-value = 0.0182
alternative hypothesis: greater

1.3.2 Conclusion

The best model is model 4

2 Wooster Temperature Series

2.1 Q1

2.1.1 Answer

2.2 Q2

2.2.1 Normalization of Time and Creation of Seasonal Contrast

  Temperature Obs       timeSq
1          23   1 0.000000e+00
2          29   2 3.002439e-07
3          19   3 1.200976e-06
4          14   4 2.702196e-06
5          27   5 4.803903e-06
6          32   6 7.506099e-06

2.2.2 wooster_df_M21

wooster_df_M21 = fevd(x = Temperature, data = wooster_df, type = "GEV")
wooster_df_M21


fevd(x = Temperature, data = wooster_df, type = "GEV")

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  7746.659 


 Estimated parameters:
  location      scale      shape 
36.0199004 18.8227904 -0.4837154 

 Standard Error Estimates:
  location      scale      shape 
0.47461203 0.36611774 0.01404148 

 Estimated parameter covariance matrix.
             location        scale         shape
location  0.225256582 -0.043944803 -0.0024123971
scale    -0.043944803  0.134042199 -0.0035349313
shape    -0.002412397 -0.003534931  0.0001971633

 AIC = 15499.32 

 BIC = 15515.85

2.2.3 wooster_df_M22

cov_CS = wooster_df_final[, c("cos", "sin")]
wooster_df_M22 = fevd(x = wooster_df_final$Temperature, data = cov_CS, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$cos + wooster_df_final$sin)
wooster_df_M22


fevd(x = wooster_df_final$Temperature, data = cov_CS, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$cos + wooster_df_final$sin)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  6690.409 


 Estimated parameters:
        mu0         mu1         mu2       scale       shape 
 37.0911102 -19.2477959  -7.8178277   9.6798709  -0.2800591 

 Standard Error Estimates:
        mu0         mu1         mu2       scale       shape 
0.240745453 0.354252792 0.316680437 0.163408356 0.007654226 

 Estimated parameter covariance matrix.
                mu0          mu1           mu2         scale         shape
mu0    0.0579583730  0.005610600  0.0001790053 -0.0024589705 -5.352029e-04
mu1    0.0056106004  0.125495041  0.0040672369  0.0100656419 -1.489703e-03
mu2    0.0001790053  0.004067237  0.1002864993  0.0021190025 -1.314995e-04
scale -0.0024589705  0.010065642  0.0021190025  0.0267022907 -8.719226e-04
shape -0.0005352029 -0.001489703 -0.0001314995 -0.0008719226  5.858718e-05

 AIC = 13390.82 

 BIC = 13418.37

2.2.4 wooster_df_M23

cov_CS = wooster_df_final[, c("cos", "sin")]
wooster_df_M23 = fevd(x = wooster_df_final$Temperature, data = cov_CS, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$cos + wooster_df_final$sin, scale.fun = ~wooster_df_final$cos + 
        wooster_df_final$sin, use.phi = TRUE)
wooster_df_M23


fevd(x = wooster_df_final$Temperature, data = cov_CS, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$cos + wooster_df_final$sin, 
    scale.fun = ~wooster_df_final$cos + wooster_df_final$sin, 
    use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  6608.966 


 Estimated parameters:
        mu0         mu1         mu2        phi0        phi1        phi2 
 37.3999250 -20.0083912  -7.8420726   2.2320384   0.2611558   0.0691065 
      shape 
 -0.3251316 

 Standard Error Estimates:
       mu0        mu1        mu2       phi0       phi1       phi2      shape 
0.24423116 0.32165568 0.30732238 0.01773834 0.02086356 0.01910594 0.01168177 

 Estimated parameter covariance matrix.
                mu0           mu1           mu2          phi0          phi1
mu0    5.964886e-02  2.873357e-02  7.367519e-03 -3.310976e-04  1.736274e-06
mu1    2.873357e-02  1.034624e-01  2.528814e-04  4.041866e-05 -2.100796e-03
mu2    7.367519e-03  2.528814e-04  9.444704e-02  1.012793e-05  4.323931e-04
phi0  -3.310976e-04  4.041866e-05  1.012793e-05  3.146487e-04  9.128023e-06
phi1   1.736274e-06 -2.100796e-03  4.323931e-04  9.128023e-06  4.352879e-04
phi2  -4.836053e-05  3.664235e-04 -2.421969e-03 -4.890931e-06  2.214474e-05
shape -1.019049e-03 -5.363805e-04 -1.631545e-04 -1.269814e-04 -4.615515e-05
               phi2         shape
mu0   -4.836053e-05 -1.019049e-03
mu1    3.664235e-04 -5.363805e-04
mu2   -2.421969e-03 -1.631545e-04
phi0  -4.890931e-06 -1.269814e-04
phi1   2.214474e-05 -4.615515e-05
phi2   3.650370e-04  7.379449e-06
shape  7.379449e-06  1.364638e-04

 AIC = 13231.93 

 BIC = 13270.5

2.2.5 wooster_df_M24

cov_CS = wooster_df_final[, c("cos", "sin")]
wooster_df_M24 = fevd(x = wooster_df_final$Temperature, data = cov_CS, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$cos + wooster_df_final$sin, scale.fun = ~wooster_df_final$cos + 
        wooster_df_final$sin, shape.fun = ~wooster_df_final$cos + wooster_df_final$sin, 
    use.phi = TRUE)
wooster_df_M24


fevd(x = wooster_df_final$Temperature, data = cov_CS, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$cos + wooster_df_final$sin, 
    scale.fun = ~wooster_df_final$cos + wooster_df_final$sin, 
    shape.fun = ~wooster_df_final$cos + wooster_df_final$sin, 
    use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  6608.755 


 Estimated parameters:
          mu0           mu1           mu2          phi0          phi1 
 37.404390667 -20.062038890  -7.799043928   2.233576971   0.252871645 
         phi2           xi0           xi1           xi2 
  0.074961147  -0.328217343   0.010644200  -0.003686381 

 Standard Error Estimates:
       mu0        mu1        mu2       phi0       phi1       phi2        xi0 
0.24472251 0.33213337 0.33120771 0.01801538 0.02480489 0.02609798 0.01288500 
       xi1        xi2 
0.01885234 0.01706744 

 Estimated parameter covariance matrix.
               mu0           mu1           mu2          phi0          phi1
mu0   5.988911e-02  2.680008e-02  7.832136e-03 -3.016861e-04 -2.004178e-04
mu1   2.680008e-02  1.103126e-01  1.229894e-03 -1.294964e-04 -1.101141e-03
mu2   7.832136e-03  1.229894e-03  1.096985e-01 -1.230052e-04  1.416380e-04
phi0 -3.016861e-04 -1.294964e-04 -1.230052e-04  3.245539e-04 -3.351989e-05
phi1 -2.004178e-04 -1.101141e-03  1.416380e-04 -3.351989e-05  6.152828e-04
phi2  2.100039e-05  2.575341e-04 -3.808998e-04 -7.110116e-06 -7.107301e-05
xi0  -1.098846e-03 -8.737056e-05 -1.381834e-06 -1.441299e-04  3.615268e-05
xi1   2.732839e-04 -1.572529e-03 -2.651869e-04  5.684301e-05 -2.377211e-04
xi2   2.703546e-05 -2.314207e-04 -2.070783e-03  1.820457e-05  3.434350e-05
              phi2           xi0           xi1           xi2
mu0   2.100039e-05 -1.098846e-03  2.732839e-04  2.703546e-05
mu1   2.575341e-04 -8.737056e-05 -1.572529e-03 -2.314207e-04
mu2  -3.808998e-04 -1.381834e-06 -2.651869e-04 -2.070783e-03
phi0 -7.110116e-06 -1.441299e-04  5.684301e-05  1.820457e-05
phi1 -7.107301e-05  3.615268e-05 -2.377211e-04  3.434350e-05
phi2  6.811048e-04  3.506728e-06  2.729585e-05 -2.959937e-04
xi0   3.506728e-06  1.660232e-04 -9.978686e-05 -3.061422e-05
xi1   2.729585e-05 -9.978686e-05  3.554106e-04  5.434540e-05
xi2  -2.959937e-04 -3.061422e-05  5.434540e-05  2.912976e-04

 AIC = 13235.51 

 BIC = 13285.1

2.2.6 wooster_df_M25

cov_time_CS = wooster_df_final[, c("time", "cos", "sin")]
wooster_df_M25 = fevd(x = wooster_df_final$Temperature, data = cov_time_CS, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$time + wooster_df_final$cos + wooster_df_final$sin, 
    scale.fun = ~wooster_df_final$time + wooster_df_final$cos + wooster_df_final$sin, 
    use.phi = TRUE)
wooster_df_M25


fevd(x = wooster_df_final$Temperature, data = cov_time_CS, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$time + wooster_df_final$cos + 
        wooster_df_final$sin, scale.fun = ~wooster_df_final$time + 
        wooster_df_final$cos + wooster_df_final$sin, use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  6605.797 


 Estimated parameters:
         mu0          mu1          mu2          mu3         phi0         phi1 
 37.39032809   0.95622381 -20.03585874  -7.64411681   2.22977001  -0.04546429 
        phi2         phi3        shape 
  0.26024699   0.05652670  -0.32430637 

 Standard Error Estimates:
       mu0        mu1        mu2        mu3       phi0       phi1       phi2 
0.24366803 0.39235255 0.32127664 0.31490728 0.01776478 0.02907870 0.02126588 
      phi3      shape 
0.02021409 0.01168878 

 Estimated parameter covariance matrix.
                mu0           mu1           mu2           mu3          phi0
mu0    5.937411e-02 -4.717652e-03  2.857087e-02  5.683796e-03 -3.280827e-04
mu1   -4.717652e-03  1.539405e-01 -1.589196e-03  2.706685e-02  6.372796e-05
mu2    2.857087e-02 -1.589196e-03  1.032187e-01 -2.570896e-04  3.109943e-05
mu3    5.683796e-03  2.706685e-02 -2.570896e-04  9.916660e-02  2.028613e-05
phi0  -3.280827e-04  6.372796e-05  3.109943e-05  2.028613e-05  3.155876e-04
phi1   1.932877e-04 -4.580152e-03  3.638727e-04 -1.528276e-03  5.429809e-06
phi2   6.972999e-06  7.505661e-05 -2.060914e-03  3.535694e-04  1.101008e-05
phi3   3.935864e-05 -1.378498e-03  4.535236e-04 -2.792578e-03 -2.508763e-06
shape -1.010893e-03 -4.370680e-05 -5.224123e-04 -1.542989e-04 -1.275369e-04
               phi1          phi2          phi3         shape
mu0    1.932877e-04  6.972999e-06  3.935864e-05 -1.010893e-03
mu1   -4.580152e-03  7.505661e-05 -1.378498e-03 -4.370680e-05
mu2    3.638727e-04 -2.060914e-03  4.535236e-04 -5.224123e-04
mu3   -1.528276e-03  3.535694e-04 -2.792578e-03 -1.542989e-04
phi0   5.429809e-06  1.101008e-05 -2.508763e-06 -1.275369e-04
phi1   8.455706e-04  9.014510e-05  1.968719e-04 -1.054310e-05
phi2   9.014510e-05  4.522375e-04  4.367623e-05 -4.788198e-05
phi3   1.968719e-04  4.367623e-05  4.086094e-04  1.877446e-06
shape -1.054310e-05 -4.788198e-05  1.877446e-06  1.366276e-04

 AIC = 13229.59 

 BIC = 13279.18

2.2.7 wooster_df_M26

cov_CS_4season = wooster_df_final[, c("cos", "sin", "winter", "spring", "summer", 
    "fall")]
wooster_df_M26 = fevd(x = wooster_df_final$Temperature, data = cov_CS_4season, threshold = wooster_df_final$wu, 
    location.fun = ~wooster_df_final$cos + wooster_df_final$sin, scale.fun = ~wooster_df_final$cos + 
        wooster_df_final$sin, shape.fun = ~wooster_df_final$winter + wooster_df_final$spring + 
        wooster_df_final$summer + wooster_df_final$fall, use.phi = TRUE)
wooster_df_M26


fevd(x = wooster_df_final$Temperature, data = cov_CS_4season, 
    threshold = wooster_df_final$wu, location.fun = ~wooster_df_final$cos + 
        wooster_df_final$sin, scale.fun = ~wooster_df_final$cos + 
        wooster_df_final$sin, shape.fun = ~wooster_df_final$winter + 
        wooster_df_final$spring + wooster_df_final$summer + wooster_df_final$fall, 
    use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  6600.417 


 Estimated parameters:
         mu0          mu1          mu2         phi0         phi1         phi2 
 37.37356004 -20.20392126  -7.58254992   2.23259684   0.23952941   0.09513814 
         xi0          xi1          xi2          xi3          xi4 
 -0.34279506   0.01242587   0.01729079  -0.06056534   0.07412727 

 AIC = 13222.83 

 BIC = 13283.44

2.3 Q3

2.3.1 wooster_df_M21

2.3.2 wooster_df_M22

2.3.3 wooster_df_M23

2.3.4 wooster_df_M24

2.3.5 wooster_df_M25

2.3.6 wooster_df_M26

2.3.7 wooster_df_M26_modified

To avoid the dummy trap problem (multcollinearity problem due to the 4-seasonal dummies), remove any one of the seasonal dummy that will be absorbed within the intercept term.

wooster_df_M26_modified


fevd(x = wooster_df_final$Temperature, data = cov_CS_4season, 
    threshold = wooster_df_final$wu, location.fun = ~wooster_df_final$cos + 
        wooster_df_final$sin, scale.fun = ~wooster_df_final$cos + 
        wooster_df_final$sin, shape.fun = ~wooster_df_final$winter + 
        wooster_df_final$spring + wooster_df_final$summer, use.phi = TRUE)

[1] "Estimation Method used: MLE"


 Negative Log-Likelihood Value:  6600.417 


 Estimated parameters:
         mu0          mu1          mu2         phi0         phi1         phi2 
 37.36959427 -20.21141060  -7.57962480   2.23260147   0.23949604   0.09504965 
         xi0          xi1          xi2          xi3 
 -0.26862186  -0.06170759  -0.05683884  -0.13493849 

 Standard Error Estimates:
       mu0        mu1        mu2       phi0       phi1       phi2        xi0 
0.24385459 0.33101373 0.32624554 0.01792067 0.02450751 0.02421316 0.02767859 
       xi1        xi2        xi3 
0.02726435 0.03843651 0.03553946 

 Estimated parameter covariance matrix.
               mu0           mu1           mu2          phi0          phi1
mu0   5.946506e-02  2.697509e-02  9.597358e-03 -3.334291e-04  1.853959e-05
mu1   2.697509e-02  1.095701e-01 -3.017465e-03  8.101789e-05 -1.557318e-03
mu2   9.597358e-03 -3.017465e-03  1.064362e-01 -6.384854e-05 -1.594680e-04
phi0 -3.334291e-04  8.101789e-05 -6.384854e-05  3.211503e-04 -3.113839e-05
phi1  1.853959e-05 -1.557318e-03 -1.594680e-04 -3.113839e-05  6.006180e-04
phi2 -6.202026e-05 -3.795283e-05 -5.763927e-04  1.373577e-05 -5.878278e-05
xi0  -1.302433e-03 -8.329608e-04  2.234729e-03 -1.346319e-04 -9.524197e-05
xi1   4.671897e-04 -1.335375e-04 -2.264335e-03  3.614068e-05 -8.163094e-05
xi2   1.027055e-04  6.331504e-04 -4.208308e-03 -1.438576e-05  1.613022e-04
xi3   4.064756e-04  2.296883e-03 -3.383713e-03 -3.204327e-05  3.611805e-04
              phi2           xi0           xi1           xi2           xi3
mu0  -6.202026e-05 -1.302433e-03  4.671897e-04  1.027055e-04  4.064756e-04
mu1  -3.795283e-05 -8.329608e-04 -1.335375e-04  6.331504e-04  2.296883e-03
mu2  -5.763927e-04  2.234729e-03 -2.264335e-03 -4.208308e-03 -3.383713e-03
phi0  1.373577e-05 -1.346319e-04  3.614068e-05 -1.438576e-05 -3.204327e-05
phi1 -5.878278e-05 -9.524197e-05 -8.163094e-05  1.613022e-04  3.611805e-04
phi2  5.862770e-04  2.282032e-04 -2.129680e-04 -4.721185e-04 -3.377576e-04
xi0   2.282032e-04  7.661045e-04 -6.647774e-04 -8.207585e-04 -7.759262e-04
xi1  -2.129680e-04 -6.647774e-04  7.433446e-04  7.653417e-04  6.369531e-04
xi2  -4.721185e-04 -8.207585e-04  7.653417e-04  1.477366e-03  1.015551e-03
xi3  -3.377576e-04 -7.759262e-04  6.369531e-04  1.015551e-03  1.263053e-03

 AIC = 13220.83 

 BIC = 13275.93

2.3.8 lr test

lr.test(wooster_df_M21, wooster_df_M26_modified)


    Likelihood-ratio Test

data:  Temperaturewooster_df_final$Temperature
Likelihood-ratio = 2292.5, chi-square critical value = 14.067, alpha =
0.050, Degrees of Freedom = 7.000, p-value < 2.2e-16
alternative hypothesis: greater

lr.test(wooster_df_M22, wooster_df_M26_modified)


    Likelihood-ratio Test

data:  wooster_df_final$Temperaturewooster_df_final$Temperature
Likelihood-ratio = 179.98, chi-square critical value = 11.07, alpha =
0.05, Degrees of Freedom = 5.00, p-value < 2.2e-16
alternative hypothesis: greater

lr.test(wooster_df_M23, wooster_df_M26_modified)


    Likelihood-ratio Test

data:  wooster_df_final$Temperaturewooster_df_final$Temperature
Likelihood-ratio = 17.099, chi-square critical value = 7.8147, alpha =
0.0500, Degrees of Freedom = 3.0000, p-value = 0.0006745
alternative hypothesis: greater

lr.test(wooster_df_M24, wooster_df_M26_modified)


    Likelihood-ratio Test

data:  wooster_df_final$Temperaturewooster_df_final$Temperature
Likelihood-ratio = 16.676, chi-square critical value = 3.8415, alpha =
0.0500, Degrees of Freedom = 1.0000, p-value = 4.433e-05
alternative hypothesis: greater

lr.test(wooster_df_M25, wooster_df_M26_modified)


    Likelihood-ratio Test

data:  wooster_df_final$Temperaturewooster_df_final$Temperature
Likelihood-ratio = 10.76, chi-square critical value = 3.8415, alpha =
0.0500, Degrees of Freedom = 1.0000, p-value = 0.001037
alternative hypothesis: greater

2.3.9 Conclusion

The best model is wooster_df_M26_modified.

Refernece

Coles, Stuart. 2001. An Introduction to Statistical Modeling of Extreme Values. London: Springer.

Gilleland, Eric, and Richard Katz. 2016a. “ExtRemes 2.0: An Extreme Value Analysis Package in R.” Journal of Statistical Software 72 (August).

Gilleland, Eric, and Richard W. Katz. 2016b. “In2extRemes: Into the R Package extRemes - Extreme Value Analysis for Weather and Climate Applications.” National Center for Atmospheric Research, NCAR/TN-523+STR, 102 pp. https://doi.org/10.5065/D65T3HP2.

Two Applications with extRemes in R package