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Introduction to the mtarm Package

Luis Hernando Vanegas

Sergio Alejandro Calderón

Luz Marina Rondón

Source: vignettes/Introduction.Rmd
Introduction.Rmd

Multivariate Threshold Autoregressive (TAR) models

The mtarm package provides a computational tool designed for Bayesian estimation, inference, and forecasting in multivariate Threshold Autoregressive (TAR) models. These models provide a versatile approach for modeling nonlinear multivariate time series and include multivariate Self-Exciting Threshold Autoregressive (SETAR) and Vector Autoregressive (VAR) models as particular cases (Vanegas, Calderón V, and Rondón 2025). The package accommodates a broad class of innovation distributions beyond the Gaussian assumption, such as Student-tt, slash, symmetric hyperbolic, Laplace, contaminated normal, skew-normal, and skew-tt distributions, thereby enabling robust modeling of heavy tails, asymmetry, and other non-Gaussian characteristics.

Installation

Install from GitHub 

remotes::install_github("lhvanegasp/mtarm")

Install from CRAN

Application: Temperature, precipitation, and two river flows in Iceland

Dataset

The data are available in the object `iceland.rf` and were obtained from (Tong 1990), who provided a detailed description of the geographical and meteorological characteristics of the rivers and analyzed each series individually. Subsequently, (Tsay 1998) conducted a bivariate analysis of the same dataset. The focus is on the bivariate time series {(Y1,t,Y2,t)⊤}t≥1\{(Y_{1,t},Y_{2,t})^{\top}\}_{t\geq 1}, where Y1,tY_{1,t} and Y2,tY_{2,t} denote the daily river flow (in cubic meters per second, m3/s{m}^3/{s}) of the Jökulsá Eystri and Vatnsdalsá rivers, respectively. The sample covers the period from 1972 to 1974, comprising 1095 observations. The exogenous variables include daily precipitation XtX_t, measured in millimeters (mm{mm}), and temperature ZtZ_t, measured in degrees Celsius (∘C^\circ\mathrm{C}), both recorded at the meteorological station in Hveravellir. Precipitation corresponds to the accumulated rainfall from 9:00 A.M. of the previous day to 9:00 A.M. of the current day.

library(mtarm)
data(iceland.rf)       
str(iceland.rf)     
#> 'data.frame':    1096 obs. of  5 variables:
#>  $ Vatnsdalsa   : num  16.1 19.2 14.5 11 13.6 12.5 10.5 10.1 9.68 9.02 ...
#>  $ Jokulsa      : num  30.2 29 28.4 27.8 27.8 27.8 27.8 27.8 27.8 27.3 ...
#>  $ Precipitation: num  8.1 4.4 7 0 0 0 1.9 1.2 0 0.1 ...
#>  $ Temperature  : num  0.9 1.6 0.1 0.6 2 0.8 1.4 1.3 2.2 0.1 ...
#>  $ Date         : Date, format: "1972-01-01" "1972-01-02" ...
summary(iceland.rf[,-5])
#>    Vatnsdalsa        Jokulsa       Precipitation     Temperature      
#>  Min.   : 3.670   Min.   : 22.00   Min.   : 0.000   Min.   :-22.4000  
#>  1st Qu.: 6.100   1st Qu.: 26.70   1st Qu.: 0.000   1st Qu.: -4.2000  
#>  Median : 7.500   Median : 31.40   Median : 0.300   Median :  0.3000  
#>  Mean   : 8.938   Mean   : 41.15   Mean   : 2.519   Mean   : -0.4407  
#>  3rd Qu.: 9.240   3rd Qu.: 50.90   3rd Qu.: 2.500   3rd Qu.:  3.9000  
#>  Max.   :54.000   Max.   :143.00   Max.   :79.300   Max.   : 13.9000
plot(ts(as.matrix(iceland.rf[,-5])), main="Iceland")

Model specification

Following (Tsay 1998), the series are modeled using a TAR(2;p=(15,15),q=(4,4),d=(2,2))\mathrm{TAR}(2; p=(15,15), q=(4,4), d=(2,2)) specification given by

Yt=∑j=12I(Zt−h∈(cj−1,cj])(ϕ0(j)+∑i=115𝛟i(j)Yt−i+∑i=14𝛃i(j)Xt−i+∑i=12δi(j)Zt−i+ϵt(j)) Y_t= \sum_{j=1}^{2} I\!\left(Z_{t-h}\in(c_{j-1},c_j]\right) \left(\! \phi_0^{^{(j)}} +\sum_{i=1}^{15}\boldsymbol{\phi}_i^{^{(j)}}Y_{t-i} +\sum_{i=1}^{4}\boldsymbol{\beta}_i^{^{(j)}}X_{t-i} +\sum_{i=1}^{2}\delta_i^{^{(j)}}Z_{t-i} +\epsilon_t^{^{(j)}} \!\right)

where ϵt(j)\epsilon_t^{^{(j)}} is the error term. The last 55 observations (from November 7 to December 31, 1974), corresponding to 5%5\% of the sample, are excluded from the estimation stage and reserved for out-of-sample forecast evaluation. The following code requests the estimation for the TAR(2;p=(15,15),q=(4,4),d=(2,2))\mathrm{TAR}(2; p=(15,15), q=(4,4), d=(2,2)) specification under Gaussian, Student-tt, and Laplace error distributions.

Parameter estimation 

set.seed(09102)
fits <- mtar_grid(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
                  data=iceland.rf, subset={Date<="1974-11-06"},                           
                  row.names=Date, nregim.min=2, nregim.max=2, p.min=15,                 
                  p.max=15, q.min=4, q.max=4, d.min=2, d.max=2,                           
                  n.burnin=5000, n.sim=4000, n.thin=2, ssvs=TRUE,
                  dist=c("Gaussian","Student-t","Laplace"),
                  plan_strategy="multisession")

fits
#> 
#> 
#> Sample size          : 1026 time points (1972-01-16 to 1974-11-06)
#> 
#> Output Series        : Jokulsa    |    Vatnsdalsa
#> 
#> Threshold Series (TS): Temperature
#> 
#> Exogenous Series (ES): Precipitation
#> 
#> Error Distribution   : Gaussian, Laplace, Student-t
#> 
#> Number of regimes    : 2
#> 
#> Deterministics       : Intercept
#> 
#> Autoregressive order : 15
#> 
#> Maximum lag for ES   : 4
#> 
#> Maximum lag for TS   : 2

Model comparison using forecast accuracy measures

Adjusted within-sample

The following code requests Deviance Information Criterion (DIC) (Spiegelhalter et al. 2002, 2014) and Watanabe-Akaike Information Criterion (WAIC) (Watanabe 2010) values.

DIC(fits)
#>                          DIC
#> Gaussian.2.15.4.2  10060.438
#> Laplace.2.15.4.2    6976.339
#> Student-t.2.15.4.2  7532.597

WAIC(fits)
#>                         WAIC
#> Gaussian.2.15.4.2  10404.091
#> Laplace.2.15.4.2    8200.455
#> Student-t.2.15.4.2  7642.630
Out-of-sample with standard hh-step-ahead forecasting

In addition, the following code provides the median of the log-score (Good 1952), the Energy Score (ES) (Gneiting et al. 2008)—a multivariate extension of the Continuous Ranked Probability Score (CRPS)(Matheson and Winkler 1976; Grimit et al. 2006)—and the Absolute Percentage Error (APE), all computed from the observed and forecasted values for the last 55 observations.

newdata <- subset(iceland.rf, Date>"1974-11-06") 
set.seed(09102)

oos <- out_of_sample(fits, newdata=newdata, n.ahead=nrow(newdata), FUN=median) 
oos[,c(1,2,5,6)]
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.797904     2.811035    5.652454       32.49836
#> Laplace.2.15.4.2    3.242485     1.833367    3.823622       15.61231
#> Student-t.2.15.4.2  3.545381     2.456398    3.676001       20.90241
Out-of-sample with rolling-origin forecasting and fixed parameters 
set.seed(09102)
oos2 <- out_of_sample(fits, newdata=newdata, n.ahead=nrow(newdata), 
                      rolling=5, FUN=median) 

for(i in 1:length(oos2)){                   
   cat("\n",i,"-step-ahead\n")   
   print(oos2[[i]][,c(1,2,5,6)]) 
}                            
#> 
#>  1 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   2.584186    1.2773363   1.5798016       7.274053
#> Laplace.2.15.4.2    1.296487    0.7563662   0.9070033       4.779227
#> Student-t.2.15.4.2  1.020333    0.7762098   0.9009402       5.917259
#> 
#>  2 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.032747     1.639617    2.301814      11.871226
#> Laplace.2.15.4.2    2.093742     1.085154    1.556983       5.577137
#> Student-t.2.15.4.2  1.968642     1.213108    1.447799       6.566068
#> 
#>  3 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.223748     1.850762    2.926733      14.934488
#> Laplace.2.15.4.2    2.378543     1.239162    2.010924       6.609058
#> Student-t.2.15.4.2  2.305891     1.466251    1.926638       8.003508
#> 
#>  4 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.333167     1.968623    3.509917      17.344735
#> Laplace.2.15.4.2    2.566196     1.361887    2.329828       6.406729
#> Student-t.2.15.4.2  2.616427     1.691429    2.414252       7.173568
#> 
#>  5 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.405086     2.079208    3.701430      19.487190
#> Laplace.2.15.4.2    2.703894     1.457582    2.616441       7.007454
#> Student-t.2.15.4.2  2.828487     1.838202    2.802228       9.591203

Summary of the best model 

summary(fits[["Laplace.2.15.4.2"]])
#> 
#> 
#> Sample size          : 1026 time points (1972-01-16 to 1974-11-06)
#> 
#> Output Series (OS)   : Jokulsa    |    Vatnsdalsa
#> 
#> Threshold Series (TS): Temperature with a estimated delay equal to 0
#> 
#> Exogenous Series (ES): Precipitation
#> 
#> Error Distribution   : Laplace
#> 
#> Number of regimes    : 2
#> 
#> Deterministics       : Intercept
#> 
#> Autoregressive orders: 15 in each regime
#> 
#> Maximum lags for ES  : 4 in each regime
#> 
#> Maximum lags for TS  : 2 in each regime
#> 
#> 
#> Thresholds (Mean, HDI.Lower, HDI.Upper)
#>                                                         
#> Regime 1 (-Inf,-0.44847] (-Inf,-0.49438] (-Inf,-0.40256]
#> Regime 2  (-0.44847,Inf)  (-0.49438,Inf)  (-0.40256,Inf)
#> 
#> 
#> Regime1:
#>      OS.lag(1) OS.lag(2) OS.lag(3) OS.lag(4) OS.lag(5) OS.lag(6) OS.lag(7)
#> SSVS         1         1      0.04      0.28      0.02      0.43      0.01
#>      OS.lag(8) OS.lag(9) OS.lag(10) OS.lag(11) OS.lag(12) OS.lag(13) OS.lag(14)
#> SSVS         0      0.02        0.2       0.02          0       0.01       0.22
#>      OS.lag(15) ES.lag(1) ES.lag(2) ES.lag(3) ES.lag(4) TS.lag(1) TS.lag(2)
#> SSVS       0.12         0         0         0         0      0.03      0.05
#> 
#> Autoregressive coefficients
#>                        Mean  2(1-PD)  HDI.Lower HDI.Upper             Mean
#> (Intercept)         4.52327   0.00001   3.51623   5.56107    |     1.21217
#> Jokulsa.lag( 1)     0.77101   0.00001   0.65580   0.87775    |    -0.08284
#> Vatnsdalsa.lag( 1)  0.29727   0.00001   0.17679   0.41320    |     1.08650
#> Jokulsa.lag( 2)    -0.00333   0.90550  -0.06202   0.05971    |     0.04302
#> Vatnsdalsa.lag( 2) -0.24739   0.00200  -0.36654  -0.12501    |    -0.21791
#>                     2(1-PD)  HDI.Lower HDI.Upper
#> (Intercept)          0.00050   0.73774   1.67706
#> Jokulsa.lag( 1)      0.00050  -0.12881  -0.03549
#> Vatnsdalsa.lag( 1)   0.00001   0.98105   1.18322
#> Jokulsa.lag( 2)      0.01150   0.00927   0.07605
#> Vatnsdalsa.lag( 2)   0.00001  -0.30260  -0.13781
#> 
#> Scale parameter (Mean, HDI.Lower, HDI.Upper)
#>            Jokulsa Vatnsdalsa      Jokulsa Vatnsdalsa      Jokulsa Vatnsdalsa
#> Jokulsa    0.15963    0.02514    . 0.12522    0.01188    . 0.20057    0.03827
#> Vatnsdalsa 0.02514    0.06126    . 0.01188    0.04583    . 0.03827    0.07652
#> 
#> 
#> Regime2:
#>      OS.lag(1) OS.lag(2) OS.lag(3) OS.lag(4) OS.lag(5) OS.lag(6) OS.lag(7)
#> SSVS         1         1      0.65         0         0         0         0
#>      OS.lag(8) OS.lag(9) OS.lag(10) OS.lag(11) OS.lag(12) OS.lag(13) OS.lag(14)
#> SSVS         0         0          0          0          0          0          0
#>      OS.lag(15) ES.lag(1) ES.lag(2) ES.lag(3) ES.lag(4) TS.lag(1) TS.lag(2)
#> SSVS          0         0         0         0         0         1         1
#> 
#> Autoregressive coefficients
#>                        Mean  2(1-PD)  HDI.Lower HDI.Upper             Mean
#> (Intercept)         1.59507   0.00200   0.69771   2.59283    |     0.56925
#> Jokulsa.lag( 1)     1.07990   0.00001   1.00905   1.15349    |    -0.00026
#> Vatnsdalsa.lag( 1)  0.57932   0.00050   0.33541   0.81015    |     1.16172
#> Jokulsa.lag( 2)    -0.20992   0.00001  -0.31271  -0.11674    |     0.00344
#> Vatnsdalsa.lag( 2) -0.51318   0.02100  -1.00275  -0.05125    |    -0.34241
#> Jokulsa.lag( 3)    -0.01469   0.58150  -0.06674   0.04094    |    -0.01004
#> Vatnsdalsa.lag( 3)  0.33952   0.00150   0.15257   0.61809    |     0.16776
#> Temperature.lag(1)  1.11876   0.00001   0.90525   1.34375    |     0.05031
#> Temperature.lag(2) -0.65916   0.00001  -0.85308  -0.45923    |    -0.05447
#>                     2(1-PD)  HDI.Lower HDI.Upper
#> (Intercept)          0.00001   0.32663   0.79163
#> Jokulsa.lag( 1)      0.98150  -0.01345   0.01326
#> Vatnsdalsa.lag( 1)   0.00001   1.09323   1.23958
#> Jokulsa.lag( 2)      0.71200  -0.01303   0.02141
#> Vatnsdalsa.lag( 2)   0.00001  -0.50961  -0.15745
#> Jokulsa.lag( 3)      0.08550  -0.02103   0.00082
#> Vatnsdalsa.lag( 3)   0.00001   0.10495   0.26135
#> Temperature.lag(1)   0.01650   0.01020   0.09275
#> Temperature.lag(2)   0.00300  -0.09647  -0.01850
#> 
#> Scale parameter (Mean, HDI.Lower, HDI.Upper)
#>            Jokulsa Vatnsdalsa      Jokulsa Vatnsdalsa      Jokulsa Vatnsdalsa
#> Jokulsa    5.55932    0.32459    . 4.60081    0.20358    . 6.52985    0.44953
#> Vatnsdalsa 0.32459    0.31072    . 0.20358    0.26144    . 0.44953    0.36323
#>

Residuals 

res <- residuals(fits[["Laplace.2.15.4.2"]])  
par(mfrow=c(1,2)) 
qqnorm(res[["full"]], pch=20, col="blue", main="") 
abline(0, 1, lty=3) 
hist(res[["full"]], freq=FALSE, xlab="Quantile-type residual",
     ylab="Density", main="") 
curve(dnorm(x), col="blue", add=TRUE)

par(mfrow=c(1,2))  
acf(res[["full"]], col="blue", main="")
pacf(res[["full"]], col="blue", main="")

Forecasting 

pred <- predict(fits[["Laplace.2.15.4.2"]], newdata=newdata,
                n.ahead=nrow(newdata), row.names=Date, credible=0.8)

head(pred$summary)
#>            Jokulsa.Mean Jokulsa.Lower Jokulsa.Upper Vatnsdalsa.Mean
#> 1974-11-07     20.90878     13.411187      28.81527        6.834375
#> 1974-11-08     21.05299      9.664738      34.06857        6.934579
#> 1974-11-09     22.72315     13.680595      33.14910        7.217351
#> 1974-11-10     23.78405     16.298941      31.40126        7.299724
#> 1974-11-11     24.83764     19.000067      31.43418        7.428469
#> 1974-11-12     25.07040     19.866111      30.16559        7.256866
#>            Vatnsdalsa.Lower Vatnsdalsa.Upper
#> 1974-11-07         5.021734         8.633821
#> 1974-11-08         4.189863        10.025282
#> 1974-11-09         4.336131        10.182200
#> 1974-11-10         4.351673        10.005718
#> 1974-11-11         4.559769        10.052750
#> 1974-11-12         4.612109         9.806307
tail(pred$summary)
#>            Jokulsa.Mean Jokulsa.Lower Jokulsa.Upper Vatnsdalsa.Mean
#> 1974-12-26     25.60095      22.71138      27.99992        5.865634
#> 1974-12-27     25.57809      22.96257      28.18443        5.862626
#> 1974-12-28     25.58518      23.04160      28.28370        5.857756
#> 1974-12-29     25.60750      22.95674      28.14790        5.865459
#> 1974-12-30     25.60785      22.88612      28.06214        5.865358
#> 1974-12-31     25.62957      22.97055      28.22797        5.842145
#>            Vatnsdalsa.Lower Vatnsdalsa.Upper
#> 1974-12-26         3.861787         8.194346
#> 1974-12-27         3.630560         8.033291
#> 1974-12-28         3.604011         7.983446
#> 1974-12-29         3.631105         8.013411
#> 1974-12-30         3.659647         8.036238
#> 1974-12-31         3.650581         7.946888

Summary statistics 

fitmcmc <- coda::as.mcmc(fits[["Laplace.2.15.4.2"]])
summary(fitmcmc)
#> 
#> 
#>  Iterations = 5001:12999
#> 
#>  Thinning interval = 2
#> 
#>  Sample size per chain = 4000
#> 
#> Thresholds:
#>                 Mean       Sd  Sd(Mean)     2.5%    25%      50%      75%
#> Threshold.1 -0.44847 0.029051 0.0012974 -0.49545 -0.472 -0.44829 -0.42508
#>                97.5%
#> Threshold.1 -0.40322
#> 
#> 
#> Regime 1
#> 
#> 
#> Autoregressive coefficients:
#>                                     Mean       Sd   Sd(Mean)       2.5%
#> Jokulsa:(Intercept)            4.5232663 0.549390 0.03784471  3.5989013
#> Vatnsdalsa:(Intercept)         1.2121652 0.241025 0.00957483  0.7403184
#> Jokulsa:Jokulsa.lag( 1)        0.7710093 0.057605 0.00348468  0.6476008
#> Vatnsdalsa:Jokulsa.lag( 1)    -0.0828399 0.024016 0.00080667 -0.1323538
#> Jokulsa:Vatnsdalsa.lag( 1)     0.2972719 0.060628 0.00144518  0.1793622
#> Vatnsdalsa:Vatnsdalsa.lag( 1)  1.0865026 0.052748 0.00229988  0.9746227
#> Jokulsa:Jokulsa.lag( 2)       -0.0033294 0.032301 0.00073881 -0.0641367
#> Vatnsdalsa:Jokulsa.lag( 2)     0.0430238 0.017330 0.00044453  0.0097343
#> Jokulsa:Vatnsdalsa.lag( 2)    -0.2473945 0.062512 0.00284661 -0.3665407
#> Vatnsdalsa:Vatnsdalsa.lag( 2) -0.2179087 0.041992 0.00200789 -0.2988589
#>                                     25%        50%       75%     97.5%
#> Jokulsa:(Intercept)            4.159868  4.4710058  4.819705  5.740136
#> Vatnsdalsa:(Intercept)         1.056449  1.2096344  1.367168  1.679998
#> Jokulsa:Jokulsa.lag( 1)        0.738650  0.7758695  0.808087  0.873081
#> Vatnsdalsa:Jokulsa.lag( 1)    -0.098745 -0.0824465 -0.066779 -0.037354
#> Jokulsa:Vatnsdalsa.lag( 1)     0.258489  0.2966336  0.335548  0.416860
#> Vatnsdalsa:Vatnsdalsa.lag( 1)  1.054176  1.0897296  1.123070  1.178829
#> Jokulsa:Jokulsa.lag( 2)       -0.023571 -0.0036904  0.016285  0.058267
#> Vatnsdalsa:Jokulsa.lag( 2)     0.031575  0.0431486  0.054271  0.077065
#> Jokulsa:Vatnsdalsa.lag( 2)    -0.288748 -0.2482934 -0.205662 -0.125011
#> Vatnsdalsa:Vatnsdalsa.lag( 2) -0.246201 -0.2181547 -0.190029 -0.133153
#> 
#> 
#> Scale parameter:
#>                           Mean        Sd   Sd(Mean)     2.5%     25%      50%
#> Jokulsa.Jokulsa       0.159631 0.0195985 0.00063185 0.120718 0.14839 0.159857
#> Jokulsa.Vatnsdalsa    0.025142 0.0067334 0.00020919 0.012864 0.02056 0.024963
#> Vatnsdalsa.Vatnsdalsa 0.061264 0.0081488 0.00030448 0.043378 0.05672 0.061545
#>                            75%    97.5%
#> Jokulsa.Jokulsa       0.171538 0.197237
#> Jokulsa.Vatnsdalsa    0.029217 0.039574
#> Vatnsdalsa.Vatnsdalsa 0.066154 0.075921
#> 
#> 
#> Regime 2
#> 
#> 
#> 
#> Autoregressive coefficients:
#>                                     Mean        Sd   Sd(Mean)       2.5%
#> Jokulsa:(Intercept)            1.5950744 0.4866782 0.01254526  0.6240658
#> Vatnsdalsa:(Intercept)         0.5692453 0.1224997 0.00295270  0.3364976
#> Jokulsa:Jokulsa.lag( 1)        1.0798968 0.0378132 0.00112223  1.0133909
#> Vatnsdalsa:Jokulsa.lag( 1)    -0.0002585 0.0069370 0.00030599 -0.0141654
#> Jokulsa:Vatnsdalsa.lag( 1)     0.5793169 0.1243080 0.00676481  0.3383020
#> Vatnsdalsa:Vatnsdalsa.lag( 1)  1.1617172 0.0386841 0.00105752  1.0961299
#> Jokulsa:Jokulsa.lag( 2)       -0.2099242 0.0479443 0.00139261 -0.3114447
#> Vatnsdalsa:Jokulsa.lag( 2)     0.0034406 0.0087519 0.00023009 -0.0128809
#> Jokulsa:Vatnsdalsa.lag( 2)    -0.5131848 0.2747070 0.04090219 -1.0067886
#> Vatnsdalsa:Vatnsdalsa.lag( 2) -0.3424101 0.1074206 0.01728081 -0.5180202
#> Jokulsa:Jokulsa.lag( 3)       -0.0146910 0.0266602 0.00174071 -0.0683328
#> Vatnsdalsa:Jokulsa.lag( 3)    -0.0100354 0.0053239 0.00030481 -0.0200775
#> Jokulsa:Vatnsdalsa.lag( 3)     0.3395239 0.1303270 0.01273838  0.1434232
#> Vatnsdalsa:Vatnsdalsa.lag( 3)  0.1677631 0.0448169 0.00567434  0.1002395
#> Jokulsa:Temperature.lag(1)     1.1187577 0.1113354 0.00323477  0.8863310
#> Vatnsdalsa:Temperature.lag(1)  0.0503070 0.0213888 0.00070939  0.0082298
#> Jokulsa:Temperature.lag(2)    -0.6591592 0.1022778 0.00300431 -0.8515034
#> Vatnsdalsa:Temperature.lag(2) -0.0544666 0.0200279 0.00045545 -0.0944949
#>                                      25%         50%        75%      97.5%
#> Jokulsa:(Intercept)            1.2784933  1.60116882  1.9243651  2.5313755
#> Vatnsdalsa:(Intercept)         0.4883506  0.56613505  0.6468881  0.8061825
#> Jokulsa:Jokulsa.lag( 1)        1.0524123  1.07711812  1.1037520  1.1610065
#> Vatnsdalsa:Jokulsa.lag( 1)    -0.0047912 -0.00016617  0.0044499  0.0129182
#> Jokulsa:Vatnsdalsa.lag( 1)     0.4956837  0.58138921  0.6646114  0.8166703
#> Vatnsdalsa:Vatnsdalsa.lag( 1)  1.1341441  1.15792894  1.1855017  1.2474463
#> Jokulsa:Jokulsa.lag( 2)       -0.2387094 -0.20861005 -0.1806942 -0.1149627
#> Vatnsdalsa:Jokulsa.lag( 2)    -0.0023298  0.00289562  0.0089215  0.0216544
#> Jokulsa:Vatnsdalsa.lag( 2)    -0.7234544 -0.54535055 -0.2675414 -0.0519351
#> Vatnsdalsa:Vatnsdalsa.lag( 2) -0.4244335 -0.37135912 -0.2316756 -0.1621547
#> Jokulsa:Jokulsa.lag( 3)       -0.0294963 -0.01349073  0.0050789  0.0397651
#> Vatnsdalsa:Jokulsa.lag( 3)    -0.0137969 -0.01098968 -0.0070192  0.0019572
#> Jokulsa:Vatnsdalsa.lag( 3)     0.2448703  0.30820270  0.4250323  0.6138676
#> Vatnsdalsa:Vatnsdalsa.lag( 3)  0.1297460  0.16243302  0.2004100  0.2599303
#> Jokulsa:Temperature.lag(1)     1.0462141  1.12098133  1.1958726  1.3269358
#> Vatnsdalsa:Temperature.lag(1)  0.0357892  0.05017823  0.0647723  0.0918519
#> Jokulsa:Temperature.lag(2)    -0.7301034 -0.66166113 -0.5907202 -0.4554723
#> Vatnsdalsa:Temperature.lag(2) -0.0678179 -0.05427415 -0.0408837 -0.0157641
#> 
#> 
#> Scale parameter:
#>                          Mean       Sd  Sd(Mean)    2.5%     25%    50%     75%
#> Jokulsa.Jokulsa       5.55932 0.498163 0.0110922 4.63424 5.21730 5.5437 5.88450
#> Jokulsa.Vatnsdalsa    0.32459 0.063650 0.0018542 0.20704 0.28026 0.3223 0.36611
#> Vatnsdalsa.Vatnsdalsa 0.31072 0.026534 0.0012651 0.26365 0.29227 0.3097 0.32780
#>                         97.5%
#> Jokulsa.Jokulsa       6.57001
#> Jokulsa.Vatnsdalsa    0.45742
#> Vatnsdalsa.Vatnsdalsa 0.36590

Convergence diagnostics

Geweke statistic 
geweke_diagTAR(fits[["Laplace.2.15.4.2"]])
#> 
#> Fraction in 1st window = 0.1
#> 
#> Fraction in 2nd window = 0.5
#> Thresholds:
#> Threshold.1 
#>     -1.0478
#> 
#> 
#> Regime 1
#> 
#> 
#> Autoregressive coefficients:
#>                      Jokulsa Vatnsdalsa
#> (Intercept)        -1.060107   -0.87372
#> Jokulsa.lag( 1)     0.386917    0.28273
#> Vatnsdalsa.lag( 1) -1.207216    0.86248
#> Jokulsa.lag( 2)    -0.033279   -1.02879
#> Vatnsdalsa.lag( 2)  1.527347    0.56702
#> 
#> 
#> Scale parameter:
#>            Jokulsa Vatnsdalsa
#> Jokulsa    0.42197     1.1215
#> Vatnsdalsa 1.12153     1.6051
#> 
#> 
#> Regime 2
#> 
#> 
#> 
#> Autoregressive coefficients:
#>                     Jokulsa Vatnsdalsa
#> (Intercept)         3.16200     1.9862
#> Jokulsa.lag( 1)    -2.50578     2.7219
#> Vatnsdalsa.lag( 1)  4.21032     1.9330
#> Jokulsa.lag( 2)     4.44698     2.8767
#> Vatnsdalsa.lag( 2) -3.74674    -3.5346
#> Jokulsa.lag( 3)    -1.61489     1.8916
#> Vatnsdalsa.lag( 3)  3.15800     2.7724
#> Temperature.lag(1)  0.92370    -2.4338
#> Temperature.lag(2)  0.53083     2.3683
#> 
#> 
#> Scale parameter:
#>             Jokulsa Vatnsdalsa
#> Jokulsa    -0.58186    -2.3230
#> Vatnsdalsa -2.32302    -2.1004
Effective sample size 
effectiveSize_TAR(fits[["Laplace.2.15.4.2"]]) 
#> Thresholds:
#> Threshold.1 
#>      501.38
#> 
#> 
#> Regime 1
#> 
#> 
#> Autoregressive coefficients:
#>                    Jokulsa Vatnsdalsa
#> (Intercept)         210.74     633.67
#> Jokulsa.lag( 1)     273.27     886.40
#> Vatnsdalsa.lag( 1) 1759.98     526.02
#> Jokulsa.lag( 2)    1911.51    1519.92
#> Vatnsdalsa.lag( 2)  482.25     437.37
#> 
#> 
#> Scale parameter:
#>            Jokulsa Vatnsdalsa
#> Jokulsa      962.1    1036.11
#> Vatnsdalsa  1036.1     716.23
#> 
#> 
#> Regime 2
#> 
#> 
#> 
#> Autoregressive coefficients:
#>                     Jokulsa Vatnsdalsa
#> (Intercept)        1504.959   1721.196
#> Jokulsa.lag( 1)    1135.324    513.953
#> Vatnsdalsa.lag( 1)  337.666   1338.098
#> Jokulsa.lag( 2)    1185.257   1446.768
#> Vatnsdalsa.lag( 2)   45.107     38.641
#> Jokulsa.lag( 3)     234.569    305.070
#> Vatnsdalsa.lag( 3)  104.674     62.381
#> Temperature.lag(1) 1184.625    909.068
#> Temperature.lag(2) 1158.977   1933.732
#> 
#> 
#> Scale parameter:
#>            Jokulsa Vatnsdalsa
#> Jokulsa     2017.0    1178.38
#> Vatnsdalsa  1178.4     439.88
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