- Email: [email protected]
Bivariate autoregressive models for the time series of significant wave height and mean period
Coastal Engineering 40 Ž2000. 297–311 www.elsevier.comrlocatercoastaleng
Bivariate autoregressive models for the time series of significant wave height and mean period C. Guedes Soares ) , C. Cunha Unit of Marine Technology and Engineering, Technical UniÕersity of Lisbon, Instituto Superior Tecnico, ´ AÕ. RoÕisco Pais, 1096 Lisbon, Portugal
Abstract This paper generalises the application of univariate models of the long-term time series of significant wave height to the case of the bivariate series of significant wave height and mean period. A brief review of the basic features of multivariate autoregressive models is presented, and then applications are made to the wave time series of Figueira da Foz, in Portugal. It is demonstrated that the simulated series from these models exhibit the correlation between the two parameters a feature that univariate series cannot reproduce. An application to two series of significant wave height from two neighbouring stations shows the applicability of this type of models to other type of correlated data sets. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Univariate models; Wave height; Figueira da Foz
1. Introduction Probabilistic models are essential for the design and for the operation planning of engineering systems. These models describe the average conditions to be expected during the time period of interest as well as their variability, both of which can be used for several predictions of engineering interest, such as, for example, extreme values during a certain time frame. For the design of statically dominated structures, like breakwaters, for example, an extreme value is normally the dominant design condition. However, for other types of
)
Corresponding author. Tel.: q351-21-8417607; fax: q351-21-8474015. E-mail address: [email protected] ŽC. Guedes Soares..
0378-3839r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 Ž 0 0 . 0 0 0 1 5 - 6
298
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
structures, like the floating ones, which are dynamically sensitive, this information is often not enough. Furthermore, for operational planning, the sequence of sea states is essential and models are required to incorporate that feature. Guedes Soares and Ferreira Ž1996., after reviewing previous applications of autoregressive models to wave climate, have demonstrated that linear autoregressive models can describe adequately time series of significant wave height at a site. These models have been applied to different sites of the North Sea and of the Portuguese coast and it was found that the same type of regression model could be applicable in the same geographical area. Models with orders up to 19 were required for a good modelling although the most significant coefficients were only up to the order 10 ŽGuedes Soares et al., 1996.. A study of non-linear autoregressive models has shown that the improved description provided by these models could only be noticed on the third and fourth statistical moments of the simulated series ŽGuedes Soares et al., 1998.. If one is interested in modelling only the first two moments of the series, as occurs in many situations, the linear models are perfectly adequate. However, in many situations, it is not enough to have a description of the significant wave height, but the associated mean period is also necessary. These two parameters of a sea state are positively correlated as demonstrated by several studies. In fact, several joint probability distributions have been proposed to describe their joint occurrence. In Rodriguez and Guedes Soares Ž1999., for example, several of the available models are compared for different types of sea states. The probability distributions describe the probability of occurrence at a random point in time of a sea state described by a given pair of values of significant wave height and mean period. The bivariate auto-regressive models presented here describe also the time evolution of the occurrence of such pair of values taking into account the memory effects that exist in the process.
2. Representing multivariate time series with ARMA models The main features of univariate autoregressive models of interest for modelling the time series of significant wave height have been described in Guedes Soares and Ferreira Ž1996., where a review of earlier literature can also be found. The main aspects of bivariate models are reviewed here but more exhaustive treatments can be found for example in Brockwell and Davis Ž1996.. The bivariate models describe jointly stationary stochastic processes. Two stochastic processes X t :t s 0,1,2, . . . 4 and Yt :t s 0,1,2, . . . 4 are said to be jointly stationary up to second order if each of them is a stationary stochastic process up to second order, which implies that their mean and the covariance should be stationary: E Ž Xt . s m x ,
t s 0,1,2, . . . ,
Ž 1a .
E Ž Yt . s m y ,
t s 0,1,2, . . . ,
Ž 1b .
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
cov Ž X t , X tqk . s gx x Ž k . , cov Ž Yt ,Ytqk . s gy y Ž k . ,
t s 0,1,2, . . . ; k s 0,1,2, . . . , t s 0,1,2, . . . ; k s 0,1,2, . . . ,
299
Ž 2a . Ž 2b.
where EŽ.. denotes expectation. Also, the cross-covariance covŽ X s ,Yt . exists and is only function of the difference of the time lag between s and t, cov Ž X t ,Ytqk . s gx y Ž k . ,
t s 0,1,2, . . . ; k s 0," 1," 2, . . .
Ž 3.
For these processes, one can define the corresponding cross-correlation function
(
r x y Ž k . s gx y Ž k . r gx x Ž 0 . ggg Ž 0 .
Ž 4.
which has the following properties:
Ž i.
< r x y Ž k . < F 1,
k s 0," 1," 2, . . .
Ž ii .
r x y Ž k . s r y x Ž yk . ,
Ž iii .
r x y Ž k . is not, in general, a symmetric function
k s 0," 1," 2, . . .
Ž 5a . Ž 5b . Ž 5c .
A k-dimensional succession X t s Ž X 1 t , X 2 t , . . . , X k t . constitutes a vector ARMAŽ p,q . process, usually denoted as VARMAŽ p,q ., when it is the solution of the equation F Ž B . Xt s Q Ž B . ´ t
Ž 6.
where F and Q are matrices, F Ž B . s I y F 1 B y F 2 B 2 y . . . yFp B p Q Ž B . s I y Qt B y Q 2 B 2 y . . . yQ q B q
Ž 7.
with Fr , r s 1,2, . . . p and Q s , s s 1,2, . . . q as k = k matrices with elements f iŽjr . and u iŽjs. and ´ s Ž ´ 1 t , ´ 2 t , . . . ´ k t . is a k-dimensional vector with average 0 s Ž0,0, . . . ,0. so that, E ´ t ,´Xu 4 s
½
0u/t Sust
5
Ž 8.
where S is a matrix of covariances of order k. A process VARMAŽ p,q . is stationary if the roots of the equation < F Ž B .< s 0 lie on the unit circle and it is invertible if < Q Ž B .< s 0 has all the roots outside of the unit circle. In the studies, already referred in the introduction, dealing with the modelling of univariate time series of significant wave height, Hs, it was shown that the first problems to deal with were the nonstationarity of the series and the gaps in the time series of observations. The problem of missing observations has been discussed in Hidalgo et al. Ž1995. where a method was proposed to deal with long gaps of observations, whenever there was information available from a neighbouring measuring station. In short, in these
300
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
cases transfer function models we developed to relate the two time series and afterwards the missing data was calculated from the complete series. In the periods where the lack of observations was simultaneous in the two stations, different situations can occur. If the gap is very small, simple interpolation is made. However, for larger gaps, forecasts and backcasts are made using autoregressive models from the two sides of the gap so as to fill them. After having the complete series, different transformations were considered, as described in Cunha and Guedes Soares Ž1999.. That work provides the justification for the transformation to be adopted here. This paper will adopt the approaches described in Hidalgo et al. Ž1995. for filling the gaps of missing observations and the transformations described in Cunha and Guedes Soares Ž1999., both of which were derived from univariate models and will be applied to the two series to be modelled. Bivariate models will then be adjusted to the data and simulations from these models will be used to assess their adequacy. 3. Univariate and bivariate models of the series from Figueira da Foz The period of observations of Tz available at Figueira da Foz is precisely the same studied before 1981–1989. The problem of the lack of observations remains in the study of the bivariate series meaning that the gaps are now in both time series. The study of the bivariate series will be conducted for a segment of the long-term series without missing observations, or, at least, with a small number of them, so that the existing gaps can be filled with linearly interpolated values, without interfering too much in the correlation structure of the series. The best period that could be found in those conditions is the one between the 0 h of 1 January 1989 and the 21 h of 15 December 1989, with 2792 observations, which are indicated in Fig. 1.
Fig. 1. Time series of Hs and Tz from January to December 1989 Žvertical axis is in meters for the series of Hs and in seconds for the series of Tz; the horizontal axis represents the sequence of observations 3 h apart.
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
301
Fig. 2. Time series of the monthly means of ln Ž Hs . and ln ŽTz . from June 1981 to March 1990.
In this study, the main objective of the model is to simulate the process under study. The transformation adopted to make the two series stationary, was adopted from the studies of Cunha and Guedes Soares Ž1999. in univariate series: Trf Ž Hst . s ln Ž Hst . y m ˆ M Žln H s t . , Trf Ž Tz t . s ln Ž Tz t . y m ˆ M ŽlnT z t . ,
t s 1,2, . . . , N t s 1,2, . . . , N
Ž 9. Ž 10 .
where m ˆ M lnŽ H s t . and mˆ M lnŽT z t . represent a Fourier model fitted to the series of monthly averages of lnŽ Hst . and lnŽTz t ..
Fig. 3. Autocorrelation ŽFAC. and partial autocorrelation functions ŽFACP. of the time series of monthly means of Hs and Tz Žconfidence limits are shown..
302
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 4. Periodogram of the time series of monthly means of Hs and Tz.
The earlier transformation studies have used only significant wave data and thus, postulating the same transformation for the period time series implies the assumption that this series has the same type of behaviour and periodicity. However, this needs to be demonstrated. Fig. 2 shows the monthly averages from June 1981 to March 1990, but only the series of April 1984–March 1990 was studied, because of the existence of too many gaps in the earlier segment, either of entire months or a very high number of gaps in the same month. The periodicity in the univariate series of the monthly averages of Hs and Tz was studied by analysing the autocorrelation functions ŽFig. 3. and the periodogram ŽFig. 4., which, for both series, show the same period of 12 months. This indicates that the same type of seasonality exists for the series of monthly averages of Hs and Tz. The models adjusted to each of the series are: m Ž 11a. ˆ M Žln H s t . s 0.632 q 0.194cos Ž v tX . y 0.341sin Ž v tX . q ´ t
m ˆ M ŽlnT z t . s 1.933 q 0.07cos Ž v tX . y 0.177sin Ž v tX . q ´ t X
Ž 11b.
where v s 2pr12 s 0.5236, t is the month corresponding to each observation t and ´ t represents the series of residuals associated to each of the models.
Fig. 5. Residuals of the models adjusted from the Fourier analysis.
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
303
Fig. 6. Original time series of the monthly means and Fourier models of ln Ž Hs . and ln ŽTz ..
White noise tests were performed on the residuals of these series ŽFig. 5.. The tests did not reject the hypothesis of normality for the monthly average series of Hs but they did so for the one of Tz. Therefore, a model ARŽ4. was adjusted to the residuals of the Fourier model of the monthly averages series of Tz: Tz s y0.23Tzy2 y 0.328Tzy4 q t t ,
t t l WN Ž 0,0.006 .
Ž 12 .
Having the two models of monthly averages of the two parameters ŽFig. 6., one can apply the transformations of the original series of Hs and Tz Žeach 3 h., as indicated in the equation. The transformed series of Hs and Tz are stationary and thus they can be modelled with bivariate and univariate ARMAŽ p,q . models ŽFig. 7.. First, two univariate models were adjusted separately to each of the two series. For the Hs series, with 1688 observations in the period previously mentioned, the most appropriate autoregressive model was an ARŽ5., which has the following coefficients and variance of the white noise residuals:
f 1 s 1.06,
f 2 s y0.05,
f 5 s y0.063 Var Ž WN . s 0.014
Ž 13 .
The next better model was an ARŽ24. with the following coefficients and variance of the white noise residuals:
f 1 s 1.05,
f 2 s y0.055,
f6 s y0.05,
f 10 s y0.031,
f 12 s 0.06,
f 13 s y0.031,
f 15 s y0.05,
f 16 s 0.103,
f 17 s y0.063,
f 20 s y0.027,
f 23 s 0.1,
Var Ž WN . s 0.0135
f 24 s y0.06,
Ž 14 .
304
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 7. A segment of the transformed series of Hs and Tz.
The quality of the model was assessed by the variance of the residuals but one can see that the difference is marginal and the ARŽ5. model can be preferred for most applications. For the time series of Tz, in the same period, an ARŽ2. model was adjusted with the following parameters: f 1 s 1.07, f 2 s y0.15 Var Ž WN . s 0.0063 Ž 15 . A shorter memory is clear in this process as compared with the significant wave height. Finally, a bivariate model was adjusted to the data. The one that was found to have the best compromise between the reduced number of parameters and the quality of the adjustment to the bivariate series Ž Hs,Tz . was an ARŽ4. with the following parameters: 1.087 y0.082 y0.121 0.162 f1 s , f2 s , y0.019 1.065 0.097 y0.181 0.072 y0.176 y0.079 0.045 f3 s , f4 s , y0.021 0.003 y0.04 0.015 0.0142 0.005 cov s , AICC s y6723.7 Ž 16 . 0.005 0.006
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
305
Fig. 8. One year of simulated series of Hs and Tz.
The series has also been adjusted by an ARŽ6. model, which resulted in:
f1
1.17 0.049
f3
y0.05682 0.02377
f5
0.01711 y0.012
y0.1286 , 1.016
f2
y0.03343 , y0.03632 0.03932 , 0.0651
y0.1757 y0.02645
f4
f6 s
0.1286 , 0.1061
0.07626 y0.01563
y0.08344 y0.00537
y0.1598 , y0.02925 0.07732 , y0.02719
AICCs y6691.7.
Ž 17 .
4. Simulations with bivariate models of the series from Figueira da Foz As indicated earlier, the main objective of modelling these series was to be able to simulate them. Thus, simulations have been made and their quality is assessed in the
Fig. 9. Autocorrelation function of simulated series of Tz and of the measured series in 1987 and 1988.
306
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 10. Cross-correlation between the univariate simulated series.
way they reproduce the second-order statistical properties of the process. These are in fact the properties that are the main assumptions for the application of the ARMA models. Simulations of autoregressive bivariate ARŽ4. and ARŽ6. and of univariate models were done, and the autocorrelation function of the resulting series was analysed. The simulated series with the univariate models ŽFig. 8., has an autocorrelation similar to the original time series. Fig. 9 shows a comparison with two different years of the original time series of Tz. It is apparent that the real data exhibit variability among different years and in this example, the autocorrelation of the simulated series lies between the autocorrelation of the original series in those years. When the transformation is inverted, the autocorrelation of the simulated series increases considerably in relation to the original one, which is due to the Fourier model of the monthly average series, which has a very high autocorrelation. The univariate simulations of Hs and Tz do not have the same cross-correlation as the original series. Since they are simulated separately one from the other, there is no information in the model about their correlation ŽFig. 10.. Therefore, only a bivariate model will be able to represent the existing correlation structure between Hs and Tz.
Fig. 11. Simulation with the bivariate model ARŽ4..
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 12. Simulation with the bivariate model ARŽ6..
Fig. 13. Autocorrelation function of simulated series of Hs Žwith bivariate models..
Fig. 14. Cross-correlation between the simulated stationary series of Hs and Tz.
307
308
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 15. Cross-correlation between the simulated nonstationary series of Hs and Tz.
The simulated series with bivariate models have the same behaviour as the simulated with the univariate models in relation to the autocorrelation, Fig. 11, which was obtained from univariate series also representative of the results obtained with bivariate models ŽFig. 12.. Fig. 13 shows the autocorrelation function of the Hs series simulated with two bivariate models composed with the ones of the years of 1987 and 1988. Where the bivariate models show a better performance is in the cross-correlation function, which they are able to reproduce as shown in Fig. 14, which represents a clear improvement over the results of Fig. 9. In the simulated series, with the transformation inverted Žseries nonstationary., the cross-correlation is maintained higher as the lags are increasing ŽFig. 15..
5. Application of bivariate models to other types of series An important application of bivariate models is the series of Hs and Tz, which are the pair of parameters required for the specification of the representative sea states for analysis of many types of marine systems. However, the same type of models can be applied to any other pair of variable that exhibits correlation. An interesting example of application can be the series of Hs in two neighbouring measurement stations. This information was used in Hidalgo et al. Ž1995. to fill the gaps in one series with information from the other one. In that example, the series of Hs from Figueira da Foz and La Corunha ŽFigs. 16 and 17. were used. Since the correlation between them is high ŽFig. 18., bivariate models were also applied in modelling those series. The transformation applied to these series was the following Yt s Ž 1 y B . X t leading to the series shown in Fig. 19.
Ž 18 .
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 16. A time series of Hs from Figueira da Foz.
Fig. 17. A time series of Hs from Corunha.
Fig. 18. Cross-correlation between Figueira da Foz and Corunha.
309
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
310
Fig. 19. Transformed y time series of Hs from Figueira da Foz.
The bivariate model adjusted to the series of Hs ŽFigueira, Corunha. was a VARŽ5. given by the following coefficients:
f1 s
y0.83 y0.052
0.946 , 0.11
f3 s
y0.463 y0.045
0.57 , y0.044
f5 s
y0.0923 y0.025
f2
0.14 , y0.105
y0.646 y0.041
f4 s
0.731 , 0.085
y0.322 y0.856
cov s
0.0886 0.096
Ž 19 .
0.335 , y0.028 0.096 , 0.1865
AICCs 1094
6. Conclusions The need to generalise the univariate autoregressive models of significant wave height to bivariate ones arises whenever it is necessary to model simultaneously the average period of the sea states. Multivariate models and in particular the bivariate ones used in the present applications capture the second-order statistical information of each time series as well as the cross-correlation between the series. The application of these models to the series of Hs and Tz was demonstrated using data from Figueira da Foz and it was shown that the correlation was preserved in the simulated series with these models — a feature that two univariate models was not capable of reproducing. It was also shown with an example of two Hs series from neighbouring stations that the bivariate models can also be applied to other type of situations, in which the univariate series exhibit some degree of correlation.
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
311
References Brockwell, P.J., Davis, R.A., 1996. Introduction to Time Series and Forecasting. Springer, New York. Cunha, C., Guedes Soares, C., 1999. On the choice of data transformation for modelling time series of significant wave height. Ocean Engineering 26, 489–506. Guedes Soares, C., Ferreira, A.M., 1996. Representation of non-stationary time series of significant wave height with autoregressive models. Probabilistic Engineering Mechanics 11, 139–148. Guedes Soares, C., Ferreira, A.M., Cunha, C., 1996. Linear models of the time series of significant wave height in the southwest coast of Portugal. Coastal Engineering 29, 149–167. Guedes Soares, C., Scotto, M., Cavaco, P., 1998. Linear and non-linear models of long-term time series of wave data. In: Proceedings of the 17th International Conference on Offshore Mechanics and Arctic Engineering. ASME, New York, pp. 98–1493. Hidalgo, O., Nieto, J.C., Cunha, C., Guedes Soares, C., 1995. Filling missing observations in time series of significant wave height. In: Guedes Soares, C. ŽEd.., Proceedings of the 14th International Conference on Offshore Mechanics and Arctic Engineering vol. II ASME, New York, pp. 9–17. Rodriguez, G.R., Guedes Soares, C., 1999. The bivariate distribution of wave heights and periods in the mixed sea states. J. Offshore Mechanics Arctic Engineering 121, 102–108.
Bivariate autoregressive models for the time series of significant wave height and mean period C. Guedes Soares ) , C. Cunha Unit of Marine Technology and Engineering, Technical UniÕersity of Lisbon, Instituto Superior Tecnico, ´ AÕ. RoÕisco Pais, 1096 Lisbon, Portugal
Abstract This paper generalises the application of univariate models of the long-term time series of significant wave height to the case of the bivariate series of significant wave height and mean period. A brief review of the basic features of multivariate autoregressive models is presented, and then applications are made to the wave time series of Figueira da Foz, in Portugal. It is demonstrated that the simulated series from these models exhibit the correlation between the two parameters a feature that univariate series cannot reproduce. An application to two series of significant wave height from two neighbouring stations shows the applicability of this type of models to other type of correlated data sets. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Univariate models; Wave height; Figueira da Foz
1. Introduction Probabilistic models are essential for the design and for the operation planning of engineering systems. These models describe the average conditions to be expected during the time period of interest as well as their variability, both of which can be used for several predictions of engineering interest, such as, for example, extreme values during a certain time frame. For the design of statically dominated structures, like breakwaters, for example, an extreme value is normally the dominant design condition. However, for other types of
)
Corresponding author. Tel.: q351-21-8417607; fax: q351-21-8474015. E-mail address: [email protected] ŽC. Guedes Soares..
0378-3839r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 Ž 0 0 . 0 0 0 1 5 - 6
298
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
structures, like the floating ones, which are dynamically sensitive, this information is often not enough. Furthermore, for operational planning, the sequence of sea states is essential and models are required to incorporate that feature. Guedes Soares and Ferreira Ž1996., after reviewing previous applications of autoregressive models to wave climate, have demonstrated that linear autoregressive models can describe adequately time series of significant wave height at a site. These models have been applied to different sites of the North Sea and of the Portuguese coast and it was found that the same type of regression model could be applicable in the same geographical area. Models with orders up to 19 were required for a good modelling although the most significant coefficients were only up to the order 10 ŽGuedes Soares et al., 1996.. A study of non-linear autoregressive models has shown that the improved description provided by these models could only be noticed on the third and fourth statistical moments of the simulated series ŽGuedes Soares et al., 1998.. If one is interested in modelling only the first two moments of the series, as occurs in many situations, the linear models are perfectly adequate. However, in many situations, it is not enough to have a description of the significant wave height, but the associated mean period is also necessary. These two parameters of a sea state are positively correlated as demonstrated by several studies. In fact, several joint probability distributions have been proposed to describe their joint occurrence. In Rodriguez and Guedes Soares Ž1999., for example, several of the available models are compared for different types of sea states. The probability distributions describe the probability of occurrence at a random point in time of a sea state described by a given pair of values of significant wave height and mean period. The bivariate auto-regressive models presented here describe also the time evolution of the occurrence of such pair of values taking into account the memory effects that exist in the process.
2. Representing multivariate time series with ARMA models The main features of univariate autoregressive models of interest for modelling the time series of significant wave height have been described in Guedes Soares and Ferreira Ž1996., where a review of earlier literature can also be found. The main aspects of bivariate models are reviewed here but more exhaustive treatments can be found for example in Brockwell and Davis Ž1996.. The bivariate models describe jointly stationary stochastic processes. Two stochastic processes X t :t s 0,1,2, . . . 4 and Yt :t s 0,1,2, . . . 4 are said to be jointly stationary up to second order if each of them is a stationary stochastic process up to second order, which implies that their mean and the covariance should be stationary: E Ž Xt . s m x ,
t s 0,1,2, . . . ,
Ž 1a .
E Ž Yt . s m y ,
t s 0,1,2, . . . ,
Ž 1b .
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
cov Ž X t , X tqk . s gx x Ž k . , cov Ž Yt ,Ytqk . s gy y Ž k . ,
t s 0,1,2, . . . ; k s 0,1,2, . . . , t s 0,1,2, . . . ; k s 0,1,2, . . . ,
299
Ž 2a . Ž 2b.
where EŽ.. denotes expectation. Also, the cross-covariance covŽ X s ,Yt . exists and is only function of the difference of the time lag between s and t, cov Ž X t ,Ytqk . s gx y Ž k . ,
t s 0,1,2, . . . ; k s 0," 1," 2, . . .
Ž 3.
For these processes, one can define the corresponding cross-correlation function
(
r x y Ž k . s gx y Ž k . r gx x Ž 0 . ggg Ž 0 .
Ž 4.
which has the following properties:
Ž i.
< r x y Ž k . < F 1,
k s 0," 1," 2, . . .
Ž ii .
r x y Ž k . s r y x Ž yk . ,
Ž iii .
r x y Ž k . is not, in general, a symmetric function
k s 0," 1," 2, . . .
Ž 5a . Ž 5b . Ž 5c .
A k-dimensional succession X t s Ž X 1 t , X 2 t , . . . , X k t . constitutes a vector ARMAŽ p,q . process, usually denoted as VARMAŽ p,q ., when it is the solution of the equation F Ž B . Xt s Q Ž B . ´ t
Ž 6.
where F and Q are matrices, F Ž B . s I y F 1 B y F 2 B 2 y . . . yFp B p Q Ž B . s I y Qt B y Q 2 B 2 y . . . yQ q B q
Ž 7.
with Fr , r s 1,2, . . . p and Q s , s s 1,2, . . . q as k = k matrices with elements f iŽjr . and u iŽjs. and ´ s Ž ´ 1 t , ´ 2 t , . . . ´ k t . is a k-dimensional vector with average 0 s Ž0,0, . . . ,0. so that, E ´ t ,´Xu 4 s
½
0u/t Sust
5
Ž 8.
where S is a matrix of covariances of order k. A process VARMAŽ p,q . is stationary if the roots of the equation < F Ž B .< s 0 lie on the unit circle and it is invertible if < Q Ž B .< s 0 has all the roots outside of the unit circle. In the studies, already referred in the introduction, dealing with the modelling of univariate time series of significant wave height, Hs, it was shown that the first problems to deal with were the nonstationarity of the series and the gaps in the time series of observations. The problem of missing observations has been discussed in Hidalgo et al. Ž1995. where a method was proposed to deal with long gaps of observations, whenever there was information available from a neighbouring measuring station. In short, in these
300
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
cases transfer function models we developed to relate the two time series and afterwards the missing data was calculated from the complete series. In the periods where the lack of observations was simultaneous in the two stations, different situations can occur. If the gap is very small, simple interpolation is made. However, for larger gaps, forecasts and backcasts are made using autoregressive models from the two sides of the gap so as to fill them. After having the complete series, different transformations were considered, as described in Cunha and Guedes Soares Ž1999.. That work provides the justification for the transformation to be adopted here. This paper will adopt the approaches described in Hidalgo et al. Ž1995. for filling the gaps of missing observations and the transformations described in Cunha and Guedes Soares Ž1999., both of which were derived from univariate models and will be applied to the two series to be modelled. Bivariate models will then be adjusted to the data and simulations from these models will be used to assess their adequacy. 3. Univariate and bivariate models of the series from Figueira da Foz The period of observations of Tz available at Figueira da Foz is precisely the same studied before 1981–1989. The problem of the lack of observations remains in the study of the bivariate series meaning that the gaps are now in both time series. The study of the bivariate series will be conducted for a segment of the long-term series without missing observations, or, at least, with a small number of them, so that the existing gaps can be filled with linearly interpolated values, without interfering too much in the correlation structure of the series. The best period that could be found in those conditions is the one between the 0 h of 1 January 1989 and the 21 h of 15 December 1989, with 2792 observations, which are indicated in Fig. 1.
Fig. 1. Time series of Hs and Tz from January to December 1989 Žvertical axis is in meters for the series of Hs and in seconds for the series of Tz; the horizontal axis represents the sequence of observations 3 h apart.
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
301
Fig. 2. Time series of the monthly means of ln Ž Hs . and ln ŽTz . from June 1981 to March 1990.
In this study, the main objective of the model is to simulate the process under study. The transformation adopted to make the two series stationary, was adopted from the studies of Cunha and Guedes Soares Ž1999. in univariate series: Trf Ž Hst . s ln Ž Hst . y m ˆ M Žln H s t . , Trf Ž Tz t . s ln Ž Tz t . y m ˆ M ŽlnT z t . ,
t s 1,2, . . . , N t s 1,2, . . . , N
Ž 9. Ž 10 .
where m ˆ M lnŽ H s t . and mˆ M lnŽT z t . represent a Fourier model fitted to the series of monthly averages of lnŽ Hst . and lnŽTz t ..
Fig. 3. Autocorrelation ŽFAC. and partial autocorrelation functions ŽFACP. of the time series of monthly means of Hs and Tz Žconfidence limits are shown..
302
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 4. Periodogram of the time series of monthly means of Hs and Tz.
The earlier transformation studies have used only significant wave data and thus, postulating the same transformation for the period time series implies the assumption that this series has the same type of behaviour and periodicity. However, this needs to be demonstrated. Fig. 2 shows the monthly averages from June 1981 to March 1990, but only the series of April 1984–March 1990 was studied, because of the existence of too many gaps in the earlier segment, either of entire months or a very high number of gaps in the same month. The periodicity in the univariate series of the monthly averages of Hs and Tz was studied by analysing the autocorrelation functions ŽFig. 3. and the periodogram ŽFig. 4., which, for both series, show the same period of 12 months. This indicates that the same type of seasonality exists for the series of monthly averages of Hs and Tz. The models adjusted to each of the series are: m Ž 11a. ˆ M Žln H s t . s 0.632 q 0.194cos Ž v tX . y 0.341sin Ž v tX . q ´ t
m ˆ M ŽlnT z t . s 1.933 q 0.07cos Ž v tX . y 0.177sin Ž v tX . q ´ t X
Ž 11b.
where v s 2pr12 s 0.5236, t is the month corresponding to each observation t and ´ t represents the series of residuals associated to each of the models.
Fig. 5. Residuals of the models adjusted from the Fourier analysis.
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
303
Fig. 6. Original time series of the monthly means and Fourier models of ln Ž Hs . and ln ŽTz ..
White noise tests were performed on the residuals of these series ŽFig. 5.. The tests did not reject the hypothesis of normality for the monthly average series of Hs but they did so for the one of Tz. Therefore, a model ARŽ4. was adjusted to the residuals of the Fourier model of the monthly averages series of Tz: Tz s y0.23Tzy2 y 0.328Tzy4 q t t ,
t t l WN Ž 0,0.006 .
Ž 12 .
Having the two models of monthly averages of the two parameters ŽFig. 6., one can apply the transformations of the original series of Hs and Tz Žeach 3 h., as indicated in the equation. The transformed series of Hs and Tz are stationary and thus they can be modelled with bivariate and univariate ARMAŽ p,q . models ŽFig. 7.. First, two univariate models were adjusted separately to each of the two series. For the Hs series, with 1688 observations in the period previously mentioned, the most appropriate autoregressive model was an ARŽ5., which has the following coefficients and variance of the white noise residuals:
f 1 s 1.06,
f 2 s y0.05,
f 5 s y0.063 Var Ž WN . s 0.014
Ž 13 .
The next better model was an ARŽ24. with the following coefficients and variance of the white noise residuals:
f 1 s 1.05,
f 2 s y0.055,
f6 s y0.05,
f 10 s y0.031,
f 12 s 0.06,
f 13 s y0.031,
f 15 s y0.05,
f 16 s 0.103,
f 17 s y0.063,
f 20 s y0.027,
f 23 s 0.1,
Var Ž WN . s 0.0135
f 24 s y0.06,
Ž 14 .
304
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 7. A segment of the transformed series of Hs and Tz.
The quality of the model was assessed by the variance of the residuals but one can see that the difference is marginal and the ARŽ5. model can be preferred for most applications. For the time series of Tz, in the same period, an ARŽ2. model was adjusted with the following parameters: f 1 s 1.07, f 2 s y0.15 Var Ž WN . s 0.0063 Ž 15 . A shorter memory is clear in this process as compared with the significant wave height. Finally, a bivariate model was adjusted to the data. The one that was found to have the best compromise between the reduced number of parameters and the quality of the adjustment to the bivariate series Ž Hs,Tz . was an ARŽ4. with the following parameters: 1.087 y0.082 y0.121 0.162 f1 s , f2 s , y0.019 1.065 0.097 y0.181 0.072 y0.176 y0.079 0.045 f3 s , f4 s , y0.021 0.003 y0.04 0.015 0.0142 0.005 cov s , AICC s y6723.7 Ž 16 . 0.005 0.006
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
305
Fig. 8. One year of simulated series of Hs and Tz.
The series has also been adjusted by an ARŽ6. model, which resulted in:
f1
1.17 0.049
f3
y0.05682 0.02377
f5
0.01711 y0.012
y0.1286 , 1.016
f2
y0.03343 , y0.03632 0.03932 , 0.0651
y0.1757 y0.02645
f4
f6 s
0.1286 , 0.1061
0.07626 y0.01563
y0.08344 y0.00537
y0.1598 , y0.02925 0.07732 , y0.02719
AICCs y6691.7.
Ž 17 .
4. Simulations with bivariate models of the series from Figueira da Foz As indicated earlier, the main objective of modelling these series was to be able to simulate them. Thus, simulations have been made and their quality is assessed in the
Fig. 9. Autocorrelation function of simulated series of Tz and of the measured series in 1987 and 1988.
306
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 10. Cross-correlation between the univariate simulated series.
way they reproduce the second-order statistical properties of the process. These are in fact the properties that are the main assumptions for the application of the ARMA models. Simulations of autoregressive bivariate ARŽ4. and ARŽ6. and of univariate models were done, and the autocorrelation function of the resulting series was analysed. The simulated series with the univariate models ŽFig. 8., has an autocorrelation similar to the original time series. Fig. 9 shows a comparison with two different years of the original time series of Tz. It is apparent that the real data exhibit variability among different years and in this example, the autocorrelation of the simulated series lies between the autocorrelation of the original series in those years. When the transformation is inverted, the autocorrelation of the simulated series increases considerably in relation to the original one, which is due to the Fourier model of the monthly average series, which has a very high autocorrelation. The univariate simulations of Hs and Tz do not have the same cross-correlation as the original series. Since they are simulated separately one from the other, there is no information in the model about their correlation ŽFig. 10.. Therefore, only a bivariate model will be able to represent the existing correlation structure between Hs and Tz.
Fig. 11. Simulation with the bivariate model ARŽ4..
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 12. Simulation with the bivariate model ARŽ6..
Fig. 13. Autocorrelation function of simulated series of Hs Žwith bivariate models..
Fig. 14. Cross-correlation between the simulated stationary series of Hs and Tz.
307
308
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 15. Cross-correlation between the simulated nonstationary series of Hs and Tz.
The simulated series with bivariate models have the same behaviour as the simulated with the univariate models in relation to the autocorrelation, Fig. 11, which was obtained from univariate series also representative of the results obtained with bivariate models ŽFig. 12.. Fig. 13 shows the autocorrelation function of the Hs series simulated with two bivariate models composed with the ones of the years of 1987 and 1988. Where the bivariate models show a better performance is in the cross-correlation function, which they are able to reproduce as shown in Fig. 14, which represents a clear improvement over the results of Fig. 9. In the simulated series, with the transformation inverted Žseries nonstationary., the cross-correlation is maintained higher as the lags are increasing ŽFig. 15..
5. Application of bivariate models to other types of series An important application of bivariate models is the series of Hs and Tz, which are the pair of parameters required for the specification of the representative sea states for analysis of many types of marine systems. However, the same type of models can be applied to any other pair of variable that exhibits correlation. An interesting example of application can be the series of Hs in two neighbouring measurement stations. This information was used in Hidalgo et al. Ž1995. to fill the gaps in one series with information from the other one. In that example, the series of Hs from Figueira da Foz and La Corunha ŽFigs. 16 and 17. were used. Since the correlation between them is high ŽFig. 18., bivariate models were also applied in modelling those series. The transformation applied to these series was the following Yt s Ž 1 y B . X t leading to the series shown in Fig. 19.
Ž 18 .
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
Fig. 16. A time series of Hs from Figueira da Foz.
Fig. 17. A time series of Hs from Corunha.
Fig. 18. Cross-correlation between Figueira da Foz and Corunha.
309
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
310
Fig. 19. Transformed y time series of Hs from Figueira da Foz.
The bivariate model adjusted to the series of Hs ŽFigueira, Corunha. was a VARŽ5. given by the following coefficients:
f1 s
y0.83 y0.052
0.946 , 0.11
f3 s
y0.463 y0.045
0.57 , y0.044
f5 s
y0.0923 y0.025
f2
0.14 , y0.105
y0.646 y0.041
f4 s
0.731 , 0.085
y0.322 y0.856
cov s
0.0886 0.096
Ž 19 .
0.335 , y0.028 0.096 , 0.1865
AICCs 1094
6. Conclusions The need to generalise the univariate autoregressive models of significant wave height to bivariate ones arises whenever it is necessary to model simultaneously the average period of the sea states. Multivariate models and in particular the bivariate ones used in the present applications capture the second-order statistical information of each time series as well as the cross-correlation between the series. The application of these models to the series of Hs and Tz was demonstrated using data from Figueira da Foz and it was shown that the correlation was preserved in the simulated series with these models — a feature that two univariate models was not capable of reproducing. It was also shown with an example of two Hs series from neighbouring stations that the bivariate models can also be applied to other type of situations, in which the univariate series exhibit some degree of correlation.
C. Guedes Soares, C. Cunhar Coastal Engineering 40 (2000) 297–311
311
References Brockwell, P.J., Davis, R.A., 1996. Introduction to Time Series and Forecasting. Springer, New York. Cunha, C., Guedes Soares, C., 1999. On the choice of data transformation for modelling time series of significant wave height. Ocean Engineering 26, 489–506. Guedes Soares, C., Ferreira, A.M., 1996. Representation of non-stationary time series of significant wave height with autoregressive models. Probabilistic Engineering Mechanics 11, 139–148. Guedes Soares, C., Ferreira, A.M., Cunha, C., 1996. Linear models of the time series of significant wave height in the southwest coast of Portugal. Coastal Engineering 29, 149–167. Guedes Soares, C., Scotto, M., Cavaco, P., 1998. Linear and non-linear models of long-term time series of wave data. In: Proceedings of the 17th International Conference on Offshore Mechanics and Arctic Engineering. ASME, New York, pp. 98–1493. Hidalgo, O., Nieto, J.C., Cunha, C., Guedes Soares, C., 1995. Filling missing observations in time series of significant wave height. In: Guedes Soares, C. ŽEd.., Proceedings of the 14th International Conference on Offshore Mechanics and Arctic Engineering vol. II ASME, New York, pp. 9–17. Rodriguez, G.R., Guedes Soares, C., 1999. The bivariate distribution of wave heights and periods in the mixed sea states. J. Offshore Mechanics Arctic Engineering 121, 102–108.