Temporal Dependence of Monthly TPN and DO at Station C70

The TPN and DO at station C70 are chosen for the study of temporal dependence. Table 12.1 lists the dataset for both temporal and spatial dependence study. We will use the water quality data before 2012 to build a copula-based Markov process model, and the water quality data of 2012 and 2013 will be used for model validation purpose.

In general, before we proceed to investigate the temporal dependence using copulas, we first evaluate whether there exists periodicity (or seasonality) in the sequence. For monthly TPN and DO, we suspect that there should exist seasonality. We can use the sample autocorrelation function plot or cumulative periodogram through spectral analysis (Box et al., 2007) to assess the seasonality.

The sample autocorrelation coefficient [γ_k]γk for time series x_txt at lag k can be written as follows:

ck=1N∑t=1N−kxt−x¯xt+k−x¯(12.1a)

γk=ckc0;c0=1N∑t=1Nxt−x¯2(12.1b)

The cumulative periodogram [C(f_k)Cfk] for time series x_txt can be written as follows:

Ifj=2n∑t=1Nxt−2πifjt=2N∑t=1Nxtcos2πfjt2+∑t=1Nxtsin2πfjt212(12.2a)

Cfk=∑j=1kIfjNσx2̂(12.2b)

In Equations (12.2a) and (12.2b), I(f_j)Ifj stands for the periodogram function; fj=jN,j=1,…⌊N2⌋; σx2̂ is the estimated variance for the time series.

Applying Equations (12.1) and (12.2), Figure 12.3 plots the sample autocorrelation function and cumulative periodogram for the TPN and DO at station C70. From the sample autocorrelation function plots in Figure 12.1, we clearly see that both DO and TPN have a 12-month cycle. From the cumulative periodogram plot for TPN at C70, we notice a discontinuity at frequency f=0.0833≈112. The discontinuity of cumulative periodogram indicates the existence of periodicity (or seasonality). From the cumulative periodogram plot for DO at C70, again we see the discontinuity at the same frequency as that of TPN; we see another very small discontinuity at frequency f = 0.1667≈1/6f=0.1667≈1/6, which means six-month period may also exist for the DO sequence. Comparatively speaking, the six-month subcycle is not significant, and we will only deal with the dominating 12-month periodicity for both TPN and DO sequences.

Figure 12.3 Autocorrelation and cumulative periodogram plots for original monthly TPN and DO series.

To remove the periodicity, we will introduce a simple but effective method (called the full deseasonalization method). For our monthly water quality study, we will actually remove the monthly average and monthly standard deviation from the water quality time series using the following:

xr,mde−season=xr,m−μ̂mσ̂m,m=1,2,…S(12.3)

In this case study, we have S = 12S=12 to show that we have monthly period. After applying Equation (12.3), we can then use the deseasonalized sequence to reevaluate whether the periodicity has been successfully removed as shown in Figure 12.4. As seen in Figure 12.4, the periodicity has been successfully removed. Table 12.2 tabulates the monthly sample mean and sample standard deviation for TPN and DO time series, respectively.

Figure 12.4 Autocorrelation and cumulative periodogram plots for deseasonalized TPN and DO series.

Table 12.2. Monthly sample mean and standard deviation of TPN and DO series.

Month	TPN (mg/L)		DO (mg/L)
	μ̂	σ̂	μ̂	σ̂
January	0.26	0.06	12.94	0.36
February	0.22	0.07	12.87	0.47
March	0.21	0.05	12.67	0.46
April	0.16	0.04	12.31	0.33
May	0.10	0.03	12.10	0.35
June	0.07	0.02	11.50	0.44
July	0.07	0.02	10.65	0.48
August	0.09	0.04	10.09	0.58
September	0.14	0.04	10.48	0.53
October	0.21	0.07	11.28	0.52
November	0.24	0.07	12.21	0.82
December	0.26	0.06	12.71	0.47

With the successful removal of periodicity, we can now proceed to study the temporal dependence using the copula-based Markov process. As stated in Chapter 9, with the application of the copula-based Markov process, the time series does not need to belong or transform to the Gaussian process. In addition, the marginals and serial dependence can be studied separately to avoid possible misidentification. Following the discussion in Sections 9.3–9.5, we will illustrate the application of the copula-based Markov process to the water quality time series. As stated in Chapter 9, the procedure involved for the copula-based Markov process is as follows:

1. 
   

   i. Identify the Markov order for the stationary time series.

   
   
2. 
   

   ii. Investigate the marginal distribution of the Markov process.

   
   
3. 
   

   iii. Study the serial dependence using copula.

   
   
4. 
   

   iv. Perform one-step ahead forecasting with the copula-based Markov process.

Identification of the Proper Markov Order for the Deseasonalized TPN and DO Time Series

The Markov order will be identified using the method discussed in Section 9.5.2. The meta-Gaussian copula is applied as the building block for the order identification purpose only. The kernel density method is applied to estimate the marginals nonparametrically. Following the order identification procedure, we obtain that the deseasonalized TPN and DO may be modeled using the first- and second-order Markov process, respectively (as listed in Table 12.3).

With the identified Markov order, we can move on to choose the best-fitted copula functions. For the deseasonalized TPN series, the most common bivariate copulas (i.e., Gumbel, meta-Gaussian, meta-Student t, and Frank) will be selected as the candidates. For the deseasonlized DO series, the D-vine copula application to time series discussed in Chapter 9 will be selected. The pseudo-MLE discussed in Section 9.5.3 is applied for parameter estimation with the use of empirical distribution estimated from kernel densities. To illustrate the empirical distribution with the use of kernel density, we selected a simple Gaussian kernel with the bandwidth of 0.3097 and 0.3507 for deseasonalized TPN and DO, respectively. As shown in Figure 12.5, the kernel density fits the histogram very well. The CDF computed from kernel density also fits the empirical CDF computed with the use of Weibull plotting-position formula very well. Figure 12.5 verifies that the kernel density may be applied to model the marginal distribution of time series.

Figure 12.5 Plots of deseasonalized TPN and DO time series, kernel density, as well as the CDF computed from kernel density.

Parameter Estimation for the Deseasonalized TPN and DO Series

Monthly TPN and DO Simulation and Forecast

Deseasonalized TPN Series

The simulation method discussed in Section 9.4.3 is applied to the first-order TPN series. Likewise, Section 9.4.4 is applied for the one-step ahead median and VaR forecasts. Using one simple example, we will show the inversion of simulated variate in the frequency domain back to the real domain.

Suppose that we simulated U_June = 0.8UJune=0.8 from the Guassian copula fitted to the first-order deseasonalized TPN series. Looking up the empirical CDF computed from the kernel density function, we see the simulated U_June = 0.8UJune=0.8 is bounded by [CDF, TPN] in {[0.787, 0.761], [0.804, 0.844]}. Applying the interpolation, we compute the simulated deseasonalized TPN as follows:

TPNsimdeseason=0.761+0.844−0.7610.804−0.7870.8−0.787=0.8245.

Adding back the monthly average and standard deviation for the month of June, we can compute the simulated TPN of June as follows:

TPN_sim = 0.8245(0.0175) + 0.0666 = 0.0811 mg/L

TPNsim=0.82450.0175+0.0666=0.0811mg/L.

Applying the one-step ahead forecast discussed in Example 9.3, we can proceed with the median forecast as well as the 95% and 5% VaR. To compute the VaR, Equation (9.22) can be rewritten as follows:

Zt+195%=Fn−1(CFnzt+1∣Fnzt0.95Fnzt;α̂(12.4a)

Zt+15%=Fn−1(CFnzt+1∣Fnzt0.05Fnzt;α̂(12.4b)

Figure 12.6 plots the comparison of simulated monthly TPN with the observed TPN. It also plots the forecasted monthly TPN, its 5% and 95% VaR versus the observed monthly TPN. Figure 12.6 indicates that (a) the simulated deseasonal TPN from the fitted Gaussian copulas well presents the lag-1 temporal dependence compared to the observed deseasonal TPN series; (b) simulated monthly TPN also well presents the dependence of the observed monthly TPN series; (c) the one-step ahead monthly TPN forecast captured the main trend of monthly TPN; and (d) though there is an obvious error for the extreme TPN values, the VaR values may help identify these extreme values. The forecasted and VaR values are listed in Table 12.6.

Figure 12.6 Simulations of deseasonal monthly, monthly TPN, and monthly TPN forecast with 95% and 5% VaRs.

Table 12.6. Forecast and VaR results computed from the fitted copula-based Markov model.

Date	TPN (mg/L)				DO (mg/L)
	Observed	Forecast	5%VaR	95%VaR	Observed	forecast	5%VaR	95%VaR
12-Jan	0.356	0.253	0.176	0.366	13.2	13.29	12.58	13.80
12-Feb	0.191	0.242	0.141	0.382	12.8	13.12	12.37	13.84
12-Mar	0.233	0.195	0.135	0.284	12.8	12.69	12.02	13.47
12-Apr	0.123	0.165	0.115	0.236	12.83	12.29	11.81	12.84
12-May	0.115	0.089	0.058	0.136	12.73	12.24	11.64	12.81
12-Jun	0.094	0.068	0.044	0.101	12.2	11.86	11.04	12.50
12-Jul	0.041	0.080	0.046	0.126	11.2	11.02	10.17	11.70
12-Aug	0.061	0.072	0.021	0.146	10.8	10.40	9.46	11.28
12-Sep	0.081	0.124	0.076	0.195	10.8	10.71	9.84	11.55
12-Oct	0.21	0.175	0.091	0.298	11.6	11.44	10.64	12.28
12-Nov	0.227	0.231	0.144	0.359	12.6	12.35	11.11	13.69
12-Dec	0.42	0.256	0.182	0.366	12	12.78	12.08	13.55
13-Jan	0.337	0.297	0.207	0.420	13.3	12.87	12.34	13.51
13-Feb	0.244	0.237	0.137	0.375	12.5	12.76	12.03	13.58
13-Mar	0.175	0.205	0.142	0.295	12.8	12.67	12.01	13.46
13-Apr	0.171	0.151	0.105	0.221	12.2	12.25	11.77	12.80
13-May	0.067	0.100	0.066	0.148	11.8	12.08	11.57	12.67
13-Jun	0.058	0.058	0.037	0.089	11.4	11.38	10.75	12.14
13-Jul	0.084	0.067	0.037	0.111	9.8	10.50	9.82	11.31
13-Aug	0.226	0.095	0.040	0.175	9.6	9.89	9.05	10.90
13-Sep	0.177	0.163	0.104	0.244	10	10.16	9.38	11.04

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