Doubly stochastic Poisson pulse model for finescale rainfall
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Abstract
Stochastic rainfall models are widely used in hydrological studies because they provide a framework not only for deriving information about the characteristics of rainfall but also for generating precipitation inputs to simulation models whenever data are not available. A stochastic point process model based on a class of doubly stochastic Poisson processes is proposed to analyse finescale point rainfall time series. In this model, rain cells arrive according to a doubly stochastic Poisson process whose arrival rate is determined by a finitestate Markov chain. Each rain cell has a random lifetime. During the lifetime of each rain cell, instantaneous random depths of rainfall bursts (pulses) occur according to a Poisson process. The covariance structure of the point process of pulse occurrences is studied. Moment properties of the time series of accumulated rainfall in discrete time intervals are derived to model 5min rainfall data, over a period of 69 years, from Germany. Secondmoment as well as thirdmoment properties of the rainfall are considered. The results show that the proposed model is capable of reproducing rainfall properties well at various subhourly resolutions. Incorporation of thirdorder moment properties in estimation showed a clear improvement in fitting. A good fit to the extremes is found at larger resolutions, both at 12h and 24h levels, despite underestimation at 5min aggregation. The proportion of dry intervals is studied by comparing the proportion of time intervals, from the observed and simulated data, with rainfall depth below small thresholds. A good agreement was found at 5min aggregation and for larger aggregation levels a closer fit was obtained when the threshold was increased. A simulation study is presented to assess the performance of the estimation method.
Keywords
Moment properties Point process Precipitation Rainfall generator Stochastic models Subhourly rainfall1 Introduction
An important challenge we face in environmental or ecological impact studies is to provide fast and realistic simulations of atmospheric variables such as rainfall at various temporal scales. Stochastic point process models provide a basis for generating synthetic rainfall input to hydrological models where the observed data at the required temporal scale are not available. They also enable us to assess the risk associated with hydrological systems. There has been a number of stochastic point process models developed by many authors over the years. Among these, the models based on Poisson cluster processes (RodriguezIturbe et al. 1987; Cowpertwait 1994; Onof and Wheater 1994; Wheater et al. 2005) utilizing either the NeymanScott or BartlettLewis processes have received much attention, since their model structure reflects well the climatological features of the rainfall generating mechanism. A good review of developments in modelling rainfall using Poisson cluster processes is provided by Onof et al. (2000). In addition, rainfall models based on Markov processes have also made a reasonable contribution to help tackle this challenge. See for example, Smith and Karr (1983), Bardossy and Plate (1991), Ramesh (1998), Onof et al. (2002) and Ramesh and Onof (2014) amongst others.
Much of the work on this topic, however, has concentrated on modelling rainfall data recorded at hourly or longer aggregation levels. Stochastic models for finescale rainfall are equally important, because in some hydrological applications there is a need to reproduce rainfall time series at fine temporal resolutions. For example, subhourly rainfall is required as input to urban drainage models and for small rural catchment studies. In addition, the study of climate change impacts on hydrology and water management initiatives requires the availability of data at fine temporal resolutions, which is usually not available from general circulation model (GCM) simulations. The best available approach to generating such rainfall currently lies in the combination of an hourly stochastic rainfall simulator, together with a disaggregator making use of downscaling techniques. There has been some recent work on modelling finescale rainfall using point process models. Rather than attempting to reproduce actual rainfall records at a finescale, using downscaling techniques or by other methods, these stochastic point process models aim to generate synthetic precipitation time series directly from the proposed stochastic model. One good example of this is provided by the work of Cowpertwait et al. (2007, 2011) who developed a BartlettLewis pulse model to study finescale rainfall structure. Their model particularly enables the reproduction of the fine timescale properties of rainfall. A class of doubly stochastic Poisson processes was employed by Ramesh et al. (2012, 2013) and Thayakaran and Ramesh (2013) to study finescale rainfall intensity using rainfall bucket tip times data. They utilised maximum likelihood methods to estimate parameters of their models.
Our main objective in this paper is to develop a simple stochastic point process model capable of reproducing finescale structure of the rainfall process. The other objective is to provide a fast and efficient way of generating synthetic finescale rainfall input to hydrological models directly from one stochastic model. In this regard, and to take this finescale rainfall modelling work further, we develop a simple point process model based on a doubly stochastic Poisson process, following the Poisson cluster pulse model approach of Cowpertwait et al. (2007). Our preliminary work on this (Ramesh and Thayakaran 2012), analysing properties of rainfall time series at subhourly resolutions, produced encouraging initial results. In this paper, we extend this work further and accommodate thirdorder moments in estimation. Mathematical expressions for the moment properties of the accumulated rainfall in disjoint intervals are derived. The proposed stochastic model is fitted to 69 years of 5min rainfall time series from Germany. The results of the analysis show that the proposed model is capable of reproducing rainfall properties well at various subhourly resolutions. Furthermore, the analysis incorporating thirdorder moments produced better results than the one that used only up to secondorder moments. Unlike Cowpertwait et al. (2007), who used superposition of two BartlettLewis pulse models to account for different storm types, we use one simple model to reproduce subhourly rainfall structure. The novel contribution of this study is the derivation of the thirdorder moment properties of the proposed model, as well as their incorporation in estimation, to reproduce finescale structure of rainfall more accurately. The proposed model provides a solid framework to generate synthetic finescale rainfall input to hydrological models directly from one stochastic point process model. In addition, the availability of a new stochastic model for the generation of finescale rainfall, at various subhourly scales, provides scientists with a useful tool for environmental or ecological impact studies.
The following section provides a background to this work, illustrates the model framework and then focuses on deriving moment properties of various component processes, such as the cell and pulse arrival processes. Properties of the aggregated rainfall sequence are studied and mathematical expressions for the thirdorder moment and the coefficient of skewness are derived. Section 3 presents the results of data analysis using 5min rainfall aggregations and compares the results of two different analyses that used second and thirdorder moments in estimation. Extremes of the historical data are compared with the simulated extremes at various resolutions. The proportion of dry intervals is also studied. A simulation study is carried out to evaluate the performance of the estimation method. Conclusions and possible further work are summarised in Sect. 4.
2 Model framework and moment properties
2.1 Background
Doubly stochastic Poisson processes (DSPP) provide a flexible set of point process models for finescale rainfall. Ramesh et al. (2012, 2013) utilised this class of processes and developed stochastic models, for a singlesite and multiple sites respectively. These models were used to analyse data collected in the form of rainfall bucket tiptime series. One of the advantages of these models, over most other point process models for rainfall, is that it is possible to write down their likelihood function which allows us to estimate the model parameters using maximum likelihood methods. However, the rainfall bucket tiptime series is not usually available for a long period of time. Most of the longer series of rainfall data are available in accumulated form, hourly or subhourly, rather than in tiptime series format. Moreover, the above DSPP models cannot be fitted directly to data collected in aggregated form, such as hourly rainfall, using the maximum likelihood method. Therefore it is useful to look for alternative models, based on doubly stochastic processes, that can be used to model subhourly data collected in aggregated form. Motivated by the performance of the above class of doubly stochastic models, we seek to develop models with the same structure that can be used to analyse accumulated rainfall sequences at fine time scales.
2.2 Model formulation
2.3 Moment properties of the pulse process
We shall now study the moment properties of the pulse arrival process and focus our attention on deriving an expression for its covariance density. These moment properties are required to derive the statistical properties of the aggregated rainfall process later in Sect. 2.4. The lifetimes \(L_{i}\) of the rain cells, under the DSP model framework, are assumed to be exponentially distributed with parameter \(\eta\) and hence we have \({\text{E}}(L_{i})= 1/\eta .\) Let us take N(t) as the counting process of pulse occurrences from all cells generated by the process M(t). An active cell generates a series of instantaneous pulses at Poisson rate \(\xi\) during its lifetime and therefore the mean number of pulses per cell is \(\xi /\eta\). As noted earlier, the arrival rate of the cell process is m and hence the mean arrival rate of the pulse process is given by \({\text{E}}(N(t))=m\xi /\eta\).
2.4 Moment properties of the aggregated rainfall
In most applications, the rainfall data are usually available in aggregated form in equally spaced time intervals of fixed length. The DSP process we have developed, however, evolves in continuous time. We now, therefore, derive mathematical expressions for the moment properties of the aggregated rainfall arising from the DSP process. These expressions are useful to describe the properties of the accumulated rainfall and can be used for model fitting and assessment.
3 Model fitting and assessment
We shall explore the application of the proposed DSP model in the analysis of finescale rainfall data and assess how well it reproduces the properties of the rainfall over a range of subhourly resolutions. We aim to do this using 69 years (1931–1999) of 5min rainfall data from Bochum in Germany.
Although we employed the method of moments to estimate the other parameters, which essentially equates the sample moments to theoretical moments from the model, the estimation can be done in different ways. In this application, we constructed an objective function as the sum of squares of differences between the sample and theoretical values of the moment properties at different aggregation levels and then minimized it using standard optimisation routines. This is carried out separately for each month by considering the data for a month as realisations of a stationary point process. Essentially the process of model fitting involves calculating the empirical mean, variance, correlation, coefficient of variation and skewness from the observed data at each aggregation level and matching these with the corresponding theoretical values, calculated using Eqs. (6–9), for a given set of parameter values. The role of the objective function and the optimiser employed was to find the best possible match using a minimum error criterion. We used the statistical software environment R for the optimisation and for the simulation of the process (R Development Core Team 2011). A number of options were available for the objective function depending on the application, but we used a form of weighted sum of squares. We employed the routines “optim” and “nlminb” in R for parameter estimation in our analysis. The following subsections describe two different methods used to estimate the first 6 parameters of the model and discuss the results produced. Once these parameter estimates were determined, Eq. (10) was used to estimate the final parameter \(\mu _{X}\).
3.1 Estimation using secondorder moments
3.2 Estimation incorporating thirdorder moments
Parameter estimates for the DSP model incorporating thirdorder moments in estimation
Month  \(\hat{\lambda }\)  \(\hat{\mu }\)  \(\hat{\phi _{1}}\)  \(\hat{\phi _{2}}\)  \(\hat{\eta }\)  \(\hat{\xi }\)  \(\hat{\mu }_{x}\) 

JAN  0.0306  2.6900  0.0998  3.4795  3.5699  263.5150  0.0085 
FEB  0.0161  1.6124  0.0819  2.9469  3.0257  264.8286  0.0079 
MAR  0.0090  5.0823  0.1282  5.6073  5.7007  289.9880  0.0102 
APR  0.0245  4.8790  0.0764  5.0201  5.1021  246.6266  0.0156 
MAY  0.0529  7.0143  0.0400  7.1321  7.1555  239.9341  0.0272 
JUN  0.0411  7.7027  0.0331  7.8086  7.8255  237.8311  0.0486 
JUL  0.0199  6.7546  0.0318  6.8875  6.8837  245.2713  0.0591 
AUG  0.0197  6.1291  0.0276  6.3391  6.3118  265.9815  0.0502 
SEP  0.0491  6.9914  0.0205  7.1563  7.1348  246.2618  0.0387 
OCT  0.0147  1.9679  0.0362  2.4563  2.4956  223.6488  0.0166 
NOV  0.1154  3.7691  0.0430  4.1023  4.1339  269.6375  0.0087 
DEC  0.0234  1.8008  0.0860  2.9616  3.0464  227.5842  0.0099 
Figures 6, 7, 8 and 9 show the corresponding results when the thirdorder moments are incorporated into the parameter estimation process. These clearly show an improvement over the earlier results of the method that used the moments up to the secondorder. Although the same model is used here, the parameter estimates are obviously different from the earlier values due to the fact that the coefficient of skewness was used in estimation for this second method. In most cases a near perfect fit was obtained and in some cases an exact fit was obtained when the thirdorder moments were incorporated. The mean, standard deviation and coefficient of variation have all been reproduced remarkably well at all aggregation levels, including those that were not used in fitting (h = 30, h = 60 min). Comparing these with earlier results reveals that incorporating thirdorder moments in estimation has certainly produced much better results for most of the properties.
3.3 Extremes and proportion of dry periods
In many hydrological applications, more emphasis is placed upon a stochastic model’s ability to reproduce the properties of extreme rainfall rather than the usual moment properties. One good example arises in urban drainage modelling within the context of flood estimation. One can find many other examples of this in analyses involving environmental data, see for example Ramesh and Davison (2002), Davison and Ramesh (2000) and Leiva et al. (2016). In addition, knowledge of the extremes enables us to assess the risk associated with hydrological systems. In view of this, we shall evaluate the performance of our proposed model in capturing the properties of extreme rainfall. In this regard, we compare the extreme values of the 69 years of observed rainfall data with those generated by the proposed DSP model.
The annual maxima of the empirical data from the observed 69 year long historical record were extracted, ordered and plotted against the corresponding Gumbel reduced variates at each aggregation level. Fifty copies of the 5min rainfall series, each 69 years long, were then simulated from the fitted model. Each copy of the simulated data was subsequently aggregated to generate 1, 12 and 24 h rainfall series. The annual maxima of each of the 50 simulated series, at each aggregation level, were extracted and ordered to make up the interval plots against the corresponding Gumbel reduced variates. These were superimposed on the corresponding Gumbel reduced variate plots of the empirical data for comparison.
Ordered empirical annual maxima for the 12 and 24h aggregated rainfall and the corresponding ordered reduced Gumbel variates
Gumbel reduced variate  12h aggregation  24h aggregation  

Empirical maxima  Simulated maxima  Empirical maxima  Simulated maxima  
Method 1  Method 2  Method 1  Method 2  
1.44  37.05  22.53  39.25  45.12  24.55  42.50 
1.52  37.70  22.86  40.10  48.96  24.91  43.37 
1.61  38.30  23.27  41.04  49.00  25.39  44.37 
1.70  39.95  23.65  41.86  49.68  25.81  45.46 
1.80  40.16  24.10  42.78  50.1  26.27  46.52 
1.90  47.32  24.65  44.29  50.81  26.87  47.54 
2.02  49.00  25.19  45.43  51.17  27.41  48.86 
2.16  49.15  25.77  46.80  52.55  27.93  50.30 
2.31  49.20  26.39  48.20  55.65  28.60  51.86 
2.48  50.81  27.07  50.24  56.58  29.40  54.14 
2.68  53.90  28.09  52.44  62.80  30.47  56.66 
2.94  55.45  29.44  55.02  63.50  31.96  59.26 
3.28  56.58  31.04  58.89  67.08  33.39  63.60 
3.78  60.50  33.47  65.08  67.90  35.68  69.42 
4.81  67.90  37.99  80.71  75.95  40.40  81.35 
Another property of interest to hydrologists is the proportion of intervals with little or no rain. This will help to quantify the proportion of dry periods. Very often the gauge rainfall data are recorded in a rounded form (to the nearest 0.1 mm) and hence, following Cowpertwait et al. (2007), we calculate the proportion of rainfall below a small threshold \(\left( \hat{p} \{Y_i^{h}<\delta \} {\text{ for }} \delta >0 \right)\) instead of the actual proportion of dry intervals (\(\delta =0\)). Therefore by choosing smaller values of \(\delta\) we can provide approximate estimates of the proportion of dry intervals.
The proportion of intervals below a given threshold were calculated for each month, using 50 samples of 69 years of simulated 5min data, for each of the 5min, 1 and 24 h aggregations (\(h=1/12, 1, 24\)). The mean of the 50 values was calculated, at each aggregation, for each calendar month. The observed proportions for the historical data and the average of the simulated values, from the fitted model, for different thresholds are given in Table 3. In order to find a good estimate of the proportion of dry periods at \(h=1/12\) aggregation level, a small threshold of \(\delta =0.05\) mm was used. For the hourly rainfall, two threshold values of \(\delta =0.05, 0.1\) mm were used. Higher threshold values (\(\delta =0.5, 2\) mm) were used for the daily rainfall, as occasional light rain during the day may cause discrepancies between the observed and simulated proportions.
Proportion of rainfall below defined thresholds at different time scales. Here \(\hat{p}_O\) and \(\hat{p}_S\) represent the estimates of the proportions for the observed and simulated series
Month  \(p_{o}\)  \(p_s\)  \(p_{o}\)  \(p_{s}\)  \(p_{o}\)  \(p_{s}\)  \(p_{o}\)  \(p_{s}\)  \(p_{o}\)  \(p_{s}\) 

JAN  0.951  0.964  0.848  0.757  0.866  0.865  0.903  0.975  0.976  0.991 
FEB  0.958  0.968  0.867  0.769  0.883  0.872  0.915  0.979  0.979  0.991 
MAR  0.961  0.972  0.867  0.841  0.892  0.907  0.919  0.971  0.980  0.995 
APR  0.960  0.972  0.878  0.821  0.893  0.877  0.917  0.974  0.979  0.991 
MAY  0.964  0.980  0.898  0.867  0.908  0.891  0.924  0.969  0.974  0.991 
JUN  0.962  0.984  0.892  0.856  0.902  0.875  0.916  0.958  0.968  0.994 
JUL  0.965  0.987  0.895  0.892  0.907  0.902  0.919  0.950  0.969  0.992 
AUG  0.968  0.986  0.906  0.886  0.916  0.898  0.927  0.957  0.969  0.993 
SEP  0.965  0.984  0.899  0.873  0.909  0.891  0.923  0.964  0.972  0.993 
OCT  0.959  0.971  0.887  0.793  0.899  0.859  0.921  0.978  0.975  0.991 
NOV  0.952  0.964  0.852  0.730  0.872  0.848  0.905  0.978  0.973  0.989 
DEC  0.949  0.966  0.849  0.748  0.869  0.875  0.902  0.975  0.972  0.992 
Threshold \(\delta\)(mm)  0.05  0.05  0.05  0.05  0.1  0.1  0.5  0.5  2  2 
Scale (h)  1/12  1/12  1  1  1  1  24  24  24  24 
3.4 Simulation study
Since the likelihood function of the model we proposed for the accumulated rainfall data was not available, we employed the moment method to estimate the parameters in our analysis. The method of moments simply equates sample moments from the observed data to the theoretical moments of the model being fitted to obtain estimates of the parameters. It is important to note that moment estimators, unlike maximum likelihood estimators, do not necessarily have the usual large sample properties leading to asymptotic results. Nevertheless, we carried out a simulation study to evaluate the statistical performance of the estimation method we employed in our analysis.

Simulate one hundred (\(n = 100\)) sample series from the fitted values of the parameters (\(\theta\)) for the month.

Aggregate the simulated data in \(h = 5, 10, 20\) min intervals.

Calculate their sample statistics: coefficient of variation \(\nu (h)\), lag 1 autocorrelation \(\rho (h)\) and coefficient of skewness \(\kappa (h)\) for each value of h.

Compute the moment estimates (\(\hat{\theta }\)) for the simulated samples, using the objective function used in Sect. 3.2.
 Compute the mean, bias and mean squared error of the moment estimates, separately for each of the parameters, using the expressionswhere n = 100 and \(\theta = \lambda\), \(\mu\), \(\phi _{1}\), \(\phi _{2 }\), \(\eta\) and \(\xi\) are the fitted parameter values of the empirical rainfall data for the month under study and \(\hat{\theta }\) is the corresponding estimate for the simulated data.$${\text{Mean}}=\frac{1}{n}\sum _{i=1}^{n} \hat{\theta }, \quad {\text{Bias}}=\frac{1}{n}\sum _{i=1}^{n} (\hat{\theta } \theta ), \quad{\text{MSE}}=\frac{1}{n}\sum _{i=1}^{n} (\hat{\theta } \theta )^2,$$
Simulation study results: values of the mean, bias and root mean squared error (\(\sqrt{\text {MSE}}\)) for the moment estimates of the parameters based on 100 simulated series of length \(n_l\) years for the month of August. True values are the fitted parameter values for the observed rainfall data for August
\(n_l\)  \(\hat{\lambda }\)  \(\hat{\mu }\)  \(\hat{\phi _{1}}\)  \(\hat{\phi _{2}}\)  \(\hat{\eta }\)  \(\hat{\xi }\)  

Mean  20  0.01964  6.21585  0.02008  6.21159  6.22667  270.2398 
40  0.01966  6.20756  0.02046  6.19146  6.20708  268.9749  
60  0.01963  6.17081  0.01971  6.16408  6.17867  269.7928  
69  0.01963  6.14516  0.01983  6.14164  6.15634  268.2950  
True values  0.01970  6.12910  0.02760  6.33910  6.31180  265.9815  
Bias  20  −6.4E−05  0.08675  −0.00752  −0.12751  −0.08513  4.25827 
40  −7.0E−05  0.07846  −0.00715  −0.14764  −0.10472  3.99343  
60  −6.9E−05  0.04171  −0.00789  −0.17502  −0.13313  3.81131  
69  −6.6E−05  0.01606  −0.00777  −0.19746  −0.15546  2.31354  
\(\sqrt{\text {MSE}}\)  20  7.80E−05  0.25363  0.00784  0.30055  0.28529  8.31461 
40  7.28E−05  0.26760  0.00732  0.32294  0.30633  8.00499  
60  6.87E−05  0.26515  0.00803  0.33909  0.32019  8.22053  
69  7.28E−05  0.27809  0.00792  0.36235  0.34201  8.37347 
The means of the simulated estimates showed that these are generally close to the corresponding true values. The means of \(\hat{\mu }\) and \(\hat{\xi }\) were converging towards their true values as \(n_l\) became large, whereas the means of \(\hat{\phi }_2\) and \(\hat{\eta }\) did not seem to exhibit this property. It is worth pointing out that the parameters \(\phi _2\) and \(\eta\) are positively correlated and when \(\phi _2\) increased \(\eta\) also increased. The reason for this is that much of the rain cells come from the high intensity state 2 and, as we are equating the rainfall moments in our estimation, increased arrival rate \(\phi _2\) of rain cells results in smaller cell lifetimes of \(1/\eta\), and viceversa. Means of \(\hat{\lambda }\) and \(\hat{\phi }_1\) do not seem to show much variation.
The bias of \(\hat{\mu }\) and \(\hat{\xi }\) became smaller when \(n_l\) became larger but both \(\hat{\phi }_2\) and \(\hat{\eta }\) seemed to show a slight negative bias. As we can see from the values, \(\hat{\lambda }\) and \(\hat{\phi }_1\) both show very small negative bias and again do not show much variation with \(n_l\). The root mean squared errors seemed to stay more or less at the same level for most of the parameters, as \(n_l\) increased, except for \(\hat{\phi }_2\) and \(\hat{\eta }\) which showed a slight increase. One possible reason for that might be that the moment estimates may be slightly biased because of the serial correlation in the data, as noted by Cowpertwait et al. (2007), especially at subhourly aggregation levels which show high autocorrelation. Ideally we would have liked the bias and MSE of the estimators to converge to zero asymptotically for all parameters, as would those of the maximum likelihood estimators which have good large sample statistical properties. However, the means of the simulated estimates were closer to their true values and the relative sizes of the bias and root mean squared errors seemed to be reasonably small for the application.
4 Conclusions and future work
We have developed a doubly stochastic point process model, using instantaneous pulses, to study the finescale structure of subhourly rainfall time series. Second and thirdorder moment properties of the aggregated rainfall for the proposed DSP model were derived. The model was used to analyse 69 years of 5min rainfall data. The empirical properties of the rainfall accumulations were shown to be in very good agreement with the fitted theoretical values over a range of subhourly time scales, including those that were not used in fitting. Although the use of secondorder moments in estimation produced very good results, incorporation of thirdorder moments showed a clear improvement in fitting. Overall, the results of our analysis suggest that the proposed stochastic model is capable of reproducing the finescale structure of the rainfall process, and hence could be a useful tool in environmental or ecological impact studies.
The simulated extreme values at daily and 12hourly aggregations are in very good agreement with their empirical counterparts. However, although the model reproduces the moment properties well, it underestimates the extremes at fine timescale. The results from the analysis of the proportion of dry periods, using intervals with rainfall depths below appropriate threshold levels, show that the model generally reproduces the observed proportions well. The simulation study shows that the estimation method used is capable of reproducing the estimates closer to the true values, although it may not have the desired large sample properties which maximum likelihood estimators exhibit. Overall, the above analyses indicate that the proposed modelling approach is able to fit data over a range of subhourly time scales and reproduce most of the properties well. It has potential application in many areas, as it provides a fast and efficient way of generating synthetic finescale rainfall input to hydrological models directly from one stochastic model.
Despite this, there is potential to develop the model further to employ a 3state doubly stochastic model for cell arrivals and also to explore its capability to handle aggregations at higher levels. The 3state model, along with further developments to study other hydrological properties of interest, may be a fruitful area for future work. Although our model based on one doubly stochastic process has performed well, another possibility is to consider superposing two doubly stochastic pulse processes, as in Cowpertwait et al. (2007), to better account for different types of precipitation, such as convective and stratiform.
Notes
Acknowledgments
The authors would like to thank the Editor, Associate Editors and two anonymous referees for their constructive comments and suggestions on earlier versions of the manuscript. Their comments have greatly improved the quality of the manuscript.
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