SOME SIMPLE MODELS FOR DISCRETE VARIATE TIME SERIES1

ABSTRACT: Simple models are presented for use in the modeling and generation of sequences of dependent discrete random variables. The models are essentially Markov Chains, but are structurally autoregressions, and so depend on only a few parameters. The marginal distribution is an intrinsic component in the specification of each model, and the Poisson, Geometric, Negative Binomial and Binomial distributions are considered. Details are also given for the introduction of time-dependence into the means of the sequences so that seaonality can be treated simply.     

PAST AND FUTURE OF ANALYSIS OF WATER RESOURCES TIME SERIES1

ABSTRACT: water resources supply and demand time series consist of several or all of the four basic characteristics: tendency, intermittency, periodicity and stochasticity. Their importance changes from one type of variables to another. Historic developments of analysis of time series in hydrology have varied significantly over the past, from the stress on search for periodicities and persistence in annual series to the emphasis on the series stochastic properties. Supply and demand series are often highly interrelated, which fact is most often neglected in planning water resources systems in general, and water storage capacities in particular. The future of series analysis in water resources will likely be by a joint use of physically-based structural analysis and the use of advanced methods of treating data by stochastic processes, statistical estimation and inference techniques. The most intriguing challenge of the future of this analysis may be the treatment of nonnormal, nonlinear and in general nonstationary hydrologic and water use time series. The proper treatment of complex multivariate processes will also challenge the specialists, especially for the purposes of transfer of information between data on variables at given points, or between data at several points of a given variable, or both.     

A MONTHLY TIME SERIES MODEL OF MUNICIPAL WATER DEMAND1

A multivariate time series model is formulated to study monthly variations in municipal water demand. The left hand side variable in the multivariate regression model is municipal water demand (gallons per connection per day) and the right hand side contains (explanatory) variables which include price (constant dollars), average temperature, total precipitation, and percentage of daylight hours. The application of the regression model to Salt Lake City Water Department data produced a high multiple correlation coefficient and F-statistic. The regression coefficients for the right hand side variables all have the appropriate sign. In an ex post forecast, the model accurately predicts monthly variations in municipal water demand. The proposed monthly multivariate model is not only found useful for forecasting water demand, but also useful for predicting and studying the impact of nonstructural management decisions such as the effect of price changes, peak load pricing methods, and other water conservation programs.     

A COMPARATIVE STUDY OF LINEAR AND NONLINEAR TIME SERIES MODELS FOR WATER QUALITY1

ABSTRACT: Surface water quality data are routinely collected in river basins by state or federal agencies. The observed quality of river water generally reflects the overall quality of the ecosystem of the river basin. Advanced statistical methods are often needed to extract valuable information from the vast amount of data for developing management strategies. Among the measured water quality constituents, total phosphorus is most often the limiting nutrient in freshwater aquatic systems. Relatively low concentrations of phosphorus in surface waters may create eutrophication problems. Phosphorus is a non-conservative constituent. Its time series generally exhibits nonlinear behavior. Linear models are shown to be inadequate. This paper presents a nonlinear state-dependent model for the phosphorous data collected at DeSoto, Kansas. The nonlinear model gives significant reductions in error variance and forecasting error as compared to the best linear autoregressive model identified.     

SOME SIMPLE MODELS FOR CONTINUOUS VARIATE TIME SERIES1

ABSTRACT: A survey is given of recently developed models for continuous variate non–Gaussian time series. The emphasis is on marginally specific models with given correlation structure. Exponential, Gamma, Weibull, Laplace, Beta and Mixed Exponential models are considered for the marginal distributions of the stationary time series. Most of the models are random coefficient, additive linear models. Some discussion of the meaning of autoregression and linearity is given, as well as suggestions for higher–order linear residual analysis for non–Gaussian models.     

APPROACHES TO MULTIVARIATE MODELING OF WATER RESOURCES TIME SERIES1

ABSTRACT: Alternative approaches suggested for modeling multiseries of water resources systems are reviewed and compared. Most approaches fall within the general framework of multivariate ARMA models. Formal modeling procedures suggest a three-stage iterative process, namely: model identification, parameter estimation and diagnostic checks. Although a number of statistical tools are already available to follow such modeling process, in general, it is not an easy task, especially if high order vector ARMA models are used. However, simpler ARMA models such as the contemporaneous and the transfer-function models may be sufficient for most applications in water resources. Two examples of modeling bivariate and trivariate streamflow series are included. Alternative modeling procedures are used and compared by using data generation techniques. The results obtained suggest that low order models, as well as contemporaneous ARMA models, reproduce quite well the main statistical characteristics of the time series analyzed. It is assumed that the same conclusions apply for most water resources time series.     

CHAOTIC FORECASTING OF DISCHARGE TIME SERIES: A CASE STUDY1

ABSTRACT: This paper considers the problem of forecasting the discharge time series of a river by means of a chaotic approach. To this aim, we first check for some evidence of chaotic behavior in the dynamic by considering a set of different procedures, namely, the phase portrait of the attractor, the correlation dimension, and the largest Lyapunov exponent. Their joint application seems to confirm the presence of a nonlinear deterministic dynamic of chaotic type. Second, we consider the so‐called nearest neighbors predictor and we compare it with a classical linear model. By comparing these two predictors, it seems that nonlinear river flow modeling, and in particular chaotic modeling, is an effective method to improve predictions.     

MULTIFRACTAL SCALING OF DAILY RUNOFF TIME SERIES IN AGRICULTURAL WATERSHEDS1

Abstract: Multifractal scaling behavior of long-term records of daily runoff time series in 32 subwatersheds covering a wide range of sizes was examined. These subwatersheds were associated with four agricultural watersheds with different climates and topography. The empirical moment scaling curves obtained using the trace moment method showed that the runoff time series exhibited a multifractal behavior, which was valid over a time scale range from one day to about three years. The multi-fractal scaling of the runoff time series was well described by the Universal Multifractal Model. The spectral analysis (β < 1) and the order of fractional integration (H ⋍; 0) indicated that the runoff time series were conservative. The multifractal parameters, α (multifractal index) and C₁ (co-dimension), were reasonably close to each other for subwatersheds within each of the watersheds and were generally similar among the four watersheds. The α values of the four watersheds were 1.10 ± 0.13, 1.61 ± 0.06,1.61 ± 0.24, and 1.63 ± 0.19. The C₁ values of four watersheds were 0.19 ± 0.01, 0.17 ± 0.01, 0.17 ± 0.04, and 0.11 ± 0.02. The multifractal analyses provided useful insight into the runoff time series, especially the occurrence and distribution of extreme events.     

TIME SERIES ANALYSIS IN PERSPECTIVE1

ABSTRACT: By employing a set of criteria for classifying the capabilities of time series models, recent developments in time series analysis are assessed and put into proper perspective. In particular, the inherent attributes of a wide variety of time series models and modeling procedures presented by the authors of the 18 papers contained in this volume are clearly pointed out. Additionally, it is explained how these models can address many of the time series problems encountered when modeling hydrologic, water quality and other kinds of time series. For instance, families of time series models are now available for modeling series which may contain nonlinearities or may follow nonGaussian distributions. Based upon a sound physical understanding of a problem and results from exploratory data analyses, the most appropriate model to fit to a data set can be found during confirmatory data analyses by following the identification, estimation and diagnostic check stages of model construction. Promising future research projects for developing flexible classes of time series models for use in water resources applications are suggested.     

AN EVALUATION OF WEEKLY AND MONTHLY TIME SERIES FORECASTS OF MUNICIPAL WATER USE1

ABSTRACT: A cascade model for forecasting municipal water use one week or one month ahead, conditioned on rainfall estimates, is presented and evaluated. The model comprises four components: long term trend, seasonal cycle, autocorrelation and correlation with rainfall. The increased forecast accuracy obtained by the addition of each component is evaluated. The City of Deerfield Beach, Florida, is used as the application example with the calibration period from 1976–1980 and the forecast period the drought year of 1981. Forecast accuracy is measured by the average absolute relative error (AARE, the average absolute value of the difference between actual and forecasted use, divided by the actual use). A benchmark forecast is calculated by assuming that water use for a given week or month in 1981 is the same as the average for the corresponding period from 1976 to 1980. This method produces an AARE of 14.6 percent for one step ahead forecasts of monthly data and 15.8 percent for weekly data. A cascade model using trend, seasonality and autocorrelation produces forecasts with AARE of about 12 percent for both monthly and weekly data while adding a linear relationship of water use and rainfall reduces the AARE to 8 percent in both cases if it is assumed that rainfall is known during the forecast period. Simple rainfall predictions do not increase the forecast accuracy for water use so the major utility of relating water use and rainfall lies in forecasting various possible water use sequences conditioned on sequences of historical rainfall data.     

WATER CONSERVATION KITS: A TIME SERIES ANALYSIS OF A CONSERVATION POLICY1

ABSTRACT: The effect on water usage of installing water conservation kits in the city of Oxnard, California, is statistically evaluated. Using binary multiple regression analysis on a large sample of residential units, water consumption was compared before and after receipt of the kit. For this sample, installers reduced water consumption by 4.2 percent each billing period. Income (wealth) and household size elasticities are reported along with the Characteristics of the installing households.     

SHORT-TERM FORECASTING OF SNOWMELT RUNOFF USING ARMAX MODELS1

ABSTRACT: Time series models of the ARMAX class were investigated for use in forecasting daily riverflow resulting from combined snowmelt/rainfall. The Snowmelt Runoff Model (Martinec-Rango Model) is shown to have a form similar to the ARMAX model. The advantage of the ARMAX approach is that analytical model identification and parameter estimation techniques are available. In addition, previous forecast errors can be included to improve forecasts and confidence limits can be estimated for the forecasts. Diagnostic checks are available to determine if the model is performing properly. Finally, Kalman filtering can be used to allow the model parameters to vary continuously to reflect changing basin runoff conditions. The above advantages result in improved flow forecasts with fewer model parameters.     

MODELING HYDROLOGIC TIME SERIES FROM THE ARCTIC1

The general intervention model is applied to hydrologic and meteorologjc time series from the Canadian Arctic. The authors show how the model is able to account for environmental interventions, missing observations in the data, changes in data collection procedures, the effects of external inputs, as well as seasonality and autocorrelation. Methods for identifying transfer functions by making use of a physical understanding of the processes involved are demonstrated and sample applications of the general intervention model to Arctic data are shown.     

THE RELATIONSHIP BETWEEN THE TIME BASES OF SIMULATION MODELS AND THEIR STRUCTURE1

ABSTRACT: The time base of a simulation model can be defined as a combination of two time intervals. One is the interval used for input and internal computations. The second is the interval used for the output and calibration of the model. The time base of a model is related on the one hand to the type of applications for which the simulated data are used, and on the other hand to the structure and complexity of the model. The latter may be represented by the number of parameters employed to specify the operation of the model. Using data typical to relatively small watersheds in a semiarid climate, the interaction between the complexity of a series of models and the time bases used by them was studied. This included the effects of the two factors, time base and complexity, on the values of the optimal parameters, prediction of mean annual flow, and general performance of the models. The main conclusion is that if the acceptable time base is longer, the model can be less complex needing fewer parameters. There is also an advantage in using a time base comprising a shorter input time interval and a longer output time interval.     

TREND ASSESSMENT OF WATER QUALITY TIME SERIES1

ABSTRACT A general methodology is described for identifying and statistically modeling trends which may be contained in a water quality time series. A range of useful exploratory data analysis tools are suggested for discovering important patterns and statistical characteristics of the data such as trends caused by external interventions. To estimate the entries in an evenly spaced time series when data are available at irregular time intervals, a new procedure based upon seasonal adjustment is described. Intervention analysis is employed at the confirmatory data analysis stage to rigorously model changes in the mean levels of a series which are identified using exploratory data analysis techniques. Furthermore, intervention analysis can be utilized for estimating missing observations when they are not too numerous. The effects of cutting down a forest upon various water quality variables and also the consequences of acid rain upon the alkalinity in a stream provide illustrative applications which demonstrate the effectiveness of the methodology.     

ALTERNATIVE MODELS FOR ESTIMATING THE TIME SERIES COMPONENTS OF WATER CONSUMPTION DATA1

ABSTRACT: Understanding the behavior of different time-series components of water consumption data is essential for a more effective analysis of economic incentive effects of alternative policy measures and also closer integration of water supply and demand management. Additive and multiplicative models are used to analyze the trend (T), cyclical (C), seasonal (S) and irregular (I) components. The stepwise regression method was applied to 187 data points (January 1960 to July 1975), each representing average daily water consumption within the service area of the Honolulu Board of Water Supply. Although statistically similar results (R² 0.95 and 0.96 and respective corresponding F-ratios 277 and 307) might suggest little difference in model performances, closer analysis of the results point to important multiplicative effects which should be taken into account in both short-run and long-run analyses.     

QUANTIFICATION FOR VISUALIZATION OF TIME TREND1

ABSTRACT: A method for quantifying fluctuations in time-series data was developed and tested to aid the process of visualization. The methodology is based on free-form sliding polynomials and identifies (a) short-period variability about the mean value, (b) a long-term trend or cycle, and (c) random errors residual to these two structured components. Consistent results were obtained for designed synthetic data and natural data from seven sites in Georgia. Statistics of fit of the analytical model for the natural data were not significant on a site-by-site basis. An unexpected finding for the study was obtained when the statistical results for the seven data sets for temperature were pooled. The smoothing model yielded consistent long-term trends even though the individual station results were not significant. Also, the correlation coefficients, while low, showed a statistically significant trend toward higher values toward the northwest and away from the Georgia coast line. This study thus supports the concept that multiple-site, and regionally based, analyses are necessary for the detection of trends. Secondarily, such consistency of results strengthens the conclusion that the proposed smoothing method is an effective procedure in the presence of varying amounts of random content in the natural data sets.     

FOURIER INFERENCE: SOME METHODS FOR THE ANALYSIS OF ARRAY AND NONGAUSSIAN SERIES DATA1

Fourier inference is a collection of analytic techniques and philosophic attitudes, for the analysis of data, wherein essential use is made of empirical Fourier transforms. This paper sets down some basic results concerning the finite Fourier transforms of stationary process data and then, to illustrate the approach, uses those results to develop procedures for: 1) estimating cloud and storm motion, 2) passive sonar and 3) fitting finite parameter models to nonGaussian time series via bispectral fitting. This last procedure is illustrated by an analysis of a stretch of Mississippi River runoff data. Examples 1), 2) refer to data having the form Y(x_j, y_j, t) for j = 1, …, J and t = 0, …, T-l say, and view that data as part of a realization of a spatial-temporal process. Such data has become common in geophysics generally and in hydrology particularly. The goal of this paper is to present some new statistical procedures pertinent to problems in the water sciences, equally it is to illustrate the genesis of those procedures and how their properties may be approximated.     

INLET CONCENTRATION TIME NOMOGRAM FOR URBAN BASINS1

ABSTRACT: A physically-based nomogram to determine inlet concentration times for composite developed basins is presented. This nomogram is constructed from previously derived kinematic wave formulations of both catchment and gutter flow components of surface runoff. With specified physical properties of a basin and rainfall intensity-duration relationships, the inlet time can be obtained from the nomogram directly.     

THRESHOLD TIME SERIES MODELING OF TWO ICELANDIC RIVERFLOW SYSTEMS1

ABSTRACT: This paper reports our experience in building time series models which connect the flows in two Icelandic rivers with the meteorological variables of precipitation and temperature. Two rivers with different hydrological characteristics were studied. In areas where precipitation may be either in the form of rain or snow linear models are inadequate to describe the relationship between the river and the meteorological variables. The methodology of threshold models recently developed seems to be well suited for taking into account the sharp difference in the relationship according to whether it is freezing or not. The possibility of identifying an alternative threshold variable is also explored.