Stochastic models of streamflow: some case studies

Abstract

Ten candidate models of the Auto-Regressive Moving Average (ARMA) family are investigated for representing and forecasting monthly and ten-day streamflow in three Indian rivers. The best models for forecasting and representation of data are selected by using the criteria of Minimum Mean Square Error (MMSE) and Maximum Likelihood (ML) respectively. The selected models are validated for significance of the residual mean, significance of the periodicities in the residuals and significance of the correlation in the residuals. The models selected, based on the ML criterion for the synthetic generation of the three monthly series of the Rivers Cauvery, Hemavathy and Malaprabha, are respectively AR(4), ARMA(2,1) and ARMA(3,1). For the ten-day series of the Malaprabha River, the AR(4) model is selected. The AR(1) model resulted in the minimum mean square error in all the cases studied and is recommended for use in forecasting flows one time step ahead.

Daily streamflow modelling using autoregressive moving average and artificial neural networks models: case study ofÇoruh basin,Turkey

“…ARMA models can be expressed as: where ϕ ( B ) = 1 − ϕ 1 B − ϕ 2 B 2 − … − ϕ p B p is the autoregressive operator, p is the number of autoregressive terms, θ ( B ) = 1 − θ 1 B − θ 2 B 2 − … − θ q B q is the moving average operator, q is the number of moving average terms, ε t is the random component (residuals) of the model, and B is the backward operator (defined as B m y t = y t − m ). ARMA models have been successfully applied to various types of series (Granger & Newbold ; Mujumdar & Kumar ). Commonly used transformations, which are needed to fulfill the normality requirement, are logarithmic and square roots (McLeod et al .…”

Analysis of cross-correlated chaotic streamflows

“…In their paper, the authors have considered the results to be interestingly contrary to the common belief that models with larger number of parameters give better forecasts. In light of the recent revealed chaotic behaviour of the hydrological time series, the results of Mujumdar & Nagesh Kumar (1990) can be interpreted differently. Although both AR(1) and AR(10), for example, are global models but when forecast is done using AR(1), only the value of the most recent observation is used for the forecast.…”

“…This argument may justify why AR(1) resulted in lower MSE than models that include more parameters. The comment of Mujumdar & Nagesh Kumar (1990) that "the simplest model is sufficient" may be reasonable if simple and more complicated models result in similar values of MSE. But the deterioration that occurs when more parameters are included can be attributed to the existence of chaos.…”

“…In a study of stochastic models of streamflows, Mujumdar & Nagesh Kumar (1990) have shown that an autoregressive model, AR(1), is superior to other ARMA models, that have more parameters, in terms of the MSE of forecasting future values. In their paper, the authors have considered the results to be interestingly contrary to the common belief that models with larger number of parameters give better forecasts.…”

Neuro-fuzzy modeling for level prediction for the navigation sector on the Magdalena River (Colombia)

“…For river flow forecasting, the stochastic models that have been most widely used belong to the class of ARIMA (Auto-Regressive Integrated Moving Average) models proposed by Box & Jenkins (1976) (e.g. Mujumdar & Kumar 1990). …”