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To apply the error modelling module, time series data are required for both the simulated and historical period at a given location, as well as the forecast time series at this location. Under normal configuration, these time series will be of the same parameter.

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The error modelling module returns two time series, an update time series for the historic period, and an updated time series for the forecast period. In principal the updated time series over the historic period is almost identical to the observed time series.

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The configuration of the error modelling module is used to determine its behaviour in establishing the statistical model of the error and how this is applied to derive the updated series

Configuration items

• Order_AR: (maximum) order of the AR component;
• Order_MA: 0;
• Order_Sel: Option to determine if the orders are to be derived automatically (with the maxima as defined above) or as given;
• Transform: Option to apply a transformation to residuals. This may either be "none", "mean" or "boxcox";
• Lambda: A required parameter for the "boxcox" transformation option.

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Three types of time series models can be distinguished, autoregressive or Ar, moving average or MA and the combined ARMA type. An ARMA(p,q) process can be written as (Priestley, 1981)

where e_n is a purely random process, thus a sequence of independent indetically distributed stochastic variables with zero mean and variance s_e². This  This process is purely AR for q=0 and MA for p=0. Any stationary stochastic process can be written as a unique AR(¥) or MA(¥) process The roots of

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are the zeros. Processes and models are called stationary if all poles are strictly within the unit circle, and they are invertible if all zeros are within the unit circle.

While (Priestley, 1981) states the ARMA formula using + on the left hand side, the formula is also well known using -, and the coefficients as reported by FEWS will solve the equation:

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AR estimation

This model type is the backbone of time series analysis in practise. Burg's method, also denoted as maximum entropy, estimates the reflection coefficients (Burg, 1967;Kay and Marple, 1981), thus making sure that the model will be stationary, with all roots of A(z) within the unit circle. Asymptotic AR order selection criteria can give wrong orders if candidate order are higher than 0.1N (N is the signal length). The finite sample criterion CIC(p) is used for model selection (see Broersen, 2000). The model with the smallest value of CIC(p) is selected. CIC uses a compromise between the finite sample estimator for the Kullbach-Leibler information (Broersen and Wensink, 1998) and the optimal asymptotic penalty factor 3 (Broersen, 2000,Broersen and Wensink, 1996).

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The Box Cox transformation (Box and Cox, 1964) can be applied in the order selection and estimation of the coefficients. The object in doing so is usually to make the residuals more homoskedastic and closer to a normal distribution:

for l not equal to zero, when l=0 T(y) =log(y) .

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Figure 2. Application of AR module to the Moesel basin using Box Cox transformation and subtraction of mean. Blue is the measured discharge (Q), red is the updated model update and forecast, green is the model simulation. Forecasts starts at t=401 hours.

Figure 2 shows the effect of additionally applying a Box Cox transformation (l=0.3). It gives slightly better predictions than without (Figure 1).

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Figure 3. Application of AR module to the Moesel basin using subtraction of mean. Blue is the measured discharge (Q), red is the updated model update and forecast, green is the model simulation. Forecasts starts at t=250 hours.

Figure 4. Application of AR module to the Moesel basin using subtraction of mean. Blue is the measured discharge (Q), red is the updated model update and forecast, green is the model simulation. Forecasts starts at t=500 hours.

Figure 3 and 4 show two applications (forecast starts at t=250 hours and at t=500 hours) of the algorithm with subtraction of mean but without Box-Cox transformation.

References

Box, G.E.P and D.R. Cox, 1964. An analysis of transformations. J. Royal Statistical Soc. (series B), vol 26, pp 211-252.

Broersen, P.M.T.; Weerts, A.H. (2005). Automatic Error Correction of Rainfall-Runoff models in Flood Forecasting Systems. Instrumentation and Measurement Technology Conference, 2005. IMTC 2005. Proceedings of the IEEE
Volume 2, Issue , 16-19 May 2005 Page(s): 963 - 968IMTC05river.pdf

Broersen, P.M.T., 2000. Finite sample criteria for Autoregressive order selection. IEEE  Trans. Signal Processing, vol 48, pp 3550-3558.

Broersen, P.M.T. Automatic spectral analysis with time series models. IEEE Instr. Meas., vol 51, pp 211-216.

Broersen, P.M.T. Matlab toolbox ARMASA (online) Available: http://www.tn.tudelft.nl/mmr.

Broersen, P.M.T. and H.E. Wensink, 1996. On the penalty factor for autoregressive order selection in finite samples, vol 44, pp 748-752.

Broersen, P.M.T. and H.E. Wensink, 1998. Autoregressive model order selection by a finite sample estimator for the Kullbach-Leibler discrepancy. IEEE Trans. Signal Processing, vol 46, pp 2058-2061.

Burg, J.P., 1967. Maximum entropy spectral analysis. Proc. 37^th Meeting Soc. Exploration Geophys., Oklahoma City, OK, pp 1-6.

Kay, S.M. and S.L. Marple, 1981. Spectrum analysis-A modern perspective. Proc IEEE, vol 69, pp 1380-1419.

Madsen, H., M.B. Butts, S.T. Khu, S.Y. Liong, 2000. Data assimilation in rainfall-runoff forecasting. Hydroinformatics 2000, 4^th Inter. Conference on Hydroinformatics, Cedar Rapids, Iowa, USA, 23-27 July 2000, 9p.

Priestely, M.B., 1981. Spectral analysis and time series. New York:Academic.