Forecasting
For forecasting future values of a time series we imagine that the time series is generated by a (possibly noisy) deterministic process such as a Mealy or a Moore machine. This leads to recurrent or auto-regressive models. Building forecasting models is e
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Forecasting
Abstract
For forecasting future values of a time series we imagine that the time series is generated by a (possibly noisy) deterministic process such as a Mealy or a Moore machine. This leads to recurrent or auto-regressive models. Building forecasting models is essentially a regression task. The training data sets for forecasting models are generated by finite unfolding in time. Popular linear forecasting models are auto-regressive models (AR) and generalized AR models with moving average (ARMA), with integral terms (ARIMA), or with local regression (ARMAX). Popular nonlinear forecasting models are recurrent neural networks.
7.1
Finite State Machines
Sequences of data play an important role in data analysis. In Chap. 2 we presented relations for sequential data and sampling schemes for continuous signals. In Chap. 3 we showed how sequences can be preprocessed by filtering. And in Chap. 4 we illustrated how sequential data can be analyzed and visualized using spectral analysis. In data analysis we not only preprocess, visualize and analyze sequential data but also try to use observed data sequences to produce reliable forecasts of future data. Finding an appropriate forecasting model begins with the assumption that the sequence is generated by a dynamic feedback system [2] with an input x 2 Rp , a hidden state s 2 Rh , and an output y 2 Rq . Notice the similarity to the three-layer MLP architecture presented in the previous chapter. At each time step k such a dynamic feedback system can be described by a state equation sk D fs .sk1 ; xk /
© Springer Fachmedien Wiesbaden 2016 T.A. Runkler, Data Analytics, DOI 10.1007/978-3-658-14075-5_7
(7.1)
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7 Forecasting
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Fig. 7.1 Finite state machines: Mealy machine (top left), Moore machine (top right), unfolded Moore machine (bottom)
and an output equation yk D fy .sk ; xk /
(7.2)
The top left view of Fig. 7.1 shows a block diagram of such a system. If the set of states that can be reached in a such system is finite, we call the system a finite state machine, for example when s 2 f0; 1gr , r 2 f1; 2; : : :g. Equations (7.1) and (7.2) describe a so-called Mealy machine [3]. Special cases of Mealy machines are Moore machines [4], where the output only depends on the state but not the current input. So a Moore machine is described by a state equation sk D fs .sk1 ; xk /
(7.3)
yk D fy .sk /
(7.4)
and an output equation
The top right view of Fig. 7.1 shows a block diagram of a Moore machine. Any Mealy machine can be translated into an equivalent Moore machine and vice versa. To realize a given behavior, Mealy machines usually require a smaller state space. However, Moore machines are often easier to design and their behavior is easier to analyze. Without loss of generality we will restrict here to Moore machines.
7.2
Recurrent Models
Forecasting uses the observed input and output sequences .x1 ; : : : ; xn g and .y1 ; : : :
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