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Abstract. A time series I:, can be transformed into another time series Vt by means of a linear transformation. Should the matrix of that transformation have an inverse, the pair (Y, Vt) is called invertible. Based on the decomposition procedure for stationary time series introduced in a previous paper it is shown that a sufficient condition for the invertibility of the pair (Y, V,) is that V~ be the first component of Y~, i.e. Vt = V~. By the invertibility property V~ can be used for forecasting, that is, predictions are made on V~ which is then transformed into Yr. This is accomplished by means of a special kind of predictor permitring to make one-step-ahead forecasts in a straightforward way. Since the first component depends on a parameter ~ i.e. V~ = V~ (a), a procedure is proposed that allows us to find the optimal parameter value, = 0b. Thus, it is shown that better forecasting accuracy may result by fitting a simple autoregression to the first component V~ (ao), than if the process Yt were described by a more elaborate model. Model building is therefore no longer a prerequisite in forecasting. The forecasting procedure is then extended so as to cope with the homogeneous nonstationary case, and exampies are given to illustrate the forecasting accuracy as compared to customary model-based approaches. In the light of these results the problem of the information conveyed by the values of the series is discussed in terms of the spreading rate concept, thus highlighting the role of the current time value, as well as that of the remote values of the series, in forecasting stationary and nonstationary time series. I IntroductionTime-dependent phenomena are widespread in biology and most of the related stochastic processes can be appropriatedly treated as either stationary or nonstationary time series. The main tool in analysing and forecasting such processes is the stochastic model of the time series, which is usually derived starting from the observed series by means of a three stages model building strategy (identification, estimation of the parameters, diagnostic checking). Though several such models which turn out to be useful in biology are currently in use, building a model for the data at hand is not an easy task. On the other hand, even if the model is carefully built-up, the adequacy of that model with respect to the data is still an open problem. As a consequence the aim of the present paper is twofold. First, to provide some new tools permitting to make accurate forecasts in the absence of a pertinent model of the series. Secondly, based on these tools to analyse the contribution of the information conveyed by the successive values of the series. To achieve this we show that the process Y, and its first component V~ are related by a linear transformation with an invertible matrix (Theorem 2.1), such that Yt and V~ can be transformed one into each other. Furthermore, as the first component is dependent on a parameter ~, i.e. V~ = V~ (a), it is shown that there exists a positive value = 0b such that the...
Abstract. A time series I:, can be transformed into another time series Vt by means of a linear transformation. Should the matrix of that transformation have an inverse, the pair (Y, Vt) is called invertible. Based on the decomposition procedure for stationary time series introduced in a previous paper it is shown that a sufficient condition for the invertibility of the pair (Y, V,) is that V~ be the first component of Y~, i.e. Vt = V~. By the invertibility property V~ can be used for forecasting, that is, predictions are made on V~ which is then transformed into Yr. This is accomplished by means of a special kind of predictor permitring to make one-step-ahead forecasts in a straightforward way. Since the first component depends on a parameter ~ i.e. V~ = V~ (a), a procedure is proposed that allows us to find the optimal parameter value, = 0b. Thus, it is shown that better forecasting accuracy may result by fitting a simple autoregression to the first component V~ (ao), than if the process Yt were described by a more elaborate model. Model building is therefore no longer a prerequisite in forecasting. The forecasting procedure is then extended so as to cope with the homogeneous nonstationary case, and exampies are given to illustrate the forecasting accuracy as compared to customary model-based approaches. In the light of these results the problem of the information conveyed by the values of the series is discussed in terms of the spreading rate concept, thus highlighting the role of the current time value, as well as that of the remote values of the series, in forecasting stationary and nonstationary time series. I IntroductionTime-dependent phenomena are widespread in biology and most of the related stochastic processes can be appropriatedly treated as either stationary or nonstationary time series. The main tool in analysing and forecasting such processes is the stochastic model of the time series, which is usually derived starting from the observed series by means of a three stages model building strategy (identification, estimation of the parameters, diagnostic checking). Though several such models which turn out to be useful in biology are currently in use, building a model for the data at hand is not an easy task. On the other hand, even if the model is carefully built-up, the adequacy of that model with respect to the data is still an open problem. As a consequence the aim of the present paper is twofold. First, to provide some new tools permitting to make accurate forecasts in the absence of a pertinent model of the series. Secondly, based on these tools to analyse the contribution of the information conveyed by the successive values of the series. To achieve this we show that the process Y, and its first component V~ are related by a linear transformation with an invertible matrix (Theorem 2.1), such that Yt and V~ can be transformed one into each other. Furthermore, as the first component is dependent on a parameter ~, i.e. V~ = V~ (a), it is shown that there exists a positive value = 0b such that the...
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