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Register a new userA simple randomized algorithm for sequential prediction of ergodic time series
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Added by
Gusztav Morvai
Publication date
2008
fields
Informatics Engineering
and research's language is
English
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We present a simple randomized procedure for the prediction of a binary sequence. The algorithm uses ideas from recent developments of the theory of the prediction of individual sequences. We show that if the sequence is a realization of a stationary and ergodic random process then the average number of mistakes converges, almost surely, to that of the optimum, given by the Bayes predictor. The desirable finite-sample properties of the predictor are illustrated by its performance for Markov processes. In such cases the predictor exhibits near optimal behavior even without knowing the order of the Markov process. Prediction with side information is also considered.
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The problem of extracting as much information as possible from a sequence of observations of a stationary stochastic process $X_0,X_1,...X_n$ has been considered by many authors from different points of view. It has long been known through the work of D. Bailey that no universal estimator for $textbf{P}(X_{n+1}|X_0,X_1,...X_n)$ can be found which converges to the true estimator almost surely. Despite this result, for restricted classes of processes, or for sequences of estimators along stopping times, universal estimators can be found. We present here a survey of some of the recent work that has been done along these lines.
The forward estimation problem for stationary and ergodic time series ${X_n}_{n=0}^{infty}$ taking values from a finite alphabet ${cal X}$ is to estimate the probability that $X_{n+1}=x$ based on the observations $X_i$, $0le ile n$ without prior knowledge of the distribution of the process ${X_n}$. We present a simple procedure $g_n$ which is evaluated on the data segment $(X_0,...,X_n)$ and for which, ${rm error}(n) = |g_{n}(x)-P(X_{n+1}=x |X_0,...,X_n)|to 0$ almost surely for a subclass of all stationary and ergodic time series, while for the full class the Cesaro average of the error tends to zero almost surely and moreover, the error tends to zero in probability.
The setting is a stationary, ergodic time series. The challenge is to construct a sequence of functions, each based on only finite segments of the past, which together provide a strongly consistent estimator for the conditional probability of the next observation, given the infinite past. Ornstein gave such a construction for the case that the values are from a finite set, and recently Algoet extended the scheme to time series with coordinates in a Polish space. The present study relates a different solution to the challenge. The algorithm is simple and its verification is fairly transparent. Some extensions to regression, pattern recognition, and on-line forecasting are mentioned.
Let ${X_n}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${cal X}$. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times $lambda_n$ along which we will be able to estimate the conditional probability $P(X_{lambda_n+1}=x|X_0,...,X_{lambda_n})$ from data segment $(X_0,...,X_{lambda_n})$ in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then $ lim_{nto infty} {nover lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $lambda_n$ is upperbounded by a polynomial, eventually almost surely.
This study concerns problems of time-series forecasting under the weakest of assumptions. Related results are surveyed and are points of departure for the developments here, some of which are new and others are new derivations of previous findings. The contributions in this study are all negative, showing that various plausible prediction problems are unsolvable, or in other cases, are not solvable by predictors which are known to be consistent when mixing conditions hold.
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