M. Casdagli, "Chaos and deterministic versus stochastic nonlinear modeling, " J. Roy. Stat. Soc. B, vol. 54, pp. 303--328, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of dynamical noise, the observed states form a discrete time, continuous state Markov Chain, and estimating interesting features of the dynamics (e.g. the map F ) can often be accomplished in part by an appeal to traditional time series techniques. Representative work can be found in references [37, 19, 11, 12, 23, 31, 26, 35]; an alternative approach to the map estimation problem is described in [30] Of interest here is the so called observational noise model, in which the available data are observations (or measurements) of an underlying deterministic system that are corrupted by additive noise. In this model our ....

Casdagli, M., Chaos and deterministic versus stochastic non-linear modeling, J. R. Stat. Soc. B, vol.54, pp.303-328, 1992.

....more predictable than white noise, and that the predictability falls off in a manner consistent with chaos [74] The interpretation of results such as these is somewhat difficult, since stochastically forced linear systems can show similar fall offs. A second approach, due originally to Casdagli [9] examines whether locally fit linear models perform better at forecasting than globally fit models. If the locally fit models are better, this provides evidence for nonlinear structure in the dynamics even nonlinear stochastic structure can be detected in this manner. This approach has been ....

Casdagli, M, "Chaos and Deterministic versus Stochastic Non-linear Modelling ", J. R. Statist. Soc. B. 54:302-328, 1991

....illustrated in Fig. 3b, we need to impose some constraints on the types of functions we will allow in constructing a function F : IR p IR. The literature suggests many possibilities for suitable forms of functions, for example, radial basis functions [5] or piecewise linear functions [6]. Another approach can be taken, though, that avoids the need to fit any function. Suppose that there is a continuous function F : IR p IR describing the time series fx i g N i=1 . If this function is continuous, then we expect that if two points x j and x k are very close together, then ....

M Casdagli "Chaos and deterministic versus stochastic non-linear modeling." J. R. Statist. Soc. B 54(2):303-328 (1992)

....and therefore it is important to investigate the nonlinear dynamical properties of speech before attempting any synthesis. This is an area that has received much interest recently with many authors reporting differing evidence for and against the existence of low dimensional attractors for speech [4, 5, 1, 6]. In a previous paper [3] the authors have shown that there is evidence that speech is a low dimensional, nonlinear, non chaotic system, and as such it should be feasible to use the dynamics as a synthesis tool. 1 Patent application number GB 9600774.5 2 This work funded by BT Labs Martlesham, ....

....In previous analyses [3] the short term prediction properties and the Lyapunov spectra for isolated vowels were explored . The short term prediction properties show that a locally linear model performs better than a globally linear model and suggesting that the system can be considered as nonlinear[4]. Further more the results suggest that the system is low dimensional and that the vowel sounds contain varying amounts of intrinsic fricative noise which must be considered in the calculation of Lyapunov spectra. A similar analysis on fricatives shows them to be modelled well by high dimensional, ....

M.Casdagli. Chaos and deterministic versus stochastic non-linear modelling. Journal of the Royal Statistical Society B, 54(2):303--328, 1991.

....problem of estimating an iterated map has previously been studied primarily in the context of smooth dynamical systems, with the ultimate goal of prediction, estimating Lyapunov exponents, or estimating the dimension of an attractor. Representative work and additional references can be found in [12, 6, 7, 19, 28, 21, 35], and the surveys [11, 18, 17] In most of this work it is assumed that the map under study is di erentiable, and that successive iterates of the map are perturbed by observational or dynamical noise. Central limit theorems for U statistics and smooth functionals of noiseless dynamical systems ....

Casdagli, M. (1992). Chaos and deterministic versus stochastic non-linear modeling. J. R. Stat. Soc. B, 54:303-328.

....in the context of chaos and non linear dynamics, where the ultimate goal is typically prediction, or the estimation of some features of the dynamics such as Lyapunov exponents or the dimension of an attractor. Representative work can be found in the papers of Farmer and Sidorowich [8] Casdagli [5, 6], Kostelich and Yorke [10] Nychka et al. 16] and Lu and Smith [12] Additional work and references can be found in the book of Tong [20] This work di ers from that in the present paper in several respects. The cited references consider continuous or, more commonly, di erentiable ....

M. Casdagli, Chaos and deterministic versus stochastic non-linear modeling. Journal of the Royal Statistical Society, series B, 54 (1992), 303-328.

....[8] which groups all the methods which defer the learning procedure until a specific query needs to be answered. In this paper we propose an application of AMB to the problem of timeseries prediction which has been already the focus of numerous studies in the classic memory based literature [9, 10, 11]. In particular, we present some experimental results obtained in the prediction of the Mackey Glass chaotic time series. 2 The adaptive memory based paradigm In a function estimation approach, the dominant criterion is the global performance of the resulting approximator over the whole input ....

M. Casdagli, "Chaos and deterministic versus stochastic non-linear modelling ", Journal of the Royal Statistical Society, vol. 55, no. 2, pp. 303-- 328, 1991.

....data sets (Theiler et al. 1991) we show that filtering techniques can give some spurious evidence for the presence of deterministic nonlinear behavior. Consequently, any predictions based on the assumption of such a process are not significantly better than those from linear stochastic models (Casdagli, 1991). Copyright 1992 by the American Geophysical Union. Paper number 92JA01459. 0148 0227 92 92JA 01459 05.00 19,113 1. Introduction Sunspots have been recorded for millennia. There are Chinese oracle bones dating from before 1000 B.C. which record sunspots [Hsu, 1972] The current record of monthly ....

.... deterministic low dimensional nonlinear process and subsequent analysis of various data sets derived from the monthly Wolf number set have been made by Kurths (1987) and by Ajmanova and Makarenko (1988) referenced in the work by Kurths and Ruzmaikin [1990] Mundt et al. 1991] Hogenson [1992] Casdagli et al. 1991a] and Casdagli [1991] Kurths and Ruzmaikin [1990] Mundt et al. 1991] Theiler et al. 1991] Casdagli et al. 1991a] Casdagli [1991] and Hogenson [1992] make predictions of future sunspot cycles using nonlinear techniques. None of the previous work has application to the basic problem of ....

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Casdagli, M., Chaos and deterministic versus stochastic nonlinear modeling, Report SFI 91-07-029, Santa Fe Inst., Santa Fe, N.M., 1991.

....the means mk and variances oe 2 k can be viewed as a kind of bootstrap estimation: the original sample y is used to generate a larger sample y which is then used to estimate some parameters related to the original sample. This is especially obvious if nearest neighbors types of predictors [14] are used for f . 5. PRELIMINARY RESULTS A preliminary series of simulations has been conducted in order to validate the proposed approach. In these preliminary experiments, a standard 1 D chaotic models is used: the logistic equation. The logistic model is defined by the ....

M. Casdagli, "Chaos and deterministic versus stochastic nonlinear modelling", J. Roy. Stat. Soc. B, vol. 54, no. 2, pp. 303--328, 1991.

....N = 6000, 2:05 (9) 2:9 (9) medicine EEG, long lived epilepsy N = 18000, 5:6 (10) 1:0 (10) biology marine plakton diatoms N = 830, 2 (11) Table 1: Estimates of correlation dimension and the largest Lyapunov exponent from real data of size N . The references in the table are: 1) [Casdagli, 1992], 2) Sano and Sawada, 1985] 3) Brandstater and Swinney, 1987] 4) Harding et al. 1990] 5) Nicolis and Nicolis, 1984] 6) Grassberger, 1986] 7) Vautard and Ghil, 1989] 8) Destexhe et al. 1988] 9) Babloyantz and Destexhe, 1986] 10) Frank et al. 1990] 11) ....

M. Casdagli. Chaos and deterministic versus stochastic nonlinear modeling. Journal of the Royal Statistical Society Series B-Methodological, 54:303 328, 1992.

.... are considered in [16,19,176,313,379,386,403,417] A radically different approach to modeling self similar phenomena relies on ideas from the theories of chaos and fractals [73,97,118 120,124,125,171,250,287,337,344,345,377] for a general discussion on chaos, probability and statistics, see [29, 46, 47] An overview of statistical inference methods for self similar models and random processes with long range dependence can be found in [22, 24] the papers [392 394] listing additional techniques. More specifically, R S analysis is discussed in [18, 24, 26, 28, 130, 200, 258, 272, 273, 286, 288, ....

M. Casdagli. Chaos and deterministic versus stochastic non-linear modeling. Journal of the Royal Statistical Society, Series B, 54:303--328, 1991.

....averages have been used extensively (e.g. 26] and analysed extensively (e.g. 15] 3. Linear models the group of nearest neighbours is used to create a local linear model. This model is then used to find the desired value for a new point. One example, which we use later, is given by Casdagli [4]: y = Theta 1 x 1 : x k 2 6 6 6 4 ff 0 ff 1 . ff k 3 7 7 7 5 (1) where ff i , i = 0; 1; are constants, k is the number of neighbours, and n is the dimension of the input vector x i , i = 1; 2; k. The parameters ff i , i = 0; 1; k are found by using ....

M. Casdagli. Chaos and deterministic versus stochastic non-linear modelling. J.R. Statistical Society B, 54(2):302--328, 1991.

....the predicted value (local AR model) 9] As k increases, the model approaches the classical AR model. In the case of pure chaotic data, best predictions are obtained for small k. In the case of nonlinear systems corrupted with noise, the best predictions are expected for moderate values of k ([10]) The estimation of the local linear maps is done with ordinary least squares (OLS) as in the case of global AR. Recently, it has been shown in [11] that regularization of OLS, e.g. using the principal component regression (PCR) concept, enhances the predictability of LLP applied to noisy data, ....

M. Casdagli. Chaos and deterministic versus stochastic nonlinear modeling. Journal of the Royal Statistical Society Series B-Methodological, 54:303 -- 328, 1992.

.... 2 For comparison purposes, we have used the TDNN (Time Delay Neural Network) architecture 2 , the Back Tsoi FIR 3 and IIR MLP architectures (Back Tsoi 1991b) where every synapse contains an FIR or IIR filter and a gain term, and the local approximation algorithm used by Casdagli (k NN LA) (Casdagli 1991) 4 . The Gamma MLP is a special case of the IIR MLP. 3 TASK 3.1 MOTIVATION Accurate speech recognition requires models which can account for a high degree of variability in the data. Large amounts of data may be available but it may be impractical to use all of the information in standard ....

Casdagli, M. (1991), `Chaos and deterministic versus stochastic non-linear modelling', J.R.

....is linear, but which has a long (in fact, infinite) coherence time. 3.3 Linear versus nonlinear modeling: an example Another way to test for nonlinearity in a time series is to compare the linear and nonlinear models to see which more accurately predicts the future. For example, Casdagli [8, 9] has described an exploratory approach in which the data is fit with local linear models using k nearest neighbors. The parameter k is swept from m 1, the minimum value required to make a local linear fit in m dimensions, up to the size N of data set itself. For k N , the model is ....

....the model itself. One finds that for the same embedding dimension m, nonlinear models fit this data better than linear models. Theiler, Linsay, and Rubin, p. 20 In particular, as seen in Fig. 8, Casdagli s plots of forecasting error as a function of number of neighbors in the local linear fit [8, 9] indicate nonlinearity in a time series, even though the system is formally speaking linear. Our intuitive explanation is that the nonlinear models are able to use information that is unavailable to the direct linear model; namely the amplitudes and relative phase of the two sine waves. So, while ....

M. Casdagli, "Chaos and deterministic versus stochastic nonlinear modeling," J. R. Stat. Soc. B 54, 303--328 (1992).

....the time series at some delayed time values to create a delay coordinate vector. This is sometimes referred to as a phase space. As a means of comparing each algorithm, we benchmark their relative performances againsta windowed input MLP, and a local approximation method developed by Casdagli [5] (a version of the nearest neighbor method) Obviously, there are many variations in which this could have been done. Our intent is to provide a reasonable means of quickly assessing the performance of these algorithms which may provide a starting point for anyone interested in considering them ....

M. Casdagli, "Chaos and Deterministic versus Stochastic Non-linear Modelling", J. R. Statist. Soc. B, 1991, 54, No. 2, pp. 303-328.

....the ranks of the data values x i , it will be pivotal. #8 When testing for nonlinearity, with an eye to the alternative of chaos, one may be interested in using fairly exotic discriminating statistics, involving fractal dimensions [15, 37 41] Lyapunov exponents [42 48] or nonlinear predictors [49 52], as well as various hybrid statistics which measure determinism without directly predicting [53 57] And it can be difficult to enforce the requirement that these discriminating statistics be pivotal. It may be easier, we argue, to make the randomization method itself pivotal (in a manner of ....

M. Casdagli, "Chaos and deterministic versus stochastic nonlinear modeling." J. R. Stat. Soc. B 54, 303--328 (1992).

....and coworkers [96] In his contribution to the Santa Fe Institute time series contest in 1991, Sauer [97] has emphasised the close interplay between phase space embedding and fitting of the dynamics. The optimal degree of locality of a locally linear modeling approach has been used by Casdagli [98] as a measure for nonlinearity in a time series. He compares the predictive quality of models fitted with using different numbers of neighbours. In the absence of nonlinearity, the globally linear fit using all available points as neighbours should give best results since it uses the largest ....

M. Casdagli, Chaos and deterministic versus stochastic nonlinear modeling, J. Roy. Stat. Soc. 54 (1991) 303.

....while the dashed line is that of a (more accurate) third predictor combinationing the two, as described in the text. Implied Chaos 13 timal, it d be nice, but computationally expensive, to choose a neighborhood with respect to the local characteristics of the underlying function. Casdagli [4, 5] has investigated the variation of predictions with k using local linear maps in a variety of circumstances. 3.4.1 Nearest neighbor(s) The nearest neighbor predictor simply chooses the point in the learning set closest to the point to be predicted and uses its image as the prediction. As shown ....

M. Casdagli. Chaos and deterministic versus stochastic non-linear modeling. J. R. Statist. Soc. B, 54(2):303--328, 1992.

....(10) which must satisfy ( t H x y n n = 1 (11) Finding t is then a function interpolation problem. Several strategies have been suggested in work concerned with the accurate short term prediction of time series believed to come from chaotic systems (Broomhead, D.S. Farmer, D. 1987, Casdagli, M. 1989, Taylor, W. Singer, A.C. These include a single global nonlinear function defined over the whole embedded state space, a piece wise linear or polynomial function and radial basis functions. For this work, the emphasis is on having a resynthesis model that is computationally simple as ....

Casdagli, M. Chaos and Deterministic versus Stochastic Nonlinear Modelling. Journal of the Royal Statistical Society B, V54, N2, pp303-328, 1992.

....more accurate than linear predictions. However, we will ignore this issue, and use the term recurrence in this paper whenever the nonlinear prediction algorithm identifies nonlinearity. The nonlinear prediction algorithm we use is a simple modification of algorithms described elsewhere [2, 3, 6, 9, 24, 26, 28], and is defined as follows. First, the EEG recording is resampled at 50Hz to save on computation time (Qualitatively similar results were obtained from a limited amount of data sampled at 200 Hz when tested for 4 step ahead nonlinear predictability using the technique of this section) Second, ....

....(b) d) RST4, LTD3 and ROF2 60min before the seizure. Fig. 6 shows how the prediction error E(k) varies with the number of neighbors k, for several EEG segments and surrogate data sets, and provides a comparison of nonlinear deterministic (small k) versus linear stochastic (large k) models [3,5]. We chose the embedding dimension m = 4; larger values of m did not lead to substantial decreases in the prediction errors E(k) Fig. 6(a) is for the EEG recording from electrode LTD1 during the ictal period shown in Fig. 2(a) Nonlinear models give significant improvements in predictive accuracy ....

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M.C. Casdagli. Chaos and deterministic versus stochastic nonlinear modeling. J. Roy. Statist. Soc. Ser. B 54 (1992) 303-328.

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M. Casdagli, "Chaos and deterministic versus stochastic nonlinear modeling, " J. Roy. Stat. Soc. B, vol. 54, pp. 303--328, 1991.

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M. Casdagli. Chaos and deterministic versus stochastic non-linear modelling. J. R. Statist. Soc. B, 54:303--28, 1991.

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Casdagli, M. (1991). Chaos and Deterministic versus Stochastic Non-linear Modelling, J. R. Statist. Soc. B, 54(2), 303--328.

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M. Casdagli. Chaos and deterministic versus stochastic non-linear modeling. J. R. Statist. Soc. B, 54(2):303-328, 1992. 85

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Casdagli, M., "Chaos and deterministic versus stochastic nonlinear modeling" Journal of the Royal Statistical Society B, 54, 1991, pp. 303

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-16. Casdagli, M. 1991 Chaos and deterministic versus stochastic non-linear modelling. J. R. Statist. Soc. B 54,

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M.Casdagli. Chaos and deterministic versus stochastic non-linear modelling. Journal of the Royal Statistical Society B, 54#2#:303#328, 1991.

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Casdagli, M. (1992). Chaos and deterministic versus stochastic non-linear modeling. J. R. Statist. Soc. B, 54, No. 2, 303-328.

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M. Casdagli, "Chaos and Deterministic versus Stochastic Non-Linear Modelling", J. Roy. Statist. Soc. 54, Series B, 303-328, 1991.

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