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Introduction

Both the correlation integral method and the maximum likelihood methods described in this chapter are based on the assumption that distances $r$ between two randomly chosen points on an attractor in phase space obey the cumulative distribution function [Grassberger and Procaccia, 1984]
\begin{displaymath}
P(r) = \phi r^{\nu} = \tilde{\phi} \exp (-d \tau K_{2}) r^{\nu}
\end{displaymath} (5.1)

It has been shown that the variance of the estimated correlation dimension using linear least squares methods is severely underestimated (see Chapter 4). Takens [Takens, 1985] and Ellner [Ellner, 1988] have derived maximum likelihood estimators of the correlation dimension, with expressions for their variances under the assumption that the finite number of distances is the only source of error. Maximum likelihood (ML) estimators are, in general, biased, but asymptotically normal with [Lehmann, 1983]
\begin{displaymath}
\sqrt{N}(t-T(\theta)) \stackrel{\cal L}{\rightarrow} p_N(0,\sigma)
\end{displaymath} (5.2)

where $N$ is the number of observations, $t$ is a vector of ML estimators of functions $T$ of parameters $\theta$, $p_N$ is the multivariate normal distribution, $\sigma^{2}$ are the asymptotic variances and $\cal L$ denotes ``convergence in probability''.

To derive maximum likelihood estimators of the dimension and entropy, we begin by finding an appropriate likelihood function. Consider a sample that consists of $N$ (identically independently distributed) distances on an attractor, with $N_{l}$ in the range $[0,r_{l}]$, $N_{s}$ in $]r_{l},r_{u}]$ and $N_{u}$ in $]r_{u},1]$. The distribution (5.1) is assumed to hold for distances in $]r_{l},r_{u}]$, which is called the scaling region (see Chapter 4). The distribution is assumed not to hold for distances $r \leq r_l$ and for distances $r > r_u$ due to noise and the finite size of the attractor respectively. Note that all distances have been divided by a reference distance so that the maximum distance on the attractor is 1. Since we know $P(r)$ for $]r_{l},r_{u}]$, we also know $P(r_l)$ and $1 - P(r_u)$, so that likelihood functions can be constructed by using ``truncated'' and ``censored'' data sets (see [Kendall and Stuart, 1979, §32.15]). A distribution is called truncated if both the number of observations and their values are unknown in certain ranges of the variate; a sample is called censored if the number of observations in certain ranges is known, but their values are unknown.

In the following sections we derive various ML correlation dimension estimators and derive expressions for their asymptotic variances using various likelihood functions. Furthermore, in section 5.3 we derive a ML estimator of the correlation entropy and an expression for its asymptotic variance. In section 5.4 we discuss generalizations, i.e. methods for estimating the whole dimension spectrum with maximum likelihood methods. In section 5.5, we discuss goodness-of-fit tests, to test whether the distribution (with estimated values for $\nu$ and $\phi$) fits to the actual data or not. The validity of the expressions for the variances is supported by Monte Carlo simulations in section 5.6. Finally, in section 5.7, we describe the current implementations of the estimation algorithms.


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Next: The estimation of the Up: MAXIMUM LIKELIHOOD METHODS Previous: MAXIMUM LIKELIHOOD METHODS   Contents
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