We studied dimension and entropy estimators and their applicability to time series obtained from models and biological sources. In this chapter we give a summary of this study with conclusions and suggestions for future research.

Nearest neighbour methods are based on the scaling behaviour of the distances between points in phase space as a function of the neighbour index () or the number of points (). We introduced a variant of this method, where the neighbour index is a function of the number of points, of the form .

- It is unclear whether this form and which values for and are optimal.
- The correction function (eq. (3.10)) should be incorporated into the function that describes the scaling behaviour of nearest neighbours including the entropy function (eq. (3.11)).
- The statistical properties of the estimators of the and should be derived.
- The k-method resembles the correlation integral method and it should be investigated how these are connected. Since we will specify a region instead of we may start by writing a likelihood function for a ``Type II doubly censored data set'' see [Kendall and Stuart, 1979, eq. 32.38].
- It should be investigated whether the k(N)-method is indeed to be preferred over the k-method with the maximum value of .
- In general, the N-method will not be optimal because the -th nearest neighbour will always enter the noise region for sufficiently large .
- It is difficult to identify a ``scaling region'', i.e. an interval for which the scaling behaviour of nearest neighbour distances is not distorted by noise or by the finite attractor size. A reasonable scaling region may result in large systematic errors in the estimated . It could be that estimating a ``local'' dimension at each value of alleviates this problem.

We studied the correlation integral method because it provides a good means for identifying the scaling region (that is, for the correlation dimension ) and that the form of the distribution for the distances is known ( ). We derived maximum likelihood estimators of the correlation dimension and entropy by using information from outside the scaling region.

- We compared the estimators that can be derived under the different conditions of truncation and censoring.
- We found expressions for the bias and variance of the ``Takens'' estimator (see section 5.2.1) for finite .
- Since the asymptotic variance of the estimator of the dimension decreases by censoring instead of truncating on the left, there is some information contained in the smallest distances. Furthermore, by using the information from all distances, an estimator for the correlation entropy could be derived.
- We derived an expression for the asymptotic variance of the maximum likelihood estimator of the correlation entropy, under the conditions that the distances are independent, that the finite number of distances is the only source of error and that the scaling regions (for two embedding dimensions) have the same slope: the correlation dimension. Monte Carlo simulations showed that this expression is (probably) correct and useful.
- Increasing the value of the parameter which is the difference between the two embedding dimensions, leads to an increasing precision of the estimated dimension and entropy.
- It is not clear how to obtain independent distances when the source is a time series. Monte Carlo simulations provide a means to study the properties of the dimension and entropy estimators. For example, Monte Carlo simulations using models of chaotic systems showed that the number of distances one can use is approximately ten times the length of the time series, for the expressions of the asymptotic variances to be useful. If one uses more distances, the estimates will become more precise, but not as precise as given by the asymptotic variance.
- It should be studied how other sources of error, for example lacunarity, affect the estimators of dimension and entropy.
- A test can be used to test whether the scaling region is straight; passing that test, however, does not imply that the slope is the correlation dimension.
- There is a need for an automatical, objective way of identifying the scaling region.
- The statistical properties of the estimators of the generalized dimensions () and entropies () should be derived.

We applied the maximum likelihood estimators of the correlation dimension and entropy to time series obtained from the measurements of the brain activity (electroencephalograms) and of the respiration (volume of breaths).

- For these sources (and others which are not reported here), we could not identify low-dimensional chaotic dynamics. Usually we could not even identify scaling regions: they were ambiguous, short, and hopping around on different length scales with increasing embedding dimension. Biological systems are probably chaotic, but with a very high dimension.
- There should be a way to discriminate between high-dimensional chaotic and ``pure stochastic'' behaviour.
- The method of reducing noise by computing singular values was not useful for the estimation of the dimension of an attractor related to the alpha rhythm.
- Without the Theiler correction, it is possible to find a spurious saturation of the dimension vs. embedding dimension curve.
- The presence of a dominant (apparently) periodic component in a signal (respiration) gives rise to a low-dimensional scaling behaviour of the scaling region at a relatively high distance range.
- Spectral analysis revealed the presence of the heart rate in the respiration signal and the hum in the electroencephalograms; our most chaotic analysis did not.
- The accuracy of the studied estimators of the dimension decreases rapidly with increasing dimension. There is a need for other, independent, ways of estimating properties of chaotic systems.
- It should be investigated how much useful information is contained in the dimension and entropy estimates when there is a lack of saturation, for the comparison between and approximation of high-dimensional chaotic systems.