RECENT results1,2 from two ice cores drilled in central Greenland have revealed large, abrupt climate changes of at least regional extent during the late stages of the last glaciation, suggesting that climate in the North Atlantic region is able to reorganize itself rapidly, perhaps even within a few decades. Here we present a detailed stable-isotope record for the full length of the Greenland Ice-core Project Summit ice core, extending over the past 250 kyr according to a calculated timescale. We find that climate instability was not confined to the last glaciation, but appears also to have been marked during the last interglacial (as explored more fully in a companion paper3) and during the previous Saale–Holstein glacial cycle. This is in contrast with the extreme stability of the Holocene, suggesting that recent climate stability may be the exception rather than the rule. The last interglacial seems to have lasted longer than is implied by the deep-sea SPECMAP record4, in agreement with other land-based observations5,6. We suggest that climate instability in the early part of the last interglacial may have delayed the melting of the Saalean ice sheets in America and Eurasia, perhaps accounting for this discrepancy.

Evidence for general instability of past climate from a 250-kyr ice-core record

A summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented in this tutorial. An examination of the underlying time series model assumed by each technique serves as the common basis for understanding the differences among the various spectrum analysis approaches. Techniques discussed include the classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods. A summary table in the text provides a concise overview for all methods, including key references and appropriate equations for computation of each spectral estimate.

Spectrum analysis&#8212;A modern perspective

Jaynes's principle of maximum entropy and Kullbacks principle of minimum cross-entropy (minimum directed divergence) are shown to be uniquely correct methods for inductive inference when new information is given in the form of expected values. Previous justifications use intuitive arguments and rely on the properties of entropy and cross-entropy as information measures. The approach here assumes that reasonable methods of inductive inference should lead to consistent results when there are different ways of taking the same information into account (for example, in different coordinate system). This requirement is formalized as four consistency axioms. These are stated in terms of an abstract information operator and make no reference to information measures. It is proved that the principle of maximum entropy is correct in the following sense: maximizing any function but entropy will lead to inconsistency unless that function and entropy have identical maxima. In other words given information in the form of constraints on expected values, there is only one (distribution satisfying the constraints that can be chosen by a procedure that satisfies the consistency axioms; this unique distribution can be obtained by maximizing entropy. This result is established both directly and as a special case (uniform priors) of an analogous result for the principle of minimum cross-entropy. Results are obtained both for continuous probability densities and for discrete distributions.

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Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy

Singular-spectrum analysis: a toolkit for short, noisy chaotic signals

With the continuing growth of modern communications technology, demand for image transmission and storage is increasing rapidly. Advances in computer technology for mass storage and digital processing have paved the way for implementing advanced data compression techniques to improve the efficiency of transmission and storage of images. In this paper a large variety of algorithms for image data compression are considered. Starting with simple techniques of sampling and pulse code modulation (PCM), state of the art algorithms for two-dimensional data transmission are reviewed. Topics covered include differential PCM (DPCM) and predictive coding, transform coding, hybrid coding, interframe coding, adaptive techniques, and applications. Effects of channel errors and other miscellaneous related topics are also considered. While most of the examples and image models have been specialized for visual images, the techniques discussed here could be easily adapted more generally for multidimensional data compression. Our emphasis here is on fundamentals of the various techniques. A comprehensive bibliography with comments is included for a reader interested in further details of the theoretical and experimental results discussed here.

Image data compression: A review

The duality between the maximum entropy method (MEM) of spectral analysis and the autoregressive (AR) representation of the data allows the application of recent advances in AR analysis to MEM in an attempt to obviate some shortcomings in this method of spectral decomposition. Specifically, this paper investigates the work of Akaike (1969a, b) on a criterion for choosing the length of the required prediction error filter and compares two methods of determining the filter coefficients. Recent work by Kromer (1970) on asymptotic properties of the AR spectral estimator is also of importance. Some preliminary results of the splitting of the normal modes of the earth are presented as an illustration of the application of MEM to geophysics.

Tad J. Ulrych

Papers

Maximum entropy spectral analysis and autoregressive decomposition