Transition Count Research Articles

Large deviations, hypotheses testing, and source coding for finite Markov chains

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{X_{n}\} n \geq 1</tex> be a finite Markov chain with transition probability matrix of strictly positive entries. A large deviation theorem is proved for the empirical transition count matrix and is used to get asymptotically optimal critical regions for testing simple hypotheses about the transition matrix. As a corollary, the error exponent in the source coding theorem for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{X_{n}\}</tex> is obtained. These results generalize the corresponding results for the independent and identically distributed case.

Markov chain analysis finds a significant influence of neighboring bases on the occurrence of a base in eucaryotic nuclear DNA sequences both protein-coding and noncoding.

Sixty-four eucaryotic nuclear DNA sequences, half of them coding and half noncoding, have been examined as expressions of first-, second-, or third-order Markov chains. Standard statistical tests found that most of the sequences required at least second-order Markov chains for their representation, and some required chains of third order. For all 64 sequences the observed one-step second-order transition count matrices were effective in predicting the two-step transition count matrices, and 56 of 64 were effective in predicting the three-step transition count matrices. The departure from random expectation of the observed first- and second-order transition count matrices meant that a considerable sample of eucaryotic nuclear DNA sequences, both protein coding and noncoding, have significant local structure over subsequences of three to five contiguous bases, and that this structure occurs throughout the total length of the sequence. These results suggested that present DNA sequences may have arisen from the duplication, concatenation, and gradual modification of very early short sequences.

Chi-square tests for markov chain analysis

Markov chain analysis has become a popular and useful technique for the evaluation of stratigraphic information. Field data on frequencies of facies transitions are first assembled in a transition count matrix. Observed frequencies can be compared statistically with frequencies expected if no order, or memory, exists in the stratigraphic sequence. Two chi-square (χ2) tests have been proposed for this purpose. One of the previously proposed χ2 tests was observed to give anomalously large values of the test statistic during another study. This unsatisfactory behavior is verified here. Limiting values of the two χ2 tests (from tables) are compared with the distribution of values for both test statistics obtained by analyzing many “random” matrices generated by a simple computer program. In all cases, the logarithmic χ2 test, suggested by several previous authors, fails to provide any meaningful assessment of the presence or absence of order in stratigraphic sequences evaluated. Further use of this test should therefore be avoided.

On Redundancy and Fault Detection in Sequential Circuits

In this correspondence we show that the well-known concepts of redundancy and undetectability of a stuck-at fault, which are equivalent in combinational circuits, are not equivalent in sequential circuits. We also show that some faults in sequential circuits, which are undetectable (by "conventional" methods of testing), are detectable by transition count testing methods.

Mechanism of the hysteresis effect in rate-dependent photomultiplier gain variations

The hysteresis effect in rate-dependent photomultiplier gain variations observed in a particular ten-stage tube was extensively studied to clarify the mechanism involved. The transition points or count rates at which abrupt gain changes take place are measured with respect to the anode and ninth dynode pulses as a function of light pulse intensity and applied high voltage. The experimental results show that the transition occurs at constant average currents and is attributed to certain physical processes on the ninth dynode. Besides, extensive measurements of the hysteresis curves are performed with respect to the ninth dynode pulse for various biases applied to the tenth dynode. The results clearly indicate that the transitions are due to sudden changes in surface potential on the ninth dynode, presumably resulting from dielectric breakdown in the dynode surface layer. In accordance with these and other measurements, a possible physical mechanism is proposed to interpret the hysteresis effect. With the proposed mechanism, temperature effect on the transition points and possible dependence of the hysteresis effect on dynode structure are briefly discussed.

Generation of Optimal Transition Count Tests

The problem of generating minimum-length transition count (TC) tests is examined for combinational logic circuits whose behavior can be defined by an n-row fault table. Methods are presented for generating TC tests of length n+2 and 2n-1 for fault detection and fault location, respectively. It is shown that these tests are optimal with respect to the class of n-row fault tables in the sense that there exist n-row fault tables that cannot be covered by shorter TC tests. The practical significance of these tests is discussed.

Transition Count Testing of Combinational Logic Circuits

Logic circuits are usually tested by applying a sequence of input patterns S to the circuit under test and comparing the observed response sequence R bit by bit to the expected response Ro. The transition count (TC) of R, denoted c(R), is the number of times the signals forming R change value. In TC testing c(R) is recorded rather than R. A fault is detected if the observed TC c(R) differs from the correct TC c(Ro). This paper presents a formal analysis of TC testing. It is shown that the degree of detectability and distinguishability of faults obtainable by TC testing is less than that obtainable by conventional testing. t is argued that the TC tests should be constructed to maximize or minimize c(Ro). General methods are presented for constructing complete TC tests to detect both single and multiple stuck-line faults in combinational circuits. Optimal or near-optimal test sequences are derived for one-and two-level circuits. The use of TC testing for fault location is examined, and it is concluded that TC tests are relatively inefficient for this purpose.

Markov chain analysis applied to an ancient alluvial plain succession

ABSTRACTMarkov chain analysis is a comparatively simple statistical technique for the detection of repetitive processes in space or time. Coal measure cyclothems or fluvial fining‐upward cycles are good examples of sedimentary successions laid down under the control of Markovian processes.Analyses of stratigraphic sections commence with a transition count matrix, a two‐dimensional array in which all possible vertical lithologic transitions are tabulated. Various probability matrices may be derived from this raw data, and these are then subjected to chi‐square tests to determine the presence or absence of the Markov property. This technique is applied to four types of stratigraphic succession which occur in the Devonian rocks of Prince of Wales Island, Arctic Canada.(1) A conglomerate succession of alluvial fan origin. Markov analysis is of little or no assistance in the interpretation of these rocks, in which only two principal lithologies are present.(2) A conglomerate‐sandstone succession. Fluvial fining‐upward cycles are detectable by visual examination of the sections and are strongly indicated by Markov analysis.(3) A sandstone‐carbonate succession, of marginal marine origin, and including both marine and non‐marine strata. Cyclicity is weak in these rocks, but analysis suggests that regressions took place much more rapidly than transgressions during their period of deposition.(4) A succession in which the relative proportions of the various lithologies vary markedly with age. The varying nature of the cyclic tendencies is emphasized in this case by dividing the succession into two subintervals, for the purpose of analysis.

Asymptotically Optimal Tests for Finite Markov Chains

A discrete time, finite Markov chain with fixed initial state and stationary transition behavior is considered. Using Whittle's formula a large deviation result (similar to Hoeffding's result for one multinomial distribution) is obtained for the transition count matrix of a path of the chain of arbitrary length. This result is then used in the asymptotic comparison of a given sequence of tests about the transition probability matrix with a suitably constructed sequence of likelihood ratio tests. It is assumed that the sizes of these tests decrease to zero at a certain rate as the length of the observed path increases. The comparison is carried out at fixed alternatives in terms of the behavior of the ratio of type-II-error probabilities.

A central limit theorem for absorbing Markov chains

SUMMARY A central limit theorem is obtained for a sequence of random variables defined on a finite absorbing Markov chain, conditional on absorption not having taken place. The transition count for such a chain when suitably scaled is found to follow a multivariate normal distribution asymptotically. In the case where the transition probability matrix of the chain is a function of a single parameter ac, a consistent estimator for oc is found; this estimator is asymptotically normally distributed about the true value of c. The result is illustrated by a simulation study of a genetic model of Moran, for the case of one-way mutation in a population of gametes. Absorbing Markov chains appear frequently in the literature as models for processes in medicine and biology, where, for example, an absorbing state may represent the disappearance of a certain characteristic in the population. Bartlett (1960) pointed out that in many cases the time to reach an absorbing state is effectively infinite, and in such cases it is relevant to consider the behaviour of the process within the transient states. This leads to the study of chains in which it is known that absorption has not taken place at a certain stage, and to the theory of quasi-stationary distributions as developed by Darroch & Seneta (1965). In this paper we consider the behaviour of a finite absorbing Markov chain, conditional on the knowledge that an absorbing state has not been reached. In ? 2 a central limit theorem for processes defined on such a chain is derived. This theorem provides the basis for the theory of statistical inference which is developed in ? 3. The results are illustrated in ? 4 by a genetic model of Moran.