Methods for the quantification of assemblage zones based on multivariate analysis of weighted and unweighted data

A general strategy for the quantification of assemblage zones is outlined. The data represent presences and absences of different species for numerous samples in n stratigraphic sections, and can be applied in raw form or converted to range through form. Next, the attributes of an index fossil, vertical range, geographic persistence, and degree of facies independence are quantified for each species. With this information, relative biostratigraphic values (RBV's) can be calculated for the taxa. The RBV's measure the amount of biostratigraphic information conveyed by the presence of a particular species. Numerous RBV's are available and the appropriate one can be selected which incorporates the required properties. Depending on the distribution of the RBV's, some or all of the taxa with low RBV's may be omitted from later stages of the analysis. The presences can be weighted by the RBV's of the species. Absences remain absences regardless of the RBV of the taxon concerned. The advantage of this technique is that each species contributes an amount of information that is proportional to its RBV. With either weighted or unweighted data, the next step is the computation of a similarity matrix between all pairs of samples. Inasmuch as a species can be absent from a sample for various reasons, absences cannot be evaluated completely for biostratigraphic data. Consequently only similarity coefficients based on mutual presences or positive matches are recommended. Various similarity coefficients, such as Jaccard, Dice, or Simpson can be employed although we prefer middle-of-the-road types such as the Dice coefficient. The last step in the analysis is to extract the main themes from the similarity matrix with methods such as cluster analysis, principal components, factor analysis, principal coordinates, multidimensional scaling, or lateral tracing. For our case studies, the best results have been obtained by cluster analysis, multidimensional scaling, and lateral tracing. A two-fold approach is adopted. First, the entire data set is treated in order to gain an overall picture and the desired correlations or zones are extracted from the clusters or ordinations. Second, the analysis focuses on pairs of adjacent sections to prepare a line of sections or a fence diagram yowing more detailed correlations. The data matrix can be transposed and an R-Mode analysis can be performed on the species with the same techniques.     

A comparison of methods for the quantification of assemblage zones

Several methods of quantitative biostratigraphy that are based on assemblage zones are examined utilizing three sets of faunal distribution data. Two of the data sets are structured simply and one is complex. Various types of cluster analysis and multidimensional scaling are applied to weighted and unweighted binary (presence/absence) data. For weighted data, the presences are multiplied by the relative biostratigraphic values (RBV) of the taxa. There are two options for calculating the RBV's. One method (RBV2) emphasizes timestratigraphic correlation, whereas the other is a compromise between time-stratigraphic correlation and biofacies correlation (RBV1). Results from lateral tracing also are examined.The case studies allow the formulation of a general strategy. Weighting is not appropriate if paleoecological groupings are sought. If biostratigraphic zonation is required, weighted data may produce clusters that are stratigraphically more homogeneous than those based on unweighted data. Also the RBV's can point out species that can be deleted from the analysis without losing significant biostratigraphic information. Range through data should be employed in most situations. Similarity matrices between samples can be calculated from various coefficients based on presences. Biostratigraphic zones are extracted from the similarity matrix by cluster analysis, multidimensional scaling and lateral tracing to produce an overall view of the data structure. Lines of correlation and fence diagrams can be constructed between the samples in adjacent stratigraphic sections using the same techniques.     

Ser,a fortran program for the seriation of biostratigraphic data

A FORTRAN program which involves simple iterative, averaging, and sorting operations is effective for seriation of biostratigraphic data. The data matrix includes the presence/absence of m taxa and n samples which are grouped in p stratigraphic sections. The basic procedure is to arrange the taxa and samples into a range chart by concentrating the presences along the diagonal of the matrix in order to minimize the range zones of the taxa. The method can calculate two types of seriation. If information on stratigraphic position of the samples is ignored, unconstrained seriation results and the samples are free to group in any order. This method usually results in sequences of taxa and samples that are allied closely to those derived from multivariate techniques such as cluster analysis. If the data on stratigraphic superposition of the samples are used, the result is constrained seriation in which the samples remain in stratigraphic order in the final matrix. Range charts derived from this type of seriation are most useful for stratigraphic correlation.     

A note on three-dimensional finite automata

This paper investigates the relationships between the accepting powers of three-dimensional six-way finite automata (3-FA's) and three-dimensional five-way Turing machines (5WTM's), where the input tapes of these automata are restricted to cubic ones. A 3-FA (5WTM) can be considered as a natural extension of the two-dimensional four-way finite automaton (two-dimensional three-way Turing machine) to three dimensions. The main results are: (1) n²logn (n³) space is necessary and sufficient for deterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's; (2) n² (n²) space is necessary and sufficient for nondeterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's.     

Set-theoretic foundations of data-structure representation

A formal set-theoretic model for the computer representation of data is developed. The key idea of the model is the concept of the distinguished urelement. A Zermelo-Fraenkel set theory ZF is the starting point. This theory contains all the fundamental sets which are required. A new Zermelo-Fraenkel theory ZF' is then formed which is identical to ZF except that it contains an additional constant a, interpreted as a urelement.Working in ZF', new definitions for the usual constructions in mathematics (e.g. permutation, projection, join, composition) are developed for the elements in ZF. These definitions are far better suited for computer representations than are the usual definitions. In particular, the new definition of ordered n-tuple possesses a symmetry not found in the usual definition.To insure rapid and accurate computer implementation of common set-theoretic operations such as union, intersection, difference, etc., it is extremely useful to have a canonical well ordering on the set of computer-representable elements. Using an extremely large class of sets from ZF' (one which contains ZF) as the basic sets, such an ordering theory is developed. Computation of the ordering involves simple checks on trees in a top-down manner.Finally, using the framework developed, a new definition of Codd's first normal form is presented. In contrast to previous definitions, this one allows sets as well as urelements as domain elements, while being perfectly consistent with the axioms of ZF'.     

ASSØCA: FORTRAN IV program for Williams and Lambert Association analysis with printed dendrograms

A general-purpose computer program implementing Williams and Lambert's method of association analysis is documented. The method embodies a hierarchical-divisive-monothetic strategy for the numerical classification of multiattribute, binary-coded data sets. The main user-controlled options permit a selection from among two different association coefficients and three termination criteria; employment of Yates' correction; presetting significance thresholds; discarding insignificant association coefficients; treatment of ambiguities in dividing attributes; and the transformation of quantitative measurements to a binary-coded form. A line-printer dendrogram plot displaying the final classification is generated as part of the output. Both normal and inverse association analysis can be performed.     

Real-time recognition of two-dimensional tapes by cellular automata

We introduce an n-dimensional cellular automaton (n-BCA) which accepts (n + 1)-dimensional tapes in real time. Here we regard an (n + 1)-dimensional tape as a time sequence of n-dimensional tapes, and we say that an (n + 1)-dimensional tape is accepted by an n-BCA if a final-state configuration of the acceptor belongs to a predetermined set of n-dimensional words. We state some features of the sets of 2-dimensional tapes accepted by deterministic 1-BCA's (1-DBCA's), including closure properties. For the unary input alphabet, we obtain that 1-DBCA's can recognize every set of tapes each of which has length represented by a polynomial in its width. For an arbitrary input alphabet, we obtain that the class of languages each of which consists of all the ith rows of 2-dimensional tapes accepted by a 1-DBCA coincides with the class of context-sensitive languages. For n ? 2, we show that a language not containing the empty word is recursively enumerable if and only if it is the set of top rows of (n + 1)-dimensional tapes accepted by an n-DBCA.     

Subjective response of standing persons exposed to fore-aft,lateral and vertical whole-body vibration

The aims of this study were to propose multiply scale factors for evaluation of discomfort of standing persons and to investigate whether there exist differences between multiplying factors used for evaluation of discomfort of standing persons and those of seated persons exposed to WBV. Twelve male subjects were exposed to twenty-seven stimuli that comprise three acceleration magnitudes (0.2, 0.4, and 0.8 m/s² r.m.s.) along fore-aft (x), lateral (y) or vertical (z) direction. The subjects with seated or standing posture on the platform of the vibration test rig rated the subjective discomfort for each stimulus that has frequency contents ranging from 1.0 Hz to 20 Hz with a constant power spectrum density. The order of presentation of the test stimuli was fully randomized and each stimulus was repeated three times. The subjective scale for discomfort was calculated by using the category judgment method. The best combinations of multiplying factors were determined by calculating correlation coefficients of regression curves in-between subjective ratings and vibration magnitudes. In all the directions, body posture significantly influenced on subjective discomfort scales. Particularly in the fore-aft and lateral direction, the upper limit of all the categories for the standing posture resulted in higher vibration acceleration magnitudes than those for the seated posture. In contrast, in the vertical direction, only the upper limit of category “1: Not uncomfortable” for standing posture was observed to be higher than that for seated posture. The best agreement for ISO-weighted vibration acceleration occurred at x factor of 1.8 and y factor of 1.8 in the standing posture and x factor of 2.8 and y factor of 1.8 in the seated posture. The results suggest that seated people respond more sensitively and severely in perception of discomfort to fore-aft and lateral vibration than standing people do while standing people respond more sensitively and severely to vertical vibration than seated people do. Thus the effects of body postures on multiplying factors should be considered in evaluation of discomfort caused by whole-body vibration.Relevance to industryThis study reports differences in subjective response of standing persons to fore-aft, lateral and vertical whole-body vibration. The results obtained in this study propose the fundamental data on the sensitivity to whole-body vibration exposed with standing posture.     

Unitary associations method: use of graph theory and computer algorithm

Two species are defined as compatible if their chronologic cooccurrence has been observed (= real association) or can be deduced from biostratigraphic data (= virtual association). An unitary association (U.A.) is a maximal set of compatible species. Each U.A. is characterized by a set of species and/or by a set of species pairs: these characteristic elements are used to identify the U.A. in fossiliferous beds. The U.A.'s which can be identified in a large geographical area are said to be reproducible. A biochronological scale is an ordered sequence of reproducible U.A.'s. The problem of constructing such a discrete “time” scale is approached from a graph-theoretic point of view.     

基于施密特正交马田系统和φs转换的灰模糊积分关联度决策模型

针对传统灰关联度不能有效处理属性间存在的交互作用问题,定义了灰模糊积分关联度的概念,并给出了利用Фs函数将属性权重与属性间的交互度转换为λ模糊测度的方法;对于属性权重的计算,提出一种利用施密特正交马田系统计算属性权重的方法,该方法不但考虑了决策者的主观偏好,而且可以消除属性间的重叠信息,从而使权重的计算更加合理;构建了灰模糊积分关联度决策模型,并给出了详细的决策步骤.最后,通过实例验证了所提出的决策模型的可行性,并分析了不同交互度对决策结果的敏感性.     

Artificial data for pattern recognition

The paper describes the generation of three types of artificial data and their use as test material in pattern recognition research. Type A data: The user defines the perfect decision surface. The classes are separable and the pdf's flat. This type is useful in two ways: (i) To investigate whether a learning procedure can achieve a minimal-cost solution. (ii) To compare the powers of two classifiers. Type B data: The user defines the optimal decision surface. The classes are not separable; the degree of overlap between the classes can be controlled by the user. The pdf's are approximately flat, except in regions close to this optimal decision boundary. This type is useful in the following ways: (i) To study the effect of varying the overlap between classes upon a learning procedure. (ii) To compare the powers of two classifiers on a random problem. Type C data: This type is a model of natural, clustered data. The user specifies the location, height, and spread of a number of “hills” in the pdf (for each class). These parameters allow us to calculate the pdf's and hence the Bayes' classification, at any given point. This provides a powerful tool for the objective evaluation of a learning classifier, operating on a realistic problem.     

Smooth interpolation over hypercubes

Multivariate interpolation problems, which occur often in scientific research, can sometimes be approached using Coons' patches. Coons' patches are smooth, local interpolants to lower dimensional data sets. (For example, they interpolate curves of data when considering 3-dimensional problems.) However, they do not interpolate as desired unless all the mixed partial derivatives, the ‘twists’, are equal. The twists are not equal in many cases of practical importance, such as for wire frame data. Gregory (1983) has developed a compatibly corrected Coons' patch for 3-dimensional surfaces. We generalize this interpolant, ‘Gregory's square’, to the case of functions of n variables. The interpolant we propose is built up inductively from one to n-dimensions, requiring, at each step, only one additional term to be defined. This is the key to the whole process and involves the definition of a general ‘twist operator’.     

A necessary and sufficient condition for fundamental periods of cascade machines to be products of the fundamental periods of their constituent finite state machines

Suppose that a finite state machine α produces a periodic sequence (whose fundamental period is A) of nonnegative integer outputs which are used to drive another finite state machine β through a subsequence of its sequence of internal states in the following way. At the jth clock tick of α's clock its integer output a(j) tells β's clock to tick a(j) times (quickly) to drive β numerous steps down its sequence of internal states to the next internal state in this subsequence of its internal states. Suppose that the sequence of outputs of β is periodic with fundamental period B. Suppose that the sum (over any list of A successive entries of the output sequence of α) of α's outputs is S. Then the subsequence of outputs of the cascade machine β ← α consisting of β, driven by α's outputs as described above, is periodic. The fundamental period of the sequence of outputs of the cascade machine β ←a consisting of β driven by α's outputs is no larger than AB. If every output of α is smaller than B, and if S is relatively prime to B, then the fundamental period of the output sequence of this cascade machine is exactly AB. Moreover, every internal state of B occurs exactly A times in each block of AB successive internal states of the cascade machine β ← α. If, on the other hand, S is not relatively prime to B, then the fundamental period of the output sequence of β ← α is less than AB.     

Lower bounds for the size of expressions for certain functions in d-ary logic

Lower bounds on the size of expressions for functions in d-ary logic, d?2, are obtained from a generalization of a theorem of Specker. Denote {0,1,…,d-1} by D and let f:Dⁿ→D be a function that can be represented by an expression of size ?c·n. For any m, if n is sufficiently large, there is a restriction f′:Dⁿ→D of f that is representable by a special class of expressions called simple combinations of chains of length m. By showing that certain functions do not have restrictions representable by simple combinations of chains, it is shown that the size of expressions representing them cannot be bounded by a linear function in the number of arguments. Examples of such functions are the mod p sum, p?3, when the size of the range of the operators used in the expression is ?p?1, the threshold function when the size of the threshold is ? the size of the range of the operators used in the expression, or the connectedness of a square pattern of 0's and 1's.     

Measurement of vorticity in the surface layer using an acoustic echo sounder array

The earth's atmospheric surface layer is usually defined as that region of the lower atmosphere (generally below about 10 m above the earth's surface) where surface friction causes vertical fluxes of heat, moisture and momentum to be constant with height. Within the surface layer, either in response to surface friction or to the atmosphere above, horizontal circular eddies often develop. These circular motions may provide the source of rotation for dust devils so often seen on hot and dry days particularly in desert regions. Also, at larger scales ($\text{～ 1 km in diameter}$) regions of warm and therefore buoyant upward moving air, called thermal plumes, may acquire rotation. These plumes may extend from the earth's surface to more than a kilometer in height on a warm afternoon. Fluid dynamicists quantify this horizontal rotation with a parameter known as the vertical component of vorticity. Vorticity is very difficult to measure in the earth's atmosphere at scales of close to a kilometer because the calculation involves wind-speed differences over horizontal distances of about 500 m. The winds must be measured quite accurately because the differences can be quite small and, therefore, the errors in these measurements are often quite large. This work describes a method of measuring vertical vorticity at scales down to 500 m using an array of three acoustic sounders about 2 m above the earth's surface which overcomes some of the accuracy problems mentioned above. We relate these vorticity measurements to other atmospheric parameters and compute temporal spectra of these quantities to help explain the relationship between vorticity, thermal plume activity, and the smaller-scale dust devils.     

fstds System: A fuzzy-set manipulation system

This paper describes an implementation of a system for fuzzy sets manipulation which is based on fstds (Fuzzy-Set-Theoretic Data Structure), an extended version of Childs's stds (Set-Theoretic Data Structure). The fstds language is considered as a fuzzy-set-theoretically oriented language which can deal, for example, with ordinary sets, ordinary relations, fuzzy sets, fuzzy relations, L-fuzzy sets, level-m fuzzy sets and type-n fuzzy sets. The system consists of an interpreter, a collection of fuzzy-set operations and the data structure, fstds, for representing fuzzy sets. fstds is made up of eight areas, namely, the fuzzy-set area, fuzzy-set representation area, grade area, grade-tuple area, element area, element-tuple area, fuzzy-set name area and fuzzy-set operator name area. The fstds system, in which 52 fuzzy-set operations are available, is implemented in fortran, and is currently running on a FACOM 230-45S computer.     

On the entropy function of degree β

Kamiński and Mikusiński (1974) formed a functional equation H(x,y,z) = H(x + y,0,z) + H(x,y,0), x ? 0, y ? 0, z ? 0, xy + yz + zx > 0, and determined its continuous and symmetric solution which is homogeneous of degree 1. This approach simplifies Fadeev's well-known characterization of Shannon's entropy. In this paper we obtain this functional equation from a generalized set of axioms and solve it under continuity and homogeneity of degree β (β ≠ 1, β > 0). This extends Kamiński and Mikusiński's approach of characterizing to degree β the measure of entropy, which has been studied by a number of workers with considerable interest.