Joint Conditional Probability Research Articles

Estimation of quartet phase sums from a new joint probability distribution of normalized structure factors

A new joint probability distribution of normalized structure factors is derived for equal-atom structures in space group P1. From this general distribution, a series expansion, the conditional joint probability distribution of the quartet phase sum is obtained, when restrictive conditions among the reciprocal vectors are imposed. The main difference from existing quartet distributions is the possibility of enclosing higher-order terms to any given order of N, although an approximation employed in the derivation limits the number of them considerably. The higher-order terms present are easily employed in the series since the determination of their explicit appearance has been automated: a computer program derives the terms up to a desired order and stores them in a coded form. In general, the incorporation of selective terms up to order N-3 appears to yield sufficient convergence. Only high |E| values or a low N value may necessitate the use of higher-order terms. Test results show that, in contrast to results from the quartet distributions of Hauptman and Giacovazzo, systematic estimation errors are hardly present, while absolute estimation errors are reduced as well.

On the consistency and finite-sample properties of nonparametric kernel time series regression, autoregression and density estimators

Kernel estimators of conditional expectations and joint probability densities are studied in the context of a vector-valued stationary time series. Weak consistency is established under minimal moment conditions and under a hierarchy of weak dependence and bandwidth conditions. Prompted by these conditions, some finite-sample theory explores the effect of serial dependence on variability of estimators, and its implications for choice of bandwidth.

Determination of the positions of anomalous scatterers: probabilistic coefficients for a Patterson synthesis

The conditional joint probability distribution of the phase ϕ = ϕh + ϕ-h given |F+|, |F-| is used in order to suggest coefficients for a Patterson synthesis for the determination of the positions of anomalous scatterers. A theoretical comparison with Rossmann's approach [Rossmann, M. G. (1981). Acta Cryst. 14, 383-388] is made.

A modified cluster sampling method for estimating the bacterial density in Xanthan gum

Abstract Suppose particles are randomly distributed in a certain medium, powder or liquid, which is conceptually divided into N cells. Let pi denote the probability that a particle falls in the ith cell and Yi denote the number of particles in the ith cell. Assume that the joint probability function of the Yi follows a multinomial distribution with cell probabilities pi respectively. Take n (≤N) cells at random without replacement and put each of the cells separately through a mixing mechanism of dilution and swirl. These n cells constitute the first stage samples and the number of particles in these cells are not observable. Now conceptually divide each of n cells into M subcells of equal size and let Xij denote the number of particles in the jth subcell of the ith cell selected in the first stage; i=1,2,…,N and j=1,2,…,M. Consequently assume that the conditional joint probability function of the Xij given Yi=yi follows a multinomial distribution with equal cell probabilities. Now take m (≤M) subcells at random from each of the cells selected in the first stage sample. Assume that the numbers of particles in M×N subcells are observable. The properties of the estimator of the particle density per sample unit are investigated under the modified two-stage cluster sampling method. A laboratory experiment for Xanthan Gum Products is analyzed in order to examine the appropriateness of the model assumed in this paper.

A modal approach for estimation in distributed parameter systems

A scheme, based on eigenmode expansion, for the estimation of system responses in distributed parameter systems with non-linear sensors is presented. The associated joint conditional probability distribution is realized by using the partition theorem and Wiener functional expansions. The system state and parameters are subsequently computed by obtaining the innovation processes from the observations.

A new determinantal theory for solving the phase problem using stereochemical information

A priori structural information is incorporated into determinantal phasing techniques to improve phase prediction accuracy in resolution ranges where atomic or isotropic group-scatterer assumptions are not valid. For this purpose a conditional joint probability distribution to triplet order for any set of normalized structure factors of space groups P1 and P\bar 1 is derived. The covariance of two normalized structure factors from the original set is calculated. A more general conditional joint probability distribution, involving covariance matrices of any order, is further derived. Numerical tests are performed employing ideal models consisting of several atomic groups of known stereochemistry but with random positions and orientations. The results indicate that the inclusion of stereochemical information improves the accuracy of phase prediction. The relative merit of this strategy in either one of or both normalization and covariance calculations for different resolutions is discussed.

JOINT and SEQU: FORTRAN routines for the analysis of observational data

In a lag analysis, the frequencies of various events that follow a target or criterion event are tallied according to the ordinal position of each event observed following the target event. For behavior analysts, lag analysis is most useful for detecting regular patterns of contingent events, since reinforcers and punishers can be both identified and evaluated by their effects on antecedent and consequent behaviors. Once contingent events have been identified it is of considerable interest to detect extended sequences of particular events or classes of events. This article describes two FORTRAN programs that can facilitate the analysis of observational data: One, JOINT, computes the simple frequencies of events and provides a lag-sequential analysis; the other, SEQU, detects predefined sequences of events. Sackett, Holm, Crowley, and Henkins (1979) have recently described a FORTRAN program, LAGS, that performs lag-sequential analysis on behavioral interaction data, and they have addressed a number of theoretical issues involved in coding observational data and in applying inferential statistics to those data. JOINT shares many common features with LAGS and is offered here only as an alternative. There are two major differences between the two programs, nonetheless. First, in JOINT, all behaviors serve as the target behaviors in the lag analysis, whereas in LAGS only one target can be specified at a time. Second, in JOINT, the maximum lag allowed is 10, whereas LAGS can handle lags of up to 800. These two differences seem to result because JOINT passes through the data only once, but LAGS passes through the data separately for each target behavior. Thus, one program requires no a priori decisions about target behaviors, and the other allows a more extended lag analysis. Program Input. The input to both JOINT and SEQU is in two parts, or meso The first me, a control file, provides a general title, the numerical codes that represent the behaviors and alphanumeric labels for each, the lags to be analyzed, and an optional format statement for reading the second me, the data file. If any of the numerical codes represent more than one behavior that is coded in the data, the recoding information is given here. The second me, the data file, begins with a line of identifying information for the session and, in sequence, the numerical codes for each behavior recorded during the session. A unique code marks the end of the session, and another code marks the end of the set of sessions to be analyzed. JOINT accepts observational data that consist of up to 80 different codes, each code being a number that represents a particular behavior. At the user's option, codes may be assigned that represent the occurrence of any I of up to 80 other uniquely coded behaviors. Thus, the raw data may comprise many codes, but the program can analyze only 80 codes at a time. Memory limitations on some computers may require program modifications that will reduce the number of codes per analysis to 20 or 40. The data to be analyzed may be drawn from a single session or may be combined across as many as 50 different sessions. The number of behaviors recorded in a session is limited by only the largest integer value the computer can store, a value usually so large as to pose no practical problems. Program Output: JOINT. JOINT computes simple frequencies and probabilities and rates for each of the codes defined in the control file. JOINT also computes joint frequencies, conditional probabilities, z scores, and rates for the codes with lags from I to 10. The lag analysis is printed in matrix form, with the entire set of codes comprising both the row (R) and column (C) behaviors. For a lag of 1, the joint frequency of Rand C, f(R & C), is the number of times Behavior C occurred immediately after Behavior R occurred. For a lag of 2, of course, feR & C) is the number of times C was the second behavior recorded after R occurred, and so on. The conditional probability, p(C/R), is the probability of the C behavior, given that the R behavior was occurring in a preceding time frame. The conditional probability of a particular Rand C is obtained by dividing feR & C) by f(R). The rate of each behavior is calculated simply from the duration of the observation periods. Likewise, the rate of each joint frequency is given, that is, the rate at which a C behavior followed an R behavior at each lag. The z scores indicate the extent to which an observed joint frequency deviated from the joint frequency that would occur if Behavior Rand Behavior C were independent of each other. Assuming independence (or only chance association), the expected

Quality Control in Metallurgy by Texture Analysis

Quality control in metallurgy practically means mapping the set of empirical qualitative| attributes onto a measurable and separable set of quantities. The paper presents an approach based on statistical texture analysis to perform this mapping. The textural features are derived from second order statistics based on the conditional joint probability function of the gray levels of photomicrographs made from the microstrueture of the material to be tested. Several methods are given partly for selecting and extracting the principal features (such procedures are discriminant analysis and discrete K-L expansion serving for clustering), partly for classifying the digitized images by similarity measures. As a new idea a phenomenological approach is given for determining the relationship between qualitative properties and textural features. Finally, the paper describes a complex computerized investigation aimed at testing the method developed, then analyzes the connection between mechanical properties determined by static bend tests and the quantitative pictorial features of carbide distribution for 6-5-2 high-speed steel

A probability theory of the cosine invariant cos(φh+ φk+ φl– φh+k+l)

A conditional joint probability distribution is derived in order to estimate the values of the cosine invariant cos (ϕh + ϕk + ϕl - ϕh + k + l) in terms of the magnitudes of Eh, Ek, El, Eh + k, Eh + l, Ek + l, Eh + k + l. The theory leads to values for the cosines which lie anywhere between - 1 and + 1. Some applications of the quartets in procedures for crystal structure determination are described.

On the theory and estimation of the cosine invariants cos(φl+ φm+ φn+ φp)

For fixed h1 + h2 + h3 = 0, and uniformly distributed k, the conditional joint probability distribution of the pair of phases ϕk, ϕh1 + k, given |E-h3 + k|, |Ek|, |Eh1 + k| is found. If l + m + n + p = 0, this distribution leads, via a suitable sampling technique, to estimates having probabilistic validity for the cosine invariant cos (ϕl + ϕm + ϕn + ϕp) in terms of the seven magnitudes |El|, |Em|, |En|, |Ep|, |El + m|, |El + n|, |El + p|.

On optimal nonlinear estimation — Part II: Discrete observation

A general representation for the joint conditional probability density of an arbitrary random signal process under discrete-time observation is obtained. This representation forms the cornerstone of the paper, and from it all other results are deduced. The conditional densities of prediction and smoothing are expressed in terms of filtering via the application of the general representation. The prediction and smoothing of a random process with linear dynamics and arbitrary a priori distribution are given to illustrate the applicability of the previous results in obtaining effectively computable formulas.

Some comments on the generalized Shannon lower bound for stationary finite-alphabet sources with memory (Corresp.)

We consider a class of strictly stationary binary sources <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{u_n, n \geq 1\}</tex> , characterized by joint conditional probability mass functions \begin{equation} Q_n(U_n = k_n \mid \theta) = \prod_{i=1}^n \theta^{1-k_i} (1 - \theta)^{k_i} \end{equation} for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \geq 1</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta</tex> is an r.v. on

A multivariate joint probability for phase determination

A multivariate conditional joint probability distribution of a set of K normalized structure factors has been developed using a novel approach. The covariance matrix of the distribution is calculated for all the space groups in terms of linear combinations of specific unitary structure factors. It is shown that if any number of off-diagonal elements are set arbitrarily to zero, an approximation is obtained to the covariance matrix which corresponds to a particular set of a priori conditions. The importance of this result in practical phase-determining methods is pointed out. Group theory is used to obtain results valid for all space groups. The multivariate distribution is used to calculate more general versions of the Cochran and Woolfson sign probability and the Karle and Hauptman tangent formulae.

Probabilistic connexion between one phase invariant and moduli of all structure factors

Recalling the `conditional joint probability law' of m normalized structure factors [Tsoucaris (1969), C.R. Acad. Sci. Paris, Sér. B, 268, 875; (1970). Acta Cryst. A26, 492] the conditional probability density is established for three moduli, assuming that the structure invariant α is known. Then, using the Bayes theorem, in order to `invert' the role of random variables and known parameters, the probability density of α is obtained under the condition that the moduli are fixed. By the application of the axiom of joint probabilities for vectors running over all reciprocal lattice points located within the observable Ewald sphere, the expression for one phase invariant α is derived as a function of all observed moduli p(α|all moduli).

Calculation of NMR Relaxation Times for Quadrupolar Nuclei in the Presence of Chemical Exchange

The correlation function for a quadrupolar nucleus which may exchange its environment between two kinds of sites in solution has been calculated from an appropriate joint conditional probability for changes in site and in electric field gradient direction with time. In the extreme narrowing limit, the resultant NMR linewidth depends both on the exchange lifetimes and also on the correlation times for rotation at each site; the NMR linewidth may thus be used to obtain information about any of these quantities. The results bear directly on the use of quadrupolar nuclei in the presence of chemical exchange of nuclei between suitable sites to obtain motional information about macromolecules in dilute solution. In particular, the proposed correlation function may account for anaomalously small calculated rotational-correlation times for large molecules previously obtained from such studies. A precise criterion for applicability of McConnell-modified Bloch equations is proposed. Other applications of the correlation function derived in this paper are discussed.

Bayesian identification of nonlinear systems

The problem considered is a given plant described by a nonrandom nonlinear vector ordinary differential equation. The observed output consists of a nonlinear function of the state vector plus white noise. The plant contains an unknown constant parameter vector. Observations are made continuously in time, starting at some initial time t 0. The initial state vector at time t 0 is also unknown. The problem is, at each time t > t 0, to make an optimal estimate of the initial state and the parameter vector, based on the entire history of observations over the interval [ t 0, t]. The estimate is made in real time and continuously updated as additional data are received. A Bayesian approach is adopted. It is assumed that there is given a joint a priori distribution for the initial state and plant parameter vectors. Furthermore, this distribution is assumed to be discrete and concentrated on a finite set of points, or else adequately approximable, by such a distribution. Using the theory of absolutely continuous measures in function space, the joint conditional probability distribution for the initial state and parameter, given the history of the observations, is derived. The optimal estimate adopted is the mean of this distribution. By use of the theory of stochastic differential equations, it is shown how the optimal recursive estimator can be realized as a linear stochastic differential equation of finite dimension, followed by some algebraic operations. The practical problem of mechanizing this estimator is discussed in some detail. It is shown how the real-time computational burden may be minimized by precomputing a certain family of functions. Finally, the relation of this scheme to other comtemporary nonlinear filtering problems is discussed, as well as the implications of the assumptions made concerning the a priori distribution.

Pattern recognition by an adaptive process of sample set construction

A pattern recognition technique is described in which a parametric representation of input signals or stimuli is employed. An input is considered as a vector, while the stimulus class is a multivariate process in the vector space. An adaptive sample set construction technique is described through which the conditional joint probability density of a class is approximated by the sum of Gaussian densities. The mean of each such density is an adaptively chosen "typical" sample of the class, and the set of samples so chosen are contained in the region of the space in which samples of the class are most populous. The decision process using the typical samples partitions the space into regions that envelop the chosen samples of a class. Arbitrary shaped and multiply connected regions can be constructed in this way, and multimodal probability densities can be approximated with a computationally simple procedure. Decision making on an incomplete set of parameters and on multiple observations of the input stimulus are discussed. This technique was successfully applied to the automatic recognition of speaker identity regardless of the spoken test. Experimental results are given.