Graphical models for statistical inference and data assimilation

In data assimilation for a system which evolves in time, one combines past and current observations with a model of the dynamics of the system, in order to improve the simulation of the system as well as any future predictions about it. From a statistical point of view, this process can be regarded as estimating many random variables which are related both spatially and temporally: given observations of some of these variables, typically corresponding to times past, we require estimates of several others, typically corresponding to future times.

Graphical models have emerged as an effective formalism for assisting in these types of inference tasks, particularly for large numbers of random variables. Graphical models provide a means of representing dependency structure among the variables, and can provide both intuition and efficiency in estimation and other inference computations. We provide an overview and introduction to graphical models, and describe how they can be used to represent statistical dependency and how the resulting structure can be used to organize computation. The relation between statistical inference using graphical models and optimal sequential estimation algorithms such as Kalman filtering is discussed. We then give several additional examples of how graphical models can be applied to climate dynamics, specifically estimation using multi-resolution models of large-scale data sets such as satellite imagery, and learning hidden Markov models to capture rainfall patterns in space and time.     

Hidden variable models for quantum theory cannot have any local part

It was shown by Bell that no local hidden variable model is compatible with quantum mechanics. If, instead, one permits the hidden variables to be entirely nonlocal, then any quantum mechanical predictions can be recovered. In this Letter, we consider general hidden variable models which can have both local and nonlocal parts. We show the existence of (experimentally verifiable) quantum correlations that are incompatible with any hidden variable model having a nontrivial local part, such as the model proposed by Leggett.     

Probability and logical structure of statistical theories

A characterization of statistical theories is given which incorporates both classical and quantum mechanics. It is shown that each statistical theory induces an associated logic and joint probability structure, and simple conditions are given for the structure to be of a classical or quantum type. This provides an alternative for the quantum logic approach to axiomatic quantum mechanics. The Bell inequalities may be derived for those statistical theories that have a classical structure and satisfy a locality condition weaker than factorizability. The relation of these inequalities to the issue of hidden variable theories for quantum mechanics is discussed and clarified.     

基于拓展分离变量法的非傅里叶传热研究

常规的分离变量方法不能求解非傅里叶传热模型问题,本文对常规分离变量法进行拓展,利用拓展后的方法求解热传导的傅里叶模型,单相位滞后（C-V）模型和双相位滞后（DPL）模型,比较利用三种模型计算温度场的差别.研究C-V模型和DPL模型中温度波动的规律,得到温度波动速度和热流量时间滞后值的关系.     

Development of particle multiplicity distributions using a general form of the grand canonical partition function

Various phenomenological models of particle multiplicity distributions are discussed using a general form of a unified model which is based on the grand canonical partition function and Feynman's path integral approach to statistical processes. These models can be written as special cases of a more general distribution which has three control parameters which are a,x,z. The relation to these parameters to various physical quantities are discussed. A connection of the parameter a with Fisher's critical exponent τ is developed. Using this grand canonical approach, moments, cumulants and combinants are discussed and a physical interpretation of the combinants are given and their behavior connected to the critical exponent τ. Various physical phenomena such as hierarchical structure, void scaling relations, Koba–Nielson–Olesen or KNO scaling features, clan variables, and branching laws are shown in terms of this general approach. Several of these features which were previously developed in terms of the negative binomial distribution are found to be more general. Both hierarchical structure and void scaling relations depend on the Fisher exponent τ. Applications of our approach to the charged particle multiplicity distribution in jets of L3 and H1 data are given.     

An Optimization Principle for Deriving Nonequilibrium Statistical Models of Hamiltonian Dynamics

A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a vector of resolved variables, selected to describe the macroscopic state of the system, a family of quasi-equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model, and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities. Specifically, a cost function is constructed to quantify the lack-of-fit to the microscopic dynamics of any feasible path of densities from the statistical model; it is an ensemble-averaged, weighted, squared-norm of the residual that results from submitting the path of densities to the Liouville equation. The path that minimizes the time integral of the cost function determines the best-fit evolution of the mean resolved vector. The closed reduced equations satisfied by the optimal path are derived by Hamilton-Jacobi theory. When expressed in terms of the macroscopic variables, these equations have the generic structure of governing equations for nonequilibrium thermodynamics. In particular, the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes. The adjustable closure parameters in the best-fit reduced equations depend explicitly on the arbitrary weights that enter into the lack-of-fit cost function. Two particular model reductions are outlined to illustrate the general method. In each example the set of weights in the optimization principle contracts into a single effective closure parameter.     

Compressing the hidden variable space of a qubit

In previously exhibited hidden variable models of quantum state preparation and measurement, the number of continuous hidden variables describing the actual state of single realizations is never smaller than the quantum state manifold dimension. We introduce a simple model for a qubit whose hidden variable space is one-dimensional, i.e., smaller than the two-dimensional Bloch sphere. The hidden variable probability distributions associated with quantum states satisfy reasonable criteria of regularity. Possible generalizations of this shrinking to an N-dimensional Hilbert space are discussed.     

A joint shape evolution approach to medical image segmentation using expectation-maximization algorithm

This study proposes an expectation–maximization (EM)-based curve evolution algorithm for segmentation of magnetic resonance brain images. In the proposed algorithm, the evolution curve is constrained not only by a shape-based statistical model but also by a hidden variable model from image observation. The hidden variable model herein is defined by the local voxel labeling, which is unknown and estimated by the expected likelihood function derived from the image data and prior anatomical knowledge. In the M-step, the shapes of the structures are estimated jointly by encoding the hidden variable model and the statistical prior model obtained from the training stage. In the E-step, the expected observation likelihood and the prior distribution of the hidden variables are estimated. In experiments, the proposed automatic segmentation algorithm is applied to multiple gray nuclei structures such as caudate, putamens and thalamus of three-dimensional magnetic resonance imaging in volunteers and patients. As for the robustness and accuracy of the segmentation algorithm, the results of the proposed EM-joint shape-based algorithm outperformed those obtained using the statistical shape model-based techniques in the same framework and a current state-of-the-art region competition level set method.     

Long range dynamics and broken symmetries in gauge models. The Stückelberg-Kibble model

The general algebraic features associated to long range dynamics like the problem of removing the infrared cutoff, the definition of the algebraic dynamics and the occurrence of variables at infinity, the essential localization (seizing of the vacuum), the effective dynamics and its covariance group (dynamical symmetry group), the generalization of Goldstone's theorem and the non-trivial Goldstone spectrum, the mass/energy gap generation by the non-trivial classical motion of the variables at infinity are explicitly shown in the Kibble model as a prototype of gauge models exhibiting the Higgs phenomenon. The relation between mass generation in the Higgs phenomenon and the plasma energy gap is also discussed.Work supported in part by INFN, Sezione di Pisa     

Reality,locality, and probability

It is frequently argued that reality and locality are incompatible with the predictions of quantum mechanics. Various investigators have used this as evidence for the existence of hidden variables. However, Bell's inequalities seem to refute this possibility. Since the above arguments are made within the framework of conventional probability theory, we contend that an alternative solution can be found by an extension of this theory. Elaborating on some ideas of I. Pitowski, we show that within the framework of a generalized probability theory, reality, locality, hidden variables, and the predictions of quantum mechanics can be maintained together. Although our principal model in this work is a spin system, there are indications that this program can be extended to more general systems.     

Axioms for quantum theory

The first three of these axioms describe quantum theory and classical mechanics as statistical theories from the very beginning. With these, it can be shown in which sense a more general than the conventional measure theoretic probability theory is used in quantum theory. One gets this generalization defining transition probabilities on pairs of events (not sets of pairs) as a fundamental, not derived, concept. A comparison with standard theories of stochastic processes gives a very general formulation of the non existence of quantum theories with hidden variables. The Cartesian product of probability spaces can be given a natural algebraic structure, the structure of an orthocomplemented, orthomodular, quasi-modular, not modular, not distributive lattice, which can be compared with the quantum logic (lattice of all closed subspaces of an infinite dimensional Hubert space). It is shown how our given system of axioms suggests generalized quantum theories, especially Schrödinger equations, for phase space amplitudes.     

A novel configuration model for random graphs with given degree sequence

Recently, random graphs in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices have attracted much attention. This paper presents a specific realization of a class of random network models in which the connection probability between two vertices (i,j) is a specific function of degrees ki and kj. In the framework of the configuration model of random graphs, we find the analytical expressions for the degree correlation and clustering as a function of the variance of the desired degree distribution. The obtained expressions are checked by means of numerical simulations. Possible applications of our model are discussed.     

Delinearization of quantum logic

The algebraic structure of the set of elementary observables of a delinearized quantal theory is described. As the delinearization procedure provides a kind of classical representation for any quantal theory, its relation to the traditional hypothesis of hidden variables is discussed.     

Hidden variables and locality

Bell's problem of the possibility of a local hidden variable theory of quantum phenomena is considered in the context of the general problem of representing the statistical states of a quantum mechanical system by measures on a classical probability space, and Bell's result is presented as a generalization of Maczynski's theorem for maximal magnitudes. The proof of this generalization is shown to depend on the impossibility of recovering the quantum statistics for sequential probabilities in a classical representation without introducing a randomization process for the hidden variables. Hidden variable theories that exclude such a randomization process are termed strict, and it is shown that the class of local hidden variable theories is included in the class of strict theories. A counterargument by Freedman and Wigner is evaluated with reference to Clauser's extension of a hidden variable model proposed by Bell.     

Standard deviations of chemical reaction rates and validity ranges of the statistical theory

Validity ranges of the statistical theory of chemical reactions are discussed. On the basis of the quantum mechanical scattering theory, the rate of the chemical reaction of the complex-formation mode is formulated. The interaction matrix elements between resonance states and scattering continua are regarded as stochastic variables. The expectation value of the reaction rate is shown to coincide with the prediction of the conventional statistical theory if the condition of overlapping resonance is fulfilled. The standard deviation is found to be inversely proportional to the number of resonance states involved in the width of the collision energy. High density of vibrational energy levels of the collision complex serves to suppress the standard deviation and to make the statistical theory accurate. The condition for the conventional statistical theory to hit the correct value with probabilities more than 99% is obtained as a relation between the number of vibrational modes and the depth of the potential energy well of the collision complex.     

A cellular-automaton fluid model with simple rules in arbitrarily many dimensions

A new cellular-automaton model for fluid dynamics is introduced. Unlike the conventional FHP-type models, the model uses easily implementable, deterministic pair interaction rules which work on arbitrary-dimensional orthogonal lattices. The statistical and hydrodynamic theory of the model is developed, and the Navier-Stokes-like hydrodynamic equations that describe the macroscopic behavior of the model are derived. It turns out that the unwanted anisotropic convection behavior can be eliminated in the incompressible limit by suitable choice of the mass density. An explicit expression for the viscosity tensor is calculated from a Boltzmann-type approximation. Unfortunately, the viscosity turns out to be anisotropic, which is a drawback as against the conventional FHP and FCHC models. Nevertheless, the new model could become interesting for fluid dynamic problems with additional variables (e.g., free surfaces), especially in two dimensions, since its simple rules could relatively easily be extended for such cases.     

基于深度信念网络与混合波长选择方法的蓝莓糖度近红外检测模型优化

利用近红外光谱技术结合组合区间偏最小二乘（SiPLS）、竞争性自适应重加权（CARS）、连续投影算法（SPA）、无信息变量消除（UVE）特征提取方法,运用深度信念网络（DBN）建立蓝莓糖度的通用检测模型,实现蓝莓糖度在线无损快速检测。采集了“蓝丰”和“瑞卡”共280个蓝莓样本的近红外光谱,采用手持折光仪测定其糖度;首先利用联合X-Y的异常样本识别方法（ODXY）检测到蓝丰和瑞卡蓝莓分别有2个和4个样本呈现异常,剔除该6个异常样本,对其余274个样本利用光谱-理化值共生距离算法（SPXY）以3∶1的比例划分出训练集和测试集;其次,对比分析卷积平滑（S-G平滑）、中心化、多元散射校正等预处理对蓝莓原始光谱的改善效果,运用SiPLS对光谱降维,筛选特征波段,利用CARS,UVE和SPA方法对特征波段进行二次筛选,以最优的特征波长建立DBN和偏最小二乘回归（PLSR）模型。结果表明,蓝莓糖度近红外检测模型的最优预处理方法为S-G平滑,SiPLS方法挑选的蓝莓糖度最优波段为593~765和1 458~1 630 nm,UVE算法从SiPLS筛选的346个变量中优选出159个最佳波长。建立蓝莓糖度DBN模型时,分析了不同隐含层数对检测模型的影响,并以交互验证均方根误差（RMSECV）作为适应度函数,利用粒子群算法（PSO）对各隐含层神经元个数在[1,100]之间寻优,发现隐含层为3层且隐含层节点数为67-43-25时,DBN模型的RMSECV达到最小,为0.397 7。无论是以全光谱还是特征波长建模,蓝莓糖度近红外DBN模型均优于常规PLSR方法;尤其以UVE方法二次筛选的特征波长建立的模型大大减少了建模变量,且模型精度更高,蓝莓糖度最优的PLSR模型测试集相关系数（R_P）为0.887 5,均方根误差（RMSEP）为0.395 9,最优DBN模型R_P为0.954 2,RMSEP为0.310 5。研究表明,利用SiPLS-UVE进行特征提取,结合深度信念网络方法建立的蓝莓糖度检测模型可以更好地完成蓝莓糖度在线精准分析,该方法有望应用于蓝莓及其他果蔬内部品质检测。     

Why god might play dice

We study the question of the existence of hidden variables within the formalism of Pitowsky. We show that probabilities admit factorizable hidden variable models iff they admit a Kolmogorovian representation. In particular, directly deduced experimental frequencies always admit a factorizable hidden variable model and thus a Kolmogorovian representation. We apply this result in the framework of Bell's inequalities. We show that a deterministic interpretation of the hidden variables associated with this situation refutes the possibility for the experimenter of choosing freely the conditions of experimentation.According to the convention introduced by Pitowsky (Pitowsky, 1989), we mean by the truth-value of an experimental outcome the probability of realization of this outcome. A deterministic truth-value is equal to 0 or 1 by definition.