Multivariate Normal Probability Distribution Research Articles

Intrinsic Information-Theoretic Models.

With this follow-up paper, we continue developing a mathematical framework based on information geometry for representing physical objects. The long-term goal is to lay down informational foundations for physics, especially quantum physics. We assume that we can now model information sources as univariate normal probability distributions N (μ, σ0), as before, but with a constant σ0 not necessarily equal to 1. Then, we also relaxed the independence condition when modeling m sources of information. Now, we model m sources with a multivariate normal probability distribution Nm(μ,Σ0) with a constant variance-covariance matrix Σ0 not necessarily diagonal, i.e., with covariance values different to 0, which leads to the concept of modes rather than sources. Invoking Schrödinger's equation, we can still break the information into m quantum harmonic oscillators, one for each mode, and with energy levels independent of the values of σ0, altogether leading to the concept of "intrinsic". Similarly, as in our previous work with the estimator's variance, we found that the expectation of the quadratic Mahalanobis distance to the sample mean equals the energy levels of the quantum harmonic oscillator, being the minimum quadratic Mahalanobis distance at the minimum energy level of the oscillator and reaching the "intrinsic" Cramér-Rao lower bound at the lowest energy level. Also, we demonstrate that the global probability density function of the collective mode of a set of m quantum harmonic oscillators at the lowest energy level still equals the posterior probability distribution calculated using Bayes' theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. While these new assumptions certainly add complexity to the mathematical framework, the results proven are invariant under transformations, leading to the concept of "intrinsic" information-theoretic models, which are essential for developing physics.

A method to integrate and classify normal distributions.

Univariate and multivariate normal probability distributions are widely used when modeling decisions under uncertainty. Computing the performance of such models requires integrating these distributions over specific domains, which can vary widely across models. Besides some special cases where these integrals are easy to calculate, there exist no general analytical expressions, standard numerical methods, or software for these integrals. Here we present mathematical results and open-source software that provide (a) the probability in any domain of a normal in any dimensions with any parameters; (b) the probability density, cumulative distribution, and inverse cumulative distribution of any function of a normal vector; (c) the classification errors among any number of normal distributions, the Bayes-optimal discriminability index, and relation to the receiver operating characteristic (ROC); (d) dimension reduction and visualizations for such problems; and (e) tests for how reliably these methods may be used on given data. We demonstrate these tools with vision research applications of detecting occluding objects in natural scenes and detecting camouflage.

Recursive Calculation Model for a Special Multivariate Normal Probability of First-Order Stationary Sequence

The consecutive hit probability of antiaircraft artillery corresponds to the multivariate normal probability distribution. The computational complexity depends on the length of the firing error sequence (i.e., the integral dimension may exceed 100). The traditional numerical integration and the Monte Carlo method are too slow for this calculation. This paper established the state equation of the firing error sequence, which was the bridge between the multivariate normal probability distribution and stochastic process theory. The recursive calculation model was given after the rigorous derivation process. The accuracy and computational complexity of the model were quantified by theoretical analysis and expressed intuitively by examples. The model shows the upper bound for the absolute error, and the computational efficiency is significantly improved.

On L. Klein’s Erroneous Assessments of J M Keynes’s Views on Probability, Mathematics, and Statistics: Reviewing His 1951 Review of Harrod’s the Life of John Maynard Keynes

Lawrence Klein was a follower of the Limiting Frequency Interpretation of probability. This concept of probability requires that probability must always be a precise, exact mathematical probability. This meant that additivity was always the case. His acceptance of Kolmogorov’s axiom of countable additivity meant that he had no idea whatsoever about the concept of interval valued probability espoused by Boole and Keynes using upper and lower probability bounds. Klein’s belief in the limiting frequency interpretation of probability meant that there can be no such things as imprecise and/or indeterminate probabilities. This view impacts Klein’s understanding of economic statistics and Econometrics. Klein builds his approach to economic statistics and Econometrics on the application of the Multivariate Normal probability distribution a la Tinbergen in 1939-40. There is no evidence that Klein was familiar with Mandelbrot’s demonstration, starting in the late 1950’s, that the Normal Distribution is not supported by any empirical evidence or Keynes’s point that it was impossible to use a Normal Distribution to account for changes in the demand (supply) of durable, physical capital goods, due to the impact of future technological change, innovation (obsolescence), and advance that impacted the expectations of those investing in such producer goods. However, Klein’s approach could be used to analyze the consumption function and inventory changes over time since these are stable, linear functions. Klein misunderstood Keynes’s comments about mathematical economics in the General Theory in the appendix to chapter 19 and chapter 21 of the General Theory. Keynes’s comments were a critique of Pigou’s use of Marshallian mathematics, based on ceteris paribus, partial equilibrium models that did not take into account the interdependencies and complex, complicated interactions between variables caused by the positive feedback effects emanating from, for one example, the interactions of the multiplier and accelerator. Klein also failed to recognize that Keynes was the creator, developer and innovator of the IS-LM (LP) model, which Keynes presented as a three equation system on pp. 298-299 of the General Theory in 1936. Hicks’s and Harrod’s versions of IS-LM were just inferior versions of Keynes’s original model that left out expectations and uncertainty considerations, which Keynes had incorporated in his supporting D-Z model of chapters 20 and 21 (there is no D-Z model in chapter 3 of the GT).

Stochastic fractional programming for minimizing variability

In situations where risk aversion is involved, minimization of the variability of the economic criterion under consideration is indispensable. This paper provides a typical stochastic fractional programming model for minimizing the absolute value of the coefficient of variation of a linear function of decision variables under the assumption that the coefficient parameters of the decision variables have a known multivariate normal probability distribution. The stochastic problem is shown to be equivalent to a deterministic problem whose solution can be obtained by solving a related quadratic programming problem.

Accounting for multiple climate components when estimating climate change exposure and velocity

Summary The effect of anthropogenic climate change on organisms will likely be related to climate change exposure and velocity at local and regional scales. However, common methods to estimate climate change exposure and velocity ignore important components of climate that are known to affect the ecology and evolution of organisms. We develop a novel index of climate change (climate overlap) that simultaneously estimates changes in the means, variation and correlation between multiple weather variables. Specifically, we estimate the overlap between multivariate normal probability distributions representing historical and current or projected future climates. We provide methods for estimating the statistical significance of climate overlap values and methods to estimate velocity using climate overlap. We show that climates have changed significantly across 80% of the continental United States in the last 32 years and that much of this change is due to changes in the variation and correlation between weather variables (two statistics that are rarely incorporated into climate change studies). We also show that projected future temperatures are predicted to be locally novel (<1·5% overlap) across most of the global land surface and that exposure is likely to be highest in areas with low historical climate variation. Last, we show that accounting for changes in the variation and correlation between multiple weather variables can dramatically affect velocity estimates; mean velocity estimates in the continental United States were between 3·1 and 19·0 km yr−1 when estimated using climate overlap compared to 1·4 km yr−1 when estimated using traditional methods. Our results suggest that accounting for changes in the means, variation and correlation between multiple weather variables can dramatically affect estimates of climate change exposure and velocity. These climate components are known to affect the ecology and evolution of organisms, but are ignored by most measures of climate change. We conclude with a set of future directions and recommend future work to determine which measures of climate change exposure and velocity are most related to biological responses to climate change.

Interventricular comparison of the energetics of contraction of trabeculae carneae isolated from the rat heart

We compare the energetics of right ventricular and left ventricular trabeculae carneae isolated from rat hearts. Using our work-loop calorimeter, we subjected trabeculae to stress-length work (W), designed to mimic the pressure-volume work of the heart. Simultaneous measurement of heat production (Q) allowed calculation of the accompanying change of enthalpy (H = W + Q). From the mechanical measurements (i.e. stress and change of length), we calculated work, shortening velocity and power. In combination with heat measurements, we calculated activation heat (Q(A)), crossbridge heat (Q(xb)) and two measures of cardiac efficiency: 'mechanical efficiency' ((mech) = W/H) and 'crossbridge efficiency' ((xb) = W/(H - Q(A))). With respect to their left ventricular counterparts, right venticular trabeculae have higher peak shortening velocity, and higher peak mechanical efficiency, but with no difference of stress development, twitch duration, work performance, shortening power or crossbridge efficiency. That is, the 35% greater maximum mechanical efficiency of right venticular than left ventricular trabeculae (13.6 vs. 10.2%) is offset by the greater metabolic cost of activation (Q(A)) in the latter. When corrected for this difference, crossbridge efficiency does not differ between the ventricles.

Cogeneration planning under uncertainty: Part I: Multiple time frame approach

Cogeneration system planning spans a multi-year time interval and is affected by various sources of uncertainty, mainly depending on the evolution of energy loads and prices. The high level of uncertainty requires assessing the convenience of adopting predefined technological alternatives in different scenarios of variation of the uncertain variables. This paper introduces an original framework based on identifying the characteristics of small-scale and large-scale uncertainties, whereby a comprehensive approach based on multiple (long-, medium- and short-term) time frames is formulated. Medium-term time periods exhibiting small variations of both electrical and thermal load patterns are grouped together and represented through electrical/thermal load and electricity price correlated random variables (RVs). A Monte Carlo simulation of the cogeneration plant operation is carried out in the short-term by extracting the RVs for each group from multivariate Normal probability distributions. Multi-year scenarios in the long-term time frame are addressed in the companion paper (Part II). The proposed approach is applied to a real energy system.

A Chance Constrained Approach to Fractional Programming with Random Numerator

This paper presents a chance constrained programming approach to the problem of maximizing the ratio of two linear functions of decision variables which are subject to linear inequality constraints. The coefficient parameters of the numerator of the objective function are assumed to be random variables with a known multivariate normal probability distribution. A deterministic equivalent of the stochastic linear fractional programming formulation has been obtained and a subsidiary convex program is given to solve the deterministic problem.

Pricing of multi-period rate of return guarantees

The basis for this paper is the pricing of multi-period rate of return guarantees. These guarantees can typically be found in life insurance and pension contracts. We derive closed form solutions, expressed by the cumulative multivariate normal probability distribution, for multi-period rate of return guarantees on both a money market account and a stock. The guarantees of Hipp (1996), Persson and Aase (1997), and Miltersen and Persson (1999) [Options for guaranteed index-linked life insurance, in: Proceedings of the AFIR 1996, 1996, pp. 1463–1483; J. Risk Insur. 64 (4) (1997) 599; Insur.: Math. Econ. 25 (3) (1999) 307] are special cases of our results.

Probability bounds given by hypercherry trees

In this article new lower and upper bounds are given for the probability of the union of events. For this purpose the new concept of hypercherry trees has been introduced. Earlier the concept of cherry tree and its application for bounding the probability of union of events was introduced by Bukszár and Prékopa. This, based on the cherry tree bound, is always an upper bound, and it can be regarded as a generalisation of the upper bound introduced by Hunter by means of maximum weight spanning trees. Later the Hunter bound was generalised by Tomescu. He used the concept of hypertrees in the framework of uniform hypergraphs and on the basis of these new hypergraph structures it became possible to define not only upper but also lower bounds on the probability of union of events. The new bounds of the paper are generalisations of Tomescu's bounds in the same sense as the upper bound by Bukszár and Prékopa was a generalisation of the Hunter bound. The efficiency of the new bounds is illustrated on some test problems according to multivariate normal probability distribution function calculations.

Improved Bounds and Simulation Procedures on the Value of the Multivariate Normal Probability Distribution Function

Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function value are given in the paper. The author's variance reduction technique was based on the Bonferroni bounds involving the first two binomial moments only. The new variance reduction technique is adapted to the most refined new bounds developed in the last decade for the estimation the probability of union respectively intersection of events. Numerical test results prove the efficiency of the simulation procedures described in the paper.

The demand for risky assets under uncertain inflation: An examination of some widely used assumptions

This paper examines the effect of the following commonly used methods of incorporating random inflation into discrete-time models of the demand for risky assets: 1) the use of a multivariate normal probability distribution for nominal asset returns and the random inflation rate, and 2) the approximation of real asset returns by the difference between nominal returns and the rate of inflation. The combination of these assumptions results in a deceptively simple version of the inflationary capital asset pricing model (CAPM). However, in an approximation-free version of this model the expected value of real wealth does not exist. While it is obvious that mean-variance analysis is not applicable in such models, we also find that the model does not satisfy Ohlson's weakened conditions for a quadratic approximation to the portfolio selection problem. Furthermore, this model is neither a member of the generalized Pareto-Levy nor log-stable class of portfolio models analyzed by Fama, Samuelson, Ohlson, and Struck.

Closure Approximations Applied to Stochastically Distributed Second-Order Reactants

The nonlinear problem of the behavior of decaying, isothermal, second-order reactions is posed stochastically. The exact solution, using an initial multivariate normal probability distribution, is compared with various closure approximations that have been employed in the study of turbulence. Kraichnan's direct-interaction hypothesis provides a satisfactory discription of the behavior of single-point functions and of the covariance of the concentration field as does the joint normal distribution hypothesis. However, the former is slightly superior even in this problem in which the energetic difficulties of the latter are not apparent.

Statistical Method of Estimating Isolines

In this paper, it is pointed out firstly that the so-called objective method of drawing isolines which is based on the method of least squares can be generalized but that it is disadvantageous in practice. As an alternative, a method based on the simultaneous confidence region of true value at each station is proposed. What has the objective meaning is the confidence region by which the existence region of isolines is provided. Especially, the confidence interval of proportional division point of two adjacent stations can be easily found by a graphical method. A definite set of isolines may be drawn arbitrarily in the restricted limit, by considering several points in each local area successively. The method is developed for two cases; the one concerns the distribution of mean values and the other the directly observed values. In the former the only assumption is that every set of random fluctuations of simultaneously observed values submits to the same multivariate normal probability distribution law, and in the latter the assumption that the covariances of observed values are known must be added.