Browsing by Author "Good, Irving John"

Now showing 1 - 5 of 5
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    Density estimation and some topics in multivariate analysis
    Gaskins, Ray Allen (Virginia Tech, 1972-05-15)
    Part I, entitled "A test of goodness-of-fit for multivariate distributions with special emphasis on multinornality", investigates a modification of the Chi-squared goodness-of-fit statistic which eliminates certain objectionable properties of other multivariate goodness-of-fit tests. Special emphasis is given to the multinormal distribution, and computer simulation is used to generate an empirical distribution for this goodness-of-fit statistic for the standardized bivariate normal density. Attempts to fit a four-parameter generalized gamma density function to this empirical distribution were only partially successful. Part II, entitled "The centroid method of numerical integration", begins with a discussion of the often slighted midpoint method of numerical integration, then, using Taylor's theorem, generalized formulae for the centroid method of numerical integration of a function of several variables over a closed bounded region are developed. These formulae are in terms of the derivatives of the integrand and the moments of the region of integration with respect to its centroid. Since most nonpathological bounded regions can be well approximated by a finite set of simplexes, formulae are developed for the moments of general as well as special simplexes. Several numerical examples are given and a comparison is made between the midpoint and Gaussian quadrature methods. FORTRAN programs are included. Part III - entitled "Non-parametric density estimation," begins with an extensive literature review of non-parametric methods for estimating probability densities based on a sample of N observations and goes on to suggest a new method which is to subtract a penalty for roughness fron the log-likelihood before maximizing. The roughness penalty is a functional of the assumed density function and the recommendation is to use a linear combination of the squares of the first and second derivatives of the square root of the density function. Many numerical examples and graphs are given and show that the estimated density function, for selected values of the coefficients in the linear expression, turns out to be very smooth even for very small sample sizes. Computer programs are not included but are available upon request. Part IV, entitles "On separation of product and error variability," surveys standard techniques of partitioning the total variance into product (or item) variance and error (or testing) variance when destructive testing makes replication over the same item impossible. The problem of negative variance estimates is also investigated. The factor-analysis model and related iterative techniques are suggested as an alternative method for dealing with this separation when three or more independent measurements per item are available. The problem of dependent measurements is discussed. Numerical examples are included.
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    Least squares mixture decomposition estimation
    Kim, Donggeon (Virginia Tech, 1995-02-13)
    The Least Squares Mixture Decomposition Estimator (LSMDE) is a new nonparametric density estimation technique developed by modifying the ordinary kernel density estimators. While the ordinary kernel density estimator assumes equal weight (l/n) for each data point, LSMDE assigns the optimized weight to each data point via the quadratic programming under the Mean Integrated Squared Error (MISE) criterion. As results, we find out that the optimized weights for a given data set are far different from l/n for a reasonable smoothing parameter and, furthermore, many data points are assigned to zero weights after the optimization. This implies that LSMDE decomposes the underlying density function to a finite mixture distribution of p (< n) kernel functions. LSMDE turns out to be more informative, especially in multi-dimensional cases when the visualization of the density function is difficult, than the ordinary kernel density estimator by suggesting the underlying structure of a given data set.
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    A New Method for Comparing Experiments and Measuring Information
    Kitchin, Patricia Lee III (Virginia Tech, 1997-09-09)
    A statistic that summarizes an entire data set without losing any information about the family of distributions or the model is often called a sufficient statistic. Generally, one would like to use the statistic that contains the most information about the parameter space. Sometimes there are several sufficient statistics. At other times the only sufficient statistic is the entire data set. A large data set can be difficult to work with. In this case, can one use a statistic that, though not sufficient, does summarize the data set somewhat? How much information would be lost? How can one compare two statistics that aren't sufficient in terms of the amount of information each provides? A new method for comparing experiments and measuring information is introduced. No assumptions are made and no conditions are required in order for this new method to measure the amount of information contained in almost any statistic. Several properties of this new method are discussed and a new characterization of sufficiency based on this new method is presented. The new method is used to evaluate the expected efficiency of a statistic in discriminating between any two values of the parameter as compared to a sufficient statistic. This new method can be self-calibrated to give this expected efficiency a meaningful scale. It is shown that this new method has some advantages over existing methods of measuring information. This new method is applied to Casino Blackjack. Several card-counting statistics are compared by the amount of information each provides in discriminating between different deck compositions as compared to a sufficient statistic. This new method provides new insight about information in card-counting statistics by putting this information on a meaningful scale.
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    A response surface approach to the mixture problem when the mixture components are categorized
    Cornell, John A. (Virginia Tech, 1968-12-05)
    A method is developed for experiments with mixtures where the mixture components are categorized (acids, bases, etc.), and each category of components contributes a fixed proportion to the total mixture. The number of categories of mixture components is general and each category will be represented in every mixture by one or more of its member components. The purpose of this paper is to show how standard response surface designs and polynomial models can be used for estimating the response to mixtures of the k mixture components. The experimentation is concentrated in an ellipsoidal region chosen by the experimenter, subject to the constraints placed on the components. The selection of this region, the region of interest, permits the exclusion of work in areas not of direct interest. The transformation from a set of linearly dependent mixture components to a set of linearly independent design variables is shown. This transformation is accomplished with the use of an orthogonal matrix. Since we want the properties of the predictor ŷ at a point w to be invariant to the arbitrary elements of the transformation matrix, we choose to use rotatable designs. Frequently, there are underlying sources of variation in the experimental program whose effects can be measured by dividing the experimentation into stages, that is, blocking the observations. With the use of orthogonal contrasts of the observations, it is shown how these effects can be measured. This concept of dividing the program of experiments into stages is extended to include second degree designs. The radius of the largest sphere, in the metric of the design variables, that will fit inside the factor space is derived. This sphere provides an upper bound on the size of an experimental design. This is important when one desires to use a design to minimize the average variance of ŷ only for a first-degree model. It is also shown with an example how with the use of the largest sphere, one can cover almost all combinations of the mixture components, subject to the constraints.
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    Speculations Concerning the First Ultraintelligent Machine
    Good, Irving John (Virginia Tech, 2005-03-05)
    The survival of man depends on the early construction of an ultraintelligent machine. In order to design an ultraintelligent machine we need to understand more about the human brain or human thought or both. In the following pages an attempt is made to take more of the magic out of the brain by means of a "subassembly" theory, which is a modification of Hebb's famous speculative cell-assembly theory. My belief is that the first ultraintelligent machine is most likely to incorporate vast artificial neural circuitry, and that its behavior will be partly explicable in terms of the subassembly theory. Later machines will all be designed by ultra-intelligent machines, and who am I to guess what principles they will devise? But probably Man will construct the deus ex machina in his own image.