independent valued minimax pathological or beneficial minimax search which is used by most game playing programs is considered pathological when deeper searches produce worse evaluations than shallower ones this phenomenon was first observed in theoretical analyses under seemingly reasonable conditions it was most commonly explained by the lack of dependence between nearby positions in the analyses if nearby positions have similar values as is typically the case in real games the pathology no longer occurs in this paper we show that the pathology can be eliminated even without position value dependence by assigning enough different values to the positions and modeling the heuristic error as normally distributed noise that is independent of the depth in the game tree this leads to the conclusion that minimax is less prone to the pathology than was previously thought and indicates the importance of the number of different position values

fuzzy linear regression with fuzzy intervals abstract in this paper we introduce a new class of fuzzy linear regression models based on tanaka s approach unlike in the tanaka model here all training data influence the estimated interval the crisp lp of tanaka s approach is generalized to a fuzzy linear programming problem with the proposed model an adaptation of the fuzzy regression equation to new data becomes possible

general fuzzy least squares abstract a model for least squares fitting of fuzzy valued data is described a previous process obtained for triangular fuzzy numbers is generalized and improved to include all fuzzy numbers represented by single maxima piecewise continuous functions with compact support the new model is compared with the previous procedure and its applicability is demonstrated

reduction to least squares estimates in multiple fuzzy regression analysis in this paper we deal with the problem of least squares multiple regression with fuzzy data the regression coefficients are assumed to be real crisp a formula for solving the regression coefficients in one variable models is derived if each independent variable is effective i e its corresponding regression coefficient is nonzero the multiple regression problem can be replaced with a 0 1 programming problem its optimal solution is easily computed finally we also propose effective algorithms to compute the regression coefficients in a general case

fuzzy regression based on asymmetric support vector machines this paper presents a modified framework of support vector machines which is called asymmetric support vector machines asvms and is designed to evaluate the functional relationship for fuzzy linear and nonlinear regression models in earlier works in order to cope with different types of input output patterns strong assumptions were made regarding linear fuzzy regression models with symmetric and asymmetric triangular fuzzy coefficients excellent performance is achieved on some linear fuzzy regression models however the nonlinear fuzzy regression model has received relatively little attention because such nonlinear fuzzy regression models having certain limitations this study modifies the framework of support vector machines in order to overcome these limitations the principle of asvms is applying an orthogonal vector into the weight vector in order to rotate the support hyperplanes the prime merits of the proposed model are in its simplicity understandability and effectiveness consequently experimental results and comparisons are given to demonstrate that the basic idea underlying asvms can be effectively used for parameter estimation

mixed effects nonlinear regression for unbalanced repeated measures repeated measures data such as clinical pharmacokinetic data growth data and dose response data are often inherently nonlinear with respect to a given response function and are frequently incomplete and or unbalanced nonlinear random effects models together with a variety of estimation procedures have been proposed for the analysis of such data this paper is concerned with a straightforward procedure for estimating and comparing the parameters of a generalized mixed effects nonlinear regression model the asymptotic properties of the proposed estimators are given and large sample tests of hypothesis provided the results are applied to in vitro data on the water transport kinetics of hemodialyzers used in the treatment of patients with chronic renal failure

bicriteria variable selection in a fuzzy regression equation abstract by considering two criteria of minimum total sum of vagueness and minimum total sum of squares in estimation this article proposes a variable selection method for a fuzzy regression equation with crisp input and fuzzy output a branch and bound algorithm is designed and the least resistance principle is adopted to determine the set of compromised solutions numerical examples are provided for illustration

fuzzy linear regression and its applications to forecasting in uncertain environment abstract linear regression has been used for many years in developing mathematical models for application in marketing management and sales forecasting in this paper two different sales forecasting techniques are discussed the first technique involves non fuzzy abstract methods of linear regression and econometrics a study of the international market sales of cameras done in 1968 by john scott armstrong utilized these non fuzzy forecasting techniques the second sales forecasting technique uses fuzzy linear regression introduced by h tanaka s uejima and k asai in 1980 in this paper a study of the computer and peripheral equipment sales in the united states is discussed using fuzzy linear regression moreover fuzzy linear regression is applied to forecasting in an uncertain environment finally some possible improvements and suggestions for further study are mentioned

estimating probability of non response to treatment using mixture distributions repeat measurements of patient characteristics are often used to assess response to treatment in this paper we discuss a normal mixture model for the observed change in the characteristic of interest in treated patients the methods described can be used to estimate the overall proportion of non response to treatment and also the probability that a patient has not responded to treatment given his or her observed change the model parameters are estimated using maximum likelihood and the delta method is used to construct a pointwise confidence band for the conditional probability that a patient is a non responder to treatment the work was initially motivated by analysis issues in the fracture intervention trial fit a randomized trial of the osteoporosis drug alendronate and the method is illustrated with data from that study we also evaluate key aspects of the estimation procedure with two simulation studies in the first the data generation model is the assumed normal mixture model and in the second the data are generated according to a shifted and scaled central t distribution model suggested by the fit data copyright 2001 john wiley sons ltd

the effect of mobility on minimaxing of game trees with random leaf values abstract random minimaxing introduced by beal and smith icca j 17 1994 3 9 is the process of using a random static evaluation function for scoring the leaf nodes of a full width game tree and then computing the best move using the standard minimax procedure the experiments carried out by beal and smith using random minimaxing in chess showed that the strength of play increases as the depth of the lookahead is increased we investigate random minimaxing from a combinatorial point of view in an attempt to gain a better understanding of the utility of the minimax procedure and a theoretical justification for the results of beal and smith s experiments the concept of domination is central to our theory intuitively one move by white dominates another move when choosing the former move would give less choice for black when it is black s turn to move and subsequently more choice for white when it is white s turn to move we view domination as a measure of mobility and show that when one move dominates another then its probability of being chosen is higher we then investigate when the probability of a good move relative to the probability of a bad move increases with the depth of search we show that there exist situations when increased depth of search is beneficial but that this is not always the case under the assumption that each move is either good or bad we are able to state sufficient conditions to ensure that increasing the depth of search increases the strength of play of random minimaxing if the semantics of the game under consideration match these assumptions then it is fair to say that random minimaxing appears to follow a reasonably intelligent strategy in practice domination does not always occur so it remains an open problem to find a more general measure of mobility in the absence of domination

a generalized fuzzy weighted least squares regression abstract a fairly general fuzzy regression technique is proposed based on the least squares approach the main concept is to estimate the modal value and the spreads separately in order to do this the interactions between the modal value and the spreads are first analyzed in detail the advantages of this new fuzzy weighted least squares regression fwlsr approach are 1 the estimation of both non interactive and interactive fuzzy parameters can be performed by the same method 2 the decision makers confidence in the gathered data and in the established model can be incorporated into the process and 3 suspicious outliers or fuzzy outliers that is data points that are obviously and suspiciously lying outside the usual range can be treated and their effects can be reduced a numerical example is provided to show that the proposed method can be an effective computational tool in fuzzy regression analysis

linear regression analysis for fuzzy input and output data using the extension principle abstract the method for obtaining the fuzzy least squares estimators with the help of the extension principle in fuzzy sets theory is proposed the membership functions of fuzzy least squares estimators will be constructed according to the usual least squares estimators in order to obtain the membership value of any given value taken from the fuzzy least squares estimator optimization problems have to be solved we also provide the methodology for evaluating the predicted fuzzy output from the given fuzzy input data

a variable spread fuzzy linear regression model with higher explanatory power and forecasting accuracy fuzzy regression models have been applied to operational research or applications such as forecasting some of previous studies on fuzzy regression analysis obtain crisp regression coefficients for eliminating the problem of increasing spreads for the estimated fuzzy responses as the magnitude of the independent variable increases however they still cannot cope with the situation of decreasing or variable spreads this paper proposes a three phase method to construct the fuzzy regression model with variable spreads to resolve this problem in the first phase on the basis of the extension principle the membership functions of the least squares estimates of regression coefficients are constructed to conserve completely the fuzziness of observations in the second phase then they are defuzzified by the center of gravity method to obtain crisp regression coefficients in the third phase the error terms of the proposed model are determined by setting each estimated spread equals its corresponding observed spread furthermore the mamdani fuzzy inference system is adopted for improving the accuracy of its forecasts compared to the previous studies the results from five examples and an application example of japanese house prices show that the proposed fuzzy linear regression model has higher explanatory power and forecasting performance

fuzzy linear regression with spreads unrestricted in sign abstract by the use of actual examples it is shown that when the two trends namely the fuzzy spread trend and the center line or modal trend are in conflict or inconsistent the results of the fuzzy linear regression flr model based on the approach of tanaka and coworkers frequently misinterprets the data to avoid this misinterpretation a modification of the flr procedure by allowing the spreads of the parameter to be unrestricted in sign is proposed it is shown that by the use of this unrestricted fuzzy linear regression uflr model correct prediction of the trends are obtained

an incremental negamax algorithm abstract in certain models of game trees with erroneous evaluation functions the minimax algorithm does not reduce errors even under favourable assumptions about the size of the errors and the frequency of their occurrence we present an incremental negamax algorithm which uses estimates of all nodes in the tree rather than only those of the leaves to determine the root value under the assumption of independently occurring and sufficiently small errors the algorithm is shown to have exponentially reduced error probabilities with respect to the height of the tree

s curve regression model in fuzzy environment abstract the purpose of this paper is to discuss the problem for least squares fitting of fuzzy valued data which are expressed as fuzzy numbers and to develop an s shaped curve regression model for fitting this type of data it is shown that the solution of the s curve regression model is equivalent to the solution of the corresponding linear equations and furthermore the solution can be explicitly obtained by solving the linear equations

a mathematical programming approach to fuzzy linear regression analysis most of previous studies on fuzzy regression analysis have a common characteristic of increasing spreads for the estimated fuzzy responses as the independent variable increases its magnitude which is not suitable for general cases in this paper fuzzy linear regression models with fuzzy crisp output fuzzy crisp input are considered and an estimated method along with a mathematical programming based approach is proposed the advantages of the proposed approach are simplicity in programming and computation and minimum difference of total spread between observed and estimated values

a linear mixed effects model with heterogeneity in the random effects population abstract this article investigates the impact of the normality assumption for random effects on their estimates in the linear mixed effects model it shows that if the distribution of random effects is a finite mixture of normal distributions then the random effects may be badly estimated if normality is assumed and the current methods for inspecting the appropriateness of the model assumptions are not sound further it is argued that a better way to detect the components of the mixture is to build this assumption in the model and then compare the fitted model with the gaussian model all of this is illustrated on two practical examples

the multi player version of minimax displays game tree pathology abstract it is widely believed that by searching deeper in the game tree the decision maker is more likely to make a better decision dana nau and others have discovered pathology theorems that show the opposite searching deeper in the game tree causes the quality of the ultimate decision to become worse not better the models for these theorems assume that the search procedure is minimax and the games are two player zero sum this report extends nau s pathology theorem to multi player game trees searched with maxn the multi player version of minimax thus two player zero sum game trees and multi player game trees are shown to have an important feature in common

fuzzy linear regression models with least square errors to estimate the parameters of fuzzy linear regression models with fuzzy output and crisp inputs we develop a mathematical programming model in this paper the method is constructed on the basis of minimizing the square of the total difference between observed and estimated spread values or in other words minimizing the least square errors the advantage of the proposed approach is its simplicity in programming and computation as well as its performance to compare the performance of the proposed approach with the other methods two examples are presented

fuzzy linear regression analysis a multi objective programming approach fuzzy regression analysis is an extension of the classical regression analysis that is used in evaluating the functional relationship between the dependent and independent variables in a fuzzy environment fuzzy regression has been criticized because it is sensitive to outliers it does not allow all data points to influence the estimated parameters and the spread of the estimated values become wider as more data are included in the model in this paper we develop a multi objective fuzzy linear regression model to overcome the above shortcomings

estimation of the nonlinear random coefficient model when some random effects are separable a method is presented for marginal maximum likelihood estimation of the nonlinear random coefficient model when the response function has some linear parameters this is done by writing the marginal distribution of the repeated measures as a conditional distribution of the response given the nonlinear random effects the resulting distribution then requires an integral equation that is of dimension equal to the number of nonlinear terms for nonlinear functions that have linear coefficients the improvement in computational speed and accuracy using the new algorithm can be dramatic an illustration of the method with repeated measures data from a learning experiment is presented

mixed fuzzy least absolute regression analysis with quantitative and probabilistic linguistic information abstract regression analysis is widely used in evaluation and prediction and fuzzy least absolute regression is preferred when data is fat tailed or out linear given that the probabilistic linguistic term set is a powerful tool in expressing evaluators complex linguistic perceptions this paper incorporates the probabilistic linguistic term set to the fuzzy least absolute regression and builds a fuzzy regression model with mixed types of inputs to achieve this goal this paper introduces the concept of double cut set of the probabilistic linguistic term set then new operations of probabilistic linguistic term sets based on the double cut sets are investigated a mixed fuzzy least absolute regression model is proposed and a linear programming is introduced to work out the fuzzy regression parameters a numerical example concerning the house lease price evaluation under the shared economy is provided to validate the applicability of the proposed model the study ends with some concluding remarks

finite mixture modeling with mixture outcomes using the em algorithm summary this paper discusses the analysis of an extended finite mixture model where the latent classes corresponding to the mixture components for one set of observed variables influence a second set of observed variables the research is motivated by a repeated measurement study using a random coefficient model to assess the influence of latent growth trajectory class membership on the probability of a binary disease outcome more generally this model can be seen as a combination of latent class modeling and conventional mixture modeling the em algorithm is used for estimation as an illustration a random coefficient growth model for the prediction of alcohol dependence from three latent classes of heavy alcohol use trajectories among young adults is analyzed

a finite mixture of nonlinear random coefficient models for continuous repeated measures data nonlinear random coefficient models nrcms for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development or growth over time in measured variables as an extension of this model this study considers the finite mixture of nrcms that combine features of nrcms with the idea of finite mixture or latent class models the efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations thus allowing the simultaneous investigation of intra individual within person variability inter individual between person variability and subpopulation heterogeneity effectiveness of this model to work under real data analytic conditions was examined by executing a monte carlo simulation study the simulation study was carried out using an r routine specifically developed for the purpose of this study the r routine used maximum likelihood with the expectation maximization algorithm the design of the study mimicked the output obtained from running a two class mixture model on task completion data

benefits of using multivalued functions for minimaxing abstract minimaxing has been very successful in game playing practice although a complete explanation of why it has been that successful has not yet been given in particular it has not been shown why it should be useful as it is in practice to use multivalued evaluation functions such functions have many distinct values as their result and can discriminate between positions according to the heuristic knowledge represented in these values in this paper we modify a basic pathological model by adding two assumptions regarding multivalued evaluation functions these assumptions non uniformity of error distribution and dependency of heuristic values directly relate to the properties of multivalued evaluation functions as used in practice simulation studies of our multivalued model have exhibited sharp error reductions for deeper searches using minimaxing this behavior corresponds to observations in practice the error reductions are primarily due to the improved evaluation quality as search depth increases this phenomenon of lower probability of static evaluation errors with increasing search depth is revealed through our model although the same evaluation function is used at all levels of the tree and although its general error probability is independent of the depth essentially with increasing search depth the evaluation function is more frequently used on such positions which can be more reliably evaluated by a multivalued function with the assumed properties this effect together with the ability to discriminate between positions of different goodness leads to the benefits of using multivalued evaluation functions of appropriate granularity for minimaxing

estimation and hypothesis testing in finite mixture models summary finite mixture models are a useful class of models for application to data when sample sizes are not large and the number of underlying densities is in question likelihood ratio tests based on joint maximum likelihood estimation of the mixing parameter x and the parameter of the underlying densities 0 are problematical our approach places a prior distribution on x and estimates 0 by maximizing the likelihood of the data given 0 with x integrated out advantages of this approach computational issues using the em algorithm and directions for further work are discussed the technique is applied to two examples

insight of a fuzzy regression model fuzzy regression a nonparametric method can be quite useful in estimating the relationships among variables where the available data are very limited and imprecise and variables are interacting in an uncertain qualitative and fuzzy way thus it may have considerably practical applications in many management and engineering problems but there is still lack of proper interpretation about fuzzy regression in this paper we provide an insight into regression intervals so that regression interval analysis data type analysis and variable selections can be analytically performed numerical examples are provided for illustration

resolution of fuzzy regression model abstract fuzzy linear regression was originally introduced by tanaka et al to cope with different types of input output information several approaches to fuzzy linear regression have been proposed in this paper a type of problem with crisp input and fuzzy output described by tanaka is considered of which a modified fuzzy least square method was proposed for solution it shows that with such an approach the predictability in the new model is better than tanaka s and its computation efficiency is better than the conventional fuzzy least square method

possibilistic linear regression with fuzzy data tolerance approach with prior information abstract we introduce the tolerance approach to the construction of fuzzy regression coefficients of a possibilistic linear regression model with fuzzy data both input and output the method is very general the only assumption is that cuts of the fuzzy data are efficiently computable we take into account possible prior restrictions of the parameters space we assume that the restrictions are given by linear and quadratic constraints the method for construction of the possibilistic regression coefficients is in a reduction of the fuzzy valued model to an interval valued model on a given cut which is further reduced to a linear time method i e running in time o n p computing with endpoints of the intervals here n is the number of observations and p is the number of regression parameters the speed of computation makes the method applicable for huge datasets unlike various approaches based on mathematical programming formulations the tolerance based construction preserves central tendency of the resulting regression coefficients in addition we prove further properties if inputs are crisp and outputs are fuzzy then the construction preserves piecewise linearity and convex shape of fuzzy numbers we derive an o n 2 p algorithm for enumeration of breakpoints of the membership function of the estimated coefficients similar results are also derived for the fuzzy input and output model we illustrate the theory for the case of triangular and asymmetric gaussian fuzzy inputs and outputs of an inflation consumption model

mixtures of conditional mean and covariance structure models models and parameters of finite mixtures of multivariate normal densities conditional on regressor variables are specified and estimated we consider mixtures of multivariate normals where the expected value for each component depends on possibly nonnormal regressor variables the expected values and covariance matrices of the mixture components are parameterized using conditional mean and covariance structures we discuss the construction of the likelihood function estimation of the mixture model with regressors using three different em algorithms estimation of the asymptotic covariance matrix of parameters and testing for the number of mixture components in addition to simulation studies data on food preferences are analyzed

on the nature of pathology in game searching abstract game playing programs usually process the estimates attached to game positions through repeated minimax operations as if these were true terminal payoffs this process introduces a spurious noise which degrades the quality of the decisions and in the extreme case may cause a pathological phenomenon the deeper we search the worse we play using a probabilistic game model this paper examines the nature of this distortion quantifies its magnitude determines the conditions when its damage is curtailed and explains why search depth pathology the extreme manifestation of minimax distortion is rarely observed in common games

asymptotic properties of levelregular decision trees with randomly evaluated leaves game trees are an important model of decision making situations both in artificial intelligence and decision analysis the model most frequently investigated in theoretical research consists of a uniform tree of heighh and a constant branching factorb where the terminal positions are assigned the values of independent identically distributed random variables 1 3 10 our paper investigates two generalizations

a fuzzy approach for multiresponse optimization an off line quality engineering problem abstract multiresponse optimization techniques are used to identify settings of process parameters that make the product s performance close to target values in the presence of multiple quality characteristics in this study a fuzzy multiresponse optimization procedure is proposed to search an appropriate combination of process parameter settings based on multiple quality characteristics or responses fuzzy regression models are first used to model the relations between process parameters and responses the possibility distributions of the prediction responses are then obtained the problem is reduced to a multiobjective optimization problem with fuzzy objectives a strategy of optimizing the most possible response values and minimizing the deviations from the most possible values is proposed we consider not only the most possible value but also the imprecision of the prediction responses through a die casting example we show how to use our approach to reach an appropriate machine setting which simultaneously optimize both casting quality and die life

when is it better not to look ahead in situations where one needs to make a sequence of decisions it is often believed that looking ahead will help produce better decisions however it was shown 30 years ago that there are pathological situations in which looking ahead is counterproductive two long standing open questions are a what combinations of factors have the biggest influence on whether lookahead pathology occurs and b whether it occurs in real world decision making this paper includes simulation results for several synthetic game tree models and experimental results for three well known board games two chess endgames kalah with some modifications to facilitate experimentation and the 8 puzzle the simulations show the interplay between lookahead pathology and several factors that affect it and the experiments confirm the trends predicted by the simulation models the experiments also show that lookahead pathology is more common than has been thought all three games contain situations where it occurs

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

score tests for dose effect in the presence of non responders when only a certain proportion of subjects respond to treatment responders or may never experience an event of interest thus cured mixture models often lead to increased understanding of the treatment or disease process this paper focuses on hypothesis testing in a dose response framework and shows that increased power is possible by using a mixture model where both the logit of the response rate and the response mean are linear functions of the dose level three score tests are developed for testing an overall effect and permutation methods are used to control the type i error extensive simulations establish the power properties of the tests and show that our proposed score test has the best performance the approach is illustrated by a multi country clinical trial of rapid acting intramuscular olanzapine copyright 2004 john wiley sons ltd