arXiv subjects
Publications and source records attributed to Akimichi Takemura.
We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere $S^n$, which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems.
We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all braid slices, including those not associated with unfolding models, are in one-to-one correspondence with the chambers of an arrangement. By identifying those which are associated with unfolding models, we find the number of ranking patterns. We also give an upper bound for the number of ranking patterns when the difference by a permutation of objects is ignored.
Markov chain models are used in various fields, such behavioral sciences or econometrics. Although the goodness of fit of the model is usually assessed by large sample approximation, it is desirable to use conditional tests if the sample size is not large. We study Markov bases for performing conditional tests of the toric homogeneous Markov chain model, which is the envelope exponential family for the usual homogeneous Markov chain model. We give a complete description of a Markov basis for the following cases: i) two-state, arbitrary length, ii) arbitrary finite state space and length of three. The general case remains to be a conjecture. We also present a numerical example of conditional tests based on our Markov basis.
We give an expository review of applications of computational algebraic statistics to design and analysis of fractional factorial experiments based on our recent works. For the purpose of design, the techniques of Gröbner bases and indicator functions allow us to treat fractional factorial designs without distinction between regular designs and non-regular designs. For the purpose of analysis of data from fractional factorial designs, the techniques of Markov bases allow us to handle discrete observations. Thus the approach of computational algebraic statistics greatly enlarges the scope of fractional factorial designs.
We propose procedures for testing whether stock price processes are martingales based on limit order type betting strategies. We first show that the null hypothesis of martingale property of a stock price process can be tested based on the capital process of a betting strategy. In particular with high frequency Markov type strategies we find that martingale null hypotheses are rejected for many stock price processes.
We study Markov bases of decomposable graphical models consisting of primitive moves (i.e., square-free moves of degree two) by determining the structure of fibers of sample size two. We show that the number of elements of fibers of sample size two are powers of two and we characterize primitive moves in Markov bases in terms of connected components of induced subgraphs of the independence graph of a hierarchical model. This allows us to derive a complete description of minimal Markov bases and minimal invariant Markov bases for decomposable models.
We propose minimum empirical divergence (MED) policy for the multiarmed bandit problem. We prove asymptotic optimality of the proposed policy for the case of finite support models. In our setting, Burnetas and Katehakis has already proposed an asymptotically optimal policy. For choosing an arm our policy uses a criterion which is dual to the quantity used in Burnetas and Katehakis. Our criterion is easily computed by a convex optimization technique and has an advantage in practical implementation. We confirm by simulations that MED policy demonstrates good performance in finite time in comparison to other currently popular policies.
We give some sufficient conditions of separation of two sets of integer points by a hyperplane. Our conditions are related to the notion of convexity of sets of integer points and are weaker than existing notions.
In this paper we propose an investing strategy based on neural network models combined with ideas from game-theoretic probability of Shafer and Vovk. Our proposed strategy uses parameter values of a neural network with the best performance until the previous round (trading day) for deciding the investment in the current round. We compare performance of our proposed strategy with various strategies including a strategy based on supervised neural network models and show that our procedure is competitive with other strategies.
We introduce a new formulation of asset trading games in continuous time in the framework of the game-theoretic probability established by Shafer and Vovk (Probability and Finance: It's Only a Game! (2001) Wiley). In our formulation, the market moves continuously, but an investor trades in discrete times, which can depend on the past path of the market. We prove that an investor can essentially force that the asset price path behaves with the variation exponent exactly equal to two. Our proof is based on embedding high-frequency discrete-time games into the continuous-time game and the use of the Bayesian strategy of Kumon, Takemura and Takeuchi (Stoch. Anal. Appl. 26 (2008) 1161--1180) for discrete-time coin-tossing games. We also show that the main growth part of the investor's capital processes is clearly described by the information quantities, which are derived from the Kullback--Leibler information with respect to the empirical fluctuation of the asset price.
We propose a sequential optimizing betting strategy in the multi-dimensional bounded forecasting game in the framework of game-theoretic probability of Shafer and Vovk (2001). By studying the asymptotic behavior of its capital process, we prove a generalization of the strong law of large numbers, where the convergence rate of the sample mean vector depends on the growth rate of the quadratic variation process. The growth rate of the quadratic variation process may be slower than the number of rounds or may even be zero. We also introduce an information criterion for selecting efficient betting items. These results are then applied to multiple asset trading strategies in discrete-time and continuous-time games. In the case of continuous-time game we present a measure of the jaggedness of a vector-valued continuous process. Our results are examined by several numerical examples.
We discuss connecting tables with zero-one entries by a subset of a Markov basis. In this paper, as a Markov basis we consider the Graver basis, which corresponds to the unique minimal Markov basis for the Lawrence lifting of the original configuration. Since the Graver basis tends to be large, it is of interest to clarify conditions such that a subset of the Graver basis, in particular a minimal Markov basis itself, connects tables with zero-one entries. We give some theoretical results on the connectivity of tables with zero-one entries. We also study some common models, where a minimal Markov basis for tables without the zero-one restriction does not connect tables with zero-one entries.
We prove game-theoretic generalizations of some well known zero-one laws. Our proofs make the martingales behind the laws explicit, and our results illustrate how martingale arguments can have implications going beyond measure-theoretic probability.
An integer feasibility problem is a fundamental problem in many areas, such as operations research, number theory, and statistics. To study a family of systems with no nonnegative integer solution, we focus on a commutative semigroup generated by a finite set of vectors in $\Z^d$ and its saturation. In this paper we present an algorithm to compute an explicit description for the set of holes which is the difference of a semi-group $Q$ generated by the vectors and its saturation. We apply our procedure to compute an infinite family of holes for the semi-group of the $3\times 4\times 6$ transportation problem. Furthermore, we give an upper bound for the entries of the holes when the set of holes is finite. Finally, we present an algorithm to find all $Q$-minimal saturation points of $Q$.
We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in Aoki and Takemura [2008a]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof in Aoki and Takemura [2008a] which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.
A new class of two-level non-regular fractional factorial designs is defined. We call this class an {\it affinely full-dimensional factorial design}, meaning that design points in the design of this class are not contained in any affine hyperplane in the vector space over $\mathbb{F}_2$. The property of the indicator function for this class is also clarified. A fractional factorial design in this class has a desirable property that parameters of the main effect model are simultaneously identifiable. We investigate the property of this class from the viewpoint of $D$-optimality. In particular, for the saturated designs, the $D$-optimal design is chosen from this class for the run sizes $r \equiv 5,6,7$ (mod 8).
We propose iterative proportional scaling (IPS) via decomposable submodels for maximizing likelihood function of a hierarchical model for contingency tables. In ordinary IPS the proportional scaling is performed by cycling through the elements of the generating class of a hierarchical model. We propose to adjust more marginals at each step. This is accomplished by expressing the generating class as a union of decomposable submodels and cycling through the decomposable models. We prove convergence of our proposed procedure, if the amount of scaling is adjusted properly at each step. We also analyze the proposed algorithms around the maximum likelihood estimate (MLE) in detail. Faster convergence of our proposed procedure is illustrated by numerical examples.
In this paper we consider exact tests of a multiple logistic regression, where the levels of covariates are equally spaced, via Markov beses. In usual application of multiple logistic regression, the sample size is positive for each combination of levels of the covariates. In this case we do not need a whole Markov basis, which guarantees connectivity of all fibers. We first give an explicit Markov basis for multiple Poisson regression. By the Lawrence lifting of this basis, in the case of bivariate logistic regression, we show a simple subset of the Markov basis which connects all fibers with a positive sample size for each combination of levels of covariates.