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Contributions of Professor Hirotugu Akaike in Statistical Science

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Contributions of Professor Hirotugu Akaike in Statistical Science J. Japan Statist. Soc. Vol. 38 No. 1 2008 119–130 CONTRIBUTIONS OF PROFESSOR HIROTUGU AKAIKE IN STATISTICAL SCIENCE Genshiro Kitagawa* Prof. Akaike made significant contributions in various fields of statistical science, in particular, in time series analysis in frequency domain and time domain, informa- tion criterion and Bayes modeling. In this article, his research contributions are described in order of launching period, frequency time domain analysis, time domain time series modeling, AICand statistical modeling, and Bayes modeling. Key words and phrases: ABIC, AIC, Bayes modeling, FPE, frequency domain anal- ysis, time series analysis, time series models, TIMSAC. 1. Introduction Professor Hirotugu Akaike was awarded the 2006 Kyoto Prize for “his major contribution to statisticalscience and modelingwith the Akaike Information Criterion (AIC)”. In 1973, he proposed the AIC as a naturalextension of the log-likelihood. The most natural way of applying the AIC is to use it as the model selection or order selection criterion. In the MAICE (minimum AIC estimation) procedure, the model with the minimum value of the AIC is selected as the best one among many possible models. This provided a versatile procedure for statistical modeling that is free from the ambiguities inherent in application of the hypothesis test procedure. However, the impact of the AIC is not limited to the realization of an au- tomatic model selection procedure, and it eventually led to a paradigm shift in statisticalscience. In conventionalstatisticalinference, the theories of estimation and test are developed under the assumption of the presence of a true model. However, in statistical modeling, the model should be constructed based on the entire knowledge such as the established theory, empirical facts, current obser- vations and even the objective of the analysis. Prof. Akaike gave a practical answer to the selection of the prior distribution of the Bayes model. Due to the development of information technologies, we can now access to huge amounts of data in various fields of science and social life. In this information and knowledge society, the Bayes modelis becoming a key technology. In this article, we shall look back at his research in five stages, namely, the launching period, frequency domain time series analysis, time series modeling, AIC and statistical modeling and Bayes modeling (Parzen et al. (1998)). It should be noted here that the reader will notice that his research was always performed based on the needs of researchers in the real-world. Accepted November 20, 2007. *The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan. 120 GENSHIRO KITAGAWA 2. Launching period: Making contacts with engineers In 1952, Professor Akaike graduated from the Department of Mathematics, The University of Tokyo and became a researcher at the Institute of Statisti- calMathematics. During the 1950’s, he publishedseveraltheoreticalpapers related to decision processes, evaluation of probability distributions, computa- tion of eigenvalues, and the Monte Carlo method for solving linear equations. Among these works, the most famous one is the convergence analysis of the opti- mum gradient method (Akaike (1959a)). In this paper, he analyzed the limiting behavior of a probability distribution when a type of transformation was repeat- edly applied to an initial distribution, and showed the convergence property of the optimum gradient method. This result became a foundation in the develop- ment of more sophisticated nonlinear optimization methods and is introduced in a standard textbook of nonlinear optimization methods (Kowarik and Osborne (1968)). During this launching period, however, he was rather interested in real-world problems and tried to develop substantial contacts with engineers in various fields of industries. Through these contacts, he realized that the conventional data analysis methods often are not able to yield interesting results for real-world problems, and that we should develop a model that fully takes into account the structure of the process. In the analysis of traffic density on a road, he considered a zero-one process and, under the assumption of the independence of the lengths of time intervals between cars, he derived the gap process for the structure of the series (Akaike (1956)). Then, in collaboration with Dr. Shimazaki of the Sericultural Experiment Station, Ministry of Agriculture, he developed a control method for silk filature production processes, based on gap process modeling (Akaike (1959b)). This method provided a reference process that could be used for the detection of abnormalities in the actual reeling process. By extending this method, Dr. Shimazaki brought significant innovation to the method of silk production in Japan. It should be emphasized here that from a very early stage in the 1950’s, he realized the importance and necessity of modeling the structure of an object. 3. Frequency domain time series analysis In the early 1960’s, however, he suddenly changed his research policy. Dur- ing his communications with industrialengineers, he became to understand that linear stationary models were used frequently and that they had a very wide range of applications. Further, he noticed that there were many unsolved impor- tant problems that statisticians should contribute to. He therefore, changed his mind and tried to develop practical analysis methods based on linear stationary models. Through collaboration with Dr. Kaneshige of the Isuzu Motor Company, he learned the Blackman-Tukey method, and realized that a practical method of spectrum estimation can be developed by using a smoothing method. In this area of spectrum smoothing, he proposed a spectralwindow with negative co- efficients so that the peak of the spectrum is not reduced significantly (Akaike CONTRIBUTIONS OF PROF. AKAIKE IN STATISTICAL SCIENCE 121 (1962)). He suggested comparing the results of smoothing by positive windows and non-positive windows. Then, by the cooperative research with Dr. Yamanouchi of the Transporta- tion TechnicalResearch Institute, he developed a practicalmethod of estimating the frequency response function by solving the problem of phase-shift by shifting the time axis of the cross-covariance function (Akaike and Yamanouchi (1962)). This method made it possible to estimate the frequency response function from observations under normalsteering without using sinusoidalinputs to systems with various frequencies. He then organized a workshop and tried to apply the method to various areas of engineering such as, vibrations, engines of cars, the roll of a ship, the response of an airplane to cross winds, hydroelectric power plants, underground structures based on micro-tremors, tsunamis, the back rush of a nonlinear system, etc. The outcome of the workshop was reported in the supplement of AISM (Annals of the Institute of Statistical Mathematics) (Akaike (1964)). However, during the joint work with these engineers, an important problem that cannot be solved by the conventional spectral analysis arose. Namely, most of the systems in the real-world such as ships, cars, airplanes, chemical plants, and economics contain some kind of feedback. In these feedback systems, an output from a subsystem A becomes an input to another subsystem B, and the output of the subsystem B becomes an input to the subsystem A. Then it is quite difficult to identify the real cause of the fluctuation in the feedback system. In particular, it cannot be solved by the conventional frequency domain approach, since in the frequency domain analysis; we cannot explicitly utilize the physical realisability of the system that imperatively exists in the real world physical system. In Akaike (1967), he pointed out this difficulty and the limitation of the frequency domain analysis. 4. Time domain time series modeling At this opportunity, he returned to the time domain analysis. At that time, Chichibu Cement Company was preparing to apply optimal control theory to the control of a rotary kiln in a cement plant instead of the conventional PID controller. Conventional spectral analysis methods were useless for the analysis of a typicalfeedback system consisting of the raw materialfeed, fuelfeed, gas damper angle, and temperature at various locations, etc. As stated in the previ- ous section, this is because in the spectral analysis, we cannot properly take into account the restrictions necessary for physical realisability. However, if we introduce a structuralmodelthat expresses the relationbe- tween variables yn1,...,yn with feedback, m (4.1) yni = αkijyn−k,j + εni, j=i k=1 where m is the maximum time lag and εni is a Gaussian white noise, then it is possible to perform causal analysis of a feedback system. Akaike (1968) used the 122 GENSHIRO KITAGAWA fact that the above structuralmodelfor a feedback system can be expressed in a multivariate autoregressive model, m (4.2) yn = Ajyn−j + wn, j=1 T T where yn =(yn1,...,yn) , wn =(εn1,...,εn) and Aj is an × AR coef- ficient matrix, therefore the above structuralmodelcan be estimated from the time series data. In the paper, based on the identified modeland the relation between the multivariate autoregressive model and the cross spectrum, the power contribution was defined by | |2 2 bij σj (4.3) rij(f)= , pii(f) where pii(f) is the power spectrum of the channel i at frequency f obtained by | |2 2 (4.4) Pii(f)= bij σj , j=1 m − −1 and B(f)=(bjk)=( j=1 aj(j, k) exp( 2πijf)) . The power contribution rij(f) expresses the proportion of the contribution of noise of the variable yj in the power of the fluctuation of the variable yi at frequency f in the feedback system. This method was used to analyze not only the cement rotary kiln, but also in the analysis of nuclear and thermal power plants, living bodies, economic and financialsystems, etc. (Akaike and Nakagawa (1988), Akaike and Kitagawa (1998)). Thus, by the use of a multivariate autoregressive model, the analysis of feedback systems became practical. However, there was a difficulty with this method. It is not so difficult to estimate the parameters of a given autoregressive model.
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