Akaike, H. (1974). A new look at the statistical model identi cation. IEEE Trans. Automat. Control 19, 716-723.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the PRESS criterion for model selection in general regression (see e.g. Cook Weisberg [8] But this procedure has some de ciencies. Shao [32] pointed out that in linear models, leave one out cross validation is asymptotically equivalent to the Akaike information criterion (AIC) Akaike [1]) the C p (Mallows [28] the jackknife, and the bootstrap (Efron [14] But these do not provide consistent model selection, meaning that they do not provide the best predictive model with probability 1, as n 1. It has been found by Stone [35] that the probability of choosing a good model ....

Akaike, H. (1974). A new look at statistical model identi cation. IEEE Transactions on Automatic Control 19, 716-723.

....bounds have in nite entropy and are of no help when choosing a model. 40 Model choice and universal principles Some procedures of model choice are universal in that they are based on a principle which is independent of any substantive knowledge about the data. Examples are Akaike s AIC (Akaike (1973, 1974, 1977, 1978 1981) Schwarz s BIC (Schwarz (1978) Bozdogan s ICOMP (Bozdogan (200) Rissanen s MDL (Rissanen (1987) and Bayes. Substantive knowledge is incorporated in the choice of the family of models and, in the Bayesian scheme, in the prior distribution. This having been done no further ....

Akaike, H. (1974). A new look at the statistical model identi cation. IEEE Transactions on Automatic Control, 19:716-723.

....with the simplest sequence of nite dimensional spaces which still has good approximation properties for F , since for us the estimation of is of primary interest, and f is a nuisance parameter. However, other choices are possible: spaces generated by splines based on an irregular partition of [0,1], wavelets or trigonometric polynomials; see Birg e and Massart (1996) for a detailed discussion. 3. Risk bounds for the penalized least squares estimator In this section we give risk bounds for the estimators of ( f) satisfying (2.2) That is, we consider k k and k k n , respectively, as ....

....= 1. 3. There exists b 0 such that E exp(jW i j=b) 4, for all i 2 f1; ng. 4. There exists c 0 such that ks vk1 c for all v 2 Sm and for all m 2 Mn . 5. There exists h 0 ; h 1 0 such that h 0 f T h 1 , where f T is the density of T with respect to , Lebesgue measure on [0, 1]. 6. l V ar(X jT )l is bounded away from zero and in nity for any l 2 R . De ne A 8(b c) B 2c. For any 0 let ( and Let C 1 9:8 q 0 2f1; 2 g (e k=1 (e , C 2 1= and c 1 C 1 C 2 . Let S be the orthogonal projection operator onto a space S and ....

[Article contains additional citation context not shown here]

Akaike, H. A new look at the statistical model identi cation. IEEE Trans. Automat. Control, 19:716-723, 1974.

....j 0 j Kn g. This approximating space is known to have good approximation properties for a range of smoothness classes to which f may belong; see, e.g. DeVore and Lorentz (1993) However, other choices are possible: spaces generated by piecewise polynomials based on an irregular partition of [0,1], wavelets or trigonometric polynomials; see Birg e and Massart (1996) for a detailed discussion. Note that this construction is needed for Case 3 and it can be particularized to the other three cases. In Case 0 only one sieve is needed, namely S I0 ;Kn; with K n; n . In Cases 1 and 2 one ....

....basis in R q0 and R , respectively, which implies that A = O q0 (q q0 ) which completes the proof of this lemma. Proof of Proposition 3.1 Let f j g j=1 be an orthonormal basis in L 2 ( for SNn de ned in Section 2. Here, by abuse of notation, we denote the restriction of to [0, 1] still by . Then, we can write any g 2 S Iq ;Nn as g(x; t) P q j=1 a j x j j=1 b j j (t) By Lemma 1, page 337 in Birg e and Massart (1998) we have that max t2[0;1] j=1 j (t) 2r 1) Nn . Also, by Assumption 1, there exists an M 0 such that jXj 2 M with probability 27 one, ....

Akaike, H. A new look at the statistical model identi cation. IEEE Trans. Automat. Control, 19:716-723, 1974.

....is dicult or impossible to formalize. There are statistical criteria (such as the Akaike, Schwarz, and Takeuchi Information Criterion and generalizations of these related to informational complexity of models) which depend not only upon estimates of the maximum likelihood estimator (MLE) [2, 6, 53]. but incorporate the number of parameters and the number of observations, for a quantitative evaluation of di erent models. Bozdogan s criterion [23] is based upon a linear combination of the lack of t (characterized by the size of an objective function) the lack of parsimony, and correlations ....

Akaike H., A new look at the statistical model identi cation, IEEE Trans. Automatic Control, 19 (1974) 716-723.

....perspective: Every method uses some prior. The Bayesian methodology, however, is the only one that makes the prior explicit. An example is model selection: whatever method we look at, there is always a best t versus complexity trade o . That is, Akaike s information criterion (AIC) see [Aka74] minimum description length (MDL) see [Ris78] minimum message length (MML) see [WF87] or likelihood ratio tests as used in frequentist statistics (see [SS71] all use a prior that prefers simple models. The same is true for statistical learning theory summarized in [Vap95] where model ....

H. Akaike. A new look at statistical model identi cation. IEEE Transactions on Automatic Control, 19:716-723, 1974.

....dim(S) is the dimension number of parameters needed to specify the model of the Bayesian network with structure given by S. It is computed as dim(S) Q n i=1 q i (r i 1) This penalization function f(N) is a non negative one. Some examples for f(N) are the Akaike s Information Criterion (AIC) [37] where f(N) 1 , and the Je reys Schwarz criterion, sometimes called the Bayesian Information Criterion (BIC) 38] where f(N) 2 log N . Following the latter criterion, the corresponding BIC score BIC(S; D) for a Bayesian network structure S constructed from a database D and containing N ....

H. Akaike. New look at the statistical model identi cation. IEEE Transactions on Automatic Control, 19(6):716-723, 1974.

....of subjective based methods are listed in [43] Although objective based methods can be implemented in practice, most have high computational cost and give good performance only at large SNR. The candidates of the objectivebased methods include the model selection methods proposed by Akaike [44], Schwartz [45] and Rissanen [46] and the information theoretic criteria developed byWax and Kailath [47] and Reddy and Biradar [48] Sano and Tsuji [49] develop an objective based method from perturbation analysis. Their method introduces a set of regularization parameters and estimates the ....

....theoretic criteria developed byWax and Kailath [47] and Reddy and Biradar [48] Sano and Tsuji [49] develop an objective based method from perturbation analysis. Their method introduces a set of regularization parameters and estimates the number of poles more accurate than the methods in [44] and [47] A set of regularization parameters, # i #, is added to (2.14) so that the regularized SVD truncated pseudoinverse is described by ## c X y R # = r X m## 1 m m vmu H m (2.15) where r is the rank of R # . If i = 0 for i = 1; 2; ###;M, and i # # for i = M ....

H. Akaike, \A new look at the statistical model identication," IEEE Trans. Automatic Control,vol. 19, pp. 716-723, Dec. 1974.

.... are: i) signal detection in the presence of noise, where certain parameters of the desired signal (e.g. amplitude, phase, Doppler shift) are unknown [9] 28] ii) pattern recognition problems like speech recognition [20] and optical character recognition [25] iii) model order selection [1], 21] for instance, estimating the order of a Markov process [16] and (iv) universal decoding in the presence of channel uncertainty [2, Chap. 2, Sect. 5] 5] 11] 14] 30] The latter application, which will receive special attention in this paper, is actually the one that motivated our ....

H. Akaike, \A new look at the statistical model identication," IEEE Trans. on Automatic Control , vol. AC{19, no. 6, pp. 716-723, December 1974.

....to the PRESS criterion for model selection in general regression (see e.g. Cook Weisberg [8] But this procedure has some de ciencies. Shao [32] pointed out that in linear models, leave one out cross validation is asymptotically equivalent to the Akaike information criterion (AIC) Akaike [1]) the C p (Mallows [28] the jackknife, and the bootstrap (Efron [14] But these do not provide consistent model selection, meaning that they do not provide the best predictive model with probability 1, as n 1. It has been found by Stone [36] that the probability of choosing a good model ....

Akaike, H. (1974). A new look at statistical model identication. IEEE Transactions on Automatic Control 19, 716-723.

....a ects the result of segmentation. In a statistical problem formulation such as the one introduced in the previous section, the use of information theoretic criteria for the problem of model determination arises as a natural choice. Two popular approaches are Akaike s information criterion (AIC) [55], and Rissanen s minimum description length (MDL) 56] Akaike proposes the selection of the model that gives the minimum AIC, that is de ned by AIC(K a ) 2 log(L( r ML ) 2K a (30) where r ML is the maximum likelihood estimate of the model parameter set r, and K a is the number of free ....

.... length (MDL) 56] Akaike proposes the selection of the model that gives the minimum AIC, that is de ned by AIC(K a ) 2 log(L( r ML ) 2K a (30) where r ML is the maximum likelihood estimate of the model parameter set r, and K a is the number of free adjustable parameters in the model [17, 55] and is given by 3K 1 for the SFNM model. The AIC selects the correct number of the image regions K 0 such that K 0 = arg min 1 K KMAX AIC(K a ) 31) Rissanen addresses the problem from a quite di erent point of view. He reformulates the problem explicitly as an information coding ....

H. Akaike, \A New Look at the Statistical Model Identication," IEEE Transactions on Automatic Control, Vol. 19, No. 6, December 1974.

....a generalisation error or uncertainty from this SVM solution. In the above techniques one of the open questions remaining is how to determine the best regularization parameter (see Wahba, 1990) in (1. 1) One method can use the model selection criteria such as VC theory (Vapnik, 1995) AIC (Akaike, 1974) and NIC (Murata et al. 1994) etc. In (Gao et al. 2000b) a variational approximation method has been proposed to deal with the SVR problem. In this paper we rst brie y review the general likelihood in regression learning problems and derive in more depth the variational Bayesian learning from a ....

Akaike, H. (1974). A new look at the statistical model identication. IEEE Transactions on Automatic Control 19 (6), 716-723.

....1992; Ryd#n, 1995) the Lagrange multiplier principle (Hamilton, 1996) Bayesian inference (Robert, Ryd#n, Titterington, 2000) and covariance function of a HMM (Zhang Stine, 2000) have been suggested for estimating the order of the Markov process. Penalized likelihood criteria, such as the Akaike (1974) information criterion (AIC) and the Bayesian information criterion (BIC) of Schwarz (1978) seem to be the easiest ones to use in practice after the computation of the likelihood function. We apply the AIC in determining both the number of states and the observation densities of the model with ....

Akaike, H. (1974). A new look at the statistical model identication. IEEE Transactions on Automatic Control, AC19, 716723.

....normalization (z) z 1=2 , as discussed in Section 2. To evaluate the ecacy of TRI in this problem we compared its performance to a number of standard model selection strategies, including: structural risk minimization, SRM [CMV97, Vap96] RIC [FG94] SMS [Shi81] GCV [CW79] BIC [Sch78] AIC [Aka74], CP [Mal73] and FPE [Aka70] We also compared it to 10 fold cross validation, CVT (a standard hold out method [Efr79, WK91, Koh95] We conducted a simple series of experiments by xing a domain distribution P X on X = IR and then xing various target functions f : IR IR. 8 (The speci c ....

....of our approach, we repeated the experiments of Section 3.2. The rst class of methods we compared against were the same model selection methods considered before: 10 fold cross validation CVT, structural risk minimization SRM [CMV97] RIC [FG94] SMS [Shi81] GCV [CW79] BIC [Sch78] AIC [Aka74], CP [Mal73] FPE [Aka70] and the metric based model selection strategy, ADJ, introduced in Section 3.3. However, since none of the statistical methods, RIC, SMS, GCV, BIC, AIC, CP, FPE, performed competitively in our experiments, we report results only for GCV which performed the best among ....

H. Akaike. A new look at the statistical model identication. IEEE Transactions on Automatic Control, 19:716-723, 1974.

.... and reversible jump MCMC methods (M uller and Rios Insua, 1998 (for regression) Andrieu et al. 1999 (for radial basis networks) More established methods (see Bishop (1995) include cross validation (Stone, 1974) and penalized likelihood methods using the Akaike Information Criterion (AIC) (Akaike, 1974), the Bayesian Information Criterion (BIC) Schwarz, 1978) or the Network Information Criterion (NIC) Murata et al. 1994) Cross validation has also been applied to variable selection, but not during a selection of model size as well. A Bayesian approach to model selection is Automatic ....

....Neural networks present the additional challenge of simultaneously choosing both the variables and the size of the network. There are many competing ideas on how to choose the best model. Some are based on maximizing the likelihood subject to some penalty, such as the Akaike Information Criterion (Akaike, 1974) and the Bayesian Information Criterion (Schwarz, 1978) Other methods attempt to use part of the data to check the model in some way, such as cross validation (Stone, 1974) The methodology of this paper originates from the Bayesian approach of choosing the model with highest posterior ....

Akaike, H. (1974). \A New Look at Statistical Model Identication." IEEE Transactions on Automatic Control , AU{19, 716-722.

....Thus one needs to balance the desire to increase the t to the data with the need to limit over tting and to increase predictive performance. Methods for model selection include cross validation (Stone 1974) as well as the penalized likelihood based methods such as the Akaike Information Criterion (Akaike 1974) and the Bayesian Information Criterion (BIC) Schwarz 1978) In Bayesian inference, one can simply pick the model with highest posterior probability. The Bayesian approach also lends itself very nicely to prediction, in the guise of model averaging. A good review article on model averaging is ....

Akaike, H. (1974). \A New Look at Statistical Model Identication." IEEE Transactions on Automatic Control , AU{19, 716-722.

....show a method which estimates the relationship between the learner s behaviors and the other agents through interactions (observation and action) using the method of system identi cation. In order to construct the local predictive model of other agents, we apply Akaike s Information Criterion(AIC) [14] to the result of Canonical Variate Analysis(CVA) 15] which is widely used in the eld of system identi cation. The local predictive model is constructed based on the observation and action of the learner (observer) We apply the proposed method to a simple soccer like game in a physical ....

H. Akaike. A new look on the statistical model identication. IEEE Trans. AC-19, pages 716-723, 1974.

....large mean value space in order to model every fMRI time series, the dimensionality of the model can be greatly reduced once a speci c series is considered. In order to do so, we sequentially excluded column vectors of the design matrix by the principle of minimizing Akaike s information criteria (Akaike 1974). For a general model P = fP ; 2 g where R d the criteria is given by AIC(P) 2 log L( 2d; where L( is the maximized likelihood function. By selecting the minimum AIC model we obtain a model that ts data well, yet are parsimoniously parametrized. The reduction ....

Akaike, H. (1974), `A new look at the statistical model identication', IEEE Trans.

....work well in practice, but the theoretical mechanism for small sample cases is not still clear. For the second approach, the asymptotic approximation is used for estimating the expectation of JG over training input points fxm g M m=1 . This approach includes Akaike s information criterion (AIC) [1] and its various derivatives. Although it is theoretically shown that AIC gives an asymptotic unbiased estimate of the generalization error, it does not work well with small samples. For the third approach, e.g. VC theory [6] a probabilistic upper bound of the generalization error is evaluated. ....

Akaike, H. (1974). A new look at the statistical model identication. IEEE Transactions on Automatic Control, AC-19(6), 716-723.

....is the dimension number of parameters needed to specify the model of the Bayesian network with structure given by S. It is computed as dim(S) Q n i=1 q i (r i 1) This penalisation function f(N) is a non negative one. Some examples for f(N) are the Akaike s Information Criterion (AIC) [1] where f(N) 1 , and the Je reys Schwarz criterion, sometimes called the Bayesian Information Criterion (BIC) 62] where f(N) 1 2 log N . Following the latter criterion, the corresponding BIC score BIC(S; D) for a Bayesian network structure S constructed from a database D and containing ....

H. Akaike, `New look at the statistical model identication', IEEE Transactions on Automatic Control, 19(6), 716-723, (1974).

....context, a penalty term is added to the likelihood function so as to limit the number of neurons; thereby avoiding over tting. Classical examples of penalty terms include the well known Akaike information criterion (AIC) Bayesian information criterion (BIC) and minimum description length (MDL) (Akaike 1974, Schwarz 1985, Rissanen 1987) Penalised likelihood has also been used extensively to impose smoothing constraints either via weight decay priors (Hinton 1987, Mackay 1992) or functional regularisers that penalise for high frequency signal components (Girosi, Jones and Poggio 1995) In the ....

....demanding. The choice of one method over the other should ultimately depend on the modelling constraints and speci cations. 5 Reversible Jump Simulated Annealing In this section, we show that traditional model selection criteria within a penalised likelihood framework, such as AIC, BIC and MDL (Akaike 1974, Schwarz 1985, Rissanen 1987) can be shown to correspond to particular hyper parameter choices in our hierarchical Bayesian formulation. As pointed out by one of the reviewers, AIC is not restricted to the situation 22 where the true model belongs to the set of model candidates. That is, we ....

[Article contains additional citation context not shown here]

Akaike, H. (1974). A new look at statistical model identication, IEEE Transactions on Automatic Control AC-19: 716-723.

....with the simplest sequence of nite dimensional spaces which still has good approximation properties for F , since for us the estimation of is of primary interest, and f is a nuisance parameter. However, other choices are possible: spaces generated by splines based on an irregular partition of [0,1], wavelets or trigonometric polynomials; see Birg e and Massart (1996) for a detailed discussion. 7 4. Risk bounds for the penalized least squares estimator In this section we give risk bounds for the estimators of ( f) satisfying (3.9) That is, we consider k k 2 and k k 2 n , ....

....= 1. 3. There exists b 0 such that E exp(jW i j=b) 4, for all i 2 f1; ng. 4. There exists c 0 such that ks vk1 c for all v 2 Sm and for all m 2 Mn . 5. There exists h 0 ; h 1 0 such that h 0 f T h 1 , where f T is the density of T with respect to , Lebesgue measure on [0, 1]. 6. l 0 V ar(X jT )l is bounded away from zero and in nity for any l 2 R q . 8 De ne A 8(b c) B 2c. For any 0 let ( A [1 (1 B 2A ) 1=2 ] and 9 2 20 [32A 2 6 AB] 1 : Let C 1 9:8 P q 0 2f1; 2 q g (e ( 8= q 0 P 1 k=1 (e ( 8= ....

[Article contains additional citation context not shown here]

Akaike, H. A new look at the statistical model identication. IEEE Trans. Automat. Control, 19:716-723, 1974.

.... As mentioned, there are many variants of this general approach, including the minimum description length principle [Ris86] Bayesian maximum a posteriori selection, structural risk minimization [Vap82, Vap96, Vap98] generalized cross validation [CW79] and a host of other statistical criteria [Aka70, Mal73, Aka74, Sch78, Shi81, FG94, Mal95]. These strategies di er in the speci c complexity values they assign and the particular tradeo function 8 . h 0 h 1 h 2 h 3 h 4 Figure 4: ....

....normalization (z) z 1=2 , as discussed in Section 2. To evaluate the ecacy of TRI in this problem we compared its performance to a number of standard model selection strategies, including: structural risk minimization, SRM [Vap96, CMV96] RIC [FG94] SMS [Shi81] GCV [CW79] BIC [Sch78] AIC [Aka74], CP [Mal73] and FPE [Aka70] We also compared it to 10 fold cross validation, CVT (a standard hold out method [Efr79, WK91, Koh95] We conducted a simple series of experiments by xing a domain distribution P X on X = IR and then xing various target functions f : IR IR. To generate training ....

[Article contains additional citation context not shown here]

H. Akaike. A new look at the statistical model identication. IEEE Transactions on Automatic Control, 19:716-723, 1974.

....the full posterior distribution in an ecient way. 1 INTRODUCTION In this paper, we show that traditional model selection criteria within a penalized likelihood framework, such as Akaike s information criterion (AIC) minimum description length (MDL) and the Bayesian information criterion (BIC) (Akaike 1974, Schwarz 1985, Rissanen 1987) can be shown to correspond to particular hyperparameter choices in a Bayesian formulation. That is, it is possible to calibrate the prior choices so that the problem of model selection within the penalized likelihood context can be mapped exactly to a problem of ....

....M s = arg min Mk :k2f0; kmaxg log(p(yjk; b ; x) P (1) where b = b 1:m ; b 1:k ; b 2 k ) is the ML estimate of for model M k . P is a penalty term that depends on the model order. Examples of ML penalties include the well known AIC, BIC and MDL information criteria (Akaike 1974, Schwarz 1985, Rissanen 1987) The expressions for these in the case of Gaussian observation noise are: P AIC = and P BIC = P MDL = 2 log(N) where denotes the number of model parameters (k(c 1) c(1 d) in the case of an RBF network) These criteria are motivated by di erent ....

Akaike, H. (1974). A new look at statistical model identication, IEEE Transactions on Automatic Control AC-19: 716-723.

....model order would lead to an undesired smoothing of the data, whereas over tting would be the result of using too high a value. AR model order determination falls within the larger class of model selection problems. In signal processing classical methods to solve these problems include e.g. AIC [1] and MDL [17] which rely on information theoretic criteria, and BIC [21] which is approximately an asymptotic Bayes factor. These methods, however, often fail when the data sets are small, and for AR processes AIC is not asymptotically consistent. They are also not easily modi ed to accommodate ....

H. Akaike. A new look at the statistical model identication. IEEE Transactions on Automatic Control, AC-19:716-723, 1974.

.... An obvious choice in this class is to use the minimum description length (MDL) criterion [10] 138] but several other model selection criteria have been proposed: Schwarz s Bayesian inference criterion (BIC) the minimum message length (MML) criterion, and Akaike s information criterion 77 (AIC) [2], 148] 167] Resampling based schemes and cross validation type approaches have also been suggested; these techniques are (computationally) much closer to stochastic algorithms than to the methods in the previous paragraph. Stochastic approaches generally involve Markov chain Monte Carlo ....

....the precious data for training which is especially undesirable when the training data set is small. To avoid this problem, a number of model selection schemes [71] have been proposed, including Bayesian methods 80 [14] minimum description length (MDL) 138] Akaike information criterion (AIC) [2] and marginalized likelihood [101, 159] Various other regularization schemes which incorporate prior knowledge about model structure and parameters have also been proposed. Structural risk minimization based on the notion of VC dimension has also been used for model selection where the best model ....

H. Akaike, \A new look at statistical model identication," IEEE Trans. on Automatic Control, vol. AC-19, pp. 716-723, 1974.

....and attempt to select the best one using various forms of complexity penalization and hold out testing. The methods we compared were: 10 fold cross validation CVT (Efron, 1979) structural risk minimization SRM (Vapnik, 1996; Cherkassky, Mulier, Vapnik, 1996) GCV (Craven Wahba, 1979) AIC (Akaike, 1974), BIC (Schwarz, 1978) FPE (Shibata, 1981) CP (Mallows, 1973) RIC (Foster George, 1994) and the metric based model selection strategy ADJ introduced in (Schuurmans, 1997) However, none of the statistical methods GCV, BIC, FPE, CP and RIC performed competetively in our experiments, so we ....

Akaike, H. (1974). A new look at the statistical model identication. IEEE Trans. Automat. Control, 19, 716-723.

....This method nds the relationships between the behaviors of the learner and the other agents through interactions (observations and actions) using the method of system identi cation. In order to construct the local predictive model of other agents, we apply Akaike s Information Criterion (AIC) [2] to the results of Canonical Variate Analysis (CVA) 13] which is widely used in the eld of system identi cation. The local predictive model is based on the observation and action of the learner (observer) We apply the proposed method to a simple soccer like game. The task of the robot is to ....

H. Akaike. A new look on the statistical model identication. IEEE Trans. AC-19, pp. 716-723, 1974.

....a method which nds the relationships between the behaviors of the learner and the other agents through interactions (observation and action) using the method of system identi cation. In order to construct the local predictive model of other agents, we apply Akaike s Information Criterion(AIC) [1] to the results of Canonical Variate Analysis(CVA) 10] which is widely used in the eld of system identi cation. The local predictive model is based on the observation and action of the learner (observer) We apply the proposed method to a simple soccer like game. The task of the robot is to ....

H. Akaike. A new look on the statistical model identi cation. IEEE Trans. AC-19, pp. 716-723, 1974.

....estimator [14] is used which estimates the relations between the learner s behaviors and the other agents through interactions (observation and action) using the method of system identi cation. In order to construct the local predictive model of other agents, Akaike s Information Criterion(AIC) [1] is applied to the result of Canonical Variate Analysis(CVA) 7] which is widely used in the eld of system identi cation. The local predictive model is based on the observation and action of the learner (observer) for the details of the estimator, see the Appendix A) Figure 1 shows an overview ....

H. Akaike. A New Look on the Statistical Model Identication. IEEE Trans. AC-19, pp. 716-723, 1974.

....and the other agents through interactions using the method of system identi cation. Here, we put our em phasis on the problem B, and we assume that the other agent does not change the strategy. In order to identify the model of each other agent, we apply Akaike s Information Criterion(AIC) (Akaike 1974) to the result of Canonical Variate Analysis(CVA) Larimore 1990) which is widely used in the eld of system identi cation. We apply the proposed method to a simple soccerlike game including two active agents. The task of the agent is to discriminate the strategy of the other agents. Here, the ....

Akaike, H. 1974. A new look on the statistical model identication. IEEE Trans. AC-19 716-723.

....are given, we can estimate the parameters by the EM algorithm. We presume that the structure of each component densities are given. Then, the remaining problem is to determine the number of the categories. To estimate the number of the categories, we can use some information criteria, such as AIC[1]. But it is hard to believe that a baby is calculating AIC when it learns phonation. There must be some kind of criteria, but it might not exactly the same as AIC. The authors assume that the number of the categories are determined by communicating with the source. Starting from the mixture model ....

H. Akaike, \A new look at the statistical model identication," IEEE Trans. Automat. Contr., vol. AC-19, pp. 716-723, Dec. 1974.

....a method which estimates the relationship between the learner s behaviors and the other agents through interactions (observation and action) using the method of system identi cation. In order to construct the local pre dictive model of other agents, we apply Akaike s Information Criterion(AIC) [13] to the result of Canonical Variate Analysis(CVA) 14] which is widely used in the eld of system identi cation. The local predictive model is constructed based on the observation and action of the learner (observer) We apply the proposed method to a simple soccer like game in a physical ....

H. Akaike. A new look on the statistical model identication. IEEE Trans. AC-19, pages 716-723, 1974.

....a method which nds the relationships between the behaviors of the learner and the other agents through interactions (observation and action) using the method of system identi cation. In order to construct the local predictive model of other agents, we apply Akaike s Information Criterion(AIC) [1] to the results of Canonical Variate Analysis(CVA) 10] which is widely used in the field of system identification. The local predictive model is based on the observation and action of the learner (observer) We apply the proposed method to a simple soccer like game. The task of the robot is to ....

H. Akaike. A new look on the statistical model identication. IEEE Trans. AC-19, pp. 716-723, 1974.

....respectively, and e t is the noise and Efe t g = 0; Efe t e g = 2 e t ; where 2 e is variance of fe t g. The problem is how to determine the dimension p and q of ARMA model. Here, we apply AIC (Akaike s Information Criterion) which is widely used in the eld of time series analysis [1] to determine p and q. For some p and q, a i and b i are calculated by prediction error method(PEM) and then AIC is computed by AIC = N ln 2 e 2(p q) 11) where N is the number of data sets, and the factors unrelated to comparison are ignored. The estimation of oe 2 e , and oe ....

H. Akaike. A New Look on the Statistical Model Identi cation. IEEE Trans. AC-19, pp. 716-723, 1974.

....a method which estimates the relationship between the learner s behaviors and the other agents through interactions (observation and action) using the method of system identi cation. In order to construct the local predictive model of other agents, we apply Akaike s Information Criterion(AIC) (Akaike, 1974) to the result of Canonical Variate Analysis(CVA) Larimore, 1990) which is widely used in the eld of system identi cation. The local predictive model is constructed based on the observation and action of the learner (observer) We apply the proposed method to a simple soccer like game in a ....

Akaike, H. (1974). A new look on the statistical model identication. IEEE Trans. AC-19, pages 716-723.

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Akaike, H. (1974). A new look at the statistical model identi cation. IEEE Trans. Automat. Control 19, 716-723.

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Akaike, H. (1985). A new look at the statistical model identi cation. IEEE Transactions on Automatic Control, 19:716-723.

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Akaike, H. #1974#. A New Look at the Statistical Model Identi#cation. IEEE Transactions on Automatic Control, AC-19, 716-723.

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H. Akaike (1978). New Look at the Statistical Model Identication. IEEE Transactions on Automatic Control 19, 6, 716-723.

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H. Akaike, New Look at the Statistical Model Identication, IEEE Transactions on Automatic Control 19, 6, (1974), 716-723.

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