Adjustment of Sampling Locations in Rail-Geometry Datasets: Using Dynamic Programming and Nonlinear Filtering

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1 Systems and Computers in Japan, Vol. 37, No. 1, 2006 Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J87-D-II, No. 6, June 2004, pp Adjustment of Samplin Locations in Rail-Geometry Datasets: Usin Dynamic Prorammin and Nonlinear Filterin Masako Kamiyama 1,2 and Tomoyuki Hiuchi 3 1 Track Technoloy Division, Railway Technical Research Institute, Kokubunji, Japan 2 Department of Statistical Science, The Graduate University for Advanced Studies (Sokendai), Tokyo, Japan 3 Department of Statistical Modelin, The Institute of Statistical Mathematics, Research Oranization of Information and Systems, Tokyo, Japan SUMMARY A track inspection car, which measures the shape of railway tracks (hereafter, rail eometry) while it is runnin on rails, discretizes the measurement results at nearly fixed spatial intervals. However, the distance between the discretized locations (spatial samplin intervals) may shorten or lenthen locally due to slippin or slidin of the car wheel, and this prevents the samplin locations from alinin with those of a dataset obtained with another measurin run. The authors developed an alorithm for approximately alinin the samplin locations of the measurement datasets obtained with different runs. First, they considered this problem as the selection of the series of data correspondin to each supervised data from a trainin dataset, which was constructed by interpolation in order to minimize the evaluation function of a number sequence representin data points. Next, they used the maximum likelihood method to identify the unknown parameters contained in the evaluation function. This problem uses two features of the evaluation function. The first is that the evaluation function is minimized by dynamic prorammin, and the obtained optimum sequence is equivalent to a maximum a posteriori (MAP) estimate in the Bayesian framework. The second is that by convertin the evaluation function to a eneral state space representation, the lo likelihood of the model that includes the parameters is obtained by a nonlinear filterin method. Also, to simplify the search for the identification, they devised a parameter search procedure for the parameters in the autoreressive (AR) model Wiley Periodicals, Inc. Syst Comp Jpn, 37(1): 61 70, 2006; Published online in Wiley InterScience ( wiley.com). DOI /scj Key words: dynamic prorammin; state space representation; nonlinear filter; maximum likelihood method. 1. Introduction The special train called a track inspection car measures the shape of railway tracks (rail eometry) reularly since this eometry varies slihtly under the load of daily passin trains. The measurement results are discretized at equidistant intervals and stored for processin by computers. To obtain temporal variations of the rail eometry, the measurement results ideally should always be discretized at the same locations on the track. However, since the pulse sinals for specifyin the samplin locations are linked with the rotation of the wheel of the track inspection car as shown in Fi. 1, the pulse eneration locations cannot be reproduced. Therefore, the samplin locations on the track Wiley Periodicals, Inc.

2 Fi. 1. Scheme for spatial discretization of measured railway track eometry accordin to wheel rotation pulses. vary for each measurin run. To alin two sets of samplin locations obtained with difference runs, we must estimate the location aps from the discretized datasets, but this is difficult. If the spatial samplin intervals between all of the data were uniform, these aps between two sets of the samplin locations could be easily estimated by calculatin the cross correlation coefficient. However, because the samplin intervals locally shorten or lenthen as shown in Fi. 2 due to the slippin or slidin of the car wheel, this method cannot be applied. (In Fi. 2, the measurement results are discretized once per rotation of the wheel to simplify the explanation.) Fiure 3 shows an example of the measurement results. The datasets (A) and (B) for the upper two raphs show the values of the track aue (spacin between the left and riht rails) for the same railway section measured 1 day Fi. 2. Scheme for variations in spatial samplin intervals accompanyin wheel rotation states. Fi. 3. Example of datasets obtained from two measurements for the same railway section and results of adjustin samplin locations. and 8 days later (0 means that the aue is equal to the reference value). Both datasets were discretized at equal intervals (approximately 30 cm; 8 measurements per wheel rotation). In addition, for both (A) and (B), the location where the first data was obtained is somewhere within one samplin interval in the car movement direction of a certain device that is fixed at a specific location on the track. Similarly, the data for the 3201st point of (A) and 3197th point of (B) were also measured within one samplin interval in the car movement direction from another location detection device. It is known empirically that the aue hardly varies over an 8-day period or so, and the waveforms for (A) and (B), which are shown in the first and second raphs from the top of Fi. 3, are actually similar. Also, the datasets close to the 1st data in (A) and (B) are obtained at similar locations. This is apparent from the difference between the two waveforms, which is shown in the third raph from the top of Fi. 3. Similarly, the fourth raph from the top of Fi. 3 shows that the datasets close to the end data in (A) and (B) are also obtained at similar locations. In other words, despite the fact that (A) and (B) are measurement values of the aue for the same railway section, the number of data points constitutin (B) is smaller than the number constitutin (A) by 4 points (correspondin to approximately 1 m). This is because the samplin interval shortened or lenthened locally due to slippin or slidin of the car wheel durin one or the other measurin runs. 62

3 We developed an alorithm that uses dynamic prorammin (DP [1]) and nonlinear filterin for a eneral state space representation to approximately alin the samplin locations for datasets of this kind. The fifth raph in Fi. 3 shows the results when the samplin locations for dataset (B) were alined with those of dataset (A) by usin the proposed alorithm. This paper describes the location adjustment alorithm in Sections 2 and 3, presents a discussion in Section 4, and presents conclusions in Section Adjustment of Samplin Locations by Usin Dynamic Prorammin and Nonlinear Filterin An overview of the adjustment alorithm that we developed is presented below: (1) Model a mechanism to be empirically considered to yield the nonuniform samplin. (2) Formulate this model in a nonlinear optimization problem that can be solved by dynamic prorammin. However, since this form contains unknown parameters, the solution cannot be uniquely obtained. (3) Represent the above nonlinear form with a eneral state space representation and identify the included unknown parameters throuh a nonlinear filterin alorithm based on the maximum likelihood method. (4) Substitute the estimated parameters to solve the optimization problem of step (2). (End) These steps are explained in detail below Modelin the problem We decided to consider this location adjustment problem as a problem for selectin the most suitable data points correspondin to individual supervised data {Y 1, Y 2,..., Y T 1, Y T } (where T is the number of the supervised data) from the trainin dataset that was interpolated so that the number of points was a multiple of α as shown in Fi. 4. Since the aue dataset is smoothed by an analo low-pass filter before the spatial discretization, the oriinal trainin dataset is discretized so that the oriinal spatial frequency component does not chane. The spatial frequency characteristic of this filter varies with time, since this characteristic chanes with the speed of the inspection car. The interpolated dataset hereafter is called the trainin dataset {X 1, X 2,..., X N 1, X N }. Note that N ( αt) is the number of trainin data after interpolation. To distinuish between the dataset as a set and the individual data, we hereafter denote {Y 1, Y 2,..., Y T 1, Y T } collectively as Y 1:T. The other variables are denoted in a similar manner. Fi. 4. Scheme for the basic idea for adjustin the discretized location aps. Let n t be a data point index of the trainin set selected to match the t-th supervised data Y t ; the value of the correspondin data is X nt. This problem becomes a problem of findin the optimal number sequence (this is denoted as n 1:T ) from amon the feasible number sequences n 1:T. Also, let Y t X nt be denoted by e t (e t ~ N(0, τ 2 )). Next, we discuss the characteristics of n 1:T. Althouh the accurate distribution of the samplin interval is unknown, the runnin distance due to slippin or slidin is empirically determined to be approximately 0.1% of the total distance or less. Therefore, we assume that n def t = n t n t 1 of n 1:T, which corresponds to the samplin interval, is α for almost every t. Also, for the rane of values of n t, this article employs 1 n t 2α 1. This is based on the empirical knowlede that was obtained from conventional runnin experiments. Add to this assumption, we decide that the probability distribution of n t would be constant for all values other than α. We also assume that the second-order difference 2 n def t = n t 2n t 1 + n t 2 of n 1:T is 0 for almost every t. This is because 2 n t = 0 is naturally zero when the wheel rotates reularly; n t = n t 1 = α. Similarly, even when the wheel slips or slides, the wheel in the course of slippin or slidin seems to continue rotatin at the same rate ( 2 n t = 0), or to recover normal rotation ( n t = α) Adjustin the location by usin dynamic prorammin From the discussion above, we model the characteristics of the number sequence n 1:T indicatin the optimal sequence of the data points as follows: n t = def n t n t 1 is α for almost every t. 2 n t = def n t 2n t 1 + n t 2 is 0 for almost every t. 63

4 e t ~ N(0, τ 2 ) (where e t = Y t X nt ). At this time, n 1:T is obtained by optimizin the followin evaluation function F(n t ): (1) Calculate (x t ) = min[(x t 1 ) + f (x t, x t 1 ) x t 1 S t 1 ] for each element x t S t (where min[ ] represents the minimum value of the elements within [ ]). (2) Increment t by 1 and return to step (1). where S def t = [S t S t 1 ] T. When (x t ) is rearded as a state that transitions as t increases, the value of (x t ) depends only on (x t 1 ), x t, and x t 1 alone. (End) Since n 1:T varies with the values of parameters µ 1 and µ 2, we should carefully determine the parameter values. where (1) 2.3. Dynamic prorammin and MAP estimation in the Bayesian framework To determine the values of µ 1 and µ 2, n 1:T, which was obtained in the previous section, should be interpreted statistically. Assume that e t = Y t X nt is a normal random variable with a zero mean and a standard deviation τ. By multiplyin (1) by 1/(2τ 2 ) and exponentiatin it, we obtain (2) (3) (4) The factors µ 1 0 and µ 2 0 are penalties that are added to the evaluation function when n t α and 2 n t 0 respectively. In addition, as mentioned above, the data points at both ends of the supervised and trainin datasets are always measured within a fixed distance from specific locations on thetrack.therefore,theboundaryconditionsforn 1:T are as follows: Assume that both the penalties n 1 and n T have uniform distributions. For obtainin the n 1:T that satisfies the conditions shown in (1) to (4), dynamic prorammin (DP) can be applied [1], since this optimization problem observes the principle of optimality. Substitute x def t = [n t n t 1 ] T (where T represents transpose) in Eq. (1) and let then (x t ) def t = Σ j=1 F(n j ) can be replaced by the followin recursive formulation: The former term of the riht side (first { }) can be interpreted as a certain multiple of the conditional density p(y 1:T n 1:T ) of the supervised dataset Y 1:T assumin normality when all n 1:T are iven. Similarly, the latter term (second { }) can also be interpreted as a certain multiple of the prior distribution p(n 1:T ) of n 1:T in the Bayesian framework. Therefore, Eq. (1) can be interpreted as a search for the n 1:T that maximizes the posterior distribution p(n 1:T Y 1:T ) p(y 1:T n 1:T )p(n 1:T ) [this is called the maximum a posteriori (MAP) estimate]. Therefore, µ 1, µ 2, and τ 2 are hyperparameters (in the Bayesian framework) that ive the prior distribution [2]. Optimal solutions that are obtained by dynamic prorammin, not just for this problem, can be interpreted as MAP estimates [3]. Hence, the hyperparameter values can be evaluated usin the lo likelihood LL(µ 1, µ 2, τ 2 ) def = lo p(y 1:T µ 1, µ 2, τ 2 ) of Y 1:T. In other words, the hyperparameters for maximizin LL (denoted by µ ~ 1, µ ~ 2, and τ ~2 ) are interpreted as the optimal hyperparameters within the maximum likelihood method. The method for computin LL is described in the next section. (5) 64

5 2.4. Estimatin hyperparameters by the maximum likelihood method Lo likelihood LL(µ 1, µ 2, τ 2 ) is obtained as the followin sum of conditional probability distributions: Therefore, to obtain the LL, we should obtain p(y t Y 1:t 1, µ 1, µ 2, τ 2 ) for all t {1, 2,..., T}. Next, to obtain p(y t Y 1:t 1, µ 1, µ 2, τ 2 ), we transform the statistical model (5) into a eneralized state space representation [4]. By lettin x t = def [n t n t 1 ] and y t = def Y t, we can obtain the statistical model as follows. [Observation model] where e t ~ N(0, τ 2 ). [System model] where since q(v t ) = 1. Fiure 5 shows sample distributions for q( ). If we assume that e t and v t are white noises, we can define the prediction and filterin operations as follows. The p(y t y 1:t 1 ) values (where t = 1, 2,..., T 1, T) are obtained as the by-product of the computation for p(n 1:T y 1:T ), and the lo likelihood LL is obtained as the sum of their loarithms [4]. Since Y t / y t = 1, p(y t y 1:t 1 ) is equal to p(y t Y 1:t 1 ). [Prediction] where v t ~ q( x t 1, µ 1, µ 2, τ 2 ) and the distribution of q( x t 1, µ 1, µ 2, τ 2 ) is as follows. When n t 1 n t 2 = α, then where since q(v t ) = 1. When n t 1 n t 2 α, then Fi. 5. Typical distribution q( ) for system noise v t when α = 5, µ 1 = 1, µ 2 = 2, and τ 2 = 1. 65

6 [Filterin] p(x 1 y 0:0 ) is a uniform distribution from 1 to (2α 1). Some filterin operations in a eneralized state space representation involve complicated numerical interation. However, since the system vector x t consists of discrete values, while y t are continuous real values in this model, p(y t y 1:t 1 ) can be computed arithmetically without interation. Note that we used a rid search to maximize LL(µ 1, µ 2, τ 2 ). We now substitute (µ ~ 1, µ ~ 2, and τ ~2 ) in Eq. (1) and denote the n 1:T obtained by usin dynamic prorammin to optimize that equation by n ~ 1:T hereafter. Fi. 6. Fixed-interval smoothin distributions p(n t y 1:T ) (closeup view of t 520) (characterizin the differences e ~ 1:T as a Gaussian white noise sequence) Results and discussion Table 1 shows the parameters that were estimated by the alorithm described above when the actual measurement values for (A) shown in Fi. 3 were taken as the supervised dataset and the values for (B) were taken as the trainin dataset (before adjustment). (Note that α = 5 is used in this paper.) Also, n ~ 1:T that was estimated by dynamic prorammin is shown as (Y 1:T X n ~ 1:T ) in the fifth raph and as n ~ t in the sixth raph from the top of Fi. 3. From the sixth raph, it is apparent that if it is assumed that the track inspection car did not slip or slide durin the measurement run for dataset (A), then it is assumed to have slid for a distance correspondin to four points (approximately 1 m) near t = 1900 durin the measurement run for dataset (B). Also, the track inspection car s runnin speed when measurin dataset (B), which is shown in the bottom raph in Fi. 3 (this is calculated back from the number of distance pulses enerated within a unit time and normally cannot be referred), reveals that the train s speed dropped unnaturally in the vicinity of where the train was assumed to have slid and suests that it is hihly likely that the wheel actually slid and the eneration of distance pulses was temporarily reduced. Next, Fi. 6 shows the results when p(x t y 1:T ) is obtained by fixed-interval smoothin as follows: and then the followin is obtained as the marinal smoothin distribution: It is apparent that the marinal smoothin distribution does not convere and the reliability of n ~ 1:T is low. The major cause is that e 1:T is assumed to be a Gaussian white noise sequence even thouh an analo low-pass filter has smoothed the actual measurement data. As shown in Fi. 7, the autocorrelation of e ~ 1:T = def Y t X n ~ t is hih. Therefore, to improve the model, we apply an autoreressive (AR) model to e 1:T in the next section. Table 1. Estimated parameters obtained usin the actual measurement values in Fi. 3 (characterizin the differences e ~ 1:T as a Gaussian white noise sequence) Fi. 7. Autocorrelation coefficients of a difference sequence e ~ 1:T (characterizin e ~ 1:T as a Gaussian white noise sequence). 66

7 3. Remodelin the Difference Sequence 3.1. Introduction of an autoreressive (AR) model Assume that the difference component e t = Y t X nt is colored noise that can be described as follows usin an autoreressive model: Fi. 8. Autocorrelation coefficients of a residual sequence w ~ 1:T (characterizin e ~ 1:T as a first-order autoreressive process). (w t ~ N(0, σ 2 ) and k is the order of the autoreressive model, which is appropriately determined). If we substitute the followin for the T t=1 (Y t X nt) 2 in Eq. (1), and then optimize it usin dynamic prorammin, n 1:T is obtained where x def t = [n t n t 1 n t k ] T. This model can be transformed into a eneralized state space representation by lettin y def t = Y t k i=1 a i Y t i and X nt 0 = 0, and the maximum likelihood of the parameters can be estimated in a similar manner as in the previous section. LL to be maximized to estimate the parameters is LL(µ 1, µ 2, a 1:k, σ 2 ). Hereafter, the estimated parameters are denoted by (µ ~ 1, µ ~ 2, a ~ 1:k, σ ~2 ). Table 2 shows the estimated parameters when the above alorithm was applied to datasets (A) and (B), which were shown in Fi. 3 when k = 1 was assumed. It is apparent that by introducin the autoreressive model, the lo likelihood LL increased sinificantly from 608 to 3294 and that the reliability of the estimates improved. Next, Fi. 8 shows the autocorrelation coefficients of the residual sequence w ~ def t = X n ~ t a ~ 1X n ~ t 1. This is closer to a white noise autocorrelation than in Fi. 7. However, by introducin the autoreressive model, the number of points at which n ~ t α increased from 4 to 42 (fiure is omitted). In other words, althouh introducin the autoreressive model increased the likelihood, an implausible n ~ 1:T based on conventional knowlede was obtained. Also, by introducin the autoreressive model, the time required for the (µ ~ 1, µ ~ 2, a ~ 1, σ ~2 ) search increased. If Table 2. Estimated parameters obtained usin the actual measurement values in Fi. 3 (characterizin the differences e ~ 1:T as a first-order autoreressive process) (G 1, G 2, G a, G σ ) denote the rid counts that were set for (µ 1, µ 2, a 1, σ 2 ), respectively, the required time is a G 1 G 2 G a G σ multiple of the time for calculatin the LL value once. In our environment (Pentium MHz), since it took approximately 100 seconds per LL calculation (when T = 3201), when all rid point counts were set to 5, introducin the autoreressive model increased the calculation time from approximately 3.5 hours to 17.4 hours. However, aue eometry datasets are massive dependin on the track lenth (hundreds of kilometers), and fast location adjustments are desirable Improvement of the estimation alorithm As we have seen in the previous section, it is apparent that the alorithm must be improved so that an n ~ 1:T that conforms to conventional knowlede is obtained and parameters havin hih likelihood can be estimated. Therefore, we used the fact that (a ~ 1:k,σ ~2 ) are autoreressive parameters representin the difference sequence e ~ t to develop the followin alorithm. (0) (1) Set the initial value a 1:k of a 1:k to 0. (2) Maximize the lo likelihood LL(µ 1, µ 2, σ 2 a 1:k = a (0) 1:k ) accordin to a rid search to obtain provisional values of µ ~ 1 and µ ~ (0) 2 (which are described as µ 1 and (0) µ 2, respectively). In addition, use (0) (µ1, µ (0) 2, a (0) 1:k ) to estimate n ~ 1:T accordin to dynamic prorammin. (3) Use the Yule Walker method to estimate a ~ 1:k and σ ~2 from the estimated difference sequence e ~ 1:k (these are (1) described as a 1:k and σ 2(1) ) [5]. (4) Substitute (a (1) 1:k, σ 2(1) ) and maximize the lo likelihood LL aain to obtain (µ (1) 1, µ (1) 2 ), and estimate n ~ 1:T accordin to dynamic prorammin. (5) Compare the newly obtained n ~ 1:T with the one obtained previously and end the calculation if all entries are equal. If any entries differ, return to step (3). (End of the alorithm) 67

8 Table 3. Transitions of parameter estimates usin the actual measurements shown in Fi. 3 (characterizin differences e ~ 1:T as a first-order autoreressive process and applyin the alternate estimate alorithm) Maximum likelihood estimates of µ 1 (0) and µ2 (0) Estimates of σ (1) and a 1 (1) by dynamic prorammin Maximum likelihood estimates of µ 1 (1) and µ 2 (1) Estimates of σ (2) and a 1 (2) by dynamic prorammin µ ~ µ ~ LL a ~ 1 0 (fixed) σ ~ Remark Estimation ended Hereafter, this alorithm is termed the alternate estimate alorithm. Note that the (µ ~ 1, µ ~ 2, a ~ 1:k, σ ~2 ) that are obtained by the alternate estimate alorithm are not maximum likelihood estimates. (µ ~ 1, µ ~ 2) are maximum likelihood estimates when (a ~ 1:k, σ ~2 ) are already known. As a result of applyin the alternate estimate alorithm to the datasets (A) and (B) shown in Fi. 3, the same n ~ 1:T was obtained as before the autoreressive model was introduced. Table 3 shows transitions in estimates for the hyperparameters (µ ~ 1, µ ~ 2, a ~ 1:k, σ ~2 ). Althouh the lo likelihood was reduced to 3247 from the 3294 of Table 2, a hiher level is maintained than the 608 of Table 1. Additionally, the second raph from the bottom of Fi. 3 shows the obtained residual sequence w ~ def t = X n ~ t a ~ 1X n ~ t 1. The autocorrelation coefficients of this residual sequence are nearly the same as those in Fi. 8. The marinal smoothin distribution of this shown in Fi. 9 also converes better than the distribution in Fi. 6. Fi. 9. Fixed-interval smoothin distribution p(n t y 1:T ) (closeup view of t 520, modelin the differences e ~ 1:T as a first-order autoreressive process and applyin the alternate estimate alorithm). Now, we discuss the required time for the rid search. If the alternate estimate alorithm is repeated M times, the sum total of the required times is 100[G 1 G 2 G α + (M 1)G 1 G 2 ]. Since M = 2 for the calculations in Table 3, when all rid point counts were set to 5, the calculation time was reduced to approximately 4.2 hours from 17.4 hours. Note that M chaned from 2 to approximately 5. Therefore, the alternate estimate alorithm seems practical for obtainin n ~ 1:T from the taret data. 4. Discussion To verify the alternate estimate alorithm, we performed the followin simulation. (1) Establish fictional aue eometry and slippin or slidin to create data sequences for the supervised and trainin datasets (before the interpolation). Let n 1:T denote true values of the data points. (2) Independently create two white noise sequences as realization from the same normal distribution to simulate measurement noise and add them respectively to the two data sequences that were created in step (1). (3) Create Z 1:T, the sum of the supervised aue eometry and the noise. Then sequentially calculate Y 1 = Z 1, Y t = Z t + a * Y t 1 (2 t T) to create the supervised dataset Y 1:T. Also create the trainin dataset in a similar manner. (These sequences are shown in the top two raphs in Fi. 10.) Note that in this paper, a * = 0.8. (4) Estimate parameters (µ ~ 1, µ ~ 2, a ~ 1, σ ~2 ) and use those parameters to estimate n ~ 1:T. (End) The two results, obtained by the maximum likelihood estimation and by alternate estimate alorithm (see Table 4), are equal, and, therefore, the n ~ 1:T were also both the same. In addition, a ~ 1 and σ ~ 2 are both close to the true values. 68

9 We believe that this is because e 1:T has been modeled correctly. However, as Fi. 10 shows, n ~ 1:T n 1:T. The results of the parameter estimations usin actual measurement data differed for maximum likelihood estimation and the alternate estimate alorithm as shown in Tables 2 and 3. We believe that this is because e 1:T cannot be described accurately by a first-order autoreressive model for the followin reasons. That is, the main causes are that the measurement data contains irreular noise (for example, in the neihborhood of t = 2300) as is apparent from Fi. 3 and the filter that is actually applied is an analo low-pass filter that varies with time. 5. Conclusions Fi. 10. The result of adjustin samplin locations of the simulated datasets. The samplin locations of two measurement datasets of a track eometry obtained with different runs of a track inspection car can be approximately alined by dynamic prorammin if we model the wheel rotation and location detection pulse. Also, usin the fact that the optimal solution accordin to dynamic prorammin is a MAP estimate in the Bayesian framework enables unknown parameters included in the model to be estimated by usin the maximum likelihood method. When a low-pass filter is employed to represent the measurement datasets, its effect can be reduced if an autoreressive (AR) model is applied to the difference sequence. REFERENCES Table 4. Parameter estimation results for the simulation in Fi. 10 Maximum likelihood estimation Alternate estimate alorithm True value µ ~ µ ~ a ~ σ ~ LL Ibaraki T. Dou-teki Keikaku-hou, RisanSaitekikahou to Aruorizumu. Iwanami Shoten; p (in Japanese) 2. Akaike H. Likelihood and the Bayes procedure. In Bernardo JM, DeGroot MH, Lindley DV, Smith AFM (editors). Bayesian Statistics. University Press; p Godsill S, Doucet A, West M. Maximum a posteriori sequence estimation usin Monte Carlo particle filters. Ann Inst Statist Math 2001;53: Kitaawa G. Jikeiretsu-moderu no Yuudo-keisan to Parameta-suitei, FORTRAN77 Jikeiretsu-kaiseki Purouraminu. Iwanami Shoten; p (in Japanese) 5. Ishiuro M. Jiko-kaiki moderu no Atehame, Jikeiretsu-kaiseki no Houhou. Asakura Shoten; p (in Japanese) 69

10 AUTHORS (from left to riht) Masako Kamiyama completed her master s course in 1992 with a specialty in Resource Enineerin at Waseda University. Currently, she is employed in the Track Technoloy Division of the Railway Technical Research Institute, and she completed her doctoral course in 2004 in the Department of Statistical Science at the Graduate University for Advanced Studies (Sokendai). She is a member of the Japan Society for Industrial and Applied Mathematics and Japan Statistical Society. Tomoyuki Hiuchi (member) completed his doctoral course in the Department of Geophysics, in 1989 at the University of Tokyo. He holds a Ph.D. deree in Science. Currently, he is a professor in the Department of Statistical Modelin and Vice Director-General at the Institute of Statistical Mathematics, Research Oranization of Information and Systems. His specialty is statistical analysis, in particular, Bayesian modelin for knowlede discovery. He is a member of the Japan Statistical Society, the Society of Geomanetism and Earth, Planetary and Space Sciences, the American Geophysical Union, and the American Statistical Association. 70