An Anomaly Detection Method for Spacecraft using Relevance Vector Learning

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1 An Anomaly Detection Method for Spacecraft using Relevance Vector Learning Ryohei Fujimaki 1, Takehisa Yairi 2, and Kazuo Machida 2 1 The Univ. of Tokyo, Aero. and Astronautics 2 The Univ. of Tokyo, RCAST Abstract. This paper proposes a novel anomaly detection system for spacecrafts based on data mining techniques. It constructs a nonlinear probabilistic model w.r.t. behavior of a spacecraft by applying the relevance vector regression and autoregression to massive telemetry data, and then monitors the on-line telemetry data using the model and detects anomalies. A major advantage over conventional anomaly detection methods is that this approach requires little a priori knowledge on the system. 1 Introduction Anomaly detection is a key issue in the development of recent advanced spacecraft. The space environment is very harsh for spacecraft due to a variety of factors such as direct radiation, great temperature difference, and so on. In addition, the space is so distant from the earth that it is extremely difficult to directly inspect and repair a damaged component. Therefore, early detection of anomalous symptoms is important to avoid disastrous situations such as loss of control. Although several anomaly detection/diagnosis methods using modern reasoning techniques have been developed, they have difficulties in acquiring accurate and complete models and knowledge of the spacecraft systems and in monitoring the system behavior exhaustively and efficiently. In this paper, we propose a new anomaly detection method for spacecraft based on data mining technique, autoregressive model and the relevance vector regression, and constructs a predictive model for each time series in the telemetry data. Then, it monitors online telemetry data and detect anomalies by checking the probability density of the observation. 2 Conventional Approaches to Anomaly Detection for Spacecraft Limit-checking checks whether the value is within the pre-defined upper and lower limits. Though the limit-checking can be applied to any types of spacecraft, it lacks flexibility and expressiveness and suffers from false alarm problem. In the model-based fault detection and diagnosis method, system models are utilized to simulate the spacecraft behavior and examine the validity of the

2 actual telemetry data. This approach would provide an ideal performance if an accurate and complete model and infinite computational power were available. In practice, however, both of them are not available. Expert systems also have been developed for this purpose. The knowledge is generally represented in the form of if-then production rules. Though the expert systems are powerful and flexible, it has a difficulty in preparing a set of accurate and complete knowledge on the spacecraft. In summary, the above methods have a common problem that they are too dependent on the knowledge of human experts. A reasonable approach to this problem is the application of data mining and machine learning techniques to the telemetry data. Actually, some researchers have developed anomaly detection methods for spacecraft using regression tree learning[7], temporal pattern clustering[8], association rule mining[9]. 3 Proposed Anomaly Detection System 3.1 Autoregressive Model Autoregressive (AR) model is the most basic data mining technique for timeseries data[3][10][11]. For the purpose of applying AR model to anomaly detection problems for spacecraft, we define AR model as τ j ARk = Θj x k (1) where j = {1,, s} represents the jth series of telemetry data, Θ j = (Θ j 0, Θj 1,k 1,..., Θ j 1,k p,..., Θj s,k 1,..., Θj s,k p ) is a row vector of AR coefficients, x k = (1, τk 1 1,..., τ k p 1,..., τ k 1 s,..., τ k p s ) is the data vector of all p previous time series, and the notation AR represent that the target value τ is based on AR model. This modified AR model implies that the value of a target time-series depends not only on the past values of itself but also on those of other series. The capability of modeling the relationships among some series is a great advantage. Fig.1 is the concept of this model. We made use of the framework of the relevance vector learning to extend this model to nonlinear and probabilistic. Fig. 1. Concept of our AR model.

3 3.2 Relevance Vector Regression The relevance vector regression (RVR) originally proposed by Tipping[5] is a state of the art kernel-based nonlinear regression learning method[1][2][4][6]. We write a sample of N training pairs as {x n, t n } N n=1 for the jth telemetry series, corresponding x k and τ j k. Hereinafter, we deal with the jth series of telemetry data and omit the notation j for simplification. The RVR model assumes that the targets are samples from a distribution model with additive independent zero-mean Gaussian noise, with variance σ 2, t = y + ɛ = Φw + ɛ (2) where t = (t 1,..., t N ), y, Φ = (φ 1,..., φ M ), w = (w 0,..., w M ) T, φ m (x) = K(x, x m ) and ɛ = (ɛ 1,..., ɛ n ) T represent the target vector, the vector of the predicted value, the N M design matrix, the weights, the kernel function, and an error vector, respectively. To achieve the sparsity, M independent hyperparameters, α = (α 1,..., α M ) T over w are indroduced. Then, after maximizing logarithm likelihood w.r.t. α and σ 2, we obtain a conditional distribution model for a new datum x as p(t t, α MP, σ 2 MP ) = N (t y, σ 2 ) (3) where y is the predict value of the new target t and σ 2 is the variance of the prediction. See [1][4][6] for more details of the sparse Bayesian learning. 3.3 Anomaly Detection System The anomaly detection system based on the proposed method operates as follows, 1. (Learning) Learn the relevance vector autoregressive model using a set of validated normal telemetry data. 2. (Prediction) Compute the next probable range of the target series. 3. (Monitoring) Obtain and check the (pseudo-)telemetry data. 4. (Alarming) Give an alarm if the data is out of the predicted range. 5. Repeat steps 2-4 The system is supposed to give many false alarms if we directly apply the redefined AR model Eq.(1) and the RVR Eq.(3). The reasons are, 1. The AR model completely cannot be modeled as the complicate spacecraft system, t (x ) = t AR (x ) = t T RUE (x ) + ɛ AR = y (x ) + ɛ (x ) p(ɛ ) = N (0, σ 2 ) (4) where ɛ AR is the modeling error of the AR model. 2. The RVR adopts the relevance vector for the prototypes of the model formed as Eq.(2) 3. 3 This is a great advantage of the RVR for the execution speed of anomaly detection.

4 As the result, some training data can be mistaken as anomaly though all training data are normal because these data have relatively so large ɛ AR that the system regards them as the data which is far from the prototype data. We evaluated the variance of the difference between another data set {x i, t i } N 2 i=1 and corresponding prediction values, y (x i ). ˆσ AR 2 = 1 N 2 (t i y (x i )) 2, N 2 i N 2 i ɛ(x i ) 0 (5) If the AR model completely describes the system behavior, (5) must be zero in theory. Therefore, we extended (3) as p(t t, α MP, σ 2 MP ) = N (y, σ 2 + ˆσ 2 AR). (6) 4 Experiment and Discussion We performed an experiment with telemetry data obtained from an orbital rendezvous simulation. This telemetry consists of 27 time-series variables in total, 13 of which are from position and attitude control subsystem, and the rest are from propulsion subsystem. In more detail, the former group consists of 12 numerical observation time-series variables regarding the position and attitude of the vehicle and one command sequence. The latter group consists of 14 discretevalued time-series variables, each of which indicates the command input to each of the 14 thruster engines. In this experiment, we assumed a scenario where the power of fourth thruster engine used for the pitch control falls to zero at time 250 [sec]. With this scenario, we performed following two scenarios. Comparison of Proposed Method with normal RV Autoregressive Model First, we have compared the proposed method with the normal relevance vector autoregressive model. Fig.2, Fig.3 show the results of anomaly detection in the series which represents the pitch angle. The solid line in the upper figure shows the pitch angle and the dotted line shows the predicted range, and the solid line in the lower figure shows the probability density of the observation. The system gives the alarm when the probability became lower than the computed limit 4. We can see that the normal model gives many false alarms. On the other hand, the proposed method correctly gives alarms after a little time delay. The proposed method also succeeded in detecting anomalies in the series representing the pitch rate as shown in Fig.4. Comparison with Conventional Limit-checking We compared the proposed method with the conventional limit-checking. We 4 In this experiment, we adopted the probability density value of σ 2 + σar 2 as limit.

5 Fig. 2. Result of anomaly detection by normal RV AR model. The upper graph shows the confidence of target series in the telemetry data. Fig. 3. Result of anomaly detection by the proposed method. The lower graph shows the probability density. set the limit on the standard deviation for the proposed method and on the maximum absolute value in the normal phase for the limit-checking. Fig.4 shows the result. The conventional limit-checking fails to detect slight anomalies like this case. On the other hand, the proposed method is capable of detecting this anomaly because it can dynamically estimate proper range of the target series. Fig. 4. Comparison of the proposed method with conventional limit-checking. The solid line represents the pitch rate and the dotted lines represent the upper and lower bounds given by each method. We only showed the results with respect to the series which included the anomalies due to limitations of space, though we have ran the experiments against all series. 5 Conclusion This paper proposed a new anomaly detection method based on the relevance vector regression and autoregressive model. First, we extended the traditional AR model (3.1) and adopted the relevance vector frameworks for learning this model. In addition, we extended this RV AR model for the purpose of removing false alarm (3.3).

6 Compared with the conventional anomaly detection method, this method has great advantages. First, the proposed method requires little a priori knowledge on the spacecraft system. Therefore, it can be applied to various kinds of spacecraft. We performed an experiment with telemetry data obtained from an orbital rendezvous simulation and confirmed the efficiency of the propos4ed method. Acknowledgements The authors would like to thank Japan Aerospace Exploration Agency (JAXA) for providing the simulation telemetry data. References 1. A.C. Faul and M.E. Tipping. Analysis of Sparse Bayesian Learning, In T.G. Dietterich, S. Becker and Z. Ghahramanim editors, Advances in Neural Information Processing Systems 14, pages , MIT Press, C.M. Bishop and M.E. Tipping, Variational Relevance Vector Machines, In C. Boutilier and M. Goldszmidt, editors, Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence, page 46-53, Morgan Kaufmann, K.-R. Muller, A.J. Smola, G.Ratsch, B. Scholkopf, J. Kohlmorgen, and V. Vapnik, Predicting time series with support vector machines. In W. Gerstner, A. Germond, M. Hasler, and J.-D. Nicoud, editors, Artificail Neural Networks, ICANN 97, page , Berlin, 1997, Springer Lecture Notes in Computer Science, Vol M.E. Tipping, Sparse Bayesian Learning and the Relevance Vector Machinem Journal of Machine Learning Researchm 1: , M.E. Tipping The Relevance Vector Machine. In S.A. Solla, T.K. Leen, and K.- R. Muller, editors, Advances in Neural Information Processing Systems 12, pages MIT Press M.E. Tipping, and A. C. Faul (2003). Fast marginal likelihood maximisation for sparse Bayesian models. In C. M. Bishop and B. J. Frey (Eds.), Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, Key West, FL, Jan Minoru Nakatsugawa, Takehisa Yairi, Naoki Ishihama, Koichi Hori and Shinichi Nakasuka Supporting Anomaly Detection from Satellite Telemetry Data by Regression Trees The 24th International Symposium on Space Technology and Science (ISTS), Takehisa Yairi, Shiro Ogasawara, Koichi Hori, Shinichi Nakasuka, and Naoki Ishihama Summarization of Spacecrafts Telemetry Data By Extracting Significant Temporal Patterns The Eighth Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD2004), pp , Takehisa Yairi, Yoshikiyo Kato and Koichi Hori, Fault Detection by Mining Association Rules from House-keeping Data, Proc. of International Symposium on Artificial Intelligence, Robotics and Automation in Space (i-sairas 2001) 10. U.U. Muller, A. Schick and W. Wefelmeyer(2004), Efficient prediction for linear and nonlinear autoregressive models, 2004, (Submitted paper) 11. W.D. Penny and S.J. Roberts, Bayesian Methods for Autoregressive Models, IEEE Workshop on Neural Networks for Signal Processing, Sydney Australia, December 2000.

ADAPTIVE LIMIT-CHECKING FOR SPACECRAFT USING RELEVANCE VECTOR AUTOREGRESSIVE MODEL. 2 RCAST, Univ.

ADAPTIVE LIMIT-CHECKING FOR SPACECRAFT USING RELEVANCE VECTOR AUTOREGRESSIVE MODEL Ryohei Fujimaki 1, Takehisa Yairi 2, and Kazuo Machida 2 1 Dept. of Aero. and Astronautics, Univ. of Tokyo, 4-6-1 Komaba