Initial estimates for R 0 and Γ0 R Γ. Γ k. State estimate R Error covariance. k+1 k+1

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1 HERA-B -32 VDET -1 CATS track fitting algorithm based on the discrete Kalman filter D. Emeliyanov 12 I. Kisel 34 1 Institute of Control Sciences, Profsoyuznaya ul., 65, Moscow, Russia 2 Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-2267, Hamburg, Germany 3 Joint Institute for Nuclear Research, Dubna, Moscow region, Russia 4 Max-Planck-Institut für Physik, Werner Heisenberg Institut, Föhringer Ring 6, D-885 München, Germany Abstract The Kalman filter is a set of mathematical equations that provides an efficient recursive solution of the least squares method. This paper provides a brief introduction to the discrete Kalman filter equations of which are derived for the case of track parameter estimation with measurements from the vertex detector. To reduce computational cost of the algorithm an optimized numerical implementation is proposed. The algorithm was tested on simulated data. Test results regarding an accuracy of estimates and computing time are presented. 1 The discrete Kalman filter The Kalman filter addresses the general problem of trying to estimate the state vector R of a discrete-time process that is governed by the linear difference equation [1] R k+1 = A k R k + ν k (1) with a measurement u that is the linear function of the vector R u k = H k R k + k : (2) The matrix A k in the difference equation (1) relates the state at step k to the state at step k +1, the matrix H k in the measurement equation (2) relates the state to the measurement u k. The random variables k and ν k represent the measurement noise and process noise, respectively. They are assumed to be independent with Gaussian distributions. Let's define b R k to be a state estimate at step k after processing measurement u k. The main idea of the Kalman filter is that the optimal (in mean-square sense) estimate b R k should be a sum of an extrapolated estimate e R k and a weighted difference between an actual measurement u k and a measurement prediction H k e Rk br k = e R k + K k (u k H k e Rk ) where e R k = A k 1 b R k 1 : The difference (u k H k e Rk ) is called the residual. The residual reflects the discrepancy between the predicted measurementvalue H k e Rk and the actual measurement u k. A residual of zero means that they are in complete agreement. 1

2 The matrix K k is called the filter gain and is chosen to minimize the sum of diagonal elements of an estimation error covariance matrix b k. By definition, h b k = E (R k R b k )(R k R b i k ) T where E denotes mathematical expectation, T denotes transposition. measurements, the minimization leads to the following formula for K k In the case of scalar K k = e k V k + H k e k where V k is the covariance of the measurement error k, e k is an extrapolated estimation error covariance matrix b k 1. The formula for e k follows from equation (1): e k = A k 1 b k 1 AT k 1 + Q k 1 where Q k 1 is the covariance matrix of the process noise ν. The new minimized value of the error covariance matrix b k is defined by the equation b k =(I K k H k ) e k where I is aunity matrix of the same dimensions as matrix. In fact, this equation describes an update of the matrix b k after each measurement. The computational algorithm of the discrete Kalman filter consists of two steps (Fig. 1): ~ ~ R Γ k+1 k+1 Initial estimates for R and Γ Prediction step Filtering step R k Γ k State estimate R Error covariance Figure 1: The operation diagram of the discrete Kalman filter. Γ 1. Prediction step extrapolation of the estimate b R and error covariance matrix b to the next step of the algorithm. 2. Filtering step er k = A k 1 b R k 1 e k = A k 1 b k 1 AT k 1 + Q k 1: (3) (a) The gain matrix calculation K k = e k V k + H k e k (4) 2

3 (b) The updated estimate br k = e R k + K k (u k H k e Rk ) (5) (c) The updated error covariance matrix b k =(I K k H k ) e k : (6) After each Prediction/Filtering pair, the process is repeated until all measurements are processed. 2 A Kalman filter for estimating track parameters In context of this paper, a track through the vertex detector is described as a straight-line motion in the presence of Gaussian disturbances: x k+1 = x k + t xk (z k+1 z k ) y k+1 = y k + t yk (z k+1 z k ) t xk+1 = t xk + xk t yk+1 = t yk + yk T where R k = x k y k t xk t yk is the vector of the track parameters taken at a plane z = zk, T denotes transposition, xk, yk are non-correlated random values normally distributed with parameters (ffx(z 2 k )), (ffy(z 2 k )). These random values are introduced in order to describe the multiple scattering which affects the track when it passes through detector planes z = z k, k = 1::: For description of the multiple scattering process we use the model shown in Fig.2. It is is (7) Y Wall Detector planes Wall σ 1 σ 2 σ 2 σ3 σ 3 σ 4 Z z min z min z max z min z max z min Figure 2: Illustration of the multiple scattering model used for track fit. similar to the model proposed in [2] and takes into account a two-sided structure of the VDS detector planes. A particle enters the detector volume on the surface of z min, produces the hit and undergoes the scattering. Then it produces hit on the other plane at z max, undergoes the scattering again and, finally, leaves the volume. In the case of insensitive walls, the particle undergoes the scattering only once on z min surface. The model of the measurements (hits) in the VDS has the following form: u k = u(z k )=x(z k )cosff k y(z k ) sin ff k + ff k k = 1::: (8) where ff k rotation angle of the strips in the plane z = z k, k a random value normally distributed with parameters ( 1), ff r.m.s of the measurement noise. 3

4 The problem is to obtain an estimation of the vector's R components using the measurements fu k g, k = 1 2 :::. It is assumed that the track recognition problem is already solved before fitting, and the measurements fu k g are collected in a downstream order: z k >z k 1, k>. To solve the stated problem, we propose to use the discrete Kalman filter, equations of which are derived below. It should be noted, that the Kalman filter is a natural choice for track fitting in the vertex detector due to the linearity of the track and measurement models in this detector. According to the track model (7), the matrix A in equation (1) has the form A k = 1 z k 1 z k C A z k = z k z k 1 k =1 2:::: The detector volumes and insensitive walls are treated as so-called thin scatterers [2]. In case of such scatterers, the non-zero elements q ij of the process noise covariance matrix Q k equal to q 33 = p 2 ff 2 MS (1+t2 x)(1 + t 2 x + t 2 y) q 1+t 2 x + t 2 y q 34 = q 43 = p 2 ff 2 MS t xt y (1+t 2 x + t 2 y) q 1+t 2 x + t 2 y q 44 = p 2 ff 2 MS (1+t2 y)(1 + t 2 x + t 2 y) q 1+t 2 x + t 2 y where p is an external estimate for inverse momentum of the particle, ffms 2 is the variance of the multiple scattering angle for 1 GeV particle. The latter parameter is obtained from ARTE GEDE and GWAL table for each material the particle passes through. According to the measurement equation (8), the matrix H is: h i H k = cos ff k sin ff k : (9) Using obtained formulas, we can write algorithm of the Kalman filter in the following form. 1. Prediction step, k =1 2:::: ex k = b x k 1 + z k b t x k 1 e xk = b y k 1 + z k b t yk 1 et x k = b t x k 1 e tyk = b t yk 1 e k = A k 1 b k 1 AT k 1 + Q k 1: 2. Filtering step, k = 1::: (a) The gain matrix of the filter K k = e k ff 2 + H k e k (b) The updated estimate of the track parameters at z = z k br k = R e k + K k (u k H e krk ) (c) The updated error covariance matrix b k =(I K k H k ) e k : 4

5 3 Computational refinements of the algorithm It should be noted that any efficient numerical implementation of the Kalman filter algorithm is always application-dependent. There are several ways of reducing computational cost of the algorithm: ffl calculation of the filter covariance matrix in the upper-triangle form, ffl elimination of multiplication by one and zero during prediction and filtering step if the track and measurement models allow such elimination, ffl optimization of the covariance matrix update procedure [3]. Following these directions, we propose a fast numerical implementation of the Kalman filter algorithm in which the number of arithmetical operations is significantly reduced. Calculating the covariance matrix in the upper-triangle form, we obtain the following equation for the prediction step: R 1 = zr b 3 R 2 = zr b 4 R 3 = R 4 = 11 = z 2 b 13 + z b = z 2 b 24 + z b = q = z b + b 23 + z b = z b = q 34 (1) 13 = z b = z b = q = z b 34 where z = z k z k 1, matrix = e k b k 1,vector R = e R k b R k 1. The most time-consuming parts of the filtering step are calculations of the gain matrix and the updated covariance matrix. Substitution of the measurement matrix (9) into (4) yields the following formula for the gain matrix K k = B k s k where vector B and scalar value s are B 1 = e 11 cos ff k e 12 sin ff k B 2 = e 12 cos ff k e 22 sin ff k B 3 = e 13 cos ff k e 23 sin ff k B 4 = e 14 cos ff k e 24 sin ff k (11) s = ff 2 + e 11 cos 2 ff k e 12 sin 2ff k + e 22 sin 2 ff k : (12) In terms of B and s, the filtering step of the filter can be simplified. The formula for updated estimate takes the form br k = e R k + C k k where k = u k e R 1 cos ff k + e R 2 sin ff k (13) and C k = s 1 k B k. The covariance matrix is updated as follows b 11 = e 11 B 1 C 1 b 22 = e 22 B 2 C 2 b 33 = e 33 B 3 C 3 b 12 = e 12 B 1 C 2 b 23 = e 11 B 2 C 3 b 34 = e 11 B 3 C 4 b 13 = e 13 B 1 C 3 b 24 = e 24 B 2 C 4 b 44 = e 44 B 4 C 4 (14) b 14 = e 14 B 1 C 4 The equations (1)-(14) describe the fast implementation of the Kalman filter algorithm for track fitting in the VDS. The number of arithmetical operations needed for the fast algorithm 5

6 Operation: Multiplication Division Summation Classical scheme Fast scheme Table 1: Computational costs of the fast and classical implementation of the algorithm. in comparison with the classical implementation defined by (3)-(6) is presented in Table 1. The state vector is assumed to be four-dimensional one. It can be easily seen that the number of operations for proposed implementation is almost 5 times less than for the classical scheme. Usually the actual implementation of the classical scheme is even more slow due to the use of libraries for matrix operations, in particular, for matrix multiplication. Note that the fast version of the algorithm needs no matrix multiplications. 4 The initialization of the Kalman filter To obtain the initial estimates of vector R b and covariance matrix b = G two space points are used. Let's their coordinates be (x I y I z I ) for the first point and (x II y II z II ) for the second one (Fig. 3). The components of vector R b can be easily obtained from an equations of the Y II Z X I α ΙΙ α Ι Figure 3: The space points used for initialization of the filter. straight line drawn through the space points: br 1 = x I(z II z ) x II (z I z ) z II z I br 2 = y I(z II z ) y II (z I z ) z II z I br 3 = x II x I z II z I b R4 = y II y I z II z I : The errors of estimates of the space point coordinates are assumed to be non-correlated random values normally distributed with parameters (ff 2 ). It is also assumed that angles ff I, ff II between strips which form the space points are approximately ±ß=2. Then the initial covariance matrix can be represented in the following upper-triangle form G 11 = G 22 = ff 2 (z (z II z I ) 2 II z ) 2 +(z I z ) 2 +2ff 2 (z II z I ) 2 6

7 G 13 = G 24 = G 33 = G 44 = ff 2 z (z II z I ) 2 II + z I 2z 2ff 2 (z II z I ) 2 +2ff2 G 12 = G 14 = G 23 = G 34 =: In these formulas, the additional terms with ff 2 are included in order to take the influence of the multiple scattering on the initial data into account. In the numerical implementation of the initialization procedure, ff is equal to 2. mrad that seems to be a quite reasonable approximation. 5 Results and discussion The developed algorithm was implemented numerically in CATS program [4] and its performance has been tested using simulated inelastic events. Some results of the test are presented below. Table 2 provides the reconstruction efficiencies (see definitions in the CATS note [4]) of reference and all tracks, the clone rate, the ghost rate and the time needed by CATS (total, reconstruction and fitting only) for different number of mixed Poisson distributed interactions for fully instrumented VDS. While mean interaction rate is expected to be 4, we investigated CATS up to 1 interactions. The reconstruction efficiency for reference tracks slowly goes down from about 98% to 89% demonstrating stability of the program. The clone rate is constantly abut 3%. The ghost rate grows from 7% up to 3%. Ghost tracks are mainly very short tracks which should be accepted or rejected at the next step of tracking using information from other subdetectors. The total time used by CATS grows almost linearly from 6 to 174 ms per event running on PC hb-af3 at DESY, Hamburg. This time is mainly determined by the reconstruction itself. The Kalman refit procedure takes about 1 ms per interaction. The rest of time spent for storing the ARTE tables. It should be emphasized that such low computing time is achieved due to optimized numerical implementation of the filter. Interactions Refset Allset Clone Ghost t, total t, reco t, fit Table 2: Results of a tracking efficiency study using different number of mixed Poisson distributed inelastic events generated in a Monte Carlo simulation. Tracking efficiency is measured in per cent and time unit is ms. For each track the algorithm for Kalman refit has estimated track parameters in two planes: z = z f and z = z l, where z f, z l are z-coordinates of the first and the last hits of a simulated track, respectively. The reliability of parameter error estimation was studied by investigating normalized residual distributions, which use the estimated errors for normalization. The normalized residual (also called the pull) of a track parameter, for instance x-coordinate at the first 7

8 hit, is defined by P(x) = xrec x MC p 11 where x REC is the reconstructed x position of a track and x MC the corresponding Monte Carlo value, while 11 is the estimate for the corresponding covariance matrix diagonal element. In the ideal case the pull distribution should be unbiased and have gaussian core of unity. The distribution of residuals and pulls for the plane of the first hit are plotted in Fig. 4 and in Fig. 5. The results of testing on Monte Carlo simulated inelastic events show that 1. The algorithm provides high accuracy of the estimation of track parameters (see Table 3). ff of residual first hit plane last hit plane x, μm y, μm t x,mrad t y,mrad Table 3: Sigma of track parameter residuals. 2. All observed residual distributions have shapes close to Gaussian without heavy tails. 3. The non-gaussian tails in the pull distributions are small, the gaussian cores of the pulls agree with unity, indicating reliable estimate of the covariance matrix. 4. The comparison with the ideal case shows that if particle momentum is known the non- Gaussian tails in the distributions completely disappear. It means that the multiple scattering model embedded in the filter corresponds quite accurately to the way the particle's interaction with the detector is simulated. 6 Summary An algorithm for track parameter estimation in the vertex detector has been developed. The algorithm is based on the Kalman filter technique and takes effects from multiple scattering into account. Due to an optimized numerical implementation the algorithm shows very low computing time. Tested on simulated data, a good performance regarding estimate of track parameters and their errors is observed. References [1] R.K. Bock et al., Data analysis techniques for high-energy physics experiments, Cambridge Univ. Press (199). [2] R. Mankel and A. Spiridonov, A Pattern Recognition Algorithm for the HERA-B Main Tracking System, Part IV: The Object-Oriented Track Fit, HERA-B Note (1998). [3] K. Brammer and G. Siffling, Kalman-Bucy Filter: Deterministische Beobachtung und Stochastische Filterung, (1974). [4] I. Kisel and S. Masciocchi, CATS: A Cellular Automaton for Tracking in Silicon for the HERA-B Vertex Detector, HERA-B Note (1999). 8

9 Mean E E / 84 Constant 39.2 Mean -.221E-4 Sigma.13E Mean.3991E E / 86 Constant Mean.1556E-4 Sigma.15E x first residual ideal y first residual ideal Mean.522E E / 97 Constant 279. Mean.4428E-5 Sigma.591E Mean -.582E-5.136E / 97 Constant Mean.1417E-4 Sigma.5826E dpx/dpz first residual ideal dpy/dpz first residual ideal Mean -.31E / 77 Constant Mean E-2 Sigma x first pull ideal Mean E / 78 Constant 37.1 Mean.1575E-2 Sigma dpx/dpz first pull ideal Mean.285E / 81 Constant 383. Mean.163E-1 Sigma y first pull ideal Mean E / 82 Constant Mean.8215E-2 Sigma dpy/dpz first pull ideal Figure 4: Residuals and normalized residuals (pulls) of tracks parameters at the first hit obtained on simulated inelastic events with known momentum value. 9

10 Mean E E / 85 Constant Mean E-4 Sigma.142E x first residual Allset Mean.622E E / 85 Constant Mean.2355E-4 Sigma.174E y first residual Allset Mean E E / 97 Constant Mean E-5 Sigma.5346E Mean E-5.147E / 97 Constant 218. Mean.925E-5 Sigma.638E dpx/dpz first residual Allset dpy/dpz first residual Allset Mean -.167E / 89 Constant Mean E-2 Sigma x first pull Allset Mean.3966E / 85 Constant Mean.225E-1 Sigma y first pull Allset Mean -.268E / 94 Constant Mean.6385E-2 Sigma Mean -.492E / 96 Constant Mean.4615E-2 Sigma dpx/dpz first pull Allset dpy/dpz first pull Allset Figure 5: Residuals and normalized residuals (pulls) of tracks parameters at the first hit obtained on simulated inelastic events with approximate momentum value. 1