State Estimation using Gaussian Process Regression for Colored Systems

Kyuman Lee Eric N. Johnson School of Aerospace Engineering School of Aerospace Engineering Georgia Institute of Technology Georgia Institute of Technology Atlanta, GA 30332 Atlanta, GA 30332 404-422-3697 404-385-2519 [email protected] [email protected]

Abstract—The goal of this study is to use Gaussian process (GP) the times of interest in the system; for brevity such errors are regression models to estimate the state of colored noise systems. called “colored” noise [1]. In audio engineering, electronics, The derivation of a Kalman filter assumes that the process noise physics, and many other fields, the color of a noise and measurement noise are uncorrelated and both white. In is generally understood to be some broad characteristic of relaxing those assumptions, the Kalman filter equations were its power . Different have signif- modified to deal with the non-whiteness of each noise source. The standard Kalman filter ran on an augmented system that icantly different properties: for example, as audio had white and other approaches were also introduced they will different to human ears, and as images they depending on the forms of the noises. Those existing methods will have a visibly different texture. In image processing, can only work when the characteristics of the colored noise are the quality of images will directly influence the accuracy perfectly known. However, it is usually difficult to model a noise of locating landmarks. During sampling and transmission, without additional knowledge of the noise statistics. When the images are often degraded by noises which may originate parameters of colored noise models are totally unknown and from a multiplicity of sources. These noises are colorful the functions of each underlying model (nonlinear dynamic and and their variances are not known beforehand. The original measurement functions) are uncertain or partially known, filter- images having the colored noise must be filtered to ensure ing using GP-Color models can perform regardless of whatever forms of colored noise. The GPs can learn the residual outputs the accuracy of location and measurement [2]. Furthermore, between the GP models and the approximate parametric models in radar and sonar signal processing, various colored noise (or between actual sensor readings and predicted measurement are: external spurious signals that, intentionally or not, jam readings), as a member of a distribution over functions, typically reception; echoes from various reflectors in the landscape; with a mean and covariance function. Lastly, a series of simu- and, in sonar, [3]. Therefore, each application lations, including Monte Carlo results, will be run to compare using such signals typically requires the noise of a specific the GP based filtering techniques with the existing methods to color and it is necessary to extend conventional Kalman handle the sequentially correlated noise. filtering approach in order to solve the problem of the color noise effectively. To deal with colored noise over several decades, the standard TABLEOF CONTENTS Kalman filter ran on an augmented system that had white 1. INTRODUCTION ...... 1 noises. Bryson et al [1] introduced the augmented filter for colored measurement noise first. For example, due to 2. MODELSAND SETUP ...... 2 the non-stationary nature of the speech signal, augmented 3. PRELIMINARIES ...... 2 Kalman filtering of colored noise for speech enhancement 4. LEARNING GPMODELSFOR COLORED NOISE was also described [4], [5]. Recently, Liu et al [6] presented SYSTEMS (GPC)...... 3 a self-tuning Kalman filter for auto regressive or moving 5. GPC-BASED BAYES FILTERS ...... 4 average (ARMA) signals with colored noise. Also, if the process noise is colored, then it is straightforward to modify 6. SIMULATION RESULTS ...... 5 the Kalman filter equations and obtain an equivalent but 7. CONCLUSION ...... 6 high-order system with white process noise [7]. On the REFERENCES ...... 6 other hand, the augmented state procedure may lead to ill- BIOGRAPHY ...... 8 conditioned computations in constructing the data processing filter. It is a singular problem as the case of no noise in some of the measurements because the correlation matrix of the measurement noise is singular, i.e., R−1 does not 1. INTRODUCTION exist, in the notation of a Kalman filter. Instead, another Estimating the state of a dynamic system is a fundamental filtering, measurement differencing [8], [9], that was capable problem at modern GNC areas. The most successful tech- of converting the received colored noise into , niques for state estimation are Bayesian filters such as a was established. The contribution of the colored part of the Kalman filter. The key assumption of the standard Kalman noise to the received noisy signal was first estimated, and filter is that the process noise and measurement noise are un- this estimate was then subtracted from the received signal. correlated and both white. These are often too restrictive as- This approach, by subtraction of the colored part of the noise, sumptions, and applications with non-white noise frequently was strictly equivalent to the whitening approach and had arise in practice. lower dimension than the augmented state filters. Simon [10] provided a survey of all previously noted methods. Details will be described in Section 3. as preliminaries. Many practical systems exist in which the correlation times of the random measurement errors are not short compared to Those existing methods can only work when the characteris- 1 tics (power and ) of colored noise are assumed observation system. to be known. However, it is typically difficult to model a noise without a priori knowledge of the noise statistics or a wk = φ(wk−1) + ηk (3) supplementary measurement. vk = ψ(vk−1) + ζk (4)

A Gaussian process (GP) is a function approximation which ηk ∼ N (0,Qk) can be thought of as a distribution over functions. The true ζ ∼ N (0,R ) function at any given point in the domain exists in the Gaus- k k T sian distribution modeled by mean and covariance functions E[ηkηj ] = Qkδk−j evaluated at the same point [11]. GPs are used in several T disciplines when some underlying function is unknown but E[ζkζj ] = Rkδk−j needed. In implementation, a GP can be formulated as the T E[ηkζj ] = 0 regression of training data (points in the state space and function evaluations) with respect to a basis function. A GP where ηk is zero-mean white noise that is uncorrelated with can be dynamically updated with new function evaluations or w and ζ is zero-mean white noise that is uncorrelated modeled once at the onset of an algorithm. k−1 k with vk−1. φ is the nonlinear colored process noise function and ψ is the nonlinear colored measurement noise function. The machine learning community has applied GP to both Both functions are totally unknown, and we don’t even know controls and estimation processes in the past. When a system if each signal has colored noise in the form of Eqs. (3) and has unknown dynamics and measurement models, a GP can (4). The covariances of the noise functions are given as be used to learn them. The GP mean provides approximates of the state transition matrix and measurement model, while T T T the GP covariance provides estimates of the process and E[wkwk−1] = E[φ(wk−1)wk−1 + ηkwk−1] measurement white noise. Estimation methods have ranged   T ∂φ T from Bayesian filtering with GP to extended and unscented = E[φ(wk−1)wk−1] ≈ |wk−1 E[wk−1wk−1] 6= 0 Kalman filtering with GP [12], [13]. Likely, when the ∂w T T T parameters of colored noise models are totally unknown and E[vkvk−1] = E[ψ(vk−1)vk−1 + ζkvk−1] the functions of each underlying model (nonlinear dynamic   and measurement functions) are uncertain or partially known, T ∂ψ T = E[ψ(vk−1)vk−1] ≈ |vk−1 E[vk−1vk−1] 6= 0 filtering using enhanced GP-Color models can perform re- ∂v gardless of whatever forms of colored noise. The GPs can learn the residual outputs between the GP models and the We see that wk or vk is colored noise because it is correlated approximate parametric models (or between prior estimated with itself at other time steps. states by the GPs and the outputs of parametric models), as a member of a distribution over functions, typically with a mean and covariance function. The Q and R noise covariance 3. PRELIMINARIES matrices in the filter can also be learned by the GP-Color prediction model and observation model, respectively. Input- Gaussian Process Regression dependent noise was described in the article of Kersting et A Gaussian process is a nonparametric tool for learning al [14], but we are not aware of previous work with colored regression functions from sample data. Consider now the case noise functions using GPs. where we have measurements of the observation which are corrupted with white noise The rest of this paper is organized as follows. The next two sections demonstrate the formulation of the problem and y = h(x ) + ν = h + ν , ∀ i = 1, ··· ,N the outline of the approach. Section 4 explains how each i i i i i model with colored noise can be learned by GPs and section −1 5 presents a novel filtering algorithm using GPs for colored where νi ∼ N (0, β ). Since the white noise is independent noisy systems. Next, the simulation environment is described of each data point, we have that and results are presented. Finally, conclusion and future work −1 are discussed. p(y1:N |h1:N ) = N (h1:N , β IN×N ) p(h1:N ) = N (0,K)

⇒ p(y1:N ) = N (0,CN ) (5) 2. MODELSAND SETUP N×N We consider discrete-time stochastic dynamics systems of the where the covariance matrix CN ∈ R and it is defined form as −1 CN = K + β IN×N x = f(x ) + w (1) k k−1 k−1 Therefore every element of the covariance matrix C will have zk = h(xk) + vk (2) the form −1 C(xi, xj) = k(xi, xj) + β δi,j n m where xk ∈ R is the state and zk ∈ R is the measurement at time step k = 1, ··· ,T0. f is the nonlinear dynamic The most widely used kernel function is the squared expo- function and h is the nonlinear measurement function. Both nential, which has the form are partially known or uncertain. θ k(x , x ) = θ exp(− 1 kx − x k2) Now suppose that we have colored process and measurement i j 0 2 i j noises. The process noise is itself the output of the dynamic system and the measurement noise is itself the output of the

2 −1 + − The hyperparameters, θ = [θ0, θ1, β ], can be learned estimation-error covariance from time (k − 1) to time k by maximizing the log marginal likelihood of the training outputs given the inputs: − + + xˆk = f(ˆxk−1) + φ(w ˆk−2) − aug + aug T aug θmax = arg max{ln p(y1:N |x1:N , θ)} (6) Pk = Fk−1Pk−1(Fk−1) + Qk−1 θ − + The goal in regression is to predict y for a new input point where superscript or denote the a priori or a posteriori N+1 estimates, respectively. At time k−, compute the follow- xN+1 given the set of training data, D = hx1:N , y1:N i. From Eq. (5), ing Jacobian matrix and augment the original measurement model as follows: p(y1:N+1) = N (0,CN+1) where aug h ∂h i H = ∂x |xˆ− 0 I  C k  k k C = N ∗ N+1 kT c ∗ Then, perform the measurement update of the state estimate −1 and estimation-error covariance where c = k(xN+1, xN+1) + β and then p(yN+1) = N (0, c). Now we can claim the conditional distribution is − + a Gaussian distribution with mean and covariance specified zˆk = h(ˆxk ) + ψ(ˆvk−1) as follows − aug T aug − aug T −1 Kk = Pk (Hk ) (Hk Pk (Hk ) ) T −1  +   −  GPµ(yN+1|D, θ) = k∗ CN y1:N (7) xˆk xˆk + − T −1 wˆk  = wˆk  + Kk(zk − zˆk) GPΣ(yN+1|D, θ) = c − k∗ CN k∗ (8) + − vˆk vˆk k ∈ N k(x , x ), k(x , x ), + − aug − where ∗ R and it has elements 1 N+1 2 N+1 Pk = Pk − KkHk Pk ..., k(xN , xN+1). The colored noise problem ends up being solved with the Enhanced GP Models above augmented extended Kalman filter (EKF). However, The GP for regression has a zero mean assumption in Eq. (7), this method can only work when the characteristics of colored and if a query state is far away from the training states, then noise are assumed to be known. the output of GP quickly tends towards zero. This makes the choice of training data for the GP very important. A parametric model is one which attempts to represent a partic- 4. LEARNING GPMODELSFOR COLORED ular phenomenon with physical equations. The disadvantage NOISE SYSTEMS (GPC) of parametric models is that substantial domain expertise is required to build these models, and even then they are often Training Data simplified representations of the actual systems. The training data is a sampling from the dynamics and observations of the system. The training data for each GP Higher prediction accuracy can be obtained by combining GP consists of a set of input-output relations. models with parametric models. We call the combination of GP and parametric models enhanced-GP (EGP) models Given x1:T0 and z1:T0 , let the approximate parametric predic- [12], [13]. The EGP models alleviate some of the problems fˆ hˆ found when using either model alone. Using this idea of EGP tion and observation models be denoted and . These are method, we can develop novel GP models for colored noise assumed to be known but not perfect based on partial knowl- systems. Details will be described in Section 4. edge of actual systems in Eqs. (1) and (2). The approximate process and measurement residuals are denoted wˆ and vˆ and Existing Methods for Colored Noise Models defined as ˆ There are a couple of ways to solve the colored noise problem wˆ1:T0−1 = x2:T0 − f(x1:T0−1) when each parameter of Eqs. (1), (2), (3), and (4) is perfectly vˆ = z − hˆ(x ) known. Here we will solve the discrete-time problem by 1:T0 1:T0 1:T0 augmenting the state. First, we augment the original dynamic model as follows: where x1:T0 = [x1, x2, ..., xT0 ] and z1:T0 = [z1, z2, ..., zT0 ].  ∂f  By Eqs (3) and (4), each residual is the function of itself at ∂x |xˆ+ I 0 k−1 previous one time step. The training data sets of each residual aug  ∂φ  F = 0 | + 0 , are given as k−1  ∂w wˆk−1   ∂ψ  0 0 |vˆ+ ∂v k−1 ˆ DW = hwˆ1:T0−2, wˆ2:T0−1i "0 0 0 # aug Dˆ V = hvˆ1:T −1, vˆ2:T i Qk−1 = 0 Qk−1 0 0 0 0 0 Rk−1 where the hat ” ˆ ” denotes a estimate or an approximate. By optimizing similar to Eq. (6), the hyperparameters θw and Then, perform the time update of the state estimate and θv can be learned.

3 GPC Models where k∗ is the vector of kernel values between the x =w ˆ vˆ The idea of GPC method is to use a GP to learn the residual query input, ∗ k−1 or k−1, and the training inputs, wˆ1:T0−2 or vˆ1:T0−1, respectively. N(= T0 − 2 or T0 − 1) output between the GP model and the approximate parametric is the number of training data, and d(= n or m) is the model. In particular, the residuals of prediction model can be approximate outputs between prior estimated states by dimensionality of the input space. the GPs and the outputs of parametric models. Also, the Eq. (11) defines the d-dimensional Jacobian vector of the GP residuals of observation model can be approximate outputs mean function for a single output dimension. The full n × between actual sensor readings and predicted measurement n Jacobian of a prediction model or m × n Jacobian of a readings. Then, from Eqs. (1) and (2), GPC models at time n m k = T + 1, ··· ,T observation model is determined by stacking or Jacobian step 0 become vectors together, one for each of the output dimensions. x = f(x ) + φ(w ) + η k k−1 k−2 k−1 We are now prepared to incorporate the GPC models into ˆ ˆ = f(xk−1) + [f(xk−1) − f(xk−1)] + φ(wk−2) + ηk−1 augmented EKF, as shown in Algorithm 1. ˆ  ˆ  ⇒ xk = f(xk−1) + GPµ wˆk−2, DW +η ˆk−1 (9) Algorithm 1 The GPC-EKF + + + + zk = h(xk) + ψ(vk−1) + ζk 1: (w ˆ , vˆ , xˆ ,P , z ) are given: T0−1 T0 T0 T0 T0+1 ˆ ˆ aug T = h(xk) + [h(xk) − h(xk)] + ψ(vk−1) + ζk Require: X = [xˆ wˆ vˆ] + +   2: wˆ = GP (w ˆ , Dˆ ) ˆ ˆ T0 µ T0−1 W ⇒ zk = h(xk) + GPµ vˆk−1, Dˆ V + ζk (10) 3: for k = (T0 + 1) : T do where 4: Time Updates: − + 5: wˆ = GP (w ˆ , Dˆ )  ˆ  k µ k−1 W ηˆk ∼ N 0, GPΣ(w ˆk−1, DW ) ˆ + ˆ 6: Qk−1 = GPΣ(w ˆk−1, DW )   ∂GP (w ˆ+ ,Dˆ ) ˆ ˆ ˆ µ k−1 W and ζk ∼ N 0, GPΣ(ˆvk−1, DV ) . 7: Φk−1 = + ∂wˆk−1 − + ˆ where all parameters of φ, ψ, η, and ζ of colored noise are un- 8: vˆk = GPµ(ˆvk−1, DV ) + known since it is typically difficult to model a noise without 9: Rˆ = GP (ˆv , Dˆ ) a priori knowledge of the noise statistics or a supplementary k−1 Σ k−1 V ∂GP (ˆv+ ,Dˆ ) ˆ µ k−1 V measurement. 10: Ψk−1 = + ∂vˆk−1 The key advantages of GPC are their modeling flexibility, − + + 11: xˆ = fˆ(ˆx ) +w ˆ their ability to provide uncertainty estimates, and their ability k k−1 k−1 to learn noise and smoothness parameters from training data.  ∂fˆ  ∂x |xˆ+ In×n 0n×m aug k−1 12: F =  ˆ  k−1  0n×n Φk−1 0n×m 5. GPC-BASED BAYES FILTERS 0m×n 0m×n Ψˆ k−1 aug ˆ ˆ Algorithm of GPC-EKF 13: Qk−1 = diag(0n×n, Qk−1, Rk−1) − aug + aug aug The following describes the integration of GPC prediction 14: P = F P (F )T + Q and observation models into the EKF. In addition to the GPC k k−1 k−1 k−1 k−1 mean and covariance estimates used in Eqs. (9) and (10), 15: Measurement Updates: the incorporation of GPC models into the EKF requires a ˆ − − 16: zˆk = h(ˆxk ) +v ˆk linearization of the GPC prediction and observation model h i in order to propagate the state and observation, respectively. aug ∂hˆ 17: Hk = ∂x |xˆ− 0m×n Im×m For the EKF, this linearization is computed by taking the first k term of the Taylor series expansion of the GP function. − aug T aug − aug T −1 18: Kk = Pk (Hk ) (Hk Pk (Hk ) )  +   −  xˆk xˆk By the derivation of Ko et al [12], the Jacobian of the GP + − mean function (7) can be expressed as 19: wˆk  = wˆk  + Kk(zk − zˆk) + − vˆk vˆk T ∂GPµ(x∗,D) ∂(k∗) + − aug − = C−1y (11) 20: Pk = Pk − KkHk Pk ∂(x ) ∂(x ) N 1:N ∗ ∗ 21: end for  ∂(k(x ,x )) ∂(k(x ,x ))  1 ∗ ··· 1 ∗ ∂(x∗[1]) ∂(x∗[d]) ∂(k∗)  . . .  where =  . .. .  ∂(x∗)   Qˆk is the additive white noise part of the process noise, ∂(k(xN ,x∗)) ··· ∂(k(xN ,x∗)) ∂(x∗[1]) ∂(x∗[d]) which corresponds directly to the GPC uncertainty. Φˆ k, ∂(k(x, x∗)) the linearization of the prediction model, is the Jacobian of = −θ1(x∗[i] − x[i]) k(x, x∗), i = 1, ··· , d the GPC mean function found in Eq. (11). Similarly, the ∂(x∗[i]) noise covariance, Rˆk, and the linearization of the observation model, Ψˆ k, are computed by using the GPC observation model. 4 GPs are typically defined for scalar outputs, and GPC based EKF only - Standard and Augmented Bayes Filters represent models for vectorial outputs by learn- Let the state vector be the position and velocity of a vehicle, ing a separate GPC for each output dimension. This forces T ˆ ˆ x(t) = [pos, vel] . The vehicle flies with following linear the resulting noise covariances, Qk and Rk, to be modeled as time-invariant dynamics with colored process noise, and a independent diagonal matrices. simple position sensor with colored measurement noise is used as follows:

6. SIMULATION RESULTS xk = Fk−1xk−1 + Φk−2wk−2 + ηk−1 (12) A variety of cases have been simulated to examine the in- yk = Hkxk + Ψk−1vk−1 + ζk (13) fluence of the color noise and to test the effectiveness of the GPC-EKF algorithm for colored-noise systems. The results where will be given by means of Monte Carlo methods. 1 ∆t F = ,H = [1 0] GPC only k 0 1 k To verify the GPC mean and variance functions given in 0.99 0  Eqs. (9) & (10) as well as the hyperparameter learning via Φk = 0 0.99 , Ψk = 0.99 optimization, one example of GPC regression is given as  0 0 ηk ∼ N 0, , ζk ∼ N (0, 1). True: yk = sin(xk) + 0.99vk−1 + ζk where ζk ∼ N (0, 0.1) 0 1

GP: yk = GPµ (xk,D) + νk where νk ∼ N (0, GPΣ (xk,D)) Initialize a filter ˆ GPC: yk = 0.9 sin(xk) + GPµ(ˆvk−1, Dˆ V ) + ζk x = [0, 0]T , xˆ+ = (x ) ˆ T0 T0 E T0 where ζk ∼ N (0, GPΣ(ˆvk−1, Dˆ V )) + + + T PT = E[(xT0 − xˆT )(xT0 − xˆT ) ] = I2×2 Figure 1 contains a sine function with color noise (black), 0 0 0 noisy samples drawn from the function (red plus), the re- The first simulation compares a standard EKF and an aug- sulting GPC(blue)/GP(green) mean function estimate, and mented EKF, introduced in Section 3., of given the colored the GPC(dotted blue)/GP(dotted green) uncertainty sigma noise system, Eqs. (12) & (13). For an ideal (but not realistic) bounds. The GPC hyperparameters are determined by op- case in this simulation only, let’s assume that we know perfect timization of the data likelihood. The uncertainty gets wide underlying functions, F and H , as well as exact values of where the data points are sparse. k k all parameters(Φk, Ψk, etc.) of each colored noise. Since the augmented EKF is complicated or requires more computa- tion, there is a question regarding why the augmented EKF is considered here. See Figure 2 for an answer.

Figure 1. GP only vs. GPC with improper parametric model

GP GPC Mean RMS error 0.5989 0.2865

Even though the GPC are learned by using an improper parametric model that has a 10% modeling error, the RMS error of the GPC is even smaller than that of the GP only. Figure 2. Standard EKF vs. Augmented EKF 5 The estimate errors of the standard EKF are not bounded algorithm of GPC-EKF approach to reliably estimate state is within 2 − σ uncertainties, and they are much bigger than the introduced. The various simulations, including Monte Carlo estimate errors of augmented EKF. Furthermore, the figure results, show why augmented filter is required here and why can give information on the existence of color-noise as well as the color noise should be handled in the GP-based filters. The on the significance of the color of noisy signals. For example, performance of a standard EKF that ignores the color of noisy if the color noise is not remarkable, the two filters may signals as well as of an augmented EKF that compensates the perform similarly. Thus, even if there is no modeling error colored noise are verified based on each estimation error. As and all parameters are perfectly known, then the augmented expected, the accuracy of GPC-EKF increases significantly EKF is necessary in this case. rather than the performance of the enhanced GP-based filter with improper white noise models for colored noise systems. GPC-EKF with Improper Parametric Models Lastly, the results validate how the GPC-EKF method is This simulation presents an execution of the GPC-EKF to the robust to an improper model and to even a white-noise system example system of the colored noise problem. as well.

If an engineer is uncertain of the underlying functions and Future work will examine the application of GPC-EKF ap- ignores the color noise, since the engineer is not even aware proach to an auto regressive or a moving average (ARMA) of whether each signal has colored noise, EGP-based EKF, signal that is considered as the parametric model of color introduced in Section 3., can be run for the given system. noise. In addition, other novel GPs can directly be incor- We can call this approach EGP-White. However, the GPC- porated into this approach. For example, heteroscedastic EKF can deal with the improper underlying models as well as GPs [14] can allow accurate inference for input-dependent the totally unknown parameters of colored noise in the same noise models and the sparse spectrum GPs [15] can be used system. to reduce the computational complexity of online-fashion GPs. Let’s assume that the approximate parametric model, Fˆ, has 10% modeling errors at each time step. From Eq. (12), REFERENCES     1 ∆t ˆ 1 ∆t [1] A. Bryson and D. Johansen, “Linear filtering for time- Fk = 0 1 ⇒ Fk = 0 0.9 varying systems using measurements containing col- ored noise,” IEEE Transactions on Automatic Control, vol. 10, no. 1, pp. 4–10, 1965. Figure 3 illustrates, under this circumstance, that the GPC- [2] X. Shenshu, Z. Zhaoying, Z. Limin, X. Changsheng, EKF outperforms EGP-White. In other words, the GPC- and Z. Wendong, “Adaptive filtering of color noise EKF will in fact result in a significant increase in estimation using the kalman filter algorithm,” in Instrumentation performance. The resulting estimation errors of GPC-EKF and Measurement Technology Conference, 2000. IMTC method are bounded by 2−σ uncertainties, and unfortunately, 2000. Proceedings of the 17th IEEE, vol. 2. IEEE, the estimate errors of EGP-White diverge as the simulation 2000, pp. 1009–1012. progresses. By comparing Figure 3(a) with Figure 2, the GPC-EKF is shown here to be suitable to the colored-noise [3] F. Le Chevalier, Principles of radar and sonar signal problem with unknown parameters. processing. Artech House, 2002. [4] J. D. Gibson, B. Koo, and S. D. Gray, “Filtering of Now we are curious about whether the GPC-EKF method can colored noise for speech enhancement and coding,” work as well in white-noise systems since we do not typically IEEE Transactions on Signal Processing, vol. 39, no. 8, know if a system has colored noise. Let’s assume a system pp. 1732–1742, 1991. where Φ and Ψ are zeros, i.e., the system has white noises. The next simulation shows how the GPC-EKF method is [5] D. C. Popescu and I. Zeljkovic, “Kalman filtering of robust and suitable to both the color-noise system and white- colored noise for speech enhancement,” in , noise system. Speech and Signal Processing, 1998. Proceedings of the 1998 IEEE International Conference on, vol. 2. IEEE, As shown in Figure 4, even if true signals do not have color- 1998, pp. 997–1000. noise, the GPC-EKF is working here as well. Obviously, [6] J. Liu and Z. Deng, “Self-tuning weighted measure- GPC-EKF estimation errors are smaller than the position ment fusion kalman filter for arma signals with colored sensor errors. That is why an appropriate estimator is needed noise,” Applied Mathematics & Information Sciences, to design although some sensors offer noisy signals of state. vol. 6, no. 1, pp. 1–7, 2012. Lastly, by comparing Figure 4 with Figure 3(a), the GPC- EKF can be considered regardless of whatever forms of [7] R. S. Bucy and P. D. Joseph, Filtering for stochastic noise. processes with applications to guidance. American Mathematical Soc., 1987, vol. 326. [8] A. BRYSON, JR and L. Henrikson, “Estimation using 7. CONCLUSION sampled data containing sequentially correlated noise.” Journal of Spacecraft and Rockets, vol. 5, no. 6, pp. This paper describes that filtering using Gaussian process 662–665, 1968. models for colored noise systems (GPC-EKF) is applicable to the color-noise problem when the characteristics of the noise [9] E. B. Stear and A. R. Stubberud, “Optimal filtering for are unknown and the underlying parametric functions are gauss-markov noise,” International Journal of Control, uncertain. Without information on the color noise, the GPC vol. 8, no. 2, pp. 123–130, 1968. model demonstrates how Gaussian process regression can be [10] D. Simon, Optimal state estimation: Kalman, H infinity, used to learn the colored noise systems. Furthermore, the and nonlinear approaches. John Wiley & Sons, 2006. 6 (a) Estimation error by Monte Carlo Method

(b) Mean Square Error

Figure 3. EGP-White vs. GPC-EKF

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BIOGRAPHY[

Kyuman Lee received his B.S. degrees in Aerospace Engineering from Korea Advanced Institute of Science and Tech- nology in 2007 and a M.S. in Aerospace Engineering in 2013 from Georgia In- stitute of Technology. He is currently a Ph.D student at Georgia Institute of Technology. His current research ac- tivities include adaptive estimation for vision-aided navigation of UAVs.

Eric N. Johnson received his B.S. de- gree in Aeronautics and Astronautics Engineering from University of Wash- ington in 1991; a M.S. in Aeronautics at the George Washington University in 1993; a S.M. in Aeronautics and As- tronautics from Massachusetts Institute of Technology in 1995; and a Ph.D. in Aerospace Engineering from Georgia Institute of Technology in 2000. He is currently a Lockheed Martin Associate Professor of Avionics Integration at Georgia Institute of Technology. His interests include: Fault Tolerant Estimation and Control as enabled by advancements in digital processing and new sensor technolo- gies; Aerospace Vehicle Design, particularly as enabled by digital avionics (micro-air vehicles, unmanned aerial vehi- cles (UAVs), autonomous vehicles, and new configurations); Digital Avionics Systems, including architectures and redun- dancy management; and Modeling/Simulation, including im- proving aerospace system design and integration processes.

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