Measure of Nonlinearity for Stochastic Systems

Transcription

1 Measure of Nonlinearity for Stochastic Systems X. Rong Li Department of Electrical Engineering, University of New Orleans New Orleans, LA 70148, U.S.A. Abstract Knowledge of how nonlinear a stochastic system is important for many applications. For example, a full-blown nonlinear filter is needed in general if the system is highly nonlinear, but a quasi-linear filter (e.g., an extended Kalman filter) is sufficient if the system is only slightly nonlinear. We first briefly survey various measures of nonlinearity for different representations of problems. Unfortunately, the conclusion of our survey is that a good quantitative measure of nonlinearity for stochastic systems is still lacking and existing measures designed for other applications are not suitable here. In view of this, we propose a general measure of nonlinearity for stochastic systems based on the idea of quantifying its deviation from linearity. It can be interpreted as a measure of the mean-square distance between a point (i.e., the given nonlinear system) and a subspace (i.e., the set of all linear systems) in a functional space. Properties and computation of this measure are explored. A numerical example is given in which the measure is applied to a target tracking problem. Keywords: measure of nonlinearity, degree of nonlinearity, stochastic system, nonlinear filtering I. INTRODUCTION A nonlinear problem in any area is usually much more difficult to deal with than a linear one. And the difficulty increases with the degree of nonlinearity (DoN). Although it is usually not hard to determine if a system is nonlinear, merely knowing that the system is nonlinear is not enough it is desirable to know how nonlinear the system is, that is, to quantify the nonlinearity of a problem. Such quantitative information about the problem reveals the root of the difficulty inherent in dealing with the problem, especially when comparing different problems. Take nonlinear filtering as an example. A number of techniques different in applicability and computational complexity (e.g., extended Kalman filters, unscented filters, and particle filters) have been developed. Knowing the DoN of the system would help the user make an informed choice of the nonlinear filters. Also, it is a common practice to approximate a nonlinear system by a linear one, which can significantly simplify the analysis. This method, however, works well only if the nonlinearity is weak and so an appropriate quantitative measure is needed. In this paper, we first provide a review of measures of nonlinearity (MoN) since knowledge of them is limited in the information fusion community. To our knowledge, the first study of MoN was reported in [4], [11] in 1960s, and later a number of MoNs have been proposed for different applications. Most existing measures form two classes: (a) Research supported in part by ONR-DEPSCoR through Grant N and LEQSF-EPS(2012)-PFUND-301. measure the nonlinearity as separation between the nonlinear function and its closest (e.g., best) linear one, and (b) use the curvature of a function at some point as a nonlinearity measure. We outline the pros and cons of these measures and conclude that they do not apply to or work well with stochastic systems because they were not intended for such applications. Then, we propose a general idea for MoN as measuring nonlinearity by how far it is from linearity. Specifically, the MoN of a nonlinear function is defined as the function s deviation from the set of all linear functions, rather than from a specific linear function. In fact, in a functional space each function is a point and the set of all linear functions forms a subspace. The deviation from linearity can be measured quantitatively by closeness between a point (i.e., the given nonlinear function) and the subspace (i.e., the set of all linear functions). This definition is conceptually more appealing than existing ones for example, it is more natural, promising, and is of a global nature. Without this important recognition, deviation from linearity can be understood in various ways. This definition will be the foundation for a more versatile measure of nonlinearity to be presented in a forthcoming paper, with which the entire linear space is reduced to a single point. Using the most popular definition of a distance between a point and a set the greatest lower bound of the distances between the point and each point in the set in this paper we propose a simple MoN for stochastic systems. It can measure the nonlinearity of both the dynamic model and the measurement model jointly. Although the computation of this measure reduces to evaluating the closeness between the nonlinear function and its probabilistically optimal linear approximation, the interpretation is different: this point-wise closeness (between two functions) is used to represent the closeness between the given nonlinear function and the subspace of linear functions. With this concept in mind, other appropriate closeness measures may also be used for different needs or applications. For example, multiple (e.g., typical) points, rather than the single best point, in the set of linear functions may be used to compute the MoN. Also, the best linear approximation of a nonlinear function is obtained by stochastic linearization. The random effect of the system state can be accounted for by mathematical expectation. Our proposed measure has several nice properties: (a) It is a relatively neutral measure [22], which is usually preferred to the existing worst-case measures; (b) the measure is invariant under invertible affine transformations of the independent variable; (c) it is a global measure, and can be readily adapted 1073

2 to serve as a local measure if desired; (d) the measure does not require evaluation of a derivative; (e) simple numerical procedures can be used to compute the measure if analytical solutions are difficult to obtain. The paper is organized as follows. A brief review of existing MoN is presented in Sec. II. A general definition of MoN is proposed and an MoN for stochastic systems is presented in Sec. III. Computation of the measure is addressed in Sec. IV. A simulation example is given in Sec. V. Conclusions are made in Sec. VI. II. EXISTING MEASURES OF NONLINEARITY A. Measures of Deviation from Closest Linear One Beale s pioneering work [4] of exploiting MoN in the context of regression analysis was the first serious work on MoN known to us. It measures the separation between a nonlinear function g and the linear function that is closest to g based on a Taylor series expansion (TSE). It is a local measure since g was linearized around some point x 0 by the firstorder TSE, and the MoN is defined as the (normalized) total separation between g and its linear approximation evaluated on multiple sample points in a small neighborhood of x 0.This local measure is somewhat heuristic, but its underlying idea of using the separation between the nonlinear function and a linear approximation forms the basis for all the measures in this class. An MoN was proposed in [6] for a nonlinear control system as a function g, givenby N =inf L g (1) L L It quantifies the difference between g and its best linear approximation L within an admissible set L of linear functions. It is superior to Beale s measure conceptually in that the difference is quantified in a functional norm and the linear approximation L is the one that minimizes this measure, rather than from TSE at a specific point. Any appropriate norm may be used. Similar measures were also proposed in [27], [8] for different applications. Instead of using only one linear approximation, [33] proposed to use two linear systems to capture the nonlinearity of a single-input single-output (SISO) system. It defines the MoN as the larger of the distances from g to its greatest-lower and smallest-upper linear boundary functions. It was shown that any two nonlinear functions g 1 and g 2 related by g 1 = g 2 + L always have the same value of this MoN for any linear function L. This was justified by arguing that adding a linear function to a nonlinear function should not alter the MoN value, which is controversial and, in our opinion, questionable. Two major problems limit the applicability of this measure are: a) it was designed for the SISO system, and its extension for a general multi-input multi-output (MIMO) system is not trivial; b) the two linear-boundary functions need not exist and can be hard to find even if the system is bounded-input-bounded-output stable. More recently, [14] proposed a relative measure of nonlinearity for deterministic control systems. Basically, it is a normalized difference between the system s nonlinear inputoutput mapping g and its best linear approximation (i.e., normalized version of (1)), measuring the DoN in the output w.r.t. the control input u and initial state x 0 : N(t) = inf L sup u,x N 0 inf x L 0 { L(u, x L 0,t) g(u, x N } 0,t) g(u, x N 0,t) where L(u, x L 0,t) is a linear approximation of the nonlinear function g(u, x N 0,t) in question, and xl 0 and xn 0 are the initial states of the linear and nonlinear systems, respectively. The best L is the one that has the minimum normalized difference to g(u, x N 0,t) in an admissible set L of linear operators. Direct computation of this measure is usually infeasible, rendering a numerical solution necessary. Rather than calculating this N(t), [13] proposed to compute the upper and lower bounds of a similar MoN based on a functional expansion, which relies on the Laplace-Borel transform and the shuffle product. Measures in this class, except Beale s measure, are derivative free they do not require evaluating function derivatives which is desirable for many applications where derivatives are hard to evaluate or even non-existent. However, several drawbacks impede their application to stochastic systems: (a) Most of them are for the worst case the overall MoN is represented by the worst case with the least favorable input and system state. In this sense, these measures are pessimistic. This may fail to reveal faithfully the typical degree of nonlinearity of the system in a normal operation. For a practical problem, it is usually rare for the system to have such an extreme input and/or state. Hence, the system is not as nonlinear as these pessimistic measures indicate in most situations. (b) Many of these measures (e.g., (2)) are difficult to compute, especially for a MIMO system. Their computation amounts to minimax optimization, which can be converted to a nonlinearly-constrained nonlinear minimization problem. As generally expected, the computation would be complicated and only local minima can be reached. (c) These measures do not account for randomness if they are applied to a stochastic system. The gap metric [7], [9] between two linear systems L 1 and L 2 was also proposed as the basis for an MoN in [34]. The idea is to linearize the nonlinear function and define the MoN as the gap metric between this linear approximation and another appropriately selected linear system. This proposal is not appealing for several reasons. First, using the gap metric, which is meant for linear systems, to measure the difference between nonlinear systems by linearization is farfetched. The result may rely heavily on the linearization methods used and the measure works only for functions with weak nonlinearity, since otherwise the linear approximation cannot represent the nonlinear function well and the results can hardly reveal the actual difference. Further, it makes little sense for an MoN to represent the nonlinear system by its linearized system, since it is the nonlinearity that is of interest. Besides, the information lost by the linearization is not accounted for in this measure. (2) 1074

3 B. Curvature-Based Nonlinearity Measures Another path to MoN is based on curvatures studied in differential geometry. [2], [3] proposed a curvature-based MoN for a regression model. For a function z = g(x), themon proposed is determined by its first and second derivatives ż l and z l at x along some direction l. They have clear physical interpretations: they are the instantaneous velocity and acceleration vectors, respectively, of the curve g(x+cl) at point x, wherec is an independent scalar variable. Clearly, the acceleration vector z l usually does not lay in the tangent plane at x. A curvature-based MoN at x is defined as N(x) =max l N l (x), wheren l (x) = z l / ż l 2 is the MoN in the direction l. Decompose z l into two orthogonal vectors z l I and z l N, which are within and orthogonal to the tangent plane, respectively. Then, define the intrinsic curvature and MoN as Nl I(x) = zn l / ż l 2 and N I (x) = max l Nl I (x), and define the parameter-effects curvature and MoN as Nl P (x) = z l I / ż l 2 and N P (x) = max l Nl P (x). The intrinsic one does not depend on the parametrization, but the parametereffects one does. Scaling invariant versions of the relative curvature and MoN were also proposed in [2]. The calculation of the intrinsic and parameter-effects curvatures was studied in [1]. These curvature-based measures (along with some other questionable measures) were applied to target tracking with either a nonlinear dynamic model [30] or a nonlinear measurement model [28], [15] for bearing-only tracking [25], ground moving target indicator radar tracking [23], and video tracking [24]. [26] studied the filter performance w.r.t. the MoN by simulation. The curvature-based measures have the following pros and cons relative to those based on deviation from linear approximation: (a) They are easier to compute given the derivatives. (b) They have clearer physical and geometric interpretations, as explained in [2], and the intrinsic curvature is invariant to parametrization, which is not true for other measures. (c) Similar to Beale s measure, they are local measures, which may be good for measuring DoN locally but not for the overall DoN. Since they are also based on quantities at a specific (expansion) point, they only measure nonlinearity within a small neighborhood of that point. Extensions to measuring the overall nonlinearity of the function is not straightforward. A quick-and-dirty solution is to use the worst-case expansion point to represent the overall nonlinearity. This is both theoretically crude and computationally demanding. (d) They need to evaluate derivatives, which may be difficult or impossible due to non-existent derivatives for many practical applications, especially discrete-valued problems. (e) They are also for the worst case (i.e., the worst direction l). As mentioned before, this is pessimistic and may differ significantly from the typical or normal case. (f) They are actually the ratio between the second and first derivatives of the nonlinear function g. Higher order terms are ignored, which cannot be well justified. In fact, all the terms higher than the first-order contribute to nonlinearity. (g) In addition, nonlinearity measures have been applied to target tracking with either a nonlinear dynamic model or a nonlinear measurement model, but not both; that is, it is still unclear how to measure nonlinearity of dynamic and measurement models combined. C. Tests of Nonlinearity Exploring nonlinearity has also been studied in time series analysis [12], [32], [31] and clinical tests [19], where several tests for the nonlinearity of the data have been proposed. A popular one is the surrogate data test [35], usually for the null hypothesis the data are from a linear Gaussian model. Surrogate data, which retain some statistical properties of the original data (e.g., power spectrum or magnitude distribution), are generated according to the null hypothesis. A significance test with level α is implemented, where discriminating statistics [29] are computed based on the original and surrogate data. However, the result of the test is binary reject or don t reject the null hypothesis. So, it is not easy to quantify the DoN in general, although the significance level reflects this degree partially, which, however, also depends on other factors. Additionally, the test relies on the deviation from a linear Gaussian model, not just from a linear model. Hence, the data distribution, rather than just the nonlinearity, would also affect the test results. D. Conclusion Measures reviewed above were meant for deterministic systems or functions of unknown but non-random parameters. Their direct application to stochastic systems would ignore the random effect of the system state. In other words, for a stochastic system the MoN should depend on not only the functional form but also the distribution of the state x. This is similar to the fact that the MoN of a deterministic function g(x) also depends on the range of the independent variable x. Stochastic systems with the same functional form but different distributions of x should have different MoN values. For example, a function should be more nonlinear if x is more likely to be in the highly-nonlinear region of the function. Given the form of a nonlinear stochastic system, a scenario affects the nonlinearity of the problem mainly through the distribution of x. In summary, nonlinearity measures particularly suitable for stochastic systems are still lacking. Development of such measures deserves more attention and effort in many areas, such as nonlinear filtering. III. PROPOSED MEASURE FOR STOCHASTIC SYSTEMS Consider a discrete-time nonlinear stochastic system, x k+1 = f k (x k )+u k + w k (3) z k = h k (x k )+v k (4) where x k is the (random) state, u k is a deterministic and known additive control input, and w k and v k are zero-mean white process noise and measurement noise independent of the initial state x 0. It is of our interest to measure the nonlinearity of the system (3) and (4) jointly. Stacking (3) and (4) together yields y k = g k (x k )+U k + η k (5) 1075

4 where y k =[x k+1, z k ], g k (x k )=[f k (x k ), h k (x k ) ] U k =[u k, 0 ], η k =[w k, v k] Since the nonlinearity between y k and x k is of the most interest and y k is linear in the control input U k and noise η k, we focus on the nonlinearity of function g k. The idea of measuring the deviation of the nonlinear function g k from the best linearity is still applicable to a stochastic system. However, we give a more general and solid definition of deviation from linearity. Denote by the functional space F the set of all functions (with a fixed dimension) of a random variable x with a specified distribution. Partition F into two subspaces the set L of all linear functions and the set G of all nonlinear functions. Given a nonlinear function g k G, its MoN can be defined as the deviation of g k from L (rather than a point L L), that is, how far a point is from the subspace L (rather than a point in L). This recognition is important for further development of useful measures for different applications since this deviation can be defined in various ways as needed. The most widely used deviation measure in this case is the greatest lower-bound of the distances between g k and each point in L. We proceed by following this definition because of its popularity and simplicity. Actually, it leads to the measure (1) if it is applied to a deterministic system with a point-wise distance defined in a functional norm. However, other appropriate choices are also possible for a specific problem at hand, including a combination of distances between g k and multiple points in L. Let the closeness between two points (i.e., functions) g 1 and g 2 in F be J. For stochastic systems it should account for the random effect of x. Therefore, a natural choice is J(g 1,g 2 )=(E[ g 1 (x) g 2 (x) 2 2 ])1/2 (6) So the closeness between g k and L (i.e., the greatest lowerbound of the distances between g k and L k L)is J k = inf L k L J(L k,g k ) = inf L k L (E[ L k(x) g k (x) 2 2 ])1/2 (7) where the expectation E is w.r.t. the random variable x k,and L is the set of all linear (actually affine) functions L(x) = Ax + b that have the same dimension as g k. J k can serve as an unnormalized MoN. We define the following normalized version J k ν k = (8) [tr(c gk )] 1/2 as the measure of nonlinearity (MoN), where C gk is the covariance matrix of g k (x). The expectations in (7) and (8) are assumed to exist. This measure is not applicable to the rare functions for which the expectation does not exist. Remark 1: Although this MoN reduces to the deviation of g k from its closest linear function ˆL k, the interpretation is different: the closeness between g k and ˆL k is used to represent the deviation of g k from the linear subspace L. The L 2 -norm is chosen for its simplicity and popularity. Actually, J is simply the square root of the mean-square error (mse) of the mse-optimal stochastic linear approximation ˆL k : J k =[mse(ˆl k )] 1/2. Since mse is perhaps the most widely used estimation criterion, many existing results can be applied to compute J readily. Other appropriate vector norms are also optional. For example, a (positive-definite) weight matrix W can be included: J(L k,g k )=(E[(L k (x) g k (x)) W (L k (x) g k (x))]) 1/2 which is more general than (6) since the weight of each component of g k is considered. This introduces no theoretical difficulty it only makes the computation more involved. Hence, for brevity of the presentation, we only consider the simpler form (6). Remark 2: This MoN is time varying in general if the system (or the distribution of x k ) is time varying. Remark 3: This MoN is derivative free, which is preferable to the curvature-based MoN for many applications. It may be evaluatedevenwheng k does not have an analytical form. Remark 4: The unnormalized MoN J is an absolute measure, which quantifies the absolute deviation of g k from L. This deviation can be intuitively understood as the pure nonlinear part of g k that cannot be accounted for by linear functions. The MoN ν quantifies the portion of this nonlinear part in g k. It has a standard range of [0, 1], as shown later in Sec. IV. Remark 5: Clearly, ν =0implies that g k is linear almost everywhere, while ν = 1 implies that ˆLk = 0, meaning roughly that g k contains no linear component at all. Remark 6: The expectation in (7) serves several purposes. First, it accounts for the random effect and the specific distribution of x k. As mentioned above, different distributions should in general have different MoN values. Second, it results in a global measure rather than one that is only for a small neighborhood of some point. Further, it leads to a relatively neutral measure, as opposed to the existing pessimistic ones that consider only the worst case. Remark 7: It is easy to verify that J(g) = J(g + L), meaning that adding a linear function L to the nonlinear function g does not alter our absolute, unnormalized measure J. This makes sense because the absolute amount of a function s nonlinear part will not be affected by adding a linear function. However, our MoN ν is and should be altered since the relative portion of the nonlinear part changes due to the normalization. Note that adding a constant to the nonlinear function alters neither J nor ν. Remark 8: J and ν are invariant w.r.t. any invertible linear (actually affine) transformation L of x (i.e., x = L(s) and L 1 exists). This can be shown as follows. It is clear that for agivenl k (x), E[ L k (x) g k (x) 2 2]=E[ L k (L(s)) g k (L(s)) 2 2] (9) So, it suffices to show that ˆL k (x) minimizing (7) also minimizes E[ L k (L(s)) g k (L(s)) 2 2], thatis, ˆL k (x) =ˆL k (L(s)) = arg min L k L E[ L k(l(s)) g k (L(s)) 2 2] (10) 1076

5 First, ˆL k (L(s)) is linear in s and hence ˆL(L(s)) L. Assume there exists a linear function Ľk(s) such that E[ ˆL k (L(s)) g k (L(s)) 2 2] >E[ Ľk(s) g k (L(s)) 2 2] Then E[ ˆL k (x) g k (x) 2 2 ] >E[ Ľk(L 1 (x)) g k (x) 2 2 ] Since Ľk(L 1 (x)) L is linear in x, this contradicts the assumption that ˆL k (x) is the solution of (7). So, (10) holds and we have E[ ˆL k (x) g k (x) 2 2 ]=E[ ˆL k (L(s)) g k (L(s)) 2 2 ] That is, J and ν are invariant under invertible linear (affine) transformation. Remark 9: Although our MoN is a global one, it can be readily modified to serve as a local measure if so desired. If the DoN in a neighborhood X around a point x 0 is of interest, simply replacing the unconditional expectation in (7) by the one conditioned on {x X}makes the measure a local one. This is still superior to the Taylor series expansion based measures since all the points in X, rather than only the expansion point x 0, are considered. Remark 10: In general, the analytical solution ˆL k of (7) may be difficult to obtain, and the expectation requires knowledge of the distribution of x k and may be hard to evaluate exactly for many applications. So, numerical techniques may be necessary. IV. COMPUTATION OF PROPOSED MEASURE In this section, the subscript k is dropped for brevity if no ambiguity arises. The exact solution ˆL of (7) can be derived from the first-order necessary condition J(L, g) =2E[Ax + b g(x)] = 0 b J(L, g) =2E[(Ax + b g(x))x ]=0 A These two equations have the solution  = C gxcx 1, ˆb = ḡ(x)  x, where( ) E[ ] is the mean, and C x = cov(x) and C gx = cov(g, x) are covariance matrices. So, ˆL(x) =ḡ(x)+c gx Cx 1 (x x) (11) (This is indeed the solution, as can be verified by the secondorder condition.) In essence, it is the stochastic linearization of g in terms of mse, and can be viewed as the linear MMSE estimator of g using observation x [21]. Plugging (11) into (7) yields J =[mse(ˆl)] 1/2 =(E[(ˆL(x) g(x)) (ˆL(x) g(x))]) 1/2 =[tr(e[(ˆl(x) g(x))(ˆl(x) g(x)) ])] 1/2 =[tr(c g C gx Cx 1 C gx )]1/2 and the MoN is ν = 1 tr(c gxcx 1 tr(c g ) C gx) (12) which has the range [0, 1] since tr(c g ) tr(c gx Cx 1 C gx) 0 for every g and x. This standard range is perfect for a measure. Remark 11: It is clear that evaluating {C g,c gx } is the key to computing ν. Numerical methods (e.g., Gaussian quadrature) can be applied if the integral is difficult to evaluate analytically. Approximating {C g,c gx } by their sample versions is also optional. Remark 12: In most applications, even if the prior distribution of the initial state x 0 is known, the exact {C g,c gx } is difficult to evaluate analytically due to the nonlinearity and dynamics of the system. Nevertheless, a sample representation of {C g,c gx } can be obtained numerically (e.g., by Markov chain Monte Carlo (MCMC) methods [10], [5]). Random samples can be drawn from the initial distribution and propagated forward to time k, resulting in a sample approximation of {C g,c gx }. Admittedly, to achieve a good accuracy this requires a large sample and hence is computationally demanding, but it is simple and can be done offline and in parallel. Remark 13: Our MoN can also be calculated online and conditioned on the set of available measurements z k [z 1,,z k ] or z k 1 : replace the unconditional expectation by the one conditioned on z k or z k 1, leading to the MoN conditioned on z k or z k 1 at time k. If the conditional expectation is difficult to compute, as for many applications, {C g,c gx } may be approximated by unscented transformation (UT) [16], [17], which is accurate at least to the second order. V. SIMULATION EXAMPLE A numerical example of target tracking is presented in this section. In this example, a target is taking a planar constant turn (CT) with a known turn rate. We consider two models for this motion and compare their measures of nonlinearity. Further, we apply a nonlinear filter an unscented filter (UF) [16], [18] to both models and compare their performance to reveal the impact of the degree of nonlinearity on estimation performance. The target state is chosen as either x c k =[x, y, ẋ, ẏ] k with the position (x, y) and the velocity (ẋ, ẏ) or as x p k =[x, y,s,φ] k with the position and the velocity (s, φ), wheres and φ are the target speed and heading angle, respectively. We consider two cases having the following CT models with the known turn rate for x c and x p respectively: sin T 1 cos T 1 0 x c k+1 = 1 cos T sin T cost sin T xc k + wc k 0 0 sint cos T (13) x+(2/)s sin(t)cos(φ + T/2) x p k+1 = y+(2/)s sin(t) sin(φ + T/2) s + w p k (14) φ + T k 1077

6 y position Position root mean square error x position (a) Target trajectory (b) Position RMSE Unnormalized MoN Measure of nonlinearity (c) Unnormalized MoN (d) MoN Conditional unnormalized MoN Conditional measure of nonlinearity (e) Average conditional unnormalized MoN (f) Average conditional MoN Figure 1: Nonlinearity measures and filter performance. Figs. 1(a) and 1(b) are, respectively, the target trajectory (one realization) and the position root-mean-square error (RMSE) of the unscented filters. Figs. 1(c) and 1(d) are the nonlinearity measures based on MCMC. Figs. 1(e) and 1(f) are the average nonlinearity measures conditioned on z k computed by unscented transformation. 1078

7 See [20] for more details. Both cases have the same nonlinear measurement model x2 +y 2 z k = x x2 +y 2 k + v k which measures the range and the direction cosine of the azimuth angle in the plane. Clearly, model (13) is preferred for this case since it is linear, while model (14) is nonlinear (and approximate). Denote by m c and m p the system models for x c and x p, respectively, along with the nonlinear measurement model. The UF was initialized by ˆx c 0 N(x; xc 0,Pc 0 ) x c 0 =[ 500, 500, 5, 8.7] P c 0 = diag(103, 10 3, 1, 1) ˆx p 0 N(x; xp 0,Pp 0 ) x p 0 =[ 500, 500, 10,π/3] P p 0 = diag(103, 10 3, 1, 10 2 ) which leads to approximately the same initial estimates (see Fig. 1(b)). The process and measurement noises have covariances Q c = diag(1, 1, 0.01, 0.01), Q p = diag(1, 1, 0.01, 0.001), andr = diag(100, 0.01). The target trajectory (in one realization) and tracking performance (from 1,000 Monte Carlo runs) are given in Figs. 1(a) and 1(b), respectively. Clearly, model m c outperforms model m p because model m p is more nonlinear than m c due to the additional nonlinearity contributed by the dynamic model of (14). This is also confirmed by all measures in Figs. 1(c) 1(f). Before any measurement is collected, the nonlinearity of the systems can be evaluated with the distributions of the initial states. The unnormalized MoN J and the MoN ν, computed by the MCMC method with 10, 000 sample points (from the initial time to k), are given in Figs. 1(c) and 1(d). They vary periodically because of the periodicity of the trigonometric functions involved in models m c and m p. The degrees of nonlinearity of these two models are low since MoN < 2%. Figs. 1(e) and 1(f) show the average (over 1,000 MC runs) unnormalized MoN and MoN at each time k conditioned on observations z k, where UT is applied to approximate the quantities needed in the measures. The conditional MoN is superior to the unconditional one since the conditional case is more tailored to the situation and thus more indicative than the prior (i.e., global average). The periodical pattern disappears for model m c but is still visible for model m p since it arises mainly from the dynamic model in m p. For this example, since both models have only weak nonlinearity, the difference in the filtering results (e.g., RMSE) is minor, and use of a more powerful nonlinear filter (e.g., a particle filter) than the UF would not make much difference. The MoN of a more nonlinear case of video tracking is presented next, where a target is making a constant turn [20] with a turn rate unknown to the tracking filter. Here, the target state is x =[x, ẋ, y, ẏ, ] and the system dynamic model is [20] x + sin(t) ẋ 1 cos(t) ẏ cos(t)ẋ sin(t)ẏ x k+1 = 1 cos(t) ẋ + y + sin(t) ẏ sin(t)ẋ +cos(t)ẏ k + w k In a case of video tracking of a target in nearly consturn turn with an unknown turn rate, the values of our unnormalized MoN and (normalized) MoN conditioned on z k are approximately in the range of [2, 12] and [0.1, 0.3]. This shows that the degree of nonlinearity for the system is significant. Therefore, different nonlinear filters exhibit significant performance differences and the EKF even diverges. (These results are not shown due to space limitation.) VI. CONCLUSIONS Measuring the nonlinearity of a stochastic system is an important problem but has not drawn enough attention yet. This is partly reflected in that good measure is not available for stochastic systems. We have proposed a more general and solid definition of the degree of nonlinearity, which is conceptually superior to existing measures. It is the closeness between the nonlinear function and the set of all linear functions. Different closeness measures can be chosen depending on specific needs. By following the most widely used definition of the closeness between a point and a set the greatest lower-bound of the distances between the point and each point in the set, we have developed a nonlinearity measure for stochastic systems. This measure is simple and has many nice properties, but it by no means excludes other appropriate choices. Numerical solutions can be used whenever analytical ones are difficult to obtain. Finally, we emphasize that the developed MoN is applicable to not only nonlinear filtering but also other nonlinear problems. REFERENCES [1] D. M. Bates, D. C. Hamilton, and D. G. Watts. Calculation of intrinsic and parameter-effects curvatures for nonlinear regression models. Communications in Statistics - Simulation and Computation, 12(4): , [2] D. M. Bates and D. G. Watts. 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