IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 28, NO. 1, JANUARY 2017 231
Multiscale Support Vector Learning With ProjectionOperator Wavelet Kernel for Nonlinear
Dynamical System IdentificationZhao Lu, Senior Member, IEEE, Jing Sun, Fellow, IEEE, and Kenneth Butts, Member, IEEE
Abstract— A giant leap has been made in the past coupleof decades with the introduction of kernel-based learning as amainstay for designing effective nonlinear computational learningalgorithms. In view of the geometric interpretation of conditionalexpectation and the ubiquity of multiscale characteristics inhighly complex nonlinear dynamic systems [1]–[3], this paperpresents a new orthogonal projection operator wavelet ker-nel, aiming at developing an efficient computational learningapproach for nonlinear dynamical system identification. In theframework of multiresolution analysis, the proposed projectionoperator wavelet kernel can fulfill the multiscale, multidimen-sional learning to estimate complex dependencies. The specialadvantage of the projection operator wavelet kernel developedin this paper lies in the fact that it has a closed-form expression,which greatly facilitates its application in kernel learning. To thebest of our knowledge, it is the first closed-form orthogonalprojection wavelet kernel reported in the literature. It providesa link between grid-based wavelets and mesh-free kernel-basedmethods. Simulation studies for identifying the parallel modelsof two benchmark nonlinear dynamical systems confirm itssuperiority in model accuracy and sparsity.
Index Terms— Composite kernel, linear programming supportvector regression (LP-SVR), multiscale modeling, nonlinearsystems identification, orthogonal projection operator, raised-cosine wavelet.
I. INTRODUCTION
KERNEL-BASED support vector (SV) learning was orig-inally proposed for solving nonlinear classification and
recognition problems, and it marked the beginning of a new erain the computational learning from examples paradigm [4]–[7].Thereafter, the rationale of SV learning has been successfullygeneralized to various fields such as nonlinear regression, sig-nal processing, integral equation, and path planning [8]–[11].When SV learning is employed for function approximationand estimation, the approaches are often referred to as theSV regression (SVR). As a typical nonparametric kernel
Manuscript received January 18, 2015; revised December 10, 2015; acceptedDecember 20, 2015. Date of publication January 5, 2016; date of currentversion December 22, 2016. This work was supported by Toyota MotorEngineering and Manufacturing North America, Inc.
Z. Lu is with the Department of Electrical Engineering, Tuskegee University,Tuskegee, AL 36088 USA (e-mail: [email protected]).
J. Sun is with the Department of Naval Architecture and Marine Engineeringand the Department of Electrical Engineering and Computer Science, Univer-sity of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]).
K. Butts is with Toyota Motor Engineering and ManufacturingNorth America, Ann Arbor, MI 48105 USA (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNNLS.2015.2513902
learning approach, SVR also provides a promising avenue tononlinear dynamical systems modeling.
Recently, a new line of research has been initiated fordeveloping novel nonstandard kernels for meshless methods insolving the partial differential equations in the realm of com-putational mathematics [12]–[14]. Even the term kernel engi-neering has been coined lately, because efficient algorithmsrequire specially tailored application-dependent kernels [14].On the other hand, although the past decade has witnessedintensive research activities on the kernel-based computationallearning methods, most of the researchers use standard kernelfunctions, such as Gaussian radial basis function (RBF)kernel and polynomial kernel. It was pointed out that thekernel machine with the widely used Gaussian RBF kernelis endowed with the capability little more than a templatematcher, and some inherited drawbacks, such as the locality ofthe kernel function, may result in the inefficiency in represent-ing highly complex functions by kernel expansion [15]. Hence,one objective of this paper is to bridge this gap by exploringthe computational capability of the nonstandard kernel inkernel-based SV learning: the closed-form orthogonal waveletis exploited to construct a multiscale projection operatorwavelet kernel for complex systems modeling and prediction.
It has been revealed in recent studies that in modelinghighly complex nonlinear dynamical systems, the multiscaleSV learning is more capable and flexible over conventionalsingle-scale SV learning [14]. In particular, in [14], it wasemphasized that the success and failure of kernel usagemay crucially depend on proper scaling. As the keystone ofnonlinear SV learning, the construction of kernel functionsplays an important role in fulfilling the efficacious multiscaleSV learning. For identifying nonlinear dynamical systems, thewavelet outperforms the (windowed) Fourier transform dueto its aptitude in capturing very short-lived high-frequencyphenomena, such as transients in signals [16]. Albeit someefforts have been made to develop wavelet kernel functionsfor SV learning [17]–[21], weaving multiresolution waveletanalysis into modern kernel learning is not a trivial task,because almost all known orthonormal wavelets, except forthe Haar and the Shannon, cannot be expressed in the closedform in terms of elementary analytical functions, such as thetrigonometric, exponential, or rational functions [22], [23].Discontinuities in the Haar wavelet and poor time localizationof the Shannon wavelet have limited their applicability inmultiscale modeling problems [23].
232 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 28, NO. 1, JANUARY 2017
The closed-form representation of the Morlet wavelet wasemployed for constructing a single-scale kernel functionin [17] and [18], but the lack of interscale orthogonality andintrascale orthogonality makes it difficult to be used for imple-menting the multiscale kernel learning in a systematic way.In addition, different from the anisotropic stationary multiscalewavelet kernel in [24], where the symmetry requirement isrelaxed and the kernel proposed is translation (but no longerrotational) invariant, the kernel function developed herein canbe viewed as a class of novel finite expansion kernels [25],which is not translation invariant. Inspired by the geometricpurport of conditional expectation [1]–[3], novel projectionoperator wavelet kernels are developed for linear programmingSV learning, and it excels in multiscale learning and modelingfor complex nonlinear dynamical systems.
In the realm of nonlinear dynamic system identification,the nonlinear autoregression with exogenous (NARX) inputmodel is widely used for representing discrete-time nonlinearsystems, and the regressor for the NARX model consists oftwo parts: 1) an autoregressive (AR) part and 2) a moving-average (MA) part. The mathematical description of theNARX model is as follows:yn = f (yn−1, yn−2, . . . , yn−P , un, un−1, . . . , un−Q+1) (1)
where un and yn are the input and output to the system at timeinstant tn , and the vectors yn−1 = [yn−1, yn−2, . . . , yn−P ]T
and un = [un, un−1, . . . , un−Q+1]T are the AR and MA parts,respectively. The AR part is a window of past system outputswith output order P , and the MA part is a window of pastand current system inputs with input order Q. The NARXmodel (1) is also called the series–parallel model, because thesystem and model are parallel with respect to un but in serieswith respect to yn .
Essentially, the identification of the NARX model canbe formulated as a nonlinear function regression problem.It amounts to modeling the conditional expectation of systemoutput yn , given the regression vector consisting of the AR partyn−1 and the MA part un , i.e., E[yn|yn−1,un] [26]. Fromthe geometric standpoint from the linear operator theory,the conditional expectations implement the projections ontolinear subspaces as best approximation in essence [2]. Thisenlightens us to conceive innovative wavelet-based projectionoperator kernels for multiscale SV learning. In this paper,by integrating the multiresolution wavelet analysis and kernellearning systems, a new computational learning approach tononlinear system identification and predictions is developed.To confirm and validate the effectiveness of the proposedlearning strategy, the identification of parallel models ofnonlinear dynamical systems is used as a touchstone for thesimulation study. Contrary to the series–parallel model (1),where the past values of the system input and the systemoutput constitute the regressor, the regressor of the parallelmodel is composed of the past values of the system input andthe model output, that is
yn = f (yn−1, yn−2, . . . , yn−P , un, un−1, . . . , un−Q+1). (2)
Model (2) can be simulated as standalone without usingthe real data as inputs. It is, however, well known that the
identification of a parallel model is much more challengingthan that for a series–parallel model due to the feedbackinvolved in the model [27], [28].
The following generic notations will be used throughoutthis paper: non-boldface symbols, such as x, y, α, . . . , referto scalar valued objects, lower case boldface symbols, such asx, y,β, . . . , refer to vector valued objects, and capital boldfacesymbols, such as K1,K2, . . . , will be used for matrices.
II. CLOSED-FORM ORTHOGONAL WAVELET
IN MULTIRESOLUTION ANALYSIS
Multiresolution analysis is conceptualized by acoarse-to-fine sequence of embedded closed linear subspaces{Vj } j∈Z ⊆ L2(R) as follows.
Definition 1: A multiresolution analysis is a decompositionof L2(R) into a chain of nested subspaces · · · ⊂ V−1 ⊂ V0 ⊂V1 ⊂ · · · Vj−1 ⊂ Vj ⊂ Vj+1 · · · such that the following holds.
1) (Separation) ∩ j∈Z Vj = V−∞ = {0}.2) (Density) ∪ j∈Z Vj = V∞ = L2(R).3) (Self-similarity in scale) f (x) ∈ V0 if and only if
f (2 j x) ∈ Vj , j ∈ Z .4) There exists a scaling function ϕ ∈ V0 whose integer-
translates span the space V0, and for which the set{ϕ(x − k), k ∈ Z} is an orthonormal basis.
Here, j is the index of resolution level. The function ϕ is calledthe scaling function, since its dilates and translates constituteorthonormal bases for all approximation subspaces Vj , andthe orthogonal complement of Vj in Vj+1, i.e., the directdifference W j = Vj+1 � Vj , is called the wavelet space ordetail space. ♦
By successively decomposing the approximation spaces asVj+1 = Vj ⊕ W j , where ⊕ denotes the orthogonal directsum, the functional space L2(R) can be decomposed as anorthogonal direct sum of wavelet spaces of different resolu-tions, i.e., ⊕ j∈Z W j = L2(R). The wavelet function ψ canbe defined, such that {ψ(x − k)}k∈Z is an orthonormal basisof W0, and W j is the span of orthonormal wavelet functions
ψ j,k(x) = 2 j/2ψ(2 j x − k), i.e., W j = span({ψ j,k}k∈Z ).Obviously, the scaling functions and a wavelet areorthogonal whenever the scaling functions are of lowerresolution [29].
Almost all the known orthonormal wavelets, except for theHaar wavelet and the Shannon wavelet, cannot be expressedin the closed form or in terms of simple analytical func-tions. Instead, they can only be expressed as the limit ofa sequence or the integral of some functions [22], [23].This has been a main stumbling block to developing waveletkernels for multiscale kernel learning and modeling. In thispaper, the type-II raised-cosine wavelet, a recently discoveredclosed-form orthonormal wavelet family [22], [30], will becapitalized on to develop innovative and effective projectionoperator wavelet kernels with multiscale and spatially varyingresolution properties for SV learning. Its advantages willbe demonstrated by identifying the parallel models of twobenchmark nonlinear dynamical systems.
As in signal reconstruction technology, the raised- cosinescaling function is derived from its power spectrum
LU et al.: MULTISCALE SV LEARNING WITH PROJECTION OPERATOR WAVELET KERNEL 233
(energy spectrum), defined as follows [22], [30]:
|ϕ(ω)|2 =
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
1, |ω| ≤ π(1 − b)1
2
[
1 + cos|ω| − π(1 − b)
2b
]
π(1 − b) ≤ |ω| ≤ π(1 + b)
0, |ω| ≥ π(1 + b)
(3)
where ϕ(ω) is the Fourier transform of scaling function ϕ(x),i.e., ϕ(ω) = ∫∞
−∞ ϕ(t)e−iωt dt . From (3), it follows that thespectrum of the scaling function involves the positive andcomplex square roots:
ϕ1(ω) =
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
1, 0 ≤ |ω| ≤ π(1 − b)
cos
[ |ω| − π(1 − b)
4b
]
π(1 − b) ≤ |ω| ≤ π(1 + b)
0, |ω| ≥ π(1 + b)
(4)
ϕ2(ω) =
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
1, 0 ≤ ω ≤ π(1 − b)1
2
[1 + ei(1/2b)(ω−π(1−b))
]
π(1 − b) ≤ ω ≤ π(1 + b)
0, ω ≥ π(1 + b).
(5)
The scaling functions can be found using the inverse Fouriertransform as follows:
ϕ1(x) = sinπ(1 − b)x + 4bx cosπ(1 + b)x
πx(1 − (4bx)2)(6)
ϕ2(x) = sinπ(1−b)x + sin π(1+b)x
2πx(1 + 2bx)= cos(πbx)
(1+2bx)sinc(πx).
(7)
The scaling functions ϕ1(x), ϕ2(x) correspond to thetype-I and type-II raised-cosine wavelets. In this paper,the type-II raised-cosine wavelet, derived from the scalingfunction ϕ2(x), is our primary concern. To derive the type-IIraised-cosine wavelet function ψ2(x) from the explicit formof ϕ2(x), one may apply Theorem 1 directly [30].
Theorem 1: Let ℘ be the set of all g ∈ L1(R), such thatg(x) ≥ 0 and supp g ⊂ [−π/3, π/3], and then g(x) is even;∫ υ−υ g(x)dx = π for some 0 < υ ≤ π/3 where supp g =
{x ∈ R|g(x) �= 0}. For each g ∈ ℘, the function ϕ(x) definedby its spectrum
ϕ(ω) = 1
2+ 1
2exp iϑ(ω) (8)
where ϑ(ω) = ∫ ω−π−ω−π g(x)dx is a real band-limited orthonor-
mal cardinal scaling function, and the corresponding motherwavelet function ψ(x) is given by
ψ(x) = 2ϕ(2x − 1)− ϕ
(1
2− x
)
. (9)
♦The rigorous proof of this theorem can be found in [30].
Evidently, the type-II raised-cosine scaling function spectrumϕ2(ω) given by (5) is in the form of (8). Hence, it follows from
Fig. 1. Type-II raised-cosine wavelet function.
Theorem 1 that the type-II raised-cosine wavelet function isin the form of [22]:
ψ2
(
x + 1
2
)
= 1
2πx[1 + 4bx] [sin 2π(1 − b)x + sin 2π(1 + b)x]
− 1
2πx[1 − 2bx] [sin π(1 − b)x + sin π(1 + b)x]
= 2 cos(2πbx)
1 + 4bxsinc(2πx)− cos(πbx)
1 − 2bxsinc(πx) (10)
where the sinc function, closely related to the spherical Besselfunction of the first kind, is defined as sinc(x) = sin x/x .Parallel to the scaling functions, the raised-cosine waveletfunctions ψ1(x) and ψ2(x) are both band-limited functions,and the type-II raised-cosine wavelet function (10) is plottedin Fig. 1.
As eigenfunctions of the Calderón–Zygmund operator [31],orthogonal wavelets have exceptional potential for modelinghigh-dimensional, multiscaled input–output maps. The motherwavelet ψ gives birth to an entire family of wavelets bymeans of two operations: 1) dyadic dilations and 2) integertranslations. Let j denote the dilation index and k representthe translation index, and each wavelet born of the motherwavelet is indexed by both of these indices
ψ j,k(x) = 2 j/2ψ(2 j x − k) (11)
for integer-valued j and k. A wavelet (which, when appro-priately dilated, forms the basis for the detail spaces) mustbe localized in time, in the sense that ψ(x) → 0 quicklyas |x | gets large [29]. Similarly, the family of scaling functions(father wavelets) takes the form of
ϕ j,k(x) = 2 j/2ϕ(2 j x − k). (12)
In the literature, constructing the stationary (translation-invariant) wavelet kernels by defining k(x, y) = ψ(x − y) is apopular approach for multiscale learning [17], [18], [20], [24].In contrast to that, the multiscale wavelet kernels are con-structed based on the notion of the orthogonal projectionoperator in Section III.
234 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 28, NO. 1, JANUARY 2017
III. PROJECTION OPERATOR WAVELET KERNEL
A projection P on an inner product space X is known asorthogonal projection or orthogonal projector if the imageof P and the null space of P are orthogonal, i.e., ifIm(P) ⊥ Null(P) [32], [33]. In multiresolution analysis, everyf ∈ L2(R) can be approximated arbitrarily accurately byits orthogonal projections projVj+1
f on the approximationspace Vj+1, and then the orthogonal projection of f onVj+1 can be decomposed as the summation of orthogonalprojections on Vj and W j [16]
projVj+1f = projVj
f + projW jf. (13)
The complement projW jf provides the details of f that appear
at the scale j but disappear at the coarser scale, and a givenfunction f ∈ L2(R) can be decomposed as the summation ofits projection on the wavelet spaces W j as follows:f =∑
j
projW jf =∑
j
∑
k
c j,kψ j,k =∑
j
∑
k
c j,kψ(2j x −k)
(14)
which is called the wavelet series expansion.By defining the integral operator kernel function Q(x, y) =
∑k ϕ(x − k)ϕ(y − k), the orthogonal projection operator
E j : L2(R) → Vj can be written in terms of kernel functionQ(x, y)
E j [ f ] = projVjf (x) =
∑
k
(∫ϕ j k(y) f (y)dy
)
ϕ j k(x)
=∫
2 j Q(2 j x, 2 j y) f (y)dy. (15)
By defining
Q j (x, y)=∑
k
ϕ j,k(x)ϕ j,k(y)=2 j∑
k
ϕ(2 j x −k)ϕ(2 j y−k)
(16)
the orthogonal projection operator E j can be simplified as theintegral operator with kernel Q j (x, y), that is
E j [ f ] = projVjf (x) =
∫Q j (x, y) f (y)dy. (17)
Analogous to the integral operator representation ofprojVj
f (x), the projection projW jf (x) of f on wavelet space
can also be represented in the form of an integral operator.To this end, Theorem 2 is needed.
Theorem 2 [32], [34]: Let Vj+1 and Vj be the closed linearsubspace of L2(R), and let E j+1 and E j be the orthogonalprojections onto Vj+1 and Vj , respectively. The differenceD j = E j+1 − E j is an orthogonal projection if and onlyif Vj ⊂ Vj+1. The range of D j is W j = Vj+1 � Vj , which isthe orthogonal complement of Vj in Vj+1. ♦
For the difference between the projection operatorE j+1 and E j , i.e., D j = E j+1−E j , it follows from Theorem 2that D j is also an orthogonal projection operator on thesubspace W j and it can be precisely given by:
D j [ f ] = projW jf (x) =
∑
k
(∫ψ j k(y) f (y)dy
)
ψ j k(x)
=∫
2 j K (2 j x, 2 j y) f (y)dy (18)
where K (x, y) =∑k ψ(x − k)ψ(y − k). If one defines
K j (x, y)=∑
k
ψ j,k(x)ψ j,k(y)=2 j∑
k
ψ(2 j x −k)ψ(2 j y−k)
(19)
the orthogonal projection operator D j can be represented bythe following integral operator with kernel K j (x, y):
D j [ f ] = projW jf (x) =
∫K j (x, y) f (y)dy. (20)
As the kernel of the projection integral operator onto W j ,K j (x, y) is called an orthogonal projection operator waveletkernel. Note that it is based on the rigorous multiresolutionanalysis framework and has an analytic expression in terms ofa raised-cosine wavelet function, thereby enabling multiscalelearning by dyadic dilation.
In the presence of irregular localized features, multiresolu-tion learning algorithms are necessary to take local as wellas global complexities of the input–output map into account.In this way, underfitting and overfitting can be avoided simul-taneously in approximating highly nonlinear functions [35].In effect, multiresolution approximation is a mathemati-cal process of hierarchically decomposing the input–outputapproximation to capture both the macroscopic and micro-scopic features of the system behavior [36]. The unknownfunction underlying any given measured input–output datacan be considered as consisting of high-frequency localinput–output variation details superimposed on the compar-atively low-frequency smooth background. At each stage,finer details are added to the coarser description, providinga successively better approximation to the input–output data.The multiscale learning strategy developed herein aims totake advantage of the multiresolution structure of waveletsto provide spatially varying resolution, and for this purpose,the orthogonal projection operator wavelet kernel (22) can beextended to multiscale kernels [25], [37], [38], according toTheorems 3 and 4.
Theorem 3 [33], [34]: Let W j ,W j+1, . . . ,W j+m bea family of closed linear subspaces of L2(R), and letD j , D j+1, . . . , D j+m be the orthogonal projections onW j ,W j+1, . . . ,W j+m , respectively. The finite sum of orthog-onal projection operators
= D j + D j+1 + · · · + D j+m (21)
is an orthogonal projection operator if and only if thesubspaces W j+k (k = 0, 1, . . . ,m) are pairwise orthog-onal. In this case, the range of operator is W j ⊕W j+1 ⊕ · · · ⊕ W j+m . ♦
Hence, the multiscale orthogonal projection operatorwavelet kernel can be constructed as the summation ofK j (x, y) as follows:
K (x, y) =∑
j
K j (x, y) =∑
j
∑
k
ψ j,k(x)ψ j,k(y) (22)
and it follows from Theorem 3 that the integral operatorwith kernel (22) fulfills the orthogonal projection onto thedirect sum of wavelet subspaces at different scales W jmin ⊕W jmin+1 ⊕ · · · ⊕ W jmax . The developed kernel (22) shares asimilar form to the finite multiscale kernel proposed in [25],
LU et al.: MULTISCALE SV LEARNING WITH PROJECTION OPERATOR WAVELET KERNEL 235
which is also called the expansion kernel in [12].However, the construction of finite multiscale kernels in [25] isbased on the superposition of shifts and scales of a sin-gle compactly supported function on grids. In contrast, theraised-cosine wavelet function used in (22) is a band-limitedwavelet [39], which is impossible to be compactly supportedin the time-domain according to the well-known duration-bandwidth theorem (uncertainty principle) [40]. Neverthe-less, the raised-cosine wavelet function is fast decaying intime (contrary to the poor time localization of the Shannonwavelet), which also makes the evaluation of the kernelinexpensive.
The addition of multiple single-scale wavelet kernels atdifferent scales provides multiscale wavelet kernels with moreflexibility than single-scale wavelet kernels [37], [38]. Fora given system, the range of level index j in kernel (22)needs to be tailored to the particular application. Furthermore,for estimating the multivariate dependencies, the multiscaleprojection operator wavelet kernels need to be extended tothe multidimensional space. Based on the fact that a waveletbasis in higher dimensions can be obtained by taking thetensor product of 1-D wavelet bases, the construction of amultidimensional wavelet kernel function can be carried outusing Theorem 4 [41], [42].
Theorem 4: Let a multidimensional set of functions bedefined by the basis functions that are the tensor productsof the coordinatewise basis functions. Then, the kernel thatdefines the inner product in the n-dimensional basis is theproduct of n 1-D kernels. ♦
Hence, as the inner product in high-dimensional space,the multidimensional multiscale orthogonal projection oper-ator wavelet kernel can be constructed as the product of1-D multiscale kernels
K (x, y) =d∏
i=1
∑
j
λ j
∑
k
ψ j,k(xi )ψ j,k(yi ) (23)
where d is the dimension, and x = [x1, x2, . . . , xd ]T and y =[y1, y2, . . . , yd ]T . To illustrate the multiscale characteristics
of the developed closed-form projection operator waveletkernel, an exemplary 2-D projection operator wavelet kernelgiven by
K (x, y) =2∏
i=1
7∑
j=−7
λ j
1∑
k=−1
ψ j,k(xi )ψ j,k(yi) (24)
is plotted in Fig. 2.
IV. LINER PROGRAMMING SVR WITH
COMPOSITE KERNEL
In quadratic programming SVR (QP-SVR), the smoothnessis used as a prior for regularizing the function to ensure thegeneralization, i.e., the smooth functions having few or smallvariations are more likely. However, recent research indicatesthat the smoothness prior alone could be problematic andinsufficient in learning highly nonlinear functions with manysteep and/or smooth variations, which characterize the kindof complex task needed for artificial intelligence (AI) [15].Hence, instead of using the smoothness prior as that inquadratic programming SV learning, linear programmingSVR (LP-SVR) takes an entirely different avenue to build themodel by the regularization technique.
A model identified through the SVR is represented as thekernel expansion on the SVs, which are the data points in aselected subset of the training data [4]–[6]. In other words,the model is represented in a data-dependent nonparametricform. In the endeavor of applying kernel learning strategiesfor identifying nonlinear dynamical systems, the idea of thecomposite kernel was conceptualized and developed for takinginto account the different cause–effect relationships of theAR and MA parts to the NARX model output instead ofassimilating them [43], [44]. The model represented by acomposite kernel expansion is in the form of
yn =N∑
i=1
βi (k1(yi−1, yn−1)+ k2(ui ,un)) (25)
236 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 28, NO. 1, JANUARY 2017
where βi is the expansion coefficient and N is the numberof sampled data. k1 and k2 are the kernel functions for theAR and MA parts respectively, and k1(yi−1, yn−1)+k2(ui ,un)is defined as the composite kernel. The composite kernelexpansion model (25) enables us to use different kernelfunctions for the AR and MA parts of the regressor in (1).
The vector pairs [(yi−1)T , (ui )
T ]T corresponding to thenonzero coefficients βi in model representation (25) arethe SVs. Consequently, the model sparsity, which is defined asthe ratio of the number of SVs to the number of all trainingdata points, plays a critical role in controlling model com-plexity and alleviating model redundancy. A kernel expansionmodel with substantial redundant terms is against the parsi-monious principle that ensures the simplest possible modelthat explains the data, and may deteriorate the generalizationperformance and increase the computational requirements sub-stantially.
The number of nonzero components in the coefficient vectorβ = [β1, β2, . . . , βN ]T largely determines the complexityof the kernel expansion model (25). In order to enforce thesparseness of the model, the linear programming SV learningis considered here, instead of QP-SVR. It employs the 1 normof the coefficient vector β in model (25) as a regularizer inthe objective function to control the model complexity andstructural risk. By introducing the ε-insensitive loss function,which is defined as
L(yn − yn) ={
0, if |yn − yn| ≤ ε
|yn − yn | − ε, otherwise(26)
the regularization problem to be solved becomes
min Rreg[ f ] = ‖β‖1 + CN∑
n=1
L(yn − yn) (27)
where the parameter C controls the extent to which theregularization term influences the solution and ε is the errortolerance. Geometrically, the ε-insensitive loss function definesa ε-tube. The idea of using the 1 norm to secure a sparserepresentation is also explored in the emerging theory ofcompressive sensing [45], [46].
By introducing the slack variables ξn , n = 1, 2, . . . , N toaccommodate otherwise infeasible constraints and to enhancerobustness, the regularization problem (27) can be trans-formed into the following equivalent constrained optimizationproblem:
min ‖β‖1 + CN∑
n=1
ξn
s.t.
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
N∑
i=1βi (k1(yi−1, yn−1)+ k2(ui ,un))− yn ≤ ε + ξn
yn−N∑
i=1βi (k1(yi−1, yn−1)+k2(ui ,un)) ≤ ε+ξn
ξn ≥ 0, n = 1, 2, . . . , N
(28)
where the constant C > 0 determines the tradeoff betweenthe sparsity of the model and the amount up to which
deviations larger than ε can be tolerated. For the purposeof converting (28) into a linear programming problem, thecomponents βi of the coefficient vector β and their absolutevalues |βi | are decomposed as follows:
βi = α+i − α−
i |βi | = α+i + α−
i (29)
where α+i , α
−i ≥ 0, and for a given βi , there is an unique
pair (α+i , α
−i ) fulfilling both the equations in (29). Note
that both the variables cannot be positive at the same time,i.e., α+
i · α−i = 0. In this way, the optimization problem (28)
can be reformulated as
minN∑
i=1
(α+
i + α−i
)+ CN∑
n=1
ξn
s.t.
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
N∑
i=1
(α+
i −α−i
)(k1(yi−1, yn−1)+k2(ui ,un))−ξn ≤ε + yn
−N∑
i=1
(α+
i −α−i
)(k1(yi−1, yn−1)+k2(ui ,un))−ξn ≤ε−yn
ξn ≥ 0, n = 1, 2, . . . , N.
(30)
Next, define the vector
c = (1, 1, . . . , 1︸ ︷︷ ︸
N
, 1, 1, . . . , 1︸ ︷︷ ︸
N
,C,C, . . . ,C︸ ︷︷ ︸
N
)T (31)
and write the 1 norm of β as
‖β‖1 = (1, 1, . . . , 1︸ ︷︷ ︸
N
, 1, 1, . . . , 1︸ ︷︷ ︸
N
)
(α+α−)
(32)
with the N-dimensional column vectors α+ and α− definedas α+ = (α+
1 , α+2 , . . . , α
+N )
T and α− = (α−1 , α
−2 , . . . , α
−N )
T,and the constrained optimization problem (30) can be cast asa linear programming problem in the following form:
min cT
⎛
⎝α+α−ξ
⎞
⎠
s.t.
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(K1 + K2 −(K1 + K2) −I
−(K1 + K2) K1 + K2 −I
)
·⎛
⎜⎝
α+
α−
ξ
⎞
⎟⎠ ≤
(y + ε
ε − y
)
α+, α− ≥ 0, ξ ≥ 0(33)
where ξ = (ξ1, ξ2, . . . , ξN )T , y = (y1, y2, . . . , yN )
T , andI is an N × N identity matrix. K1 and K2 are the kernelmatrices with entries defined as (K1)in = k1(yi−1, yn−1)and (K2)in = k2(ui ,un). The calculation of the vectors α+,α− and the SVs selection can be accomplished by solvingthe linear optimization problem (33) using the well-knownsimplex algorithm or the primal–dual interior point algorithm.With the solution to linear programming problem (33), thecoefficients of the composite kernel expansion model (25) canbe calculated using (29), and thereby model (25) can be builtas follows:
yn =∑
i∈SV
βi (k1(yi−1, yn−1)+ k2(ui ,un)). (34)
LU et al.: MULTISCALE SV LEARNING WITH PROJECTION OPERATOR WAVELET KERNEL 237
Fig. 3. Model in series–parallel configuration.
This composite kernel expansion on the selected SVs is forrepresenting the nonlinear dynamics underlying the time series{ui , yi }, i = 1, 2, . . . , N . Contrary to the QP-SVR where allthe data points not inside the ε-tube are selected as SVs, theLP-SVR still can generate a sparse solution even when the ε isset to be zero.
Most of the preceding works applying SV learning tononlinear system identification treat system identification as ageneral regression problem, where the AR and MA parts areconsolidated in the regressor [47]–[50]. However, the chosensingle kernel function might be ineffective in characterizingdifferent cause–effect relationships of the AR and MA partsto the model output. Modeling the different dependenciesby heterogeneous kernel functions is the main motivationfor using the composite kernel, which provides new degreesof freedom in representing nonlinear dynamics. The use ofcomposite kernel also makes the model more amenable tocontrol law design, which also blazes a new path to control-oriented sparse modeling.
V. APPLICATION TO NONLINEAR DYNAMICAL
SYSTEM IDENTIFICATION
Machine learning is centered around making predictions,based on already identified trends and properties in the trainingdata set. Forecasting future behavior based on past observa-tions has also been a long standing topic in system identi-fication and time series analysis. According to the differentregression vectors used, the identification model for nonlineardynamical systems can be categorized as the series–parallelmodel (or NARX model), and parallel model (or nonlinearoutput error model) [27], [28]. For the series–parallel model(see Fig. 3), the past values of the input and output of theactual system form the regression vector in order to producethe estimated output yn at time instant tn . For the parallelmodel depicted in Fig. 4, however, the regression vector iscomposed of the past values of the input and output ofthe identification model. Thus, without coupling to the realsystems, the parallel models are emancipated from relying onthe outputs of the actual systems. In effect, the parallel modelis a recurrent NARX model, whose computational capability isequivalent to a Turing machine [51], [52]. The identificationof the series–parallel model amounts to building a one-step
Fig. 4. Model in parallel configuration.
ahead predictor, while the identification of the parallel modelis for the long-term prediction.
In the realm of nonlinear systems identification, there isgeneral consensus that one of the most formidable technicalchallenges is building a model usable in parallel configuration,which is much more intractable than building a series–parallelmodel due to the feedback involved [27], [28]. However,a multitude of applications, e.g., fault detection and diagnosis,predictive control, simulation, and so on, require a parallelmodel, since the prediction of many steps into the future isneeded.
In theory, long-term predictions can be obtained from ashort-term predictor, for example, a one-step ahead predictor,simply by applying the short predictor many times (steps)in an iterative way. This is called iterative prediction, andlays the foundation for obtaining a parallel model by trainingin the series–parallel configuration [53]–[55]. Another waycalled direct prediction provides a once-completed predictorwith a long-term prediction step, and the specified multistepprediction can then be obtained directly from the establishedpredictor in a manner similar to computing one-step predic-tions [53], [54]. The main downside of the direct modelingapproach is that it requires different models for different stepsahead prediction. It is generally believed that the iterativeprediction approach is in most cases more efficient than thedirect approach assuming that the dynamics underlying thetime series are correctly specified by the model [53].
In this simulation study, to demonstrate the superiority andeffectiveness of the proposed novel kernel function for nonlin-ear dynamical systems modeling, the LP-SVR learning algo-rithm with multiscale orthogonal projection operator waveletkernel is used to build parallel models for the benchmarkhydraulic robot arm data set and Box and Jenkins’ data set inthe spirit of the iterative prediction approach. Although thesedata sets have been widely used for the performance evaluationof various system identification methods in [17], [48], [49],and [56]–[58], most of the work reported in the literaturefocuses on the identification of the series–parallel models andtheir parallel models have rarely been studied.
Partitioning the benchmark data sets into training andvalidation subsets, the identification procedure includestwo phases. First, the one-step ahead predictor, i.e., the
238 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 28, NO. 1, JANUARY 2017
series–parallel model, is identified on the training data set inthe series–parallel configuration as that in Fig. 3, and thenin the second phase, the attained one-step ahead predictoris iteratively used in parallel configuration for the long-termprediction on the validation data set, as shown in Fig. 4.
For the sake of comparison, several commonly used kernelfunctions are employed for modeling on the same data sets aswell, such as the Gaussian RBF kernel defined by
k(x, z) = exp
(−‖x − z‖2
2σ 2
)
(35)
the polynomial kernel defined by
k(x, z) = (1 + 〈x, z〉)q (36)
the inverse multiquadric kernel defined by
k(x, z) = 1√‖x − z‖2 + c2
(37)
the B-spline kernel defined by
k(x, z) =d∏
i=1
B2J+1(xi − zi ) (38)
and the Morlet wavelet kernel given by
k(x, z) =d∏
i=1
φ
(xi − zi
δ
)
(39)
where φ(x) = cos(1.75x) exp(−x2/2) and δ, σ , q , c,and J are all the adjustable parameters of the above kernelfunctions. For the B-spline kernel, B-spline function B (·)represents a particular example of a convolutional basis andcan be expressed explicitly as [59]
B (x) = 1
! +1∑
r=0
( + 1
r
)
(−1)r(
x + + 1
2− r
)
+(40)
where the function (·)+ is defined as the truncated powerfunction, that is
x+ ={
x, for x > 0
0, otherwise.(41)
A. Hydraulic Robot Arm Dynamical System Identification
For the hydraulic robot arm dynamical system, the positionof a robot arm is controlled by a hydraulic actuator. The con-trol input un represents the size of the valve opening throughwhich oil flows into the actuator, and the output yn is a mea-sure of the oil pressure that determines the robot arm position.In modeling this dynamical system, for fair comparison, thesame regressor [yn−1, yn−2, yn−3, un−1, un−2] and the data setpartition scheme as those in the literature [17], [48], [49], [56]are adopted herein. The first half of the data set containing511 training data pairs is used for training in series–parallelconfiguration, and the other half for validation data in parallelconfiguration.
In the training phase, model (34) with yn−1 = [yn−1,yn−2, yn−3] and un = [un−1, un−2] is learned by LP-SVRto attain the one-step ahead approximator. Upon training
Fig. 5. Training in series–parallel configuration for model (34) of robotarm by LP-SVR with projection operator wavelet kernel (44) and (45).Solid line: actual system output. Dotted line: model output.
completion, our objective is to provide satisfactory multistepprediction without using the actual system output yn , i.e., tovalidate the model in parallel configuration as follows:
yn =∑
i∈SV
βi (k1(yi−1, yn−1)+ k2(ui ,un)) (42)
where yn−1 = [yn−1, yn−2, yn−3]. The approximation accura-cies on the training and validation data sets are evaluated bycalculating the root mean square error (RMSE)
Erms =√√√√ 1
M
M∑
n=1
[yn − yn]2 (43)
where yn is the estimated output of the model and M is thenumber of data in the data set for evaluation. The validationaccuracy is crucial in assessing the generalization performanceof the model. In applying SVR with kernel functions to trainthe model, manual tuning of the kernel parameters as well asε and C for optimum results is required.
The parameters used for learning are ε = 0.06 and C = 5,and the projection operator wavelet kernels k1(yi−1, yn−1) andk2(ui ,un) in the composite kernel expansion model take thefollowing forms respectively:
k=−6 ϕ j,k(xi )ϕ j,k(zi ) with the kernel parameter 0.001. Thetraining result based on the multiscale projection operatorwavelet kernel (44) and (45) is illustrated in Fig. 5, and thetraining RMSE is 0.0745. The attained model is subsequently
LU et al.: MULTISCALE SV LEARNING WITH PROJECTION OPERATOR WAVELET KERNEL 239
TABLE I
ROBOT ARM PARALLEL MODEL IDENTIFICATION BY LP-SVR WITH DIFFERENT COMPOSITE KERNEL FUNCTIONS
TABLE II
ROBOT ARM PARALLEL MODEL IDENTIFICATION BY QP-SVR WITH DIFFERENT COMPOSITE KERNEL FUNCTIONS
Fig. 6. Validation in parallel configuration for mode (42) of robot armby LP-SVR with projection operator wavelet kernel (44) and (45).Solid line: actual system output. Dotted line: model output.
validated on the validation data set in parallel configurationfor long-term/mid-term prediction, and the validation result isshown in Fig. 6 and Table I.
Following the same procedure, the kernelfunctions (35)–(39) are also applied to train model (34)by LP-SVR and QP-SVR. After tuning the parameters foroptimum results, the RMSEs on the training and validationdata sets together with model sparsity obtained by thesecomparative models are listed in Table I for LP-SVR andTable II for QP-SVR. The corresponding plots can be foundin [24].
Measured by the SV ratio, the sparsity of the model withthe closed-form projection operator wavelet kernel is com-mensurate with the models adopting other kernel functions;moreover, it is very evident that the projection operator waveletkernel considerably outperforms other kernel functions interms of validation accuracy in parallel configuration, whichimplies excellent generalization performance.
In parallel configuration, the errors for the sth-step pre-diction are the accumulation of the errors of the previous(s − 1) steps. In general, the longer the forecasting horizon,the larger the accumulated errors are and the less accurate theiterative method is. Hence, it is remarkable that, while usingthe identical regressor on the same training and validation datasets, this parallel model validation accuracy is even better thansome of those obtained in the series–parallel configuration by
240 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 28, NO. 1, JANUARY 2017
TABLE III
GAS FURNACE PARALLEL MODEL IDENTIFICATION BY LP-SVR WITH DIFFERENT COMPOSITE KERNEL FUNCTIONS
TABLE IV
GAS FURNACE PARALLEL MODEL IDENTIFICATION BY QP-SVR WITH DIFFERENT COMPOSITE KERNEL FUNCTIONS
other popular learning strategies. For example, the RMSEwas 0.467 for a one-hidden-layer sigmoid neural network caseand 0.579 for a wavelet network case [56].
In terms of computing time for training, LP-SVR isaround seven times faster than QP-SVR on this data set(Intel Core i5 processor), and the computing resource requiredby QP-SVR might become prohibitively expensive whenincreasing the size of the training data set. It is also notablewhen comparing the model sparsities in Tables I and II thatthe LP-SVR substantially exceeds the QP-SVR in producingsuccinct model representations.
B. Box and Jenkins’ Identification Problem
The Box and Jenkins’ gas furnace data set was recordedfrom a combustion process of a methane-air mixture. Theoriginal data set consists of 296 input–output data pairs thatwere recorded at a sampling rate of 9 s. The gas combustionprocess has one input variable, gas flow rate un , andone output variable, the concentration of carbon dioxide (CO2)in the outlet gas, yn . The instantaneous value of the outputyn can be regarded as being influenced by ten variablesyn−1, yn−2, . . . , yn−4 and un−1, un−2, . . . , un−6 [57], [58].In modeling this dynamical system, the regressor[yn−1, yn−2, un−2, un−3, un−4] is employed herein.
The first 150 data pairs are used for training in theseries–parallel configuration, and the subsequent 90 datapairs are used for validation in parallel configuration. Dueto the different distribution and magnitude order of themeasurements in the data set, proper data rescaling isnecessary.
In training model (34) with yn−1 = [yn−1, yn−2] andun = [un−2, un−3, un−4], the kernel functions k1(yi−1, yn−1)and k2(ui ,un) take the following forms, respectively:
K1(x, z) =2∑
j=−9
� j (x1, z1)
4∑
j=1
� j (x2, z2) (46)
K2(x, z)=2∑
j=0
� j (x1, z1)
3∑
j=2
� j (x2, z2)
2∑
j=1
� j (x3, z3) (47)
where � j (xi , zi ) = ∑0k=−2 ψ j,k(xi)ψ j,k(zi ) and the kernel
parameters are 1.22 and 0.013, respectively. The trainingresult based on the multiscale projection operator waveletkernel (46) and (47) is illustrated in Fig. 7, and the correspond-ing RMSE is 0.2298. Subsequently, the model is validated inparallel configuration
yn =∑
i∈SV
βi (k1(yi−1, yn−1)+ k2(ui ,un)) (48)
LU et al.: MULTISCALE SV LEARNING WITH PROJECTION OPERATOR WAVELET KERNEL 241
Fig. 7. Training in series–parallel configuration for model (34) of gasfurnace by LP-SVR with projection operator wavelet kernel (46) and (47).Solid line: actual system output. Dotted line: model output.
Fig. 8. Validation in parallel configuration for model (48) of gas furnaceby LP-SVR with projection operator wavelet kernel (46) and (47).Solid line: actual system output. Dotted line: model output.
where yn−1 = [yn−1, yn−2]. The validation results are plottedin Fig. 8, and the corresponding RMSE is 0.5148. For con-firming the superiority of the proposed kernel function, themodel is also trained with other kernel functions (35)–(39)by the LP-SVR and the QP-SVR. Together with the modelsparsity, the training RMSE and the validation RMSE are listedin Tables III and IV. It can be remarkably found from the tablesthat the validation accuracy obtained by using the multiscaleprojection operator wavelet kernel is dramatically improvedand, in particular, the validation RMSE is even better than thetraining RMSEs from the Gaussian RBF kernel, polynomialkernel, and inverse multiquadric kernel. The correspondingplots can be found in [24].
Due to the ubiquity of transient characteristics andmultiscale structures in nonlinear dynamics, the refinablekernel functions capable of taking account of local as wellas global complexity are highly desired. Compared withthe conventional single-scale kernel functions, the multiscaleclosed-form wavelet kernel functions display its main strength
in capturing the localized temporal and frequency informationof rapidly changing signals.
VI. CONCLUSION
The triumph of kernel methods largely depends on thecapability of kernel functions. Confronted with the more andmore challenging learning tasks, such as nonlinear dynamicsystems identification, nonlinear time series prediction, andcomputer vision, the kernel machines are expected to be ableto cope with the multiscale nature of complex systems.
Most of the kernel functions used in the literature, includ-ing the non-Mercer kernel in [48], are single-scale kernels.Criticized as template matchers, the commonly usedtranslation-invariant kernels and rotation-invariant kernels maylimit the performance that SV learning can achieve. In thispaper, by leveraging the closed-form raised-cosine orthogonalwavelets to fulfill the finite expansion kernel, multiscale kernellearning was implemented in the framework of multiresolutionanalysis. In view of the geometric notion of the integral oper-ator in function space, the developed finite multiscale expan-sion kernels are conceptualized as the multiscale projectionoperator wavelet kernel, thereby overcoming the limitation ofthe commonly used translation-invariant kernels and rotation-invariant kernels.
By focusing on control-oriented nonlinear dynamicalsystems modeling, the developed projector operator waveletkernels are used to construct the composite kernel, and thesparsity inherited in linear programming SV learning ensuresthe lacunary kernel expansion representation for modelingnonlinear dynamic systems. Two examples have demonstratedthe utility and effectiveness of the proposed projection operatorwavelet kernel in representing nonlinear dynamic models inparallel configuration. The potential of the proposed kernellearning algorithm in hyperspectral image analysis, multi-scale computer vision [60], and linear operator equations willbe investigated further. On the theoretical aspect, the pro-posed multiscale projection operator wavelet kernels also shedlight on the unexpected confluence of kernel regression andresolvent-type kernel-based nonuniform sampling [61], [62],which will enable us to explore the essence of SV selection inthe LP-SVR from the perspective of modern sampling theory.
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LU et al.: MULTISCALE SV LEARNING WITH PROJECTION OPERATOR WAVELET KERNEL 243
Zhao Lu (M’08–SM’15) received the M.S. degree incontrol theory and engineering from Nankai Univer-sity, Tianjin, China, in 2000, and the Ph.D. degree inelectrical engineering from the University ofHouston, Houston, TX, USA, in 2004.
He was a Post-Doctoral Research Fellow with theDepartment of Electrical and Computer Engineering,Wayne State University, Detroit, MI, USA, and theDepartment of Naval Architecture and Marine Engi-neering, University of Michigan, Ann Arbor, MI,USA, respectively, from 2004 to 2006. Since 2007,
he has been a Faculty Member with the College of Engineering, TuskegeeUniversity, Tuskegee, AL, USA, where he is currently an Associate Professorwith the Department of Electrical Engineering. His current research interestsinclude machine learning, computational intelligence, and nonlinear controltheory.
Jing Sun (M’89–SM’00–F’04) received the B.S. andM.S. degrees from the University of Scienceand Technology of China, Hefei, China, in 1982and 1984, respectively, and the Ph.D. degree fromthe University of Southern California, Los Angeles,CA, USA, in 1989.
She was an Assistant Professor with the Depart-ment of Electrical and Computer Engineering,Wayne State University, Detroit, MI, USA, from1989 to 1993. She joined the Ford Research Labo-ratory, Dearborn, MI, USA, in 1993, where she was
with the Powertrain Control Systems Department. After spending almost tenyears with the industry, she returned to academia and joined as a FacultyMember with the College of Engineering, University of Michigan, Ann Arbor,MI, USA, in 2003, where she is currently a Professor with the Department ofNaval Architecture and Marine Engineering and the Department of ElectricalEngineering and Computer Science. She holds over 30 U.S. patents andhas co-authored the textbook entitled Robust Adaptive Control. Her currentresearch interests include system and control theory and its applications tomarine and automotive propulsion systems.
Prof. Sun was a recipient of the 2003 IEEE Control System TechnologyAward.
Kenneth Butts (M’10) received the B.E. degreein electrical engineering from Kettering Univer-sity, Flint, MI, USA, the M.S. degree in electri-cal engineering from the University of Illinois atUrbana–Champaign, Champaign, IL, USA, and thePh.D. degree in electrical engineering from the Uni-versity of Michigan, Ann Arbor, MI, USA.
He is currently an Executive Engineer with thePowertrain and Regulatory Division, Toyota MotorEngineering and Manufacturing North America,Ann Arbor, where he is investigating methods to
improve engine control development productivity. He has been involved inthe field of automotive electronics and control since 1982, almost exclusivelyin research and advanced development of powertrain controls.
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