Journal of Machine Learning Research 2 (2001) 97-123 Submitted 3/01; Published 12/01

Kernel Partial Least Squares Regression in Reproducing Kernel Hilbert Space

Roman Rosipal^{∗} rrosipal@mail.arc.nasa.gov Applied Computational Intelligence Research Unit University of Paisley Paisley PA1 2BE, Scotland and Laboratory of Neural Networks Institute of Measurement Science, SAS Bratislava 842 19, Slovak Republic

Leonard J. Trejo ltrejo@mail.arc.nasa.gov Computational Sciences Division NASA Ames Research Center Moffett Field, CA 94035-1000

Editors: Nello Cristianini, John Shawe-Taylor and Bob Williamson

Abstract

A family of regularized least squares regression models in a Reproducing Kernel Hilbert Space is extended by the kernel partial least squares (PLS) regression model. Similar to principal components regression (PCR), PLS is a method based on the projection of input (explanatory) variables to the latent variables (components). However, in contrast to PCR, PLS creates the components by modeling the relationship between input and output variables while maintaining most of the information in the input variables. PLS is useful in situations where the number of explanatory variables exceeds the number of observations and/or a high level of multicollinearity among those variables is assumed. Motivated by this fact we will provide a kernel PLS algorithm for construction of nonlinear regression models in possibly high-dimensional feature spaces.

We give the theoretical description of the kernel PLS algorithm and we experimentally compare the algorithm with the existing kernel PCR and kernel ridge regression techniques. We will demonstrate that on the data sets employed kernel PLS achieves the same results as kernel PCR but uses significantly fewer, qualitatively different components.

1. Introduction

Rosipal and Trejo

In this paper we will focus our attention on least squares regression models in a Reproducing Kernel Hilbert Space (RKHS). The models are derived based on a straightforward connection between a RKHS and the corresponding feature space representation where the input data are mapped. In our previous work (Rosipal et al., 2000a, 2001, 2000b) we proposed the kernel principal components regression (PCR) technique and we also made theoretical and experimental comparison to kernel ridge regression (RR) (Saunders et al., 1998, Cristianini and Shawe-Taylor, 2000). In this work we extend the family with a new nonlinear kernel partial least squares (PLS) regression method.

∗. Current address: NASA Ames Research Center, Mail Stop 269-3, Moffett Field, CA 94035-1000 \begin{matrix}c \\ ◯ \end{matrix}2001 Roman Rosipal and Leonard J Trejo.

Classical PCR, PLS and RR techniques are well known shrinkage estimators designed to deal with multicollinearity (see, e.g., Frank and Friedman, 1993, Montgomery and Peck, 1992, Jolliffe, 1986). The multicollinearity or near-linear dependence of regressors is a serious problem which can dramatically influence the effectiveness of a regression model. Multicollinearity results in large variances and covariances for the least squares estimators of the regression coefficients. Multicollinearity can also produce estimates of the regression coefficients that are too large in absolute value. Thus the values and signs of estimated regression coefficients may change considerably given different data samples. This effect can lead to a regression model which fits the training data reasonably well, but generalizes poorly to new data (Montgomery and Peck, 1992). This fact is in a very close relation to the argument stressed in (Smola et al., 1998), where the authors have shown that choosing the flattest linear regression function^{1} in a feature space can, based on the smoothing properties of the selected kernel function, lead to a smooth nonlinear function in the input space.

The PLS method (Wold, 1975, Wold et al., 1984) has been a popular regression techniques in its domain of origin—Chemometrics. The method is similar to PCR where principal components determined solely from explanatory variables creates orthogonal, i.e. uncorrelated, input variables in a regression model. In contrast, PLS creates orthogonal components by using the existing correlations between explanatory variables and corresponding outputs while also keeping most of the variance of explanatory variables. PLS has proven to be useful in situations when the number of observed variables (N) is significantly greater than the number of observations (n) and high multicollinearity among the variables exists. This situation when N ≫ n is common in chemometrics and gave rise to the modification of classical principal component analysis (PCA) and linear PLS methods to their kernel variants (Wu et al., 1997a, Rännar et al., 1994, Lewi, 1995). However, rather than assuming a nonlinear transformation into a feature space of arbitrary dimensionality the authors attempted to reduce computational complexity in the input space. Motivated by these works we propose a more general nonlinear kernel PLS algorithm.^{2}

There exist several nonlinear versions of the PLS model. In (Frank, 1990, 1994) approaches based on fitting the nonlinear input-output dependency by providing the extracted components as inputs to smoothers and spline-based additive nonlinear regression models were proposed. Another nonlinear PLS model (Malthouse, 1995, Malthouse et al., 1997) is based on relatively complicated artificial neural network modeling of nonlinear PCA and consequent nonlinear PLS. From that point our approach differs in the sense that the original input data are nonlinearly mapped to a feature space F where a linear PLS model is created. Good generalization properties of the corresponding nonlinear PLS model are then achieved by appropriate estimation of regression coefficients in F and by the selection of an appropriate kernel function. Moreover, utilizing the kernel function corresponding to the canonical dot product in F allows us to avoid the nonlinear optimization involved in the above approaches. In fact only linear algebra as simple as in a linear PLS regression is required.

1. The flatness is defined in the sense of penalizing high values of the regression coefficients estimate.

2. In the following, where it is clear, we will not stress this nonlinear essence of the proposed kernel PLS

regression model and will use the kernel PLS notation.

Kernel Partial Least Squares Regression in RKHS

In Section 2 a basic definition of a RKHS and formulation of the Representer theorem is given. Section 3 describes the “classical” PLS algorithm. In Section 4 the kernel PLS method is given. Some of the properties of kernel PLS are shown using a simple example. Kernel PCR and kernel RR are briefly described in Section 5. Section 6 describes the used model selection techniques. The results are given in Section 7. Section 8 provides a short discussion and concludes the paper.

2. RKHS and Representer Theorem

The common aim of support vector machines, regularization networks, Gaussian processes and spline methods (Vapnik, 1998, Girosi, 1998, Williams, 1998, Wahba, 1990, Cristianini and Shawe-Taylor, 2000) is to address the poor generalization properties of existing nonlinear regression techniques. To overcome this problem a regularized formulation of regression is considered as a variational problem in a RKHS H

min_{f∈H} R_{reg}(f) = \frac{1}{n} ∑_{i=1}^{n} V (y_{i}, f(x_{i})) + ξ∥f∥_{H}^{2} .  (1)

We assume a training set of regressors {x_{i}}_{i=1}^{n} to be a subset of a compact set X ⊂ R^{N} and {y_{i}}_{i=1}^{n} ∈ R to be a set of corresponding outputs. The solution to the problem (1) was given by Kimeldorf and Wahba (1971), Wahba (1999) and is known as the

Representer theorem (simple case): Let the loss function V (y_{i}, f) be a functional of f which depends on f only pointwise, that is, through {f(x_{i})}_{i=1}^{n}—the values of f at the data points. Then any solution to the problem: find f ∈ H to minimize (1) has a representation of the form

f(x) = ∑_{i=1}^{n} c_{i}K(x_{i}, x) ,  (2)

where {c_{i}}_{i=1}^{n} ∈ R.

In this formulation ξ is a positive number (regularization coefficient) to control the tradeoff between approximating properties and the smoothness of f. ∥f∥_{H}^{2} is a norm (sometimes called “stabilizer” in the regularization networks domain) in a RKHS H defined by the positive definite kernel K(x, y); i.e. a symmetric function of two variables satisfying the Mercer theorem conditions (Mercer, 1909, Cristianini and Shawe-Taylor, 2000). The fact that for any such positive definite kernel there exists a unique RKHS is well established by the Moore-Aronszjan theorem (Aronszajn, 1950). The form K(x, y) has the following reproducing property

f(y) = ⟨f(x), K(x, y)⟩_{H} ∀f ∈ H ,

where ⟨., .⟩_{H} is the scalar product in H. The function K is called a reproducing kernel for H.

It follows from Mercer’s theorem that each positive definite kernel K(x, y) defined on a compact domain X × X can by written in the form

K(x, y) = ∑_{i=1}^{M} λ_{i}ϕ_{i}(x)ϕ_{i}(y) M ≤ ∞ ,  (3)

where {ϕ_{i}(.)}_{i=1}^{M} are the eigenfunctions of the integral operator Γ_{K} : L_{2}(X) → L_{2}(X)

(Γ_{K}f)(x) = ∫_{X} K(x, y)f(y)dy ∀f ∈ L_{2}(X)

and {λ_{i} > 0}_{i=1}^{M} are the corresponding positive eigenvalues. The sequence {ϕ_{i}(.)}_{i=1}^{M} creates an orthonormal basis of H and we can express any function f ∈ H as f(x) = ∑_{i=1}^{M} a_{i}ϕ_{i}(x) for some a_{i} ∈ R. This allows us to define a scalar product in H:

⟨f(x), h(x)⟩_{H} = ⟨∑_{i=1}^{M} a_{i}ϕ_{i}(x), ∑_{i=1}^{M} b_{i}ϕ_{i}(x)⟩_{H} ≡ ∑_{i=1}^{M} \frac{a_{i}b_{i}}{λ_{i}}

and the norm ∥f∥_{H}^{2} = ∑_{i=1}^{M} \frac{a_{i}^{2}}{λ_{i}} .

Rewriting (3) in the form

K(x, y) = ∑_{i=1}^{M} \sqrt{λ_{i}}ϕ_{i}(x)\sqrt{λ_{i}}ϕ_{i}(y) = (Φ(x).Φ(y)) = Φ(x)^{T} Φ(y) ,

it becomes clear that any kernel K(x, y) also corresponds to a canonical (Euclidean) dot product in a possibly high-dimensional space F where the input data are mapped by

\begin{aligned}Φ : & X → F \\ & x → (\sqrt{λ_{1}}ϕ_{1}(x), \sqrt{λ_{2}}ϕ_{2}(x), …, \sqrt{λ_{M}}ϕ_{M}(x)) . \end{aligned}

The space F is usually denoted as a feature space and \{\{\sqrt{λ_{i}}ϕ_{i}(x)\}_{i=1}^{M}, x ∈ X \} as feature mappings. The number of basis functions ϕ_{i}(.) also defines the dimensionality of F. It is worth noting, that we can also construct a RKHS and a corresponding feature space by choosing a sequence of linearly independent functions (not necessary orthogonal) {ψ_{i}(x)}_{i=1}^{M} and positive numbers α_{i} to define a series (in the case of M = ∞ absolutely and uniformly convergent) K(x, y) = ∑_{i=1}^{M} α_{i}ψ_{i}(x)ψ_{i}(y). This also gives the connection between the RKHS and Gaussian processes where the K is assumed to represent the correlation function of a zero-mean Gaussian process evaluated at points x and y (Wahba, 1990).

Until now, we assumed that K is a positive definite kernel. However, the above results can be extended even for the case when K is a positive semidefinite. Is such a case a RKHS H contains a subspace of functions f with a zero norm ∥f∥_{H}^{2} (the null space). Kimeldorf and Wahba (1971) showed that in such a case the solution of (1) leads to a more general form of the Representer theorem:

f(x) = ∑_{i=1}^{n} c_{i}K(x_{i}, x) + ∑_{j=1}^{l} b_{j}υ_{j}(x) ,

where the functions {υ_{j}(.)}_{j=1}^{l} span the null space of H and the coefficients {c_{i}}_{i=1}^{n}, {b_{j}}_{j=1}^{l} are again given by the data. In this paper we will consider only the case when l = 1 and υ_{1}(x) = const ∀ x.

3. Partial Least Squares Regression

PLS regression is a technique for modeling a linear relationship between a set of output variables (responses) {y_{i}}_{i=1}^{n} ∈ R^{L} and a set of input variables (regressors) {x_{i}}_{i=1}^{n} ∈ R^{N}. In the first step, PLS creates uncorrelated latent variables which are linear combinations of the original regressors. The basic point of the procedure is that the weights used to determine these linear combinations of the original regressors are proportional to the covariance among input and output variables (Helland, 1988). A least squares regression is then performed on the subset of extracted latent variables. This leads to a biased but lower variance estimate of the regression coefficients comparing to the Ordinary Least Squares (OLS) regression.

In the following X will represent the (n×N) matrix of n inputs and Y will stand for the (n × L) matrix of the corresponding L-dimensional responses. Further we assume centered input and output variables; i.e. the columns of X and Y are zero mean.

There exist several different modifications (see Martens and Naes, 1989, Manne, 1987, Helland, 1988, de Jong, 1993) of the basic algorithm for PLS regression originally developed by Wold (1975). In its basic form a special case of the nonlinear iterative partial least squares (NIPALS) algorithm (Wold, 1966) is used. NIPALS is a robust procedure for solving singular value decomposition problems and is closely related to the power method (Golub and van Loan, 1996). After an initial random estimate of the latent vector t the following two steps are repeated until convergence of t and the loadings vector p:

1. p = X^{T} t

2. t = Xp, t ← t/∥t∥ .

After the extraction of t and p vectors the matrix X is deflated by t

X ← X − tt^{T} X

and by repeating the whole procedure we may extract a new pair of vectors t and p which are by construction orthogonal to the previous one. It is worth noting that in the case that N < n the normalization of the N-dimensional vector p after the first step is computationally advantageous in comparison to the normalization of the n-dimensional vector t. However, the normalization of t allows us to adapt the NIPALS algorithm to extract the latent vectors from the kernel matrices XX^{T} (Lewi, 1995):

1. p = XX^{T} t

2. t = p, t ← t/∥t∥ .

The deflation of the XX^{T} matrix is given by

XX^{T} ← (X − tt^{T} X)(X − tt^{T} X)^{T} .

Wold et al. (1984) applied the NIPALS algorithm to the PLS regression with the aim to sequentially extract the latent vectors t, u and weight vectors w, c from X and Y matrices in decreasing order of their corresponding singular values. What follows is a modification of the “classical” NIPALS-PLS algorithm in the sense that normalization of the latent vectors t, u rather than normalization of the vectors of weights w, c is used (Lewi, 1995):

1. randomly initialize u

2. w = X^{T} u

3. t = Xw, t ← t/∥t∥

4. c = Y^{T} t

5. u = Yc, u ← u/∥u∥

6. repeat steps 2. – 5. until convergence

7. deflate X,Y matrices: X ← X − tt^{T} X, Y ← Y − tt^{T} Y .

The PLS regression is an iterative process; i.e. after extraction of one component the algorithm starts again using the deflated matrices X and Y computed in step 7. Thus we can achieve the sequence of the models up to the point when the rank of X is reached. However, in practice the cross-validation technique is usually used to avoid underfitting or overfitting caused by the use of too small or too large dimensional models. After the extraction of the p components we can create the (n×p) matrices T, U, the (N ×p) matrix W and the (L × p) matrix C consisting of the columns created by the vectors {t_{i}}_{i=1}^{p}, {u_{i}}_{i=1}^{p}, {w_{i}}_{i=1}^{p} and {c_{i}}_{i=1}^{p}, respectively, extracted during the individual iterations.

The PLS regression model can be written in matrix form as (Manne, 1987, Rännar et al., 1994)

Y = XB + F ,

where B is an (N ×L) matrix of the of the regression coefficients and F is an (n×L) matrix of residuals. This equation is identical to that used in other regression models; multiple linear regression, ridge regression and principal components regression, however, in contrast to these models the matrix B has the form (Manne, 1987, Rännar et al., 1994)

B = W(P^{T} W)^{−1}C^{T} ,  (4)

where P is the (N × p) matrix consisting of loadings vectors {p_{i} = X^{T} t_{i}/(t_{i}^{T} t_{i})}_{i=1}^{p}. Due to the fact that p_{i}^{T} w_{j} = 0 for i > j and in general p_{i}^{T} w_{j} ≠ 0 for i < j (Höskuldsson, 1988) the matrix P^{T} W is upper triangular and thus invertible. Moreover, using the fact that t_{i}^{T} t_{j} = 0 for i ≠ j and t_{i}^{T} u_{j} = 0 for j > i Rännar et al. (1994) derived the following equalities^{3}

W = X^{T} U  (5)
P = X^{T} T(T^{T} T)^{−1}  (6)
C = Y^{T} T(T^{T} T)^{−1} .  (7)

Substituting (5–7) into (4) and using the orthogonality of the T matrix columns we can write the matrix B in the following form

B = X^{T} U(T^{T} XX^{T} U)^{−1}T^{T} Y .  (8)

It is worth noting that different scalings of the individual latent vectors {t_{i}}_{i=1}^{p} and {u_{i}}_{i=1}^{p} do not influence this estimate of the matrix B.

3. In our case T^{T} T is the p-dimensional identity matrix. This is simply a consequence of normalization of

the individual latent vectors {t_{i}}_{i=1}^{p}.

4. Kernel Partial Least Squares Regression in RKHS

Assume a nonlinear transformation of the input variables {x_{i}}_{i=1}^{n} into a feature space F; i.e. mapping Φ : x_{i} ∈ R^{N} → Φ(x_{i}) ∈ F. Our goal is to construct a linear PLS regression model in F. Effectively it means that we can obtain a nonlinear regression model in the space of the original input variables. Denote by Φ an (n × M) matrix of regressors whose i-th row is the vector Φ(x_{i}). Depending on the nonlinear transformation Φ(.) the feature space can by high-dimensional, even infinite dimensional when the Gaussian kernel function is used. However, in practice we are working only with n observations and we have to restrict ourself to finding the solution of the linear regression problem in the span of the points {Φ(x_{i})}_{i=1}^{n}. This situation is analogous to the case when the input data matrix X has more columns than rows; i.e. we are dealing with more variables than measured objects. This motivated Rännar et al. (1994) to introduced the (input space) kernel PLS algorithm to speed up the computation of the components for a linear PLS model. The idea is to compute the components from the (n×n) XX^{T} matrix rather than the (N ×N) X^{T} X matrix when n ≪ N. The same approach can be also used for the computation of the principal components (see Wu et al., 1997a) or the nonlinear version (Schölkopf et al., 1998).

Now, motivated by the theory of RKHS described in Section 2 we derive the algorithm for the (nonlinear) kernel PLS model. From the previous section we can see that by the connection of 2. and 3. step and by using the Φ matrix of mapped input data we can modify the NIPALS-PLS algorithm into the form

1. randomly initialize u

2. t = ΦΦ^{T} u, t ← t/∥t∥

3. c = Y^{T} t

4. u = Yc, u ← u/∥u∥

5. repeat steps 2. – 5. until convergence

6. deflate ΦΦ^{T},Y matrices: ΦΦ^{T} ← (Φ − tt^{T} Φ)(Φ − tt^{T} Φ)^{T} , Y ← Y − tt^{T} Y .

Applying the so-called “kernel trick”; i.e. the fact that Φ(x_{i})^{T} Φ(x_{j}) = K(x_{i}, x_{j}), we can see that ΦΦ^{T} represents the (n × n) kernel Gram matrix K of the cross dot products between all mapped input data points {Φ(x_{i})}_{i=1}^{n}. Thus, instead of an explicit nonlinear mapping, the kernel function can be used. The deflation of the ΦΦ^{T} = K matrix after extraction of the t component is now given by

K ← (I − tt^{T})K(I − tt^{T}) = K − tt^{T} K − Ktt^{T} + tt^{T} Ktt^{T} ,  (9)

where I is an n-dimensional identity matrix. We would like to point out that a similar kernel PLS algorithm can be also derived by the nonlinear modification of the (linear) kernel PLS algorithm described in (Rännar et al., 1994). This modification leads to the extraction of the t, u components from the KYY^{T} and YY^{T} matrices, however this approach can be more fruitful when the multivariate kernel PLS model is assumed (L > 1) as compared to the NIPALS-PLS algorithm described above.

Similarly we can see that the matrix of the regression coefficients B (8) will have the form

B = Φ^{T} U(T^{T} KU)^{−1}T^{T} Y  (10)

and to make prediction on training data we can write

Ŷ = ΦB = KU(T^{T} KU)^{−1}T^{T} Y = TT^{T} Y ,  (11)

where the last equality follows from the fact that the matrix of the components T may be expressed as T = ΦR where R = Φ^{T} U(T^{T} KU)^{−1} (de Jong, 1993, Helland, 1988). It is important to stress that during the iterative process of the estimation of the components {t_{i}}_{i=1}^{p} we made the deflation of the K matrix after each step. Effectively it means that T ≠ KU. Thus, for predictions made on testing points {x_{i}}_{i=n+1}^{n+n_{t}} the matrix of regression coefficients (10) have to be used; i.e.

Ŷ_{t} = Φ_{t}B = K_{t}U(T^{T} KU)^{−1}T^{T} Y ,  (12)

where Φ_{t} is the matrix of the mapped testing points and consequently K_{t} is the (n_{t} × n) “test” matrix whose elements are K_{ij} = K(x_{i}, x_{j}) where {x_{i}}_{i=n+1}^{n+n_{t}} and {x_{j}}_{j=1}^{n} are the testing and training points, respectively.

At the beginning of the previous section we assumed a centralized PLS regression problem. To centralize the mapped data in a feature space F we can simply applied the following procedures (Schölkopf et al., 1998, Wu et al., 1997b)

K = (I − \frac{1}{n}1_{n}1_{n}^{T})K(I − \frac{1}{n}1_{n}1_{n}^{T})  (13)
K_{t} = (K_{t} − \frac{1}{n}1_{n_{t}}1_{n}^{T} K)(I − \frac{1}{n}1_{n}1_{n}^{T}) ,  (14)

where I is again an n-dimensional identity matrix and 1_{n}, 1_{n_{t}} represent the vectors whose elements are ones, with length n and n_{t}, respectively.

In conclusion we would like to make several remarks about the interpretation of the kernel PLS model. For simplicity we will consider the univariate kernel PLS regression case (L = 1) and we denote the (n × 1) vector d = U(T^{T} KU)^{−1}T^{T} Y. Now we can represent the solution of the kernel PLS regression as

f(x, d) = ∑_{i=1}^{n} d_{i}K(x, x_{i}) ,

which agrees with the solution of the regularized formulation of regression (2) given by the Representer theorem in Section 2. Using equation (11) we may also interpret the kernel PLS model as a linear regression model of the form (for more detailed interpretation of linear PLS models we refer the reader to Garthwaite, 1994, Höskuldsson, 1988)

f(x, c) = c_{1}t_{1}(x) + c_{2}t_{2}(x) + … + c_{p}t_{p}(x) = c^{T} t(x) ,

where the {t_{i}(x)}_{i=1}^{p} are the projections of the data point x onto the extracted p components and c is the vector of weights given by (7).

Figure 1: First three principal components (PC) extracted by kernel PCA (left) and components (C) extracted by kernel PLS (right). The curves represents the (principal) component values. Sinc function is shown as a solid line.

4.1 Example

In this section we would like to demonstrate some of the properties of the kernel PLS when applied to approximating the sinc(x) function defined as

f(x) = sinc(x) = \frac{sin|x|}{|x|} .

We generated 100 uniformly spaced samples in the range [−10, 10] and computed the corresponding values of the sinc(.) function which were subsequently centralized. We used the Gaussian kernel function with width equal to 1. In Figure 1 the first 3 principal components (left) computed from the centralized Gram matrix K are compared with the components extracted by the proposed kernel PLS algorithm (right). We can see the qualitative difference between the principal components and components extracted by kernel PLS where also the correlations between the input and output data are used.

In the next step we added the white Gaussian noise with standard deviation 0.2 to the outputs. This corresponds to the ratio between the standard deviation of noise and signal equal to 56%. We generated an additional 80 uniformly spaced testing samples from the same range [−10, 10], however not identical to the previous ones. Using these data points we computed noise-free outputs. The results obtained on training and testing parts of the data are depicted in Figure 2. We can see that although the kernel PLS method fits more precisely noisy training data by the appropriate selection of the components we can avoid the overfitting effect and achieve the same performance on the testing set as with the kernel PCR method. The number of components in the kernel PLS case is significantly smaller. Using the number of components on which minimum test error occurred, in Figure 3, we plotted approximating functions computed on noisy training data. We achieved qualitatively similar results in the case when the Gaussian noise with standard deviation equal to 0.05 and 0.1 was added to the outputs.

Figure 2: Sinc function approximation. Dependence of the training and testing error of kernel PLS (solid line) and kernel PCR (dashed line) on the number of extracted components. Gaussian noise with standard deviation equal to 0.2 was added to the training data set outputs. Error is evaluated in terms of mean squared error (MSE).

Figure 3: Comparison of kernel PLS (solid line) and kernel PCR (dash-dotted line) on noisy sinc function approximation. The number of used (principal) components was selected based on plots in Figure 2 (right) and was 1 and 10 in kernel PLS and kernel PCR, respectively. Gaussian noise with standard deviation equal to 0.2 was added to the data set outputs (dots). The true function is shown dashed.

5. Kernel PCR and Kernel RR in RKHS

In this section, we briefly describe and compare another two kernel-based regularized least squares models; kernel PCR and kernel RR. However, we start with the kernel PCA method for the extraction of nonlinear principal components used in the kernel PCR model.

5.1 Kernel PCA

The PCA problem in a high-dimensional feature space F can be formulated as the diagonalization of an n-sample estimate of the covariance matrix

Ĉ = \frac{1}{n} ∑_{i=1}^{n} Φ(x_{i})Φ(x_{i})^{T} = \frac{1}{n}Φ^{T} Φ ,

where Φ now denotes the (n × M) matrix of the centered nonlinear mappings of the input variables {x_{i}}_{i=1}^{n} ∈ R^{N}. The diagonalization represents a transformation of the original data to new coordinates defined by orthogonal eigenvectors u. We have to find eigenvalues λ ≥ 0 and non-zero eigenvectors u ∈ F satisfying the eigenvalue equation

λu = Ĉu .

When n ≪ M, similar to linear kernel PCA (see, e.g., Wu et al., 1997a), we may transform this eigenvalue problem to the problem of diagonalization of the (n × n) matrix ΦΦ^{T} = K; i.e. solving the eigenvalue problem as described in (Schölkopf et al., 1998, Sirovich, 1987)

Kũ = nλũ = λ̃ũ .  (15)

The eigenvectors {u^{k}}_{k=1}^{n} are then given by

u^{k} = (nλ_{k})^{−1/2}Φ^{T} ũ^{k} = λ̃_{k}^{−1/2}Φ^{T} ũ^{k} ,

where ũ^{k} is the k-th principal component extracted by solving (15) and nλ_{k} = λ̃_{k} the corresponding eigenvalue. Finally, we can compute the projection of Φ(x) onto the k-th nonlinear principal component by

β_{k}(x) = Φ(x)^{T} u^{k} = λ̃_{k}^{−1/2} ∑_{i=1}^{n} ũ_{i}^{k}K(x_{i}, x) .  (16)

Re-writing this projection into the matrix form we can write for the projection of training data points {x_{i}}_{i=1}^{n}

P = ΦΦ^{T} ŨΛ^{−1/2} = KŨΛ̃^{−1/2} = ŨΛ̃^{1/2} ,  (17)

where the columns of Ũ are created by the eigenvectors {ũ^{i}}_{i=1}^{n} and Λ̃ is a diagonal matrix diag(λ̃_{1}, λ̃_{2}, …, λ̃_{n}). Similarly for the projection of testing points {x_{i}}_{i=n+1}^{n+n_{t}} we have

P_{t} = Φ_{t}Φ^{T} ŨΛ̃^{−1/2} = K_{t}ŨΛ̃^{−1/2} .

Note that the assumption of the centralized nonlinear mappings is again transformed to the “centralization” of the K and K_{t} matrices given by (13) and (14), respectively.

5.2 Kernel PCR

Consider the standard regression model in feature space F

y = Φw + ϵ ,  (18)

where y is a vector of n observations of the dependent variable, Φ now represents an (n×M) matrix of zero-mean regressors {Φ(x_{i})}_{i=1}^{n}, w is a vector of regression coefficients and ϵ is the vector of error terms whose elements have equal variance σ^{2}, and are independent of each other. Φ^{T} Φ is proportional to the sample covariance matrix and kernel PCA can be performed to extract its M eigenvalues {λ̃_{i}}_{i=1}^{M} and corresponding eigenvectors {u^{i}}_{i=1}^{M}^{4} (15). Having the eigensystem {λ̃_{i}, u^{i}}_{i=1}^{M} the spectral decomposition (Jolliffe, 1986) of Φ^{T} Φ has the form

Φ^{T} Φ = ∑_{i=1}^{M} λ̃_{i}u^{i}(u^{i})^{T} .

The projection of Φ(x) onto the k-th nonlinear principal component is given by (16). By projection of all original regressors onto the principal components we can rewrite (18) as

y = Bv + ϵ ,  (19)

where B = ΦU is now an (n × M) matrix of transformed regressors and U is an (M × M) matrix whose k-th column is the eigenvector u^{k}. The columns of the matrix B are now orthogonal and the least squares estimate of the coefficients v becomes

v̂ = (B^{T} B)^{−1}B^{T} y = Λ̃^{−1}B^{T} y ,

where Λ̃ = diag(λ̃_{1}, λ̃_{2}, …, λ̃_{M}). It is worth noting that PCR, as well as other biased regression techniques, is not invariant to the relative scaling of the original regressors (Frank and Friedman, 1993). However, similar to OLS regression, the solution of (19) does not depend on a possibly different scaling in individual eigendirections used in the kernel PCA transformation. Further, the results obtained using all principal components for the projection of the original regressor variables for (19) is equivalent to that obtained by least squares using the original regressors. In fact we can express the estimate ŵ of the original model (18) as

ŵ = Uv̂ = U(B^{T} B)^{−1}B^{T} y = (Φ^{T} Φ)^{−1}Φ^{T} y = ∑_{i=1}^{M} λ̃_{i}^{−1}u^{i}(u^{i})^{T} Φ^{T} y

and its corresponding variance-covariance matrix (Jolliffe, 1986) as

cov(ŵ) = σ^{2}U(B^{T} B)^{−1}U^{T} = σ^{2}UΛ̃^{−1}U^{T} = σ^{2} ∑_{i=1}^{M} λ̃_{i}^{−1}u^{i}(u^{i})^{T} ,  (20)

where we used the fact that y ∼ N (Φw, σ^{2}I). Similarly to PLS, to avoid the problem of multicollinearity, PCR uses only some of the principal components. It is clear from (20) that the influence of small eigenvalues can significantly increase the overall variance of the estimate. PCR simply deletes the principal components corresponding to small values of the eigenvalues λ̃_{i}. The penalty we have to pay for the decrease in variance of the regression coefficient estimate is bias in the final estimate. However, if multicollinearity is a serious problem the introduced bias can have a less significant effect than a high variance estimate. If the elements of v corresponding to deleted regressors are zero, an unbiased estimate is achieved (Jolliffe, 1986).

4. For the moment, we are theoretically assuming that n > M. Otherwise we have to deal with a singular

case (n ≤ M) allowing us to extract only up to n − 1 eigenvectors corresponding to non-zero eigenvalues.

Using the first p nonlinear principal components (16) to create a linear model based on orthogonal regressors in feature space F we can formulate the kernel PCR model as (Rosipal et al., 2000a, 2001)

f(x, a) = ∑_{k=1}^{p} v_{k}β_{k}(x) + b = ∑_{k=1}^{p} v_{k} ∑_{i=1}^{n} λ̃_{k}^{−1/2}ũ_{i}^{k}K(x_{i}, x) = ∑_{i=1}^{n} a_{i}K(x_{i}, x) + b ,

where {a_{i} = ∑_{k=1}^{p} v_{k}λ̃_{k}^{−1/2}ũ_{i}^{k}}_{i=1}^{n} and b is a bias term. Similar to kernel PLS and kernel RR (see below) we can assume a centralized regression model leading to a zero bias term b.

We have shown that by removing the principal components whose variances are very small we can eliminate large variances of the estimate due to multicollinearities. However, if the orthogonal regressors corresponding to those principal components have a large correlation with the dependent variable y such deletion is undesirable (experimentally demonstrated in Rosipal et al., 2000b). There are several different strategies for selecting the appropriate orthogonal regressors for the final model (see Jolliffe, 1986, 1982, and references therein). In Section 6 we discuss approaches used in our experiments.

5.3 Kernel Ridge Regression

Kernel RR is another technique to deal with multicollinearity by assuming the linear regression model (18) whose solution is now achieved by minimizing

R_{rr}(w) = ∑_{i=1}^{n}(y_{i} − f(x_{i}, w))^{2} + ξ∥w∥^{2} ,  (21)

where f(x, w) = w^{T} Φ(x) and ξ is a regularization coefficient. The least squares estimate of w is

ŵ = (Φ^{T} Φ + ξI)^{−1}Φ^{T} y ,

which is biased but has lower variance compared to an OLS estimate. To make the connection to the kernel PCR case we express the estimate ŵ in the eigensystem {λ̃_{i}, u^{i}}_{i=1}^{M}

ŵ = ∑_{i=1}^{M}(λ̃_{i} + ξ)^{−1}u^{i}(u^{i})^{T} Φ^{T} y

and corresponding variance-covariance matrix as (Jolliffe, 1986)

cov(ŵ) = σ^{2} ∑_{i=1}^{M} λ̃_{i}(λ̃_{i} + ξ)^{−2}u^{i}(u^{i})^{T} .

We can see, that in contrast to kernel PCR (20), the variance reduction in kernel RR is achieved by giving less weight to small eigenvalue principal components via the factor ξ.

In practice we usually do not know the explicit mapping Φ(.) or its computation in the high-dimensional feature space F may be numerically intractable. Using the dual representation of the linear RR model, Saunders et al. (1998) derived a formula for estimation of the weights w for the linear RR model in a feature space F; i.e. (nonlinear) kernel RR. Again, using the fact that K(x, y) = Φ(x)^{T} Φ(y) we can express the final kernel RR estimate of (21) in the dot product form (Saunders et al., 1998, Cristianini and Shawe-Taylor, 2000)

f(x) = c^{T} k = y^{T} (K + ξI)^{−1}k ,

where K is again an (n × n) Gram matrix and k is the vector of dot products of a new mapped input example Φ(x) and the vectors of the training set; k_{i} = (Φ(x_{i}).Φ(x)). It is worth noting that the same solution to the RR problem in the feature space F can also be derived based on the dual representation of the regularization networks minimizing the regularized risk functional (1) using the quadratic loss function V (y_{i}, f(x_{i})) = (y_{i} − f(x_{i}))^{2} (Girosi et al., 1993, 1995, Haykin, 1999) or through the techniques derived from Gaussian processes (Williams, 1998, Cristianini and Shawe-Taylor, 2000).

In this paper we assume centralized kernel RR (Rosipal et al., 2000b); i.e. we assume the sample mean of the mapped data Φ(x_{i}) and outputs y_{i} to be zero. The centralization of the individual mapped data points is again accomplished by the “centralization” of K and k given by the equations (13) and (14), respectively.

6. Model Selection

To determine unknown parameters in all regression models, cross validation (CV) techniques were used (Stone, 1974). While in kernel RR, a regularization coefficient and parameters of the kernel function have to be estimated, in kernel PLS and kernel PCR it is mainly the problem of appropriate selection of (principal) components.

For a comparison of models using particular values of estimated parameters, the prediction error sum of squares (PRESS) statistic was used.

PRESS = ∑_{i=1}^{n}(y_{i} − f(x_{i}))^{2} ,

where f(x_{i}) represents the prediction of the measured response y_{i}. PRESS was summed over all CV subsets.

6.1 Kernel PLS

In kernel PLS the number of components gradually increases until the model reaches some optimal dimension. For example we can use CV to determine the adequacy of the individual components to enter the final model (Wold, 1978) or use CV for the comparison of whole models of certain dimensionality 1, 2, …, p. In our study we used the second approach and the validity of individual models was compared in terms of PRESS.

6.2 Kernel PCR

The situation is more difficult in the case of kernel PCR than in kernel PLS because principal components are extracted solely based on the description of the input space without using any existing correlations with the outputs. The influence of individual principal components regressors can be consequently measured by the t-test for regression coefficients (Montgomery and Peck, 1992). By assuming a centralized regression model (19) for which the design matrix B satisfies B^{T} B = I, we can write for the t^{2} statistic of k-th regressor t_{k}^{2} ≡ (β_{k}^{T} Y)^{2}, where β_{k} represents the (n×1) vector of the projections of input data onto the k-th principal component. The condition B^{T} B = I simply means sphering of the projected data which can be achieved on the training data by taking P = Ũ in equation (17).

There are several different situations that can occur in PCR. First, the principal directions with large eigenvalues and significant values of t^{2} should always be used in the final model. The principal directions with high eigenvalues and insignificant values of t^{2} should also be included in the final model due to the fact that a significant amount of variability of the input data can be lost. The principal directions with low eigenvalues and insignificant values of t^{2} should always be deleted. The most difficult problems arise when some of the directions with small eigenvalues have a significant contribution to prediction. This situation on two data sets used was already demonstrated in (Rosipal et al., 2000b, 2001). Moreover, in Figure 4 we also give one of the examples we observed on the used data sets. Comparing the left and right graphs we can see that some of the small eigenvalues principal components may have relatively high prediction properties. In contrast we can see that t^{2} values of some high-eigenvalue principal components indicate their low contribution to the overall prediction abilities of the regression model. For further discussion on the topic of principal components selection we refer the reader to (Jolliffe, 1986, Stone and Brooks, 1990).

First, we would like to stress that as a consequence of the orthogonality of regressors the individual single variable models have an independent contribution to the overall regression model. This significantly simplifies the selection of individual regressors and in our study we decided on the following model selection strategy. We were iteratively increasing the number of large eigenvalue principal components entering the model without considering their values of the t^{2} statistic. The criterion employed was the amount of described variance. The rest of the principal components were ordered based on the t^{2} statistic. As with kernel PLS, CV was used to compare the whole models of particular dimension. However, in contrast to kernel PLS the PRESS statistics were used to select the final model over all possible arrangements of the final models; i.e. for a different, fixed number of principal components with large eigenvalues entering the final model.

7. Results

The present work was carried out with two types of kernels. Gaussian kernels K(x, y) =

e^{−( \frac{∥x−y∥^{2}}{d})}, where d determines the width of the Gaussian function. The Gaussian kernel possesses good smoothness properties (suppression of the higher frequency components) and in the case we do not have a priori knowledge about the regression problem we would prefer a smooth estimate (Girosi et al., 1993, Smola et al., 1998). The polynomial kernels

Figure 4: Example of the estimated ordered eigenvalues (left) and t^{2} statistic (right) of the regressors created by the projection onto corresponding principal components. Vertical dashed lines indicate a number of principal components describing 95% of the overall variance of the data. One of the training data partitions of subject D from the regression problem described in Section 7.4 was used.

K(x, y) = ((x.y)+1)^{a} of different orders a were also employed. In the case of a = 1 we used the K(x, y) = (x.y) = x^{T} y kernel leading to the construction of linear regression models. The results were evaluated in terms of R^{2} (coefficient of determination) (Montgomery and Peck, 1992) or in terms of normalized root mean squared error (NRMSE).

7.1 Corn Data

This data set consists of 80 samples of corn measured on 3 different near-infra-red spectrometers (in our study spectra from instrument m5 were used) and is electronically available

at http://www.eigenvector.com/Data/Data sets.html. The wavelength range is 1100-

2498nm at 2 nm intervals (N = 700 channels). The moisture, oil, protein and starch values represent four output variables (L = 4). As the first principal component described 99% of the overall variance this indicated high multicollinearity among the input variables. We have used the spectra to form the matrix of regressors X, however, instead of modeling the real responses we generated four different outputs as follows

y_{1} = exp(\frac{x^{T} x}{2m})  y_{2} = exp(\frac{x^{T} A^{−1}x}{2m_{1}})
y_{3} = (\frac{x^{T} x}{m})^{3} exp(\frac{x^{T} x}{2m}) y_{4} = 0.3y_{1} + 0.25y_{2} − 0.7y_{3} . (22)

A is a symmetric matrix with off-diagonal elements set to 0.8 and diagonal elements set to 1.0. m and m_{1} are averages of {x_{i}^{T} x_{i}}_{i=1}^{700} and {x_{i}^{T} A^{−1}x_{i}}_{i=1}^{700}, respectively. The first 60 samples were used to create a training data set and the remaining 20 samples created a testing set. To the outputs computed on training data we added white noise with normal distribution and with different levels corresponding to ratios of the standard deviation of the noise and the clean output variables. We denote this ratios n/s. For each noise level 25 different training sets were generated.

Leave-one-out cross validation (LOO) applied on the training partition was used to select desired parameters for individual kernel regression models. In the case of kernel PCR the first principal component was always included into the regression models and only the first 30 principal components covering almost the whole data variance were investigated.

Both multivariate and univariate regression models were studied. Multivariate kernel PLS tries to find components that are good predictors for all response variables. In the final model the same components are used for the prediction of individual responses. These components are determined based on the PRESS statistic computed as the summation of squared errors over all response variables. Multivariate kernel PLS might be advantageous especially when predicted response variables are highly multicollinear. We observed that both approaches lead to the same results on data sets with a smaller noise level, however, for noise levels equal to n/s = 60% and n/s = 90% multivariate kernel PLS provides superior predictions on the test set. Multivariate RR approach was discussed by Frank and Friedman (Frank and Friedman, 1993). It was shown that if the original response variables are correlated it might be profitable to assume separate RR or PCR regression models on decorrelated outputs rather than on the original responses. However, by applying this method we did not observe an improvement on test set predictions.

The goodness of prediction of the univariate kernel RR, kernel PCR and kernel PLS regression models on the test set are summarized in Table 1. We may observe comparable performance of all three kernel regression techniques employed. However, the results achieved with multivariate kernel PLS regression were superior to the kernel PCA and kernel RR models in the case of a higher noise level n/s = 90%. In this case multivariate kernel PLS resulted in R^{2} values equal to 0.95 and 0.93 for the prediction of y_{1} and y_{2} response variables on the test set. Increasing the noise level leads to the selection of a smaller number of (principal) components. Although we are losing the possibility of finer approximation of the function those components, especially in the case of kernel PLS, may still give relatively good performance even in the situation where there is a higher noise level. However, we have to note that this is due to the relatively simple nonlinear dependencies that we constructed by (22). In the nonlinear case, on average, univariate kernel PLS uses less than 80% of the components used by kernel PCR. However, in the case of linear regression (i.e. using the polynomial kernel K(x, y) = (x.y)) the number of used components is approximately the same.

We would like to note that using the original response variables (moisture, oil, protein, starch) we did not achieve significantly better results using polynomial kernels of higher orders compared to linear regression.

7.2 Polymer Data

This data set is taken from a polymer test plant (Ungar, 1995). There are 10 input variables (N = 10), measurements of controlled variables in a polymer processing plant (temperatures, feed rates, etc.), and 4 output variables (L = 4) are measures of the output of that plant. It is claimed that this data set is particularly good for testing the robustness of nonlinear modeling methods to irregularly spaced data.

We first took 41 samples as training and the remaining 20 samples as testing data. LOO applied on the training partition was used to select the desired parameters for individual

Noise  kernel PLS  kernel PCR  kernel RR
level y_{1} y_{2} y_{3} y_{4} y_{1} y_{2} y_{3} y_{4} y_{1} y_{2} y_{3} y_{4}

K(x, y) = (x.y)

15% .96 .95 .76 .75 .95 .94 .76 .72 .96 .95 .76 .74
30% .94 .93 .74 .70 .92 .93 .75 .73 .94 .94 .74 .71
60% .86 .87 .75 .73 .81 .89 .67 .71 .88 .89 .74 .73
90% .77 .75 .70 .73 .69 .77 .62 .60 .78 .81 .69 .73

K(x, y) = ((x.y) + 1)^{2}

15% .98 .99 .96 .97 .98 .97 .93 .94 .98 .98 .96 .97
30% .96 .96 .91 .94 .96 .94 .87 .88 .97 .97 .91 .94
60% .87 .89 .77 .85 .85 .88 .71 .76 .90 .92 .84 .85
90% .73 .77 .70 .79 .70 .78 .55 .70 .78 .84 .80 .82

K(x, y) = ((x.y) + 1)^{3}

15% .99 .99 .98 .98 .99 .99 .96 .97 .99 .99 .97 .98
30% .98 .97 .94 .95 .97 .97 .89 .91 .98 .97 .95 .96
60% .93 .89 .88 .88 .88 .88 .77 .82 .93 .90 .90 .91
90% .85 .77 .85 .85 .72 .73 .61 .79 .83 .77 .86 .86

Table 1: Goodness of prediction in R^{2} terms. The values correspond to an average of 25 different simulations. Noise level is represented as the ratio between the standard deviation of the added Gaussian noise and the standard deviation of the generated response variables y_{1}, y_{2}, y_{3}, y_{4} (22).

kernel regression models. Polynomial kernels of different orders were used. The indication of high multicollinearity (the ratio between the first and fourth eigenvalues of the covariance matrix of the response variables equals 628) among the response variables suggests that assuming a multivariate regression approach may by profitable in this case.

Table 2 compares the goodness of prediction of the multivariate kernel PLS on the test set with the kernel RR and kernel PCR regression models used on decorrelated outputs. The results applying univariate regression models on the same data set are summarized in Table 3.

We can see that all three regression models achieved comparable results when the best predictions on individual response variables are compared. We may see that univariate kernel PLS provides better prediction on the responses y_{1} and y_{2} while its multivariate modification seems to be better for the prediction of y_{3} and y_{4}. Univariate kernel RR and kernel PCR are better on the prediction of y_{2}, y_{3} and y_{4}, while decorrelation of the outputs may improve the prediction of y_{1}. However, we have to say that these results may also depend on the model selection criterion used. Further experiments using different data sets and model selection criteria will have to be employed to provide a more valuable comparison. Results also suggest that on this data set kernel PLS provides the most consistent results over the range of different orders of the polynomial kernel. Simulations with Gaussian kernels of different widths did not lead to better performance.

Order  kernel PLS  kernel PCR  kernel RR
a  y_{1} y_{2} y_{3} y_{4} y_{1} y_{2} y_{3} y_{4} y_{1} y_{2} y_{3} y_{4}
1  .43 .06 .73 .77 .23 .68 .83 .79 .36 .21 .79 .79
2  .90 .30 .89 .88 0^{∗} .0^{∗} .63 .62 .0^{∗} .0^{∗} .68 .86
3  .84 .57 .90 .90 .89 .33 .60 .68 .86 .0^{∗} .58 .54
4  .79 .69 .87 .87 .84 0^{∗} .74 .74 .79 .30 .85 .83
5  .72 .65 .65 .57 .80 .62 .76 .81 .70 .37 .73 .71

Table 2: Goodness of prediction of the y_{1}, y_{2}, y_{3}, y_{4} in terms of R^{2}. Bold numbers indicate the best achieved prediction on individual response variables. Polynomial kernels of different orders a were used. 0^{∗} represents a case when averaged performance of the model is worse than assuming a linear model equal to the mean of a response variable.

Order  kernel PLS  kernel PCR  kernel RR
a  y_{1} y_{2} y_{3} y_{4} y_{1} y_{2} y_{3} y_{4} y_{1} y_{2} y_{3} y_{4}
1  .21 .0^{∗} .71 .78 .39 .0^{∗} .77 .78 .28 .0^{∗} .76 .76
2  .90 .04 .85 .87 .82 .41 .84 .84 .79 .45 .90 .90
3  .91 .59 .85 .86 .87 .63 .55 .81 .52 .0^{∗} .67 .22
4  .93 .75 .68 .66 .81 .58 .88 .88 .68 .71 .78 .75
5  .85 .79 .58 .81 .81 .79 .83 .84 .60 .74 .66 .66

Table 3: Goodness of prediction of the y_{1}, y_{2}, y_{3}, y_{4} in terms of R^{2}. Bold numbers indicate the best achieved prediction on individual response variables. Polynomial kernels of different orders a were used. 0^{∗} represents a case when averaged performance of the model is worse than assuming a linear model equal to the mean of a response variable.

7.3 Chaotic Mackey-Glass Time-Series

The chaotic Mackey-Glass time-series is defined by the differential equation

\frac{ds(t)}{dt} = −b_{1}s(t) + b_{2} \frac{s(t − τ)}{1 + s(t − τ)^{10}}

with b_{1} = 0.1, b_{2} = 0.2. The data were generated with τ = 17 and using a second-order Runge-Kutta method with a step size 0.1. Training data is from t=200 to t=3200 while test data is in the range t= 5000 to 5500. To this generated time-series we added white noise with normal distribution and with different levels corresponding to ratios of the standard deviation of the noise and the clean Mackey-Glass time-series.

The training data partitions were constructed by moving a “sliding window” over the 3000 training samples in steps of 250 samples. This window had a size of 500 samples. The validation set was then created by using the following 250 data points. This created ten partitions of size 500/250 (training/validation) samples.

All regression models were trained to predict the value sampled 85 steps ahead (L = 1) from inputs at time t, t − 6, t − 12, t − 18 (N = 4).

The Gaussian kernel was used. We estimated the variance of the overall clean training set and based on this estimate σ̂^{2} ≐ 0.05 the CV technique was used to find the optimal width d from the range ⟨0.01, 0.2⟩ using the step size 0.01. A fixed test set of size 500 data points was used in all experiments. The performance of the regression models to make predictions on “clean” test set of 500 data points was evaluated in terms of NRMSE.

The results achieved using the individual regression models are summarized in Table 4. We can see that there are no significant differences among the methods employed. However, comparing kernel PLS and kernel PCR we can observe a significant reduction in the number of components used in the case of kernel PLS regression. In some cases kernel PLS uses less than 10% of the number of components used by kernel PCR.

Increasing the value of d leads to a faster decay of the eigenvalues (see, e.g., Williamson et al., 1998) and to the potential loss of the fine structure due to a smaller number of nonlinear principal components describing the same percentage of all the data variance. Increasing levels of the noise has the tendency to increase the optimal value for the d parameter which coincides with the intuitive assumption about smearing out the local structure (for the discussion on this topic see Rosipal et al., 2001). In contrast small values of d will lead to “memorizing” of the training data structure. Thus, in Figure 5 we also compared the results on the noisy time series (n/s = 22%) and their dependence on the width d of the Gaussian kernel. Similar behavior was observed for data with n/s = 11%. We may observe a smaller range of the d values on which kernel PLS and kernel PCR achieves the optimal results on the testing set compared to kernel RR. However, the results also suggest a smaller variance in the case of latent variable projection methods; i.e. kernel PLS and kernel PCR.

Noise  kernel PLS  kernel PCR  kernel RR
level  NRMSE # of C NRMSE # of PC NRMSE
n/s=0.0% 0.048  155  0.046  383  0.044
(0.031) (38) (0.030)  (78)  (0.027)
n/s=11%  0.322  7  0.327  79  0.321
(0.030)  (2)  (0.030)  (35)  (0.041)
n/s=22%  0.455  6  0.462  48  0.451
(0.021)  (2)  (0.031)  (24)  (0.029)

Table 4: The comparison of the approximation errors (NRMSE) of prediction, the number of used components (C) and principal components (PC). The values represent an average of 10 simulations. The corresponding standard deviations are presented in parentheses.

7.4 Human Signal Detection Performance Monitoring

In this study eight male Navy technicians experienced in the operation of display systems performed a signal detection task. Event related potentials (ERP) and performance data from an earlier study (Trejo and Shensa, 1999, Trejo et al., 1995, Koska et al., 1997) were used. The performance of the subjects was measured in terms of PF1 measure (L = 1)

Figure 5: Comparison of the results achieved on the noisy Mackey-Glass (n/s=22%) time series with the kernel PLS (top), kernel PCR (middle) and kernel RR (bottom) methods. Ten different training sets of size 500 data points were used. The performance for different widths (d) of the Gaussian kernel is compared in normalized root mean squared error (NRMSE) terms. The error bars represent the standard deviation on results computed from ten different runs.

based on their accuracy, confidence and reaction time to detect relevant stimuli. For details on the experimental setting see (Rosipal et al., 2001).

The results achieved on individual subjects in our former studies informed our choice of the Gaussian kernel. For each individual subject we split the data into 10 different 55% and 45% training and testing partitions. Eleven-fold CV to estimate desired parameters was applied on each training partition. After CV a final model was tested on an independent testing partition. This was repeated 10 times for each training and testing data pair. The validity of the models was measured in terms of R^{2}.

Table 5 summarizes the results achieved on eight subjects (A to H), using kernel PLS, kernel PCR and kernel RR methods. As with results reported on the Mackey-Glass time series prediction we can not observe any significant differences among the kernel regression models. The number of components used in the case of kernel PLS is on average 10 times lower when compared to kernel PCR.

Subject  kernel PLS  kernel PCR  kernel RR
R^{2}  # of C R^{2}  # of PC R^{2}
A  0.841 27.9  0.841 373.1  0.841
(891 ERP) (0.025) (16.3) (0.023) (30.9)  (0.025)
B 0.883 33.9 0.882 224.4 0.883
(592 ERP) (0.027) (12.3) (0.026) (32.2)  (0.026)
C  0.741 15.5  0.740 134.8  0.747
(417 ERP) (0.060) (4.1)  (0.044) (17.9)  (0.044)
D  0.870 24.8  0.870 241.2  0.874
(702 ERP) (0.011) (6.0)  (0.010) (50.3)  (0.010)
E  0.942 42.4  0.941 274.4  0.943
(734 ERP) (0.006) (20.1) (0.006) (42.2)  (0.007)
F  0.884 24.6  0.875 186.6  0.886
(614 ERP) (0.023) (9.9)  (0.025) (61.4)  (0.024)
G  0.895 23.6  0.893 323.3  0.895
(868 ERP) (0.018) (13.5) (0.018) (53.2)  (0.017)
H  0.827 19.7  0.825 280.4  0.827
(776 ERP) (0.022) (7.0)  (0.022) (49.0)  (0.022)

Table 5: The comparison of R^{2} and the number of used components (C) and principal components (PC), respectively, for subjects A to H. The values represent an average of 10 different simulations and the corresponding standard deviations are presented in parentheses.

8. Discussion and Conclusions

On several different regression tasks we compared the proposed kernel PLS method with kernel PCR and kernel RR techniques. We show that kernel PLS provides the same results as kernel PCR and kernel RR. However, in comparison to kernel PCR, the kernel PLS method uses a much smaller number of qualitatively different components. As demonstrated in Section 4.1, using the existing correlations between outputs and mapped inputs, kernel PLS may provide components which follow more closely the investigated nonlinear function. However, as with kernel PCA, the interpretation of these components in the input space may be difficult.

There exists a large body of literature comparing standard OLS regression with PLS, PCR and RR (see, e.g., Frank and Friedman, 1993, Stone and Brooks, 1990). Assuming a construction of regularized linear regression models in a RKHS we can make some conclusions by using the analogy with the reported observations. First, in the situation where high multicollinearity among regressors exists OLS leads to the unbiased but high variance estimate of regression coefficients. PLS, PCR and RR are designed to shrink the solution to the regression from the areas of the low data spread resulting in biased but lower variance estimates. Second, there exist real world regression problems where the number of observed variables N significantly exceeds the number of samples (observations) n—a situation quite common in chemometrics. Moreover, we may usually also observe that the real rank of the matrix of regressors is significantly lower than n and N. The projection of the original regressors to the “real” latent variables is the main advantage of methods such as PLS or PCR. This is also similar to the situation where the input variables are corrupted by a certain amount of noise (the situation with noisy Mackey-Glass time series and ERP data sets). By the projection of original data to the components with higher eigenvalues we may usually discard the noise component contained in the original data (assuming kernel PCR as discussed in Rosipal et al., 2001). We hypothesize that both situations are also quite common when a kernel-type of regression is used. Usually we nonlinearly transform the original data to the high-dimensional space whose dimension M is in many cases much higher than the number of observations M ≫ n. Although, assuming high-dimensional feature spaces, we cannot diagnose multicollinearity by the inspection of the sample covariance matrix, the indication of strong multicollinearity may by detected from a large ratio between maximal and minimal eigenvalues of the covariance matrix. The eigenvalues can by estimated by the kernel PCA method. Values greater than 100 usually indicate strong multicollinearity among regressors (Montgomery and Peck, 1992). In many cases, in our data sets we observed that the ratio between the larger and smaller eigenvalues exceeded 1000, indicating severe multicollinearity.^{5}

The proposed kernel PLS uses the NIPALS procedure to iteratively estimate the desired components. We have already pointed out that the NIPALS algorithm is very similar to the power method and as with this method was found to be very robust for solving eigenvalue-eigenvector related problems where dominant eigenvectors are calculated one at a time. The rate of the convergence of both algorithms is given by the ratio of two largest eigenvalues (Malthouse, 1995). Both the NIPALS-PLS procedure described in Section 4 and the deflation procedure (9) used after the extraction of the individual components scale as O(n^{2}). The need to repeat these procedures increases the computational costs in direct proportion to the number of desired components. However, on the employed data sets we have demonstrated that the best results are achieved with the number of components p ≪ n. Moreover, investigating the curves of the PRESS statistics computed on validation sets we observed that we usually do not need to investigate a wide spectrum of components as a rather strong increase of PRESS occurs after extracting the optimal number of components. Assuming n > L, the procedure (11) for the estimation of the desired training data set output values scales as O(pnL). Using the procedure (12) to make the prediction computed on a single test data point, the complexity scales as O(pn^{2} + p^{3}), where the second term associated with the inversion of the (p × p) matrix may be neglected in the case that p ≪ n. Moreover, in this procedure we do not even need to invert the whole (n × n) Gram matrix. In fact using the kernel PLS method on large scale regression tasks we may avoid storing the whole Gram matrix K. Re-computation of its elements may, however, significantly slow down the whole algorithm.

5. We investigated the situation by using the Gaussian kernel with different width parameters and polyno-

mial kernels of different orders.

Acknowledgments

We would like to thank Bogdan Gabrys for helpful discussions on model selection techniques. ERP data were obtained under a grant from the US Navy Office of Naval Research (PE60115N), monitored by Joel Davis and Harold Hawkins. R. Rosipal is funded by a research grant for the project “Objective Measures of Depth of Anaesthesia”; University of Paisley and Glasgow Western Infirmary NHS trust, and is partially supported by Slovak Grant Agency for Science (grant No. 2/1136/21). L.J. Trejo is supported by the U.S. National Aeronautics and Space Administration, Aerospace Operations Systems Program, and by the Intelligent Systems Program.

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