Applied Mathematics
Vol.08 No.04(2017), Article ID:75585,23 pages
10.4236/am.2017.84037
The Principal Component Transform of Parametrized Functions
Ilia Zabrodskii, Arcady Ponosov
Department of Science and Technology, Norwegian University of Life Sciences, Å s, Norway
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: February 23, 2017; Accepted: April 21, 2017; Published: April 24, 2017
ABSTRACT
Many advanced mathematical models of biochemical, biophysical and other processes in systems biology can be described by parametrized systems of nonlinear differential equations. Due to complexity of the models, a problem of their simplification has become of great importance. In particular, rather challengeable methods of estimation of parameters in these models may require such simplifications. The paper offers a practical way of constructing approximations of nonlinearly parametrized functions by linearly parametrized ones. As the idea of such approximations goes back to Principal Component Analysis, we call the corresponding transformation Principal Component Transform. We show that this transform possesses the best individual fit property, in the sense that the corresponding approximations preserve most information (in some sense) about the original function. It is also demonstrated how one can estimate the error between the given function and its approximations. In addition, we apply the theory of tensor products of compact operators in Hilbert spaces to justify our method for the case of the products of parametrized functions. Finally, we provide several examples, which are of relevance for systems biology.
Keywords:
Principal Component Analysis, Discretization of Functions, Metamodeling, Latent Parameters
1. Introduction
This study is closely related to applications in the so-called “metamodeling” of differential equations, where a “proper” model of an e.g. complex biological process is replaced by its approximation which contains “most information” about the model, but which is simpler. In particular, the true parameters of the model are replaced by “the latent parameters”, which makes the model linear with respect to the latter and hence enables the usage of the (if necessary, partial) least-squares regression. This explains why this idea proved to be efficient in parameter estimation (see e.g. [1] ). This also justifies the high numerical efficiency of metamodeling, which has been widely used in statistics [2] , chemometrics [3] , biochemstry [1] , genetics [4] [5] [6] , infrared spectroscopy [7] to simplify theoretical and computational analysis of the “true” models.
Let be a function, where and , being a space of parameters and be a given number. The kth Principal Component Transform (PCT) is a specially constructed parametrized function of the form . The image is constructed to yield the minimum distance (in some sense) between and all possible approximations of of the form . The distance is chosen to ensure an efficient way to estimate the deviation of from .
Geometrically, the parametrized function may be regarded as a curve in a separable Hilbert space. Then can be inter- preted as a projection of this curve onto an -dimensional subspace, which is chosen in such a way that the image gives a best possible individual fit to among all -dimensional subspaces. As we will see in Subsection 3.1, this necessarily leads to nonlinearity of the mapping PCT.
As we will see in Subsection 3.3, discretizing the function and its PCT yields matrices and the projections onto their first principal compo- nents, respectively. This explains our terminology: PCT can be regarded as a functional analog of the principal component analysis (PCA) of matrices. This terminology was suggested by Prof. E. Voit in a private talk with the second author during his seminar lecture in Oslo in 2014.
All the papers cited above concentrate on efficiency of the metamodeling approach and disregard mathematical properties of PCT and their justification, which is, for instance, quite important for understanding the limitations of the method and describing the exact conditions under which the method is applicable. In particular, the convergence properties of the sequence of metamodels to the original model has not been studied in the available literature. In our paper we try to fill this gap suggesting a rigorous mathematical approach to PCT and analysis of its basic properties. More precisely, we demonstrate how the theory of compact operators in separable Hilbert spaces can be used to provide such an analysis.
The paper is organized as follows. In Section 2 we introduce the distance in the space of parametrized functions, formulate the theorem on the best indivi- dual fit in terms of PCT of functions (Subsection 2.1) and provide some examples relevant for systems biology (Subsection 2.2). In Section 3 we study mathematical properties of PCT: nonlinearity (Subsection 3.1), continuity (Subsection 3.2) and show relations of PCT and PCA via discretization of functions (Subsections 3.3 and 3.4). In Section 4 we study PCT of products of parametrized functions which are interpreted as elements of the tensor product of two or several Hilbert spaces (Subsection 4.1). We aslo show that PCT pre- serves the tensor products and therefore the product of parametrized functions (Subsection 4.2) and give some examples (Subsection 4.3). In Appendix 5 we offer short proofs of some auxiliary results used in the paper: Allahverdiev’s theorem (Subsection 5.1) and some propositions related to tensor products of linear compact operators in Hilbert spaces (Subsection 5.2).
2. The Best Individual Fit Theorem
In this section we define the distance in the space of parametrized functions and describe how best individual fits to a given function can be obtained using the theory of compact operators in Hilbert spaces. We also prove nonlinearity and continuity of PCT and give some specific examples.
2.1. The Distance in the Space of Parametrized Functions
Let be a compact subset of and be a compact subset of We consider the separable Hilbert spaces and with the standard scalar products and the norms .
Suppose we are given a measurable, square integrable function , i.e.
(1)
The aim is to find a best possible approximation of in the class of all functions of the form , where and .
To explain better the nature of topology we use in this case let us have a look at finite dimensional Hilbert, i.e. Euclidean, spaces. Let be an -matrix, for instance, a discretized function where . In this case, the best approximation to in the class of -matrices of rank not greater than is given by the first terms in the singular value decomposition of :
(2)
where and are the normalized eigenvectors of the matrix and is the conjugate (transpose) of a matrix . In other words,
(3)
The matrix norm is defined as , where is the Euclidean norm in .
Now we will look at arbitrary real separable Hilbert spaces which are denoted by and and which are equipped with the scalar products and and the corresponding norms and , respectively. Assume that
is a linear compact operator. Its norm is again defined as .
Put
(4)
We want to find an operator for which . The construction of is very close to the singular value decomposition of matrices.
Assume that is the adjoint of . Then the linear compact operators are self-adjoint and positive-definite.
Let be all positive eigen- values of the operator , the associated normalized eigenvectors being , respectively:
(5)
It is well-known that can always be chosen to be orthogonal: and for any there is a unique set , and a unique for which and, moreover, Now, the operator can be represented as
(6)
where and the convergence is understood in the sense of the norm in the space . The truncated versions of this representation is defined by
(7)
The following result, a short proof of which is offered in Appenix 5.1, is known as Allahverdiev’s theorem, see e.g. [8, Chapter II, p. 28]:
Theorem 1. For any linear compact operator
(8)
The functions in numerical calculations are usually replaced by their discreti- zations, which in the case of parametrized functions gives matrices. That is why, the distance in the space of the parametrized functions should be consistent with the distance in the space of matrices, so that we can get all the advantages of the finite dimensional singular value decomposition as well as Allahverdiev’s theorem. To define the distance in the space of matrices we have to interpret matrices as linear operators between two Euclidean spaces. Analo- gously, we have to interpret parametrized functions as operators between suitable Hilbert spaces, and define the distance accordingly.
Let us therefore go back to the spaces , , where , as before, is a compact subset of and is a compact subset of We denote the norm in both spaces as Consider the integral operator
(9)
Under the assumptions of the square integrability of the kernel the operator becomes compact and linear from the space to the space (see e.g. [9] , Chapter 7, p. 202]).
The distance between two square integrable parametrized functions and can be now defined in the following way:
(10)
where is defined in (9) and The norm of the linear operators acting from to is defined in the standard way.
Remark 1. Evidently,
(11)
for some constant . Therefore, -convergence of the sequence implies the convergence in the sense of the distance dist.
Let be the adjoint of , so that
(12)
Now, the self-adjoint and positive-definite integral operators
(13)
can be written as follows:
(14)
and
(15)
respectively. Let, as before,
(16)
be all positive eigenvalues of the integral operator (14) associated with its normalized and mutually orthogonal eigenfunctions , i.e.
(17)
From Theorem 1 we immediately obtain the Best Individual Fit Theorem.
Theorem 2. For a given function satisfying (1) the best approximation of in the class of all functions of the form , where and , is given by
(18)
where are the normalized, mutually orthogonal eigenfunctions of the operator (14) and . Moreover, for all natural .
In other words,
(19)
Remark 2. The functions have the following properties (which we do not use in this paper):
• for all ;
• for all ;
• for all .
Definition 1.
• The kth Principal Component Transform (PCT) of the function is defined as
(20)
• The Full Principal Component Transform of the function is given by
(21)
We will also write
We remark that none of these transforms is uniquely defined: even if all are all different, we have always a choice between two normalized eigenfunctions . However, the distance between and any is independent of the projection we use. On the other hand, this means that the properties of PCT should be formulated with a care.
2.2. Examples of PCT
In this subsection we consider three examples which are of importance in systems biology.
Example 1. Let
(22)
Assume that Then, using Formulas (14) and (15), we obtain the following representations of the kernels and
(23)
(24)
Therefore the normalized eigenfunctions can be obtained from the equation
(25)
The functions can be alternatively found from the equations
(26)
The parametrized power function is of crucial importance in the bioche- mical system theory, where represents the concentration of a metabolite, while stands for the kinetic order. In the case of several metabolites, one gets products of such power functions, which, in turn, are included into the right- hand side of the so-called “synergetic system”, see (e.g. [10] , Chapter 2, p. 51) and the references therein. The products of parametrized power functions are considered in Section 4.
Example 2. Consider the function
(27)
Assume that Then, using Formulas (14) and (15), we obtain the following representations of the kernels and
(28)
(29)
We denote for simplicity
(30)
and get
(31)
Therefore the normalized eigenfunctions can be obtained from the equation
(32)
The functions can be also obtained from the equations
(33)
The function is often used in the neural field models, where it serves as the simplest example of the so-called “connectivity functions” describing the interactions between neurons, see e.g. [11] and the references therein.
Example 3. Consider the Hill function
(34)
Assume that , , Putting and we obtain
(35)
and
(36)
The Hill function plays central role in the theory of gene regulatory networks, where it stands for the gene activation function, being the gene concentra- tion and being the activation threshold, see e.g. [12] and the references therein.
3. Some Properties of PCT
The Principal Component Transform is not uniquely defined. That is why, we will use a special notation when comparing PCT of different func- tions, namely, we will write if there exist coinciding versions of PCT of and .
3.1. PCT Is Homogeneous, But Not Additive
Theorem 3.
1. for any and
2. In general, is different from
Proof.
1. The case is trivial. We assume therefore that . Let and , see (21). By definition, are normalized, mutually orthogonal eigenfunctions of the ope- rator and . Let . Then
(37)
so that are the same for and . On the other hand, and
(38)
2. Before constructing an example illustrating nonlinearity of PCT we remark that this statement, in its more precise formulation, says that there are no
versions of , , , for which
Let and the functions satisfy
(39)
We put
(40)
To calculate PCT we observe that both operators have a 2-dimensional image in . Using the representation where we reduce the operators and to the matrices
so that
(41)
where and are row and column vectors, respectively.
Matrices and are symmetric. Then and . The first eigenpairs of and are and , respectively. There- fore the best rank 1 approximations of and are
so that and which both are operators with an 1-dimensional image. However, their sum
(42)
has a 2-dimensional image, as its representation in the basis is given by the non-singular matrix . Therefore cannot coincide with any version of .
3.2. PCT Is Continuous
Let us consider a sequence of parametrized, square integrable functions .
Theorem 4. Let and for some parame- trized, square integrable functions . Then for any version there are versions such that
(43)
Proof. Let , . We define the compact linear integral operators using the kernels , respectively. By the definition of the dist we immediately get that
Let be the normalized, mutually orthogonal eigenfunctions of the operator corresponding to its first eigenvalues . Since converges to the operator in norm, we can always choose a sequence of the eigenfunctions such that
(44)
In this case
(45)
Therefore which implies
(46)
The above theorem can be reformulated in terms of robustness of PCT.
Corollary 1. Let and be a parametrized, square inte- grable function and . Then given an there is a such that for every parametrized, square integrable function the follow- ing holds true:
(47)
for some suitable versions of PCT.
3.3. Discretization of Functions
In the papers [5] [6] , which are aimed at applying the metamodeling approach to gene regulatory networks, the approximations of the parametrized sigmoidal functions are performed numerically by using discretization and SVD of the resulting matrices. The continuity of PCT, proved in the previous subsection, can now be used to justify this analysis and, in particular, the results on the number of the principal components ensuring the prescribed precision.
In this subsection we suppose that all functions are continuous, which is sufficient for most applications. The general case is, however, unproblematic as well if we slightly adjust the approximation procedure.
Let be a continuous function on a compact set where
For all is divided into measurable subsets :
(48)
We define the sequence of the functions as follows:
(49)
where is an arbitrary point in
Lemma 1. Let be a continuous function on . Then
(50)
provided that as .
Proof. The function is continuous on the compact set , therefore is uniformly continuous on . Then for all there is such that
(51)
On the other hand, there is a number for which as long as . Let be an arbitrary point from . Then for any there is such that . Taking now an arbitrary we obtain
(52)
so that , where is the Lebesgue measure of the set .
Hence
Corollary 2. Let and be a parametrized, continuous function, be a sequence of discrete approximations satisfied the assump- tions of Lemma 1. Then for any version there are versions such that
Finally, we observe that if are defined as , where for any and are measurable partitions of and , respectively, and
, then PCT of the discrete functions coincide with the - truncated SVD of the matrix . In the next subsection we provide an example of such approximation stemming from the biochemical systems theory.
3.4. Examples of Discrete Approximations
In this subsection we study the parametrized power function defined on the interval with the parameter values To approximate this function we construct a matrix as follows: we divide into parts: Similarly, we divide the interval into parts. Every entry of the matrix will be given by the values :
(53)
The corresponding discretization of will be then given by the matrix
(54)
The vectors and can be obtained from the singular value decompo- sition of the matrix
(55)
where the rows of the scores matrix consists of the numbers and the columns of the loadings matrix are the vectors . As an example, let us consider the case , , , . Then
(56)
so that the Expression (54) becomes
(57)
Assume now that . This value corresponds to row in the matrix . We find a number as follows:
(58)
This yields
and hence
(59)
where are the columns in the loadings matrix , see Figure 1.
The Figure 1 depicts the power function vs. its PCT with 4 components; ; the error is estimated as and the Hill function vs. its PCT with 12 components; ; the error is estimated as . The Figure 2 depicts the cumulative normal distribution function vs. its PCT with 27 components and ; the error is estimated as and the normal distribution function vs. its PCT with 25 PCs; ; the error is estimated as .
(a) (b)
Figure 1. (a) The power function and its PCT; (b) The Hill function and its PCT.
(a) (b)
Figure 2. (a) The cumulative normal distribution function and its PCT; (b) The normal distribution function and its PCT.
4. PCT of Products of Functions
To calculate PCT of products of parametrized functions we need to apply the theory of tensor products of Hilbert spaces and compacts operators. Appendix 5.2 includes all the necessary details we need in this section.
Below we use the following notation (where ):
• , are compact sets;
• , ;
• , , , ;
• , , are square integrable functions and
• so that ;
• so that .
4.1. Products of Parametrized Functions
Theorem 5. In the above notation:
• ,
•
Proof. We use the definition of the tensor product from Appendix 5.2.
Let have an orthonormal basis so that any can be represented as
(60)
where
We prove now that the set is an orthonormal basis in the space . Its orthonormality follows directly from its definition. It remains therefore to check that the set of all linear combinations of the elements from is dense in . Indeed, the set of continuous functions, and hence the set of polynomials , on is dense in . On the other hand, the set of polynomials of the form spans the set and, finally, the set spans the set . Thus, spans and we have proved that any can be represented as the -convergent series
(61)
for some set satisfying
(62)
Defining
(63)
and comparing the Representation (61) with the Formula (94) proves the equality . The equality can be checked similarly.
Let us now prove the last formula of the theorem. First of all, we remark that the Definition (63) implies
(64)
for any .
By the assumptions on the kernels, the operators in this equality are linear and bounded. Therefore, it is sufficient to check the equality for (see Appendix 5.2).
(65)
due to (64). Hence . Comparing this for- mula with the Definition (100) completes the proof of the theorem.
4.2. PCT Preserves Tensor Products
The main theoretical result of this subsection is the following theorem:
Theorem 6.
(66)
Proof. For we have by definition
(67)
where are normalized, mutually orthogonal eigenvectors of the operator corresponding to the eigenvalues and .
Put and . Using the properties of the tensor product listed in Appendix 5.2 we obtain
(68)
where
(69)
This proves that are normalized, mutually orthogonal eigenvectors of the operator corresponding to the eigenvalues .
On the other hand,
(70)
Therefore,
(71)
which proves the theorem.
Remark 3. Theorem 6 is only valid for the full PCT. The truncated versions of PCT are not necessarily valid, as the order of the singular values depends on the magnitude of the eigenvales and .
4.3. Examples of Products of Parametrized Functions
In this subsection we describe the kernels of the integral operators related to products of parametrized functions from Subsection 0. These examples are of importance in systems biology.
Example 1. Consider the following function
(72)
Assume that Then, using Formulas (14) and (15), we obtain the following representations of the kernels and
Example 2. Consider the function
(73)
Assume that Then, using Formulas (14) and (15), we obtain the following representations of the kernels and
Example 3. For the Hill function we obtain
(74)
Assume that
Putting and Then, using Formu- las (14) and (15), we obtain the following representations of the kernels and
(75)
(76)
Remark 4. The eigenfunctions of the integral operators with the kernels that are products of parametrized functions are, according to Subsection 5.2, also products of the respective eigenfunctions of the factors.
5. Conclusions
The main results of the paper can be summarized as follows. We defined the distance in the space of parameterized functions. We defined the -th Principal Component Transform (PCT) and the Full Principal Component Transform of functions . The kth PCT is the best approximation of the given function, i.e. it minimizes . We proved that if the sequence of functions converge to the continuous function , then the sequence of the PCT of will converge to the PCT of . Some properties of PCT were considered. These results can also serve as theoretical background for the design of some metamodels. Using the theory of the tensor product of Hilbert spaces and compact operators we calculated the PCT of products of functions. We provided several examples of the discrete approximations and products of the parametrized functions.
We will emphasize that our study is related to systems biology. In future works we aim to investigate the problem of “sloppiness” in nonlinear models [1] and create an effective parameter estimation method for the “S-systems” ( [10] , Chapter 2, p. 51).
Acknowledgements
The work of the second author has been partially supported by the Norwegian Research Council, grant 239070.
Cite this paper
Zabrodskii, I. and Ponosov, A. (2017) The Principal Component Transform of Parametrized Functions. Applied Mathematics, 8, 453-475. https://doi.org/10.4236/am.2017.84037
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Appendix
1. Allahverdiev’s theorem
Let and be two real separable Hilbert spaces, equipped with the scalar products and and the corresponding norms and , respectively. Assume that is a linear compact operator. Its norm is
defined as .
Put
We want to find an operator for which . This construction is very close to the finite dimensional singular value decomposition.
Assume that is the adjoint of . Then the linear compact operators are self-adjoint and positive-definite. Let , be all positive eigenvalues of the operator , the associated normalized eigenvectors being , respectively:
(77)
It is well-known that can always be chosen to be orthogonal: By the Hilbert-Schmidt theorem, for any there is a
unique set , and a unique for which and, moreover, Thus, the operator can be represented as
(78)
where , and the convergence is understood in the sense of the norm in the space . We define the linear bounded operators by
(79)
The following result is known as Allahverdiev’s theorem, see e.g. [8]:
Proposition 7. For any linear compact operator
(80)
Proof. First of all, we prove that . By definition,
(81)
From (79) and (78) we get
(82)
We calculate the norm of using (81), (82):
(83)
because
(84)
and
(85)
As , ( ) and , we obtain . As for all ,
(86)
if and .
Hence,
(87)
Secondly, we prove that
(88)
Let be a basis in . Then there exist some from H such that
(89)
We want to prove that
(90)
If then
If then
Therefore
(91)
This homogeneous system has unknowns and equations, so that there is such that and . Therefore
(92)
as for
2. Tensor product of operators in Hilbert spaces
Let and be real separable Hilbert spaces, where
• has an orthonormal basis
• has an orthonormal basis
• has an orthonormal basis
• has an orthonormal basis
Let
(93)
Now, we define the tensor product of the spaces and as the real separable Hilbert space, which has the basis consisting of all ordered pairs , and we put By definition, any can be uniquely represented as
(94)
Definition 2. The scalar product in is defined as
(95)
where .
Evidently, the set is an orthonormal basis of the space and therefore
(96)
is the norm on . The series
converges in this norm. It is also straightforward to check that
(97)
for all , .
Let us consider two compact linear operators
(98)
For all we have
(99)
We define the tensor product of and as
(100)
where is given by (94).
Proposition 8. If are linear compact ope- rators, then so is the operator .
Proof. Linearity of follows directly from the definition. Taking an arbitrary satisfying (94) we obtain
(101)
Therefore is bounded, and in particular,
(102)
To prove compactness we choose an arbitrary and linear bounded finite dimensional operators for which .
Evidently,
(103)
Using (102) we obtain
(104)
Therefore, the operator can be approximated in norm by finite dimensional operators of the form with an arbitrary precision. Thus, is compact.
Proposition 9. For all linear compact operators and we have
(105)
Proof. The set of linear combinations is dense in , i.e. for all there is a sequence of linear combinations of which converges to in the norm. As the operators and are linear and bounded, it is sufficient to prove the equality in the lemma for the special case of , where we by definition have the formula
(106)
Let . where and . Then
(107)
Hence .
Proposition 10. If is the eigenpair of the operator ( ), then is the eigenpair of the operator .
Proof.
(108)
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