New approaches to regression in financial mathematics and life sciences by generalized additive models

This paper introduces into and improves the theoretical research done by the authors in the last two years in the applied area of GAMs (generalized additive models) which belong to the modern statistical learning, important in many areas of prediction, e.g., in financial mathematics and life science...

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Дата:	2008
Автори:	Taylan, P., Weber, G.-W.
Формат:	Стаття
Мова:	English
Опубліковано:	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України 2008
Теми:	Математичні методи, моделі, проблеми і технології дослідження складних систем
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/11995
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	New approaches to regression in financial mathematics and life sciences by generalized additive models / P. Taylan, G.-W. Weber // Систем. дослідж. та інформ. технології. — 2008. — № 3. — С. 101-118. — Бібліогр.: 21 назв. — англ.

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spelling	irk-123456789-119952013-02-13T03:08:02Z New approaches to regression in financial mathematics and life sciences by generalized additive models Taylan, P. Weber, G.-W. Математичні методи, моделі, проблеми і технології дослідження складних систем This paper introduces into and improves the theoretical research done by the authors in the last two years in the applied area of GAMs (generalized additive models) which belong to the modern statistical learning, important in many areas of prediction, e.g., in financial mathematics and life sciences, e.g., computational biology and ecology. These models have the form ψ(x) = β0 + Σj=1^m fj(xj), where ψ are functions of the predictors, and they are fitted through local scoring algorithm using a scatterplot smoother as building blocks proposed by Hastie and Tibshirani (1987). Aerts, Claeskens and Wand (2002) studied penalized spline generalized additive models to derive some approximations. We present a mathematical modeling by splines based on a new clustering approach for the input data x, their density, and the variation of the output data y. We bounding (penalizing) second order terms (curvature) of the splines, we include a regularization of the inverse problem, contributing to a more robust approximation. In a first step, we present a refined modification and investigation of the backfitting algorithm previously applied to additive models. Then, by using the language of optimization theory, we initiate future research on solution methods with mathematical programming. Описываются теоретические результаты, полученные авторами за последние два года в прикладной области GAM (обобщенных аддитивных моделей), которые принадлежат к статистическому обучению и важны во многих случаях получения предсказаний, например, в финансовой математике или в науках о жизни (например, в вычислительной биологии и экологии). Эти модели имеют вид ψ(x) = β0 + Σj=1^m fj(xj) где ψ — предсказывающие функции. Они фильтруются алгоритмами локального выигрыша с использованием рассеянного сглаживания, предложенного Hastie и Tibshirani (1987 г.). Aerts, Claeskеns і Wand (2002 г.) использовали сплайновые обобщенные аддитивные модели со штрафом, чтобы получить некоторые аппроксимации. Мы предлагаем математическое моделирование со сплайнами, основанное на новом кластерном подходе к входным данным х, их плотности и вариации выходных данных у. Ограничивая (штрафом) члены второго порядка (кривизну) сплайнов, включаем регуляризацию обратных задач, получая более грубую аппроксимацию. На первом этапе представляем улучшенную модификацию и исследуем алгоритм обратных шагов, который ранее применялся к аддитивным модулям. Затем с использованием языка теории оптимизации инициируем будущие исследования методов решения с использованием математического программирования. Описано теоретичні результати, отримані авторами за останні два роки у прикладній області GAM (узагальнених адитивних моделей), що належать до статистичного навчання і важливі для багатьох випадків одержання прогнозу, наприклад, у фінансовій математиці або у науках про життя (наприклад, у обчислювальній біології та екології). Ці моделі мають вигляд ψ(x) = β0 + Σj=1^m fj(xj), де ψ — прогнозуючі функції. Вони фільтруються алгоритмами локального виграшу із використанням розсіяного згладжування, запропонованого Hastie і Tibshirani (1987 р.). Aerts, Claeskеns і Wand (2002 р.) використали сплайнові узагальнені адитивні моделі із штрафом, аби одержати деякі апроксимації. Ми пропонуємо математичне моделювання із сплайнами, яке базується на новому кластерному підході до вхідних даних х, їх густини та варіації вихідних даних у. Обмеживши (штрафом) члени другого порядку (кривизну) сплайнів, включаємо регуляризацію зворотних задач одержуючи більш грубу апроксимацію. На першому етапі пропонуємо покращену модифікацію і досліджуємо алгоритм зворотних кроків, який раніше застосовувався до адитивних модулей. Потім із використанням мови теорії оптимізації, ініціюємо майбутні дослідженя методів розв’язання із використанням математичного програмування. 2008 Article New approaches to regression in financial mathematics and life sciences by generalized additive models / P. Taylan, G.-W. Weber // Систем. дослідж. та інформ. технології. — 2008. — № 3. — С. 101-118. — Бібліогр.: 21 назв. — англ. 1681–6048 http://dspace.nbuv.gov.ua/handle/123456789/11995 519.95 en Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Математичні методи, моделі, проблеми і технології дослідження складних систем Математичні методи, моделі, проблеми і технології дослідження складних систем
spellingShingle	Математичні методи, моделі, проблеми і технології дослідження складних систем Математичні методи, моделі, проблеми і технології дослідження складних систем Taylan, P. Weber, G.-W. New approaches to regression in financial mathematics and life sciences by generalized additive models
description	This paper introduces into and improves the theoretical research done by the authors in the last two years in the applied area of GAMs (generalized additive models) which belong to the modern statistical learning, important in many areas of prediction, e.g., in financial mathematics and life sciences, e.g., computational biology and ecology. These models have the form ψ(x) = β0 + Σj=1^m fj(xj), where ψ are functions of the predictors, and they are fitted through local scoring algorithm using a scatterplot smoother as building blocks proposed by Hastie and Tibshirani (1987). Aerts, Claeskens and Wand (2002) studied penalized spline generalized additive models to derive some approximations. We present a mathematical modeling by splines based on a new clustering approach for the input data x, their density, and the variation of the output data y. We bounding (penalizing) second order terms (curvature) of the splines, we include a regularization of the inverse problem, contributing to a more robust approximation. In a first step, we present a refined modification and investigation of the backfitting algorithm previously applied to additive models. Then, by using the language of optimization theory, we initiate future research on solution methods with mathematical programming.
format	Article
author	Taylan, P. Weber, G.-W.
author_facet	Taylan, P. Weber, G.-W.
author_sort	Taylan, P.
title	New approaches to regression in financial mathematics and life sciences by generalized additive models
title_short	New approaches to regression in financial mathematics and life sciences by generalized additive models
title_full	New approaches to regression in financial mathematics and life sciences by generalized additive models
title_fullStr	New approaches to regression in financial mathematics and life sciences by generalized additive models
title_full_unstemmed	New approaches to regression in financial mathematics and life sciences by generalized additive models
title_sort	new approaches to regression in financial mathematics and life sciences by generalized additive models
publisher	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
publishDate	2008
topic_facet	Математичні методи, моделі, проблеми і технології дослідження складних систем
url	http://dspace.nbuv.gov.ua/handle/123456789/11995
citation_txt	New approaches to regression in financial mathematics and life sciences by generalized additive models / P. Taylan, G.-W. Weber // Систем. дослідж. та інформ. технології. — 2008. — № 3. — С. 101-118. — Бібліогр.: 21 назв. — англ.
work_keys_str_mv	AT taylanp newapproachestoregressioninfinancialmathematicsandlifesciencesbygeneralizedadditivemodels AT webergw newapproachestoregressioninfinancialmathematicsandlifesciencesbygeneralizedadditivemodels
first_indexed	2025-07-02T14:06:21Z
last_indexed	2025-07-02T14:06:21Z
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fulltext	© P. Taylan, G.-W. Weber, 2008 Системні дослідження та інформаційні технології, 2008, № 3 101 УДК 519.95 NEW APPROACHES TO REGRESSION IN FINANCIAL MATHEMATICS AND LIFE SCIENCES BY GENERALIZED ADDITIVE MODELS P. TAYLAN, G.-W. WEBER This paper introduces into and improves the theoretical research done by the authors in the last two years in the applied area of GAMs (generalized additive models) which belong to the modern statistical learning, important in many areas of predic- tion, e.g., in financial mathematics and life sciences, e.g., computational biology and ecology. These models have the form ∑ = += m j jj xfx 10 )()( βψ , whereψ are functions of the predictors, and they are fitted through local scoring algorithm using a scatterplot smoother as building blocks proposed by Hastie and Tibshirani (1987). Aerts, Claeskens and Wand (2002) studied penalized spline generalized additive models to derive some approximations. We present a mathematical modeling by splines based on a new clustering approach for the input data x, their density, and the variation of the output data y. We bounding (penalizing) second order terms (curva- ture) of the splines, we include a regularization of the inverse problem, contributing to a more robust approximation. In a first step, we present a refined modification and investigation of the backfitting algorithm previously applied to additive models. Then, by using the language of optimization theory, we initiate future research on solution methods with mathematical programming. 1. INTRODUCTION 1.1. Learning and Models In the last decades, learning from data has become very important in every field of science, economy and technology, for problems concerning the public and the private life as well. Modern learning challenges can for example be found in the fields of computational biology and medicine, and in the financial sector. Learn- ing enables for doing estimation and prediction. There are regression, mainly based on the idea of least squares or maximum likelihood estimation, and classifi- cation. In statistical learning, we are beginning with deterministic models and, then, we turn to the more general case of stochastic models where uncertainties, noise or measurement errors are taken into account. For a closer information we refer to the book Hastie, Tibshirani, Friedman [10]. In classical models, the ap- proach to explain the recorded data y consists of one unknown function only; the introduction of additive models (Buja, Hastie, Tibshirani 1989 [4]) allowed an “ansatz” with a sum of functions which have separated input variables. In our pa- per, we figure out clusters of input data points x (or entire data points ),( yx ), and assign an own function that additively contributes to the understanding and learning from the measured data. These functions over domains (e.g., intervals) depending on the cluster knots are mostly assumed to be splines. We will intro- duce an index useful for deciding about the spline degrees by density and varia- tion properties of the corresponding data in x and y components, respectively. In a P. Taylan, G.-W. Weber ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 102 further step of refinement, aspects of stability and complexity of the problem are implied by keeping the curvatures of the model functions under some chosen bounds. The corresponding constrained least squares problem can, e.g., be treated as a penalized unconstrained minimization problem. In this paper, for the general- ized (penalized) problem, we specify (modify) the backfitting algorithm which was investigated and applied for additive models. Our new investigation of gen- eralized additive models is introduced in the stochastic case and closer presented in the deterministic case. This paper contributes to both the m-dimensional case of input data sepa- rated by the model functions and, as our new alternative, to 1-dimensional input data clustered. Dimensional generalizations of the second interpretation and a combination of both interpretations are possible and indicated. Applicability for data classification is noted. We point out advantages and disadvantages of the concept of backfitting algorithm. By all of this, we initiate future research with a strong employing of optimization theory. This paper is related with our research as initiated the papers [16, 17, 19, 20]. 1.2. A Motivation of Regression This paper has been motivated by the approximation of finanical data points ),( yx , e.g., coming from the stock market. Here, x represents the input constella- tion, while y stands for the observed data. The discount function, denoted by ( )xδ , is the current price of a risk free, zero coupon bond paying unit of money at time x. We use ( )xy to denote the zero-coupon yield curve and to ( )xf to denote the instantaneous forward rate curve. These are related to the discount function by ( ) ( )( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −=−= ∫ dssfxxy x x 0 expexpδ . (1.1) The term interest rate curve can be used to refer to any one of these three re- lated curves. In a world with complete markets and no taxes or transaction, absence of ar- bitrage implies that the price of any coupon bond can be computed from an inter- est rate curve. In particular, if the principal and interest payment of a bond is jc units of money at time jx ( mj ,...,1= ), the pricing equation for the bond is ( ) ( )( ) ( ) ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ −=−= ∫∑∑∑ === dssfcxyxcxc jxm j jjj m j jj m j j 0111 expexpδ . (1.2) The interest rate curve can be estimated if given a set of bond prices. For this reason, let NiiB ,...,1)( = comprise the bonds, mXXX <<< ...21 be the set of dates at which principal and interest payments occur, let ijc be the principal and inter- est payment of the ith bond on date jX , and iP be the observed price of the ith bond. The pricing equation is ˆ i i iP P ε= + , (1.3) New approaches to regression in financial mathematics and life … Системні дослідження та інформаційні технології, 2008, № 3 103 where iP̂ is defined by ( )j m j iji XcP δ∑ = = 1 ˆ [18]. The curves of discount ( )xδ , yield ( )xy and forward rate ( )xf can be extracted via linear regression, regression with splines, smoothing splines, etc., using prices of coupon bond. For example, assuming ( )TNPP ,...,: 1=P and )(: ijc=C , Ni ,,1= , mj ,,1…= to be known, denoting the vector of errors or residuals (i.e., noise, inaccuracies and data un- certainties) by ( )TNεεε ,,: 1 …= and writing ( )TmXXX )(),...,()(: 1 δδδβ == , then the pricing equation looks as follows: εβ +=CP . (1.4) Thus, the equation (1.4) can be seen as linear model with the unknown pa- rameter vector ( ) βδδ =T mXX )(),...,( 1 . If we use linear regression methods or maximum likelihood estimation and, in many important cases, just least squares estimation, then we can extract ( )Xδ . For introductory and closer information about these methods from the viewpoints of statistical learning or the theory of inverse problems, we refer to the books of Hastie, Tibshirani, Friedman [10] and Aster, Borchers, Thurber [2], respectively. While the papers [16, 17] refer to the financial sector, the works [19, 20] ad- dress the areas of computational biology, environmental protection and the inter- faces between both. Actually, finance — the world of prosperity, and develop- ment are related with the gene-environment networks. 1.3. Regression 1.3.1. Linear Regression Provided an input vector ( )TmXXX ,...,1= of (random) variables and an output variable Y, our linear regression model has the form εββε ++=+= ∑ = m j jjm XXXYEY 1 01 ),...,( . (1.5) The linear model either assumes that the regression function )\|( XYE is lin- ear or that linearity means a reasonable approximation. Here, jβ are unknown parameters or coefficients, the error ε is a Gaussian random variable with expec- tation zero and variance 2σ , written ),0( 2σε N∼ , and the variables jX can be from different sources. Typically we have a set of training data ( ) ( )NN yxyx ,,...,, 11 from which we estimate the parameters jβ . Here, each T imiii xxxx ),...,,( 21= is a vector of feature measurements for the ith case. The most popular estimation method is “least squares” which determines the cofficient vector ( )Tmββββ ,...,, 10= to minimize the residual sum of squares ( ) ( )∑ = −= N i T ii xy 1 2 :RSS ββ or ( ) ( ) ( )XβYXβYβ −−= TRSS . (1.6) P. Taylan, G.-W. Weber ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 104 Here, X is the )1( +× mN matrix with each row being an input vector (with a 1 in the first position), and Y is the N vector of outputs in the training set. The second equation in (1.6) is a quadratic function in 1+m unknown parameters. If 1+≥mN and X has full rank, then vector β which mimimizes RSS is ( ) yXXX TT 1ˆ − =β . The predicted values at an input vector 0x are given by )(ˆ 0xf ; the fitted values at the training inputs are ( ) yXXXXXy TT 1ˆˆ − == β , where )(ˆˆ ixfy = . 1.3.2. Regression with Splines In the above regression problems, sometimes ),...,()( 1 mXXYEXf = can be nonlinear and nonadditive. Since, however, a linear model is easy to interpret, we want to represent )(Xf by a linear model. Thus, an approximation by a first- order Taylor approximation to )(Xf can be used and sometimes even needs to be done. In fact, if N is small or m large, a linear model might be all we are able to use for data fitting without overfitting. As in classification, a linear, Bayes- optimal, decision boundary [10] implies that some monotone transformation of )1(Pr XY = is linear in X . Regression with splines is a very popular method as for moving beyond line- arity [10]. Here, we expand or replace the vector of inputs X with additional variables, which are transformations of X and, then, we use linear models in this new space of derived input features. Let X vector of inputs and IRIRh m j →: be the j-th transformation of X or basis function ( Mj ,,2,1 …= ). Then, )(Xf is modelled by ∑ = = M j jj XhβXf 1 )()( , a linear basis expansion in X . Herewith, the model has become linear in these new variables and the fitting proceeds as in the case of a linear model. In fact, the estimation of β is =β̂ ( ) ( )( ) ( )YxHxHxH TT 1− = , where 1,..., ; 1,..., ( ) ( ( ))j i i m j M H x h x = = = is the matrix of basis functions evaluated at the input data. Hence, f becomes estimated by .ˆ)()(ˆ βXhXf T= For the special case jj XXh =)( ( Mj ,,2,1 …= ) the linear model is recovered. Generally, in one dimension ( 1=m ), an order M spline with knots κξ ( K,,2,1 …=κ ) is piecewise polynomial of degree 1−M , and has con- tinuous derivatives up to order 2−M . A cubic spline has 4=M . Any piecewise constant function is an order 1 spline, while the continuous piecewise linear func- tion is an order 2 spline. Likewise the general form for the truncated-power basis set would be ),...,2,1()( 1 MjXXh j j == − and ),,...,2,1( Kl = where +•)( stands for the positive part of a value [10]. New approaches to regression in financial mathematics and life … Системні дослідження та інформаційні технології, 2008, № 3 105 1.4. Additive Models 1.4.1. Classical Additive Models We stated that regression models, especially, linear ones, are very important in many applied areas. However, the traditional linear models often fail in real life, since many effects are generally nonlinear. Therefore, flexible statistical methods have to be used to characterize nonlinear regression effects; among these methods is non-parametric regression [6]. By using the common assumption of linearity, it gives information to explore the data more flexibly, uncovering structure in the data that might otherwise be missed. Many nonparametric methods do not per- form well when there is a large number of independent variables in the model. The sparseness of data in this setting inflates the variance of the estimates. The problem of rapidly increasing variance for increasing dimensionality is sometimes referred to as the “curse of dimensionality”. Interpretability is another problem with nonparametric regression based on kernel and smoothing spline esti- mates [11]. To overcome these difficulties [15] proposed additive models. These models estimate an additive approximation of the multivariate regression func- tion. Here, the estimation of the individual terms explains how the dependent variable changes with the corresponding independent variables. We refer to Hastie and Tibshirani (1986) [8] for basic elements of the theory of additive models. If we have data consisting of N realizations of random variable Y at m design values, then the additive model takes the from ( ) ( )∑ = += m j ijjmiii xfxxYE 1 01,... β . (1.7) Here, the functions jf are estimated by a smoothing on a single coordinate, and standard convention is to assume at the knots ijx : ( )( ) 0=ijj xfE [9] (we shall give a justification below). Additive models have a strong motivation as a useful data analytic tool. Each variable is represented separately in (1.7) and the model has an important interpretation feature of some “linear model”: Each of the vari- ables separately effects the response surface and that effect does not depend on the other variables. For this reason, if once an additive model can be fit to data, we can plot the m coordinate functions separately to examine the roles of the vari- ables in predicting the response. Each function is estimated by an algorithm pro- posed by Friedman and Stuetzle (1981) [7] and called backfitting algorithm. As the estimator for 0β̂ , the arithmetic mean (average) of the output data is used: .)1(:),...,1\|(ave 1∑ = == N i ii yNNiy This procedure depends on the partial residual against ijx : ( )ikjk kiij xfyr ∑ ≠ −−= ˆ 0β (1.8) and consists of estimating each smooth function by holding all the other ones fixed. In a framework of cycling from one to the next iteration, this means the fol- lowing [9]: initialization ),...,1\|(ˆ 0 Niyave i ==β , jixf ijj ,,0)(ˆ ∀≡ ; P. Taylan, G.-W. Weber ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 106 cycle ,...,1,,...,1,,...,1 mmj = ( ) Nixfyr m jk ikkiij ,...,1,ˆˆ 0 =−−= ∑ ≠ β , jf̂ is updated by smoothing the partials residuals, ( ) Nixfyr m jk ikkiij ,...,1,ˆˆ 0 =−−= ∑ ≠ β , against ijx ; until the functions almost do not change. The backfitting procedure is also called Gauss-Seidel algorithm. To prove its convergence [4] reduced the problem to the solution of a corresponding homoge- neous system, analyzed by a linear fixed point equation of the form f = fT̂ . In fact, to represent the effect on the homogeneous equations of updating the jth component under Gauss-Seidel algorithm, the authors introduced the linear trans- formation NmNm j IRIRT →:ˆ , ( )( )TT mjk k T j T ffSff …… ∑ ≠ −1 . A full cycle of this algorithm is determined by 11 ˆˆˆˆ TTTT mm …−= ; then, lT̂ correspond l full cycles. If all smoothing splines jS are symmetric and have eigenvalues in [ ]1,0 , then the backfitting algorithm always converges. In Subsec- tion 2.6, we will come back closer to the algorithm and the denotation used here. 1.4.2. Additive Models Revisited In our paper, we allow a different and new motivation: In addition to the approach given by a separation of the variables jx done by the functions jf , now we per- form a clustering of the input data of the variable x by a partitioning of the do- main into cubes jQ or, in the 1-dimensional case: intervals jI , and a determina- tion of jf with reference to the knots lying in jQ (or jI ), respectively. In any such a case, a cube or interval is taking the place of a dimension or coordinate axis. We will mostly refer to the case of one dimension; the higher dimensional case can then be treated by a combination of separation and clustering. That clus- tering can incorporate any kind of periods of seasons assumed, any comparability or correspondence of successive time intervals, etc. Herewith, the functions jf are more considered as allocated to sets jI (or jQ ) rather than depending on some special, sometimes arbitrary elements of those sets (input data) or output values associated. This new interpretation and usuage of additive models (or gen- eralized ones, introduced next) is a key step of this paper. 2. GENERALIZED ADDITIVE MODELS To extend the additive model to a wide range of distribution families, Hastie and Tibshirani (1990) [11] proposed generalized additive models (GAM) which are New approaches to regression in financial mathematics and life … Системні дослідження та інформаційні технології, 2008, № 3 107 among the most practically used modern statistical techniques. These models en- able the mean of the dependent variable to be an additive predictor through a link function. Many widely used statistical models belong to this general class; they include additive models for Gaussian data, nonparametric logistic models for bi- nary data, and nonparametric log-linear models for Poisson data. 2.1. Definition of a Generalized Additive Model If we have mXX ,...,1 , being m covariates comprised by the m -tuple ( )TmXXX ,...,1= , then, in our regression setting, a generalized additive model has the form ( )∑ = +== m j jj XfβX)XG( 1 0)()( ψµ . (2.1) Here, the function jf are unspecified (“nonparametric”) and ( …,, 10 fβθ = )Tmf,… is the unknown parameter to be estimated; G is the link function. The incorporation 0β as some average outcome allows us to assume ( )( ) 0=ijj xfE ( mj ,,1…= ). Often, the unknown functions jf are elements of a finite dimen- sional space consisting, e.g., of splines and these functions depending on the clus- ter knots are mostly assumed to be splines; the spline orders (or degrees) are suitably choosen depending on the density and variation properties of the corre- sponding data in x and y components, respectively. Then, our problem of specify- ing θ becomes a finite-dimensional parameter estimation problem. 2.2. Clustering of Input Data 2.2.1. Introduction Clustering is the process of organizing objects into mII ,...,1 groups or, higher dimensionally: mQQ ,...,1 , whose elements are similar in some way. A cluster is therefore a collection of objects which are “similar” between them and are “dis- similar” to the objects belonging to other clusters. For example, we can easily identify some cluster out of a finite number of clusters into which the data can be divided with respect to the similarity criterion distance. We put two or more ob- jects belonging to the same cluster if they are “close” according to a given dis- tance (in this case, geometrical distance). Differently from usual clustering, in this paper, we understand clustering al- ways as being accompanied by a partitioning of the (input) space, including space coverage. In other words, it will mean a classification in the absense of different labels or categories. Especially, the clusters shall not be overlapping, and the par- titions containing the clusters shall also be pairwise disjoint, except at the bounda- ries. Instead of a general introduction into cluster and classification methods, we give the following information only. 2.2.2. Clustering for Generalized Additive Models Financial markets have different kinds of trading activities. These activities work with considerably long horizons, ranging from days and weeks to months and P. Taylan, G.-W. Weber ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 108 years. For this reason, we may have any kind of data. These data can sometimes be problematic for being used at the models, for example, given a longer horizon with sometimes less frequent data recorded, but to other times highly frequent measurements. In those cases, by the differences in data density and, possibly, data variation, the underlying reality and the following model will be too unstable or inhomogeneous. The process may be depending on unpredictable market be- haviour or external events like naturally calamity. Sometimes, the structure of data is has particular properties. These may be a larger variability or a handful of outliers. Sometimes we do not have any meaningful data. For instance, share price changes will not be available when stock markets are closed at weekends or holi- days. The following three parts of fig. 1 are showing some important cases of input data distribution and clustering: the equidistant case (fig. 1,a)) where all points can be put into one cluster (or interval) 1I , the equidistant case with regu- lar breaks (weekends, holidays, etc.; (fig. 1,b) where the regularly neighbouring points and the free days could be put in separate cluster intervals jI , and the gen- eral case (cf. (fig. 1,c) ) where there are many interval jI of different interval lengths and densities. We remark that we could also include properties of the out- put data y into this clustering; for the ease of exposition, however, we disregard this aspect. In the following, we will take into account the data variation; to get and im- pression of this, please have a look at fig. 2. I1 a b I1 I2 I3 I4 a b I5 I6 I7 I8 I2 I4I3 I5 I6I1 a b a b c Fig. 1. Three important cases of input data distribution and its clustering: a — equidis- tance, b — equidistance with breaks, and c — general case New approaches to regression in financial mathematics and life … Системні дослідження та інформаційні технології, 2008, № 3 109 For the sake of simplicity, we assume from now on that the number jN of input data points ijx in each cluster jI is the same, say, ),...,1( mjNN j =≡ . Otherwise there will be no approximation need at data points missing and the re- siduals of our approximation were 0 there. Furthermore, given the output data ijy we denote the aggregated value over the all the ith output values of the clusters by ).,...,1(: 1 Niyy m j iji ==∑ = In the example of case (fig. 1,b), this data summation refers to all the days i from monday to friday. Herewith, the cluster can also have a chronolocial mean- ing. By definition, up to the division by m , the values iy are averages of the out- put values ijy . Before we come to a closer understanding of data density and variation, we proceed with our introduction of splines. In fact, the selection of the splines or- ders, degrees and classes will essentially be influenced by indices based on densi- ties and variations (Subsection 2.5). 2.3. Splines Let jNjj xxx ,...,, 21 be N distinct knots of [ ]ba, , where …<<≤ jj xxa 21 bx jN ≤<… . The function )(xfk on the interval [ ]ba, (or in R ) is a spline of some degree k relative to the knots ijx if [ ] kxxk IPf jiij ∈ +1, (polynomial of degree k≤ ; 11,..., −= Ni ), (2.2) [ ]baCf k k ,1−∈ . (2.3) To characterize a spline of degree k, [ ]jiij xxkik ff 1,, : + = can be represented by [ ]( )jiij k l l ijliik xxxxxgxf 1 0 , ,)()( + = ∈−=∑ . For a closer information about spline we refer to [5, 12]. a b . . .. ...... .. . ... . ....... .. Fig. 2. Example of a data (scatterplot); here, we refer to case (fig. 1,c) P. Taylan, G.-W. Weber ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 110 2.4. Variation and Density Density is a measure of mass per unit of volume. The higher an object’s density, the higher its mass per volume. Let us assume that we have mII ,...,1 intervals; then, the density of the input data ijx in the j-th interval jI is defined by =:jD jjij IIx oflength )in point of(number . This definition can be directly gener- alized to the higher dimensional case of cubes jQ rather than intervals jI , by referring to the cubes’ volumes. Variation is a quantifiable difference between individual measurements. Every repeatable process exhibits variation. If over the interval jI we have the data ),( ,,),( 11 NjNjjj yxyx … , then the variation of these data refers to the output dimension y and it is defined as ∑ − = + −= 1 1 1 : N i jijij yyV . If this value is big, at many data points the rate of change of the angle between any approximating curve and its tangent would be big, i.e., its curvature could be big. Otherwise, the curvature could be expected to be small. In this sense, high curva- ture over an interval can mean a highly oscillating behaviour. The occurrence of outliers ijy may contribute to this very much and mean instability of the model. 2.5. Index of Data Variation Still we assume that pII ,...,1 (or mQQ ,...,1 ) are the intervals (or cubes) ac- cording to the data grouped. For each interval jI (cube jQ ), we define the associated index of data variation by VDInd j =: or, more generally, =:jInd )()(: jjjj VvDd= , where jd , jv are some positive, strongly monotonically in- creasing functions selected by the modeller. In fact, from both the viewpoints of data fitting and complexity (or stability), cases with a high variation distributed over a very long intervall are very much less problematic than cases with a high variation over a short intervall. The multiplication of variation terms with density terms due to each interval found by clustering is representing this difference. We determine the degree of the splines jf with the help of the numbers jInd . If such an index is low, then we can choose the spline degree (or order) to be small. In this case, the spline may have a few coefficients to be determined and we can find these coefficients easily using any appropriate solution method for the corresponding spline equations. If the number jInd is big, then we must choose a high degree of the spline. In this case, the spline may have a more com- plex structure and many coefficients have to be determined; i.e., we may have many system equations or a high dimensional vector of unknows; to solve this could become very difficult. Also, a high degree of splines mfff ,...,, 21 , respec- tively, causes high curvatures or oscillations, i.e., there is a high “energy” im- plied; this means a higher (co)variance or instability under data perturbations. As the extremal case of high curvature we consider nonsmoothness meaning an in- stantaneous movement at a point which does not obey to any tangent. New approaches to regression in financial mathematics and life … Системні дослідження та інформаційні технології, 2008, № 3 111 The previous words introduced a model-free element into our explanations. Indeed, as indicated in Subsection 2.4, the concrete determining of the spline de- gree can be done adaptively by the implementer who writes the code. From a close mathematical perspective we propose to introduce discrete thresholds νγ and to assign to all the intervals of indices ),[ 1+∈ νν γγInd the same specific spline degrees. This determination and allocation has to base on the above reflec- tions and data (or residuals) given. For the above reasons, we want to impose some control on the oscillation. To make the oscillation smaller, the curvature of each spline must be bounded by the penalty parameter. We introduce a penalty parameter into the criterion of minimizing RSS, called penalized sum or squares PRSS now: [ ]∑ ∫∑ ∑ == = + ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ −−= m j j b a jjj N i m j ijjim1 dttfxfyff 1 2'' 2 1 1 00 )()(:),...,,(PRSS µββ . (2.4) The first term measures the goodness of data fitting, while the second term is a penalty term and defined by means of the functions’ curvatures. Here, the inter- val ],[ ba is the union of all the intervals jI . In the case of separation of vari- ables, the interval bounds may also depend on j , i.e., they are ],[ jj ba . For the sake of simplicity, we sometimes just write ∫ and refer to the interval limits given by the context. There are also further refined curvature measures, espe- cially, one with the input knot distribution implied by Gaussian bell-shaped den- sity functions; these appear as additional factors in the integrals and have a cut- ting-off effect. For the sake of simplicity, we shall focus on the given standard one now and turn to the sophisticated model in a later study. In (2.4), 0≥jµ are tuning or smoothing parameters and they represent a tradeoff between first and second term. Large values of jµ yield smoother curves, smaller values result in more fluctuation. It can be shown that the mini- mizer of PRSS is an additive spline model: Each of the functions jf is a spline in the component jX , with knots at ijx ( Ni ,,1…= ). However, without further re- strictions on the model, the solution is not unique. The constant 0β is not identi- fiable since we can add or substract any constants to each of the functions jf , and adjust 0β accordingly. For example, one standard convention is to assume that ∑ = = m j ijj xf 1 0)( i∀ , the function average being zero over the corresponding data (e.g., of mondays, tuesdays, ... , fridays, respectively). This can be achieved by means of a bias (intercept) 0β in front of the sum ∑ = m j ijj xf 1 )( in the additive approach. In this case, ),1,2,.=\|(aveˆ 0 Niyi …=β , as can be seen easily. P. Taylan, G.-W. Weber ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 112 We firstly want to have 0)( 2 1 1 0 ≈ ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ −−∑ ∑ = = N i m j ijji xfy β and, secondly, [ ] 0)( 1 2'' ≈∑∫ = m j jjj dttf or being sufficiently small, at least bounded. In the back- fitting algorithm, these approximations, considered as equations, will give rise to expected or update formulas. For these requests, let us introduce 2 1 1 00 )(:),( ∑ ∑ = = ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ −−= N i m j ijji xfyfF ββ and [ ] jjjjj Mdttffg −= ∫ 2'' )(:)( , where ( )Tmffff ,...,, 21= . The terms )( fg j can be interpreted as curvature in- tegral values minus some prescribed upper bounds 0>jM . Now, the combined standard form of our regression problem subject to the constrained curvature con- dition takes the following form: ).,...,1(0)( subject to),( Minimize 0 mjfgfF j =≤β (2.5) Now, PRSS can be interpreted in Lagrangian form as follows: ( )( ) [ ]∑ ∫∑ ∑ == = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −+ ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ −−= m j jjjjj N i m j ijji MdttfxfyfL 1 2'' 2 1 1 00 )()(,, µβµβ , (2.6) where T m ),...,(: 1 µµµ = . Here, 0≥jµ are auxilary penalty parameters intro- duced in [3]. In the light of our optimization problem, they can now be seen as Lagrange multipliers associated with the constraints 0≤jg . The Lagrangian dual problem takes the form ( )µ µ ),,(minmax ),(0 0 fβL fβ≥ . (2.7) The solution of this optimization problem (2.7) will help us for determining the smoothing parameters jµ and, in particular, the functions jf will be found, likewise their bounded curvatures [ ] jjj dttf 2'' )(∫ . Herewith, a lot of future re- search is initialized which can become an alternative to the backfitting algorithm concept. In this paper, we go on with refining and discussing the backfitting con- cept for the generalized additive model. 2.6. Modified Backfitting Algorithm for Generalized Additive Model 2.6.1. Generalized Additive Model Revisited For the generalized additive model (cf. Section 2.1), we will modify the backfit- ting algorithm used before for fitting additive model (cf. Subsection 1.3). For this reason, we will use the following theoretical setting in term of conditional expec- tation (Buja, Hastie and Tibshirani (1989) [4]), where mj ,,1…= : New approaches to regression in financial mathematics and life … Системні дослідження та інформаційні технології, 2008, № 3 113 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−= ∑∑ ≠≠ jk jkk jk kkjjj XXfYEXfYPXf )(:)()( 00 ββ . (2.8) Here, )(⋅jP denote the conditional expectation value ( )jXE ⋅ . Now, to find )( jj Xf in our generalized addive model, let us add the term ( )[ ] kkk m k k dttf 2'' 1 ∫∑ = − µ to equation (2.8). In this case, (2.8) will become the update formula ( )[ ] ←+ ∫ jjjjjj dttfXf 2'')( µ ( )[ ] = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−← ∫∑∑ ≠≠ kkk m jk k jk kkj dttfXfYP 2'' 0 )( µβ ( )[ ] ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−= ∫∑∑ ≠≠ kkk m jk k jk jkk dttfXXfYE 2'' 0 )( µβ , (2.9) where ( )[ ] jkkk m jk k cdttf =∫∑ ≠ 2''µ (constant, i.e., not depending on the knots); the functions jf̂ are unknown and to be determined in the considered iteration. There- fore, we can write equation (2.9) as ( )[ ] ←+ ∫ j 2 j '' jjjj dttfXf µ)( [ ] ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +−−← ∑ ∫ ≠ j jk kk '' kkkk XdttfXfYE 2 0 )()( µβ . If we denote ( )[ ] kk '' kkkkkk dttfXfXZ 2 )()( ∫+= µ (the same for j ), then we get the update formula ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −−← ∑ ≠ j jk kkjj XXZYEXZ 0)( β . (2.10) For random variables ( )XY , , the conditional expectation ( ) =xf ( )xXYE == minimizes ( )( )2XfYE − over all 2L functions f [4]. If this idea is applied to our generalized additive model, then the minimizer of ( )2)(Xψ-YE will give the closest additive approximation to ( )XYE . Equivalently, the follow- ing system of normal equations is necessary and sufficient for =Z ( )TmZZZ ,...,, 21= to minimize ( )2)(Xψ−YE (for the formula without intercept 0β , we refer to [4]): P. Taylan, G.-W. Weber ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 114 ( ) ( ) ( ) ( ) ( ) ( )⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − − = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ e e e 0 02 01 22 11 22 11 β β β YP . . YP YP XZ . . XZ XZ I..PP ..... ..... P..IP P..PI mmmmm , (2.11) where e is the N -vector or entries 1; or, in short, ( )0βY −= QPZ . Here, P and Q represent the matrix and vector of operators, respectively. If we want to apply normal equation to any given discrete experimental data, we must change the variables ),( XY in the (2.11) by their realizations ),( iiy x , =ix T imii xxx ),...,,( 21= , and the conditional expectations ( )jj XEP ⋅= by smoothers jS on jx , ⎟⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − − = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ )ˆ( . . )ˆ( )ˆ( . . .. ..... ..... .. .. 0 02 01 2 1 22 11 ey ey ey β β β mmmm S S S ISS SIS SSI z z z . (2.12) In estimation notation (2.12) equation can be written ( ) 10 ˆ:ˆˆˆ yβ Q-yQP ==z . Here, Nj Niijlj xhS ,...,1 ,...,1))(( = == are smoothing matrices of type NN × , jz are N -vectors representing the spline function ( )[ ] jjjjj dttff 2''ˆˆ ∫+ µ in a canonical form, i.e., ∑ = N l jljl Xh 1 )(θ (with the number of unknown equal to the number of conditions). In this notation, without loss of generality we already changed from lower spline degrees jd to a maximal one d , and to the order N . Furthermore, (2.12) is an ( NmNm× )-system of normal equations. The solutions to (2.12) sat- isfy )( jSℜ∈jz , where )( jSℜ is the range of the linear mapping jS , since we update by ( ).βS jkj ∑ ≠ −−← kj zz ey 0 ˆ In case we want to emphasize 0β̂ among the unknowns, i.e., ( ) ,,...,,ˆ 10 TT m TTβ zz again equation (2.12) can equivalently be written for this situation. There is a variety of efficient methods for solving the system (2.12), which depend on both the number and typs of smoother used. If the smoother matrix jS is a NN × nonsingular matrix, then the matrix P̂ will be a nonsingular ( NmNm× )-matrix; in this case, the system 1 ˆˆ yQP =z has a unique solution. If the smoother matrices jS are not guaranteed to be invertible (nonsingular) symmet- ric, but just arbitrary ( NN × )-matrices, we can use a generalized inverses − jS New approaches to regression in financial mathematics and life … Системні дослідження та інформаційні технології, 2008, № 3 115 )i.e.,( jjjj SSSS =− and −P̂ . For closer information about generalized solution and matrix calculus we refer to [12]. 2.6.2. Modified Backfitting Algorithm Gauss-Seidel method, applied to blocks consisting of vectorial component mzzz ,...,, 21 , exploits the special structure of (2.12). It coincides with the backfit- ting algorithm. If in the algorithm we write ( )[ ] jjjjjj dttffz 2''ˆˆˆ ∫+= µ (in fact, the functions jf̂ are unknowns), then, the lth iteration in the backfitting or Gauss- Seidel includes the additional penalized curvature term. Not forgetting the step- wise update of the penalty parameter jµ but not mentioning it explicitely, then the framework of the procedure looks as follows: 1. initialize 0ˆ ,1ˆ 1 0 ≡= ∑ = j N i i fy N β ˆ 0jz⇒ ≡ , j∀ ; 2. cycle ,,2,1 mj …= ( ) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ −−← =≠ ∑ N ijk ikkijj xzySz 1 0 ˆˆˆ β . This iteration is done until the individual functions do not change: Here, in each iterate, jẑ is by the spline function related with the knots ijx and found by the values ( ) ),...,2,1(ˆˆ 0 Nixzy jk ikki =−− ∑ ≠ β . In the other words, by the other functions kẑ and, finally, by the functions kf̂ and the penalty (smoothing) parameter kµ . Actually, since by definition it holds ( )[ ] jjjjjj dttffz 2''ˆˆˆ ∫+= µ , throughout the algorithm we must have a book keeping about both jf̂ and the curvature effect ( )[ ] jjj dttf 2 j ''ˆ∫µ controlled by the penalty parameter jµ which we can update from step to step. This book keeping is guaranteed since jf̂ and the curvature ( )[ ] jjj dttf 2''ˆ∫ can be determined via jẑ . Since the value of ( )[ ] jjjj dttf 2''ˆ∫µ is constant, the second order derivative of jẑ is =)(ˆ '' jj tz )(ˆ '' jj tf= ; this yields [ ] [ ] jjjjj 2 jjj tdtzdttf ∫∫ = 2'''' )(ˆ:)(ˆ µµ and, herewith, ( )[ ] jjjjjj dttfzf 2''ˆˆ:ˆ ∫−= µ . 2.6.3. Discussion about Modified Backfitting Algorithm If we consider our optimization problem on (2.4) (see also (2.7)) as fixed with respect to jµ , then we can carry over the convergence theory about additive P. Taylan, G.-W. Weber ISSN 1681–6048 System Research & Information Technologies, 2008, № 3 116 models (see Section 1.3) to the present generalized additive model, replacing the functions jf̂ by jẑ . However, at least approximatively, we have to guarantee fea- sibility also, i.e., [ ] jjjj Mdttf ≤∫ 2'' )(ˆ )1,...,( mj = . If [ ] jjjj Mdttf ≤∫ 2'' )(ˆ , then we preserve the value of jµ for 1+← ll ; oth- erwise, we increase jµ . But this update changes the values of jẑ and, herewith, the convergence behaviour of the algorithm. What is more, the modified backfit- ting algorithm bases on both terms in the objective function to be approximated by 0; too large an increase of jµ can shift too far away from 0 the corresponding penalized curvature value in the second term. The iteration stops if the functions jf become stationary, i.e., not changing very much and, if we request it, if 2 1 1 0 )(∑ ∑ = = ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ −− N i m j ijji xfy β becomes sufficiently small, i.e., lying under some error threshold ε , and, in particular, [ ] jjjj Mdttf ≤∫ 2'' )(ˆ ( mj ,,2,1 …= ). 2.7. On a Numerical Example Numerical applications arise in many areas of science, technology, social life and economy with, in general, very huge and firstly unstructured data sets; in particu- lar, they may base on data from financial mathematics. These data can be got, e.g., from Bank of Canada (http://www.bankofcanada.ca/en/rates/interest- look.html) as daily, weekly and monthly; they can be regularly partioned, which leads to a partitioning (clustering) of the (input) space, and indices of data varia- tion can be assigned accordingly. Then, we decide about the degrees of the spline depending of the location of the indices between thresholds νγ . In this entire process, the practitioner has to study the structure of the data. In particular, the choice on the cluster approach at all, or of the approach on separation of vari- ables, or of a combination of both, has to be made at an early stage and in close collaboration between the financial analyst, the optimizer and the computer engi- neer. At Institute of Applied Mathematics of METU, we are in exchange with the experts of its Department of Financial Mathematics, and this application is initi- ated. Using the splines which we determine by the modified backfitting algorithm, an approximation for the unknown functions of the additive model can be itera- tively found. There is one adaptive element remaining in this iterative process: the update the penalty parameter, in connection with the observation of the conver- gence behaviour. Here, we propose the use and implementation of our algorithm and, to overcome its structural frontiers given by the choice of the penalty pa- rameter in the course of the program, a use of conic quadratic programming with interior point algorithm applied. A comparison and possible combination of these two algorithmic strategies is what we recommend in this pioneering paper. New approaches to regression in financial mathematics and life … Системні дослідження та інформаційні технології, 2008, № 3 117 3. CONCLUDING This basic and more theoretical paper has given a contribution to the discrete ap- proximation or regression of data in 1- and multivariate cases. Generalized addi- tive models have been investigated, input data grouped by clustering, its density measured, data variation quantified, spline classes selected by indices and their curvatures bounded with the help of penalization. Then, the backfitting algorithm which is also applicable for data classification has become modified and the fur- ther utilization of modern optimization recommended [14]. By this we have con- tributed to a better understanding of data from the financial world and life sci- ences, to a more refined instrument of prediction. In the paper [23], we extended our approach by spline from discrete or Gaussian approximation to the continuous type of Chebychev approximation, by this representing the occurence of errors and uncertainy in modern technology, decision making and negotiations. In the work, we made a connection to CO2 emission control, visualizing dynamics and simulations also. There is a lot of work waiting in future research and application, and we cordially invite to this. Acknowledgement: The authors express their gratitude to Prof. Dr. Alexan- der Makarenko and Prof. Dr. Peter I. Bidyuk for encouragement and help, and to the referee for valuable notes and proposals. This study carried out my postdoct duration at the Middle East Technical University. REFERENCES 1. Aerts M., Claeskens G. and Wand M.P. Some theory for penalized spline generalized additive models, J. Statist. Planning and Inference 103 (2002). 2. Aster A., Borchers B. and Thurber C. Parameter Estimation and Inverse Problems, Academic Press, 2004. 3. Boyd S. and Vandenberghe L. Convex Optimization. — Cambridge University Press, 2004. 4. Buja A., Hastie T. and Tibshirani R. Linear smoothers and additive models —The Ann. Stat. 17, 2 (1989). 5. De Boor C. 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Weber G.-W., Tezel A., Taylan P., Soyler A. and Cetin M. On dynamics and optimi- zation of gene-environment networks. To appear in the special issue of Optimiza- tion in honour of the 60th birthday of Prof. Dr. H.Th. Jongen. 20. Weber G.-W., Uğur Ö., Taylan P. and Tezel A. On optimization, dynamics and un- certainty: a tutorial for gene-environment networks. To appear in the special is- sue of Discrete Applied Mathematics “Networks in Computational Biology”. 21. Weber G.-W., Alparslan-Gök S.Z. and Söyler B. A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics, invited paper submitted to Environmental Modeling & Assessment. Received 26.03.2007 From the Editorial Board: the article corresponds completely to supmitted manu- script.

New approaches to regression in financial mathematics and life sciences by generalized additive models

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