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# Parameter Estimation for Semiparametric Models with CMARS and Its Applications

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AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 11.

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### Parameter Estimation for Semiparametric Models with CMARS and Its Applications

1. 1. 5th International Summer School Achievements and Applications of Contemporary Informatics, Mathematics and Physics National University of Technology of the Ukraine Kiev, Ukraine, August 3-15, 2010 PARAMETER ESTIMATION FOR SEMIPARAMETRIC MODELS WITH CMARS AND ITS APPLICATIONS Fatma YERLIKAYA-ÖZKURT Institute of Applied Mathematics, METU, Ankara,Turkey Gerhard-Wilhelm WEBER Institute of Applied Mathematics, METU, Ankara,Turkey Faculty of Economics, Business and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal Universiti Teknologi Malaysia, Skudai, Malaysia Pakize TAYLAN Department of Mathematics, Dicle University, Diyarbakır, Turkey
2. 2. Outline • Introduction • Estimation for Generalized Linear Model (GLM) • Generalized Partial Linear Model (GPLM) • Estimation for GPLM – Least-Squares Estimation with Tikhonov Regularization – CMARS Method • Penalized Residual Sum of Squares (PRSS) for GLM with MARS • Tikhonov Regularization for GLM with MARS • An Alternative Solution for Tikhonov Regularization Problem with CQP • Solution Methods • Application • Conclusion
3. 3. Introduction The class of Generalized Linear Models (GLMs) has gained popularity as a statistical modeling tool. This popularity is due to: • The flexibility of GLM in addressing a variety of statistical problems, • The availability of software (Stata, SAS, S-PLUS, R) )to fit the models. The class of GLM is an extension of traditional linear models allows:  The mean of a dependent variable to depend on a linear predictor by a nonlinear link function......  The probability distribution of the response, to be any member of an exponential family of distributions.  Many widely used statistical models belong to GLM: o linear models with normal errors, o logistic and probit models for binary data, o log-linear models for multinomial data.
4. 4. Introduction Many other useful statistical models such as with • Poisson, binomial, • Gamma or normal distributions, can be formulated as GLM by the selection of an appropriate link function and response probability distribution. A GLM looks as follows: i  H ( i )  xiT  ; • i  E(Yi ) : expected value of the response variable Yi , • H: smooth monotonic link function, • xi : observed value of explanatory variable for the i-th case, •  : vector of unknown parameters.
5. 5. Introduction • Assumptions: Yi are independent and can have any distribution from exponential family density Yi ~ fY ( yi ,i , ) i  y   b ( )   exp  i i i i  ci ( yi , )  (i  1, 2,..., n),  ai ( )  • ai , bi , ci are arbitrary “scale” parameters, and i is called a natural parameter. • General expressions for mean and variance of dependent variable Yi : i  E (Yi )  bi' (i ), Var (Yi )  V ( i ) , V ( i )  bi" (i ) i , ai ( ) :  / i .
6. 6. Estimation for GLM • Estimation and inference for GLM is based on the theory of • Maximum Likelihood Estimation • Least–Squares approach: n l (  ) :  (  y  i 1 i i i  bi (i )   ci ( yi , )). • The dependence of the right-hand side on  is solely through the dependence of the i on  .
7. 7. Generalized Partial Linear Models (GPLMs) • Particular semiparametric models are the Generalized Partial Linear Models (GPLMs) : They extend the GLMs in that the usual parametric terms are augmented by a single nonparametric component:  E Y X , T   G X T    T  ;          m  is a vector of parameters, T • and    is a smooth function, which we try to estimate by CMARS. • Assumption: m-dimensional random vector X which represents (typically discrete) covariates, q-dimensional random vector T of continuous covariates, which comes from a decomposition of explanatory variables. Other interpretations of    : role of the environment, expert opinions, Wiener processes, etc..
8. 8. Estimation for GPLM • There are different kinds of estimation methods for GPLM. • Generally, the estimation methods for model  E Y X , T   G X T    T  ; is based on kernel methods and test procedures on the correct specification of this model. • Now, we will try to concentrate on special types of GPLM estimation based on ------- Newly developed data mining method CMARS and ------- Least –Squares estimation with Tikhonov regularization.
9. 9. Least-Squares Estimation with Tikhonov Regularization • The general model  E Y X , T   G X T    T  ;  can be considered as semiparametric generalized linear model and can be written as follows: m H (  )   ( X , T )  X    T    X j  j   T  . T j 1 observation values yi , xi , ti (i  1, 2,..., n) . i  G(i ) and i  H ( i )  xiT     ti  and    is a smooth function. • For the estimation of parametric part, we apply the linear least squares with Tikhonov regularization.
10. 10. The Least-Squares Estimation with Tikhonov Regularization The process is as follows: Firstly, we apply the linear least squares on the given data to find a vector  preproc : (*) Y  X T  preproc   Equivalently, the model form is m y  0   X j  j   . j 1 The method of least squares is used for estimating the regression coefficients, m   ( 0 , 1 ,  2 ,...,  m ) y  0   X j  j   preproc T in , j 1 to minimize the residual sum of squares (RSS).
11. 11. The Least-Squares Estimation with Tikhonov Regularization • Tikhonov Regularization proposed an approximate solution to (*) equation by minimizing the quadratic functional 2 2 (**) min y  X  preproc   L preproc ,  preproc 2 2 where  is a regularization parameter between the first and the second part. The terms y and  preproc represents the response vector and unknown coefficients. • They are obtained by solving Tikhonov regularization problem (**). • Generally, Tikhonov regularization involves higher-order regularization terms which can be solved using generalized singular value decomposition (GSVD).
12. 12. The Least-Squares Estimation with Tikhonov Regularization • After getting the regression coefficients, we subtract the linear least- square model (without intercept) from corresponding responses: m y   X j  j  y. ˆ j 1 • Doing this at the input data, the resulting values ( ˆ y ) are our new responses. • Then, based on these new data, we find find the knots for nonparametric part with MARS. • Again consider model : i  H ( i )  xiT     ti  and    is a smooth function which we try to estimate by CMARS which is an alternative technique to the well-known data mining tool multivariate adaptive regression splines (MARS).
13. 13. CMARS Method • What is MARS? • Multivariate adaptive regression splines (MARS) is developed in 1991 by Jerome Friedman. • MARS builds flexible models by introducing piecewise linear regressions. • MARS uses expansions in piecewise linear basis functions of the form c + ( x, ) = [( x   )] , c- ( x, ) = [( x   )] , [q ] : max 0, q . • Set of basis functions:    : ( X j   )  , (  X j )  |   x1, j , x2, j ,..., xN , j , j  1, 2,..., .p  y                    c- (x,)=[ x)] ( c+(x,)=[ x  ( )]  x Basic elements in the regression with MARS.
14. 14. CMARS Method • Thus, we can represent   ti  by a linear combination which is successively built up by basis functions and the intercept 0 , such that M i  H (i )  x   0  m m (ti ) . T i m 1 Here,  m (m  1, 2,..., M ) are basis functions from  or products of two or more such functions, interaction basis functions are created by multiplying an existing basis function with a truncated linear function involving new variable.  m are the unknown coefficients for the mth basis function (m  1, 2,..., M ) or for the constant 1 (m  0). • Provided the observations represented by the data ti (i  1, 2, ..., N ) : Km  m (t ) : [ s  (t    )] m j m j m j . j 1
15. 15. CMARS Method The MARS algorithm for estimating the model function consists of two algorithms: I. Forward stepwise algorithm: • Search for the basis functions. • At each step, the split that minimized some “lack of fit” criterion from all the possible splits on each basis function is chosen. • The process stops when a user-specified value M max is reached. At the end of this process we have a large expression in Y . • This model typically overfits the data; so a backward deletion procedure is applied. II. Backward stepwise algorithm: • Prevents from over-fitting by decreasing the complexity of the model without degrading the fit to the data. • Proposition: We do not employ the backward stepwise algorithm to estimate the function. At its place, as an alternative we propose to use penalty terms in addition to the least-squares estimation in order to control the lack of fit from the viewpoint of the complexity of the estimation.
16. 16. The Penalized Residual Sum of Squares for GLM with MARS • Let we consider equation • i  H (i )  xiT     ti  , where     , 1 ,...,  M  and   ti      ti    ti   . Let us use the penalized residual sum T T of squares with M max basis functions having been accumulated in the forward stepwise algorithm. For the GLM model with MARS, PRSS has the following form: M max      N 2 PRSS   i  x     ti     2 2  T i m   Dr,s m (t m )  dt m , 2 m  i 1 m 1  1 r s  (1 , 2 ) r , sV ( m )  V (m) :  m | j  1, 2,..., K m j  t m := (tm1 , tm2 ,..., tm K )T m   (1 ,  2 )  : 1   2 , where 1 ,  2  0,1 ,  Dr,s m (t m ) :  m 1 trm 2 tsm (t m ) . 
17. 17. The Penalized Residual Sum of Squares for GLM with MARS • Our optimization problem bases on the tradeoff between both accuracy, i.e., a small sum of error squares, and not too high a complexity. • This tradeoff is established through the penalty parameters m . • Let use in approximate PRSS by discretizing the high-dimensional integration: Then, PRSS becomes  i  xiT    (di )   N 2 PRSS  i 1 ( N 1) Km  2     Dr,s m (tˆim )  tˆ im. M max     2  2 m m   1    m 1 i 1   ( , ) rs   1 2 r , sV ( m )     M    di   1, 1 (t ),..., M (t ), M 1 (t  T 1 i i M i M 1 ),..., M max (t i M max ) max , ( j ) j1,2,..., Km   0,1, 2,..., N  1  Km M 1 M 2 di : (t , t ,..., t , t 1 i i 2 i M i ,t i ,..., t i M max T )    Km  ˆ t   tl , m , tl , m ,..., tl , m  , m t    tl , m  tl , m  ˆ m i i .   j  j 1    j 1  m j m j m j m j m j  j  j  j
18. 18. Tikhonov Regularization for GLM with MARS • For a short representation, we can rewrite the approximate relation as M max ( N 1) K m   2 PRSS    X    (d )  m L2  m , im 2 2 m 1 i 1 1  2   2   Lim    2  (d )   (d1 ),..., (d N )    Dr,s m (tˆim )   tˆ im  . T   1     (1 , 2 )  r s r , sV ( m )     • We can write PRSS as M max ( N 1) K m PRSS    X  * * 2 2   m m 1  i 1 L2  m , im 2 where X * =  X  (d )  is a block matrix constructed by ( N  p) -matrix X and ( N  (Μ max  1))  (d ) ,  * =    ,     T matrix is a vector constructed and  vectors. • Then, we deal with the linear systems equations of   X * * , approximately.
19. 19. Tikhonov Regularization for GLM with MARS • We approach our problem PRSS as a Tikhonov regularization problem by using the same penalty parameter   m (:  2 ) for each derivative: * 2 * 2 PRSS    X  *   L * . 2 2 Here, high dimentional matrix  L* = R* L  , where R is an * (M max  1)  p matrix with entries being first or second derivatives of  . • We can easily note that our Tikhonov regularization problem has multiple objective functions through * 2 * 2 a linear combination of X  * and X  * . We select the solution such that it minimizes both 2 2 2 2 first objective function   X * * 2 and second objective X * * 2 in the sense of a compromise (tradeoff) solution.
20. 20. An Alternative Solution for Tikhonov Regularization Problem with CQP • We can solve Tikhonov regularization problem for MARS by continuous optimization techniques, especially, conic quadratic programming. • We formulate PRSS as a CQP problem: min z, z , *  I * subject to   X *  * 2  z, L*  *  M . 2 In general : min cT x , subject to Di x  di 2  piT x  qi (i  1, 2,..., k ). (CQP ) x c  (1, 0T max 1 p )T , u  ( z,  *T )T , D1  (0 N , X * ), d1   , p1  (1, 0,..., 0)T , q1  0, M D2  (0M Max 1 , L* ), d2  0M max 1 , p2  0M max  p2 and q2   M .
21. 21. An Alternative Solution for Tikhonov Regularization Problem with CQP • We first reformulate  I *  as a Primal Problem: min z, z , *  0N X *   z     such that      1 0T 1 p    *   0  ,      M max   0 M max 1 L*   z   0 M max 1     *    ,  0 T     M  0 M max 1 p       LN 1 ,   LM max  2 , with ice-cream (or second order or Lorentz) cones:  LN  1  x  ( x1 , x2 ,..., xN 1 )T  R N 1 | xN+1  x12  x2  ...  xN 2 2  ( N  1).
22. 22. An Alternative Solution for Tikhonov Regularization Problem with CQP • The corresponding Dual Problem is  max ( T , 0) 1  0T max 1 ,  M  2 M   0T 1   0T max 1 0   1      2   N M such that  *T X ,  0 M max 1 p  1  L*T  0M max 1 p   0 M max 1 p    1  LN 1 ,  2  L M max  2 .
23. 23. Solution Methods • CQPs belong to the well-structured convex problems. • Interior Point algorithms: – We use the structure of problem. – Yield better complexity bounds. – Exhibit much better practical performance.
24. 24. Application • GLPMs with CMARS and the parameter estimation for them have been presented and investigated in detail. Now, a numerical example for this study will be given. • Two data sets are used in our applications: • Concrete Compressive Strength Data Set • Concrete Slump Test Data Set • Data sets are obtained from UCI Machine Learning Repository (http://archive.ics.uci.edu/ml/). •
25. 25. Application • Salford Systems is used for MARS application, for CMARS a code is written by using MATLAB and in order to solve the CQP problem in CMARS, MOSEK software is preferred. For Tikhonov Regularization Regularization Toolbox in MATLAB is used. • All test data sets are also compared according to the performance measures such as Root Mean Square Error (RMSE), Correlation Coefficient (r), R2, Adjusted R2. •
26. 26. Application • To compare the performances of Tikhonov regularization, CMARS and GPLM models, let us look at the performance measure values for both data sets. WORSE BETTER Evaluation of the models based on performance values: • CMARS performs better than Tikhonov regularization with respect to all the measures for both data sets. • On the other hand, GLM with CMARS (GPLM) performs better than both Tikhonov regularization and CMARS with respect to all the measures for both data sets.
27. 27. Outlook • Important new class of GPLs: having written the Tikhonov regularization task for GLM using MARS as a CQP problem, we will call it CGLMARS: E Y X , T    G XT   T  , e.g., GPLM (X ,T ) = LM (X ) + MARS (T )
28. 28. References [1] Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004. [2] Craven, P., and Wahba, G., Smoothing noisy data with spline functions, Numer. Math. 31, Linear Models, (1979), 377-403. [3] De Boor, C., Practical Guide to Splines, Springer Verlag, 2001. [4] Dongarra, J.J., Bunch, J.R., Moler, C.B., and Stewart, G.W., Linpack User’s Guide, Philadelphia, SIAM, 1979. [5] Friedman, J.H., Multivariate adaptive regression splines, (1991), The Annals of Statistics 19, 1, 1-141. [6] Green, P.J., and Yandell, B.S., Semi-Parametric Generalized Linear Models, Lecture Notes in Statistics, 32 (1985). [7] Hastie, T.J., and Tibshirani, R.J., Generalized Additive Models, New York, Chapman and Hall, 1990. [8] Kincaid, D., and Cheney, W., Numerical Analysis: Mathematics of Scientific computing, Pacific Grove, 2002. [9] Müller, M., Estimation and testing in generalized partial linear models – A comparive study, Statistics and Computing 11 (2001) 299-309, 2001. [10] Nelder, J.A., and Wedderburn, R.W.M., Generalized linear models, Journal of the Royal Statistical Society A, 145, (1972) 470-484. [11] Nemirovski, A., Lectures on modern convex optimization, Israel Institute of Technology http://iew3.technion.ac.il/Labs/Opt/opt/LN/Final.pdf.
29. 29. References [12] Nesterov, Y.E , and Nemirovskii, A.S., Interior Point Methods in Convex Programming, SIAM, 1993. [13] Ortega, J.M., and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [14] Renegar, J., Mathematical View of Interior Point Methods in Convex Programming, SIAM, 2000. [15] Sheid, F., Numerical Analysis, McGraw-Hill Book Company, New-York, 1968. [16] Taylan, P., Weber, G.-W., and Beck, A., New approaches to regression by generalized additive and continuous optimization for modern applications in finance, science and technology, Optimization 56, 5-6 (2007), pp. 1-24. [17] Taylan, P., Weber, G.-W., and Liu, L., On foundations of parameter estimation for generalized partial linear models with B-splines and continuous optimization, in the proceedings of PCO 2010, 3rd Global Conference on Power Control and Optimization, February 2-4, 2010, Gold Coast, Queensland, Australia. [18] Weber, G.-W., Akteke-Öztürk, B., İşcanoğlu, A., Özöğür, S., and Taylan, P., Data Mining: Clustering, Classification and Regression, four lectures given at the Graduate Summer School on New Advances in Statistics, Middle East Technical University, Ankara, Turkey, August 11-24, 2007 (http://www.statsummer.com/). [19] Wood, S.N., Generalized Additive Models, An Introduction with R, New York, Chapman and Hall, 2006.
30. 30. Thank you very much for your attention!