Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.1/30

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1 Model Validation in Non-Linear Continuous-Discrete Grey-Box Models Jan Holst, Erik Lindström, Henrik Madsen and Henrik Aalborg Niels Division of Mathematical Statistics, Centre for Mathematical Sciences Lund Institute of Technology, Lund, Sweden Informatics and Mathematical Modeling Technical University of Denmark, Kongens Lyngby, Denmark Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.1/30

2 General model: Class of models dx (t) = (t, u(t), X (t), θ)dt + (t, u(t), X (t), θ)dw (t) Y (t k ) = h(t k, u(t k ), X (t k ), θ) + e k where (t, u(t), X (t), θ) : [0, T ] R l R n R d R n The integrals are interpreted in the sense of Itô. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.2/30

3 General model: Class of models dx (t) = (t, u(t), X (t), θ)dt + (t, u(t), X (t), θ)dw (t) where Y (t k ) = h(t k, u(t k ), X (t k ), θ) + e k (t, u(t), X (t), θ) : [0, T ] R l R n R d R n (t, u(t), X (t), θ) : [0, T ] R l R n R d R n m The integrals are interpreted in the sense of Itô. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.2/30

4 General model: Class of models dx (t) = (t, u(t), X (t), θ)dt + (t, u(t), X (t), θ)dw (t) where Y (t k ) = h(t k, u(t k ), X (t k ), θ) + e k (t, u(t), X (t), θ) : [0, T ] R l R n R d R n (t, u(t), X (t), θ) : [0, T ] R l R n R d R n m dw (t) is a m-dimensional Wiener process. The integrals are interpreted in the sense of Itô. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.2/30

5 General model: Class of models dx (t) = (t, u(t), X (t), θ)dt + (t, u(t), X (t), θ)dw (t) where Y (t k ) = h(t k, u(t k ), X (t k ), θ) + e k (t, u(t), X (t), θ) : [0, T ] R l R n R d R n (t, u(t), X (t), θ) : [0, T ] R l R n R d R n m dw (t) is a m-dimensional Wiener process. h(t, u(t), X (t), θ) : [0, T ] R l R n R d R p The integrals are interpreted in the sense of Itô. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.2/30

6 General model: Class of models dx (t) = (t, u(t), X (t), θ)dt + (t, u(t), X (t), θ)dw (t) where Y (t k ) = h(t k, u(t k ), X (t k ), θ) + e k (t, u(t), X (t), θ) : [0, T ] R l R n R d R n (t, u(t), X (t), θ) : [0, T ] R l R n R d R n m dw (t) is a m-dimensional Wiener process. h(t, u(t), X (t), θ) : [0, T ] R l R n R d R p e k N (0, S), e k R p The integrals are interpreted in the sense of Itô. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.2/30

7 Overview Three tools on model validation in continuous-discrete time grey-box models: Dependence identification. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.3/30

8 Overview Three tools on model validation in continuous-discrete time grey-box models: Dependence identification. Model structure identification. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.3/30

9 Overview Three tools on model validation in continuous-discrete time grey-box models: Dependence identification. Model structure identification. Generalized Gaussian residuals. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.3/30

10 Dependence identification It is often reasonable, under some regularity conditions, to approximate the system as Gaussian. This means that filter methods can be used to calculate conditional mean and covariance. Define the one-step prediction error: r k = Y t k E[Y tk F k 1 ] V [Ytk F k 1 ], where F k 1 is the available information at time t k 1. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.4/30

11 Dependence identification It is often reasonable, under some regularity conditions, to approximate the system as Gaussian. This means that filter methods can be used to calculate conditional mean and covariance. Define the one-step prediction error: r k = Y t k E[Y tk F k 1 ] V [Ytk F k 1 ], where F k 1 is the available information at time t k 1. Non-equidistantly sampled data is normalized. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.4/30

12 Dependence identification, cont. The squared multiple correlation coefficient is given by 2 0(1...k) = V [Y ] V [Y X 1,..., X k ]. V [Y ] Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.5/30

13 Dependence identification, cont. The squared multiple correlation coefficient is given by 2 0(1...k) = V [Y ] V [Y X 1,..., X k ]. V [Y ] Assuming normality and maximum likelihood estimates yields R 2 0(1...k) = SS 0 SS 0(1...k) SS 0, where SS 0 = (y i (y i /N )) 2 and SS 0(1...k) is calculated conditional on X 1,..., X k. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.5/30

14 Dependence identification, cont. 2 is very similar to squared Sample AutoCorrelation Function! Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.6/30

15 Dependence identification, cont. 2 is very similar to squared Sample AutoCorrelation Function! Let f k (r) = E[r k r k n = r] and define Lag Dependent Function as LDF(k) = sign (ˆfk (r max ) ˆf ) k (r min ) (R0(k) 2 ) +. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.6/30

16 Dependence identification, cont. 2 is very similar to squared Sample AutoCorrelation Function! Let f k (r) = E[r k r k n = r] and define Lag Dependent Function as LDF(k) = sign (ˆfk (r max ) ˆf ) k (r min ) (R0(k) 2 ) +. The SACF is obtained as a special case of the LDF if a linear model is used to calculate R0(k) 2. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.6/30

17 Dependence identification, cont. We can define Partial Lag Dependent Functions in a similar fashion. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.7/30

18 Dependence identification, cont. We can define Partial Lag Dependent Functions in a similar fashion. The partial square correlation coefficient is given by 2 (0k) (1...k 1) = V [Y X 1,..., X k 1 ] V [Y X 1,..., X k ]. V [Y X 1,..., X k 1 ] Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.7/30

19 Dependence identification, cont. We can define Partial Lag Dependent Functions in a similar fashion. The partial square correlation coefficient is given by 2 (0k) (1...k 1) = V [Y X 1,..., X k 1 ] V [Y X 1,..., X k ]. V [Y X 1,..., X k 1 ] Warning: Defining the Partial Lag Dependent Functions for general non-linear autoregressive models X (n) = g(x (n 1),..., X (n k)) + (n) is not feasible for large k. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.7/30

20 Dependence identification, cont. Confidence intervals are obtained by bootstrap under the hypothesis of a i.i.d. process (for LDF and PLDF). Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.8/30

21 Dependence identification, cont. Confidence intervals are obtained by bootstrap under the hypothesis of a i.i.d. process (for LDF and PLDF). The distribution for the LDF will be symmetric about zero. Hence, an upper confidence limit for LDF(k) or PLDF(k) is to be approximated. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.8/30

22 Dependence identification, cont. Confidence intervals are obtained by bootstrap under the hypothesis of a i.i.d. process (for LDF and PLDF). The distribution for the LDF will be symmetric about zero. Hence, an upper confidence limit for LDF(k) or PLDF(k) is to be approximated. Confidence intervals will in general be wider than corresponding intervals for linear models (SACF and/or SPACF). Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.8/30

23 Dependence identification, cont observations from a non-linear MA(1) model: X t = e t + 2 cos e t 1, Sample Autocorrelation (left) and Lag dependent function (right) SACF LDF Lag Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.9/30 Lag

24 Model structure identification Most physical, chemical or biological systems are thought of having a complex determistic behavior and only small state noise. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.10/30

25 Model structure identification Most physical, chemical or biological systems are thought of having a complex determistic behavior and only small state noise. The system noise can sometimes be interpreted as model deficiency. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.10/30

26 Model structure identification Most physical, chemical or biological systems are thought of having a complex determistic behavior and only small state noise. The system noise can sometimes be interpreted as model deficiency. Filtering techniques allow us to separate the state noise from the measurement noise. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.10/30

27 Model structure identification Most physical, chemical or biological systems are thought of having a complex determistic behavior and only small state noise. The system noise can sometimes be interpreted as model deficiency. Filtering techniques allow us to separate the state noise from the measurement noise. Idea: Use the estimated elements in the diffusion term to pinpoint model deficiencies. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.10/30

28 Model structure identification, cont. Pseudoalgorithm: Test the residuals for dependence using e.g. LDF or PLDF etc. Repeat until no significant dependence remains. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.11/30

29 Model structure identification, cont. Pseudoalgorithm: Test the residuals for dependence using e.g. LDF or PLDF etc. Extending the state space, guided by large elements in the diffusion term. Repeat until no significant dependence remains. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.11/30

30 Model structure identification, cont. Pseudoalgorithm: Test the residuals for dependence using e.g. LDF or PLDF etc. Extending the state space, guided by large elements in the diffusion term. Pinpoint dependence by non-parametric regression. Repeat until no significant dependence remains. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.11/30

31 Model structure identification, cont. Pseudoalgorithm: Test the residuals for dependence using e.g. LDF or PLDF etc. Extending the state space, guided by large elements in the diffusion term. Pinpoint dependence by non-parametric regression. Expand the state space model to model dependence. Repeat until no significant dependence remains. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.11/30

32 Model structure identification, cont. We can test parameter dependence by extending the state space. This can also be generalized to other functional relations r(t) = (t, u(t), X (t), θ). Example: Let r(t) = θ (j). We test this by extending the state space dx (t) dr(t) = ( ) 0 dt + ( ) 0 0 r dw (t) dw (t) We expect the added state (a fixed parameter) to be independent of t, u(t) and X (t).. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.12/30

33 Model structure identification, cont. Example: Simulated model of a fed-batch bioreactor. dx (t) = ( ds(t) = ( dv (t) = F(t)dt + (S(t))X (t) F(t)X (t)/v (t) ) dt + 11dW (t) (1) (S(t))X (t)/y (t) + F(t)(S F (t) S(t))/V (t) ) dt + 33dW (t) (3) 22dW (t) (2) where X (t) is biomass concentration S(t) is substrate concentration V (t) is the volyme F(t) is the feed flow rate, S F (t) is the feed concentration of substrate and Y (t) is the yield coefficient of biomass. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.13/30

34 Model structure identification, cont. Finally, (S) is the biomass growth rate given by (S) = max S K 2 S 2 + S + K 1. Measurement equation: y 1 y 2 = y 3 k X S V k + e k where e k N (0, S), S = S S S 33. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.14/30

35 Model structure identification, cont. Test the model ( (S) = ) where we have extended the state space. dx (t) = ( ds(t) = ( dv (t) = F(t)dt + d (t) = 0dt + (t)x (t) F(t)X (t)/v (t) ) dt + 11dW (t) (1) (t)x (t)/y (t) + F(t)(S(t) F S(t))/V (t) ) dt + 33dW (t) (3) 44dW (t) (4) 22dW (t) (2) The parameters and states are estimated using an Extended Kalman filter. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.15/30

36 Model structure identification, cont. Parameter estimates (Quasi Maximum Likelihood): Parameter True Estimate Std dev t-score Significant? X E E Yes S E E Yes V E E Yes E E Yes E E No E E No E E No E E Yes S E E Yes S E E Yes S E E Yes Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.16/30

37 Model structure identification, cont. Partial dependence plot of ˆk k vs. ˆX k k and ˆk k vs. Ŝk k µ t t 0 µ t t X t t S t t Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.17/30

38 Generalized Gaussian residuals Univariate but state dependence in diffusion term is no theoretical problem. dx (t) = (t, u(t), X (t), θ)dt + Y (t k ) = X (t k ). (t, u(t), X (t), θ)dw (t), where (t, u(t), X (t), θ) : [0, T ] R l R R d R Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.18/30

39 Generalized Gaussian residuals Univariate but state dependence in diffusion term is no theoretical problem. dx (t) = (t, u(t), X (t), θ)dt + Y (t k ) = X (t k ). (t, u(t), X (t), θ)dw (t), where (t, u(t), X (t), θ) : [0, T ] R l R R d R (t, u(t), X (t), θ) : [0, T ] R l R R d R Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.18/30

40 Generalized Gaussian residuals Univariate but state dependence in diffusion term is no theoretical problem. dx (t) = (t, u(t), X (t), θ)dt + Y (t k ) = X (t k ). (t, u(t), X (t), θ)dw (t), where (t, u(t), X (t), θ) : [0, T ] R l R R d R (t, u(t), X (t), θ) : [0, T ] R l R R d R dw (t) is the increment of a Wiener process. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.18/30

41 Generalized Gaussian residuals, cont. How do we define residuals? Discrete time models X (n + 1) = f (X (n), θ) + g(x (n), θ) (n + 1). The i.i.d. sequence { (n)} is the natural choice. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.19/30

42 Generalized Gaussian residuals, cont. How do we define residuals? Discrete time models X (n + 1) = f (X (n), θ) + g(x (n), θ) (n + 1). The i.i.d. sequence { (n)} is the natural choice. Continuous time models X (n+1) = X (n)+ (s, u(s), X (s), θ)ds+ No natural, general choice! (s, u(s), X (s), θ)dw (s Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.19/30

43 Generalized Gaussian residuals, cont. We follow the approach taken in (Pedersen, 1994) but will define Generalized Gaussian Residuals. The reason is threefold Uncorrelated Gaussian independent random variables! Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.20/30

44 Generalized Gaussian residuals, cont. We follow the approach taken in (Pedersen, 1994) but will define Generalized Gaussian Residuals. The reason is threefold Uncorrelated Gaussian independent random variables! Better outlier detection. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.20/30

45 Generalized Gaussian residuals, cont. We follow the approach taken in (Pedersen, 1994) but will define Generalized Gaussian Residuals. The reason is threefold Uncorrelated Gaussian independent random variables! Better outlier detection. More powerful distributional tests. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.20/30

46 Generalized Gaussian residuals, cont. Pseudoalgorithm: Define x = F 1 X (n) X (n 1) (y) = inf{x : F X (n) X (n 1)(x) y} as the generalized inverse. It follows that U (n) = F X (n) X (n 1) (X (n)) U (0, 1) U (n) is the generalized (uniform) residuals introduced by (Pedersen, 1994). This is the inverse method! We will define the sequence {Y (n)} as the Generalized Gaussian Residuals. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.21/30

47 Generalized Gaussian residuals, cont. Pseudoalgorithm: Define x = F 1 X (n) X (n 1) (y) = inf{x : F X (n) X (n 1)(x) y} as the generalized inverse. It follows that U (n) = F X (n) X (n 1) (X (n)) U (0, 1) U (n) is the generalized (uniform) residuals introduced by (Pedersen, 1994). Y (n) = 1 (U (n)) N (0, 1). This is the inverse method! We will define the sequence {Y (n)} as the Generalized Gaussian Residuals. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.21/30

48 Algorithm, cont. Calculation of F X (n) X (n 1) (x) can be done by using: Monte Carlo techniques. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.22/30

49 Algorithm, cont. Calculation of F X (n) X (n 1) (x) can be done by using: Monte Carlo techniques. Solving a Partial Differential Equation (Fokker-Planck). Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.22/30

50 Algorithm, cont. Calculation of F X (n) X (n 1) (x) can be done by using: Monte Carlo techniques. Solving a Partial Differential Equation (Fokker-Planck). Other approaches, such as binomial trees, path integration etc. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.22/30

51 Algorithm, cont. It turns out that simulation is not well suited for this problem as U (n) = P(X (n) x(n) X (n 1)) = 1 N N j=1 1 {X (n) (j) x(n)}, which is a discrete approximation of the distribution function with low accuracy in the tails. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.23/30

52 Algorithm, cont. It turns out that simulation is not well suited for this problem as U (n) = P(X (n) x(n) X (n 1)) = 1 N N j=1 1 {X (n) (j) x(n)}, which is a discrete approximation of the distribution function with low accuracy in the tails. Applying 1 will amplify the errors leading to a bad approximation (possibly 1 (0) = ). Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.23/30

53 Algorithm, cont. It turns out that simulation is not well suited for this problem as U (n) = P(X (n) x(n) X (n 1)) = 1 N N j=1 1 {X (n) (j) x(n)}, which is a discrete approximation of the distribution function with low accuracy in the tails. Applying 1 will amplify the errors leading to a bad approximation (possibly 1 (0) = ). This problem is magnified by heavy tails, c.f. Jarque-Bera tests. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.23/30

54 Algorithm, cont. The Fokker-Planck (or Kolmogorov forward) equation for a diffusion dx (t) = (t, u(t), X (t), θ)dt + (t, u(t), X (t), θ)dw (t), is given by: p t = A p(x ti 1, t i 1 ; x ti, t i ), where the operator A is defined as A f = x ( (t, u(t), x, θ)f ) The equation is solved by 2 ( x 2 Finite difference approximation of the derivatives 2 (t, u(t), x, θ)f ). Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.24/30

55 Algorithm, cont. The Fokker-Planck (or Kolmogorov forward) equation for a diffusion dx (t) = (t, u(t), X (t), θ)dt + (t, u(t), X (t), θ)dw (t), is given by: p t = A p(x ti 1, t i 1 ; x ti, t i ), where the operator A is defined as A f = x ( (t, u(t), x, θ)f ) The equation is solved by 2 ( x 2 2 (t, u(t), x, θ)f ). Finite difference approximation of the derivatives Pade approximation of matrix exponentials Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.24/30

56 Algorithm, cont. Finally, F X (n) X (n 1) (x) = x p(s, x s; t, y)dy is calculated by a Gauss-Cotes method of the same order of accuracy as the finite difference approximation. The calculation of the Generalized Gaussian residuals is closely connected to Maximum Likelihood estimation. It is therefore suggested that the model is identified by Estimating the parameters in an approximative discrete time model. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.25/30

57 Algorithm, cont. Finally, F X (n) X (n 1) (x) = x p(s, x s; t, y)dy is calculated by a Gauss-Cotes method of the same order of accuracy as the finite difference approximation. The calculation of the Generalized Gaussian residuals is closely connected to Maximum Likelihood estimation. It is therefore suggested that the model is identified by Estimating the parameters in an approximative discrete time model. Using these estimates as initial values for the Maximum Likelihood estimator. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.25/30

58 Example, Cox-Ingersoll-Ross Model: dr t = 0.17 (0.05 r t )dt r t dw t. 500 observations, t=1. (Simulated data) Probability Normal Probability Plot Data lag lag Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.26/30

59 Example, Cox-Ingersoll-Ross Generalized Gaussian residuals Probability Normal Probability Plot Data lag lag Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.27/30

60 Example, CKLS Model: dr t = ( r t )dt + rt dw t. Weekly observations, US 3-month T-bill Feb83 Aug85 Feb88 Aug90 Feb93 Aug95 Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.28/30

61 Example, CKLS Generalized Gaussian residuals. Parameter estimated using an approximative Maximum Likelihood estimator Feb83 Feb87 Feb91 Feb95 Probability Normal Probability Plot Data lag Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.29/ lag

62 Summary We have presented three tools Lag dependent functions to identify dependence. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.30/30

63 Summary We have presented three tools Lag dependent functions to identify dependence. Non-parametric regression and the estimated state diffusion term to identify model defiencies. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.30/30

64 Summary We have presented three tools Lag dependent functions to identify dependence. Non-parametric regression and the estimated state diffusion term to identify model defiencies. Generalized Gaussian Residuals as a definition of residuals to examine the true residuals. Model Validation in Non-Linear Continuous-Discrete Grey-Box Models p.30/30