Data Driven Modelling for Complex Systems

Transcription

1 Data Driven Modelling for Complex Systems Dr Hua-Liang (Leon) Wei Senior Lecturer in System Identification and Data Analytics Head of Dynamic Modelling, Data Mining & Decision Making (3DM) Lab Complex Systems & Signal Processing Research Group Department of Automatic Control & System Engineering University of Sheffield Sheffield, UK July 6-8, 017 HLW, Data Based Modelling, Slide 1 of 45 Data Driven Modelling Similar Terminologies: Black-box Modelling Data-based Modelling Learning from Data System Identification, etc. July 6-8, 017 HLW, Data Based Modelling, Slide of 45 1

2 Outline of the Talk 1) A quick selective review of system models Model types and generalised linear models ) Three classes of models White, grey and black-box models 3) Data-driven modelling Letting data speak and learning from data 4) Examples Using data-driven modelling techniques July 6-8, 017 HLW, Data Based Modelling, Slide 3 of 45 PART 1 A Quick Selective Review of System Models July 6-8, 017 HLW, Data Based Modelling, Slide 4 of 45

3 Linear vs Nonlinear Linear Systems Nonlinear Systems Linear systems are simple systems whose structure and behavior are regular and easy to understand. Nonlinear systems are complex systems whose structure and behavior are normally difficult to analyse. July 6-8, 017 Linear vs Nonlinear Models Linear Models Continuous-time model Continuous-time state space Continuous-time transfer function Discrete-time model Discrete-time state space Discrete-time transfer function (e.g. Z-transfer function) Nonlinear Models Continuous-time NL model Continuous-time NL state space Continuous-time transfer function for a nonlinear system? (No, not really!) Discrete-time NL model Discrete-time NL state space Discrete-time transfer function for nonlinear systems? (No, not really!) July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 6 of 45 3

4 Parametric & Nonparametric Models Parametric models ODE s (lumped parameter model) PDE s (distributed parameter model) AR(X) (AutoRegressive with exogenous inputs) ARMA (AautoregRessive Moving Average) NAR(X) (Nonlinear Autoregressive with exogenous inputs) NARMA(X) (Nonlinear Autoregressive Moving Average with exogenous inputs) Non-parametric approaches Correlation analysis (auto- and cross-correlation) FFT and Spectral analysis Time-frequency analysis Wavelet transform PCA, ICA, Bayesian, Gaussian, kernel methods, etc. July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 7 of 45 Linear-In-The-Parameters vs Linear-In-The-Variables Models Let x, z be independent ( input ) variables, y be the dependent variable, and e the noise signal; also let a, b, c be the model parameters. Then, y = a + bx y = a + bx + cx y = a + x b y = a + ln(x b /z c ) (linear in both) (linear-in-the-parameters) (linear-in-the-variables) (linear in neither) July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 8 of 45 4

5 Regression Model (1) The Simplest Case The simplest case of linear regression is used for line fitting y = ax + b (1) The unbiased estimates of a and b can be calculated using given data points (x k, y k ) (k=1,,, N). July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 9 of 45 Regression Model () The Multiple Regression Multiple linear regression model has a form below y f ( x1, x,, xn) e aixi e where x i is the ith input (independent) variable, a i is the associated model parameter. Given observational data points (x k, y k ) (k=1,,, N), the regression model becomes y k f ( x 1, k, x, k,, xn, k ) n i0 e k n i0 a x i i, k e k July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 10 of 45 5

6 Regression Model (3) The Vector and Matrix Form X 1, x1,1, x,1,, xm,1 y1 1, x1,, x,,, x m, y, y 1, x1, N, x, N,, x y m, N N a a a 0 1 T 1 T aˆ ( X X ) ( X y) m y y y 1 N a 0 a 0 a a x 0 1 a x 1 1,1 1, a x 1 1, N a a x a x,1, x a, N m a x m a m,1 x m m, x e e m, N 1 e N Normal equation y Xa e T T ( X X ) a X y T 1 T aˆ ( X X ) X y Least squares estimator July 6-8, 017 Regression Model (4) The General Form In general sense, regression model can be expressed in a linear-in-the-parameters form y a00 ( x1,, xn) a1 1( x1,, xn) amm ( x1,, xn) e a x) a ( x) a ( ) e 0 0 ( 1 1 m m x m i0 a i (x) e i whereφ i (x) are called model regressors or model terms formed by the n model variables through some linear or nonlinear manners. July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 1 of 45 6

7 Regression Model (5) The General Form: Examples Consider the model y = β 0 + β 1 logx 1 + β logx + ε φ 0 (x 1, x )=1 φ 1 (x 1, x )=log x 1 φ (x 1, x )=log x Consider a nonlinear IO system described by the model y( k) a1 y( k ) a y( k 1) u( k 1) a3 u ( k ) 3 a4 y ( k 1) a5 y( k ) u ( k ) e(k) July 6-8, 017 φ 1 (x(k)) φ (x(k)) φ 3 (x(k)) φ 4 (x(k)) φ 5 (x(k)) x 1 (k)= y(k-1) x (k)= y(k-) x 3 (k)= u(k-1) x 4 (k)= u(k-) x(k)=[x 1 (k), x (k), x 3 (k), x 4 (k)] T Regression Model (6) The General Form: Parameter Estimation Define the design matrix 0 ( x(1)) 0 ( x()) 0( x( N)) 1( x(1)) 1( x()) ( x( N)) 1 m( x(1)) ( ()) m x m( x( N)) Parameters can be estimated by least squares estimator T 1 T θˆ ( ) ( y) July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 14 of 45 7

8 Regression Model (7) Least Squares Method Is BLUE Least squares (LS) is BLUE Best (Minimum Variance) Linear Unbiased Estimator LS provides a unique solution if the design matrix is full rank in column. LS is equivalent to maximum-likelihood if noise is Gaussian. July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 15 of 45 PART Three Levels of Modelling for Complex Systems July 6-8, 017 HLW, Data Based Modelling, Slide 16 of 45 8

9 Three Classes of Boxes (Systems): White, Grey and Black-Boxes White-Box Grey-Box Black-Box All necessary information is completely known System information is partly known (the remaining part is unknown) No a priori information of the system is available July 6-8, 017 HLW, Data Based Modelling, Slide 17 of 45 White-, Grey- and Black-Box Modelling Approaches White-box Grey-box Black-box First principles e.g. physical, mechanical, and chemical laws First principles & data driven modelling techniques Data driven modelling (let data speak for themselves) July 6-8, 017 HLW, Data Based Modelling, Slide 18 of 45 9

10 x y Two Trivial Examples inputs 1 output z x +y z inputs x y z f (x,y) z (output) f(x,y) =??? x y z x y z=f(x,y) July 6-8, 017 White-Box Modelling: An Example - Pendulum Using Newton s second law of motion, along the tangential direction, we have ml mg sin( ) kl Taking the state variables as x 1, x we have the system state model Pictures above and below were from Richard Seto s lecture on Physics 000. x 1 x f ( x, x ) 1 1 g k f( x1, x) sin( x1 ) x L m x Using this model, we can do simulation and analysis etc. July 6-8,

11 Length Grey-Box Modelling A Simple Example - Spring a b c d e f g Experimental data Record Force (N) Length (in) 1, a , b , c , d , e , f , g d c a b e f Force Assume that the change in length of the spring is proportional to the force applied (Hooke s law),i.e., Length = a + b Force. With the known model structure, we can estimate the unknown model parameters a and b using some standard algorithm. July 6-8, 017 g When F = 5, L=? Grey-Box Modelling A Simple Example Parameter Estimation F L Data Model a =? L a b F b =? e1 L1 a 1 b F1 e e 1 L a 1 b F e e 3 a e4 b e L7 a1 b F7 e e6 a ˆ e T 1 T 7 ( A A) ( A L) bˆ L A θ e The suggested linear model L= F can only approximately represent the data, that is why there is an error term e in each of the above equations. July 6-8,

12 Black-Box Modelling White-Box: EVERYTHING of the system is known (including model structure, parameters, relevant governing laws and rules etc). Black-Box: NOTHING (or little) of the TRUE system model is known or can be known. What is available are recoded data of the system behaviour of interest. The objective is to learn a model or a set of models from data. Data Black-Box July 6-8, 017 HLW, Data Based Modelling, Slide 3 of 45 PART 3 Uncovering A Black-Box Letting Data Speak and Learning from Data July 6-8, 017 HLW, Data Based Modelling, Slide 4 of 45 1

13 ... Data Driven Modelling Black-Box Modelling: To model black-box systems, what we can do is to let data speak and learn from data. System Identification: A science of generating models from data with no (or very limited) a priori knowledge of the inherent system dynamics. Data Model Data Black-Box July 6-8, 017 Identification Toolkit Common Model Classes: Linear vs Nonlinear Parametric vs Nonparametric Continuous vs Discrete time Lumped vs Distributed Common Model Types: Regression, Differential eqns, Difference eqns, Artificial neural nets, Statistical models, Input-Output Models (1) Input signal Output signal Assumption: a) System input and output signals are measurable. b) The true system model structure is NOT known. Main Objective and Task: To generate a model that well represents the relationship and reveals the dynamics between the input and output. July 6-8, 017 HLW, Data Based Modelling, Slide 6 of 45 13

14 ... Input-Output Models () x1 () t x() t... x () n t y( t) f ( x ( t), x ( t),, x ( t)) e( t) 1 n yt () System identification aims to find a model that represents the relationship between system input and output such that e(t) is noise or error that inevitably exists in any data-driven modelling f( ) can be any arbitrary function but often is chosen to be those that are easily interpreted, or that have good properties and performance. x 1 (t), x (t),, x n (t) are called input variables (also called independent variables, explanatory variables, or simply predictors). y(t) is the output variable (also known as dependent variable). July 6-8, 017 HLW, Data Based Modelling, Slide 7 of 45 Input-Output Models (3) - ANNs Artificial Neural Networks (ANNs): ANNs are among the most popular approaches to data driven modelling. Appropriately trained ANNs can have a very good generalization property (ie prediction capability) However, ANN models are opaque and cannot be written down, and are difficult to interpret. ANN models may not be applicable in many real application scenarios. July 6-8, 017 HLW, Data Based Modelling, Slide 8 of 45 14

15 I/O Models (4) Linear Regression Linear regression models are of the form y( t) f ( x ( t), x ( t),, x ( t)) e( t) 1 a a x ( t) a x ( t)... a x ( t) e( t) n Note that in many data-based modelling tasks, the parameters a 0, a 1, a,, a n are unknown and need to be estimated from observed data. The estimates of these parameters can often be obtained by means of a least squares (LS) method. n n July 6-8, 017 HLW, Data Based Modelling, Slide 9 of 45 I/O Models (5): Generalized Linear Regression Consider a simple case with only 3 explanatory variables: x 1 (t), x (t), x 3 (t). A generalized linear model for such a case can be chosen as: y f ( x, x, x ) e 1 3 a a x a x a x a x x a x x a x x a x x a x x a x x a x x x a x x x e e is modelling error All linear model terms All cross product model terms of degree All cross product model terms of degree 3 All model terms of higher nonlinear degrees July 6-8, 017 HLW, Data Based Modelling, Slide 30 of 45 15

16 I/O Models (4) Linear Difference Eqns...u(t) Input signal y(t) Output signal In dynamic system modelling, the explanatory variables x 1 (t), x (t),, x n (t) are defined as the lagged system input and output variables; in this case, x1 ( t) y( t 1) y( t) f [ x1( t), x( t),, xn( t)] e( t)... a0 a1 y( t 1) a y( t ) a p y( t p) xp ( t) y( t p) 1 b1u ( t 1) bu( t ) bqu( t q) xp ( t) u( t 1)... et ( ) xpq( t) u( t q) This is usually referred to as an AutoRegressive with exogenous (ARX) model, where p and q are called model orders. July 6-8, 017 HLW, Data Based Modelling, Slide 31 of 45 I/O Models (5) Nonlinear Difference Eqns Consider a simple nonlinear difference equation: y( t) f [ y( t 1), y( t ), u( t 1)] e( t) We can use polynomials to approximate the nonlinear function f[ ] as below: All linear model y( t) a a y( t 1) a y( t ) a u( t 1) terms a y( t 1) y( t 1) a y( t 1) y( t ) a y( t 1) u( t 1) a y( t ) y( t ) a y( t ) u( t 1) a u( t 1) u( t 1) All model terms of higher nonlinear degrees All cross product model terms of degree et ( ) This is usually referred to as a Nonlinear AutoRegressive with exogenous input model (NARX). July 6-8, 017 HLW, Data Based Modelling, Slide 3 of 45 16

17 PART 4 Application Examples of Data-Driven Modelling July 6-8, 017 HLW, Data Based Modelling, Slide 33 of 45 Example 1: Bioethanol Production (1) Experimental Data Time Glucose Ethanol (hr) (mm) (mm) July 6-8, 017 Glucose and Ethanol (mm) Glucose Ethanol Hour We have applied the NARX modelling approach to the biomedical data, aiming to reveal the relationship between ethanol product and glucose, without using any a priori. 17

18 Example 1: Bioethanol Production () The identified NARX Model is Ethanol(t) = Ethanol(t-1) Glucose(t-) Glucose(t) Glucose(t-3) + e(t) Ethanol (mm) Measurement Model prediction Prediction error This is a simple model that can perfectly link the system output (ethonal) to the input (glucose) Hour July 6-8, 017 HLW, Data Based Modelling, Slide 35 of 45 Example : High Tide Forecast at the Venice Lagoon (1) A multiscale Cardinal B-spline NARMAX model was employed For a case study, the hourly water level data for year 199 were used for model estimation The identified model was used to predict water levels of year 1993 July 6-8, 017 HLW, Data Based Modelling, Slide 36 of 45 18

19 Example 3: (cont.) High Tide Forecast at the Venice Lagoon () Water Level [cm] Time [hr] 4 hours ahead prediction of water level at the Venice Lagoon in 1993 Thin solid line: measurements; thick dashed: model prediction July 6-8, 017 Example 3: Dst Index Prediction (1) Output signal y(t) = Dst(t) (disturbance storm time, [nt]) index Input variable u(t) = VBs(t) (solar wind rectified electric field [mv/m]) Training data Hourly Dst index and VBs data, March 1-31, 1979 No. of samples = 744 Test data Hourly Dst index and VBs data, April 1-30, 1979 No. of samples = 70 Identified model 3 y ( t) y ( t 1) y ( t 1) u( t 1) y ( t 1) y ( t 3) u( t ) y( t 1) u ( t 1) u( t ) July 6-8, 017 HLW, Data Based Modelling, Slide 38 of 45 19

20 Example 4: Dst Index Prediction () Training data: Hourly Dst index and VBs data, March 1-31, 1979 (~ 744) Test data: Hourly Dst index and VBs data, April 1-30, 1979 (~ 70) July 6-8, 017 HLW, Data Based Modelling, Slide 39 of 45 Example 4: Dst Index Prediction (3) Storm 1: PE = 91.48% (PE: Prediction Efficiency) Storm : PE = 9.17% All test data: PE = 93.34% July 6-8, 017 HLW, Data Based Modelling, Slide 40 of 45 0

21 The Understanding of Complex Systems Needs Data-Driven Modelling July 6-8, 017 HLW, Data Based Modelling, Slide 41 of 45 System Identification Has Many Applications for the Analysis of Complex Systems Chertsey BBC, Surry (9.04 am 13 th Feb 014) In practice, theoretical models are very difficult, if not impossible, to obtain (as they need first principles). Data-driven modelling provides a complementary but powerful tool for understanding complex systems. July 6-8, 017 1

22 Tips for Data Driven Modelling and Concluding Remarks July 6-8, 017 HLW, Data Based Modelling, Slide 43 of 45 Tips for Data Driven Modelling Always try the simplest possible models first (e.g. linear). If a simpler model works, then forget complex models. Keep in mind the main purpose of your modelling task. Transparent, parsimonious and easily interpretable models (e.g. regression models) are desirable if you are aiming to reveal dependency and interaction relationships between different explanatory variables/factors. If the modelling task is merely focused on prediction or classification, then either parametric models (e.g. linear regressons) or complicated opaque models (such as ANN models) can be an option. July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 44 of 45

23 Concluding Remarks System identification and data driven modelling, as powerful state-of-the-art approaches, have been widely applied to various areas of science and engineering. No particular modelling methods are always the best and/or universal for all applications, and therefore it is not possible or appropriate to claim that one approach is always better than all the others for solving all problems. July 6-8, 017 Dr HL Wei, Data Based Modelling, Slide 45 of 45 Key References 1. Wei, H.L., Billings, S.A., & Liu, J. (004) Term and variable selection for nonlinear system identification, International Journal of Control, 77, Wei, H.L., Billings, S.A., & Balikhin, M.A. (004) Prediction of the Dst index using multiresolution wavelet models Journal of Geophysical Research, 109, A Wei, H.L., & Billings, S.A. (006) Long term prediction of nonlinear time series using multiresolution models, International Journal of Control, 79, Wei, H.L., & Billings, S.A. (006) An efficient nonlinear cardinal B-spline model for high tide forecasts at the Venice Lagoon, Nonlinear Processes in Geophysics,13, Wei, H.L., Billings, S.A., & Balikhin, M.A.(006) Wavelet based nonparametric NARX models for nonlinear input-output system identification, International Journal of Systems Science, 37(15), Wei, H.L, Zhu, D., Billings, S.A., & Balikhin, M.A. (007) Forecasting the geomagnetic activity of the Dst index using multiscale radial basis function networks, Advances in Space Research, 40, Billings, S.A., Wei, H.L., & Balikhin, M.A. (007) Generalised multiscale radial basis function networks, Neural Networks, 0(10), Billings, S.A., & Wei, H.L.(008) An adaptive search algorithm for model subset selection and nonlinear system identification, International Journal of Control, 81, Wei, H.L., & Billings, S.A. (008) Model structure selection using an integrated forward orthogonal search algorithm assisted by squared correlation and mutual information, International Journal of Modelling, Identification and Control, 3, Billings, S.A. (013) Nonlinear system identification: NARMAX methods in the time, frequency, and spatio-temporal domains: John Wiley & Sons. 11. Ayala Solares, J. R., Wei, H.L., Boynton, R. J., Walker, S. N., & Billings, S. A. (016) Modelling and Prediction of Global Magnetic Disturbance in Near-Earth Space: a Case July 6-8, Study 017 for Kp Index using NARX Models, Space Weather, 14(10),

24 Acknowledgement We gratefully acknowledge that part of this work was supported by: EC Horizon 00 Research and Innovation Action Framework Programme (Grant No and grant title PROGRESS ). Engineering and Physical Sciences Research Council (EPSRC) (Grant No EP/I011056/1) EPSRC Platform Grant (Grant No EP/H00453X/1) July 6-8, 017 Thank You! Any Questions? July 6-8, 017 HLW, Data Based Modelling, Slide 48 of 45 4