Vulnerability of economic systems

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1 Vulnerability of economic systems Quantitative description of U.S. business cycles using multivariate singular spectrum analysis Andreas Groth* Michael Ghil, Stéphane Hallegatte, Patrice Dumas * Laboratoire de Météorologie Dynamique Ecole Normale Supérieure, Paris Centre International de Recherche sur l Environnement et le Développement, Nogent-sur-Marne, France Seminar at the Potsdam Institute for Climate Impact Research

2 Motivation Given Short data sets Different scales in space and time Contain observation errors Repetition not always possible (maybe under control of parameters) Wanted Distinguish between regular deterministic behavior cycles and irregular behavior noise Extract a skeleton of the underlying system Understand and model the underlying mechanism Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 2/30

3 Motivation U.S. macroeconomic data from the Bureau of Economic Analysis Trend residuals after Hodrick-Prescott detrending Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 3/30

4 Motivation Two competing business cycle theories Exogenous Exogenous shocks yield to fluctuations of an otherwise stable system Real business cycle theory Endogenous Intrinsic dynamics yield to cyclical behavior Endogenous business cycle theory Real business cycle theory is the dominating one in science and denies the need to stabilize the economy (by e.g. government) U.S. macroeconomic data from the Bureau of Economic Analysis Trend residuals after Hodrick-Prescott detrending Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 3/30

5 Outline 1 Univariate Singular Spectrum Analysis (SSA) General idea of time-delayed embedding Combination with Principal Component Analysis (PCA) Properties and interpretation 2 Multivariate Singular Spectrum Analysis (M-SSA) Extraction of shared behavior Reconstruct skeleton of attractor Local variance fraction Attraction of limit cycle 3 Verification by Monte Carlo (M)SSA Problem of short and noisy observations Alternative hypotheses of oscillation-like fluctuations Vulnerability 4 Conclusions Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 4/30

6 Univariate Singular Spectrum Analysis (SSA) 1 Univariate Singular Spectrum Analysis (SSA) General idea of time-delayed embedding Combination with Principal Component Analysis (PCA) Properties and interpretation 2 Multivariate Singular Spectrum Analysis (M-SSA) Extraction of shared behavior Reconstruct skeleton of attractor Local variance fraction Attraction of limit cycle 3 Verification by Monte Carlo (M)SSA Problem of short and noisy observations Alternative hypotheses of oscillation-like fluctuations Vulnerability 4 Conclusions Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 5/30

7 General idea of time-delayed embedding Problem Underlying dynamical systems ẏ = F (y) is unknown We obtain a scalar measurement x(t) with t = 1... N from this system x(t) = H(y(t)) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 6/30

8 General idea of time-delayed embedding Problem Idea Underlying dynamical systems ẏ = F (y) is unknown We obtain a scalar measurement x(t) with t = 1... N from this system x(t) = H(y(t)) Build a new M-dimensional time series from x(t) and its lagged copies x(t) = (x(t), x(t + 1),..., x(t + M 1)) These delay coordinates x share key topological properties with y (Whitney 1936; Mañé 1981; Takens ) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 6/30

9 General idea of time-delayed embedding Problem Idea Underlying dynamical systems ẏ = F (y) is unknown We obtain a scalar measurement x(t) with t = 1... N from this system x(t) = H(y(t)) Build a new M-dimensional time series from x(t) and its lagged copies x(t) = (x(t), x(t + 1),..., x(t + M 1)) Question These delay coordinates x share key topological properties with y (Whitney 1936; Mañé 1981; Takens ) How to extract those properties from short and noisy time series? Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 6/30

10 Combination with Principal Component Analysis (PCA) Idea Apply PCA to x(t) in order to extract the entire attractor of the (nonlinear) system (Broomhead and King 1986) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 7/30

11 Combination with Principal Component Analysis (PCA) Idea Apply PCA to x(t) in order to extract the entire attractor of the (nonlinear) system (Broomhead and King 1986) However An original motivation to interpret the results in terms of attractor dimensions fails even in relatively simple cases! (Vautard and Ghil 1989) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 7/30

12 Combination with Principal Component Analysis (PCA) Idea Apply PCA to x(t) in order to extract the entire attractor of the (nonlinear) system (Broomhead and King 1986) However An original motivation to interpret the results in terms of attractor dimensions fails even in relatively simple cases! (Vautard and Ghil 1989) Anyway The idea to reconstruct the skeleton of the attractor, e.g. the most robust, albeit unstable limit cycles by means of PCA remains promising (Vautard and Ghil 1989; Ghil and Vautard 1991) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 7/30

13 Combination with Principal Component Analysis (PCA) Idea Apply PCA to x(t) in order to extract the entire attractor of the (nonlinear) system (Broomhead and King 1986) However An original motivation to interpret the results in terms of attractor dimensions fails even in relatively simple cases! (Vautard and Ghil 1989) Anyway The idea to reconstruct the skeleton of the attractor, e.g. the most robust, albeit unstable limit cycles by means of PCA remains promising Singular Spectrum Analysis (Vautard and Ghil 1989; Ghil and Vautard 1991) Time-delayed embedding of scalar time series + PCA Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 7/30

14 Eigenvalues and eigenvectors 1 Delay embedding of x(t) x(t) = (x(t), x(t + 1),..., x(t + M 1)) R M Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 8/30

15 Eigenvalues and eigenvectors 1 Delay embedding of x(t) x(t) = (x(t), x(t + 1),..., x(t + M 1)) R M 2 Estimate covariance matrix C of x (of size M M) with elements c ij = N i j 1 x(t)x(t + i j ) N i j t=1 Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 8/30

16 Eigenvalues and eigenvectors 1 Delay embedding of x(t) x(t) = (x(t), x(t + 1),..., x(t + M 1)) R M 2 Estimate covariance matrix C of x (of size M M) with elements c ij = N i j 1 x(t)x(t + i j ) N i j t=1 3 Determine eigenvalues and eigenvectors (λ k, e k ) of C Ce k = λ k e k k = 1... M E C E = Λ in matrix notation Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 8/30

17 Eigenvalues and eigenvectors 1 Delay embedding of x(t) x(t) = (x(t), x(t + 1),..., x(t + M 1)) R M 2 Estimate covariance matrix C of x (of size M M) with elements c ij = N i j 1 x(t)x(t + i j ) N i j t=1 3 Determine eigenvalues and eigenvectors (λ k, e k ) of C Ce k = λ k e k k = 1... M E C E = Λ in matrix notation λ k equals the variance of x in the direction of e k Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 8/30

18 Eigenvalues and eigenvectors 1 Delay embedding of x(t) x(t) = (x(t), x(t + 1),..., x(t + M 1)) R M 2 Estimate covariance matrix C of x (of size M M) with elements c ij = N i j 1 x(t)x(t + i j ) N i j t=1 3 Determine eigenvalues and eigenvectors (λ k, e k ) of C Ce k = λ k e k k = 1... M E C E = Λ in matrix notation λ k equals the variance of x in the direction of e k Sum of all eigenvalues equals the total variance of the original time series x Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 8/30

19 Spectrum of eigenvalues Eigenvalues 1 4 explain 50% of variance Idea Separate signal from noise Find a break point in the spectrum of eigenvalues BUT Such a break point is not obvious for complex time series! Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 9/30

20 Principal components and reconstructed components Project x onto eigenvectors gives principal components (PC) M A k (t) = x(t + j 1)e k (j) j=1 Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 10/30

21 Principal components and reconstructed components Project x onto eigenvectors gives principal components (PC) M A k (t) = x(t + j 1)e k (j) j=1 Reconstruct that part of x that belongs to certain eigenvectors r k (t) = 1 M M A k (t j + 1)e k (j) j=1 gives reconstructed components (RC) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 10/30

22 Principal components and reconstructed components Project x onto eigenvectors gives principal components (PC) M A k (t) = x(t + j 1)e k (j) j=1 Reconstruct that part of x that belongs to certain eigenvectors r k (t) = 1 M M A k (t j + 1)e k (j) j=1 gives reconstructed components (RC) Especially, sum of all RCs gives original time series! Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 10/30

23 Principal components and reconstructed components RCs 1 4 explain already 50% of total variance RCs 1 2 and 3 4 form oscillatory pairs (Vautard and Ghil 1989) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 11/30

Principal components and reconstructed components Stepwise reconstruction of the dynamical behavior Reconstruct ghost limit cycles of dynamical systems (albeit

24 Principal components and reconstructed components Stepwise reconstruction of the dynamical behavior Reconstruct ghost limit cycles of dynamical systems (albeit unstable, the orbit is visited by system s trajectory) (Kimoto and Ghil 1993; Ghil and Yiou 1996) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 12/30

25 Properties and interpretation SSA Wavelet analysis Analyzing function Eigenvectors e k Mother wavelet Ψ Basis function e k data-adaptive Ψ chosen a priori Decomposition M j=1 x(t + j 1)e k(j) x(t)ψ(t b/a)dt Scale M a Epoch t b (after Ghil et al. 2002) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 13/30

26 Properties and interpretation SSA Wavelet analysis Analyzing function Eigenvectors e k Mother wavelet Ψ Basis function e k data-adaptive Ψ chosen a priori Decomposition M j=1 x(t + j 1)e k(j) x(t)ψ(t b/a)dt Scale M a Epoch t b (after Ghil et al. 2002) Eigenvectors are symmetric/antisymmetric Eigenvectors can form pairs, analog to sine/cosine-pairs in Fourier analysis Eigenvectors can be considered as FIR-filters (Finite Impulse Response), which extract a narrow frequency band of x Therefore, assign a frequency to each eigenvector Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 13/30

27 Properties and interpretation Eigenvectors Data-adaptive filters Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 14/30

28 Properties and interpretation Advantages The basis of decomposition is not chosen a priori This gives more flexibility to the analysis However, requires more post analysis in order to interpret and understand the outcome Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 15/30

29 Multivariate Singular Spectrum Analysis (M-SSA) 1 Univariate Singular Spectrum Analysis (SSA) General idea of time-delayed embedding Combination with Principal Component Analysis (PCA) Properties and interpretation 2 Multivariate Singular Spectrum Analysis (M-SSA) Extraction of shared behavior Reconstruct skeleton of attractor Local variance fraction Attraction of limit cycle 3 Verification by Monte Carlo (M)SSA Problem of short and noisy observations Alternative hypotheses of oscillation-like fluctuations Vulnerability 4 Conclusions Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 16/30

30 Motivation U.S. macroeconomic data from the Bureau of Economic Analysis Trend residuals after Hodrick-Prescott detrending Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 17/30

31 Extraction of shared behavior Multivariate SSA is a natural extension of SSA It extract principal patterns in time as well as space (Broomhead and King 1986b) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 18/30

32 Extraction of shared behavior Multivariate SSA is a natural extension of SSA It extract principal patterns in time as well as space Starting point Multivariate time series (Broomhead and King 1986b) X(t) = (x 1 (t), x 2 (t),..., x D (t)), t = 1... N Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 18/30

33 Extraction of shared behavior Multivariate SSA is a natural extension of SSA It extract principal patterns in time as well as space Starting point Multivariate time series (Broomhead and King 1986b) X(t) = (x 1 (t), x 2 (t),..., x D (t)), t = 1... N Covariance matrix of all pairs of time series C 1,1 C 1,2... C 1,D C 2,1 C 2,2... C 2,D C =.. C d,d. C D,1 C D,2... C D,D of size D M D M Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 18/30

34 Multivariate SSA Same procedure as before Eigenelements: Determine eigenvalues and eigenvectors of C Cẽ k = λ k ẽ k, k = 1... D M Principal components: Project X(t) onto ẽ k A k (t) = D d=1 j=1 M x d (t + j 1)e d k (j) } {{ } SSA Reconstructed components: Combine PCs with ẽ k r d k (t) = 1 M M A k (t j + 1)e d k (j) j=1 Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 19/30

35 Extraction of shared behavior Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 20/30

36 Reconstruct skeleton of attractor RCs 1 2 form oscillatory pair with period of 5.2 years Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 21/30

37 Local variance fraction Attraction of limit cycle V (t) = A2 1 (t) + A2 2 (t) DM k=1 A2 k (t) (Plaut and Vautard 1994) During periods of recession the 5.2-year cycle is more attractive than during periods of growth (Groth et al. 2010) On the other hand: The system is more volatile during periods of growth and therefore more vulnerable?! (Fluctuation dissipation) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 22/30

38 Verification by Monte Carlo (M)SSA 1 Univariate Singular Spectrum Analysis (SSA) General idea of time-delayed embedding Combination with Principal Component Analysis (PCA) Properties and interpretation 2 Multivariate Singular Spectrum Analysis (M-SSA) Extraction of shared behavior Reconstruct skeleton of attractor Local variance fraction Attraction of limit cycle 3 Verification by Monte Carlo (M)SSA Problem of short and noisy observations Alternative hypotheses of oscillation-like fluctuations Vulnerability 4 Conclusions Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 23/30

39 Problem of short and noisy observations Warning Time series are short and noisy Only a single observation is available Experiment cannot be repeated (maybe under control of parameters) GDP Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 24/30

40 Problem of short and noisy observations Warning Time series are short and noisy Only a single observation is available Experiment cannot be repeated (maybe under control of parameters) GDP Random walk (detrended) Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 24/30

41 Alternative hypotheses of oscillation-like fluctuations Surrogate data Generate similar time series without oscillations but otherwise close to x 1 Bad: White noise far from x, obviously rejected 2 Better: Red noise, here in terms of AR(1) process Closer to x z(t) = a z(t 1) + σξ(t) ξ white noise Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 25/30

42 Alternative hypotheses of oscillation-like fluctuations Surrogate data Generate similar time series without oscillations but otherwise close to x 1 Bad: White noise far from x, obviously rejected 2 Better: Red noise, here in terms of AR(1) process Closer to x z(t) = a z(t 1) + σξ(t) ξ white noise Monte Carlo SSA Measure resemblance of z with x (Allen and Smith 1996) 1 Estimate covariance matrix of z: C z 2 Project onto eigenvectors of x: Λ z = E C z E 3 Compare statistics on diagonal elements of Λ z with original eigenvalues Λ Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 25/30

43 Advantages of multivariate analysis SSA of GDP M-SSA of 5 time series Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 26/30

44 Advantages of multivariate analysis SSA of GDP M-SSA of 5 time series Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 26/30

45 Advantages of multivariate analysis SSA of GDP M-SSA of 5 time series Single time series cannot be distinguished from red noise Eigenvalues of GDP lie within natural variability of red noise Multivariate time series have significant oscillatory pair that exceeds red noise M-SSA helps to reveal ghost limit cycle Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 26/30

46 Vulnerability Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 27/30

47 Conclusions Advantages SSA and M-SSA help to distinguish between regular deterministic behavior cycles and irregular behavior noise Data-adaptive method with no a-priori basis functions Reconstruct skeleton of attractor M-SSA helps to reveal ghost limit cycles with higher confidence than univariate analysis Stability analysis Attraction of limit cycles Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 28/30

48 Conclusions Advantages SSA and M-SSA help to distinguish between regular deterministic behavior cycles and irregular behavior noise Data-adaptive method with no a-priori basis functions Reconstruct skeleton of attractor M-SSA helps to reveal ghost limit cycles with higher confidence than univariate analysis Stability analysis Attraction of limit cycles Verification of results Check robustness of pairs by changing embedding dimension M Apply statistical significance tests, Monte Carlo (M)SSA Do not consider (M)SSA as stand-alone method Apply additional methods (Fourier, Wavelet) and their tests Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 28/30

49 Future work Check other alternative hypothesis Other sources of oscillation-like behavior Other sources of period dependent volatility Model free surrogate data Constrained randomization Assimilate extracted cycles into model Endogenous business cycle model under study shows similar vulnerability behavior with respect to exogenous shocks (e.g., Schreiber 1998) (Hallegatte et al. 2005) But Model and data are still far away! Model is too simple to describe full complexity M-SSA can help to reduce this gap Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 29/30

50 References Review Ghil et al. (2002): Advanced spectral methods for climatic time series, Reviews of Geophysics, 40(1), SSA Broomhead & King (1986a): Extracting qualitative dynamics from experimental data, Physica D, 20(2-3), M-SSA Broomhead & King (1986b): On the qualitative analysis of experimental dynamical systems, in Nonlinear Phenomena and Chaos, ed. by S. Sarkar, pp Adam Hilger, Bristol, England. MC-SSA Allen & Smith (1996): Monte Carlo SSA: Detecting irregular oscillations in the Presence of Colored Noise. Journal of Climate, 9, Ghil & Vautard (1991): Interdecadal oscillations and the warming trend in global temperature time series, Nature, 350(6316), Plaut & Vautard (1994): Spells of Low-Frequency Oscillations and Weather Regimes in the Northern Hemisphere, Journal of the Atmospheric Sciences, 51(2), Vautard & Ghil (1989): Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series, Physica D, 35(3), Vulnerability of economic systems Andreas Groth, LMD, ENS, Paris 30/30

Impacts of natural disasters on a dynamic economy

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