arXiv subjects

D. Sornette

Publications and source records attributed to D. Sornette.

At least 163 records · Page 9Linked to original sources

arXiv2003

On the Occurrence of Finite-Time-Singularities in Epidemic Models of Rupture, Earthquakes and Starquakes

We present a new kind of critical stochastic finite-time-singularity, relying on the interplay between long-memory and extreme fluctuations. We illustrate it on the well-established epidemic-type aftershock (ETAS) model for aftershocks, based solely on the most solidly documented stylized facts of seismicity (clustering in space and in time and power law Gutenberg-Richter distribution of earthquake energies). This theory accounts for the main observations (power law acceleration and discrete scale invariant structure) of critical rupture of heterogeneous materials, of the largest sequence of starquakes ever attributed to a neutron star as well as of earthquake sequences.

cond-mat.stat-mech↗

arXiv2003

On Statistical Methods of Parameter Estimation for Deterministically Chaotic Time-Series

We discuss the possibility of applying some standard statistical methods (the least square method, the maximum likelihood method, the method of statistical moments for estimation of parameters) to deterministically chaotic low-dimensional dynamic system (the logistic map) containing an observational noise. A ``pure'' Maximum Likelihood (ML) method is suggested to estimate the structural parameter of the logistic map along with the initial value $x_1$ considered as an additional unknown parameter. Comparisons with previously proposed techniques on simulated numerical examples give favorable results (at least, for the investigated combinations of sample size $N$ and noise level). Besides, unlike some suggested techniques, our method does not require the a priori knowledge of the noise variance. We also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed Bayesian approaches. We note the trade-off between the need of using a large number of data points in the ML analysis to decrease the bias (to guarantee consistency of the estimation) and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition. The method of statistical moments for the estimation of the parameter of the logistic map is discussed. This method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically (not only numerically).

physics.data-an↗

arXiv2003

Evidence of a Worldwide Stock Market Log-Periodic Anti-Bubble Since Mid-2000

Following our previous investigation of the USA Standard and Poor index anti-bubble that started in August 2000, we analyze thirty eight world stock market indices and identify 21 anti-bubble. An ``anti-bubble'' is defined as a self-fulfilling decreasing price created by positive price-to-price feedbacks feeding overall pessimism and negative market sentiment further strengthened by inter-personal interactions. We mathematically characterize anti-bubbles by a power law decrease of the price (or of the logarithm of the price) as a function of time and by decelerating/expanding log-periodic oscillations. The majority of European and Western stock market indices as well as other stock indices exhibit practically the same log-periodic power law anti-bubble structure as found for the USA S&P500 index. These anti-bubbles are found to start approximately at the same time, August 2000, in all these markets. This shows a remarkable degree of synchronization worldwide. The descent of the worldwide stock markets since 2000 is thus an international event, suggesting the strengthening of globalization.

cond-mat.stat-mech↗

arXiv2003

Renormalization Group Analysis of the 2000-2002 anti-bubble in the US S&P 500 index: Explanation of the hierarchy of 5 crashes and Prediction

We propose a straightforward extension of our previously proposed log-periodic power law model of the ``anti-bubble'' regime of the USA market since the summer of 2000, in terms of the renormalization group framework to model critical points. Using a previous work by Gluzman and Sornette (2002) on the classification of the class of Weierstrass-like functions, we show that the five crashes that occurred since August 2000 can be accurately modelled by this approach, in a fully consistent way with no additional parameters. Our theory suggests an overall consistent organization of the investors forming a collective network which interact to form the pessimistic bearish ``anti-bubble'' regime with intermittent acceleration of the positive feedbacks of pessimistic sentiment leading to these crashes. We develop retrospective predictions, that confirm the existence of significant arbitrage opportunities for a trader using our model. Finally, we offer a prediction for the unknown future of the US S&P500 index extending over 2003 and 2004, that refines the previous prediction of Sornette and Zhou (2002).

physics.soc-ph↗

arXiv2003

2000-2003 Real Estate Bubble in the UK but not in the USA

In the aftermath of the burst of the ``new economy'' bubble in 2000, the Federal Reserve aggressively reduced short-term rates yields in less than two years from 6.5% to 1.25% in an attempt to coax forth a stronger recovery of the US economy. But, there is growing apprehension that this is creating a new bubble in real estate, as strong housing demand is fuelled by historically low mortgage rates. Are we going from Charybdis to Scylla? This question is all the more excruciating at a time when many other indicators suggest a significant deflationary risk. Using economic data, Federal Reserve Chairman A. Greenspan and Governor D.L. Kohn dismissed recently this possibility. Using the theory of critical phenomena resulting from positive feedbacks in markets, we confirm this view point for the US but find that mayhem may be in store for the UK: we unearth the unmistakable signatures (log-periodicity and power law super-exponential acceleration) of a strong unsustainable bubble there, which could burst before the end of the year 2003.

physics.soc-ph↗

arXiv2003

Evidence of Fueling of the 2000 New Economy Bubble by Foreign Capital Inflow: Implications for the Future of the US Economy and its Stock Market

Previous analyses of a large ensemble of stock markets have demonstrated that a log-periodic power law (LPPL) behavior of the prices constitutes a qualifying signature of speculative bubbles that often land with a crash. We detect such a LPPL signature in the foreign capital inflow during the bubble on the US markets culminating in March 2000. We detect a weak synchronization and lag with the NASDAQ 100 LPPL pattern. We propose to rationalize these observations by the existence of positive feedback loops between market-appreciation / increased-spending / increased-deficit-of-balance-of-payment / larger-foreign-surplus / increased-foreign-capital-inflows and so on. Our analysis suggests that foreign capital inflow have been following rather than causing the bubble. We then combine a macroeconomic analysis of feedback processes occurring between the economy and the stock market with a technical analysis of more than two hundred years of the DJIA to investigate possible scenarios for the future, three years after the end of the bubble and deep into a bearish regime. We also detect a LPPL accelerating bubble on the EURO against the US dollar and the Japanese Yen. In sum, our analyses is in line with our previous work on the LPPL ``anti-bubble'' representing the bearish market that started in 2000.

cond-mat.stat-mech↗

arXiv2003

Slider-Block Friction Model for Landslides: Application to Vaiont and La Clapiere Landslides

Accelerating displacements preceding some catastrophic landslides have been found empirically to follow a time-to-failure power law, corresponding to a finite-time singularity of the velocity $v \sim 1/(t_c-t)$ [{\it Voight}, 1988]. Here, we provide a physical basis for this phenomenological law based on a slider-block model using a state and velocity dependent friction law established in the laboratory and used to model earthquake friction. This physical model accounts for and generalizes Voight's observation: depending on the ratio $B/A$ of two parameters of the rate and state friction law and on the initial frictional state of the sliding surfaces characterized by a reduced parameter $x_i$, four possible regimes are found. Two regimes can account for an acceleration of the displacement. We use the slider-block friction model to analyze quantitatively the displacement and velocity data preceding two landslides, Vaiont and La Clapière. The Vaiont landslide was the catastrophic culmination of an accelerated slope velocity. La Clapière landslide was characterized by a peak of slope acceleration that followed decades of ongoing accelerating displacements, succeeded by a restabilizing phase. Our inversion of the slider-block model on these data sets shows good fits and suggest to classify the Vaiont (respectively La Clapière) landslide as belonging to the velocity weakening unstable (respectively strengthening stable) sliding regime.

cond-mat.stat-mech↗

arXiv2003

Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?

A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent close to 3. We revisit this results and use standard tests as well as develop a battery of new non-parametric and parametric tests (in particular with stretched exponential (SE) distributions) to characterize the distributions of empirical returns of financial time series, with application to the 100 years of daily return of the Dow Jones Industrial Average and over 1 years of 5-minutes returns of the Nasdaq Composite index. Based on the discovery that the SE distribution tends to the Pareto distribution in a certain limit such that the Pareto (or power law) distribution can be approximated with any desired accuracy on an arbitrary interval by a suitable adjustment of the parameters of the SE distribution, we demonstrate that Wilks' test of nested hypothesis still works for the non-exactly nested comparison between the SE and Pareto distributions. The SE distribution is found significantly better over the whole quantile range but becomes unnecessary beyond the 95% quantiles compared with the Pareto law. Similar conclusions hold for the log-Weibull model with respect to the Pareto distribution. Our main result is that the tails ultimately decay slower than any stretched exponential distribution but probably faster than power laws with reasonable exponents. Implications of our results on the ``moment condition failure'' and for risk estimation and management are presented.

physics.soc-ph↗

arXiv2003

Towards Landslide Predictions: Two Case Studies

In a previous work [Helmstetter, 2003], we have proposed a simple physical model to explain the accelerating displacements preceding some catastrophic landslides, based on a slider-block model with a state and velocity dependent friction law. This model predicts two regimes of sliding, stable and unstable leading to a critical finite-time singularity. This model was calibrated quantitatively to the displacement and velocity data preceding two landslides, Vaiont (Italian Alps) and La Clapière (French Alps), showing that the former (resp. later) landslide is in the unstable (resp. stable) sliding regime. Here, we test the predictive skills of the state-and-velocity-dependent model on these two landslides, using a variety of techniques. For the Vaiont landslide, our model provides good predictions of the critical time of failure up to 20 days before the collapse. Tests are also presented on the predictability of the time of the change of regime for la Clapière landslide.

physics.geo-ph↗

arXiv2003

The US 2000-2003 Market Descent: Clarifications

In a recent comment (Johansen A 2003 An alternative view, Quant. Finance 3: C6-C7, cond-mat/0302141), Anders Johansen has criticized our methodology and has questioned several of our results published in [Sornette D and Zhou W-X 2002 The US 2000-2002 market descent: how much longer and deeper? Quant. Finance 2: 468-81, cond-mat/0209065] and in our two consequent preprints [cond-mat/0212010, physics/0301023]. In the present reply, we clarify the issues on (i) the analogy between rupture and crash, (ii) the Landau expansion, ``double cosine'' and Weierstrass-type solutions, (iii) the symmetry between bubbles and anti-bubbles and universality, (iv) the condition of criticality, (v) the meaning of ``bullish anti-bubbles'', (vi) the absolute value of t_c-t, (vii) the fractal log-periodic power law patterns, (viii) the similarity between the Nikkei index in 1990-2000 and the S&P500 in 2000-2002 and (ix) the present status of our prediction.

cond-mat.stat-mech↗

arXiv2003

Summation of Power Series by Self-Similar Factor Approximants

A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the self-similar renormalization to the latter rather to the former. This results in self-similar factor approximants extrapolating the sought functions from the region of asymptotically small variables to their whole domains. The method of constructing crossover formulas, interpolating between small and large values of variables is also analysed. The techniques are illustrated on different series which are typical of problems in statistical mechanics, condensed-matter physics, and, generally, in many-body theory.

cond-mat.stat-mech↗

arXiv2003

Critical Market Crashes

This review is a partial synthesis of the book ``Why stock market crash'' (Princeton University Press, January 2003), which presents a general theory of financial crashes and of stock market instabilities that his co-workers and the author have developed over the past seven years. The study of the frequency distribution of drawdowns, or runs of successive losses shows that large financial crashes are ``outliers'': they form a class of their own as can be seen from their statistical signatures. If large financial crashes are ``outliers'', they are special and thus require a special explanation, a specific model, a theory of their own. In addition, their special properties may perhaps be used for their prediction. The main mechanisms leading to positive feedbacks, i.e., self-reinforcement, such as imitative behavior and herding between investors are reviewed with many references provided to the relevant literature outside the confine of Physics. Positive feedbacks provide the fuel for the development of speculative bubbles, preparing the instability for a major crash. We demonstrate several detailed mathematical models of speculative bubbles and crashes. The most important message is the discovery of robust and universal signatures of the approach to crashes. These precursory patterns have been documented for essentially all crashes on developed as well as emergent stock markets, on currency markets, on company stocks, and so on. The concept of an ``anti-bubble'' is also summarized, with two forward predictions on the Japanese stock market starting in 1999 and on the USA stock market still running. We conclude by presenting our view of the organization of financial markets.

cond-mat.stat-mech↗

arXiv2003

Finite-Time Singularity Signature of Hyperinflation

We present a novel analysis extending the recent work of Mizuno et al. [2002] on the hyperinflations of Germany (1920/1/1-1923/11/1), Hungary (1945/4/30-1946/7/15), Brazil (1969-1994), Israel (1969-1985), Nicaragua (1969-1991), Peru (1969-1990) and Bolivia (1969-1985). On the basis of a generalization of Cagan's model of inflation based on the mechanism of ``inflationary expectation'' or positive feedbacks between realized growth rate and people's expected growth rate, we find that hyperinflations can be characterized by a power law singularity culminating at a critical time $t_c$. Mizuno et al.'s double-exponential function can be seen as a discrete time-step approximation of our more general nonlinear ODE formulation of the price dynamics which exhibits a finite-time singular behavior. This extension of Cagan's model, which makes natural the appearance of a critical time $t_c$, has the advantage of providing a well-defined end of the clearly unsustainable hyperinflation regime. We find an excellent and reliable agreement between theory and data for Germany, Hungary, Peru and Bolivia. For Brazil, Israel and Nicaragua, the super-exponential growth seems to be already contaminated significantly by the existence of a cross-over to a stationary regime.

physics.soc-ph↗

arXiv2003

VaR-Efficient Portfolios for a Class of Super- and Sub-Exponentially Decaying Assets Return Distributions

Using a family of modified Weibull distributions, encompassing both sub-exponentials and super-exponentials, to parameterize the marginal distributions of asset returns and their multivariate generalizations with Gaussian copulas, we offer exact formulas for the tails of the distribution $P(S)$ of returns $S$ of a portfolio of arbitrary composition of these assets. We find that the tail of $P(S)$ is also asymptotically a modified Weibull distribution with a characteristic scale $χ$ function of the asset weights with different functional forms depending on the super- or sub-exponential behavior of the marginals and on the strength of the dependence between the assets. We then treat in details the problem of risk minimization using the Value-at-Risk and Expected-Shortfall which are shown to be (asymptotically) equivalent in this framework.

physics.soc-ph↗

arXiv2002

Self-similar Approximants of the Permeability in Heterogeneous Porous Media from Moment Equation Expansions

We use a mathematical technique, the self-similar functional renormalization, to construct formulas for the average conductivity that apply for large heterogeneity, based on perturbative expansions in powers of a small parameter, usually the log-variance $σ_Y^2$ of the local conductivity. Using perturbation expansions up to third order and fourth order in $σ_Y^2$ obtained from the moment equation approach, we construct the general functional dependence of the transport variables in the regime where $σ_Y^2$ is of order 1 and larger than 1. Comparison with available numerical simulations give encouraging results and show that the proposed method provides significant improvements over available expansions.

cond-mat.stat-mech↗

arXiv2002

Endogenous versus Exogenous Crashes in Financial Markets

We perform an extended analysis of the distribution of drawdowns in the two leading exchange markets (US dollar against the Deutsmark and against the Yen), in the major world stock markets, in the U.S. and Japanese bond market and in the gold market, by introducing the concept of ``coarse-grained drawdowns,'' which allows for a certain degree of fuzziness in the definition of cumulative losses and improves on the statistics of our previous results on the existence of ``outliers'' or ``kings.'' Then, for each identified outlier, we check whether log-periodic power law signatures (LPPS) are present and take the existence of LPPS as the qualifying signature for an endogenous crash: this is because a drawdown outlier is seen as the end of a speculative unsustainable accelerating bubble generated endogenously. In the absence of LPPS, we are able to identify what seems to have been the relevant historical event, i.e., a new piece of information of such magnitude and impact that it is seems reasonable to attribute the crash to it, in agreement with the standard view of the efficient market hypothesis. Such drawdown outliers are classified as having an exogenous origin. Globally over all the markets analyzed, we identify 49 outliers, of which 25 are classified as endogenous, 22 as exogeneous and 2 as associated with the Japanese anti-bubble. Restricting to the world market indices, we find 31 outliers, of which 19 are endogenous, 10 are exogenous and 2 are associated with the Japanese anti-bubble. The combination of the two proposed detection techniques, one for drawdown outliers and the second for LPPS, provides a novel and systematic taxonomy of crashes further subtantiating the importance of LPPS.

cond-mat.stat-mech↗

arXiv2002

Predictability of catastrophic events; a new approach for Structural Health Monitoring

We present a non-traditional general methodology for the prediction of breakdown of engineering structures, based on a global top-down monitoring of the health of the structure modelled by the theory of critical rupture. The new Acceptance Test Plan integrating the resulting Predictive Acoustic Emission procedure has been successfully applied to the prediction of the breakdown of composite over-wrapped high pressure vessels which are currently produced for the Ariane 5 launcher by EADS Launch Vehicles. This new approach should be an efficient tool more generally in Structural Health Monitoring service, especially for the prediction of the residual margins and lifetimes of heterogeneous structures which constitutes the main issue of SHM in civil infrastructures.

cond-mat.stat-mech↗

arXiv2002

Collective Origin of the Coexistence of Apparent RMT Noise and Factors in Large Sample Correlation Matrices

Through simple analytical calculations and numerical simulations, we demonstrate the generic existence of a self-organized macroscopic state in any large multivariate system possessing non-vanishing average correlations between a finite fraction of all pairs of elements. The coexistence of an eigenvalue spectrum predicted by random matrix theory (RMT) and a few very large eigenvalues in large empirical correlation matrices is shown to result from a bottom-up collective effect of the underlying time series rather than a top-down impact of factors. Our results, in excellent agreement with previous results obtained on large financial correlation matrices, show that there is relevant information also in the bulk of the eigenvalue spectrum and rationalize the presence of market factors previously introduced in an ad hoc manner.

cond-mat.stat-mech↗