Coping with Nonstationarity by Overembedding

Hegger, Rainer; Kantz, Holger; Matassini, Lorenzo; Schreiber, Thomas

doi:10.1103/physrevlett.84.4092

Cited by 105 publications

(92 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Ergodicity (or quasi-ergodicity) is the prerequisite needed to introduce statistical ensembles. Systems out of equilibrium or, more generally, nonstationary systems still pose fundamental challenges [1][2][3][4]. Complex systems -the term "complex" is used in a broad sense -show a wealth of different aspects which can be traced back to non-stationarity [5,6].…”

mentioning

confidence: 99%

Non-stationarity in financial time series: Generic features and tail behavior

Schmitt

Chetalova

Schäfer

et al. 2013

EPL

101

View full text Add to dashboard Cite

Financial markets are prominent examples for highly non-stationary systems. Sample averaged observables such as variances and correlation coefficients strongly depend on the time window in which they are evaluated. This implies severe limitations for approaches in the spirit of standard equilibrium statistical mechanics and thermodynamics. Nevertheless, we show that there are similar generic features which we uncover in the empirical return distributions for whole markets. We explain our findings by setting up a random matrix model. The great success of statistical mechanics and thermodynamics is borne out by their ability to characterize, in the equilibrium, large systems with many degrees of freedom in terms of a few state variables, for example temperature and pressure. Ergodicity (or quasi-ergodicity) is the prerequisite needed to introduce statistical ensembles. Systems out of equilibrium or, more generally, nonstationary systems still pose fundamental challenges [1][2][3][4]. Complex systems -the term "complex" is used in a broad sense -show a wealth of different aspects which can be traced back to non-stationarity [5,6]. Financial markets are presently in the focus, because they demonstrated their non-stationarity in a rather drastic way during the recent years. To assess a financial market as a whole, the correlations between the prices of the individual stocks are of crucial importance [7][8][9][10]. They fluctuate considerably in time, e.g., because the market expectations of the traders change, the business relations between the companies change, particularly in a state of crisis, and so on. The motion of the stock prices is in this respect reminiscent of that of particles in many-body systems such as heavier atomic nuclei. Depending on the excitation energy, the motion of the individual particles can be incoherent, i.e., uncorrelated in the above terminology, or coherent (collective), i.e., correlated, or even somewhere in-between [11][12][13]. This non-stationarity on the energy scale leads to very different spectral properties, [12,13]. Such an analogy can be helpful, but we do not want to overstretch it.Here, we want to show that the non-stationarity, namely the fluctuation of the correlations, induces generic features in financial time series. These become visible when looking at quantities which measure the stock price changes for the entire market. We have four goals. First, we carry out a detailed data analysis revealing the generic features. Second, we set up a random matrix model to explain them. Third, we demonstrate that the non-stationarity of the correlations leads to heavy tails. Fourth, we argue that our approach maps a noninvariant situation to an effectively invariant one. For an economic audience we discuss the consequences for portfolio management elsewhere [14].Consider K companies with stock prices S k (t), k = 1, . . . , K as functions of time t. The relative price changes over a fixed time interval ∆t, i.e., the returnsare well-known to have distributions with heavy tails, the ...

show abstract

mentioning

confidence: 99%

Non-stationarity in financial time series: Generic features and tail behavior

Schmitt

Chetalova

Schäfer

et al. 2013

EPL

101

View full text Add to dashboard Cite

show abstract

“…Hegger et al [10] presented an over-embedding technique, using increased embedding dimensions to treat timevarying parameters as state parameters which removed the non-stationarity on its physical cause. Wang et al [11] used other higher-dimensional embedding such as 'the support vector machine' to predict some non-stationary processes and obtained better prediction proficiency.…”

mentioning

confidence: 99%

A novel approach in predicting non-stationary time series by combining external forces

et al. 2011

View full text Add to dashboard Cite

In this paper, we investigate a novel technique that reconstructs the observed time series and incorporates driving forces. Furthermore, to illustrate and test the technique, we consider a couple of predictive experiments using ideal time series provided by the logistic and Lorenz systems with specific driving forces. The preliminary results show this approach can improve prediction proficiency to some extent, and the external forces play a similar role to that of state variables. Short-term prediction is one of the difficult issues in climate change studies. Because of the complexity of short-term climate processes and lack of general knowledge underpinning its mechanisms, proficiency in prediction remains at a stand-still [1,2]. As for climate change in the 21st century, global warming is the most important factor, with many studies devoted to the topic. The possible causes range from change in natural forces (such as solar activity and volcanic emission) to human activities (such as green-house gases, sulfate emission in aerosols, and land use). In addition, the internal variability of climate systems should not be ignored either. However, no matter the cause of climate change, global warming in fact gives notice that climate processes are non-stationary. So far many climate predictive theories formulated in statistics and in nonlinear science are based on the hypothesis that the process is stationary, thus ergodicity theory can be used under such conditions to reconstruct the dynamics from an observed time series and establish predictive equations constructed from it.This reconstruction is at variance with the basic behavior of climate process. As is well-known for any weather or climate system, it is impossible to state the initial conditions incontrovertible. In fact, except for global warming, recent work has addressed the proper characterization of non-stationarity using weather and climate data. For instance, Tsonis [3] analyzed low-frequency (decadal to multi-decadal) variation in global precipitation over the past century and found that fluctuations about the global mean have increased significantly, implying that the second-order moment of the precipitation has changed on those scales, or that global precipitation process is non-stationary in the past century. In another case, Trenberth [4] found that the observed winter Pacific mean sea-level pressure underwent an abrupt change towards the end of the 70s; such a change in regime highlights the decline in the stationary behavior of the system. They noted that the behavior of these quantities is non-stationary. Most real world time series have some degree of non-stationarity due to external perturbations of the observed system [5].However, there is as yet no general theory about nonstationary processes. What scientists can do for the moment

show abstract

“…All the other experimental conditions are as described above. Each experimental point is calculated over a time window of 4 s. We took care of the (slight) nonstationarity using the ''over-embedding" technique (Hegger et al, 2000). In each single trial (not shown here) a discontinuity in the D 2 behavior can be easily detected; the discontinuity is taken as a reference point (time t = 0 s) to build up the average shown in the graph.…”

Section: Resultsmentioning

confidence: 99%