Quantifying Risk and Anomalous Behaviour Within Monitoring and Investment Problems. Peter Grindrod and David Waters, QuantiSci, Chiltern House, 45 Station Road, Henley-on-Thames, UK
Whether monitoring financial or economic variables, analyzing the behaviour of industrial plants, communications networks, or account behaviour, there is a need to consider multiple time series and identify what component is deterministic behaviour and what could not be foreseen or predicted: the risk element.
The analysis of large, complex, open systems does not need to
rely on predefined models (requiring calibration): instead
multiple time series data can "speak for itself" by
searching for the characteristic patterns held within it. This is
known as an inverse problem. We view the observed behaviour as
output from a large unknown dynamical system, incorporating all
processes and "players", responding to various
systematic or random inputs. Although potentially vast, the
number of inherent degrees of freedom may be very limited once
constraints, standard practices, and large scale operational
stability are accounted for.
We discuss the application of state space embedding techniques in identifying the deterministic part (structure/pattern) of time series behaviour and removing the component within which there are no discernable patterns -- an objective, data-driven, definition of noise. Such methods are known to be sensitive enough to separate deterministic chaos from background noise with no a priori modelling assumptions. This leads to a natural definition of risk: the level of nonsystematic behaviour observed in time and across all observable series.
We discuss how this approach can be applied to financial commodities by allowing historical data to yield its own optimal set of behavioural templates. Incoming deviations representing a change in the system response can be recognised and used for critical decision support.
INTRODUCTION
Consider the observation of noisy time series data which has been generated by an unknown, potentially large, complex and nonlinear dynamical process: for example, a set of market prices or indicators, or some monitored outputs from a logistical network or industrial plant. The description of the system processes themselves may invoke a considerable amount of uncertainty, yet there is often a need to make decisions based upon estimates of transient system behaviour. Adopting a modelling methodology, postulating and verifying various process and subprocess components, integrating and calibrating them, is too uncertain. Here we propose to take an inverse approach. This is supported by two key ideas which have emerged over the last decade in the theory of dynamical systems.
First, it is known that even very large systems with many, possibly infinitely many, degrees of freedom, exhibit behaviour which can be explained in terms of the dynamics of a reduced set of variables. Such systems are driven by the response of an internal engine, sometimes called an inertial manifold, which may be hidden within conventional model state descriptions. The identification of such generic behaviour is at the heart of nonlinear systems theory. Second, there is a wide interest in reconstructing a picture of the underlying dynamics from observed times series. In theory at least [1], such state space reconstruction is possible provided that enough time series data is available, and that one embeds the data in a space with enough degrees of freedom. In practice this can often be achieved according to an algorithm given in [2], described in the next section.
STATE SPACE EMBEDDING AND PATTERN RECOGNITION
Given a single or multiple time series we may use the method of [2] in order to obtain a signature of the dynamics, by passing a moving window of length n along the time series. This is called state space embedding, and uses principal component analysis to describe the multiseries lag-covariance structure. This yields data-adaptive filters separating the time series into statistically independent components, or patterns [2]. To each pattern is assigned a constant, called a spectral value, which is a measure of its strength. The more often a pattern re-occurs within the windows the greater the spectral value. The set of spectral values, or spectrum, say, provides an ordering to the patterns. If truly random data with no preferential patterns were used the spectrum would be constant. In practice, for real data, the noisy part of the behaviour corresponds to the least relevant patterns, with low spectral values representing a background plateau. By projecting the dynamics onto the subspace spanned by the principal patterns one obtains an enhanced image. For example, in Figure 1 we show three reconstructions of delicate chaotic behaviour in the presence of various amounts of added random noise.
Figure 1: State space reconstruction of a chaotic attractor from time series in the presence of increasing added noise.
MODEL INDEPENDENT RISK ESTIMATES
Risk is a measure of the nonpredictability of behaviour. Rather than fitting a model (making assumptions about the distribution of noise) and measuring the variance around the predicted behaviour, we identify the deterministic component of the data, obtaining a smoother underlying series, and define the risk to be a measure of the spectral values which are discarded as showing no preferential structure. In Figure 2 we show the results from state space embedding 300 trading days data on three (normalised) option prices. Here n=25, and we depict the spectra as well as the raw and smoothed data. In each case from the behaviour of the spectrum (on a log scale), we project onto the most significant 9 patterns and use the spectral values for patterns 10 through 25 to define risk.
These ideas are important where risk is bought and sold, or compared between different series, since it is model independent: we allow the data to take-up an appropriate number of deterministic degrees of freedom. The important point is that this is achieved here without a priori assumptions about the model structure.
Figure 2: Raw and smoothed data for three normalised option prices (above), with risk estimates and spectra (below, on a log scale).
What about the patterns? These contain information about the dynamical structure prevalent in the various series. In Figure 3 we show the 9 key-patterns for the three options used in Figure 2. The patterns are shown as horizontal grey scale bars of window length 25, they become more complex, and less significant, moving upwards in the block. The first four patterns are the same in each case: then they start to get shuffled with respect to each other, indicating the different priorities exhibited. Thus not only is Option 3 less noisy, it has a structure recognisably different from the others.
Figure 3: Patterns recognised for the options in Figure 2. The patterns are shown as horizontal grey scale bars of window length 25, they become more complex, and less significant, moving upwards in the block. Highlighted patterns show exchange for different data.
MONITORING DYNAMICAL BEHAVIOUR
How do series change over time? By segmenting data we may
embed each segment separately and use the spectrum, 
for the ith
consecutive segment, as a signature of the prevalent dynamics in
each segment. We wish to know when and how things are changing,
and when and if the normal mode of operation has been restored
following some excursion.
For example in Figure 4 we analyze the DM/£ through the
period before and after the £ left the ERM. Clearly there is an
abrupt change of altitude, yet comparing the spectra obtained for
consecutive segments we also detect a shift in the dynamics: the
£/DM is much more risky, with a higher plateau for the spectrum
post ERM. By contrast in Figure 4 we also show a similar analysis
for the $/£. Although there is an altitude change as the £ left
the ERM, it is not accompanied by a change in . Much later there is a
significant change (mid 1995) which corresponds to a period where
the $ was supported by US intervention fighting the markets'
willingness to sell it downwards (we found this change for
ourselves!).
Figure 4: DM/£ and $/£ pre and post departure from ERM. Changes in the spectra,
IN CONCLUSION
We have shown how techniques suited to the statistical analysis of nonlinear systems may be of use in recognising the deterministic patterns in observed behaviour. All of the ideas here extend easily to multiple time series data. We have shown how a data driven definition of risk can be employed without relying upon model assumptions or sophistication. We have also indicated how to identify changes within underlying dynamics driving large unknown systems. The examples given here are illustrative, and the techniques could be applied more widely.
Monitoring (pattern recognition) and measurements of risk are key issues in many commercial and industrial fields where present models are inadequate or unavailable. Prediction of deterministic trends (for example, in trading) is also an important application.
Often decisions must be made against a background of noise and dynamical turbulence, by recognising and analyzing patterns within an open methodology. It is the aim to learn more about the processes at work, and support and rationalize existing expertise.
References
[1] Ruelle, D. Chaotic Evolution and Strange Attractors, The
statistical analysis of time series for deterministic nonlinear
systems, Lezioni Lincee, CUP, 1989.
[2] Broomhead, D.S. and King, G., Extracting qualitative
dynamics from experimental data, Physica D, 20, 217-326,
1986.