Delay Embedding as Regime Signal
Infantino and Itzhaki, in their 2010 thesis Developing High-Frequency Equities Trading Models, utilize a regime switching signal based upon time delay embedding. The intuition underlying this signal and use for regime discovery are unexpectedly interesting.
Conceptually, their signal is framed within the context of a two-state switching regime (interpreted in the classic technical sense): “momentum” or “mean reversion”. With high frequency equities portfolio data, they informally observe the familiar volatility-regime correlation: high volatility implies momentum (e.g. herd effects), low volatility implies mean reversion (e.g. market making).
In their words: “As the short-term changes in appeared to be more pronounced — identified by very narrow peaks in the time series — cumulative returns from the basic mean-reverting strategy seemed to decrease” (p. 44). Note is a measure of cross-sectional volatility on dimensionally reduced returns (i.e. standard deviation of returns projected on dominant PCA eigenvectors). This relationship is illustrated in the right graphic.
How they translate this intuitive volatility-regime correlation into a switching signal is the fun part. They define the difference of as , then define the following distance metric (illustrated in left image):
This is an interesting starting point, as dynamical systems reminds us that is a phase space reconstruction for , given is the delayed chain of discrete i.i.d. steps walking backwards in time for . In other words, the following are vectors reconstructing the volatility from which mutual distance is being measured for each observed time :
From which they define a binary regime signal as the positive first difference of :
From which the regime switch is defined: indicates volatility is increasing and thus a “momentum” regime is appropriate. On the contrary, indicates volatility is decreasing and thus a “mean-reverting” regime is appropriate.
This signal is quite interesting when considered within the larger context of several familiar time series analysis traditions:
- Time delay embedding: is a delay embedding of , and thus benefits from classic theorems of Takens, Mañé, and Sauer et al.
- Frequency analysis: delay embedding hints at potential applicability of frequency techniques from signal processing, such as singular spectrum analysis
- Distance metrics: is indeed the familiar Euclidean distance metric, and thus begs consideration of non-Euclidean metric spaces and reframing the notion of temporal distance such as via dynamic time warping
- Markov chains: interesting questions arise when considering the structure of , such as whether it is Markovian and thus may benefit from corresponding Markov chain / HMM machinery
Undoubtedly not by accident, the authors conveniently omit their choice of embedding dimension . Such is presumably left as an exercise for the reader, as selecting optimal embedding dimension is indeed well-known to be one of the most significant challenges in reconstruction.