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Delay Embedding as Regime Signal

February 24, 2011

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 \sigma_D appeared to be more pronounced — identified by very narrow peaks in the \sigma_D time series — cumulative returns from the basic mean-reverting strategy seemed to decrease” (p. 44). Note \sigma_D 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 \sigma_D as \psi , then define the following distance metric (illustrated in left image):

     E_H (t) = \sqrt{\sum\limits_{i=1}^H{[\psi(t - i)]^2}} = \sqrt{\sum\limits_{i=1}^H{[\sigma_D(t - i) - \sigma_D (t - i - 1)]^2}}

This is an interesting starting point, as dynamical systems reminds us that \psi(t - i) is a phase space reconstruction for \sigma_D , given \psi is the delayed chain of discrete i.i.d. steps walking backwards in time for \sigma_D . In other words, the following are vectors reconstructing the volatility from which mutual distance is being measured for each observed time t :

[ \psi(t - 1), \psi(t - 2), \cdots , \psi(t - H) ]

From which they define a binary regime signal \omega as the positive first difference of E_H(t) :

    \omega = [ E_H(t) - E_H(t - 1) ] > 0

From which the regime switch is defined: \omega > 0 indicates volatility is increasing and thus a “momentum” regime is appropriate. On the contrary, \omega \le 0 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:

Undoubtedly not by accident, the authors conveniently omit their choice of embedding dimension H . 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.

11 Comments leave one →
  1. Aleksey permalink
    March 1, 2011 3:42 pm

    I really find this approach interesting, and I decided to experiment with it a bit. If I achive anything interesting, I will share with you via email. Thanks for the nice blog!

    • March 1, 2011 9:55 pm

      @Aleksey: glad you found it useful; I look forward to hearing if you have similarly positive results. Also, I am drafting a follow-up post describing the cross-sectional volatility metric in more detail, including drilling into the principal component space from which it originates.

  2. Scott Locklin permalink
    April 12, 2011 7:25 pm

    I’ve not looked at SSD, though its use by climatologists make it suspect to me (HH-transforms are on my short list of things to look at though). On the other hand, I have been using this “trick” for a while now, with some success. Carol Alexander wrote about it in one of her books, but I kind of thought of it because I wrote a dissertation which talked a bit about the trace formula, and I thought it was an obvious thing to try.

    • quantivity permalink*
      April 12, 2011 10:25 pm

      @Scott: Curious to hear what problem domains you have found success with embedding, as it’s a rather general technique.

      Re SSD: am I correct in presuming from context that you mean SSA (as first step of SSA is embedding, and climatology research is a frequent user), rather than SSD? I have found the necessity of a priori parameter specification to be an annoyance with applying SSA to trading. While specifying window length is modestly annoying (but not conceptually different than selecting trading clock and time-series sample length), having to evaluate separability for use in X_i groupings gives it an unpleasant whiff of data snooping.

  3. Maxim permalink
    June 29, 2012 12:51 pm

    Thank you for initiating this discussion. I have one question. It seems to me that your definition of E_H(t) is different from the paper’s. Or, to be precise, your interpretation of Psi(t-i). You define Psi(t-i)=sigma_D(t) – sigma_D(t-i), when the paper defines it simply as Psi=d sigma_D / dt, which I believe implies Psi(t-i)=sigma_D(t-i) – sigma_D(t-i-1).
    Am I missing something here?

    • quantivity permalink*
      June 30, 2012 9:25 am

      @Maxim: correct, thanks for comment. Post updated accordingly.

      • Maxim permalink
        July 2, 2012 7:24 am

        @quantivity: thank you. Do you have any idea for the reasonable range for H?

      • quantivity permalink*
        July 2, 2012 9:21 pm

        Having not optimized on live data, my speculation is optimal H varies dynamically by tick velocity.

  4. October 8, 2015 7:28 pm

    The authors state in the paper that the value of H is same as the accumulation parameter they chose before for the regression.


  1. Cross Sectional Volatility « Quantivity
  2. Inflexible regime, inflexible prices | Portfolio Probe | Generate random portfolios. Fund management software by Burns Statistics

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