In a previous post regarding a modified wavelet feature extraction model I talked about replacing the DWT/FFT part with a customized Hilbert Transform for the purpose of capturing cyclic behavior of the price data. Here I discuss this in more detail and introduce the first of 3 Hilbert Transformer variants I am excited about.

For those not too familiar with phasors please have a look at the following links:

Simply put a phasor is an alternative way of representing a time-domain signal using a complex set of measures namely InPhase and Quadature. Recall that Hilbert Transformed signal takes the form \small\text{a + ib} where \small\text{a} is the “real” or InPhase part and \small\text{b} is the “imaginary” or Quadrature part. Plotting \small\text{a} and \small\text{b} on the complex plane and tracing out the trajectory formed by the phasor is supposed to reveal, among other things, the cyclic components of the signal.

There are many variants of the Hilbert Transform, but they all have a common structure which was popularized by [Ehlers]. For future reference we shall refer to Ehlers’ model as Variant 1.

HilT_var1.png

 

Applying a chirp signal to the input:

ChrResHT2.png

 

cplxPhasVar1.pngDue to the phase differential between the InPhase and Quadrature signals it becomes possible to trace out the trajectory taken by the phasor on the complex plane. The figure to the left shows this for a signal with frequency 20bars/cycle. Now if we superimpose a shortShowCycVar1_2.png term cycle on a longer one we see a change in the trajectory due to the presence of the new cycle. The figure to the right shows this. Notice that the cycle only shows up on the phasor plot with a lag of 4 bars. We were expecting a shift after the 10th bar, but only appeared after the 14th bar.

 

Short term cycles show up as a distortion of circles with bigger diameter and it is possible to use this feature to visualize underlying behavior in price data or any financial time series for that matter. I simulated a 31 step Brownian motion with drift = 10, diffusion = 10 and evaluated it’s complex phasor. We can clearly see how short term cycles influence longer term cycles, which is not obvious by inspecting the time series alone.

 

brnMotPhasor.png

Inspecting the phasor plot we can see the presence of cycles as represented by circular features being formed by individual points. The following cycles exist:

  1. Cycle 1: On Phasor plot points 7, 8, 9 => 3 bar cycle. On time series plot this cycle pertains to points 3, 4, 5.
  2. Cycle 2: On Phasor plot points 12, 13, 14, 15, 16, 17 => 6 bar cycle. On time series plot this cycle pertains to points 8, 9, 10, 11, 12, 13.
  3. Cycle 3: On Phasor plot points 25, 26, 27, 28, 29, 30, 31 => incomplete cycle. On time series plot this cycle pertains to points 21, 22, 23, 24, 25, 26, 27.

We also make the following observations:

  • Cycles 1 and 2 could actually be noise. We can only confirm this if we define a minimum diameter for a circular feature to qualify as a cycle and ignore all those with diameter less than the minimum.
  • Cycle 1 & 2 distort cycle 3.
  • Cycle 3 is incomplete but is the dominant one. The only way can predict its frequency is to count the number of points in the 2nd quadrant (points 26, 27, 28, 29, 30) and multiply by 4. Given that there are 5 points we predict that Cycle 3 is roughly a 20 bar cycle.

 

Key Points:

  • The Complex Phasor plot provides an excellent way of visualizing cycles and how they interact with each other. A cycle is one complete circular feature whose frequency is equal to the number of complex price points forming that circle.
  • Due to the lag constraint of Hilbert Transformer model used in this test (Variant 1) it is only possible to view cycles with a lag of 4 bars. In future I shall discuss Variants 2 and 3 which attempt to detect cycles with less lag.
  • It is possible to develop a trading strategy that exploits the cycle information obtained via Hilbert Transform. For instance we could use the InPhase vector as an anchor to decide when to buy or sell depending on whether pre-defined lines/points on the complex plane have been crossed. I have high hopes for such a method, particularly in an algorithmic trading setting.
  • It may actually be usefull to filter the input data before evaluating its complex phasor, but this may distort any micro-structure that may exist in the time series. It all depends how much filtering is acceptable.

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