Signal Analysis Applied to Investing
Every investor wants an edge when to maximize profits. This analysis will give you insight to some of the advanced mathematical tools.
Fourier Series decomposes a signal into a series of sine and cosine functions. Typically the signal under study is periodic. Each sine and cosine frequency is a whole number multiple of the first frequency (harmonic), and may have a phase shift as well. Each frequency has a different amplitude as well; however, the amplitude is constant for a given frequency. When analyzing the frequency content of periodis signals with sharp edges such a sawtooth or square wave, Fourier series will produce a huge error. This is known as Gibbs phenomena. The market is not periodic because F(t) != F(t+nT) where n is any integer and T is the period. Even if a finite period of time were taken and turned into a periodic signal, the predictive value of Fourier series is worthless due to Gibbs phenomena; therefore Fourier series is a poor choice of analysis.
Kalman filtering is even more advanced and obscure than Fourier Series. It is also called an adaptive filter. The filter estimates the natural response of the system to be controlled and then adjusts the control algorithm. For adaptive filters to work properly, the natural response of a system must remain nearly constant over time. The Federal Reserve commissioned a study to use adaptive filters for predicting the economy. What the Fed concluded was the predictive ability was fairly good over the course of 3 to 6 months, but degraded with length of forecast time. What the "Consultant" or "Exepert" never mentioned in the report was the implications of this negative outcome. THE NATURAL RESPONSE OF THE MARKETS CHANGES WITH TIME!!! If I were commissioning another study of the economy using Kalman Filters, I would focus on how the natural response changes with time rather than the observable states that make up the economy.
Bandpass filters are the same of the level of complexity as Fourier Series. Like Fourier Series, bandpass filters decompose a signal into its different frequencies by letting only a few frequencies through and suppressing those outside the band. This is like tuning a radio, you are actually changing the center of the bandpass filter. When you come to a radio station, you can hear music. To generate the spectrum of the signal, the signal is passed through the bandpass filter thousands of times. Each pass through the center of the band pass is adjusted. All those outputs can be added together to indicate where the market is going. For signals that are periodic, the amplitude of a frequency will be constant. When I sent through 100 days of closing market prices, I noticed the amplitude varied for a frequency. Sometimes the amplitude grew over a several cycles, sometimes it tapered. This is an envelop of a lower frequency. I also noticed cycles would lengthen or contract a little. I'm still in preliminary stages of a developing the bandpass filters and software to "listening to the market" and perform the frequency analysis. There are some limitations to this technique.
1.) The filter must be primed with at least 30 pieces of data before the outputs have predictive ability.
2.) The analog to digital conversion process does not involve a low pass filter, so high frequency signals are part of the digital signal. When sending the digital signal through a digital bandpass filter on the computer, these high frequencies can appear as lower frequencies. This is known as aliasing.
3.) The rate of sampling is critical to the frequency under study. The desired frequency must be sampled at least twice as
fast. This is the Shannon Sampling theorem. ( for a frequency of once every 30 days, data muct be collected at least once every 15 days.)
4.) Data is not collected consistently enough. Markets are closed at the end of the day, weekends, and holidays. How are emotional levels, the driving forces of markets, measured when markets are closed?
Given these limitations there is less risk using bandpass filters for frequencies with a period greater than 2 months. If minute by minute data is available as on the market floor, bandpass filtering should yield good results from 10:30 AM to 4 PM. The first 1/2 hour has uneven opening, and it would take a good 30 samples to prime the filter.
