(Revised June 27, 2006)

**Periodogram Spectrum -- AAPL****Wavelet Spectrum -- AAPL****Periodogram Spectrum -- MSFT****Wavelet Spectrum -- MSFT****Higher-Order Statistics**

According to the theory of **stationary
stochastic time series**, the properties of the time series are described by
the **mean** and the **autocovariance sequence**, which are constant in
time since the series is **stationary**. The time series in question
here is the series of **returns**, or daily (logarithmic) price
changes. In the classical **Random Walk** model of stock prices, the
returns series is just Gaussian white noise, which means that the **correlation**
between any two different days of returns is zero, and the **mean** and **variance**
are constant. However, this is known to be only a crude approximation for
financial time series. So the goal is to find small departures from
randomness, or in other words **correlation**, in the returns series, and to
exploit this** correlation** to construct profitable **Trading Rules**.

For
a **stationary** time series, the **autocovariance sequence** is uniquely
determined by the **power spectrum** and vice-versa. This is called the
**Wiener-Khinchin theorem** -- the two are the **Fourier transform** of
each other. From the **autocovariance sequence** or the **power
spectrum** it is then possible to compute the coefficients of a **Linear
Prediction** filter, to compute the best estimate of the **future returns**
of the price series, or in other words a **Price Projection**. If the **Random
Walk** model holds, then the **autocovariance sequence** is zero for time
lags greater than zero and is just the (constant) ** variance** for zero time
lag. (The constant **mean** of the returns is just the constant **trend**
of the **Random Walk**.) The corresponding **power spectrum** is a constant -- **white
noise**. If there is any departure from a constant **power spectrum**,
this indicates the presence of **correlation** in the time series, and hence
the possibility of making a partial prediction or estimate of the future
returns. (Note: The **correlation** of two quantities is the **covariance**
divided by the **standard deviation** of the two quantities.) For
financial time series, since we are only working with daily returns, it is
reasonable to assume that the data are **stationary** over a period of 1024
days -- about 4 years. Then we can utilize the theory of **stationary**
time series as a reasonable approximation to the actual **non-stationary**
financial time series.

For a representative **Periodogram** power spectrum, we can consider the
case of **AAPL** stock. Here is the **Periodogram** display for this
stock:

The top pane is the
standard **Periodogram**, as described in most textbooks on **Time Series
Analysis**. The horizontal axis represents frequency, corresponding to a
period of 1024 days on the left side to 2 days on the right. The middle of
the graph corresponds to a period of 4 days, half the maximum (Nyquist)
frequency. So the entire right half of the graph corresponds to periods of
4 trading days or less. This half of the graph is probably just stochastic
noise, and that is certainly the way it appears in this graph. Clicking on
the Random button, you can compare the graph to one generated from Gaussian
random numbers, and there is not much difference at first glance.
According to the standard theory, each frequency component of the **Periodogram**
is a random variable with a standard deviation of 100% -- completely
random. The only way to get a meaningful result, therefore, is by *smoothing*
the **Periodogram**. You can set the smoothing time period to a wide
range of values -- here it is set to 6 days. In the present case, however,
the spectrum appears random, for any smoothing, and this is verified by the **Kolmogorov-
Smirnov** test, which compares the cumulative spectrum to a straight line and
measures the maximum deviation. Since most of the spectrum is
high-frequency in nature, we must conclude that this high-frequency spectrum is
essentially *stochastic noise*.

However, notice the very low frequency
end of the power spectrum. This shows a distinctive spike at the lowest
frequencies, corresponding to a **long-term trend**. This is just the
kind of behavior that is expected from the power spectrum of a **fractionally
differenced stochastic process**, obeying **fractal statistics**.
People like Benoit Mandelbrot (the "inventor" of fractals) and Edgar
Peters have written about this, postulating that financial time series are
actually of this type. This would also agree with the idea of **trend
persistence**, which is an important principle of **Technical Analysis**.
The idea is that the **long-term trend** that existed in the past should
persist into the future, or that there is a positive **correlation** between
the **long-term trend** in the past and that in the future. In other
words, the low-frequency components of the spectrum should display a positive **correlation**,
and that in turn means that the spectrum should display a positive
"spike" at low frequencies (according to the standard theory of a **stationary
stochastic process**). So it is just this kind of positive spike at low
frequencies that we want to see if there is to be **trend persistence** and **correlation**
between low-frequency **technical indicators** based on past data, with **future
returns**.

From the **Periodogram** spectrum, it is possible to calculate
directly the coefficients of a **Linear Prediction** filter, and some of the
filters used by ** QuanTek** use this technique. On the other
hand, the

Another way to compute the power spectrum is to use the **Discrete Wavelet
Transform (DWT)** instead of the **Discrete Fourier Transform (DFT)** --
otherwise known as the **Fast Fourier Transform (FFT)**. The **Fourier
Transform** decomposes the signal using a basis of "infinite" sine
waves of a single precise frequency. The **Wavelet Transform**, on the
other hand, uses a basis consisting of localized (in time) "wave
packets" which are of finite extent in both the time and frequency
domains. Thus the **Wavelet** decomposition of the signal has
coefficients that depend on both the frequency (octave) as well as time, as
opposed to the **Fourier** decomposition whose coefficients depend only on
frequency.

For a representative **Wavelet** power spectrum, consider once
again the case of **AAPL **stock:

The
upper graph shows the **DWT** power spectrum (the square of the **DWT**
coefficients), which have been *averaged* within each octave over all time
values. Before this averaging is done, there are 512 **DWT**
coefficients, the same number as for the **DFT**, but after the time
averaging there are only 9 values of the power spectrum, one for each frequency *octave*.
These octaves can be seen above, in which the frequency range has been split
into regions which differ from the previous region by a factor of two (an *octave*).
The lower graph shows the same spectrum, which has been smoothed using Chebyshev
polynomials, for use in the **Wavelet Linear Prediction **filters.
(Note: The yellow band represents the one-standard-error bars for each frequency
octave. The band is wider for the lower frequency octaves because there
are fewer **DWT** components in these octaves.)

For a statistical test of
randomness, a **Chi-Square** test was adopted. In the above example,
the spectrum appears completely random according to this test. This may be
due to the fact that after the time average is taken, there are only 9
components of the power spectrum left. However, when the **Chi-Square**
test is performed on the full 512 time-dependent components of the **DWT**
spectrum, the test gives an amazing 22+ standard deviations away from a random
result. This may be due to correlation in the spectrum, which is averaged
out when the time average is taken, or perhaps due to the fact that the
distribution of the returns is *non-Gaussian*, or perhaps both. But
this result, which is consistent for all stocks, implies that perhaps an **adaptive
filter** based on **non-stationary** time-dependent statistics could be
developed, based on this **Wavelet** decomposition. We are still
working on this problem.

As in the case of the **Periodogram**, the
low-frequency spike is clearly visible in the **Wavelet spectrum**. In
fact, the main goal in adopting **Wavelets** was to display this
low-frequency behavior more clearly. This particular case would provide
evidence for **fractal statistics** with **long-memory**, or in other
words a **fractionally differenced** process with *positive* **fractional
difference **(or **fractal dimension**) **parameter**. Notice that
the measured fractal dimension in this case is *positive*, approximately
+0.08. The positive spike
in the lowest frequency octaves implies that there should be a *positive*
correlation between the **long-term trend** and *future* long-term **returns**.
This correlation, although small, can lead to significant gains if
properly utilized in an appropriate set of **Trading Rules**.

By way of contrast, let us now look at the **Periodogram** for **MSFT**
stock:

According to the **Kolmogorov-Smirnov**
test, the confidence level for a non-random spectrum is a respectable 99.71%,
much higher than for **AAPL** stock. Looking at the graphs, it appears
that the power spectrum is concentrated more in the second half of the frequency
range, corresponding to periods from about 5 days down to 2 days. So maybe
there is a certain amount of predictability in the short-term trading rules for
this stock. There is another concentration of spectral power about a
quarter of the way from the low-frequency end, which would correspond to about a
two week period. However, a major difference from the case of **AAPL**
is that, in this case, there is no sharp peak at the very low frequency
end. Instead we see a pronounced *dip* at this end. When the
spectral power starts with a low value at the low frequency end and then rises
with increasing frequency, this indicates *anti-correlation *between
past returns and future returns. So if the stock went up in the past, it
is more likely than not to go down in the future, and vice-versa. In other
words, what we have is a

This is seen more clearly by looking at the **Wavelet Spectrum** for this
stock:

From the **DWT**
spectrum, it is very clear that the spectral power at the lowest frequencies is
deficient, and rises with increasing frequency. This is just the kind of
behavior that is expected from a **fractionally differenced** process with *negative*
**fractional
difference **(or **fractal dimension**) **parameter**. And indeed,
that is the result of calculating this parameter, as shown. This kind of
spectrum seems to be typical at the present point in time, which seems to
indicate that the overall market is in a **return to the mean** mode or **trading
range**. Back in the late '90s the market was in a **trend mode**,
and in this case most stocks would have probably displayed a pronounced peak at
the low frequency end, and a *positive* **fractional
difference **(or **fractal dimension**) **parameter**. This is
probably the basis of the claims that have been made that this is the generic
condition for the market, but more probably the **fractional difference
parameter** changes with time, as the market moves from a **trending** to a
**trading** market, and back again.

As in the case of **AAPL**, the **Chi-Square**
test for the 9 time-averaged **Wavelet spectrum** octaves does not indicate a
significant departure from randomness, but the **Chi-Square** test for the
full set of 512 **Wavelet spectrum** components does show an amazing 97+
standard deviations away from randomness. Once again, this is either due
to correlation that is averaged over in the time average, or the fact that the
distribution of returns is non-Gaussian, or both. This once again seems to
indicate **non-stationary** statistics, so maybe this could be utilized to
advantage in an **Adaptive Wavelet Linear Prediction** filter.

The **Periodogram Spectrum** and **Wavelet Spectrum** measure the power
spectrum of the **auto-correlation** sequence of the (logarithmic) price
returns. These **correlations**, or more precisely the **covariance**
and **variance**, together with the **mean** of the returns, idealized as
a **stationary stochastic process**, constitute **second-order statistics**
of this stationary process. It is thought that at the second-order level,
the returns are essentially uncorrelated white noise, but we are finding some
significant **correlation** after *filtering* out the high-frequency
components, leaving only the low-frequency components. From the above
displays, it can be seen that the high-frequency components constitute most of
the power spectrum, which explains why the low-frequency correlation is so
easily masked by the high-frequency stochastic noise. Of course, there may
be (time-dependent) correlation in the high-frequency components as well, but
our idealization that the process is stationary over a period of 1024 days is
unable to find this high-frequency correlation. This is why an **Adaptive
Wavelet Linear Prediction** filter would be desirable.

However, there is
also the possibility of **higher-order statistics**. It is known
already that there is significant **autocorrelation** in the sequence of
daily **volatility** (absolute value of the returns) and in the squares of
the returns. These are **non-linear** functions of the returns, and to
make a projection of these quantities requires **non-linear** filtering
techniques. But these **higher-order auto-correlations** and **auto-covariances**
can be described by their power spectrum just as in the case of **second-order
auto-correlation**, and in the case of the **cross-covariance**, the
corresponding spectrum is called the **cross-spectrum**. So similar,
though more complex, filtering techniques can be used to project these
higher-order correlations as are used in the second-order case. In
general, several of these sequences would be projected at once using **multivariate**
prediction filters, rather than the simpler **univariate** prediction filters
used for the basic **Price Projection**. Once again, the absence of **correlation**
would be indicated by a flat **power spectrum** (**white noise**), while **correlation**
would show up as a non-flat **power spectrum**. The filter coefficients
could once again be derived from these power spectra, at least in the **stationary**
case.

Utilizing **higher-order statistics** would provide the connection
between **Signal Processing** and traditional **Technical Analysis**.
The whole objective of **Technical Analysis** is to find complicated
functions of past price data, which are supposed to be **correlated** with
future returns. Some of the indicators of traditional **Technical
Analysis** are highly complex patterns or functions of the past price history,
and highly **nonlinear** functions. So these complex technical
indicators would utilize **higher-order statistics **of a very high
order! The problem with this is that the higher the order, the more
cross-spectra there are to estimate, and there is only a limited amount of data
in most financial time series. So to actually measure statistically the
effectiveness of such complex patterns of such high order is virtually
impossible. So this is a good reason to focus attention on **second-order
statistics** and maybe **fourth-order statistics**, and construct simple
oscillator-type **technical indicators** based upon these.

As always, "**Past performance is no guarantee of future results**."

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Last modified 11/13/2006 .

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