(Revised June 27, 2006)

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.

*return to ***
Demonstrations*** page*