# The Frequency Effect Part Deux: Shifting Time at Frequency Zero For Better Trading Performance

Animation 1: The out-of-sample performance over 60 trading days of a signal built using an optimized time-shift criterion. With 5 trades and 4 successful, the ROI is nearly 40 percent over 3 month.

What is an optimized time-shift? Is it important to use when building successful financial trading signals? While the theoretical aspects of the frequency zero and vanishing time-shift can be discussed in a very formal and mathematical manner,  I hope to answer these questions in a more simple (and applicable) way in this article. To do this, I will give an informative and illustrated real world example in this unforeseen continuation of my previous article on the frequency effect a few days ago. I discovered something quite interesting after I got an e-mail from Herr Doktor Marc (Wildi) that nudged me even further into my circus of investigations in carving out optimal frequency intervals for financial trading (see his blog for the exact email and response).  So I thought about it  and soon after I sent my response to Marc, I began to question a few things even further at 3am in the morning while sipping on some Asian raspberry white tea (my sleeping patterns lately have been as erratic as fiscal cliff negotiations), and came up with an idea. Firstly, there has to be a way to include information about the zero-frequency (this wasn’t included in my previous article on optimal frequency selection). Secondly, if I’m seeing promising results using a narrow band-pass approach after optimizing the location and distance, is there anyway to still incorporate the zero-frequency and maybe improve results even more with this additional frequency information?

Frequency zero is an important frequency in the world of nonstationary time series and model-based time series methodologies as it deals with the topic of unit roots, integrated processes,  and (for multivariate data) cointegration. Fortunately for you (and me), I don’t need to dwell further into this mess of a topic that is cointegration since typically, the type of data we want to deal with in financial trading (log-returns) is closer to being stationary (namely close to being white noise, ehem, again, close, but not quite). Nonetheless, a typical sequence of log-return data over time is never zero-mean, and full of interesting turning points at certain frequency bands. In essence, we’d somehow like to take advantage of that and perhaps better locate local turning points intrinsic to the optimal trading frequency range we are dealing with.

The perfect way to do this is through the use of the time-shift value of the filter. The time-shift is defined by the derivative of the frequency response (or transfer) function at zero. Suppose we have an optimal bandpass set at $(\omega_0, \omega_1) \subset [0,\pi]$ where $\omega_0 > 0$. We can introduce a constraint on the filter coefficients so as to impose a vanishing time-shift at frequency zero. As Wildi says on page 24 of the Elements paper: “A vanishing time-shift is highly desirable because turning-points in the filtered series are concomitant with turning-points in the original data.” In fact, we can take this a step further and even impose an arbitrary time-shift with the value $s$ at frequency zero, where $s$ is any real number. In this case, the derivative of the frequency response function (transfer function) $\hat{\Gamma}(\omega)$ at zero is $s$. As explained on page 25 of Elements,  this is implemented as $\frac{d}{d \omega} |_{\omega=0} \sum_{l=0}^{L-1} b_j \exp(-i j \omega) = s$, which implies $b_1 + 2b_2 + \cdots + (L-1) b_{L-1} = s$.

This constraint can be integrated into the MDFA formulation, but then of course adds another parameter to an already full-flight of parameters.  Furthermore, the search for the optimal $s$ with respect to a given financial trading criterion is tricky and takes some hefty computational assistance by a robust (highly nonlinear) optimization routine, but it can be done. In iMetrica I’ve implemented a time-shift turning point optimizer, something that works well so far for my taste buds, but takes a large burden of computational time to find.

To illustrate this methodology in a real financial trading application, I return to the same example I used in my previous article, namely using daily log-returns of GOOG and AAPL from 6-3-2011 to 12-31-2012 to build a trading signal. This time to freshen things up a but, I’m going to target and trade shares of Apple Inc. instead of Google.  Quickly, before I begin, I will swiftly go through the basic steps of building trading signals. If you’re already familiar, feel free to skip down two paragraphs.

As I’ve mentioned in the past, fundamentally the most important step to building a successful and robust trading signal is in choosing an appropriate preliminary in-sample metric space in which the filter coefficients for the signal are computed. This preliminary in-sample metric space represents by far the most critically important aspect of building a successful trading signal and is built using the following ingredients:

• The target and explanatory series (i.e. minute, hourly, daily log-returns of financial assets)
• The time span of in-sample observations (i.e. 6 hours, 20 days, 168 days, 3 years, etc.)

Choosing the appropriate preliminary in-sample metric space is beyond the scope of this article, but will certainly be discussed in a future article.  Once this in-sample metric space has been chosen, one can then proceed by choosing the optimal extractor (the frequency bandpass interval) for the metric space. While concurrently selecting the optimal extractor, one must  begin warping and bending the preliminary metric space through the use of the various customization and regularization tools (see my previous Frequency Effect article, as well as Marc’s Elements paper for an in-depth look at the mathematics of regularization and customization). These are the principle steps.

Now let’s look at an example. In the .gif animation at the top of this article, I featured a signal that I built using this time-shift optimizer and a frequency bandpass extractor heavily centered around the frequency $\pi/12$, which is not a very frequent trading frequency, but has its benefits, as we’ll see. The preliminary metric space was constructed by an in-sample period using the daily log-returns of GOOG and AAPL and AAPL as my target is from 6-4-2011 to 9-25-2012, nearly 16 months of data. Thus we mention that the in-sample includes many important news events from Apple Inc. such as the announcement of the iPad mini, the iPhone 4S and 5, and the unfortunate sad passing of Steve Jobs. I then proceeded to bend the preliminary metric space with a heavy dosage of regularization, but only a tablespoon of customization¹. Finally, I set the time-shift constraint and applied my optimization routine in iMetrica to find the value $s$ that yields the best possible turning-point detector for the in-sample metric space. The result is shown in Figure 1 below in the slide-show. The in-sample signal from the last 12 months or so (no out-of-sample yet applied) is plotted in green, and since I have future data available (more than 60 trading days worth from 9-25 to present), I can also approximate the target symmetric filter (the theoretically optimal target signal) in order to compare things (a quite useful option available with the click of a button in iMetrica I might add). I do this so I can have a good barometer of over-fitting and concurrent filter robustness at the most recent in-sample observation. Figure 1 in the slide-show below, the trading signal is in green, the AAPL log-return data in red, and the approximated target signal in gray (recall that if you can approximate this target signal (in gray) arbitrarily well, you win, big).

Notice that at the very endpoint (the most challenging point to achieve greatness) of the signal in Figure 1, the filter does a very fine job at getting extremely close. In fact, since the theoretical target signal is only a Fourier approximation of order 60, my concurrent signal that I built might even be closer to the ‘true value’, who knows. Achieving exact replication of the target signal (gray) elsewhere is a little less critical in my experience. All that really matters is that it is close in moving above and below zero to the targeted intention (the symmetric filter) and close at the most recent in-sample observation. Figure 2 above shows the signal without the time-shift constraint and optimization. You might be inclined to say that there is no real big difference. In fact, the signal with no time-shift constraint looks even better. It’s hard to make such a conclusion in-sample, but now here is where things get interesting.

We apply the filter to the out-of-sample data, namely the 60 tradings days. Figure 3 shows the out-of-sample performance over these past 60 trading days, roughly October, November, and December, (12-31-2012 was the latest trading day), of the signal without the time-shift constraint. Compare that to Figure 4 which depicts the performance with the constraint and optimization. Hard to tell a difference, but let’s look closer at the vertical lines. These lines can be easily plotted in iMetrica using the plot button below the canvas named Buy Indicators. The green line represents where the long position begins (we buy shares) and the exit of a short position. The magenta line represents where selling the shares occurs and the entering of a short position. These lines, in other words, are the turning point detection lines. They determine where one buys/sells (enter into a long/short position). Compare the two figures in the out-of-sample-portion after the light cyan line (indicated in Figure 4 but not Figure 3, sorry).

Figure 3: Out-of-sample performance of the signal built without time-shift constraint The out-of-sample period beings where the light cyan line is from Figure 4 below.

Figure 4: Out-of-sample performance of the signal built with time-shift constraint and optimized for turning point-detection, The out-of-sample period beings where the light cyan is.

Notice how the optimized time-shift constraint in the trading signal in Figure 4 pinpoints to a close perfection where the turning points are (specifically at points 3, 4,and 5).  The local minimum turning point was detected exactly at 3, and nearly exact at 4 and 5. The only loss out of the 5 trades occurred at 2, but this was more the fault of the long unexpected fall in the share price of Apple in October. Fortunately we were able to make up for those losses (and then some) at the next trade exactly at the moment a big turning point came (3).  Compare this to the non optimized time-shift constrained signal (Figure 3), and how the second and third turning points are a bit too late and too early, respectively. And remember, this performance is all out-of-sample, no adjustments to the filter have been made, nothing adaptive. To see even more clearly how the two signals compare, here are gains and losses of the 5 actual trades performed out-of-sample (all numbers are in percentage according to gains and losses in the trading account governed only by the signal. Positive number is a gain, negative a loss)

Without Time-Shift Optimization              With Time-Shift Optimization

Trade 1:                              29.1 -> 38.7 =  9.6                          14.1 -> 22.3  =  8.2
Trade 2:                              38.7 -> 32.0  = -6.7                         22.3 -> 17.1  = -5.2
Trade 3:                              32.0 -> 40.7  =  8.7                         17.1  -> 30.5  = 13.4
Trade 4:                              40.7 -> 48.2  =  7.5                         30.5 -> 41.2   = 10.7
Trade 5:                              48.2 -> 60.2  = 12.0                        41.2 -> 53.2   = 12.0

The optimized time-shift signal is clearly better, with an ROI of nearly 40 percent in 3 months of trading. Compare this to roughly 30 percent ROI in the non-constrained signal. I’ll take the optimized time-shift constrained signal any day. I can sleep at night with this type of trading signal. Notice that this trading was applied over a period in which Apple Inc. lost nearly 20 percent of its share price.

Another nice aspect of this trading frequency interval that I used is that trading costs aren’t much of an issue since only 10 transactions (2 transaction each trade) were made in the span of 3 months, even though I did set them to be .01 percent for each transaction nonetheless.

To dig a bit deeper into plausible reasons as to why the optimization of the time-shift constraint matters (if only even just a little bit), let’s take a look at the plots of the coefficients of each respective filter. Figure 5 depicts the filter coefficients with the optimized time-shift constraint, and Figure 6 shows the coefficients without it.  Notice how in the filter for the AAPL log-return data (blue-ish tinted line) the filter privileges the latest observation much more, while slowly modifying the others less. In the non optimized time-shift filter, the most recent observation has much less importance, and in fact, privileges a larger lag more. For timely turning point detection, this is (probably) not a good thing.  Another interesting observation is that the optimized time-shift filter completely disregards the latest observation in the log-return data of GOOG (purplish-line) in order to determine the turning points. Maybe a “better” financial asset could be used for trading AAPL? Hmmm…. well in any case I’m quite ecstatic with these results so far.  I just need to hack my way into writing a better time-shift optimization routine, it’s a bit slow at this point.  Until next time, happy extracting. And feel free to contact me with any questions.

Figure 5: The filter coefficients with time-shift optimization.

Figure 6: The filter coefficients without the time-shift optimization.

¹ I won’t disclose quite yet how I found these optimal parameters and frequency interval or reveal what they are as I need to keep some sort of competitive advantage as I presently look for consulting opportunities 😉 .

# The Frequency Effect: How to Infer Optimal Frequencies in Financial Trading

Animation 1: Click to view animation. Periodogram and Various Frequency Intervals.

Animation 2: Click to view the animation. The in-sample performance of the trading signal for each frequency sweep shown in the animation above.

When constructing signals for buy/sell trades in financial data, one of the primary parameters that should be resolved before any other parameters are regarded is the trading frequency structure that regulates all the trades. The structure should be robust and consistent during all regimes of behavior for the given traded asset, namely during times of high volatility, sideways, or bull/bear markets. In the MDFA approach to building trading signals, the trading structure is mostly determined by the characteristics of the target transfer function, the $\Gamma(\omega)$ function that designates the areas of pass and stop-band frequencies in the data. As I argue in this article, I demonstrate that there exists an optimal frequency band in which the trades should be made, and the frequency band is intrinsic to the financial data being analyzed. Two assets do not necessarily share the same optimal frequency band. Needless to say, this frequency band is highly dependent on the frequency of the observations in the data (i.e. minute, hourly, daily) and the type of financial asset.  Unfortunately, blindly seeking such an optimal trading frequency structure is a daunting and challenging task in general. Fortunately, I’ve built a few useful tools in the iMetrica financial trading platform to seamlessly navigate towards carving out the best (optimal or at least near optimal) trading frequency structure for any financial trading scenario. I show how it’s done in this article.

We first briefly summarize the procedure for building signals with a targeted range of frequencies in the (multivariate) direct filter approach, and then proceed to demonstrate how it is easily achieved in iMetrica. In order to construct signals of interest in any data set, a target transfer function must first be defined. This target filter transfer function $\Gamma(\omega)$ defined on $\omega \in [0,\pi]$ controls the frequency content of the output signal through the computation of the optimal filter coefficients. Defining $\hat{\Gamma}(\omega) = \sum_{j=0}^{L-1} b_j \exp(i j \omega)$ for some collection of filter coefficients $b_j, \, j=0,\ldots,L-1$, recall that in the plain-vanilla (univariate) direct filter approach (for ‘quasi’ stationary data), we seek to find the $L$ coefficients such that $\int_{-\pi}^{\pi} |\Gamma(\omega) - \hat{\Gamma}(\omega)|^2 H(\omega) d\omega$ is minimized, where $H(\omega)$ is a ‘smart’ weighting function that approximates the ‘true’ spectral density of the data (in general the periodogram of the data, or a function using the periodogram of the data). By defining $\Gamma(\omega)$ as a function that takes on the value of one or less for a certain range of values in $[0,\pi]$ and zero elsewhere, we pinpoint exotic frequencies where we wish our filter to extract the features of the data. The characteristics of the generated output signal (after the resulting filter has been applied to the data) are those intrinsic to the selected frequencies in the data. The characteristics found at other frequencies are (in a perfect world) disregarded from the output signal. As we show in this article, the selection of the frequencies when defining $\Gamma(\omega)$ provides the utmost in importance when building financial trading signals, as the optimal frequencies in regards to trading performance vary with every data set.

As mentioned, much emphasis should be applied to the construction of this target $\Gamma(\omega)$ and finding the optimal one is not necessarily an easy task in general. With a plethora of other parameters that are involved in building a trading signal, such as customization and regularization (see my article on financial trading parameters), one could just simply select any arbitrary frequency range for $\Gamma(\omega)$ and then proceed to optimize the other parameters until a winning trading signal is found. That is, of course, an option. But I’d like to be an advocate for carving out the proper frequency range that’s intrinsically optimal for the data set given, namely because I believe one exists, and secondly because once in the proper frequency range for the data, other parameters are much easier to optimize. So what kind of properties should this ‘optimal’ frequency range possess in regards to the trading signal?

• Consistency. Provides out-of-sample performance akin to in-sample performance.
• Optimality. Generates in-sample trade performance with rank coefficient above .90.
• Robustness. Insensitive to small changes in parameterization.

Most of these properties are obvious when first glancing at them, but are completely nontrivial to obtain. The third property tends to be overlooked when building efficient trading signals as one typically chooses a parameterization for a specific frequency band in the target $\Gamma(\omega)$, and then becomes over-confident and optimistic that the filter will provide consistent results out-of-sample. With a non-robust signal, small change in one of the customization parameters completely eradicates the effectiveness and optimality of the filter. An optimal frequency range should be much less sensitive to changes in the customization and regularization of the filter parameters. Namely, changing the smoothing parameter, say 50 percent in either direction, will have little effect on the in-sample performance of the filter, which in turn will produce a more robust signal.

To build a target transfer function $\Gamma(\omega)$, one has many options in the MDFA module of iMetrica. The approach that we will consider in this article is to define $\Gamma(\omega)$ directly by indicating the frequency pass-band and stop-band structure directly. The simplest transfer functions are defined by two cutoff frequencies: a low cutoff frequency $\omega_0$ and a high-cutoff frequency $\omega_1$.  In the Target Filter Design control panel (see Figure 1), one can control every aspect of the target transfer function $\Gamma(\omega)$ function, from different types of step functions, to more exotic options using modeling. For building financial trading signals, the Band-Pass option will be sufficient. The cutoff frequencies $\omega_0$ and $\omega_1$ are adjusted by simply modifying their values using the slider bars designated for each value, where three different ways of modifying the cutoff frequency values are available. The first is the direct designation of the value using the slider bar which goes between values of $(0,\pi)$ by changes of .01. The second method uses two different slider bars to change the values of the numerator $n$ and denominator $d$ where $\omega_0$ and/or $\omega_1$ is written in fractional form $2\pi n/d$, a form commonly used for defining different cycles in the data. The third method is to simply type in the value of the cutoff in the designated text area and then press Enter on the keyboard, where the number must be a real number in the interval $(0,\pi)$ and entered in decimal form (i.e. 0.569, 1.349, etc).  When the Auto checkbox is selected, the new direct filter and signal will be computed automatically when any changes to the target transfer function are made. This can be a quite useful tool for robustness verification, to see how small changes in the frequency content affect the output signal, and consequently the trading performance of the signal.

Figure 1: Target filter design panel.

Although cycling through multiple frequency ranges to find the optimal frequency bands for in-sample trading performance can be seamlessly accomplished by just sliding the scrollbars around (as shown in Animations 1 and 2 at the top of the page), there is a much easier way to achieve optimality (or near optimality) automatically thanks to a Financial Trading Optimization control panel featured in the Financial Trading menu at the top of the iMetrica interface. Once in the Financial Trading interface, optimization of both the customization parameters for timeliness and smoothness, along with optimization of the $\Gamma(\omega)$ frequency bands can be accomplished by first launching the Trading Optimization panel (see Figure 2), and then selecting the optimization criteria desired (maximum return, minimum loss, maximum trade success ratio, maximum rank coefficient,… etc).  To find the optimal customization parameters, simply select the optimization criteria from the drop-down menu, and then click either the Simulated Annealing button, or Grid Search button (as the name implies, ‘grid search’ simply creates a fine grid of customization values $\lambda$ and smoothing expweight $\alpha$ and then chooses the maximal value after sweeping the entire grid – it takes a few seconds depending on the length of the filter. The method that I prefer for now).  After the optimal parameters are found, the plotting canvas in the optimization panel paints a contour plot of the values found in order to give you an idea of the customization geometry, with all other parameterization values fixed. The frequency bandwidth of the target transfer function can then be optimized by a quick few millisecond grid search by selecting the checkbox Optimize bandwidth only. In this case the customization parameters are held fixed to their set values, and the optimization proceeds to only vary the frequency parameters. The values of the optimization function produced during the grid-search are then plotted on the optimization canvas to yield the structure from the frequency domain point-of-view. This can be helpful when comparing different frequency bands in building trading signals. It can also help in determining the robustness of the signal, by looking at the near neighboring values found at the optimal value.

Figure 2: The financial trading optimization panel. Here the values of the optimization criteria are plotted for all the different frequency intervals. The interval with the maximum value is automatically chosen and then computed.

We give a full example of an actual trading scenario to show how this process works in selecting an optimal frequency range for a given set of market traded assets. The outline of my general step-by-step approach for seeking good trading filters goes as follows.

1. Select the initial frequency band-pass by first initializing the $(\omega_1, \omega_2)$ interval to $(0, \omega_2)$. Setting $\omega_2$ to .10-.15 is usually sufficient. Set the checkbox Fix-Bandpass width in order to secure the bandwidth of the filter.
2. In the optimization panel (Figure 2), click the checkbox Optimize Bandwidth only and then select the optimization criteria. In these examples, we choose to maximize the rank coefficient, as it tends to produce the best out-of-sample trading performance. Then tap the Grid Search button to find the frequency range with the maximum rank coefficient. This search takes a few milliseconds.
3. With the initialization of the optimal bandwidth, the customization parameters can now be optimized by deselecting the Optimize Bandwidth only and then tapping the Grid Search button once more. Depending on the length of the filter $L$ and the number of addition explaining series, this search can take several seconds.
4. Repeat steps 2 and 3 until a combination is found of customization and filter bandwidth that produces a rank coefficient above .90. Also, test the robustness of the trading signal by slightly adjusting the frequency range and the customization parameters by small changes. A robust signal shouldn’t change the trading statistics too much under slight parameter movement.

Once content with the in-sample trading statistics (the Trading Statistics panel is available from the Financial Trading Menu), the final step is to apply the filter to out-of-sample data and trade away. Provided that sufficient regularization parameters have been selected prior to the optimization (regularization selection is out of the scope of this article however) and the optimized trading frequency bandwidth was robust enough, the out-of-sample performance of the signal should perform akin to in-sample. If not, start over with different regularization parameters and filter length, or seek options using adaptive filtering (see my previous article on adaptive filtering).

In our example, we trade on the daily price of GOOG by using GOOG log-return data as the target data and first explanatory series, along with AAPL daily log-returns as the second explanatory series. After the four steps taken above, an optimal frequency range was found to be $(.63,.80)$, where the in-sample period was from 6-3-2011 to 9-21-2012. The post-optimization of the filter, showing the MDFA trading interface, the in-sample trading statistics, and the trading optimization is shown in Figure 3. Here, the in-sample maximum rank coefficient was found to be at .96 (1.0 is the best, -1.0 is pitiful), where the trade success ratio is around 67 percent, a return-on-investment at 51 percent, and a maximum loss during the in-sample period at around 5 percent.  Applying this filter out-of-sample on incoming data for 30 trading days, without any adjustments to the filter, we see that the performance of the signal was very much akin to the performance in-sample (see Figure 5). At the end of the 30 out-of-sample trading days after the in-sample period, the trading signal gives a 65 percent return for a total of a 14 percent return-on-investment in 30 trading days. During this period, there were 6 trades made (3 buys and 3 sell shorts), and 5 of them were successful (with a .1 percent transaction cost for any trade), which amounts to, on average, one trade per week.

Figure 3. After in-sample optimization on both the customization and filter frequency band.

Figure 4: After applying the constructed filter on the next 30 days out-of-sample.

The other filter parameters (customization, regularization, and filter length $L$) have been blurred-out on purpose for obvious reasons. However, interested readers can e-mail me and I’ll send the optimal customization and regularization parameters, or maybe even just the filter coefficients themselves so you can apply them to data future GOOG and AAPL data and experiment.)  We then apply the filter out-of-sample for 30 days and make trades based on the output of the trading signal. In Figure 4, the blue-to-pink line represents the performance of the trading account given by the percentage returns from each trade made over time. The grey line is the log-price of GOOG, and the green line is the trading signal constructed from the filter just built applied to the data. It signals a ‘buy’ when the signal moves above the zero line (the dotted line) and a sell (and short-sell) when below the line. Since the data are the daily log-returns at the end each market trading period, all trades are assumed to have been made near or at the end of market hours.

Notice how successful this chosen frequency range is during the times of highest volatility for Google being in this example the first 60 day period of the in-sample partition (roughly September-October 2011). This in-sample optimization ultimately helped the 30 days out-of-sample period where volatility increased again (with even an 8 percent drop on October 17th, 2012). Out of all the largest drops in the price of Google in both the in-sample and out-of-sample period, the signal was able to anticipate all of them due to the smart choice of the frequency band and then end up making profits by short-selling.

To summarize, during an out-of-sample period in which GOOG lost over 10 percent of their stock price, the optimized trading signal that was built in this example earned roughly 14 percent. We were able to accomplish this by investigating the properties of the behavior of different frequency intervals in regard to not only the optimization criteria, but also areas of robustness in both the values of the filter frequency intervals as well as customization controls (see the animations at the top of this article). This is mostly aided by the very efficient and fast (this is where the gnu-c language came in handy) financial trading optimization panel as well as the ability in iMetrica to make any changes to the filter parameters and instantaneously see the results.  Again, feel free to contact me for the filter parameters that were found in the above example, the filter coefficients, or any questions you may have.

Happy New Year and Happy Extracting!