# 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!

# Dream within a dream: How science fiction concepts from the movie Inception can be accomplished in real life (via MDFA)

“Careful, we may be in a model…within a model.” (From an Inception movie poster.)

Have you ever seen the movie Inception and wondered, “Gee, wouldn’t it be neat if I could do all that fancy subconscious dream within a dream manipulation stuff”? Well now you can (in a metaphorical way) using MDFA and iMetrica. I explain how in this article.

Before I begin, may I first draw your attention to a brief introduction of the context in which I am speaking, and that is real-time signal extraction in (nonlinear, nonstationary) information flow. The principle goal of filtering and signal extraction in real-time data analysis for whatever purpose necessary (financial trading, risk analysis, real-time trend detection, seasonal adjustment) is to detect and pinpoint as timely as possible a desired sequence of events in an incoming flow of data observations. We emphasize that this detection should be fast, in that the desired signal, or sequence of events, should be so robust in its timeliness and accuracy so as to detect turning points or actions in targeted events as they happen, or even become so awesome that it manages to anticipate what will happen in the future. Of course, this is never an exact science nor even always possible (otherwise we’d all be billionaires right?) and thus we rely on creative ways to cope with the unknown.

We can also think of signal extraction in more abstract terms. Real-time signal extraction entails the construction of a ‘smart’ illusion, an alternative to reality, where reality in this context is a time series, the information flow, the raw data. This ‘smart’ illusion that is being constructed is the signal, the vital information that has been extracted from an abundance of “noise” embedded in the reality. And the signal must produce important underlying secrets to satisfy the needs of the user, the signal extractor. How these signals are extracted from reality is the grand challenge. How are they produced in a robust, fast, and feasible manner so as to be effective in the real-time flow of information? The answer is in MDFA, or in other words as I’ll describe in this article, penetrating the subconscious state of reality to gain access to hidden treasures.

After recently re-watching the Christopher Nolan opus entitled Inception starring Leonardo DiCaprio and what seems like most of the cast from the Dark Knight Trilogy, I began to see some similarities between the main concepts entertainingly presented in the movie (using some pimped-up CGI), and the mathematics of  signal extraction using the multivariate direct filtering approach (MDFA). In this article I present some of these interesting parallels that I’ve managed to weave together.  My ultimate goal with this article is to hopefully paint a vivid picture of some interesting details stemming from the mathematics of the direct filtering approach by using the parallels that I’ve contrived between the two. Afterwards, hopefully you’ll be on your way to entering the realm of ‘dreaming within dreaming’, and extracting pertinent hidden secrets embedded in a flurry of noise.

The film introduces a slick con man by the name of Cobb (played by DiCaprio), and his team of super well-dressed con artists with leather jackets and slicked back hair (the classic con man look right?). The catchy idea that resides in the premise of the film is that these aren’t ordinary con men: they have a unique way of manipulating reality: by entering the dreams (subconscious ) of their targets (or marks as they call them in the film) and manipulate their subconscious dream state under the goal of extracting a desired idea or hidden secret. Like any group of con men, they attempt to construct a false reality by creating a certain architecture and environment in the target’s dream. The effectiveness of this ‘heist’ to capture the desired signals in the dream relies on the quality of the architecture and environment of the dream.

So how does all this relate to the mathematics of the MDFA for signal extraction. My vision can be seen as follows. In manipulating the target’s subconscious , Cobb’s group basically involves a collection of four components. Each one can be associated with a mathematical concept embedded in the MDFA.

The Target – At the highest level, we have reality. The real world in which the characters, and the target (victim), live. The target victim has an abundance of hidden information among the large capacity of mostly noise, from which Cobb’s group wish to manipulate and extract a hidden secret, the signal. In the MDFA world, we can associate or represent the information flow, the time series on which we perform the signal extraction process as the target victim in the real world. This is the data that we see, the reality. This data of course is non-deterministic,  namely we have no idea what the target victim has in mind for the future. The process of extracting the hidden thoughts or ideas from this target victim is akin to, in the MDFA world, the signal extraction process. The tools used to do the extracting are as follows.

The Extractor – The extractor is depicted in Inception as a master con man, a person who knows how to manipulate a subject (the target) in their subconscious dreaming world into revealing their deepest mental secrets. As the extractor’s goal is manipulation of the subconscious of a target to reveal a certain signal buried within reality, the extractor must transform the real-world conscious mental state of the target from reality into the dreaming subconscious world, by inducing a dream state. The multivariate direct filtering process of transforming the data (reality) into spectral frequency space (the subconscious ) via the Fourier transform to reveal the signal given the desired target data is metaphorically very similar to this process. The Inception extractor can be seen as being parallel to the process of transforming the data from reality into a subconscious world, the spectral frequency domain. It’s in this dreaming subconscious world, the frequency domain, where the real manipulation begins, using an architect.

The Architect – The Inception architect is the designer of the dream who constructs and builds the subconscious world into which the extractor brings the subject, or target. Just as the architect manipulates real world architecture and physics in order to create paradoxes like an endless staircase, folding buildings, smooth transitions from one place to another and other various phenomena otherwise impossible in the real world, the architect in the filtering world is the toolkit of filtering parameters that render the finite-dimensional metric space in which one constructs the filter coefficients to produce the desired signal. This includes the extraction rules (namely the symmetric target filter), customization for timeliness and speed, and regularization to warp and bend the finite dimensional filter metric space. Just as many different paths in the subconscious world toward the manipulation of the target subject exist and it is the architect’s job to create the optimal environment for extracting the desired signal, the architect in the direct filtering world uses the wide ranging set of filter parameters to bend and manipulate the metric space from which the filter coefficients are built and then used in the signal extraction process. Just as changing dynamics in the Inception real world (like the state of free-falling) will change the physics of the dreamt subconscious world (like floating in hotel elevator shafts while engaging in physical combat, Matrix style),  changing dynamics in the information flow will alter the geometry of the consequent architecture being built for the filter. And furthermore, just as the dream architect must be highly skilled in order to manipulate correctly, the MDFA architect must be highly skilled in order to construct the appropriate space in which the optimal signal is extracted (hint hint, call me or Marc, we’re the extractors and architects).

Dream within a dream – As one of the more fascinating concepts introduced in Inception, the concept of the dream within a dream was also the main trick to their success in dream manipulation. Starting from reality, each level of the dreaming subconscious state can be further transposed into another level of subconscious , namely dreaming within a dream. The dream within a dream process puts you into a deeper state of dreaming. The deeper you go, the further one’s mind is removed from reality. This is where the subject of dynamic adaptive filtering comes into play (see my previous article here for an intro and basics to dynamic adaptive filtering in iMetrica). In the direct filtering world, dynamic adaptive filtering is akin to the dream within a dream concept: Once in a level of subconscious (the spectral frequency space in MDFA), and the architect has created the dream used for manipulation (the metric space for the filter coefficients), a new level of subconscious can then be entered by introducing a newly adapted metric space based on the information extracted from the first level of subconscious.

In the dream within a dream, time is the other factor. The deeper you go into a dream state, the faster your mind is able to imagine and perceive things within that dream state. For example, one minute in reality can seem like one hour in the dream state. At the next level of subconscious, at each level in the subconscious , the element of time speeds up exponentially. A similar analogy can be extracted (no pun intended) in the concept of dynamic adaptive filtering. In dynamic adaptive filtering, we first begin by extracting a signal with the desired filter architecture at the first level transformation from reality to the spectral frequency space. When new information is received and our extracted signal is not behaving how we desire, we can build a new filter architecture for manipulating the signal with the newly provided information, with all the filter parameters available to control the desired filter properties. We are inherently building a new updated filter architecture on top of the old filter architecture, and consequently building a new signal from the output of the old signal by correcting (manipulating) this old signal toward our desired goals. This is akin to the dream within a dream concept. And just like the idea of time passing much faster at each subconscious level, the effects of filter parameters for controlling regularization and speed occur at a much faster rate since we are dealing with less information, a much shorter time frame (namely the newly arrived information) at each subsequent filtering level. One can even continue down the levels of subconscious, building a new architecture on top of the previous architecture, continuously using the newly provided information at each level to build the next level of subconsciousness; dream within a dream within a dream.

To summarize these analogies, I’ll be adding a graphic soon to this article that explains in a more succinct manner these parallels described above between Inception and MDFA. In the meantime, here are the temporary replacements.

Haters gonna hate… extractors gonna extract.

Nolan, why you leavin’ Leo out?

# Hierarchy of Financial Trading Parameters

Figure 1: A trading signal produced in iMetrica for the daily price index of GOOG (Google) using the log-returns of GOOG and AAPL (Apple) as the explanatory data, The blue-pink line represents the account wealth over time, with a 89 percent return on investment in 16 months time (GOOG recorded a 23 percent return during this time). The green line represents the trading signal built using the MDFA module using the hierarchy of parameters described in this article. The gray line is the log price of GOOG from June 6 2011 to November 16 2012.

In this article, we give an in-depth look at the hierarchy of financial trading parameters involved in building financial trading signals using the powerful and versatile real-time multivariate direct filtering approach (MDFA, Wildi 2006,2008,2012), the principle method used in the financial trading interface of iMetrica.  Our aim is to clearly identify the characteristics of each parameter involved in constructing trading signals using the MDFA module in iMetrica as well as what effects (if any) the parameter will have on building trading signals and their performance.

With the many different parameters at one’s disposal for computing a signal for virtually any type of financial data and using any financial priority profile, naturally there exists a hierarchy associated with these parameters that all have well-defined mathematical definitions and properties. We propose a categorization of these parameters into three levels according to the clarity on their effect in building robust trading signals. Below are the four main control panels used in the MDFA module for the Financial Trading Interface (shown in Figure 1). They will be referenced throughout the remainder of this article.

Figure 2: The interface for controlling many of the parameters involved in MDFA. Adjusting any of these parameters will automatically compute the new filter and signal output with the new set of parameters and plot the results on the MDFA module plotting canvases.

Figure 3: The main interface for building the target symmetric filter that is used for computing the real-time (nonsymmetric) filter and output signal. Many of the desired risk/reward properties are controlled in this interface. One can control every aspect of the target filter as well as spectral densities used to compute the optimal filter in the frequency domain.

Figure 4: The main interface for constructing Zero-Pole Combination filters, the original paradigm for real-time direct filtering. Here, one can control all the parameters involved in ZPC filtering, visualize the frequency domain characteristics of the filter, and inject the filter into the I-MDFA filter to create “hybrid” filters.

Figure 5: The basic trading regulation parameters currently offered in the Financial Trading Interface. This panel is accessed by using the Financial Trading menu at the top of the software. Here, we have direct control over setting the trading frequency, the trading costs per transaction, and the risk-free rate for computing the Sharpe Ration, all controlled by simply sliding the bars to the desired level. One can also set the option to short sell during the trading period (provided that one is able to do so with the type of financial asset being traded).

The Primary Parameters:

• Timeliness of signal. The timeliness of the signal controls the quality of the phase characteristics in the real-time filter that computes the trading signal. Namely, it can control how well turning points (momentum changes) are detected in the financial data while minimizing the phase error in the filter. Bad timeliness properties will lead to a large delay in detecting up/downswings in momentum. Good timeliness properties lead to anticipated detection of momentum in real-time. However, the timeliness must be controlled by smoothness, as too much timeliness leads to the addition of unwanted noise in the trading signal, leading to unnecessary unwanted trades. The timeliness of the filter is governed by the $\lambda$ parameter that controls the phase error in the MDFA optimization. This is done by using the sliding scrollbar marked $\lambda$ in the Real-Time Filter Design in Figure 2. One can also control the timeliness property for ZPC filters using the $\lambda$ scrollbar in the ZPC Filter Design panel (Figure 4).
• Smoothness of signal.  The smoothness of the signal is related to how well the filter has suppressed the unwanted frequency information in the financial data, resulting in a smoother trading signal that corresponds more directly to the targeted signal and trading frequency. A signal that has been submitted to too much smoothing however will lose any important timeliness advantages, resulting in delayed or no trades at all. The smoothness of the filter can be adjusted through using the $\alpha$ parameter that controls the error in the stop-band between the targeted filter and the computed concurrent filter. The smoothness parameter is found on the Real-Time Filter Design interface in the sliding scrollbar marked $W(\omega)$ (see Figure 2) and in the sliding scrollbar marked $\alpha$ in the ZPC Filter Design panel (see Figure 4).
• Quantization of information.   In this sense, the quantization of information relates to how much past information is used to construct the trading signal. In MDFA, it is controlled by the length of the filter $L$ and is found on the Real-Time Filter Design interface (see Figure 2). In theory, as the filter length $L$ gets larger. the more past information from the financial time series is used resulting in a better approximation of the targeted filter. However, as the saying goes, there’s no such thing as a free lunch: increasing the filter length adds more degrees of freedom, which then leads to the age-old problem of over-fitting. The result: increased nonsense at the most concurrent observation of the signal and chaos out-of-sample. Fortunately, we can relieve the problem of over-fitting by using regularization (see Secondary Parameters). The length of the filter is controlled in the sliding scrollbar marked Order-$L$ in the Real-Time Filter Design panel (Figure 2).

As you might have suspected, there exists a so-called “uncertainty principle” regarding the timeliness and smoothness of the signal. Namely, one cannot achieve a perfectly timely signal (zero phase error in the filter) while at the same time remaining certain that the timely signal estimate is free of unwanted “noise” (perfectly filtered data in the stop-band of the filter).   The greater the timeliness (better phase error), the lesser the smoothness (suppression of unwanted high-frequency noise). A happy combination of these two parameters is always desired, and thankfully there exists in iMetrica an interface to optimize these two parameters to achieve a perfect balance given one’s financial trading priorities. There has been much to say on this real-time direct filter “uncertainty” principle, and the interested reader can seek the gory mathematical details in an original paper by the inventor and good friend and colleague Professor Marc Wildi here.

The Secondary Parameters

Regularization of filters is the act of projecting the filter space into a lower dimensional space,reducing the effective number of degrees of freedom. Recently introduced by Wildi in 2012 (see the Elements paper), regularization has three different members to adjust according to the preferences of the signal extraction problem at hand and the data. The regularization parameters are classified as secondary parameters and are found in the Additional Filter Ingredients section in the lower portion of the Real-Time Filter Design interface (Figure 2). The regularization parameters are described as follows.

• Regularization: smoothness. Not to be confused with the smoothness parameter found in the primary list of parameters, this regularization technique serves to project the filter coefficients of the trading signal into an approximation space satisfying a smoothness requirement, namely that the finite differences of the coefficients up to a certain order defined by the smoothness parameter are kept relatively small. This ultimately has the effect that the parameters appear smoother as the smooth parameter increases. Furthermore, as the approximation space becomes more “regularized” according to the requirement that solutions have “smoother” solutions, the effective degrees of freedom decrease and chances of over-fitting will decrease as well. The direct consequences of applying this type of regularization on the signal output are typically quite subtle, and depends clearly on how much smoothness is being applied to the coefficients. Personally, I usually begin with this parameter for my regularization needs to decrease the number of effective degrees of freedom and improve out-of-sample performance.
• Regularization: decay. Employing the decay parameter ensures that the coefficients of the filter decay to zero at a certain rate as the lag of the filter increases. In effect, it is another form of information quantization as the trading signal will tend to lessen the importance of past information as the decay increases. This rate is governed by two decay parameter and higher the value, the faster the values decrease to zero. The first decay parameter adjusts the strength of the decay. The second parameter adjusts for how fast the coefficients decay to zero. Usually, just a slight touch on the strength of the decay and then adjusting for the speed of the decay is the order in which to proceed for these parameters. As with the smoothing regularization, the number of effective degrees of freedom will (in most cases) decreases as the decay parameter decreases, which is a good thing (in most cases).
• Regularization: cross correlation.  Used for building trading signals with multivariate data only, this regularization effect groups the latitudinal structure of the multivariate time series more closely, resulting in more weighted estimate of the target filter using the target data frequency information. As the cross regularization parameter increases, the filter coefficients for each time series tend to converge towards each other. It should typically be used in a last effort to control for over-fitting and should only be used if the financial time series data is on the same scale and all highly correlated.

The Tertiary Parameters

• Phase-delay customization. The phase-delay of the filter at frequency zero, defined by the instantaneous rate of change of a filter’s phase at frequency zero, characterizes important information related to the timeliness of the filter. One can directly ensure that the phase delay of the filter at frequency zero is zero by adding constraints to the filter coefficients at computation time. This is done by setting the clicking the $i2$ option in the Real-Time Filter Design interface. To go further, one can even set the phase delay to an fixed value other than zero using the $i2$ scrollbar in the Additional Filter Ingredients box. Setting this value to a certain value (between -20 and 20 in the scrollbar) ensures that the phase delay at zero of the filter reacts as anticipated. It’s use and benefit is still under investigation. In any case, one can seamlessly test how this constraint affects the trading signal output in their own trading strategies directly by visualizing its performance in-sample using the Financial Trading canvas.
• Differencing weight. This option, found in the Real-Time Filter Design interface as the checkbox labeled “d” (Figure 2), multiplies the frequency information (periodogram or discrete Fourier transform (DFT)) of the financial data by the weighting function $f(\omega) = 1/(1 - \exp(i \omega)), \omega \in (0,\pi)$, which is the reciprocal of the differencing operator in the frequency domain. Since the Financial Trading platform in iMetrica strictly uses log-return financial time series to build trading signals, the use of this weighting function is in a sense a frequency-based “de-differencing” of the differenced data. In many cases, using the differencing weight provides better timeliness properties for the filter and thus the trading signal.

In addition to these three levels of parameters used in building real-time trading signals, there is a collection of more exotic “parameterization” strategies that exist in the iMetica MDFA module for fine tuning and constructing boosting trading performance. However, these strategies require more time to develop, a bit of experimentation, and a keen eye for filtering. We will develop more information and tutorials about these advanced filtering techniques for constructing effective trading signals in iMetrica in future articles on this blog coming soon. For now, we just summarize their main ideas.

• Forecasting and Smoothing signals. Smoothing signals in time series, as its name implies, involves obtaining a smoother estimate of certain signal in the past. Since the real-time estimate of a signal value in the past involves using more recent values, the signal estimation becomes more symmetrical as past and future values at a point in the past are used to estimate the value of the signal. For example, if today is after market hours on Friday, we can obtain a better estimate of the targeted signal for Wednesday since we have information from Thursday and Friday. In the opposite manner, forecasting involves projecting a signal into the future. However, since the estimate becomes even more “anti-symmetric”, the estimate becomes more polluted with noise. How these smoothed and forecasted signals can be used for constructing buy/sell trading signals in real-time is still purely experimental. With iMetrica, building and testing strategies that improve trading performance using either smoothed and forecasted signals (or both), is available.To produce either a smoothed or forecasted signal, there is a lag scrollbar available in the Real-Time Filter Design interface under Additional Filter Ingredients that enables one to compute either a smooth or forecasted signal. Setting the lag value $k$ in the scrollbar to any integer between -10 and 10 and the signal with the set lag applied is automatically computed. For negative lag values $k$, the method produces a $k$ step-ahead forecast estimate of the signal. For positive values, the method produces a smoothed signal with a delay of $k$ observations.
• Customized spectral weighting functions. In the spirit of customizing a trading signal to fit one’s priorities in financial trading, one also has the option of customizing the spectral density estimate of the data generating process to any design one wishes. In the computation of the real-time filter, the periodogram (or DFTs in multivariate case) is used as the default estimate of the spectral density weighting function. This spectral density weighting function in theory is supposed to serve as the spectrum of the underlying data generating process (DGP). However, since we have no possible idea about the underlying DGP of the price movement of publicly traded financial assets (other than it’s supposed to be pretty darn close to a random walk according to the Efficient Market Hypothesis), the periodogram is the best thing to an unbiased estimate a mortal human can get and is the default option in the MDFA module of iMetrica. However, customization of this weighting function is certainly possible through the use of the Target Filter Design interface. Not only can one design their target filter for the approximation of the concurrent filter, but the spectral density weighting function of the DGP can also be customized using some of the available options readily available in the interface. We will discuss these features in a soon-to-come discussion and tutorial on advanced real-time filtering methods.
• Adaptive filtering. As perhaps the most advanced feature of the MDFA module, adaptive filtering is an elegant way to achieve building smarter filters based on previous filter realizations. With the goal of adaptive filtering being to improve certain properties of the output signal at each iteration without compensating with over-fitting, the adaptive process is of course highly nonlinear. In short, adaptive MDFA filtering is an iterative process in which a one begins with a desired filter, computes the output signal, and then uses the output signal as explanatory data in the next filtering round. At each iteration step, one has the freedom to change any properties of the filter that they desire, whether it be customization, regularization, adding negative lags, adding filter coefficient constraints, applying a ZPC filter, or even changing the pass-band in the target filter. The hope is to improve on certain properties of filter at each stage of the iterative process. An in-depth look at adaptive filtering and how to easily produce an adaptive filter using iMetrica is soon to come later this week.

# iMetrica: Economic and Financial Data Control

### Data Control Interface

We begin this iMetrica blog entry by first giving an overview of the basic components featured in the Data Control module. Figures 1 and 2 show the interface and all the major components labeled. Here, a collection of simulated time series are being plotted together.

Figure 1. The major components of the data control module.

Figure 2. The major components of the data control module, showing the target series editor.

1. Main plotting canvas. This is where the time series data is plotted. Up to 10 different time series can be loaded into the data control at a time, and all of them can be plotted using the plot control in panel 2. When all the data is plotted together, to highlight a particular series, go to the main Data Control menu in the top left corner and place the mouse on any one the series names, the respective series will then be highlighted.
2. Plot control panel. The time series that are uploaded into the module can be viewed by toggling their respective check box inside the plot control panel. This is helpful when different time series are scaled different and/or have different means. One can also log-transform the data, rescale the data to have unit standard deviations, or compare data using cross-correlations. Note that the log and rescale check box actions will only apply to the data that is currently being plotted. Furthermore, to plot the cross-correlations, only two time series can be chosen at a time. When one time series is chosen, the auto-correlation plot is drawn. Here, the “Target $X(t)$ indicates a weighted aggregation of the data. To edit this, use the  “Target Series” in 3. To delete all of the data stored in the data control module, simply press the “Delete” button. Careful, there’s no going back once deleted.
3. Simulated and Target Series Panels. The simulated time series data interfaces to simulate a multitude of different time series. Simulating time series can be helpful when wanting to either learn, practice, or explore the different modules and capabilites of iMetrica, learn more about time series analysis, or learn about the dynamics of time series modules. The different types of models include (S)ARIMA models, GARCH models, correlated cycle models, trend models, multivariate factor stochastic volatility models, and HEAVY models. From simulating data and toggling the parameters, one can visualize instantly the effects of the each parameter on the simulated data. The data can then be exported to any of the modules for practicing and honing one’s skills in hybrid modeling, signal extraction, and forecasting.  Each model has a “parameter” button (see 4) that controls the dimensions, innovation distributions, or parameter values. When changes are made, the simulated series is recomputed automatically and replotted on their respective plotting canvas (see 4).
4. Simulated Data Control.  Once the parameters have been selected, and a desired simulated series has been achieved to one’s liking, it can be added to the main data control plotting canvas by clicking the “Add” button. The new simulated series is now ready to be exported to any of the modules. One can also change the random seed that controls the “burn-in” of the innovation sequence (random effects that govern the initialization and trajectory of the data). In some of the models, one can “integrate” the data to render stationary data nonstationary.
5. Parameter Controls.  Once the “Parameters” button has been clicked, an additional panel will pop up where controls for all the model’s parameters can be toggled. Once any parameter has been changed using the sliders, scrollbars, or combo boxes, the simulated data is automatically recomputed and plotted, making it a great tool to understand time series model dynamics.
6. Target Series Construction. The target series is used to construct a univariate time series that is a weighted sum of one or more time series (given by the $X_i(t)$ for $i=1,\ldots,10$ series). In modules that only deal with univariate time series data (the uSimX13, EMD, and State Space Modeling), the constructed target series is the series that gets exported for analysis. For the MDFA module, this is the series that is being filtered for constructing a signal, with the other time series acting as the explanatory time series. In the BayesCronos module, this target series is ignored and only the supporting time series data $X_i(t)$ are used.  In these up and down slider controls, one can adjust for the weight associated with that specific series, and the aggregate target series will be automatically recomputed as it is adjusted.
7. Series Checkboxes. To ignore the series entirely in the computation of the target series, simply click the check box “off” in the associated “computed in target” check box. This will eliminate it from the target sum. In the case one is constructing data for the MDFA module, one has the option of utilizing a series in the target series, but not using it as an explaining time series variable, and vice-versa.

Within this main data control hub, one can import univariate or multivariate time series data from a multitude of file formats, as well as download financial time series data directly from Yahoo! finance or another source such as Reuters for higher-frequency financial data.  To load data from a file, simply click on the “Data Input/Export” menu when in the Data Control module and select one of the “Load” data options. The “Load Data” option pop up a “file select” panel and from there, the data file can be selected. The format of the data in this “Load Data” case is simple: a single column of data for each series. If more than one series is present, the data column must be separated by a space.  In the “Load CSV” data, this assumes the file is stored in a CSV format. See Figure 3 for the menu options of the Data Control module.

Figure 3. Showing the different options for importing data into the data control module.

• Symbols(s) – In this text box, type the market ticker symbol of the desired financial series in all CAPS. Each ticker symbol must be seperated only by one space and nothing else. Up to 10 ticker symbols can be entered.
• Start Date – This indicates the year, month, and day from which the financial time series begins. This date must obviously be in the past. If the day falls on a non-traded day such as a weekend or holiday, the nearest date after that date will be chosen. The time series will then be loaded to the most recent date available for that asset.
• Hours –  This indicates the time period in which the frequency of the data is selected. In most cases, this should simply be set to “US Market Hours”.
• Frequency – The frequency of the data. The options are Second, Minute, 3,5,10,15,30-Minute, Hourly, Daily, Weekly, Monthly.
• New Data Set – Deletes all the data already stored in the data control module and uploads as new data.
• Log Returns – Download the data in log-return format. This is usually the case when using the data to build financial trading strategies using the MDFA module. However, in addition to the log-return data, it will also download the log-transformed raw time series data of the first asset in the Symbols(s) box. This is generally used for gauging financial trading accounts in the financial trading interface of iMetrica. When Financial Trading is turned on in the data control menu this is automatically set on.
• Volume Data – In addition to the asset time series data, the volume (of trades) data associated for the given frequency will also be downloaded for each market ticker symbol given in Symbols(s).
• Yahoo! Source – The financial data will be downloaded from Yahoo! finance (thus you need an internet connection). If this box is not checked, then the downloader will assume a Reuters financial database (but of course for this you need an account with Reuters).