# TWS-iMetrica: The Automated Intraday Financial Trading Interface Using Adaptive Multivariate Direct Filtering

Figure 1: The TWS-iMetrica automated financial trading platform. Featuring fast performance optimization, analysis, and trading design features unique to iMetrica for building direct real-time filters to generate automated trading signals for nearly any tradeable financial asset. The system was built using Java, C, and the Interactive Brokers IB API in Java.

### Introduction

I realize that I’ve been MIA (missing in action for non-anglophones) the past three months on this blog, but I assure you there has been good reason for my long absence. Not only have I developed a large slew of various optimization, analysis, and statistical tools in iMetrica for constructing high-performance financial trading signals geared towards intraday trading which I will (slowly) be sharing over the next several months (with some of the secret-sauce-recipes kept to myself and my current clients of course), but I have also built, engineered, tested, and finally put into commission on a daily basis the planet’s first automated financial trading platform completely based on the recently developed FT-AMDFA (adaptive multivariate direct filtering approach for financial trading). I introduce to you iMetrica’s little sister, TWS-iMetrica.

Coupled with the original software I developed for hybrid econometrics, time series analysis, signal extraction, and multivariate direct filter engineering called iMetrica, the TWS-iMetrica platform was built in a way to provide an easy to use yet powerful, adaptive, versatile, and automated trading machine for intraday financial trading with a variety of options for building your own day trading strategies using MDFA based on your own financial priorities.  Being written completely in Java and gnu c, the TWS-iMetrica system currently uses the Interactive Brokers (IB) trading workstation (TWS) Java API in order to construct the automated trades, connect to the necessary historical data feeds, and provide a variety of tick data. Thus in order to run, the system will require an activated IB trading account. However, as I discuss in the conclusion of this article, the software was written in a way to be seamlessly adapted to any other brokerage/trading platform API, as long as the API is available in Java or has Java wrappers available.

The steps for setting up and building an intraday financial trading environment using iMetrica + TWS-iMetrica are easy. There are four of them. No technical analysis indicator garbage is used here, no time domain methodologies, or stochastic calculus. TWS-iMetrica is based completely on the frequency domain approach to building robust real-time multivariate filters that are designed to extract signals from tradable financial assets at any fixed observation of frequencies (the most commonly used in my trading experience with FT-AMDFA being 5, 15, 30, or 60 minute intervals). What makes this paradigm of financial trading versatile is the ability to construct trading signals based on your own trading priorities with each filter designed uniquely for a targeted asset to be traded. With that being said, the four main steps using both iMetrica and TWS-iMetrica are as follows:

1. The first step to building an intraday trading environment is to construct what I call an MDFA portfolio (which I’ll define in a brief moment). This is achieved in the TWS-iMetrica interface that is endowed with a user-friendly portfolio construction panel shown below in Figure 4.
2. With the desired MDFA portfolio, selected, one then proceeds in connecting TWS-iMetrica to IB by simply pressing the Connect button on the interface in order to download the historical data (see Figure 3).
3. With the historical data saved, the iMetrica software is then used to upload the saved historical data and build the filters for the given portfolio using the MDFA module in iMetrica (see Figure 2). The filters are constructed using a sequence of proprietary MDFA optimization and analysis tools. Within the iMetrica MDFA module, three different types of filters can be built 1) a trend filter that extracts a fast moving trend 2) a band-pass filter for extracting local cycles, and 3) A multi-bandpass filter that extracts both a slow moving trend and local cycles simultaneously.
4. Once the filters are constructed and saved in a file (a .cft file), the TWS-iMetrica system is ready to be used for intrady trading using the newly constructed and optimized filters (see Figure 6).

Figure 2: The iMetrica MDFA module for constructing the trading filters. Features dozens of design, analysis, and optimization components to fit the trading priorities of the user and is used in conjunction with the TWS-iMetrica interface.

In the remaining part of this article, I give an overview of the main features of the TWS-iMetrica software and how easily one can create a high-performing automated trading strategy that fits the needs of the user.

### The TWS-iMetrica Interface

The main TWS-iMetrica graphical user interface is composed of several components that allow for constructing a multitude of various MDFA intraday trading strategies, depending on one’s trading priorities. Figure 3 shows the layout of the GUI after first being launched. The first component is the top menu featuring TWS System, some basic TWS connection variables which, in most cases, these variables are left in their default mode, and the Portfolio menu. To access the main menu for setting up the MDFA trading environment, click Setup MDFA Portfolio under the Portfolio menu. Once this is clicked, a panel is displayed (shown in Figure 4) featuring the required a priori parameters for building the MDFA trading environment that should all be filled before MDFA filter construction and trading is to take place. The parameters and their possible values are given below Figure 4.

Figure 3 – The TWS-iMetrica interface when first launched and everything blank.

Figure 4 – The Setup MDFA Portfolio panel featuring all the setting necessary to construct the automated trading MDFA environment.

1. Portfolio – The portfolio is the basis for the MDFA trading platform and consists of two types of assets 1) The target asset from which we construct the trading signal, engineer the trades, and use in building the MDFA filter 2) The explanatory assets that provide the explanatory data for the target asset in the multivariate filter construction. Here, one can select up to four explanatory assets.
2. Exchange – The exchange on which the assets are traded (according to IB).
3. Asset Type – If the input portfolio is a selection of Stocks or Futures (Currencies and Options soon to be available).
4. Expiration – If trading Futures, the expiration date of the contract, given as a six digit number of year then month (e.g. 201306 for June 2013).
5. Shares/Contracts – The number of shares/contracts to trade (this number can also be changed throughout the trading day through the main panel).
6. Observation frequency – In the MDFA financial trading method, we consider uniformly sampled observations of market data on which to do the trading (in seconds). The options are 1, 2, 3, 5, 15, 30, and 60 minute data. The default is 5 minutes.
7. Data – For the intraday observations, determines the nature of data being extracted. Valid values include TRADES, MIDPOINT, BID, ASK, and BID_ASK. The default is MIDPOINT
8. Historical Data – Selects how many days are used to for downloading the historical data to compute the initial MDFA filters. The historical data will of course come in intervals chosen in the observation frequency.

Once all the values have been set for the MDFA portfolio, click the Set and Build button which will first begin to check if the values entered are valid and if so, create the necessary data sets for TWS-iMetrica to initialize trading. This all must be done while TWS-iMetrica is connected to IB (not set in trading mode however). If the build was successful, the historical data of the desired target financial asset up to the most recent observation in regular market trading hours will be plotted on the graphics canvas. The historical data will be saved to a file named (by default) “lastSeriesData.dat” and the data will be come in columns, where the first column is the date/time of the observation, the second column is the price of the target asset, and remaining columns are log-returns of the target and explanatory data. And that’s it, the system is now setup to be used for financial trading. These values entered in the Setup MDFA Portfolio will never have to be set again (unless changes to the MDFA portfolio are needed of course).

Continuing on to the other controls and features of TWS-iMetrica, once the portfolio has been set, one can proceed to change any of the settings in main trading control panel. All these controls can be used/modified intraday while in automated MDFA trading mode. In the left most side of the panel at the main control panel (Figure 5) of the interface includes a set of options for the following features:

Figure 5 – The main control panel for choosing and/or modifying all the options during intraday trading.

1. In contracts/shares text field, one enters the amount of share (for stocks) or contracts (for futures)  that one will trade throughout the day. This can be adjusted during the day while the automated trading is activated, however, one must be certain that at the end of the day, the balance between bought and shorted contracts is zero, otherwise, you risk keeping contracts or shares overnight into the next trading day.Typically, this is set at the beginning before automated trading takes place and left alone.
2. The data input file for loading historical data. The name of this file determines where the historical data associated with the MDFA portfolio constructed will be stored. This historical data will be needed in order to build the MDFA filters. By default this is “lastSeriesData.dat”. Usually this doesn’t need to be modified.
3. The stop-loss activation and stop-loss slider bar, one can turn on/off the stop-loss and the stop-loss amount. This value determines how/where a stop-loss will be triggered relative to the price being bought/sold at and is completely dependent on the asset being traded.
4. The interval search that determines how and when the trades will be made when the selected MDFA signal triggers a buy/sell transaction. If turned off, the transaction (a limit order determined by the bid/ask) will be made at the exact time that the buy/sell signal is triggered by the filter. If turned on, the value in the text field next to it gives how often (in seconds) the trade looks for a better price to make the transaction. This search runs until the next observation for the MDFA filter. For example, if 5 minute return data is being used to do the trading, the search looks every seconds for 5 minutes for a better price to make the given transaction. If at the end of the 5 minute period no better price has been found, the transaction is is made at the current ask/bid price. This feature has been shown to be quite useful during sideways or highly volatile markets.

The middle of the main control panel features the main buttons for both connecting to disconnecting from Interactive Brokers, initiating the MDFA automated trading environment, as well as convenient buttons used for instantaneous buy/sell triggers that supplement the automated system. It also features an on/off toggle button for activating the trades given in the MDFA automated trading environment. When checked on, transactions according to the automated MDFA environment will proceed and go through to the IB account. If turned off, the real-time market data feeds and historical data will continue to be read into the TWS-iMetrica system and the signals according to the filters will be automatically computed, but no actual transactions/trades into the IB account will be made.

Finally, on the right hand side of the main control panel features the filter uploading and selection boxes. These are the MDFA filters that are constructed using the MDFA module in iMetrica. One convenient and useful feature of TWS-iMetrica is the ability to utilize up to three direct real-time filters in parallel and to switch at any given moment during market hours between the filters. (Such a feature enhances the adaptability of the trading using MDFA filters. I’ll discuss more about this in further detail shortly).  In order to select up to three filters simultaneously, there is a filter selection panel (shown in bottom right corner of Figure 6 below) displaying three separate file choosers and a radio button corresponding to each filter. Clicking on the filter load button produces a file dialog box from which one selects a filter (a *.cft file produced by iMetrica). Once the filter is loaded properly, on success, the name of the filter is displayed in the text box to the right, and the radio button to the left is enabled. With multiple filters loaded, to select between any of them, simply click on their respective radio button and the corresponding signal will be plotted on the plot canvas (assuming market data has been loaded into the TWS-iMetrica using the market data file upload and/or has been connected to the IB TWS for live market data feeds). This is shown in Figure 6.

Figure 6 – The TWS-iMetrica main trading interface features many control options to design your own automated MDFA trading strategies.

And finally, once the historical data file for the MDFA portfolio has been created, up to three filters have been created for the portfolio and entered in the filter selection boxes, and the system is connected to Interactive Brokers by pressing the Connect button, the market and signal plot panel can then be used for visualizing the different components that one will need for analyzing the market, signal, and performance of the automated trading environment. In the panel just below the plot canvas features and array of checkboxes and radiobuttons. When connected to IB and the Start MDFA Trading has been pressed, all the data and plots are updated in real-time automatically at the specific observation frequency selected in the MDFA Portfolio setup. The currently available plots are as follows:

Figure 8 – The plots for the trading interface. Features price, log-return, account cumulative returns, signal, buy/sell lines, and up to two additional auxiliary signals.

• Price – Plots in real-time the price of the asset being traded, at the specific observation frequency selected for the MDFA portfolio.
• Log-returns – Plots in real-time the log-returns of the price, which is the data that is being filtered to construct the trading signal.
• Account – Shows the cumulative returns produced by the currently chosen MDFA filter over the current and historical data period (note that this does not necessary reflect the actual returns made by the strategy in IB, just the theoretical returns over time if this particular filter had been used).
• Buy/Sell lines – Shows dashed lines where the MDFA trading signal has produced a buy/sell transaction. The green lines are the buy signals (entered a long position) and magenta lines are the sell (entered a short position).
• Signal – The plot of the signal in real-time. When new data becomes available, the signal is automatically computed and replotted in real-time. This gives one the ability to closely monitory how the current filter is reacting to the incoming data.
• Aux Signal 1/2 – (If available) Plots of the other available signals produced by the (up to two) other filters constructed and entered in the system. To make either of these auxillary signals the main trading signal simply select the filter associated with the signal using the radio buttons in the filter selection panel.

Along with these plots, to track specific values of any of these plots at anytime, select the desired plot in the Track Plot region of the panel bar. Once selected, specific values and their respective times/dates are displayed in the upper left corner of the plot panel by simply placing the mouse cursor over the plot panel. A small tracking ball will then be moved along the specific plot in accordance with movements by the mouse cursor.

With the graphics panel displaying the performance in real-time of each filter, one can seamlessly switch between a band-pass filter or a timely trend (low-pass) filter according to the changing intraday market conditions. To give an example, suppose at early morning trading hours there is an unusual high amount of volume pushing an uptrend or pulling a downtrend. In such conditions a trend filter is much more appropriate, being able to follow the large-variation in log-returns much better than a band-pass filter can. One can glean from the effects of the trend filter on the morning hours of the market. After automated trading using the trend filter, with the volume diffusing into the noon hour, the band-pass filter can then be applied in order to extract and trade at certain low frequency cycles in the log-return data. Towards the end of the day, with volume continuously picking up, the trend filter can then be selected again in order to track and trade any trending movement automatically.

I am in the process of currently building an automated algorithm to “intelligently” switch between the uploaded filters according to the instantaneous market conditions (with triggering of the switching being set by volume and volatility. Otherwise, for the time being, currently the user must manually switch between different filters, if such switching is at all desired (in most cases, I prefer to leave one filter running all day. Since the process is automated, I prefer to have minimal (if any) interaction with the software during the day while it’s in automated trading mode).

### Conclusion

As I mentioned earlier, the main components of the TWS-iMetrica were written in a way to be adaptable to other brokerage/trading APIs. The only major condition is that the API either be available in Java, or at least have (possibly third-party?) wrappers available in Java. That being said, there are only three main types of general calls that are made automated to the connected broker 1) retrieve historical data for any asset(s), at any given time, at most commonly used observation frequencies (e.g. 1 min, 5 min, 10 min, etc.), 2) subscribe to automatic feed of bar/tick data to retrieve latest OHLC and bid/ask data, and finally 3) Place an order (buy/sell) to the broker with different any order conditions (limit, stop-loss, market order, etc.) for any given asset.

If you are interested in having TWS-iMetrica be built for your particular brokerage/trading platform (other than IB of course) and the above conditions for the API are met, I am more than happy to be hired at certain fixed compensation, simply get in contact with me. If you are interested seeing how well the automated system has performed thus far, interested in future collaboration, or becoming a client in order to use the TWS-iMetrica platform, feel free to contact me as well.

Happy extracting!

# High-Frequency Financial Trading on FOREX with MDFA and R: An Example with the Japanese Yen

Figure 1: In-sample (observations 1-250) and out-of-sample performance of the trading signal built in this tutorial using MDFA. (Top) The log price of the Yen (FXY) in 15 minute intervals and the trades generated by the trading signal. Here black line is a buy (long), blue is sell (short position). (Bottom) The returns accumulated (cash) generated by the trading, in percentage gained or lost.

In my previous article on high-frequency trading in iMetrica on the FOREX/GLOBEX, I introduced some robust signal extraction strategies in iMetrica using the multidimensional direct filter approach (MDFA) to generate high-performance signals for trading on the foreign exchange and Futures market. In this article I take a brief leave-of-absence from my world of developing financial trading signals in iMetrica and migrate into an uber-popular language used in finance due to its exuberant array of packages, quick data management and graphics handling, and of course the fact that it’s free (as in speech and beer) on nearly any computing platform in the world.

This article gives an intro tutorial on using R for high-frequency trading on the FOREX market using the R package for MDFA (offered by Herr Doktor Marc Wildi von Bern) and some strategies that I’ve developed for generating financially robust trading signals. For this tutorial, I consider the second example given in my previous article where I engineered a trading signal for 15-minute log-returns of the Japanese Yen (from opening bell to market close EST).  This presented slightly new challenges than before as the close-to-open jump variations are much larger than those generated by hourly or daily returns. But as I demonstrated, these larger variations on close-to-open price posed no problems for the MDFA. In fact, it exploited these jumps and made large profits by predicting the direction of the jump. Figure 1 at the top of this article shows the in-sample (observations 1-250) and out-of-sample (observations 251 onward) performance of the filter I will be building in the first part of this tutorial.

Throughout this tutorial, I attempt to replicate these results that I built in iMetrica and expand on them a bit using the R language and the implementation of the MDFA available in here.  The data that we consider are 15-minute log-returns of the Yen from January 4th to January 17th and I have them saved as an .RData file given by ld_fxy_insamp. I have an additional explanatory series embedded in the .RData file that I’m using to predict the price of the Yen. Additionally, I also will be using price_fxy_insamp which is the log price of Yen, used to compute the performance (buy/sells) of the trading signal. The ld_fxy_insamp will be used as the in-sample data to construct the filter and trading signal for FXY. To obtain this data so you can perform these examples at home, email me and I’ll send you all the necessary .RData files (the in-sample and out-of-sample data) in a .zip file. Taking a quick glance at the ld_fxy_insamp data, we see log-returns of the Yen at every 15 minutes starting at market open (time zone UTC). The target data (Yen) is in the first column along with the two explanatory series (Yen and another asset co-integrated with movement of Yen).
 > head(ld_fxy_insamp) [,1]           [,2]          [,3] 2013-01-04 13:30:00  0.000000e+00   0.000000e+00  0.0000000000 2013-01-04 13:45:00  4.763412e-03   4.763412e-03  0.0033465833 2013-01-04 14:00:00 -8.966599e-05  -8.966599e-05  0.0040635638 2013-01-04 14:15:00  2.597055e-03   2.597055e-03 -0.0008322064 2013-01-04 14:30:00 -7.157556e-04  -7.157556e-04  0.0020792190 2013-01-04 14:45:00 -4.476075e-04  -4.476075e-04 -0.0014685198 

Moving on, to begin constructing the first trading signal for the Yen, we begin by uploading the data into our R environment, define some initial parameters for the MDFA function call, and then compute the DFTs and periodogram for the Yen.

load(paste(path.pgm,&quot;ld_fxy_in15min.RData&quot;,sep=&quot;&quot;))    #load in-sample log-returns of Yen
load(paste(path.pgm,&quot;price_fxy_in15min.RData&quot;,sep=&quot;&quot;)) #load in-sample log-price of Yen

in_samp_lenprice_insample&lt;-price_fxy_insamp

#setup some MDFA variables
x&lt;-ld_fxy_insamp
len&lt;-length(x[,1])
shift_constraint&lt;-rep(0,length(x[1,])-1)
weight_constraint&lt;-rep(0,length(x[1,])-1)
d&lt;-0
plots&lt;-T
lin_expweight&lt;-F

# Compute DFTs and periodogram for initial analysis
spec_obj&lt;-spec_comp(len,x,d)
weight_func&lt;-spec_obj$weight_func K&lt;-length(weight_func[,1])-1 fxy_periodogram&lt;-abs(spec_obj$weight_func[,1])^2


As I’ve mentioned in my previous articles, my step-by-step strategy for building trading signals always begin by a quick analysis of the periodogram of the asset being traded on. Holding the key to providing insight into the characteristics of how the asset trades, the periodogram is an essential tool for navigating how the extractor $\Gamma$ is chosen. Here, I look for principal spectral peaks that correspond in the time domain to how and where my signal will trigger buy/sell trades. Figure 2 shows the periodogram of the 15-minute log-returns of the Japanese Yen during the in-sample period from January 4 to January 17 2013. The arrows point to the main spectral peaks that I look for and provides a guide to how I will define my $\Gamma$ function. The black dotted lines indicate the two frequency cutoffs that I will consider in this example, the first being $\pi/12$ and the second at $\pi/6$. Notice that both cutoffs are set directly after a spectral peak, something that I highly recommend.  In high-frequency trading on the FOREX using MDFA, as we’ll see, the trick is to seek out the spectral peak which accounts for the close-to-open variation in the price of the foreign currency. We want to take advantage of this spectral peak as this is where the big gains in foreign currency trading using MDFA will occur.

Figure 2: Periodogram of FXY (Japanese Yen) along with spectral peaks and two different frequency cutoffs.

In our first example we consider the larger frequency as the cutoff for $\Gamma$ by setting it to $\pi/6$ (the right most line in the figure of the periodogram). I then initially set the timeliness and smoothness parameters, $lambda$ and expweight to 0 along with setting all the regularization parameters to 0 as well. This will give me a barometer for where and how much to adjust the filter parameters. In selecting the filter length $L$, my empirical studies over numerous experiments in building trading signals using iMetrica have demonstrated that a ‘good’ choice is anywhere between 1/4 and 1/5 of the total in-sample length of the time series data.  Of course, the length depends on the frequency of the data observations (i.e. 15 minute, hourly, daily, etc.), but in general you will most likely never need more than $L$ being greater than 1/4 the in-sample size. Otherwise, regularization can become too cumbersome to handle effectively. In this example, the total in-sample length is 335 and thus I set $L= 82$ which I’ll stick to for the remainder of this tutorial. In any case, the length of the filter is not the most crucial parameter to consider in building good trading signals. For a good robust selection of the filter parameters couple with appropriate explanatory series, the results of the trading signal with $L= 80$ compared with, say, $L= 85$ should hardly differ. If they do, then the parameterization is not robust enough.

After uploading both the in-sample log-return data along with the corresponding log price of the Yen for computing the trading performance, we the proceed in R to setting initial filter settings for the MDFA routine and then compute the filter using the IMDFA_comp function. This returns both the i_mdfa& object holding coefficients, frequency response functions, and statistics of filter, along with the signal produced for each explanatory series. We combine these signals to get the final trading signal in-sample. All this is all done in R as follows:


cutoff&lt;-pi/6 #set frequency cutoff
Gamma&lt;-((0:K)&lt;(cutoff*K/pi)) #define Gamma

grand_mean&lt;-F
Lag&lt;-0
L&lt;-82
lambda_smooth&lt;-0
lambda_cross&lt;-0
lambda_decay&lt;-c(0.,0.) #regularization - decay

lambda&lt;-0
expweight&lt;-0
i1&lt;-F
i2&lt;-F
# compute the filter for the given parameter definitions
i_mdfa_obj&lt;-IMDFA_comp(Lag,K,L,lambda,weight_func,Gamma,expweight,cutoff,i1,i2,weight_constraint,
lambda_cross,lambda_decay,lambda_smooth,x,plots,lin_expweight,shift_constraint,grand_mean)

# after computing filter, we save coefficients
bn&lt;-i_mdfa_obj$i_mdfa$b

# now we build trading signal
trading_signal&lt;-i_mdfa_obj$xff[,1] + i_mdfa_obj$xff[,2]


The resulting frequency response functions of the filter and the coefficients are plotted in the figure below.

Figure 3: The Frequency response functions of the filter (top) and the filter coefficients (below)

Notice the abundance of noise still present passed the cutoff frequency. This is mollified by increasing the expweight smoothness parameter. The coefficients for each explanatory series show some correlation in their movement as the lags increase. However, the smoothness and decay of the coefficients leaves much to be desired. We will remedy this by introducing regularization parameters. Plots of the in-sample trading signal and the performance in-sample of the signal are shown in the two figures below. Notice that the trading signal behaves quite nicely in-sample. However, looks can be deceiving. This stellar performance is due in large part to a filtering phenomenon called overfitting. One can deduce that overfitting is the culprit here by simply looking at the nonsmoothness of the coefficients along with the number of freezed degrees of freedom, which in this example is roughly 174 (out of 174), way too high. We would like to get this number at around half the total amount of degrees of freedom (number of explanatory series x L).

Figure 4: The trading signal and the log-return data of the Yen.

The in-sample performance of this filter demonstrates the type of results we would like to see after regularization is applied.  But now comes for the sobering effects of overfitting. We apply these filter coeffcients to 200 15-minute observations of the Yen and the explanatory series from January 18 to February 1 2013 and compare with the characteristics in-sample. To do this in R, we first load the out-of-sample data into the R environment, and then apply the filter to the out-of-sample data that I defined as x_out.

load(paste(path.pgm,&quot;ld_fxy_out15min.RData&quot;,sep=&quot;&quot;))
load(paste(path.pgm,&quot;price_fxy_out15min.RData&quot;,sep=&quot;&quot;))
x_out&lt;-rbind(ld_fxy_insamp,ld_fxy_outsamp) #bind the in-sample with out-of-sample data
xff&lt;-matrix(nrow=out_samp_len,ncol=2)

#apply filter built in-sample
for(i in 1:out_samp_len)
{
xff[i,]&lt;-0
for(j in 2:3)
{
xff[i,j-1]&lt;-xff[i,j-1]+bn[,j-1]%*%x_out[335+i:(i-L+1),j]
}
}
trading_signal_outsamp&lt;-xff[,1] + xff[,2]     #assemble the trading signal out-of-sample
trade_outsamp&lt;-trading_logdiff(trading_signal_outsamp,price_outsample,.0005)  #compute the performance


The plot in Figure 5 shows the out-of-sample trading signal. Notice that the signal is not nearly as smooth as it was in-sample. Overshooting of the data in some areas is also obviously present. Although the out-of-sample overfitting characteristics of the signal are not horribly suspicious, I would not trust this filter to produce stellar returns in the long run.

Figure 5 : Filter applied to 200 15 minute observations of Yen out-of-sample to produce trading signal (shown in blue)

Following the previous analysis of the mean-squared solution (no customization or regularization), we now proceed to clean up the problem of overfitting that was apparent in the coefficients along with mollifying the noise in the stopband (frequencies after $\pi/6$).  In order to choose the parameters for smoothing and regularization, one approach is to first apply the smoothness parameter first, as this will generally smooth the coefficients while acting as a ‘pre’-regularizer, and then advance to selecting appropriate regularization controls. In looking at the coefficients (Figure 3), we can see that a fair amount of smoothing is necessary, with only a slight touch of decay. To select these two parameters in R, one option is to use the Troikaner optimizer (found here) to find a suitable combination (I have a secret sauce algorithmic approach I developed for iMetrica for choosing optimal combinations of parameters given an extractor $\Gamma$ and a performance indicator, although it’s lengthy (even in GNU C) and cumbersome to use, so I typically prefer the strategy discussed in this tutorial).   In this example, I began by setting the lambda_smooth to .5 and the decay to (.1,.1) along with an expweight smoothness parameter set to 8.5. After viewing the coefficients, it still wasn’t enough smoothness, so I proceeded to add more finally reaching .63, which did the trick. I then chose lambda to balance the effects of the smoothing expweight (lambda is always the last resort tweaking parameter).

lambda_smooth&lt;-0.63
lambda_cross&lt;-0.
lambda_decay&lt;-c(0.119,0.099)
lambda&lt;-9
expweight&lt;-8.5

i_mdfa_obj&lt;-IMDFA_comp(Lag,K,L,lambda,weight_func,Gamma,expweight,cutoff,i1,i2,weight_constraint,
lambda_cross,lambda_decay,lambda_smooth,x,plots,lin_expweight,shift_constraint,grand_mean)

bn&lt;-i_mdfa_obj$i_mdfa$b    #save the filter coefficients

trading_signal&lt;-i_mdfa_obj$xff[,1] + i_mdfa_obj$xff[,2]  #compute the trading signal
trade&lt;-trading_logdiff(trading_signal[L:len],price_insample[L:len],0) #compute the in-sample performance


Figure 6 shows the resulting frequency response function for both explanatory series (Yen in red). Notice that the largest spectral peak found directly before the frequency cutoff at $\pi/6$ is being emphasized and slightly mollified (value near .8 instead of 1.0). The other spectral peaks below $\pi/6$ are also present. For the coefficients, just enough smoothing and decay was applied to keep the lag, cyclical, and correlated structure of the coefficients intact, but now they look much nicer in their smoothed form. The number of freezed degrees of freedom has been reduced to approximately 102.

Figure 6: The frequency response functions and the coefficients after regularization and smoothing have been applied (top). The smoothed coefficients with slight decay at the end (bottom). Number of freezed degrees of freedom is approximately 102 (out of 172).

Along with an improved freezed degrees of freedom and no apparent havoc of overfitting, we apply this filter out-of-sample to the 200 out-of-sample observations in order to verify the improvement in the structure of the filter coefficients (shown below in Figure 7).  Notice the tremendous improvement in the properties of the trading signal (compared with Figure 5). The overshooting of the data has be eliminated and the overall smoothness of the signal has significantly improved. This is due to the fact that we’ve eradicated the presence of overfitting.

Figure 7: Out-of-sample trading signal with regularization.

With all indications of a filter endowed with exactly the characteristics we need for robustness, we now apply the trading signal both in-sample and out of sample to activate the buy/sell trades and see the performance of the trading account in cash value. When the signal crosses below zero, we sell (enter short position) and when the signal rises above zero, we buy (enter long position).

The top plot of Figure 8 is the log price of the Yen for the 15 minute intervals and the dotted lines represent exactly where the trading signal generated trades (crossing zero). The black dotted lines represent a buy (long position) and the blue lines indicate a sell (and short position).  Notice that the signal predicted all the close-to-open jumps for the Yen (in part thanks to the explanatory series). This is exactly what we will be striving for when we add regularization and customization to the filter. The cash account of the trades over the in-sample period is shown below, where transaction costs were set at .05 percent. In-sample, the signal earned roughly 6 percent in 9 trading days and a 76 percent trading success ratio.

Figure 8: In-sample performance of the new filter and the trades that are generated.

Now for the ultimate test to see how well the filter performs in producing a winning trading signal, we applied the filter to the 200 15-minute out-of-sample observation of the Yen and the explanatory series from Jan 18th to February 1st and make trades based on the zero crossing. The results are shown below in Figure 9. The black lines represent the buys and blue lines the sells (shorts). Notice the filter is still able to predict the close-to-open jumps even out-of-sample thanks to the regularization. The filter succumbs to only three tiny losses at less than .08 percent each between observations 160 and 180 and one small loss at the beginning, with an out-of-sample trade success ratio hitting 82 percent and an ROI of just over 4 percent over the 9 day interval.

Figure 9: Out-of-sample performance of the regularized filter on 200 out-of-sample 15 minute returns of the Yen. The filter achieved 4 percent ROI over the 200 observations and an 82 percent trade success ratio.

Compare this with the results achieved in iMetrica using the same MDFA parameter settings. In Figure 10, both the in-sample and out-of-sample performance are shown. The performance is nearly identical.

Figure 10: In-sample and out-of-sample performance of the Yen filter in iMetrica. Nearly identical with performance obtained in R.

#### Example 2

Now we take a stab at producing another trading filter for the Yen, only this time we wish to identify only the lowest frequencies to generate a trading signal that trades less often, only seeking the largest cycles. As with the performance of the previous filter, we still wish to target the frequencies that might be responsible to the large close-to-open variations in the price of Yen. To do this, we select our cutoff to be $\pi/12$ which will effectively keep the largest three spectral peaks intact in the low-pass band of $\Gamma$.

For this new filter, we keep things simple by continuing to use the same regularization parameters chosen in the previous filter as they seemed to produce good results out-of-sample. The $\lambda$ and expweight customization parameters however need to be adjusted to account for the new noise suppression requirements in the stopband and the phase properties in the smaller passband. Thus I increase the smoothing parameter and decreased the timeliness parameter (which only affects the passband) to account for this change. The new frequency response functions and filter coefficients for this smaller lowpass design are shown below in Figure 11. Notice that the second spectral peak is accounted for and only slightly mollified under the new changes. The coefficients still have the noticeable smoothness and decay at the largest lags.

Figure 11: Frequency response functions of the two filters and their corresponding coefficients.

To test the effectiveness of this new lower trading frequency design, we apply the filter coefficients to the 200 out-of-sample observations of the 15-minute Yen log-returns. The performance is shown below in Figure 12. In this filter, we clearly see that the filter still succeeds in predicting correctly the large close-to-open jumps in the price of the Yen. Only three total losses are observed during the 9 day period. The overall performance is not as appealing as the previous filter design as less amount of trades are made, with a near 2 percent ROI and 76 percent trade success ratio. However, this design could fit the priorities for a trader much more sensitive to transaction costs.

Figure 12: Out-of-sample performance of filter with lower cutoff.

#### Conclusion

Verification and cross-validation is important, just as the most interesting man in the world will tell you.

The point of this tutorial was to show some of the main concepts and strategies that I undergo when approaching the problem of building a robust and highly efficient trading signal for any given asset at any frequency. I also wanted to see if I could achieve similar results with the R MDFA package as my iMetrica software package. The results ended up being nearly parallel except for some minor differences. The main points I was attempting to highlight were in first analyzing the periodogram to seek out the important spectral peaks (such as ones associate with close-to-open variations) and to demonstrate how the choice of the cutoff affects the systematic trading.  Here’s a quick recap on good strategies and hacks to keep in mind.

Summary of strategies for building trading signal using MDFA in R:

• As I mentioned before, the periodogram is your best friend. Apply the cutoff directly after any range of spectral peaks that you want to consider. These peaks are what generate the trades.
• Utilize a choice of filter length $L$ no greater than 1/4. Anything larger is unnecessary.
• Begin by computing the filter in the mean-square sense, namely without using any customization or regularization and see exactly what needs to be approved upon by viewing the frequency response functions and coefficients for each explanatory series.  Good performance of the trading signal in-sample (and even out-of-sample in most cases) is meaningless unless the coefficients have solid robust characteristics in both the frequency domain and the lag domain.
• I recommend beginning with tweaking the smoothness customization parameter expweight and the lambda_smooth regularization parameters first. Then proceed with only slight adjustments to the lambda_decay parameters. Finally, as a last resort, the lambda customization. I really never bother to look at lambda_cross. It has seldom helped in any significant manner.  Since the data we are using to target and build trading signals are log-returns, no need to ever bother with i1 and i2. Those are for the truly advanced and patient signal extractors, and should only be left for those endowed with iMetrica 😉

If you have any questions, or would like the high-frequency Yen data I used in these examples, feel free to contact me and I’ll send them to you. Until next time, happy extracting!

# High-Frequency Financial Trading with Multivariate Direct Filtering I: FOREX and Futures

Animation 1: Click to see animation of the Japanese Yen filter in action on 164 hourly out-of-sample observations.

I recently acquired over 300 GBs of financial data that includes tick data for over 7000 financial assets traded on multiple markets for the past 5 years up until February 1st 2013. This USB drive packed with nearly every detail of world financial markets coupled with iMetrica gave me an opportunity to explore at any fashion to my desire the ability of multivariate direct filtering to produce high performance financial trading signals on nearly any high-frequency. Let me begin this article with saying that I am more than ecstatic with the results, as I hope you will too after reading this article.  In this first article in a series of high-frequency trading with MDFA and iMetrica that I plan to write, I provide some initial experiments with building and extracting financial trading signals for high-frequency intraday observations on foreign exchange (FOREX) data, and by high-frequency in the context of this article, I mean higher frequencies than the daily log-returns I’ve been working with in my previous articles. In the first part of this high-frequency series, I begin by exploring hourly, 30 minute, and 15 minute log-returns, and test different strategies, mostly using low-pass and the recently introduced multi-bandpass (MBP) filter to deduce the best approach to tackle the problem of building successful trading signals in higher frequency data.

In my previous articles, I was working uniquely with daily log-return data from different time spans from a year to a year and a half. This enabled the in-sample period of computing the filter coefficients for the signal extraction to include all the most recent annual phases and seasons of markets, from holiday effects, to the transitioning period of August to September that is regularly highly influential on stock market prices and commodities as trading volume increases a significant amount. One immediate question that is raised in migrating to higher-frequency intraday data is what kind of in-sample/out-of-sample time spans should be used to compute the filter in-sample and then for how long do we apply the filter out-of-sample to produce the trades? Another question that is raised with intraday data is how do we account for the close-to-open variation in price? Certainly, after close, the after-hour bids and asks will force a jump into the next trading day. How do we deal with this jump in an optimal manner? As the observation frequency gets higher, say from one hour to 30 minutes, this close-to-open jump/fall should most likely be larger. I will start by saying that, as you will see in the results of this article, with a clever choice of the extractor $\Gamma$ and explanatory series, MDFA can handle these jumps beautifully (both aesthetically and financially). In fact, I would go so far as to say that the MDFA does a superb job in predicting the overnight variation.

One advantage of building trading signals for higher intraday frequencies is that the signals produce trading strategies that are immediately actionable. Namely one can act upon a signal to enter a long or short position immediately when they happen. In building trading signals for the daily log-return, this is not the case since the observations are not actionable points, namely the log difference of today’s ending price with yesterday’s ending price are produced after hours and thus not actionable during open market hours and only actionable the next trading day. Thus trading on intraday observations can lead to better efficiency in trading.

In this first installment in my series on high-frequency financial trading using multivariate direct filtering in iMetrica, I consider building trading signals on hourly returns of foreign exchange currencies. I’ve received a few requests after my recent articles on the Frequency Effect in seeing iMetrica and MDFA in action on the FOREX sector. So to satisfy those curiosities, I give a series of (financially) satisfying and exciting results in combining MDFA and the FOREX. I won’t give all my secretes away into building these signals (as that would of course wipe out my competitive advantage), but I will give some of the parameters and strategies used so any courageously curious reader may try them at home (or the office). In the conclusion, I give a series of even more tricks and hacks.  The results below speak for themselves  So without further ado, let the games begin.

#### Japanese Yen

Frequency: One hour returns
30 day out-of-sample ROI: 12 percent
Trade success ratio: 92 percent

Yen Filter Parameters: $\lambda$ = 9.2 $\alpha$ = 13.2, $\omega_0 = \pi/5$
Regularization: smooth = .918, decay = .139, decay2 = .79, cross = 0

In the first experiment, I consider hourly log-returns of a ETF index that mimics the Japanese Yen called FXY. As for one of the explanatory series, I consider the hourly log-returns of the price of GOLD which is traded on NASDAQ. The out-of-sample results of the trading signal built using a low-pass filter and the parameters above are shown in Figure 1.  The in-sample trading signal (left of cyan line) was built using 400 hourly observations of the Yen during US market hours dating back to 1 October 2012. The filter was then applied to the out-of-sample data for 180 hours, roughly 30 trading days up until Friday, 1 February 2013.

Figure 1: Out-of-sample results for the Japanese Yen. The in-sample trading signal was built using 400 hourly observations of the Yen during US market hours dating back to October 1st, 2012. The out-of-sample portion passed the cyan line is on 180 hourly observations, about 30 trading days.

This beauty of this filter is that it yields a trading signal exhibiting all the characteristics that one should strive for in building a robust and successful trading filter.

1. Consistency: The in-sample portion of the filter performs exactly as it does out-of-sample (after cyan line) in both trade success ratio and systematic trading performance.
2. Dropdowns: One small dropdown out-of-sample for a loss of only .8 percent (nearly the cost of the transaction).
3. Detects the cycles as it should: Although the filter is not able to pinpoint with perfect accuracy every local small upturn during the descent of the Yen against the dollar, it does detect them nonetheless and knows when to sell at their peaks (the magenta lines).
4. Self-correction: What I love about a robust filter is that it will tend to self-correct itself very quickly to minimize a loss in an erroneous trade. Notice how it did this in the second series of buy-sell transactions during the only loss out-of-sample. The filter detects momentum but quickly sold right before the ensuing downfall. My intuition is that only frequency-based methods such as the MDFA are able to achieve this consistently. This is the sign of a skillfully smart filter.

The coefficients for this Yen filter are shown below. Notice the smoothness of the coefficients from applying the heavy smooth regularization and the strong decay at the very end.  This is exactly the type of smooth/decay combo that one should desire. There is some obvious correlation between the first and second explanatory series in the first 30 lags or so as well. The third explanatory series seems to not provide much support until the middle lags .

Figure 2: Coefficients of the Yen filter. Here we use three different explanatory series to extract the trading signal.

One of the first things that I always recommend doing when first attempting to build a trading signal is to take a glance at the periodogram. Figure 2 shows the periodogram of the log-return data of the Japanese Yen over 580 hours.  Compare this with the periodogram of the same asset using log-returns of daily data over 580 days, shown in Figure 3.  Notice the much larger prominent spectral peaks at the lower frequencies in the daily log-return data. These prominent spectral peaks renders multibandpass filters much more advantageous and to use as we can take advantage of them by placing a band-pass filter directly over them to extract that particular frequency (see my article on multibandpass filters). However, in the hourly data, we don’t see any obvious spectral peaks to consider, thus I chose a low-pass filter and set the cutoff frequency at $\pi/5$, a standard choice, and good place to begin.

Figure 3: Periodogram of hourly log-returns of the Japanese Yen over 580 hours.

Figure 4: Periodogram of Japanese Yen using 580 daily log-return observations. Many more spectral peaks are present in the lower frequencies.

#### Japanese Yen

Frequency: 15 minute returns
7 day out-of-sample ROI: 5 percent
Trade success ratio: 82 percent

Yen Filter Parameters: $\lambda$ = 3.7 $\alpha$ = 13, $\omega_0 = \pi/9$
Regularizationsmooth = .90, decay = .11, decay2 = .09, cross = 0

In the next trading experiment, I consider the Japanese Yen again, only this time I look at trading on even high-frequency log-return data than before, namely on 15 minute log-returns of the Yen from the opening bell to market close.  This presents slightly new challenges than before as the close-to-open jumps are much larger than before, but these larger jumps do not necessarily pose problems for the MDFA. In fact, I look to exploit these and take advantage to gain profit by predicting the direction of the jump.  For this higher frequency experiment, I considered 350 15-minute in-sample observations to build and optimize the trading signal, and then applied it over the span of 200 15-minute out-of-sample observations. This produced the results shown in the Figure 5 below. Out of 17 total trades out-of-sample, there were only 3 small losses each less than .5 percent drops and thus 14 gains during the 200 15-minute out-of-sample time period.  The beauty of this filter is its impeccable ability to predict the close-to-open jump in the price of the Yen. Over the nearly 7 day trading span, it was able to correctly deduce whether to buy or short-sell before market close on every single trading day change. In the figure below, the four largest close-to-open variation in Yen price is marked with a “D” and you can clearly see how well the signal was able to correctly deduce a short-sell before market close. This is also consistent with the in-sample performance as well, where you can notice the buys and/or short-sells at the largest close-to-open jumps (notice the large gain in the in-sample period right before the out-of-sample period begins, when the Yen jumped over 1 percent over night.  This performance is most likely aided by the explanatory time series I used for helping predict the close-to-open variation in the price of the Yen. In this example, I only used two explanatory series (the price of Yen, and another closely related to the Yen).

Figure 5: Out-of-sample performance of the Japanese Yen filter on 15 minute log-return data.

We look at the filter transfer functions to see what frequencies they are being privileged in the construction of the filter. Notice that some noise leaks out passed the frequency cutoff at $\pi/9$, but this is typically normal and a non-issue. I had to balance for both timeliness and smoothness in this filter using both the customization parameters $\lambda$ and $\alpha$. Not much at frequency 0 is emphasized, with more emphasis stemming from the large spectral peak found right at $\pi/9$.

Figure 6: The filter transfer functions.

#### British Pound

Frequency: 30 minute returns
14 day out-of-sample ROI: 4 percent
Trade success ratio: 76 percent

British Pound Filter Parameters: $\lambda$ = 5 $\alpha$ = 15, $\omega_0 = \pi/9$
Regularizationsmooth = .109, decay = .165, decay2 = .19, cross = 0

In this example we consider the frequency of the data to 30 minute returns and attempt to build a robust trading signal for a derivative of the British Pound (BP) on this higher frequency. Instead of using the cash value of the BP, I use 30 minute returns of the BP Futures contract expiring in March (BPH3). Although I don’t have access to tick data from the FOREX, I do have tick data from GLOBEX for the past 5 years.  Thus the futures series won’t be an exact replication of the cash price series of the BP, but it should be quite close due to very low interest rates.

The results of the out-of-sample performance of the BP futures filter are shown in Figure 7. I constructed the filter using an initial in-sample size of 390 30 minute returns dating back to 1 December 2012. After pinpointing a frequency cutoff in the frequency domain for the $\Gamma$ that yielded decent trading results in-sample, I then proceeded to optimize the filter in-sample on smoothness and regularization to achieve similar out-of-sample performance. Applying the resulting filter out-of-sample on 168 30-minute log-return observations of the BP futures series along with 3 explanatory series, I get the results shown below. There were 13 trades made and 10 of them were successful. Notice that the filter does an exquisite job at triggering trades near local optimums associated with the frequencies inside the cutoff of the filter.

Figure 7: The out-of-sample results of the British Pound using 30-minute return data.

In looking at the coefficients of the filter for each series in the extraction, we can clearly see the effects of the regularization: the smoothness of the coefficients the fast decay at the very end. Notice that I never really apply any cross regularization to stress the latitudinal likeliness between the 3 explanatory series as I feel this would detract from the predicting advantages brought by the explanatory series that I used.

Figure 8: The coefficients for the 3 explanatory series of the BP futures,

#### Euro

Frequency: 30 min returns
30 day out-of-sample ROI: 4 percent
Trade success ratio: 71 percent

Euro Filter Parameters: $\lambda$ = 0, $\alpha$ = 6.4, $\omega_0 = \pi/9$
Regularizationsmooth = .85, decay = .27, decay2 = .12, cross = .001

Continuing with the 30 minute frequency of log-returns, in this example I build a trading signal for the Euro futures contract with expiration on 18 March 2013 (UROH3 on the GLOBEX). My in-sample period, being the same as my previous experiment, is from 1 December 2012 to 4 January 2013 on 30 minute returns using three explanatory time series.  In this example, after inspecting the periodogram, I decided upon a low-pass filter with a frequency cutoff of $\pi/9$. After optimizing the customization and applying the filter to one month of 30 minute frequency return data out-of-sample (month of January 2013, after cyan line) we see the performance is akin to the performance in-sample, exactly what one strives for. This is due primarily to the heavy regularization of the filter coefficients involved. Only four very small losses of less than .02 percent are suffered during the out-of-sample span that includes 10 successful trades, with the losses only due to the transaction costs. Without transaction costs, there is only one loss suffered at the very beginning of the out-of-sample period.

Figure 9 : Out-of-sample performance on the 30-min log-returns of Euro futures contract UROH3.

As in the first example using hourly returns, this filter again exhibits the desired characteristics of a robust and high-performing financial trading filter. Notice the out-of-sample performance behaves akin to the in-sample performance, where large upswings and downswings are pinpointed to high-accuracy. In fact, this is where the filter performs best during these periods. No need for taking advantage of a multibandpass filter here, all the profitable trading frequencies are found at less than $\pi/9$.  Just as with the previous two experiments with the Yen and the British Pound, notice that the filter cleanly predicts the close-to-open variation (jump or drop) in the futures value and buys or sells as needed.  This can be seen from many of the large jumps in the out-of-sample period (after cyan line).

One reason why these trading signals perform so well is due to their approximation power of the symmetric filter. In comparing the trading signal (green) with a high-order approximation of the symmetric filter (gray line) transfer function $\Gamma$ shown in Figure 10, we see that trading signal does an outstanding job at approximating the symmetric filter uniformly. Even at the latest observation (the right most point), the asymmetric filter hones in on the symmetric signal (gray line) with near perfection. Most importantly, the signal crosses zero almost exactly where required.  This is exactly what you want when building a high-performing trading signal.

Figure 10: Plot of approximation of the real-time trading signal for UROH3 with a high order approximation of the symmetric filter transfer function.

In looking at the periodogram of the log-return data and the output trading signal differences (colored in blue), we see that the majority of the frequencies were accounted for as expected in comparing the signal with the symmetric signal. Only an inconsequential amount of noise leakage passed the frequency cutoff of $\pi/9$ is found.  Notice the larger trading frequencies, the more prominent spectral peaks, are located just after $\pi/6$. These could be taken into account with a smart multibandpass filter in order to manifest even more trades, but I wanted to keep things simple for my first trials with high-frequency foreign exchange data.  I’m quite content with the results that I’ve achieved so far.

Figure 11: Comparing the periodogram of the signal with the log-return data.

#### Conclusion

I must admit, at first I was a bit skeptical of the effectiveness that the MDFA would have in building any sort of successful trading signal for FOREX/GLOBEX high frequency data. I always considered the FOREX market rather ‘efficient’ due to the fact that it receives one of the highest trading volumes in the world.  Most strategies that supposedly work well on high-frequency FOREX all seem to use some form of technical analysis or charting (techniques I’m particularly not very fond of), most of which are purely time-domain based. The direct filter approach is a completely different beast, utilizing a transformation into the frequency domain and a ‘bending and warping’ of the metric space for the filter coefficients to extract a signal within the noise that is the log-return data of financial assets.  For the MDFA to be very effective at building timely trading signals, the log-returns of the asset need to diverge from white noise a bit, giving room for pinpointing intrinsically important cycles in the data. However, after weeks of experimenting, I have discovered that building financial trading signals using MDFA and iMetrica on FOREX data is as rewarding as any other.

As my confidence has now been bolstered and amplified even more after my experience with building financial trading signals with MDFA and iMetrica for high-frequency data on foreign exchange log-returns at nearly any frequency, I’d be willing to engage in a friendly competition with anyone out there who is certain that they can build better trading strategies using time domain based methods such as technical analysis or any other statistical arbitrage technique.  I strongly believe these frequency based methods are the way to go, and the new wave in financial trading.  But it takes experience and a good eye for the frequency domain and periodograms to get used to. I haven’t seen many trading benchmarks that utilize other types of strategies, but i’m willing to bet that they are not as consistent as these results using this large of an out-of-sample to in-sample ratio (the ratios in these experiments were between .50 and .80).  If anyone would like to take me up on my offer for a friendly competition (or know anyone that would), feel free to contact me.

After working with a multitude of different financial time series and building many different types of filters, I have come to the point where I can almost eyeball many of the filter parameter choices including the most important ones being the extractor $\Gamma$ along with the regularization parameters, without resorting to time consuming, and many times inconsistent, optimization routines.  Thanks to iMetrica, transitioning from visualizing the periodogram to the transfer functions and to the filter coefficients and back to the time domain to compare with the approximate symmetric filter in order to gauge parameter choices is an easy task, and an imperative one if one wants to build successful trading signals using MDFA.

Here are some overall tips and tricks to build your own high performance trading signals on high-frequency data at home:

• Pay close attention to the periodogram. This is your best friend in choosing the extractor $\Gamma$. The best performing signals are not the ones that trade often, but trade on the most important frequencies found in the data. Not all frequencies are created equal. This is true when building either low-pass or multibandpass frequencies.
• When tweaking customization, always begin with $\alpha$, the parameter for smoothness. $\lambda$ for timeliness should be the last resort. In fact, this parameter will most likely be next to useless due to the fact that the log-return of financial data is stationary. You probably won’t ever need it.
• You don’t need many explanatory series. Like most things in life, quality is superior to quantity. Using the log-return data of the asset you’re trading along with one and maybe two explanatory series that somewhat correlate with the financial asset you’re trading on is sufficient. Anymore than that is ridiculous overkill, probably leading to over-fitting (even the power of regularization at your fingertips won’t help you).

In my next article, I will continue with even more high-frequency trading strategies with the MDFA and iMetrica where I will engage in the sector of Funds and ETFs. If any curious reader would like even more advice/hints/comments on how to build these trading signals on high-frequency data for the FOREX (or the coefficients built in these examples), feel free to get in contact with me via email. I’ll be happy to help.

Happy extracting!