The attention of researchers in the eld of signal analysis has recently been captured by several kinds of reassignment techniques. Among the reallocation methods the Synchrosqueezing approach maps a continuous wavelet transform from the time-scale to the time-frequency plane, allowing a much more de nite and consistent representation of the frequency content of a signal. Its mathematical foundations prove that the Synchrosqueezing Wavelet Transform is directly related to the Empirical Mode Decomposition - EMD, since both methods allow the decomposition of a signal having nite energy in its intrinsic mode functions, each having time-varying frequency and amplitude. We develop a fast1 C++ implementation of the Synchrosqueezed Wavelet Transform SST suitable for the synchronic extrusion of instantaneous frequency information from univariate time series. Such module, totally con gurable and adaptable to input datasets having di erent statistical properties, can be coupled with a predictor system to the purpose of real value forecasting. We project such a composite application and plan to research the forecasting accuracy achievable, using di erent types of non parametric statistical estimators or neural regressors.
In the last few years signal decomposition has been often employed to improve
the forecasting capabilities of predictor systems. Among all the time-frequency
1 ISO/IEC 14882:2014 conformant C++ implementation compiled with speed
maximization optimization, inline functions expansion and intrinsic functions enabled,
target machine x64.
analysis models suitable to be applied to univariate data series, one of the most
simple yet e ective method is the Empirical Mode Decomposition, researched in
the late 90s by Huang et al. [
EMD is an empirical method, based on a sifting procedure aimed to identi
cate a signal's components, each of them featured by both amplitude and phase
modulation. Sometimes it is compared to other analysis methods, such as Fourier
Transforms or Wavelet decomposition. This approach is suitable to study signals
which exhibits non-linearities and non-stationarity properties. The procedure
allows signal decomposition in completely and closely orthogonal basis which may
be expressed as the sum of the components of the form A(t)cos( (t)), plus a
residual series. Such decomposition can be considered as a generalized Fourier
series, in which the components phases and amplitudes are not constrained to
be constant. Interest in such approach motivated Wu and Huang [
In recent years, Daubechies et al. [
The authors are planning to develop and develop a prediction system suitable for the analysis of streaming datasets such as the search engine query trends. We develop a fast C++ implementation of the Synchrosqueezing Wavelet Transform (see footnote 1), such system being con gurable according to the speci c properties of di erent types of input datasets.
This paper is organized as follows: Section 2 provides the theoretical foundations of the SST, section 3 provides a description of several aspects of the implementation, referencing the features of the SST algorithm. Also, it depicts plannings for the most important machine learning aspects of the forecasting system. The last section provides the authors conclusions and portrays further research directions. 2
A Continuous Wavelet Tranform - CWT allows the time-varying analysis of the frequency content of a signal f (t) having nite energy. Choosing an appropriate mother wavelet function (t) such that its Fourier transform ^( ) = 0; < 0, (hence a complex wavelet), the CWT of f (t) is de ned as a series of parametric convolutions of the signal and the complex conjugate of scaled versions of the wavelet function. After the reassignment of the CWT, the vector of frequencies ! can be partitioned in equivalence classes having a chosen bandwidth, and the reconstruction is a piecewise performed function, thus creating the set of intrinsic mode functions mi(t) meant to be the original series components. From each mi(t), instantaneous amplitudes and phases can be recovered. Our actual position is to explore the possible machine learning approaches based on an input series preprocessing operation performed via the SST.
For the scope of the present demonstration we develop a fast C++ implementation of the SST algorithm, in which the complex wavelet function (t) can be selected via parametric polymorphism. In order to be able to e ectively use the system for prediction purposes, given the evident heaviness of the stepwise calculations, e ciency is considered paramount. Also, considering the structure of the CWT algorithm, hyperthreading is possible and, depending on the target machine features, compiled code could run even more e ciently. Note that a concurrent implementation of the CWT could be feasible even implementing it using alternative algorithms; as such, a parallel implementation of the CWT portion of the SST is already under development.
Several search engine query trends series are downloaded for automated testing and debugging purposes. We perform extensive analysis and synthesis tests on the gathered datasets in order to compare the reconstruction accuracies. Finally we are going to choose the wavelet function which gives the lower reconstruction error results. The system must be able to read streaming datasets such as the above mentioned search trends chunked with a xed length window, the latter sliding from the beginning of the series to the last known data sample.
Following the obtained empirical results we can consider feasible the creation of a prediction system based on learning the features extracted by a SST preprocessing module.
IMFs can be used as a training set for a prediction system such as a group of cooperating neural networks, hence forecasting the one step ahead value of each IMF using neural regression and summing up all the di erent contributions to synthesize the nal real value forecast. Also, a di erent approach could be to train a set of Nf detached perceptrons to forecast the one-step ahead instantaneous frequencies of the series.
Let us recall that a Synchrosqueezed Wavelet Tranforms outputs a matrix Tf (!; b) of size fNf ; N g, where Nf is the size of the frequencies vector F , and N is the size of an unpadded (i.e. raw) input series. Hence the number of independent neural regressors would be con ned to be Nf . Each perceptron atom would be responsible to learn the !f (a; b) rate of change in time direction, i.e. @@t !f (a; b), in order to be able to analytically nd the one-step ahead !f (a; b+1) value having empirically read the previous !f (a; b).
Having presented di erent possibile development directions, it should be now clear that the use of the SST as an assisting module to a time series prediction system might introduce many innovation features in the machine learning eld. We are going to explore all the above mentioned implementation paths and test them using the Bitcoin search engine queries datasets. Also, since it is possible to nd in the literature several benchmark methods to the Bitcoin price prediction, in order to focus on the estimate of the forecasting accuracy of the system, we plan to test the latter using the Bitcoin price series (the hourly close data of the Bitcoin - US Dollar exchange rates - BTCUSD).
The Synchrosqueezing Wavelet Transform allows an adaptive signal decomposition on time-frequency plane and it is useful for analyzing multicomponent signals. After a decomposition, signals can be written as a sum of intrinsic modes, this means that the SST can be used for prediction purposes. It is our opinion that such transform paves the way to novel machine learning approaches that focus on learning the rates of change of time series instead of tting their values; such approaches could easily become a very active area of research in the next few years. Advantages to have such a type of system, totally con gurable and exible, are multiple. It is a completely dedicated implementation, customizable and reliable enough to be adapted to every kind of input dataset. Furthermore it allows to change or modify the algorithm code according to speci c requirements. Independent graphics modules help us to detect errors and misconceptions during the development phase as well as when testing the selected datasets for research purposes. The combination of the SST and a prediction system might open the path to di erent and novel research directions. Actually we are investigating about the applicability of the system in several interesting domains: In ation forecasting, smart cities energy consumption, in uenza prediction, economic uctuations forecasts.
Acknowledgments. We would like to express gratitude to Eugene Brevdo for his contribution to determine several theoretical misconceptions related to the SST, allowing us to nd both poisonous and subtle bugs on the rst implementation of the system preprocessing module.