Non-Oscillatory Pattern Learning
for Non-Stationary Signals
This paper proposes a novel non-oscillatory pattern (NOP) learning scheme for several oscillatory data analysis problems including signal decomposition, super-resolution, and signal sub-sampling. To the best of our knowledge, the proposed NOP is the first algorithm for these problems with fully non-stationary oscillatory data with close and crossover frequencies, and general oscillatory patterns. Even in stationary cases with trigonometric patterns, numerical examples show that NOP admits competitive or better performance in terms of accuracy and robustness than several state-of-the-art algorithms.
Non-Oscillatory Pattern Learning
for Non-Stationary Signals
Jieren Xu, Yitong Li, David Dunson, Ingrid Daubechies, Haizhao Yang Duke University, National University of Singapore jieren.xu,yitong.li,dunson,firstname.lastname@example.org,email@example.com
noticebox[b]Preprint. Work in progress.\end@float
This paper concerns oscillatory data defined on a time domain of the form:
Here and are the latent instantaneous amplitude and phase functions of the th component, which are assumed to be smooth over time. The derivative of a phase function is called an instantaneous frequency function and denoted as . is a periodic shape function with periodicity one satisfying that and a unit -norm on .
Oscillatory data in Model (1) arises in numerous applications HauBio2 (); Pinheiro2012175 (); 7042968 (); Crystal (); LuWirthYang:2016 (); Eng2 (); ME (); 0957-0233-28-3-035102 (); Canvas (); Canvas2 (); GeoReview (); SSCT (); 977903 (); 4685952 (); MUSIC (); SuperResolution:Candes (); gruber1997statistical (); burg1972relationship (); roy1989esprit () and data analysis of this kind has been an active research field for decades. Usually only (and sometimes ) is available and the goal is to estimate , (or ), and from . Hence, this is a general problem including and generalizing sub-problems like adaptive time-frequency analysis Auger1995 (); Daubechies1996 (); YANG2017 (), empirical mode decomposition Huang1998 (); Wu2009 (); Wu2009EEMD (), super-resolution MUSIC (); SuperResolution:Candes (); gruber1997statistical (); burg1972relationship (); roy1989esprit (), pattern recognition zhu2013locally (), etc. In spite of many successful algorithms for solving these sub-problems, to the best of our knowledge, there is no existing algorithm fulfilling the ultimate goal of estimating , (or ), and when is fully non-stationary with close and crossover frequencies, and general shape functions. Many existing algorithms require high sampling rate for better accuracy and robustness, which is not practical due to the limit of battery capacity of mobile devices that collect oscillatory data, e.g. portable health monitors.
Fig. 1 shows a synthetic example of Model (1) when , , , , , and . is in fact a superposition of infinitely many deformed planewaves due to the Fourier series expansion of shape functions. Hence, close and crossover frequencies are unavoidable. In terms of frequencies, due to Heisenberg uncertainty principle, time-frequency analysis methods Auger1995 (); Daubechies1996 (); YANG2017 () are not able to estimate these instantaneous frequencies; due to the fully non-stationary nature of , existing super-resolution methods MUSIC (); SuperResolution:Candes (); gruber1997statistical (); burg1972relationship (); roy1989esprit () are not able to estimate frequencies . Moreover, existing GP based methods wilson2013gaussian (); pan2017prediction (); parra2017spectral (); ulrich2015gp (); quia2010sparse () cannot produce reliable source separation in Model (1). Non-compact support of in the Fourier domain (Fig. 2(b)) creates particular challenges. Some other regression methods have also been designed to estimate shape functions xu2018recursive (); MMD (); FMMD (), but they assume that phase functions are known.
In order to solve the challenge inverse problem implicit to Model (1), it is natural to choose priors within a Bayesian model for the latent components. While there are many possible stochastic process choices, Guassian Processes (GPs) are appealing in enforcing smoothness providing easy inclusion of prior information and leading to computational tractability. This paper proposes a non-oscillatory pattern (NOP) learning scheme, the first framework that can estimate , (or ), and simultaneously from based on GP. NOP repeatedly applies a two-stage iteration until convergence. In the first stage, fixing rough estimations of shapes, a novel non-stationary GP regression is proposed to estimate amplitude and phase functions. A novel set of auxiliary points, referred to as the pattern inducing points are introduced for this purpose. As oppose to traditional stationary kernels, non-stationary kernel functions in our GP model approximately transform a non-stationary data analysis problem into a stationary one, greatly reduce the regression difficulty. Furthermore, we propose to embed the information of rough shape estimation into the GP regression model, reducing a deep GP regression problem zhu2013locally (); damianou2013deep (); dai2015variational () with a composition of two latent variables (e.g. the composition of a shape function and a phase function) into a simpler problem with only one latent variable, the phase functions. This significantly reduce the computational cost and difficulty in the regression.
In the second stage, fixing rough estimations of amplitudes and phases, a non-stationary GP regression is proposed or other interative regression methods in FMMD (); xu2018recursive (); 1DSSWPT (); MMD () is applied to estimate shapes. The main difference of the proposed non-stationary GP regression in the second stage to existing methods is that, the variance of the point estimate of shape functions is simple to derive.
2 Estimation of the instantaneous information
In this section, we assume the shape functions are known and aim at estimating the phase and amplitude functions and , respectively.
2.1 Main ideas
We start with a simple case when the signal . Assume a non-stationary GP , and a stationary GP , where is the automatic relevance determination (ARD) rasmussen2006gaussian () squared exponential (SE) kernel:
with kernel parameters and . Now we consider the phase function as the latent variable of the GP , and claim the resulting GP is periodic and stationary in the domain . This is because , by plugging in the new input , the corresponding output is , i.e., any point lies on the curve . seems to be a deep/hierarchical GP model dai2015variational (), but in fact this ‘unwarping’ process directly removes the first GP layer while keeping the second layer stationary. This is a crucial idea for success of our NOP. And we write the one layer GP with input as for .
Next, we introduce a point estimate of the phase function as a latent variable. Suppose 111We will use bold font for vectors. is the observations (contaminated by white-noise with standard deviation ) of at time locations . Denote and . As can be seen, a direct inference of the latent , even when is given, is not trivial. This motivates us to introduce a new set of auxiliary points for the GP to retrieve the latent input variable , and to reveal the underlying patterns for Model 1 when is not known. We refer to this novel set of auxiliary points as the pattern inducing points, denoted as , where is the input variable and is the respective output evaluated at the inducing locations . Usually we uniformly discretize and fix at phase locations .
As for sparse inducing points snelson2006sparse (); titsias2009variational (), the pattern inducing points are treated as variational parameters, and can be updated. They possess the benefits of both sparse inducing points and training points rasmussen2006gaussian (), but aim to reveal the true underlying pattern of Model 1 and are endowed with specific meanings. The intuition behind the pattern inducing points is that they serve as certain landmarks, shown as the green dots (input as the green triangles) in Fig. 2, of the true underlying pattern (shaded green line). The latent input (red right triangle in Fig. 2) can drive the dataset (red dots in Fig. 2) across the phase domain (in Fig. 2(a) from (top) the initial wrong to (middle) to (bottom) the correct ), to match the underlying pattern (in green) by these landmarks. The that matches the dataset with the landmark points is the correct since for . So this process is also referred to as a pattern driven method for Model 1. Note that setting of with the SE kernel, is much more stable than with the periodic kernel rasmussen2006gaussian (), which is highly non-convex and can be trapped at many local minima.
Let be the variational distribution to approximate the posterior distribution of the inducing variable . For the purpose of computational efficiency, we adopt , where is the mean of and the variance matrix of is diagonal. In this section since has been observed, and . Given , the latent input that is encoded in the GP kernel can start the data-pattern matching process by the following standard GP setting as when auxiliary/training points are set. Conditioning on and , the likelihood of becomes
By further marginalizing out and out, we have
Here, is an identity matrix of size , is an covariance matrix, 222Denote as the th entry of a vector, and as the th entry of a matrix. for , and similarly for , . Hence, a point estimate of the latent variable conditioning on can be computed by Bayesian approach using Eqn. (4).
In the case of multi-components as in Model (1), we denote , , and for the th component, and . Assuming the independence among variables of the same type, e.g., among , among , etc., Eqn. (4) can be generalized to
where consists of latent variables as the th column, , , , and are defined for covariance matrices of the th component similarly to the case of one component.
where has colums .
To accelerate the computation, instead of applying the Bayesian approach with Eqn. (6), we directly fit the mean value of the Gaussian process with the observations via a least square (LS) problem:
where has been encoded in the kernel matrices in the minimization objective function.
2.2 Adaptive local estimation
The optimization problem in (7) is non-convex and there is no strong prior of and to provide a good initialization. Note that in most applications, and usually smoothly vary in time. This motivates us to generate signal patches and locally parametrize the phase and amplitude functions with low-degree polynomials in Eqn. (7). Within each patch, the non-stationary signal becomes much more stationary. Hence, our numerical experiments show that degree is sufficiently good.
For each observation path (also denoted as ), let and be the matrices consisting of the coefficients of the -degree polynomials representing the amplitude and phase functions, respectively. Then we can specify the amplitude and phase functions with and from the following LS problem:
where is the entry-wise dot product, and has been absorbed in the covariance matrices.
Once the amplitude and phase for the signal patch have been specified, they provide a local point estimate of the amplitude and phase functions, which can be used to obtain a global point estimate via a robust curve fitting algorithm garcia2010robust (); garcia2011fast (). A standard moving average can be applied to estimate the variance of the point estimate.
3 Estimation of shape functions
In this section, we assume the phase and amplitude functions and are known and estimate the shape functions . We do not aim at closed formulas for . As introduced in Section 2, it is sufficient to estimate the pattern inducing variables and to represent .
When amplitude and phase functions are given, shape function estimation methods have been studied previously in FMMD (); xu2018recursive (); 1DSSWPT (); MMD (). These methods achieves high accuracy when amplitude and phase function estimation is close to the ground truth. There is no quantitative criteria to measure how well the shape function estimation performs when the amplitude and phase function estimation is not very good. This motivates us to apply variational inference, following titsias2009variational (), that explicitly expresses the distribution of the pattern inducing variables in a joint form. Please refer to Supplemental Material (SM) Section A and B for details of these three well-established approaches.
4 Overview of NOP
NOP repeatedly applies a two-stage iteration until convergence. We have introduced the first stage in Section 2: given rough estimations of shapes, a non-stationary GP regression is applied to estimate amplitude and phase functions. The second stage has been introduced in Section 3: given rough estimations of amplitudes and phases, several regression methods can be adopted to estimate shapes. Hence, a complete algorithm description can be summarized in Algorithm 1 below. The respective prediction formulation of Algorithm 1 is derived in SM Section C.
In many applications, e.g. ECG and photoplethysmogram data analysis, heuristic properties of the physical system are available and we know the rough range of instantaneous frequencies . Hence, we can apply a band-pass filter to , and then estimate and of in a certain frequency band using traditional time-frequency analysis methods Auger1995 (); Daubechies1996 (); YANG2017 () to initialize and following the method in 1DSSWPT ().
Another simple initialization is to let . Since we adopt local patch analysis in Section 2, components after Fourier series expansion , , become approximately orthogonal to each other in a short time domain. Hence, the LS in (7) is able to recover the amplitude and phase functions corresponding to since they usually have the largest magnitude.
The LS problems are non-convex and hence we cannot guarantee convergence to the global minimizer. However, the iterative scheme seems to provide very good results and the algorithm generally converges after only a few iterations. The good performance of NOP might come from the fact that once the amplitude and phase function estimates are roughly good (might not be the global minimizer of LS problems), the shape estimation step can quickly provide very good estimation of shape functions, which can be guaranteed if we adopt methods in FMMD (); xu2018recursive (); MMD (). The global convergence analysis would be an interesting future work.
In this section, we provide numerical examples to demonstrate the performance of NOP, especially in the case of super-resolution and adaptive time-frequency analysis. LS problems in all examples are solved by Adam goodfellow2016deep () aiming at better local minimizers. We choose degree-(or degree- when specified) polynomials to approximate local amplitude and phase functions in these LS problems. The hyperparameters of NOP are set as follows: noise level , , and . In the local patch analysis, we generate signal patches such that each patch contains approximately to periods. In the tests for super-resolution, we repeat the same test with noise realizations for the purpose of using the expectation and variance of estimation error to measure the performance of different algorithms.
5.1 Super-resolution spectral estimation
There has been substantial research for the super-resolution problem that aims at estimating time-invariant amplitudes and frequencies in a signal with , , and are very close. Among many possible choices, the baseline models might be MUSIC schmidt1986multiple (), ME burg1972relationship (); georgiou2001spectral (); georgiou2002spectral (), and ESPRIT roy1989esprit (). Hence, we will compare NOP with these methods333Code from http://people.ece.umn.edu/~georgiou/files/HRTSA/SpecAn.html. to show the advantages of NOP. Although the Fourier transform usually fails schmidt1986multiple () to identify and , we use its results as the initialization for NOP.
|(a) Clean signal||(b)||(c)||(d)|
Accuracy and robustness with different spectral gaps
In this experiment, we use , where the two instantiations of / are / (red lines in Fig. 3) and / as in Fig. 1 (pink lines in Fig. 3). Here and , varies from to , and different noise variance are . The sampling rate is Hz and the number of samples is in this example. Fig. 3 shows the frequency estimation accuracy of NOP, MUSIC, ESPRIT, and ME. As we can see, NOP achieves machine accuracy in the noiseless case and is much more accurate than other methods in all noisy cases. It also reaches an approximate same accuracy for the trigonometric (red) and shaped(pink) instantiations in and .
Accuracy and robustness with different sampling rates
In this experiment, we set with , , and . The sampling rate of this signal is still Hz and the numbers of samples are , , , and to generate four sets of test data. There are two different kinds of noise to generate noisy test data: 1) white Gaussian noise is directly added to ; 2) a stochastic process in with i.i.d. uniform distribution in is added to phase functions . Fig. 4 summarizes the results of frequency estimation in this experiment. ESPRIT and MUSIC lose accuracy in all tests. NOP and ME achieve high accuracy when the number of samples is large and NOP is slightly better than ME in terms of accuracy and estimation bias.
5.2 Estimation of time-variant frequencies
In this section, we show the capacity of NOP for estimating close and crossover time-varying instantaneous frequencies. An adaptive time-frequency analysis algorithm, ConceFT Daubechies20150193 ()), is used as a comparison. And local approximation degree is set to in this section.
Close frequencies and phase estimation error
We use , where the two instantiations of / are / (Fig. 5(b)) and / as in Fig. 1 (Fig. 5(c)). varies from to . The white noise has standard deviation . The sampling rate is Hz and samples are involved. The initialization of is set to a little random perturbation of an STFT output. Result is summarized in Fig. 5.
Fig. 5(a) is the ground truth time-frequency representation of ten tested signals with different value of on . The difference between (green line) and (red line) are pretty difficult to be detected by existing time-frequency methods. The log error of the point-wise averaged frequency estimation is shown in the first row of of Fig. 5 (b) and (c) on different noise levels . The log error of point-wise averaged phase estimation (bottom row) is consistently small as changes. Under large noise case with , NOP controls the phase error approximate or below the level of . Compared to the existing time-frequency methods, NOP has no accumulated phase error.
Close and crossover frequencies
In this experiment, we generate a signal consisting of two components with close instantaneous frequencies and a signal with two crossover instantaneous frequencies. Fig. 6 visualizes the ground truth instantaneous frequencies, the time-frequency distribution by ConceFT, the initialization and the estimation results of NOP. ConceFT cannot visualize the instantaneous frequencies even if in the noiseless case. We average out the energy distribution of ConceFT to obtain the initialization of NOP. Although the initialization is very poor, NOP is still able to estimate the instantaneous frequencies with a reasonably good accuracy no matter in clean or noisy cases. Remark that a non-distinguishable issue can arise if two component are initialized with a same frequency. However, this problem can be solved by an easy ad hoc trick and is detailed in SM Section D.
5.3 Estimation of amplitudes, phases, and shapes simultaneously
Finally, we apply NOP to estimate amplitudes, phases, and shapes simultaneously from a single record. First, we generate a synthetic example , where , , and the shapes are visualized in Fig. 7. The sampling rate for this signal is Hz and we sample it at locations. The shape estimates are initialized as and for the first and second components, respectively (see Fig. 7 (b) and (c)). The frequency estimates are initialized as one constant centered in the peak spectrogram by ConceFT (see Fig. 7 (a)). As we can see in Fig. 7 (b) and (c), NOP is able to estimate shape functions with a reasonably good accuracy and the reconstructed components match the ground truth components very well.
In the second example, we apply NOP to a real signal from photoplethysmogram (PPG) (see Fig. 8 (b)). The shape estimates are still initialized as and for the two components, and samples are involved. The PPG signal contains two components corresponding to the health condition of the heart and lung in a human body.
This paper proposes a novel non-oscillatory pattern (NOP) learning scheme for several oscillatory data analysis problems including signal decomposition, super-resolution, and signal sub-sampling. To the best of our knowledge, the proposed NOP is the first algorithm for these problems with fully non-stationary oscillatory data with close and crossover frequencies, and general oscillatory patterns. Numerical examples have shown the advantage of NOP over several state-of-the-art algorithms and NOP is able to handle complicated examples for which existing algorithms fail. NOP could be a very useful tool for pattern analysis for oscillatory data. Although we cannot prove the global convergence of NOP, NOP seems to provide very good results in all of our tests. It is interesting to study the global convergence of NOP in the future.
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