Automated Bayesian model development for frequency detection in biological time series
- Emma Granqvist^{1, 2},
- Giles ED Oldroyd^{2} and
- Richard J Morris^{1}Email author
DOI: 10.1186/1752-0509-5-97
© Granqvist et al; licensee BioMed Central Ltd. 2011
Received: 23 November 2010
Accepted: 24 June 2011
Published: 24 June 2011
Abstract
Background
A first step in building a mathematical model of a biological system is often the analysis of the temporal behaviour of key quantities. Mathematical relationships between the time and frequency domain, such as Fourier Transforms and wavelets, are commonly used to extract information about the underlying signal from a given time series. This one-to-one mapping from time points to frequencies inherently assumes that both domains contain the complete knowledge of the system. However, for truncated, noisy time series with background trends this unique mapping breaks down and the question reduces to an inference problem of identifying the most probable frequencies.
Results
In this paper we build on the method of Bayesian Spectrum Analysis and demonstrate its advantages over conventional methods by applying it to a number of test cases, including two types of biological time series. Firstly, oscillations of calcium in plant root cells in response to microbial symbionts are non-stationary and noisy, posing challenges to data analysis. Secondly, circadian rhythms in gene expression measured over only two cycles highlights the problem of time series with limited length. The results show that the Bayesian frequency detection approach can provide useful results in specific areas where Fourier analysis can be uninformative or misleading. We demonstrate further benefits of the Bayesian approach for time series analysis, such as direct comparison of different hypotheses, inherent estimation of noise levels and parameter precision, and a flexible framework for modelling the data without pre-processing.
Conclusions
Modelling in systems biology often builds on the study of time-dependent phenomena. Fourier Transforms are a convenient tool for analysing the frequency domain of time series. However, there are well-known limitations of this method, such as the introduction of spurious frequencies when handling short and noisy time series, and the requirement for uniformly sampled data. Biological time series often deviate significantly from the requirements of optimality for Fourier transformation. In this paper we present an alternative approach based on Bayesian inference. We show the value of placing spectral analysis in the framework of Bayesian inference and demonstrate how model comparison can automate this procedure.
Background
Pattern recognition is central to many scientific disciplines and is often a first step in building a model that explains the data. In particular, the study of periodic phenomena and frequency detection has received much attention, leading to the well-established field of spectral analysis.
Biology is rich with (near) periodic behaviour, with sustained oscillations in the form of limit cycles playing important roles in many diverse phenomena such as glycolytic metabolism, circadian rhythms, mitotic cycles, cardiac rhythms, hormonal cycles, population dynamics, epidemiological cycles, etc. [1]. A conventional method for frequency detection is Fourier analysis. It is based on the fact that it is possible to represent any integrable function as an infinite sum of sines and cosines. The Fourier Transform (FT) uses this property to reveal the underlying components that are present in a signal [2]. Fourier theory has given rise to a wide range of diverse developments and far-reaching applications, demonstrating the theory's undisputed importance and impact. For frequency detection, however, it is known that the FT works optimally only for uniformly sampled, long, stationary time series. Furthermore, common procedures of pre-processing the data can cause problems. Time series can contain low frequency background fluctuations or drift that are unrelated to the signal of interest. For the FT, it is then necessary to remove the trends using detrending techniques. As has been shown previously, this detrending leads to convolution of the signal that can both remove evidence for periodicity and add false patterns [3]. Another known problem is aliasing. If a signal containing high frequencies is recorded with a low sampling rate, peaks of high frequencies can fold back into the frequency spectrum, appearing as low frequencies [2]. The Gibbs phenomenon provides another example where spurious peaks appear in a FT. It occurs at points of discontinuity in a periodic function, and results in so-called ringing artefacts around the "true" frequency peak [4]. As for the accuracy of the frequency estimate, no direct information of this is given by the output from a FT, since the sharpness of the peaks depends on a combination of factors such as noise levels and the length of the time series. For further details, see the extensive FT literature (e.g. [2, 5]).
Wavelet Transforms [6–10] offer an attractive alternative to Fourier Transforms. The main difference is that they are localised in both the time and frequency domain. This property makes wavelets better adapted to problems with truncated data. Wavelets have found wide-ranging applications and have proven to be particularly powerful for image processing and data compression [11–13].
Bayesian inference provides another approach for analysing data (for an introduction to Bayesian analysis, see [14]). It addresses additional aspects of the problem, such as the inherent uncertainty of the analysis and the effects of external noise. Using this framework [3], the method of Bayesian Spectrum Analysis (BSA) was developed by Bretthorst [15] and applied to Nuclear Magnetic Resonance (NMR) signals and parameter estimation with great success [16, 17].
There are several advantages of the Bayesian approach, including an inherent mechanism for estimating the accuracy of the result and all parameters, as well as the ability to compare different hypotheses directly. Focus is shifted to the question of interest by integrating out other parameters and noise levels. Initial knowledge of the system can be incorporated in the analysis and expressed in the prior probability distributions. There has been a recent flood of Bayesian papers with some convincing applications and promising developments in systems biology (see [18–30], and many others). The Bayesian approach to time series analysis has proven its value in fields such as NMR and ion cyclotron resonance analysis (e.g. [31] and [32]).
In this paper, we describe the development, implementation and testing of Bayesian model development coupled with BSA and Nested Sampling, in a biological context. We present a comparison of this approach with the FT, applied to a number of simulated test cases and two types of biological time series that present challenges to accurate frequency detection. We first present some necessary background, upon which we build to develop to our approach.
Bayesian inference
where P(D|H, I) is the probability of observing the data given the hypothesis and the prior, P(H|I) is the prior probability of the hypothesis, and P(D|I) is the probability of the data given the prior. When the hypothesis is the variable and the data is held constant, P(D|H, I) is called the likelihood function, and when the hypothesis is constant it is called the probability of obtaining a specified outcome (data). When evaluating only one hypothesis, P(D|I) is a normalising constant, but when investigating more than one hypothesis this term plays a key role and is called the evidence [33].
Bayesian Spectrum Analysis
Our presentation in this section follows closely that of Bretthorst [15]. The aim is to infer the most probable frequency (or frequencies), ω, from the given data. We start by building a model (the hypothesis H) for the observed data, parameterised by angular frequency, ω, and amplitudes, c, and then use Bayes' rule to compute the posterior probability of the parameters, P(ω, c|D, H, I). By assigning priors to the model parameters c and integrating over these, we arrive at the posterior probability for the parameter of interest, ω, . This is referred to as the marginal posterior probability of ω. We note that ω is an r-tuple, {ω_{1}, ω_{2}, ..., ω_{ r } }, with as many elements as there are distinct frequencies in the data.
in which are the expansion coefficients.
where are the background model function expansion coefficients.
where is the mean-square of the data, , and Φ _{ jk } is the matrix of the model functions, .
The goal of the analysis is to compute the posterior probability for frequencies in the data, i.e. to go from the joint probability distribution to a posterior probability of ω, independent of the other parameters. By integrating over all possible values of the parameters σ and c, the remainder is the marginal posterior of the parameters of interest, ω={ω_{1}, ω_{2}, ..., ω_{ r } }. This is an essential advantage of the Bayesian framework, allowing the analysis to focus on estimating the parameters of interest, regardless of the values of the others. If necessary, the other parameters can be estimated at a later point.
To integrate over the σ and c values, priors must first be assigned to them. We chose uniform priors for c and ω, representing complete lack of knowledge. We know that σ is continuous and must be positive, and in such cases a Jeffreys prior is appropriate, P(σ|I) = 1/σ. Both the uniform distribution over continuous variables and the Jeffreys prior are known as improper priors if bounds are not specified as they cannot be normalised. For more information on prior assignment see [3, 15].
where h is the projection of the data onto the orthonormal model functions, , and is the mean-square of the h_{ j } , , [15]. This expression of the posterior allows us to identify the strongest frequencies present in the data. For a good model, there will be a high probability peak in the posterior distribution at that ω = {ω_{1}, ω_{2}, ..., ω_{ r } }.
Results and Discussion
We employed the framework developed by Jaynes [3] and Bretthorst [15] to investigate the frequency components in a number of biological time series.
Model comparison
The probability of the data given our prior information, P(D|H_{ i } , I), which was a normalisation constant in equation (6), will now vary between models, and is called the evidence. It evaluates the fit of the data to the model, whilst penalising models that include more parameters. Each additional model parameter should be followed by a significant increase in probability, otherwise the simpler model is preferred. Thus, Bayesian model comparison naturally follows the principle of Occam's razor [33, 34].
Model development
It will often not be obvious which function to choose to model trends in the data, so an approach using basis functions and expanding these to different orders will be of advantage, as in equation (4). Each expansion represents a different model, H_{ i } , and these can be compared using inference techniques. Likewise, different functions for capturing the signals in the data and modelling a different number of signals correspond to different models for data. Following [3, 14, 33], we use the posterior ratio to evaluate different models.
where δ, γ and σ are the prior variances for amplitudes, frequencies and noise, respectively, R_{ δ } , R_{ γ } and R_{ σ }are the ratios of the integral bounds for these variances, is the mean-square projection of the data onto the orthonormal model functions at the maximum likelihood point for model H_{ n } , is the mean-square of the ω value that maximises the likelihood, , and r is the number of ω parameters, ω= {ω_{1}, ..., ω_{ r } }. The Jacobian is obtained by orthogonalising the Taylor-expansion of around the maximum-likelihood point, . See Bretthorst for further details [15]. For cases in which the number of frequencies in the data exceeds the dimension of omega, for instance multiple frequency data with a single frequency model, the above approximation for the evidence is ill-suited as the posterior will cease to be unimodal. For such scenarios, either multiple expansions or MCMC offer attractive solutions to marginalisation. For comparison we have included results from Nested Sampling [14, 35] as a means to perform the integration and compute the evidence. Nested Sampling is a variant of MCMC that employs a likelihood based sorting of sample points to efficiently guide the search strategy of the posterior distribution [14, 35].
When the model ratio, H_{ n } /H_{ n }_{+}_{1}, becomes greater than 1, the simpler model, H_{ n } , is favoured over H_{n+1}[33]. Adding more background functions than are justified by the data (based on the posterior model ratio) may lead to a lower probability for the frequency and in some cases possibly a location shift.
Model ratios for number of frequencies in the data
Case | N _{ ω } | Models | Model ratio |
---|---|---|---|
A | 1 | H_{1ω}/H_{2ω} | 66936 |
B | 2 | H_{1ω}/H_{2ω} | 0 |
B | 2 | H_{2ω}/H_{3ω} | 686 |
We point out that the proposed method stops once the current best model has been found but is not guaranteed to find the global maximum from a predefined set of models. The procedure is thus part of model development rather than model selection. If the set of hypotheses are known in advance then the posterior ratios over the full set should be used to find the best model.
Testing
We first show the power of the BSA approach on test cases using simulated data. In these tests, we sought to recover known input parameters from the simulated data, to validate the BSA approach. We employed sines and cosines as model functions (ψ_{ j } in equation (3)). For comparison, Discrete Fourier Transforms were computed using Fast Fourier Transforms (FFT) [36]. In the test cases, we varied key parameters such as noise levels, trace length, sampling intervals, amplitudes, frequencies, background trends and shape of oscillations.
BSA and FFT results from simulated harmonic data with noise and background trends
No. | ω | e_{ a }(%) | e_{ p }(%) | b |
| σ _{ FFT } |
| σ _{ BSA } |
| σ _{ BSA-NS } | s-n |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.5 | 1 | - | - | 0.49 | 0.06 | 0.5 | 0 | 0.5 | 0.0002 | 70 |
2 | 0.5 | 10 | - | - | 0.49 | 0.20 | 0.5 | 0.0002 | 0.5 | 0.0004 | 6.5 |
3 | 0.5 | 40 | - | - | 0.49 | 0.54 | 0.5 | 0.0005 | 0.5 | 0.0011 | 1.9 |
4 | 0.5 | 10 | 10 | - | 0.49 | 0.27 | 0.5 | 0 | 0.5 | 0.0003 | 4.2 |
5 | 0.5 | 10 | 40 | - | 0.49 | 0.57 | 0.5 | 0.0002 | 0.5 | 0.0007 | 2.2 |
6 | 0.5 | 100 | 40 | - | 0.49 | 0.89 | 0.5 | 0.0006 | 0.5 | 0.0020 | 0.7 |
7 | 0.3, 0.5 | 10 | 10 | - | 0.29, 0.51 | 0.14 | 0.3, 0.5 | 0.0003 | 0.34 | 0.0832 | 1 |
8 | 0.5 | 10 | - | -t | 0 | 0.15 | 0.5 | 0.0002 | 0.5 | 0.0002 | 110 |
9 | 0.5 | 10 | - | -t^{2} | 0 | 0.19 | 0.5 | 0.0002 | 0.5 | 0.0002 | 90 |
10 | 0.5 | 10 | - | -t^{3} | 0.02 | 0.24 | 0.5 | 0.0003 | 0.5 | 0.0002 | 35 |
BSA has a clear advantage over FFT when the data is non-uniformly sampled. FFT requires uniform sampling, whilst BSA is less stringent and delivers the correct result with higher precision. Bretthorst also noted that non-uniformly sampled data removes aliases from the frequency domain, another significant advantage [15]. Five further distinct cases emerged from the tests in which BSA delivers superior results to FFT: time series which have background trends, few data points, high noise levels, multiple frequencies, and non-harmonic oscillations.
Background trends
Automated model development
Models | Model ratio |
---|---|
H_{0ζ}/H_{1ζ} | 1.9459e-06 |
H_{1ζ}/H_{2ζ} | 1.0256e-167 |
H_{2ζ}/H_{3ζ} | 622.5 |
H_{3ζ}/H_{4ζ} | 566.3 |
H_{4ζ}/H_{5ζ} | 501.8 |
H_{5ζ}/H_{6ζ} | 99.2 |
Examples 8-10 in Table 2 also include trends, and without pre-processing FFT cannot pick out the correct frequency. In contrast, BSA includes background functions in the model signal and delivers the desired result. Including background functions, however, results in over-estimation of the signal-to-noise ratio.
Short time series
Additional file 2, Figure S2, shows the results from analysing a short time series. The FFT power spectrum is very broad (Additional file 2B, Figure S2B), which comes as no surprise given the FFT dependence on the number of data points. BSA estimates the correct frequency sharply, but the maximum probability drops compared to longer time series (Additional file 2C, Figure S2C). This demonstrates the higher uncertainty associated with fewer time points.
High noise levels
Multiple frequencies
Example 7 in Table 2 has two frequencies present in the data. Both BSA and FFT show these two peaks in the resulting plots. Although BSA can be used in this manner with a one-dimensional ω to scan through frequency space and estimate the number of frequencies in the data and their location, if more than one frequency is present, the model should be extended to reflect this. Without this extension the integration procedure around a single point is not well suited, so we employed Nested Sampling to compute the marginalisation in these cases. For the extension approach, when the posterior probability over ω= {ω_{1}} reveals two strong frequencies, then a better model would be ω= {ω_{1}, ω_{2}}. For example, Additional file 3, Figure S3, shows BSA and FFT results for a test case that includes higher harmonics which give rise to multiple peaks in the log P plot. If more than one peak in the resulting posterior probability emerges, then the model can be extended further. One peak in the posterior probability over the number of modelled frequencies signifies that the correct number of frequencies has been captured.
As another example, Additional file 4, Figure S4, shows the result of a two-frequency search. The BSA posterior probability distribution is now two-dimensional with a peak at the two correct frequencies (Additional file 4C, Figure S4C). The FFT results are also shown (Additional file 4B, Figure S4B).
Additional file 5A, Figure S5A, shows a time series with a high noise level and two very close frequencies of 0.498 and 0.505 rad/s. FFT cannot distinguish them and shows only one peak (Additional file 5B, Figure S5B). BSA breaks the resolution and precision limitations inherent to FFT by introducing a continuous probability distribution instead of the fixed number of points and can therefore sample the posterior more finely in areas of high probability. This approach gives rise to a high-resolution probability plot in which two distinct frequencies emerge (Additional file 5D, Figure S5D). The peaks have a larger variance at this local level, but the qualitative information of two underlying frequencies is revealed.
Non-harmonic oscillations
This highlights the differences between frequencies in the data and spike intervals. ISI are a common way of characterizing spike data, however, multiple ISI need not correspond to multiple frequencies in the data. Of the four strong ISI shown here, both BSA and FFT identify only one of these as a regular period.
Summary
After extensive test cases we find that BSA delivers superior results in cases where the FFT assumptions are too constraining, most notably in the five cases above. BSA is a flexible method allowing the underlying hypothesis to be changed depending on the focus of the analysis, and to directly compare the validity of different hypotheses. It can handle non-uniformly sampled data and has no need for pre-processing procedures. The price of these superior results comes at a computational cost that ranged from tens to hundreds of seconds for the examples shown here.
Calcium spiking data
The first biological data set comes from intracellular signalling in plant-microbe interactions. Symbiotic bacteria induce calcium oscillations, called Ca^{2+} spiking, in legume root cells (for a review, see [39]). These are non-stationary and often noisy time series, causing problems in identifying periodicity. One hypothesis for signal transduction in this system is via frequency encoding [40], so concluding whether there is underlying periodicity, and at what frequency, is of great interest.
BSA on calcium data
Cell | BSA Period ± σ(s) | BSA-NS Period ± σ(s) |
---|---|---|
1 | 97.4 ± 0.23 | 97.3 ± 0.15 |
2 | 80.9 ± 0.63 | 75.2 ± 10.1 |
3 | 74.6 ± 0.19 | 74.6 ± 0.85 |
4 | 123.8 ± 0.16 | 124.2 ± 1.18 |
5 | 88.9 ± 0.22 | 123.9 ± 0.61 |
6 | 74.6 ± 0.21 | 113.7 ± 16.16 |
7 | 121.9 ± 0.22 | 146.1 ± 21.53 |
8 | 74.4 ± 0.92 | 75.2 ± 2.69 |
9 | 48.2 ± 0.3 | 64.5 ± 13.94 |
Circadian data
The second biological data set shows gene expression of so-called clock genes. Many processes in plants follow a circadian rhythm (for reviews see e.g. [43] or [44]). A number of genes in Arabidopsis thaliana have been shown to regulate circadian rhythms, and time series of RNA levels show how these clock genes are expressed in cycles [45]. Time series with only a couple of cycles are common in biology and provide another suitable test case.
BSA on circadian data
Gene | Genotype | BSA Period ± σ(h) | BSA-NS Period ± σ(h) |
---|---|---|---|
TOC1 | fri;flc | 22.75 ± 0.18 | 22.58 ± 0.43 |
TOC1 | FRI;FLC | 23.36 ± 0.20 | 23.26 ± 0.42 |
CCA1 | fri;flc | 23.58 ± 0.15 | 23.67 ± 0.94 |
CCA1 | FRI;FLC | 23.98 ± 0.16 | 24.23 ± 0.72 |
GI | fri;flc | 22.39 ± 0.14 | 22.54 ± 0.86 |
GI | FRI;FLC | 23.41 ± 0.16 | 23.61 ± 0.83 |
LHY | fri;flc | 23.84 ± 0.16 | 23.82 ± 1.54 |
LHY | FRI;FLC | 25.74 ± 0.19 | 24.03 ± 1.23 |
Conclusions
Bayesian inference offers a powerful way of analysing biological time series. Despite the undisputed value of Fourier theory, there are cases when the necessary requirements for its optimality for time series analysis are not met. This is a consequence of the underlying assumptions of a Fourier Transform, causing it to work optimally only for uniformly sampled, long, stationary, harmonic signals that have either no or white noise. In biology these requirements are rarely fulfilled, requiring pre-processing of the data, such as noise reduction and detrending techniques, with the risk of convoluting the signal and losing valuable information.
By placing the problem of frequency extraction in the framework of Bayesian inference, the known and well-documented problems of Fourier analysis can be overcome. This approach also breaks the resolution and precision limitations inherent to the FFT by introducing a continuous probability distribution instead of the fixed number of points maintained by the discrete Fourier Transform. As we demonstrated here, BSA coupled with automated model development can give superior results to the FFT when faced with short, noisy time series, non-stationarity and non-harmonic signals. The suggested automated model development worked well in our hands but must be used with caution in practice as the approach is not guaranteed to find a global optimum in model space. Alternate models should be explored and compared using posterior probability ratios or approximations thereof. We found Nested Sampling [14] to provide a powerful means of estimating evidences for cases in which a single peak could not be identified. Other MCMC techniques such as simulated annealing running in parameter exploration mode or standard Metropolis-Hastings algorithms offer attractive alternatives [33].
BSA calculates signal-to-noise ratios, provides parameter precision estimates, and can handle high noise levels as well as background trends and therefore has no need for pre-processing. More importantly, the Bayesian framework offers flexibility in the underlying model and enables direct comparison of hypotheses. The work presented here is a merely a first step in this direction. We have employed conservative priors (uniform, Jeffreys, Gaussian) that make an analytical treatment tractable but in some cases more information could warrant a different choice of prior that might require substantial alternations to our approach to handle the numerics of marginalisation.
There are many known examples in biology in which oscillations play a key role and methods for their detection will be of value, especially in cases where subtle differences are of importance and for short, noisy time series. In the presented examples, we demonstrated the improvements that can be gained from employing this approach. Although in these cases, the biological conclusions would not have changed, one can envision scenarios in which a higher accuracy in frequency detection may allow subtle changes to be detected, which may otherwise have been swamped by noise and less powerful techniques. We believe that the presented methodology offers an attractive alternative to other approaches and will be a useful addition to the toolbox of systems biologists.
Methods
All programming was done using Octave [46], which is freely available and compatible with the widely used MATLAB^{®}. The Octave code is freely available from the authors upon request.
FT
The DFT was computed using the fft function in Octave. The results are presented in a power spectrum, to analyse which component carries the most power. This is also know as a periodogram , i.e. the squared absolute value of the FFT of the data d, normalised over the number of data points N[2]. For time series with strong trends, detrending was done before the FFT, using the moving average method [47].
There are a number of sophisticated FT methods beyond the standard FFT, developed to avoid specific problems. For example, we also present results from the multitaper method (MTM), short-time Fourier Transforms (STFT) and wavelet analysis. For the MTM, only the MTM spectrum is presented, but it should be noted that the Singular Spectrum Analysis - MultiTaper Method (SSA-MTM) toolkit provides additional features such as significance levels of the frequencies, relative to the estimated noise levels [42]. The STFT power spectrums were computed with the specgram function in Octave. The wavelet results were computed using software provided by Dr. C. Torrence and Dr. G. Compo, and is available online http://atoc.colorado.edu/research/wavelets/. This wavelet software also provides additional tools such as significance levels [37].
BSA
The next step is to specify the frequency domain of interest. This domain is then sampled with a chosen interval, and the posterior probability is computed at each frequency. Since the ω values are sampled over the frequency domain of interest with a chosen interval, the most probable frequency from this set may have a close neighbour with even higher probability, but which fell between sampling points. To avoid this, the Nelder-Mead optimisation technique was used to find the maximum of equation (10) [48]. Subsequently, the area surrounding this peak is finely sampled, to achieve a better representation of the posterior probability distribution of ω. The number of maxima should be checked and if proceeding with a multimodal distribution, an MCMC technique such as Nested Sampling should be used instead of the described marginalisation method. The outputs from the BSA algorithm are the posterior probability distribution of ω, a signal-to-noise ratio distribution and a power spectrum.
Declarations
Acknowledgements
We thank Dr David Richards and Nick Pullen for critical reading of the manuscript and useful suggestions. We thank Dr Jongho Sun for the provision of calcium spiking data. Thanks are also due to four anonymous reviewers for their detailed, constructive criticism and insightful comments. We would like to thank the Free Software Foundation and all authors of software packages who generously make their tools freely available (LATEX, gnuplot, emacs, Octave, gcc, and many many others). EG acknowledges PhD funding from the John Innes Foundation. RJM and GEDO are grateful for support from the BBSRC.
Authors’ Affiliations
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