Granqvist Emma, Oldroyd Giles E D, Morris Richard J
Department of Computational & Systems Biology, John Innes Centre, Norwich Research Park, Norwich NR4 7UH, UK.
BMC Syst Biol. 2011 Jun 24;5:97. doi: 10.1186/1752-0509-5-97.
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.
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.
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.
构建生物系统数学模型的第一步通常是分析关键量的时间行为。时间域和频率域之间的数学关系，如傅里叶变换和小波变换，通常用于从给定的时间序列中提取关于潜在信号的信息。从时间点到频率的这种一一映射本质上假设两个域都包含系统的完整知识。然而，对于具有背景趋势的截断、有噪声的时间序列，这种唯一映射会失效，问题就简化为识别最可能频率的推理问题。
在本文中，我们基于贝叶斯频谱分析方法进行构建，并通过将其应用于多个测试案例（包括两种类型的生物时间序列）来证明其优于传统方法。首先，植物根细胞中钙对微生物共生体响应的振荡是非平稳且有噪声的，这给数据分析带来了挑战。其次，仅在两个周期内测量的基因表达的昼夜节律突出了长度有限的时间序列问题。结果表明，贝叶斯频率检测方法可以在傅里叶分析可能无信息或产生误导的特定领域提供有用的结果。我们展示了贝叶斯方法在时间序列分析中的进一步优势，例如不同假设的直接比较、噪声水平和参数精度的固有估计，以及无需预处理即可对数据进行建模的灵活框架。
系统生物学中的建模通常基于对时间相关现象的研究。傅里叶变换是分析时间序列频率域的便捷工具。然而，该方法存在众所周知的局限性，例如在处理短而有噪声的时间序列时会引入虚假频率，以及对均匀采样数据的要求。生物时间序列往往显著偏离傅里叶变换的最优要求。在本文中，我们提出了一种基于贝叶斯推理的替代方法。我们展示了将频谱分析置于贝叶斯推理框架中的价值，并演示了模型比较如何使这一过程自动化。
