## Time series analysis

### What is time series analysis?

A mathematical methodology for studying records of various dynamical systems, including stock market prices; the magnitude of earthquakes; a number of species in an ecosystem; and the recorded variation of sensors.

### Why is it important to study?

In many cases, a time series is the only source of information about a complex system, where there is no known analytical model, or the complexity is so high that detailed modelling is not feasible. A daily temperature record is a good example of a time series, and the dynamical system it represents is the Earth climate system. The complexity of the Earth system, with all subsystems, interactions, and feedback loops produces the series, which includes an enormous amount of information as a composite. Despite this complexity, it is possible to model and forecast the time series dynamics, using various models and with various degrees of accuracy.

### How is the system described mathematically?

The system can be described by a stochastic model, where there is a known deterministic component (often the derivative of an analytical expression, the so-called potential) and a stochastic (noise) component. The potential component defines, in a broad sense, the basin of attraction of the system fluctuations, and is related (given quasi-stationarity of the series) to the probability distribution of the system (via the Fokker-Planck equation). For instance, if the time series has a histogram with one mode (unimodal), then the system potential has one well (similar to a parabolic function) and if the histogram is bimodal, the potential has two wells, with two local minima and one local maximum. Analytically, such a potential is described by a polynomial of 4th order. If the number of potential wells is n then the potential will be described by a 2n-degree polynomial.

### What about the system noise?

The stochastic component is often assumed to be so-called 'white noise', which denotes Gaussian noise with no memory (a 'white' flat power spectrum with scaling exponent beta=0). This is the simplest, 'well-behaved' noise, which is stationary and can be described by a few statistical parameters, like the mean value and standard deviation. In reality, however, the noise can be red (correlated, with memory effects), blue (anti-correlated), and even more complicated (multifractal, nonlinear with complex seasonal effects and non-stationarities). The assumption of 'whiteness' is often made, but in reality some information about the system is split between the stochastic and the deterministic components. It is important to remember that any model (including stochastic models) is only an approximation of a system. Moreover, the same system may be described by several models of different origin - the question is which one is more accurate and has more power in describing and forecasting system dynamics.

### Why is time series analysis useful?

Stochastic models used in time series analysis are computationally light (unlike deterministic models with highly detailed differential equations, which in many complex systems take many hours to perform equivalent calculations). Stochastic models are very flexible in describing a particular system with a few parameters. Some scientists say that stochastic models lack a physical interpretation of the system; others argue that the mathematics of many diverse systems are the same or similar even though the physics is different - and if the forecast is accurate, this says something about the uniform nature of our knowledge.

### How is time series analysis relevant to metrology?

Metrology quantifies and standardises measurements; time series analysis provides basis for quantitative methods in studying dynamical systems. For example, the tipping point toolbox (see also article in *The Cryosphere*) is a methodology based on time series techniques, and it allows one to obtain early warning signals, detection points and stochastic forecasts of various tipping points in diverse dynamical systems. It is to be developed into a metrological tool, with uncertainty estimation.

**Last Updated: 8 Apr 2014**

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