Data analysis

Classical object in climatology and meteorology is time series data representing a succession of observations ranked in time (or other variable). Examples of time series data are the records of air surface temperature, precipitation, or river runoff measured at equally spaced time intervals. A time series as experimental (input) data can be used for studying various problems, for example: prediction of values in the future basing on known values in the past, construction of a model generating the series, description of features of the time series or a set of time series in a compact form. Here we consider some methods of analysis of time series and their sets.

The mathematical model of a time series depends on the type of the problem under solution, and on the data. The general model of time series data can be introduced as

y(t_i) = f(t_i) + e (t_i),      i=(1, 2, ... I),

where

t_i is a discrete variable (for example, time);
y(t_i) is an observation;
f(t_i) is a determinate function;
e (t_i) is a noise component.

A determinate function can depend on some arguments, which can be known or unknown values. In the last case it is called a regression function.

One can consider two types of a determinate function: slowly varying function and cyclic function. Slowly varying function can be represented through polynomials of a various degree, including the linear dependence on discrete variable (time). For studying the cyclic dependence it is possible to use the spectral analysis. Smoothing and denoising of time series data can be implemented using a technique of averaging in a running window. A powerful method of time series analysis is the multifactor analysis. It allows to extract slowly varying and cyclic components, as well as to implement smoothing and denoising.

For analysis of variability of time series, estimation of their likeness and distinction, the correlation analysis can be used.

Many natural phenomena can be described as fractals. The time series of long-duration measurements of temperature, river runoff and other climatic parameters reveal fractal properties in some range of scales. The fractal tools are useful for quantitative characterization of statistically self-similar time series, which can be treated as something intermediate between pure random noise and determinate function. The examples of application of fractal analysis for studying temperature time series and multifractal analysis for studying river runoff time series are provided.

 Fractal and multifractal analysis of the time series. Nowadays it is clear that many natural systems have a complex dynamics. In order to characterize this dynamics, one should study variability of the parameters of the dynamic system. The time series, produced by many dynamic systems can not be described completely in frames of any regular model or in frames of statistical models (random noise, Gaussian, Markoff, Poisson process or etc.). Frequently natural systems reveal some intermediate state between the above mentioned. It's the state of so called 'ordered chaos'. In many cases the system reveals statistically self-similar structure (of time series) in some range of scales, i.e. the system has a fractal order. The approach to description of the dynamic system with such properties is based on fractal analysis. We apply fractal-theory based methods for the description of the properties of the climatic time series in order to characterize the dynamics of the climatic system. The fractal characteristics allow us to estimate variability of time series in a range of temporal scales, degrees of order and chaos in the time series, self-similar properties.

Multifactor analysis of the time series.
The multifactor analysis of the time series ( MAS) is a powerful method of analysis of time series. It is possible to interpret MAS as decomposition of the time series using the system of orthogonal basis functions, which are obtained on the basis of the original time series. The MAS aims at decomposition of the original series into a sum of a small number of interpretable components such as a slow varying trend, oscillatory components and a 'structureless' noise. Possible applications of MAS are very wide and include climatology, meteorology, signal processing and physics.