Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 2689429, 7 pages

http://dx.doi.org/10.1155/2016/2689429

## The Application of Recurrence Quantification Analysis in Detection of Abrupt Climate Change

^{1}College of Atmospheric Sciences, Lanzhou University, Lanzhou 730000, China^{2}Tongda College, Nanjing University of Posts & Telecommunications, Nanjing 210003, China^{3}National Climate Center, China Meteorological Administration, Beijing 100081, China^{4}College of Physical Science & Technology, Yangzhou University, Yangzhou 225002, China^{5}College of Math & Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, China

Received 25 December 2015; Accepted 24 April 2016

Academic Editor: Yi Wang

Copyright © 2016 Wen Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper explores the possible application of recurrence quantification analysis in the detection of abrupt change of the dynamic structure of the climate system. It is discovered in the recurrence quantification analysis of the typical chaotic system-logistic model that the method may well distinguish the state of logistic model with different parameters, demonstrating its potential value in identifying the dynamic change of the system. When recurrence quantification analysis is later applied to the detection of abrupt change of average daily precipitation of all regions in China, the result indicates that the abrupt change of the dynamic structure corresponding to the precipitation of China in recent 50 years occurred in the late 1970s and the early 1980s. It is in agreement with the Chinese commonly recognized years of abruption; therefore the effectiveness is further demonstrated regarding the recognition of complexity of dynamical system.

#### 1. Introduction

It is well known that climate system is a nonlinear and nonstationary dynamic system, the change of which may encounter discontinuity, and such discontinuity is generally called the abrupt climate change. The study of the phenomenon and theory of the abrupt climate change is an emerging field in the modern climatology and has been attracting more and more attention. Conducting the quantitative analysis of historical climate materials by various detection technologies of the abrupt change to determine the occurrence time of abrupt climate change and further find the mechanism of abrupt change by material diagnosis or numerical simulation is very significant for the accurate forecasting of climate in the future.

Abrupt climate change may be roughly divided into two major types: abrupt change of variables of the climate state in the statistical significance and change of the dynamic structure of the climate system. The first type exists in various climate observation materials and involves multiple time scales. Such abrupt change is not necessarily related to the change of the dynamic equation of the climate system but is a self-adaption process of the climate system, such as the conversion between cold and warm phases. The first type is mainly judged by whether there is obvious change of statistical features of variables of the climate state in the statistical significance, such as the mean value, variance, frequency distribution, and change trend [1]. Methods involved include sliding -test, Crammer Method, Yamamoto Method, Mann-Kendall Method, Wavelet Analysis, and Heuristic Segmentation Algorithm [2–6]. The second type, that is, the detection of abrupt change of the dynamic structure, is an emerging field in recent years, such as conditional entropy [7] and approximate entropy method [8]. Livina et al. [9–11] have put forward many new detection methods of the abrupt change of time series on the basis of the long-term memory of time series. These methods are quite useful for the detection of abrupt change of the structure of climate dynamics.

Although great achievements have been made in detection methods of the abrupt climate change, no method is perfect. For example, the detection technology of abrupt change based on the long-term memory of time series and the method of power law tail exponent both require the fractal feature of time series. The traditional detection technology of abrupt climate change depending on the change of statistical features of variables of the climate state is also quite limited. On the one hand, the technology needs a large amount of samples and many methods require the time series to meet the normal distributional assumption; on the other hand, it is difficult to detect the abrupt change of the dynamic structure of the system by using those methods. Therefore, developing different detection technologies of the abrupt change contains quite important scientific significance and value of application.

The studies related to the interaction between time and space is involved in various fields of life, such as zoology and botany [12, 13]. In the process of research, the recurrence plot method, firstly proposed by Eckmann et al. [14] in 1987, is a nonlinear technology to observe the features of the system state in the phase space. Recurrence plot method may expose some future knowledge about time series visually through the graph, such as the similarity contained in time series and the predictability of information. To quantitatively describe the physical significance exposed by the recurrence plot, Webber Jr. et al. [15–18] have proposed the recurrence quantification analysis, which can measure the complexity of time series effectively, assess, and classify the space recurrence graph of each time to determine differences and similarities of the dynamic structure, so as to detect the abrupt change of structure of the climate dynamics by the evolution of space recurrence graph. This method is advanced in the low requirement on the quantity of samples in the computation and strong noise resistance. As a result, this method has been widely applied to medical science [19], hydromechanics [20], and diagnosis of faults of the mechanical equipment [21–23].

Even if the recurrence quantification analysis has been widely applied to many disciplines and fields, it is seldom applied to climatology and almost has not been applied to the detection of abrupt climate change. Therefore, this paper studies the feasibility and effectiveness of recurrence quantification analysis in the detection of the abrupt change of the dynamic structure of time series. In the beginning, this paper verifies the effectiveness of recurrence quantification analysis in identifying the complexity of structure of the dynamic system by the classical chaotic model-logistic model and then applies the analysis to the detection of abrupt climate change of daily precipitation materials of all stations of China. The result indicates that recurrence quantification analysis is able to distinguish and identify well the dynamic nature of the system, and its application to the average daily precipitation materials in areas of China further proves its practicability and reliability.

#### 2. Introduction to Methods

##### 2.1. Recurrence Plot

Regarding one-dimensional original time series , in the Takens Embedding Theorem, when the embedding dimension is and the delay time is , the vector of phase space after the reconstruction isand in the formula, . Defining the distance between any two vectors in the reconstructed phase space and selecting as the threshold value, it may result in the recurrence matrix , in which is the step function and may be expressed as . The expression of isWhen , depicting points at in the two-dimensional coordinates and the result is the recurrence plot.

##### 2.2. Selection of Parameters

The selection of three parameters, namely, threshold value , embedding dimension , and delay time , requires certain technique. In view of existing experience, the selection of parameters may adopt the following principle: threshold value is generally 0.15~0.2 times of the standard deviation of the series, and this paper selects 0.18; embedding dimension may be obtained according to the correlation dimension; delay time may be determined by the autocorrelation function method. As the selected length of sliding window in the time series is fixed in the detection of abrupt change, this paper selects the fixed embedding dimension and delay time for the convenience of comparing the complexity of subsequences with the same length; that is, and .

##### 2.3. Recurrence Quantification Analysis

To quantify the recurrence of the system demonstrated in the recurrence plot, Webber Jr. et al. have proposed some parameters of the recurrence quantification analysis, such as recurrence rate (RR), determinism (DET), and recurrence entropy. These parameters describe features of the recurrence plot from difference perspectives. This study only selects two parameters, RR and DET, to describe features of the recurrence plot.

RR is percentage between recurrence points in the recurrence plan and total points of the recurrence plan; that is,DET is percentage between recurrence points forming diagonal segments and all recurrence points; that is,and in the formula, is quantity of segments in the length of and it is only counted when the length of diagonal segments is longer than the set lower limit . is generally an integer not smaller than 2 and too large may deteriorate the indicative significance of DET. DET will distinguish isolated recurrence points from recurrence points forming diagonal segments in the recurrence plot. The clearer the lines along the principal diagonal in the recurrence plot are, the stronger the determinism of the system is.

#### 3. Test of Ideal Value

For the purpose of verifying the identification of different dynamic systems by the recurrence plot method, this paper analyzes three groups of signals, namely, periodic signals, random signals, and chaotic signals in the beginning by the recurrence plot method. Each group of time series includes 1,000 samples. Periodic signals come from the periodic function below: ; random signals are generated by the random function. Chaotic series comes from the classical logistic model expressed as , in which is 3.9 and integration step is 1 and 1,000 steps iterated initially are abandoned to eliminate the influence brought by the initial value. As the evolution of recurrence plot of chaotic signals and random signals is more complicated than that of periodic signals and it contains certain self-similarity, only a part of the recurrence plot is selected in Figures 1(b) and 1(c) to demonstrate its features more clearly.