Interval Prediction and Stability Analysis of Time Series (Part I: Theory)

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Time series of different performance attributes are produced in the process of runtime of products, and any time series contain a large amount of information about the system evolution, so the information of the future evolution can be extracted from the time series to make a forecast or stability analysis. In this paper, based on the grey bootstrap method to establish a grey bootstrap distribution of time data sequences, the interval prediction of performance signal can be obtained by the given confidence level; Then according to the fuzzy-set theory, the fuzzy similar relation of engineering practice is changed into the fuzzy equivalence relation of space vector, and the stability analysis of time series is acquired by the given λ threshold. Two sets of models can effectively assess the change trend and performance evolution signs of time series, helping us to timely grasp the work performance situation. Introduction Time series for a certain performance attribute of product are formed by the order of time measurement value. According to the analogy or extension of the development process, the direction, trend and dynamic running level of time series can be achieved by predicting or analyzing next period time. If we can make full use of the temporal information, the fault diagnosis of product performance can be effectively completed [1-2]. The information mining of time series is attached highly attention by academia and engineering field, especially the aerospace, finance, economy, astronomy and geology [3-6]. While the early analysis models of time series are all almost linear, at present, more and more found that the nonlinear models can reasonably explain the practical engineering problem with the increasing requirement of the product quality indicators [7]. The range of time series is directly related weather performance characteristics can full play to its during working periods. Time series are usually accompanied by inherent evolution rule of product, so we can extract useful information to analyze the stability variation or to predict product performance weather is in good standing in the future. Based on nonlinear time series of performance data, the interval forecasting model is established by the grey bootstrap method [8], to analyze the range situation of interval evolution and promptly identify wave information of product performance signal; Then based on the fuzzy-set theory to analyze the stability of time series [9], evolution signs of time data sequences are evaluated by segment handling the raw data, and the stability of the product during operation is comprehensively discussed. The Interval Prediction of Time Series The Grey Bootstrap Prediction Model of Time Series. Suppose the vector of time series is expressed as )) ( ),..., ( ),..., 2 ( ), 1 ( ( N x n x x x X  (1) where X(n) is the nth data of the raw data; N is the number of the data in X. The first bootstrap samples ψ1 is obtained by an equiprobable sampling N times with replacement from Eq. (1). And B simulation samples can be acquired by repeating B times in a row as follows: 6th International Conference on Sensor Network and Computer Engineering (ICSNCE 2016) © 2016. The authors Published by Atlantis Press 192 ) ,..., ,..., , ( 2 1 B b Ψ Ψ Ψ Ψ Ψ  (2) where ψb is the bth sample of bootstrap samples and can be expressed as )) ( ),..., ( ),..., 2 ( ), 1 ( ( N n Ψ b b b b b      (3) According to the grey system theory, suppose the first-order accumulated generating operator (1-AGO) of ψb is given by )) ( ),..., ( ),..., 2 ( ), 1 ( ( N n Γ b b b b b      (4)