Statistical methods time-series analyses

Percival, D.B., Walden, A.T. (2006). Wavelet Methods for Time Series Analysis, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York. 

By executing the steps of the analytical process, we can take advantage of most of the basic methods of chemometrics, e.g., statistics including analysis of variance, experimental design and optimization, regression modeling, and methods of time series analysis. 

Until recently mathematical methods of time series analysis in the environmental sciences have only been used quite rarely the methods have mostly been applied in economic science. Consequently, the mathematical fundamentals of time series analysis are mainly described in textbooks and papers dealing with statistics and econometrics [FORSTER and RONZ, 1979 COX, 1981 SCHLITTGEN and STREITBERG, 1989 CHATFIELD, 1989 BROCKWELL and DAVIS, 1987 BOX and JENKINS, 1976 FOMBYet al., 1984 METZLER and NICKEL, 1986 PANDIT and WU, 1990], This section explains the basic methods of time series analysis and their applicability in environmental analysis. 

It is also a form of time series analysis, which is a growth point in statistics. It is to be expected that as improved and novel statistical methods are developed, accumulated data will be used increasingly for the purposes of quality control and forecasting performance. 

Scenario V—Univariate and multivariate (temporal) In the univariate, temporal case one indicator variable of interest Zjix) is measured at sites jc with i = 1, 2,. .., n, where n represents the number of observations and time t is measured repeatedly at times t= 1,2,..., v. Observations are considered independent in space. Potential methods to analyze such datasets are provided by time series analysis and variants that are based on an empirical statistical approach. In the simplest case the measured time series is treated as a stationary process Z(t) ... 

Measurements that are dependent on each other provide correlated data. Typically, time-dependent processes, such as the time series of glucose concentrations in blood, are of this type of data. Correlated measurements caimot be characterized by the same methods used for description of random independent observations. They require methods of time series analysis where it is assumed that the measurements are realizations of a stochastic process and where they are statistically dependent (Section 3.2). 

If data of the real system is available, the developed simulation model can be tested for similarity with the real system in a qnantitative way (bottom-right cell in Table 4.8). For this purpose, a lot of statistical procedures can be applied depending on the specific object to be tested. Typically, regression techniqnes, distribution tests, or time series analysis methods are used. A reliable qnantitative approach is to generate a forecast of the near future by means of the simulation model which is then compared with the real systems behaviour after the forecast period has expired. This is called predictive validation A mixture of trace analysis and fixed-value test is the trace-driven simulation where a historical situation is simulated. The model s output is compared with the historical records then. 

The time series analysis is a statistical method to reveal the dynamic law of a system through dynamic data (Box 2005). Its core idea is that, with the finite number of data in the series, a model could be created to precisely reflect the dynamic relationship hided in the time series, and then to forecast the future. 

Temporal Data Mining often involves processing time series, typically sequences of data, which measure values of the same attribute at a sequence of different time points. Pattern matching using such data, where we are searching for particular patterns of interest, has attracted considerable interest in recent years. In addition to traditional statistical methods for time series analysis, more recent work on sequence processing has used association rules developed by the database community. In addition Temporal Data Mining may involve exploitation of efficient... 

Basically, four main areas of methods for gear fault detection have been published. Signal processing techniques. Statistical analysis (ANDRADE, ESAT, BADI, 2001 BAYDAR eta/., 2001 TUMER HUFF, 2003 HE, KONG, YAN, 2007), Time-series analysis (ZHAN JARDINE, 2005 ZHAN, MAKIS, JAR-DINE, 2006) and Artificial neural networks AYA ESAT, 1997 SAMANTA, 2004 SANZ, PERERA, HUERTA, 2007 RAFIEE et cd., 2007). 

Barnett V, Lewis T (1994) Outliers in statistical data, 3rd edn. Wiley, Chichester Bloomfield P (2000) Fourier analysis of time series an introduction, 2nd edn. Wiley, New York Box GE, Jenkins GM (1970) Time series analysis, forecasting, and control. Holden-Day, Oakland Box GE, Hunter WG, Hunter IS (1978) Statistics for experimenters an introduction to design, data analysis, and model building. Wiley, New York Brillo J (2007) Excel for scientists and engineers numerical methods. Wiley, Hoboken Chambers JM, Cleveland WS, Kleiner B, Tukey P (1983) Graphical methods for data analysis. 

ProbabiUstic methods are based on time series analysis and synthesis. They combine deterministic and statistical analysis, and they synthesize a time (or space) series of stochastic variables and the effects of a limited number of data. It is assumed that the series represents both definable causes and an unknown number of stochastic causes, and that the stochastic causes are reasonably independent. With these methods, jumps, trends and outliers of the data set can be adequately taken into account. It is emphasized that the data used in probabilistic evaluations are based on actual measurements or variables. As with deterministic methods, probabilistic methods should be used in conjunction with engineering judgement when it is feasible, they should be checked by the use in parallel of a simplified deterministic analysis. 

Data and information gathered was exploited within DaCoTA for the estimation of road traffic fatahties based on time-series analysis, as it is important to know in what direction the annual casualties are developing, and how fast this development is expected to go. The methods applied to achieve the forecasts are sophisticated statistical tools, not easily understood by non-experts [THO 13], The forecast resnlts, however, are of direct interest for road safety practitioners with all levels of statistical expertise, therefore it was decided not only to develop a technical description of the forecasting model and of the process that led to its selection for each conntry, bnt also the Country Forecast Fact Sheets pUP 12], The forecast factsheets are meant to give a relatively non-technical description of the past development of the fatalities (and of the exposure if available). The toad traffic fatalities, the traffic volume and the fatality risks are forecasted to 2020 and also forecasts according to mobihty scenarios are carried out for all 30 European countries, with exposure as most important ejqrlaining variable. If known, the (possible) reasons for the developments are shortly described. Forecasts of the road safety situation in every country include a description of the method adopted to produce these forecasts. 

Statistical Analysis of Coatings Degradation by Time-Series Methods... 

In an attempt to address these questions, a modern method of statistical physics was recently applied by Varotsos et al. (2007) to C02 observations made at Mauna Loa, Hawaii. The necessity to employ a modern method of C02 data analysis stems from the fact that most atmospheric quantities obey non-linear laws, which usually generate non-stationarities. These non-stationarities often conceal existing correlations between the examined time series and therefore, instead of applying the conventional Fourier spectral analysis to atmospheric time series, new analytical techniques capable of eliminating non-stationarities in the data should be utilized (Hu et al., 2001 Chen et al., 2002 Grytsai et al., 2005). 

In this section the notion of an allometric relation is generalized to include measures of time series. In this view, y is interpreted to be the variance and x the average value of the quantity being measured. The fact that these two central measures of a time series satisfy an allometric relation implies that the underlying time series is a fractal random process and therefore scales. It was first determined empirically that certain statistical data satisfy a power-law relation of the form given by Taylor [17] in Eq. (1), and this is where we begin our discussion of the allometric aggregation method of data analysis. 

The analysis of available and relevant data typically requires a set of statistical methods. They are applied to extract deeper knowledge about the processes to be modelled. The set of methods is vast and the choice of methods depends on the needs of the speciflc project. Time series methodology is one prominent branch of methods. 

The topic of time-series is relevant for bioimpedance and especially bioelectricity analysis, but is not covered in this chapter on statistical methods. The methods covered are applicable for time-series after a suitable approach is employed for parameterization. All the examples in the text are given for bioimpedance measurements. 

Semi-automatic software These are moderately priced but they require the user to have some basic knowledge of forecasting principles and fechniques. Here, fhe user has to select an appropriate forecasting technique based on the analysis of time series data. The software will then compute the optimal parameters for the chosen method using some measure of forecasf error. It also gives the forecasts and all the statistics, such as MAD, MAPE, MSE, Bias, etc. The software makes no recommendation as to which forecasting technique is appropriate for the given data. 

The tools of chemometrics encompass not only the familiar (univariant) methods of statistics, but especially the various multivariant methods, together with a package of pattern-recognition methods for time-series analyses and all the known models for signal detection and signal processing. Chemometric methods of evaluation have now become an essential part of environmental analysis, medicine, process analysis, criminology, and a host of other fields. 

Muscolino G, Palmeri A (2005) Maximum response statistics of MDOF linear structures excited by non-stationary random processes. Comput Method Appl Mech Eng 194 1711-1737 Priestley MB (1999) Spectral analysis and time series. Academic, London... 

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