Signal processing time series analysis

There are numerous books on digital signal processing (DSP) and Fourier transforms. Unfortunately, many of the chemically based books are fairly technical in nature and oriented towards specific techniques such as NMR however, books written primarily by and for engineers and statisticians are often quite understandable. A recommended reference to DSP contains many of the main principles [29], but there are several similar books available. For nonlinear deconvolution, Jansson s book is well known [30]. Methods for time series analysis are described in more depth in an outstanding and much reprinted book written by Chatfield [31]. 

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). 

B. Henry, N. Lovell, and F. Camacho, Nonlinear dynamics time series analysis, M. Akay (ed.), in Nonlinear Biomedical Signal Processing, Vol. II, pp. 1-39, IEEE Press, New York, 2000. 

A model which has found application in many areas of time series processing, including audio restoration (see sections 4.3 and 4.7), is the autoregressive (AR) or allpole model (see Box and Jenkins [Box and Jenkins, 1970], Priestley [Priestley, 1981] and also Makhoul [Makhoul, 1975] for an introduction to linear predictive analysis) in which the current value of a signal is represented as a weighted sum of P previous signal values and a white noise term ... 

Sequential signals are surprisingly widespread in chemistry, and require a large number of methods for analysis. Most data are obtained via computerised instruments such as those for NIR, HPLC or NMR, and raw information such as peak integrals, peak shifts and positions is often dependent on how the information from the computer is first processed. An appreciation of this step is essential prior to applying further multivariate methods such as pattern recognition or classification. Spectra and chromatograms are examples of series that are sequential in time or frequency. However, time series also occur very widely in other areas of chemistry, for example in the area of industrial process control and natural processes. 

The analysis of process signals may be facilitated if the time series data can be cast into a symbolic form. The relevant trends and generic data features can then be extracted and monitored using this qualitative representation. Such a transformation is often carried out by defining a set of primitives (alphabet) that define a visual characteristic of the signal [78, 142, 247]. Here, the methodology proposed by Stephanopoulos and coworkers is discussed [9, 34, 35]. They treated the problem of trend representation graphically... 

The determination of arrival time is important to reconstruct the time series of velocity in the data processing stage. The duration of the signal, termed the residence or transit time of the particle, is also important for the data processing as described below. The accuracy requirement on these two quantities lies considerably below that of the signal frequency estimation. In PDA however, the transit time is often used to indirectly estimate the measurement volume size (Saffman 1987 b)) and for this a higher accuracy is required. Several refined techniques have therefore been proposed, also based on spectral analysis (Qiu and Sommerfeld 1992). 

---