CAPÍTULO V. ANÁLISIS Y DISCUSIÓN DE LOS RESULTADOS
5.1. Análisis de Objetivo General
5.1.2. Análisis de Objetivo Específico N° 02
The functions of the data processing and control system include: 1) control and display of the operation of the data acquisition system; 2) pre-processing of the raw signals received from the data acquisition system; 3) data archive into a database or storage media; 4) post-processing of the data; and 5) viewing the data.
3.5.1 Data Acquisition Control
A large-scale SHM system comprises various types of data acquisition hardware. Therefore, centralised data acquisition control is preferable. As described above (Section 3.3.4), the signals can be collected in a long-term or short-term mode. Therefore the data acquisition control unit should be flexible in handling both continuous monitoring mode and scheduled trigger modes. In practical SHM systems, the centralised control unit is located in the central control office and operated by the users for carrying out the communication with the local acquisition facilities via a graphical user’s interface.
The graphical user’s interface program is an interface between the data acquisition hardware and the hardware driver software. It controls the DATS’s operation, such as how and when the DATS collect data, and where to transmit. It provides the users with an easy interface as the details of the hardware are very complicated for most users, for example, civil engineers.
3.5.2 Signal Pre-processing
The collected raw signals are pre-processed prior to permanent storage. The data pre-processing has two primary functions: 1) transforming the digital signals into the monitored physical data; and 2) removing abnormal or undesirable data. Signal transforming is simply done by multiplying the corresponding calibration factor or sensitivities. There are several data-elimination criteria for removal of typical abnormalities associated with various types of statistical data. The source of abnormal data is possibly derived from malfunction of the measurement instrument. There are a few criteria defining abnormal data.
3.5.2.1 Data with Abnormal Magnitude
Extremely large or small data are regarded as abnormal. For example, the ambient temperature of Hong Kong must be within a certain range, and an extremely high or low temperature outside this range recorded by the temperature sensor does not have any physical meaning.
3.5.2.2 Data with Significant Fluctuation
The second criterion to eliminate the abnormal data is set in terms of difference between the maximum and minimum values in a specific period. For example, the change in ambient temperature of one day is within a certain range and an extremely significant increase or drop of temperature is suspicious. This criterion is
xmax−xmin >ε (3.1)
where | | means the absolute value, xmax and xmin denote the maximum and
minimum value, respectively; ε is a real number which defined the limit of the difference between the maximum and minimum values. The adopted values of ε vary for different types of measurement data recorded at different locations.
3.5.2.3 Data with no Variation
It is also observed that some data from vibration measurements are associated with zero standard deviation. A zero standard deviation physically indicates a steady measurement without any fluctuation within the statistical period. As a result, the statistical values of mean, maximum and minimum have the same magnitude. Having a perfectly flat signal might be considered as an abnormal measurement. Correspondingly, the third criterion to eliminate the abnormal data is set in terms of the zero standard deviation of measured records and is given by
xstd=0 (3.2)
where xstd is the standard deviation of measurement in a specific period.
3.5.3 Signal Post-processing and Analysis
The pre-processed signal will be saved into a database system for future management or storage media like hard disks and tapes after proper packaging and tagging. The stored data are processed for various uses. Here we only exemplify a few basic data processing techniques. Other techniques and more advanced analysis can be found in the following chapters or other books.
3.5.3.1 Data Mining
In a long-term SHM system, a huge number of data are recorded from the sensor system. How to extract important features or information is critical to effective use of SHM system for structural condition evaluation. The data mining is a bridge between the data and specific patterns (or features) for decision. The data mining technologies have attracted a great deal of attention in the artificial intelligence community, in which a wider term is knowledge discovery (Fayyad et al., 1996a). Actually data mining can be regarded as a knowledge discovery process.
The data mining is defined as “the nontrivial process of identifying valid, novel, potentially useful, and ultimately understandable patterns in data” (Fayyad
et al., 1996b). The data mining mainly has the following several functions (Fayyad et al., 1996a; Duan and Zhang, 2006):
• Regression: identify the relationships between a set of variables;
• Classification: classify a data item into one of several predefined classes;
• Clustering: identify a finite set of categories or clusters to describe the data
without predefined class labels;
• Summarization: find a compact description for a subset of data;
• Outlier detection: detect data which do not comply with the general
behaviour or model of the data in a database.
A wide variety of data mining methods exist, from conventional statistical methods such as regression analysis, clustering analysis, and principal component analysis to more advanced machine learning methods such as Support Vector Machine, Genetic Algorithm, Bayes Belief theory, Artificial Neural Networks (ANNs), and Colony Algorithms. Sohn et al. (2003) reviewed some applications of these techniques to structural damage detection. Here only the regression
analysis is introduced as it is widely used for preliminary data processing in the context. Interested readers may refer to many textbooks in the area, for example, Kottegoda and Rosso (1997).
The regression analysis investigates the relationship between one variable and one or more other variables. The simplest relation is the linear regression as
y=β0+βxx+εy (3.3)
where x represents the explanatory variable, y is the response variable, β0 (intercept) and βx (slope) are regression coefficients, and εy is the error. With least- squares fitting, the regression coefficients and confidence bounds can be obtained. To examine goodness of fit of the linear relation between x and y, the correlation coefficient, ρ, is defined by
( )
y x y x Cov σ σ ρ= , (3.4)where σ and Cov are standard deviation and covariance, respectively. A higher correlation coefficient implies a good linear relation between the two variables. The linear regression can be easily extended to the multiple linear regression where the equation contains more than one explanatory variable.
3.5.3.2 Frequency Domain Analysis
Frequency domain analysis allows one to examine the data in the frequency domain, rather than in the time domain. It presents the frequency components of a signal and the contributions from each frequency to the signal. Usually the signal can be converted between the time and frequency domains with a pair of transform, for example, Fourier transform and inverse Fourier transform.
Quite often in the SHM, one is interested in the frequency spectrum of loading signals and response signals of a bridge to view their frequency components. For example, acceleration responses of a bridge can reveal the natural frequencies of the bridge. Further, its important vibration characteristics (frequencies, damping, and mode shapes) can be obtained via modal analysis tools, which will be described in Chapter 4.
In practice, signals are captured at a discrete set of times, say, 1/fs, 2/fs, …,
n/fs where n is the total number of data and fs is the sampling rate. Accordingly the discrete Fourier transform is used in signal processing, which transforms a series of signal x(0), x(1), …, x(n-1) into n complex numbers as
∑
− = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 1 0 2 exp ) ( 1 ) ( n k n jk i k x n j F π (3.5)where i is the imaginary unit, and j=0, 1, 2, …, n-1, and the inverse discrete Fourier transform takes the form
∑
− = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 1 0 2 exp ) ( ) ( n j n jk i j F k x π (3.6)The above two equations indicate that complex numbers F(j) represent the amplitude and phase of the different sinusoidal components of signal x(k) while signal x(k) is a sum of sinusoidal components. The amplitude or phase of F(j) represents the spectrum of the time series x(k). Due to the symmetric property, usually only the first half spectrum is of interest. It is noted that the jth item is
associated with the physical frequency (in Hertz) of jfs/n (or circular frequency of 2πjfs/n).
The squared amplitude of the Fourier transform, or power, can be obtained as: 2 ) ( ) (j F j P = (3.7)
The resulting plot is referred to as a power spectrum, indicating the averaged power over the entire frequency range. More common in signal processing, one considers power spectrum density, i.e., the power component of a signal in an infinitesimal frequency band. According to the Wiener–Khinchin theorem, the power spectrum density is the Fourier transform of the autocorrelation function of the signal (theoretically a random signal does not obey the Dirichlet condition and its Fourier transform does not exist, whereas its autocorrelation function obeys the Dirichlet condition and the Fourier transform is valid). That is,
∑
− = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 1 0 2 exp ) ( 1 ) ( n k xx xx n jk i k R n j S π (3.8)where Sxx is the auto-power spectrum density and Rxx is the autocorrelation function taking the form of
Rxx(k)=E
[
x(j)x(j−k)]
(3.9)Similarly, for two discrete signals x and f, their cross-power spectrum can be obtained as the Fourier transform of the cross-correlation function of the two signals. When f and x refer to the input force and output response, respectively, the commonly used frequency response functions (FRFs) can be obtained as
) ( ) ( ) ( j S j S j H ff fx = (3.10)
where Sff and Sfx refer to the auto-power spectrum density of the input force and
cross-power spectrum density between the input force and response, respectively. It is recommended that a window is used to minimise the leakage problem during the transform unless the signal is transient and dies away within the record length. The Hanning window function is commonly used while the exponential window function is suggested for an impact testing (Avitabile, 2001).
The above mentioned frequency analyses, including the Fourier transform, power spectrum, and frequency response functions, are standard techniques and available in spectral analysers.
3.5.3.3 Time-frequency Domain Analysis
An important assumption of the Fourier transform is that the signal is stationary. Some signals in real world are nonstationary. i.e., the signal statistical characteristics vary with time. Examples of this include wind speed signal and earthquake ground motions. Recent advances in the field of signal processing have allowed characterization of the time-frequency properties of nonstationary signals. There are a few well-known time-frequency distributions and analysis tools such as short-time Fourier transform, Wavelet transform, and Hilbert-Huang transform.
The short-time Fourier transform computes the time-dependent Fourier transform of a signal using a sliding window (Oppenheim and Schafer, 1989). The method splits the original signal into overlapping segments and applies the
discrete-time Fourier transform to each segment to produce an estimate of the short-time frequency content of the signal over the given time period.
The Wavelet transform is a new tool in mathematics and has broad applications (Daubechies, 1992). Wavelet functions are composed of a family of basis functions that are capable of describing a signal in a localized time (or space) domain and frequency (or scale) domain. Therefore using wavelets can perform local analysis of a signal, i.e. zooming on any interval of time or space.
The Hilbert-Huang transform was proposed by Huang et al. (1998). It decomposes a signal into a series of intrinsic mode functions with the empirical mode decomposition, and then uses the Hilbert spectral analysis to obtain instantaneous frequency data. The Hilbert-Huang transform is particularly designed for nonstationary and nonlinear processes.
3.5.3.4 Data Fusion
An SHM system usually includes various types of sensors located in different spatial positions. Different types of sensors located in the same position may capture different signals. Spatially distributed sensors may also demonstrate different features of the structure. In addition, different methods and different users may reach different results. Therefore, integration of data from different sensors and integration of results made by different algorithms are important to a robust monitoring exercise (Chan et al., 2006). Data fusion is an important data processing tool to achieve this.
Data fusion is a process that integrates data and information from multiple sources in order to achieve improved information than could be achieved by use of single information alone (Hall, 1992). Fusion processes are often categorized as low, intermediate, and high levels fusion depending on the processing stage at which fusion takes place (Hall and Llinas, 1997). The low level fusion combines raw data from multiple sensors to produce new data that is expected to be more informative and synthetic than the inputs. In the intermediate level fusion or feature level fusion, features are extracted from multiple sensors’ raw data and various features are combined into a concatenated feature vector that may be used by further processing. The high level fusion, also called decision fusion, combines decisions coming from several experts to reach a consistent conclusion. Techniques involved in feature/decision-level data fusion include a wide range of areas such as artificial intelligence, pattern recognition, statistical estimation, information theory. Detailed description of these techniques such as Neural Networks, Bayesian inference, Dempster-Shafer’s methods can be found in corresponding references.