Statistical Pattern Recognition for Decision Making

Based on the values of the extracted damage-sensitive features or CPs, a decision has to be made about the condition of the structure. More specifically, the damage feature values have to be mapped to particular states of the structure, which are alternatively called classes. In the simplest case, these classes are the "healthy" state and the "damaged" state. Further cases include different structural states, extents of damage or different damage locations. The process of recognizing underlying relationships between damage feature values and structural states, and, subsequently, assigning specific structural states to damage feature values is called pattern recognition. Pattern recognition comprises three theoretical frameworks: the statistical, the syntactic (or structural) and the neural network-based framework [21]. The statistical and the neural network-based frameworks are the two most commonly used in SHM. Some neural network-based approaches for damage detection were presented briefly in section 1.3.5. Syntactic pattern recognition is applied when patterns are difficult to quantify as feature vectors. Instead, it enables describing large sets of complex patterns by using small sets of simple patterns. Some application fields for syntactic pattern recognition are computer vision and image processing.

Statistical pattern recognition (SPR) stems from the fields of statistics and machine learning. Machine learning can be supervised or unsupervised, depending on whether the true state (or class) of the data is available in training. For instance in terms of damage detection, in supervised machine learning, healthy and damaged datasets are used in training. In unsupervised machine learning, on the other hand, only data from the normal condition or the baseline state of the structure is used in training. Supervised machine learning applies to all the four SHM levels, i.e., damage detection, localization, quantification and prognosis, while unsupervised machine learning applies to level I and sometimes to level II. Ideally, SPR algorithms should return their decision with a confidence interval. Depending on the available type of data and information, and, subsequently, the type of machine learning, the following three types of SPR algorithms can be employed: (i) novelty detection algorithms, (ii) classification algorithms and (iii) regression algorithms.

Novelty detection algorithms use unsupervised learning and are employed in two-class problems, where classes are either "healthy" or "damaged". The output indicates deviations from the baseline state or the normal operational condition. Novelty detection can be carried out by defining decision boundaries on damage feature distribution obtained in training and by performing hypothesis testing. Decision boundaries may be demonstrated in control charts based on the

methodology of statictical process control (SPC). Usually, the decision is returned with a confidence interval. Further approaches, which can be employed for novelty detection are one-class support vector machines (SVMs) and novelty indices based on distance measures, such as the Euclidean distance and the Mahalanobis squared distance (MSD).

Classification algorithms use supervised learning and are used in two-class or multi-class problems. In this case, the output is the class label (e.g., "healthy" or "damaged"). It is common practice to perform hypothesis testing by setting decision boundaries based on the probabilities of the individual classes. Further classification algorithms include SVM, neural networks and decision trees [75].

Regression algorithms use supervised learning and output continuous variables. Regression analysis includes linear models, polynomial models and AR models, as well as support vector machines (SVMs) and neural networks [16]. Since the present work focuses on novelty detection and classification problems, only a brief overview of the state of the art for these two types of algorithms is given.

In an earlier section, it was shown that distance measures can be used for data normalization. In the following bibliographical references, distance measures are used to build novelty indices. Surace et al. use the Euclidean distance between transmissibility functions to detect damage on an offshore platform model and a finite element model of an aircraft wing [76]. The MSD of AR parameters is used for feature classification in [58], where a three-story frame structure is tested under operational and environmental variability. In [77], the MSD of an error deriving from principal component analysis (PCA) is used for damage detection on a footbridge. In [16], MSD is used to build a hybrid condition parameter (CP). Multidimensional CPs, which are composed of one-dimensional CPs, are evaluated based on the MSD for detecting damage on the 3.2 kW LANL wind turbine.

Häckell et al. employs statistical process control and hypothesis testing in the third tier of a modular SHM framework based on unsupervised learning. The upper and lower control limits (UCL and LCL) of control charts are set for various CPs and a series of confidence intervals before hypothesis testing [36]. SPC is also employed in [78]. Control charts are used to observe the AR coefficients during quasi-static cyclic tests of reinforced-concrete bridge columns. An extended reference to hypothesis testing in unsupervised mode is given in [11]. The null hypothesis is defined based on the PDF of damage features and current datasets are evaluated with respect to that for damage detection and localization. This is shown for a series of parametric and non parametric approaches, such as the PSD-based method, the FRF magnitude-based method, the modal parameter based method and the model residue-based method.

One-class SVMs are variations of SVMs which can be used in unsupervised machine learning. One-class SVMs allow the classification of new data points based only on data from one class. In SHM, this implies the classification of a current dataset based on information from the healthy state only. The advantage of this approach is that one-class SVMs are able to create nonlinear decision boundaries

by employing kernel functions. In [79], the concept is described and verified on a steel frame structure. On the other hand, SVMs can be used for supervised damage detection. Kernel functions transform points into other spaces allowing for nonlinear classifiers and decision boundaries. The application of SVMs is presented in [80] for damage detection on bridges, and in [81] for damage classification on ball bearings, in order to distinguish between different damage scenarios, and damage localization on a two-dimensional cantilever truss structure.