# Model-Free Data Mining of Families of Rotating Machinery
### Background
#### Model-free  
for us means that we do not have any knowledge of the harmonic structure of the machines or bearing information, we only assume a periodic movement at some operating frequency. We propose a feature space that is independent of the operating frequency of the machine without any assumptions on the harmonic structure and statistically determine the significant features. Model-free is often associated with neural networks and deep learning [7]. Deep Learning (DL) approaches often operate on the time-domain data directly, combining steps (1-3), e.g., [8] for bearing fault diagnostics, and [9] which presents a general review of time-series classification techniques. DL methods are quite accurate [8] but due to the \"black box\" nature of ML and DL it does not provide engineers any insight as to which features of the data are particularly important to classifications. We perform a model-free approach, identifying frequency patterns by performing statistical analysis in the frequency domain. More specifically, we introduce the notion of the _Harmonic Feature Space_ (HFS), an operating frequency independent domain. In this Harmonic Feature Space we perform statistical reductions [10], including principal component analysis (PCA) [11], resulting in significant dimensionality reduction. We are driven to identify components that maximize the variance. We point out that feature vectors are computed from raw sensor data without any prior understanding of the harmonic structures, fractional harmonics, resonances, or other fault frequencies. Support Vector Machines (SVM) are a go-to tool in classification and work very well for many applications [12] including fault diagnostics [13]. Multi-class SVM suffer from the problem of being sensitive to/requiring class-specific scaling [14]. Additionally, SVM are a supervised learning process, they do not directly provide new insights similarities between machines. From a classification standpoint, there is significant literature comparing neural network approaches to SVM, e.g., [13]. We take and agree that with a well-designed feature extraction SVM performs very well and often outperforms neural networks as a classification tool. While we demonstrate that the proposed harmonic feature space allows for high accuracy training/recall of SVM classifications, we focus on clustering methods. The goal of clustering is to compute subsets of the data, the clusters, grouping data that is similar as defined by some measurement function, usually a Euclidean distance [15; 16; 17]. Several questions are studied, including algorithmic aspects, what is the \"best\" distance measure and what is the \"best\" number of clusters. We are particularly interested in the latter; given a distance measure how many clusters should we group the data into, and how many will represent the natural clusterings of the data? This is not a trivial question. As mentioned previously, we use a hierarchical approach to obtain insights into the machine's family, but the \"correct\" family groupings are open to interpretation. Known methods of determining the optimal number of clusters include the Elbow method (which we use for the components later in this paper), information criterion approaches, and information-theoretic approaches. We find our answer by exploring the conditioned or Bayesian entropy. Entropy-type measures for the heterogeneity of clusters are well known [18]. As we have labels for the data set we can derive clustering goals: recordings of the same machine should be in the same cluster, and recordings corresponding to different labels should be in different clusters. Note that we do necessarily want to have clusterings identical to the labels, it isacceptable, if not ideal, to have a certain class of machine split into more than one cluster as that gives us insight into the nature of the data. The conditions are hence derived based on prior information about the clustering we want.