# Research on Data-Driven Performance Assessment and Fault Early Warning of Marine Diesel Engine
## 2 Introduction to Performance Assessment and Fault Warning Methods
### Operational Condition Classification Algorithm  
Clustering algorithms demonstrate advantages in analyzing unlabeled marine diesel engine monitoring data through their capacity to automatically determine optimal cluster numbers via performance metrics, eliminating the need for manual intervention. The grid-based CLIQUE (Clustering In QUEst) algorithm, also recognized as a high-dimensional clustering approach, exhibits particular suitability for this study due to its inherent strengths in autonomous processing of high-dimensional datasets and independence from operator-specific domain expertise. The fundamental principle of CLIQUE involves partitioning the multidimensional data space into non-overlapping rectangular cells of equal size. Cluster formation operates through density-based evaluation, where a cell exceeding a predefined density threshold qualifies as a dense unit. Cell density is defined by the number of data points contained within it. The final clusters emerge as maximally connected regions composed of adjacent dense cells [22; 23]. A Priori Properties of the CLIQUE Clustering Algorithm:  
**Theorem 1**. : _Anti-Monotonicity: If a k-dimensional spatial unit is dense, its projection onto any (k - 1)-dimensional subspace is also dense._  
**Theorem 2**. : _If any projection of a k-dimensional spatial unit onto a (k - 1)-dimensional subspace contains one or more sparse units, the k-dimensional unit is necessarily non-dense._  
Three-Step Procedure for CLIQUE Clustering in Multidimensional Space:  
Step 1: Identification of All Dense Subspaces Containing ClustersThe multidimensional space is partitioned into disjoint rectangular units. Dense units are identified using the a priori properties via a bottom-up approach:  
The multidimensional space is first partitioned into disjoint rectangular units, and the a priori properties are then utilized to determine whether these units are dense. The identification of dense units employs a \"bottom-up\" approach, which incrementally processes data from lower to higher dimensions. For instance, once the set of dense units \(D_{k-1}\) in the \((\mathrm{k}-1)\)-dimensional space is identified, the candidate dense unit set \(D_{k}t\) for the \(\mathrm{k}\)-dimensional space can be derived using the a priori properties. If \(D_{k}t\) is empty, the highest-dimensional subspace \(S_{k-1}=D_{1}\times D_{2}\times\ldots\times D_{k-1}\) containing clusters is obtained. If \(D_{k}t\) is non-empty, the \(\mathrm{k}\)-dimensional data are traversed again to eliminate non-dense units in \(D_{k}t\) using the a priori properties, yielding the dense unit set \(D_{k}\) for the \(\mathrm{k}\)-dimensional space. Subsequently, the candidate dense unit set \(D_{k+1}t\) for the \((\mathrm{k}+1)\)-dimensional space is generated. This process iterates until the candidate dense unit set for a certain dimension becomes empty. Step 2: Cluster Identification  
Input: A set D of dense units in a \(\mathrm{k}\)-dimensional subspace. Output: A partition \(\{D_{1},D_{2},\ldots D_{q}\}\) of D into connected components, where no two distinct components \(u_{i}\in D_{i},u_{j}\in D_{j}(i\
eq j)\) are interconnected. This step is analogous to searching for connected components in a graph, where the units in the multidimensional space are treated as vertices, and an edge exists between two units if they are connected. Step 3: Cluster Description Generation  
Input: A dense unit set in a \(\mathrm{k}\)-dimensional subspace, forming a cluster \(\mathrm{C}\). Output: A collection \(\mathrm{R}\) within the same subspace, where \(\mathrm{R}\in\mathrm{C}\), and every unit in \(\mathrm{C}\) belongs to at least one member of \(\mathrm{R}\). For each cluster, generate concise descriptions using the minimal covering hyper-rectangles of \(\mathrm{R}\). From the analysis of the above clustering process, the CLIQUE algorithm implementation for marine diesel engine operational mode classification proceeds through the following systematized workflow:  
1.