Performance metrics (error measures) are vital components of the evaluation frameworks in various fields. The intention of this study was to overview of a variety of performance metrics and approaches to their classification. The main goal of the study was to develop a typology that will help to improve our knowledge and understanding of metrics and facilitate their selection in machine learning regression, forecasting and prognostics. Based on the analysis of the structure of numerous performance metrics, we propose a framework of metrics which includes four (4) categories: primary metrics, extended metrics, composite metrics, and hybrid sets of metrics. The paper identified three (3) key components (dimensions) that determine the structure and properties of primary metrics: method of determining point distance, method of normalization, method of aggregation of point distances over a data set.
A non-vector-based dissimilarity measure is proposed by combining vector-based distance metrics and set operations. This proposed compound dissimilarity measure (CDM) is applicable to quantify similarity of collections of attribute/feature pairs where not all attributes are present in all collections. This is a typical challenge in the context of e.g., fingerprinting-based positioning (FbP). Compared to vector-based distance metrics (e.g., Minkowski), the merits of the proposed CDM are i) the data do not need to be converted to vectors of equal dimension, ii) shared and unshared attributes can be weighted differently within the assessment, and iii) additional degrees of freedom within the measure allow to adapt its properties to application needs in a data-driven way. We indicate the validity of the proposed CDM by demonstrating the improvements of the positioning performance of fingerprinting-based WLAN indoor positioning using four different datasets, three of them publicly available. When processing these datasets using CDM instead of conventional distance metrics the accuracy of identifying buildings and floors improves by about 5% on average. The 2d positioning errors in terms of root mean squared error (RMSE) are reduced by a factor of two, and the percentage of position solutions with less than 2m error improves by over 10%.
Minimax distance measures provide an effective way to capture the unknown underlying patterns and classes of the data in a non-parametric way. We develop a general-purpose framework to employ Minimax distances with any classification method that performs on numerical data. For this purpose, we establish a two-step strategy. First, we compute the pairwise Minimax distances between the objects, using the equivalence of Minimax distances over a graph and over a minimum spanning tree constructed on that. Then, we perform an embedding of the pairwise Minimax distances into a new vector space, such that their squared Euclidean distances in the new space are equal to their Minimax distances in the original space. We also consider the cases where multiple pairwise Minimax matrices are given, instead of a single one. Thereby, we propose an embedding via first summing up the centered matrices and then performing an eigenvalue decomposition. We experimentally validate our framework on different synthetic and real-world datasets.
Measuring similarities between unlabeled time series trajectories is an important problem in many domains such as medicine, economics, and vision. It is often unclear what is the appropriate metric to use because of the complex nature of noise in the trajectories (e.g. different sampling rates or outliers). Experts typically hand-craft or manually select a specific metric, such as Dynamic Time Warping (DTW), to apply on their data. In this paper, we propose an end-to-end framework, autowarp, that optimizes and learns a good metric given unlabeled trajectories. We define a flexible and differentiable family of warping metrics, which encompasses common metrics such as DTW, Edit Distance, Euclidean, etc. Autowarp then leverages the representation power of sequence autoencoders to optimize for a member of this warping family. The output is an metric which is easy to interpret and can be robustly learned from relatively few trajectories. In systematic experiments across different domains, we show that autowarp often outperforms hand-crafted trajectory similarity metrics.
We investigate the use of Minimax distances to extract in a nonparametric way the features that capture the unknown underlying patterns and structures in the data. We develop a general-purpose framework to employ Minimax distances with many machine learning methods that perform on numerical data. For this purpose, first, we compute the pairwise Minimax distances between the objects, using the equivalence of Minimax distances over a graph and over a minimum spanning tree constructed on that. Then, we perform an embedding of the pairwise Minimax distances into a new vector space, such that their squared Euclidean distances in the new space equal to the pairwise Minimax distances in the original space. In the following, we study the case of having multiple pairwise Minimax matrices, instead of a single one. Thereby, we propose an embedding via first summing up the centered matrices and then performing an eigenvalue decomposition. Finally, we perform several experimental studies to illustrate the effectiveness of our framework.