Metric (mathematics)

About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.

Data Construction and Data Analysis for Survey Research

Abstract: The Nature of Survey Data The Measurement of Variables Error in Measurement The Data Matrix Statistical Procedures for Analysing the Data Matrix Tables and Charts for Categorical Variables Tables and Charts for Metric Variables Data Reduction for Categorical Variables Data Reduction for Metric Variables Statistical Inference for Categorical Variables Statistical Inference for Metric Variables Testing Hypotheses and Explaining Relationships Strategies for Analysing a Data Matrix Analysing Qualitative Data

Fuzzy Clustering of Short Time-Series and Unevenly Distributed Sampling Points

Abstract: This paper proposes a new algorithm in the fuzzy-c-means family, which is designed to cluster time-series and is particularly suited for short time-series and those with unevenly spaced sampling points. Short time-series, which do not allow a conventional statistical model, and unevenly sampled time-series appear in many practical situations. The algorithm developed here is motivated by common experiments in molecular biology. Conventional clustering algorithms based on the Euclidean distance or the Pearson correlation coefficient are not able to include the temporal information in the distance metric. The temporal order of the data and the varying length of sampling intervals are important and should be considered in clustering time-series. The proposed short time-series (STS) distance is able to measure similarity of shapes which are formed by the relative change of amplitude and the corresponding temporal information. We develop a fuzzy time-series (FSTS) clustering algorithm by incorporating the STS distance into the standard fuzzy clustering scheme. An example is provided to demonstrate the performance of the proposed algorithm.

Hyperspectral Remote Sensing Image Subpixel Target Detection Based on Supervised Metric Learning

Abstract: The detection and identification of target pixels such as certain minerals and man-made objects from hyperspectral remote sensing images is of great interest for both civilian and military applications. However, due to the restriction in the spatial resolution of most airborne or satellite hyperspectral sensors, the targets often appear as subpixels in the hyperspectral image (HSI). The observed spectral feature of the desired target pixel (positive sample) is therefore a mixed signature of the reference target spectrum and the background pixels spectra (negative samples), which belong to various land cover classes. In this paper, we propose a novel supervised metric learning (SML) algorithm, which can effectively learn a distance metric for hyperspectral target detection, by which target pixels are easily detected in positive space while the background pixels are pushed into negative space as far as possible. The proposed SML algorithm first maximizes the distance between the positive and negative samples by an objective function of the supervised distance maximization. Then, by considering the variety of the background spectral features, we put a similarity propagation constraint into the SML to simultaneously link the target pixels with positive samples, as well as the background pixels with negative samples, which helps to reject false alarms in the target detection. Finally, a manifold smoothness regularization is imposed on the positive samples to preserve their local geometry in the obtained metric. Based on the public data sets of mineral detection in an Airborne Visible/Infrared Imaging Spectrometer image and fabric and vehicle detection in a Hyperspectral Mapper image, quantitative comparisons of several HSI target detection methods, as well as some state-of-the-art metric learning algorithms, were performed. All the experimental results demonstrate the effectiveness of the proposed SML algorithm for hyperspectral target detection.

On the Convergence of the Variable Metric Algorithm

Abstract: The variable metric algorithm is a frequently used method for calculating the least value of a function of several variables. However it has been proved only that the method is successful if the objective function is quadratic, although in practice it treats many types of objective functions successfully. This paper extends the theory, for it proves that successful convergence is obtained provided that the objective function has a strictly positive definite second derivative matrix for all values of its variables. Moreover it is shown that the rate of convergence is super-linear.