Modified Mahalanobis Distance Research Articles

Sequential covariate-adjusted randomization via hierarchically minimizing Mahalanobis distance and marginal imbalance.

In comparative studies, covariate balance and sequential allocation schemes have attracted growing academic interest. Although many theoretically justified adaptive randomization methods achieve the covariate balance, they often allocate patients in pairs or groups. To better meet the practical requirements where the clinicians cannot wait for other participants to assign the current patient for some economic or ethical reasons, we propose a method that randomizes patients individually and sequentially. The proposed method conceptually separates the covariate imbalance, measured by the newly proposed modified Mahalanobis distance, and the marginal imbalance, that is the sample size difference between the 2 groups, and it minimizes them with an explicit priority order. Compared with the existing sequential randomization methods, the proposed method achieves the best possible covariate balance while maintaining the marginal balance directly, offering us more control of the randomization process. We demonstrate the superior performance of the proposed method through a wide range of simulation studies and real data analysis, and also establish theoretical guarantees for the proposed method in terms of both the convergence of the imbalance measure and the subsequent treatment effect estimation.

Effect of the nature of uncertainty on the optimization of collision avoidance maneuvers

With the soaring number of space debris, collision avoidance maneuvers have become crucial to ensure the safety of any operating satellite. In real-world scenario, the problem gets a little more complicated due to the presence of uncertainty in the values of the states of the space objects. Also, the uncertainty distribution assigned to a given state variable evolves over time. This paper studies the effects of linear and nonlinear propagation of the uncertainty distributions on the collision avoidance maneuver optimization. These different methods for error propagation lead to differing shapes and sizes of the error distribution, resulting in differing regions to be avoided during the encounter, thus resulting in different optimal Δv requirements. In the method using nonlinear propagation of uncertainty distribution, the uncertainty of the passive object is propagated using Monte Carlo samples while that of the operating spacecraft is evolved using Unscented Transform. The sigma points from the Unscented Transform of the operating spacecraft are required to lie outside the stretch of most of the debris uncertainty while implementing cumulative density function. This also results in defining a new probabilistic measure which is like a modified Mahalanobis distance but for a non-Gaussian distribution. The other approach uses linearly propagating the error covariances and a combined error ellipsoid to ensure the maintenance of a desired Mahalanobis distance of the hard body from the uncertainty distribution to ascertain safety. Both the approaches consider nonlinear state dynamics while incorporating the perturbing forces to find the optimal thrust profile and the corresponding steer angles. The approach implementing nonlinear propagation of uncertainty and using cumulative density to ensure that the sigma points of the satellite avoid the debris’ uncertainty distribution is found to require lower Δv compared to the other method while also representing the error propagation better. However, the method implementing linear propagation of uncertainty and using Mahalanobis distance of the hard body from the combined error distribution to frame the avoidance constraint proves to be computationally much faster.

Double-suppressed possibilistic fuzzy Gustafson–Kessel clustering algorithm

Possibilistic fuzzy c-means clustering (PFCM) is an unsupervised hybrid clustering algorithm, which can partly inherit the stability of fuzzy c-means clustering (FCM) algorithm and the noise robustness of possibilistic c-means clustering (PCM) algorithm. However, there are still several limitations in the PFCM when clustering complex data with multiple characteristics: sensitivity to strong noise, coincident clustering problem in multi-class clustering, and disability to deal with strong correlations of feature components. Therefore, a double-suppressed possibilistic fuzzy Gustafson–Kessel​ clustering algorithm (DS-PFGK) is presented. Firstly, we design a new objective function with a modified Mahalanobis distance and an improved weight parameter setting method based on the stability of fuzzy memberships and the anti-noise ability of possibilistic memberships. Moreover, combined with the absolute attribute of possibilistic memberships and the covariance matrix in the modified Mahalanobis distance, an ellipsoidal cluster core which can depict the distribution of sample data is generated to divide each cluster into different regions for selecting appropriate learning objects and identifying noisy data adaptively. Then, a “double-suppressed competitive learning” strategy is designed for the selected learning objects to solve the coincident clustering problem in multi-class clustering by suppressing the nonwinner possibilistic memberships, and to reduce the iteration number by suppressing nonwinner fuzzy memberships and rewarding winner fuzzy memberships. Finally, a segmentation algorithm on noisy color images based on the DS-PFGK is proposed based on the generated cluster cores. Experiments on synthetic datasets, real datasets, and color images show the good performance of the proposed methods compared with several classical and state-of-the-art clustering algorithms.

A Modified Mahalanobis Distance Metric Used to Optimize Mahalanobis Space and Improve Classification Performance of MTS

The Mahalanobis–Taguchi system (MTS) is a diagnostic and forecasting technique in a multidimensional system that integrates the Mahalanobis distance (MD) and robust engineering of Taguchi. To implement MTS, a set of observations from a normal group is selected to construct the Mahalanobis space (MS). With this MS as a reference, new observations from an unknown group can be judged to be normal or abnormal, and the degree of abnormality can be determined. MD is very sensitive to data changes, so the data quality of normal samples used to construct the MS directly affects the accuracy of classification. In practical applications, the selection of normal samples depends on the experience and subjective judgment of experts and lacks an objective selection mechanism. In this paper, a modified MD metric is proposed, which is combined with the individual control chart to obtain a robust MS. First, the initial MS is constructed according to the normal samples selected by experts, and the MD of each normal sample is calculated by the initial MS. Then, the MD of each normal sample in the corresponding reduced MS is computed, and the incremental MD is used as the new distance metric to establish the individual control charts. The stability rules of the control chart are employed to eliminate abnormal points, and the MS of the stable state is obtained. To evaluate the effectiveness of the modified MD metric, a numerical simulation experiment is implemented, and the results show that the proposed method is effective and improves the classification performance of MTS. Finally, the improved MTS method is applied to a real medical diagnosis case.

Ridge rerandomization: An experimental design strategy in the presence of covariate collinearity

Randomization ensures that observed and unobserved covariates are balanced, on average. However, randomizing units to treatment and control often leads to covariate imbalances in realization, and such imbalances can inflate the variance of estimators of the treatment effect. One solution to this problem is rerandomization – an experimental design strategy that randomizes units until some balance criterion is fulfilled – which yields more precise estimators of the treatment effect if covariates are correlated with the outcome. Most rerandomization schemes in the literature utilize the Mahalanobis distance, which may not be preferable when covariates are high-dimensional or highly correlated with each other. As an alternative, we introduce an experimental design strategy called ridge rerandomization, which utilizes a modified Mahalanobis distance that addresses collinearities among covariates. This modified Mahalanobis distance has connections to principal components and the Euclidean distance, and – to our knowledge – has remained unexplored. We establish several theoretical properties of this modified Mahalanobis distance and our ridge rerandomization scheme. These results guarantee that ridge rerandomization is preferable over randomization and suggest when ridge rerandomization is preferable over standard rerandomization schemes. We also provide simulation evidence that suggests that ridge rerandomization is particularly preferable over typical rerandomization schemes in high-dimensional or high-collinearity settings.

Feature Linkage Weight Based Feature Reduction using Fuzzy Clustering Method

In this paper, a novel Feature-Reduction Fuzzy C-means (FRFCM) with Feature Linkage Weight (FRFCM-FLW) algorithm is introduced. By the combination of FRFCM and feature linkage weight, a new feature selection model is developed, called a Feature Linkage Weight Based FRFCM using fuzzy clustering. The larger amounts of features are superior to the complication of the problem, and the larger the time that is exhausted in creating the outcome of the classifier or the model. Feature selection has been established as a high-quality method for preferring features that best describes the data under certain criteria or measure. The proposed method presents three stages namely, 1) Data Formation: The process of data collection and data cleaning; 2) FRFCM-FLW. The proposed method can decrease feature elements routinely, and also construct excellent clustering results. The proposed method calculates a novel weight for every feature by combining modified Mahalanobis distance with feature δm variance in FRFCM algorithm; 3) Fuzzy C-means (FCM) cluster. The proposed FRFCM-FLW method proves high Accuracy Rate (AR), Rand Index (RI) and Jaccard Index (JI) ratio when compared to other feature reduction algorithms like WFCM, EWKM, WKM, FCM and FRFCM algorithms.

Local anomaly detection and quantitative analysis of contaminants in soybean meal using near infrared imaging: The example of non-protein nitrogen.

The melamine scandal indicates that traditional targeted detection methods only detect the specifically listed forms of contamination, which leads to the failure to identify new adulterants in time. In order to deal with continually changing forms of adulterations in food and feed and make up for the inadequacy of targeted detection methods, an untargeted detection method based on local anomaly detection (LAD) using near infrared (NIR) imaging was examined in this study. In the LAD method, with a particular size of window filter and at a 99% level of confidence, a specific value of Global H (GH, modified Mahalanobis distance) can be used as a threshold for anomalous spectra detection and quantitative analysis. The results showed an acceptable performance for the detection of contaminations with the advantage of no need of building a 'clean' library. And, a high coefficient of determination (R2LAD = 0.9984 and R2PLS-DA = 0.9978) for the quantitative analysis of melamine with a limit of detection lower than 0.01% was obtained. This indicates that the new strategy of untargeted detection has the potential to move from passive to active for food and feed safety control.

Anomaly detection of tripod shafts using modified Mahalanobis distance

The tripod shafts used in high-speed trains transmit the power of a motor to axle reduction units. A tripod shaft is designed as a shaft with a protective fuse because the load applied to the reduction units is very large during operation. Fuse failure can lead to accidents and economic loss, such as train service interruption and repair problems at the accident site. Therefore, detecting the faults of fuses used in tripod shafts by using sensors is necessary. In this study, torsion fatigue tests for the failure modes were conducted, and strain data were collected by strain sensors from around the fuse of the shaft. The modified Mahalanobis distance was employed as the health indicator, and the fuse thresholds for the anomaly detection (AD) were defined using the three-sigma method. The AD performance of the tripod shafts was evaluated by comparing the results with real crack detection times.

Root cause diagnosis of quality-related faults in industrial multimode processes using robust Gaussian mixture model and transfer entropy

Abstract Modern complex industrial processes often have multiple operating modes due to various factors, such as different manufacturing strategies, alterations of feedstock and compositions, etc. In this paper, a practical technology or solution of quality-related fault diagnosis is put forward for industrial multimode processes. Different from traditional data-based fault diagnosis methods, the alternative approach is focused more on root cause diagnosis. The new scheme addresses the quality-related fault detection issue with a developed robust Gaussian mixture model and modified Mahalanobis distance. Then, a Bayesian inference-based robust Gaussian mixture contribution index is designed to analyze the potential root-cause variables. Meanwhile, a combination of transfer entropy and direct transfer entropy-based cause and effect extraction methodologies is proposed for root cause diagnosis of quality-related faults. Finally, the whole proposed framework is applied to a real industrial multimode finishing mill process, where the performance and effectiveness are further demonstrated from real industrial data.

Clustering Accuracies on Concepts of Mathematics

Fuzzy c-means algorithm (FCM) based on Euclidean distance function converges to a local minimum of the objective function, which can only be used to detect spherical structural clusters. Gustafson-Kessel clustering algorithm and Gath-Geva clustering algorithm were developed to detect non-spherical structural clusters. However, Gustafson-Kessel clustering algorithm needs added constraint of fuzzy covariance matrix, Gath-Geva clustering algorithm can only be used for the data with multivariate Gaussian distribution. In GK-algorithm, modified Mahalanobis distance with preserved volume was used. However, the added fuzzy covariance matrices in their distance measure were not directly derived from the objective function. Improved Normalized Mahalanobis Clustering Algorithm Based on FCM by taking a new threshold value and a new convergent process is proposed. The experimental results of real data sets show that our proposed new algorithm has the best performance. Not only replacing the common covariance matrix with the correlation matrix in the objective function in the Normalized Mahalanobis Clustering Algorithm.

Clustering with fuzzy supervised algorithm

In GK-algorithm, modified Mahalanobis distance with preserved volume was used. However, the added fuzzy covariance matrices in their distance measure were not directly derived from the objective function. A Fuzzy C-Means algorithm based on Mahalanobis distance (FCM-M) was proposed to improve those limitations of GG and GK algorithms, but it is not stable enough when some of its covariance matrices are not equal. In this paper, an improved Supervised Clustering Algorithm Based on FCM by taking a new threshold value and a new convergent process is proposed. The experimental results of real data sets show that our proposed new algorithm has the best performance. Not only replacing the common covariance matrix with the correlation matrix in the objective function in the Supervised Clustering Algorithm.

Latent variables and space-time models for environmental problems

During the conference of the Italian Statistical Society on ‘‘Advances in Latent Variables’’ held in Brescia, Italy, June 19–21, 2013, the working group on environmental statistics, named GRASPA (http://www.graspa.org), organized the special track on ‘‘Space and space-time models: methods and environmental applications’’. This scientific activity was intended to celebrate the new condition as a permanent group of the Italian statistical society and renew GRASPA’s long life, which dates back to 1989. The idea of a special issue on the track main theme arose considering not only the number and scientific quality of the track contributions, but also the general interest of the topic, which exceeds the conference coverage. The result, raising from both track attendance and space-time statistical scientific community, is an interesting blend of various topics, which amounts at fourteen papers covering various environmental problems. To a large extent climate problems are considered, including upper atmosphere monitoring, rain simulation, air and water temperature and sea currents. Moreover, applications considered water body ecology, seismology and radon emissions. From the methodological point of view, areas covered by the special issue are scalar and vector-valued stochastic processes with continuous index over space or over spacetime. Moreover, functional data, point processes and time series methods are taken into account. Latent variables enter in a natural way in many of these papers either as spatial or temporal components. This methodological key is loosely used in the rest of this editorial to briefly review the special issue contributions. As for spatial processes methods, the work by RuizMedina and Porcu (2014) puts the theoretical basis for understanding the equivalence of Gaussian measures that index multivariate Gaussian fields. Their work will open the lines for research in estimation of Gaussian fields through tapering techniques as well as for assessing the properties of estimators of spatial dependence under infill asymptotics. For binary spatial data, which are built on the basis of a truncated latent Gaussian model, Bevilacqua et al. (2014) explore the possibility of building Euclidean likelihoods, obtaining high computational benefits and ensuring a reasonable level of statistical efficiency. In the same framework of spatial data, Verdin et al. (2014) introduce a stochastic weather generator for the variables of minimum and maximum temperature, as well as precipitation occurrence. In particular, temperature variables are modeled in a vector autoregressive framework, conditional on precipitation occurrence, whilst this last arises via a probit model. Both temperature and occurrence are spatially correlated using spatial Gaussian processes. Fontanella et al. (2014) consider a generalized latent-spatialquantile regression (GLSQR) model for the understanding of indoor radon gas monitoring in central Italy. Vallejo et al. (2014) consider a method for image landscape classification based on the assumption that the vector of image bands is a spatial multivariate process. They build such classification using the divergence of a modified Mahalanobis distance, given by the codispersion matrix. Methods for space-time processes are faced with great detail in this issue. If spatial design represents a critical issue, the paper by Stehlik et al. (2014) gives a clear picture A. Fasso (&) University of Bergamo, Viale Marconi 5, 24044 Dalmine BG, Italy e-mail: alessandro.fasso@unibg.it

A multiparameter aerosol classification method and its application to retrievals from spaceborne polarimetry

AbstractClassifying observed aerosols into types (e.g., urban‐industrial, biomass burning, mineral dust, maritime) helps to understand aerosol sources, transformations, effects, and feedback mechanisms; to improve accuracy of satellite retrievals; and to quantify aerosol radiative impacts on climate. The number of aerosol parameters retrieved from spaceborne sensors has been growing, from the initial aerosol optical depth (AOD) at one or a few wavelengths to a list that now includes AOD, complex refractive index, single scattering albedo (SSA), and depolarization of backscatter, each at several wavelengths, plus several particle size and shape parameters. Making optimal use of these varied data products requires objective, multidimensional analysis methods. We describe such a method, which makes explicit use of uncertainties in input parameters. It treats an N‐parameter retrieved data point and its N‐dimensional uncertainty as an extended data point, E. It then uses a modified Mahalanobis distance, DEC, to assign an observation to the class (cluster) C that has minimum DEC from the point. We use parameters retrieved from the Aerosol Robotic Network (AERONET) to define seven prespecified clusters (pure dust, polluted dust, urban‐industrial/developed economy, urban‐industrial/developing economy, dark biomass smoke, light biomass smoke, and pure marine), and we demonstrate application of the method to a 5 year record of retrievals from the spaceborne Polarization and Directionality of the Earth's Reflectances 3 (POLDER 3) polarimeter over the island of Crete, Greece. Results show changes of aerosol type at this location in the eastern Mediterranean Sea, which is influenced by a wide variety of aerosol sources.

A multivariate geostatistical approach for landscape classification from remotely sensed image data

This paper proposes a methodology to address the classification of images that have been acquired from remote sensors. One common problem in classification is the high dimensionality of multivariate characteristics. The methodology we propose consists of reducing the dimensionality of the spectral bands associated with a multispectral satellite image. Such dimensionality reduction is accomplished by the use of the divergence of a modified Mahalanobis distance. Instead of using the covariance matrix of a multivariate spatial process, the codispersion matrix is considered which have some desirable asymptotic properties under very precise conditions. The consistency and asymptotic normality hold for a general class of processes that are a natural extension of the one-dimensional spatial processes for which the asymptotic properties were first established. The results allow the selection of a set of spectral bands to produce the highest value of divergence. Then, a supervised maximum likelihood method using the selected spectral bands is employed for landscape classification. An application with a real LANDSAT image is introduced to explore and visualize how our method works in practice.

Clustering Accuracies on Concepts of Nursing

The purpose of this study is to integrate fuzzy clustering algorithm based on common Mahalanobis distance. Fuzzy clustering could distinguish characteristics of concept structures on nursing. Finally, some limitations and suggestions of this study are discussed. It shows that knowledge structures will be feasible for remedial instruction and help students get more chance to get professional certification on nurses. Based on the findings and results, combined with fuzzy clustering algorithm based on normalized Mahalanobis distance could be very feasible for cognition diagnosis in the future. The algorithm provided by Bezdek (1) is used in this study. Fuzzy c-mean algorithm based on Euclidean distance function converges to a local minimum of the objective function, which can only be used to detect spherical structural clusters. Gustafson-Kessel clustering algorithm and Gath-Geva clustering algorithm were developed to detect non-spherical structural clusters. Gustafson-Kessel clustering algorithm needs added constraint of fuzzy covariance matrix, whereas Gath-Geva clustering algorithm can only be used for the data with multivariate Gaussian distribution. In GK-algorithm, modified Mahalanobis distance with preserved volume was used. However, the added fuzzy covariance matrices in their distance measure were not directly derived from the objective function. The fuzzy covariance matrices in the Mahalanobis distance can be directly derived by minimizing the objective function. II. LITERATURE REVIEW FCM can only work well for spherical shaped clusters. In the objective function the distances between data points to the centers of the clusters are calculated by Euclidian distances. To overcome the above drawback, we could try to extend the distance measure to Mahalanobis distance (MD). However, Krishnapuram and Kim (1999) pointed out that the Mahalanobis distance can not be used directly in clustering algorithm. Gustafson and Kessel (1979) extended the Euclidian distances of the standard FCM by employing an adaptive norm, in order to detect clusters of different geometrical shape without changing the clusters' sizes in one data set. Gath-Geva (GG) fuzzy clustering algorithm is an extension of Gustafson-Kessel (GK) fuzzy clustering algorithm, and also takes the size and density of clusters for classification. Hence, it has better behaviors for irregular features. For improving the limitation of GK algorithm and GG algorithm, we added a regulating factor of covariance matrix to each class in the objective function, and deleted the constraint of the determinant of covariance matrices. We can obtain the Fuzzy C-Means based on adaptive Mahalanobis distance (FCM-M) as following (2, 3). For improving the stability of the clustering results, we replace all of the covariance matrices with the same common covariance matrix in the objective function in the FCM-M algorithm, and then, an improved fuzzy clustering method, called the Fuzzy C-Means algorithm based on common Mahalanobis distance (FCM-CM) is proposed.

Supervised Clustering Algorithm for University Student Learning Algebra

The popular fuzzy c-means algorithm based on Euclidean distance function converges to a local minimum of the objective function, which can only be used to detect spherical structural clusters. Gustafson-Kessel clustering algorithm and Gath-Geva clustering algorithm were developed to detect non-spherical structural clusters. However, Gustafson-Kessel clustering algorithm needs added constraint of fuzzy covariance matrix, Gath-Geva clustering algorithm can only be used for the data with multivariate Gaussian distribution. In GK-algorithm, modified Mahalanobis distance with preserved volume was used. However, the added fuzzy covariance matrices in their distance measure were not directly derived from the objective function. In this paper, an improved Normalized Supervised Clustering Algorithm Based on FCM by taking a new threshold value and a new convergent process is proposed. The experimental results of real data sets show that our proposed new algorithm has the best performance. Not only replacing the common covariance matrix with the correlation matrix in the objective function in the Normalized Supervised Clustering Algorithm.

Using Mahalanobis Clustering Algorithm for College Student Learning Fundamental Mathematics

The popular fuzzy c-means algorithm based on Euclidean distance function converges to a local minimum of the objective function, which can only be used to detect spherical structural clusters. Gustafson-Kessel clustering algorithm and Gath-Geva clustering algorithm were developed to detect non-spherical structural clusters. However, Gustafson-Kessel clustering algorithm needs added constraint of fuzzy covariance matrix, Gath-Geva clustering algorithm can only be used for the data with multivariate Gaussian distribution. In GK-algorithm, modified Mahalanobis distance with preserved volume was used. However, the added fuzzy covariance matrices in their distance measure were not directly derived from the objective function. In this paper, an improved Normalized Mahalanobis Clustering Algorithm Based on FCM by taking a new threshold value and a new convergent process is proposed. The experimental results of real data sets show that our proposed new algorithm has the best performance. Not only replacing the common covariance matrix with the correlation matrix in the objective function in the Normalized Mahalanobis Clustering Algorithm Based on FCM.

Data Association in Multi-target Tracking Using Belief Function

In this paper, we present a new method for data association in multi-target tracking. The representation and the fusion of the information in our method are based on the use of belief function. The proposal generates the basic belief mass assignment using a modified Mahalanobis distance. While the decision making process is based on the extension of the frame of hypotheses. Our method has been tested for a nearly constant velocity target and compared with both the nearest neighbor filter and the joint probabilistic data associations filter in highly ambiguous cases. The results demonstrate the feasibility of the proposal and show improved performance compared to the aforementioned alternative commonly used methods.

The influence of different tempers on the composition of pottery

The modification of the raw clay by the potter to produce a paste suitable for the intended purposes adds a layer of obfuscation to the problem of provenancing the original clay source by chemical fingerprinting. By preparing different pastes from the same commercially available raw clay and their chemical analysis by Instrumental Neutron Activation Analysis, an experiment under controlled conditions (tempers, mixing ratios, firing temperatures and sampling methods) sheds light on the influence of different tempers. The results show that two different sampling procedures (drilling and grinding) have almost no influence on the chemical fingerprint with the exception of the elemental concentrations of As, Zr, and Hf. This may be due to the volatility of the compounds (As) or the presence of zircon crystals (containing Zr and Hf) which are partly lost during drilling. Three different firing temperatures show no significant influence as well. The application of the modified Mahalanobis distance introduced by Beier and Mommsen in 1995 as a statistical filter and the introduction of a ‘dilution factor’ to the raw data show that the influence of quartz-dominated tempers can be filtered out of the data, resolving the underlying chemical fingerprint of the original clay source. At the same time, by mathematically removing the additional spread introduced by dilution, even subtle differences between similar pastes can be resolved by standard multivariate statistical means.

Clustering composition vectors using uncertainty information

AbstractIn the biological and environmental sciences, interest often lies in using multivariate observations to discover natural clusters of objects. In this manuscript, the incorporation of measurement uncertainty information into a cluster analysis is discussed. This study is motivated by a problem involving the clustering of composition vectors associated with each of several chemical species. The observed abundance of each component is available along with its estimated uncertainty (measurement error standard deviation). An approach is proposed for converting the abundance vectors into composition (relative abundance) vectors, obtaining the covariance matrix associated with each composition vector, and defining a Mahalanobis distance between composition vectors that are suitable for cluster analysis. The approach is illustrated using particle size distributions obtained near Houston, Texas in 2000. Computer simulation is used to compare the performance of Mahalanobis‐distance‐based and Euclidean‐distance‐based clustering approaches. The use of a modified Mahalanobis distance along with Ward's method is recommended for use. Copyright © 2007 John Wiley & Sons, Ltd.