A New Generalized Variance Approach for Measuring Multidimensional Inequality and Poverty

The paper suggests a new generalized variance concept for measuring multidimensional inequality of a stratified society, based on multivariate statistical methods, where the members of society form a cloud in the oblique space of dimensions of inequality, such as income, expenditure and property. The cloud presents the multidimensional inequality capsulized in the cloud. The goal is to condense all the inequality information embodied by the cloud into a composite compact metric characterizing both the shape and the inner structure of the cloud. Contrary to the conventional literature that considers multidimensionality as a unidimensional weighted combination of the dimensions, our new composite index measures the inequality of the configuration of the points in the cloud. Our aim is twofold. First, we introduce the Inequality Covariance Matrix (ICM) assigned to the cloud, with elements measuring the correlations among dimensions. Having ICM, we propose the Generalized Variance (GV) of ICM to measure the composite Generalized Variance Inequality (GVI) level. Second, to evaluate the stratum-specific structure of the overall inequality, we suggest a new two-stage procedure. In the first stage, we divide the total GVI into between-groups and within-groups effects. Then, in the second stage the contributions of the strata to the within-groups inequality and, the contributions of the dimensions to the between-groups inequality are calculated. This GVI approach is sensitive to the correlation system, decomposable into stratum effects and, the number of dimensions is not limited. Moreover, including the log-dimensions in the analysis, GVI yields an Entropy Covariance Matrix giving a new Generalized Variance Entropy index. Finally, the GVI of censored poverty indicators means multidimensional poverty measurement. This special complex task is not yet solved in the traditional literature so far.

SELECTION OF THE PARAMETERS OF THE GENERALIZED DISPERSION ALGORITHM FOR MULTIVARIATE STATISTICAL CONTROL OF THE PROCESS SCATTERING

To control the stability of the functioning of aviation equipment units based on the results of monitoring a group of indicators, methods of statistical processes control can be used. In the presence of significant correlations between performance indicators, multivariate methods are used. In this case, the control of the average level of the process is carried out on the basis of the Hotelling algorithm, the control of multivariate scattering is carried out using the generalized variance algorithm. If, according to the conditions of the process, it is necessary to ensure the fastest detection of a violation, then the optimization problem of finding such values of the sample size, sampling frequency and position of the control boundaries is solved that minimizes the average time of the unstable state of the process. The initial data are the number of process indicators monitored, the target value of the generalized variance (estimated from experimental data), the characteristic of the permissible increase in scattering, the intensity of process disturbances (parameter of the Poisson distribution); time to search for a violation after its detection and time to calculate the sample element.

OPTIMIZATION OF PARAMETERS OF GENERALIZED DISPERSION ALGORITHM AT STATISTICAL PROCESS CONTROL

When monitoring a real production process using statistical methods, the question of early detection of violations arises. In most cases, several indicators are monitored simultaneously in the production process, and a change in the values of some indicators leads to a change in others. If there is a dependence of indicators for their monitoring, multivariate statistical control tools are used, in particular generalized variance chart. By varying the parameters of the chart, its efficiency can be significantly increased, this allows minimizing the time the process is in an unstable state.Applying the approach of A. Duncan, which he developed for Shewhart charts, a formula for the expectation of the duration of an unstable state of a process was obtained and a Python program was developed to minimize it. To test the set optimization problem, the calculation of the data of two process indicators is given and the optimal parameters of the generalized variance chart are obtained, at which the duration of the process in an unstable state is minimal.

THE GENERALIZED VARIANCE ALGORITHM APPLICATION IN MULTIDIMENSIONAL STATISTICAL CONTROL OF THE PROCESS BASED ON INDIVIDUAL OBSERVATIONS

In practice, the use of instantaneous samples for statistical control of the process is often impossible due to technical conditions or is not economically feasible. In this case, the process is monitored by individual observations. To control the average level of a multiparametric process based on the Hotelling algorithm, appropriate methods have been developed, but when controlling multidimensional scattering based on the generalized variance algorithm, a number of problems arise. Various approaches to the estimation of the generalized variance are proposed in the article, and their comparative analysis is carried out. The practical application of the proposed algorithms is also considered.

Detection of Landmarks by Mobile Autonomous Robots Based on Estimating the Color Parameters of the Surrounding Area

Remote detection of landmarks for navigation of mobile autonomous robots in the absence of GPS is carried out by low-power radars, ultrasonic and laser rangefinders, night vision devices, and also by video cameras. The aim of the chapter is to develop the method for landmarks detection using the color parameters of images. For this purpose, the optimal system of stochastic differential equations was synthesized according to the criterion of the generalized variance minimum, which allows to estimate the color intensity (red, green, blue) using a priori information and current measurements. The analysis of classical and nonparametric methods of landmark detection, as well as the method of optimal estimation of color parameters jumps is carried out. It is shown that high efficiency of landmark detection is achieved by nonparametric estimating the first Hilbert-Huang modes of decomposition of the color parameters distribution.

THE EFFICIENCY ANALYSIS OF GENERALIZED VARIANCE ALGORITHM UNDER THE TREND OF MULTIDIMENSIONAL SCATTERING

The generalized variance algorithm is used to control the multidimensional scattering of the process. The generalized variance is understood as the determinant of the covariance matrix of the process. The control efficiency is estimated by the average length of the series, that is the number of observations from the moment of violation of the process to the moment of detection of this violation. Both analytical and numerical methods can be used to estimate the average length of the series for different control types. One of the most common types of the process violation is an abrupt or gradual increase in its dispersion. The aim of the article is to estimate the average length of the series of the generalized dispersion map under the trend of multidimensional scattering.

Multi-asset generalized variance swaps in Barndorff-Nielsen and Shephard model

This paper proposes swaps on two important new measures of generalized variance, namely, the maximum eigenvalue and trace of the covariance matrix of the assets involved. We price these generalized variance swaps for Barndorff-Nielsen and Shephard model used in financial markets. We consider multiple assets in the portfolio for theoretical purpose and demonstrate our approach with numerical examples taking three stocks in the portfolio. The results obtained in this paper have important implications for the commodity sector where such swaps would be useful for hedging risk.

Pengembangan Bagan Kendali 3 Sigma Berbasis Simulasi Untuk Pengawasan Variabilitas Proses Multivariat

This article describes a method in developing control charts for generalized variance as a quality statistics in terms of process variability through simulation. Mathematical equation that maps sample size n and number of quality variables p onto constant multiplier of standard deviation K is obtained thorough least square method using simulated data. The constant K for a certain n and p is used for control charts with the upper control limits of UCL=μ+Kσ where μ and σ are the mean and the standard deviation of the generalized variance, respectively. The simulation of finding the constant K is used with the constraint of 3 sigma paradigm.

PENERAPAN IMPROVED GENERALIZED VARIANCE PADA PENGENDALIAN KUALITAS PAVING BLOCK SEGIENAM

In quality assurance of hexagonal paving block products, quality control is needed so the products that produced are in accordance with the specified standards. Quality control carried out involves two interconnected quality characteristics, that is thickness and weight of hexagonal paving blocks, so multivariate control chart is used. Improved Generalized Variance control chart is a tool used to control process variability in multivariate manner. Variability needs to be controlled because of in a production process, sometimes there are variabilities that caused by engine problems, operator errors, and deffect in raw materials that affect the process. The purpose of this study is to apply Improved Generalized Variance control chart in controlling the quality of hexagonal paving block products and calculating the capability of production process to meet the standards. Based on the assumption of multivariate normal distribution test, it can be seen that the data of quality characteristics of hexagonal paving blocks have multivariate distribution. While based on the correlation test between variables it can be concluded that the characteristics of the quality of thickness and weight correlate with each other. The result of the control using these control chart shows that the process is statistically in control. The results of process capability analysis show that the production process has been running according to the standard because the process capability index value is generated using a weighting of 0.5 for each quality characteristic that is 1.01517. Keywords: Paving Block, Quality Control, Variability, Improved Generalized Variance, Process Capability Analysis