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Related Concept Videos

Median01:08

Median

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Besides mean, the median is a widely used measure of central tendency. Typically, median is defined as the central or middle value of a data set, measured by arranging the data elements in an increasing or decreasing order. Since this middle value is not affected by the precise numerical values of the outliers or fluctuations, it is insensitive to them. Hence, in cases where a data set may have outliers or the extreme values are not known, the median is a better measure of the central tendency...
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Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
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In multiple dimensions, the conservation of momentum applies in each direction independently. Hence, to solve collisions in multiple dimensions, we should write down the momentum conservation in each direction separately. To help understand collisions in multiple dimensions, consider an example.
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It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a...
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The measures of central tendency calculated from a data set may not reveal much about its intrinsic distribution. If a plot is made of the data set’s values, the mean and the median may not only differ, but also the plot may have more values on one side of the central tendencies. Such a data set is said to be skewed towards that side.
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Survival trees are a non-parametric method used in survival analysis to model the relationship between a set of covariates and the time until an event of interest occurs, often referred to as the "time-to-event" or "survival time." This method is particularly useful when dealing with censored data, where the event has not occurred for some individuals by the end of the study period, or when the exact time of the event is unknown.
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Depth functions and mutidimensional medians on minimal spanning trees.

Mengta Yang1, Reza Modarres1, Lingzhe Guo1

  • 1Department of Statistics, The George Washington University, Washington, D.C., USA.

Journal of Applied Statistics
|June 16, 2022
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Summary

This study introduces novel data depth and median concepts for multivariate data using minimal spanning trees (MST). These methods offer new ways to analyze complex datasets and understand data distribution.

Keywords:
62G3062G35Data depthdepth functiongraphminimal spanning treemultidimensional median

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Area of Science:

  • Multivariate Data Analysis
  • Computational Statistics
  • Graph Theory Applications

Background:

  • Traditional methods for assessing data depth and centrality in multivariate settings have limitations.
  • The minimal spanning tree (MST) provides a structural representation of data point relationships based on pairwise distances.
  • Exploring graph-based approaches can reveal novel insights into data geometry and distribution.

Purpose of the Study:

  • To introduce new definitions of data depth and medians for multivariate data.
  • To leverage the structure of the minimal spanning tree (MST) for these new statistical measures.
  • To investigate the properties, robustness, and computational aspects of the proposed MST-based methods.

Main Methods:

  • Construction of the minimal spanning tree (MST) for a given multivariate data set.
  • Definition of data depth functions based on the connectivity and edge weights within the MST.
  • Identification of multidimensional medians derived from the MST-based depth.

Main Results:

  • Novel MST-based data depth functions were successfully defined for multivariate data.
  • The study characterized key properties of these new depth functions.
  • Initial investigation into the robustness and computational complexity of MST-based medians was conducted.

Conclusions:

  • MST-based data depth offers a promising new perspective for analyzing multivariate data structures.
  • The proposed multidimensional medians warrant further investigation regarding their statistical properties and applications.
  • This approach provides a valuable tool for understanding data distribution and identifying central tendencies.