# CentralFeature

CentralFeature[{x1,x2,}]

gives the central feature of the elements .

CentralFeature[{x1v1,x2v2,}]

gives the vi corresponding to the central feature .

CentralFeature[data]

gives the central feature for several different forms of data.

# Details and Options  • CentralFeature is a location measure. It gives a point in the data with the minimum total distance to every other point.
• CentralFeature finds the element that minimizes the sum of distances for the unweighted case and for the weighted case.
• • The data data has the following forms and interpretations:
•  {data1,data2,…} list of data of different formats including numerical, geospatial, textual, visual, dates and times, as well as combinations of these {data1,data2,…}{v1,v2,…} data with indices {v1,v2,…} {data1,data2,…}Automatic take the vi to be successive integers i GeoPosition[…] array of geodetic positions WeightedData[…] data with weights
• The following option can be given:
•  DistanceFunction Automatic the distance metric to use
• The setting for DistanceFunction can be any distance or dissimilarity function or a function f defining a distance between two points.
• By default, the following distance functions are used for different types of elements:
•  EuclideanDistance numeric data ImageDistance images JaccardDissimilarity Boolean data EditDistance text and nominal sequences Abs[DateDifference[#1,#2]]& dates and times ColorDistance colors GeoDistance geospatial data Boole[SameQ[#1,#2]]& nominal data HammingDistance nominal vector data WarpingDistance numerical sequences
• All images are first conformed using ConformImages when the option DistanceFunction is Automatic.
• By default, when data elements are mixed-type vectors, distances are computed independently for each type and combined using Norm.

# Examples

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## Basic Examples(2)

Find the central feature in a list of vectors:

Find the central feature in a list of vectors with given weights:

## Scope(9)

Same inputs with different output formats:

Central feature works with WeightedData:

Central feature of a large array:

Weighted central feature:

Find the central feature of data involving quantities:

Find the central feature of a list of images:

List of pictures:

List of 3D images:

Compute the central feature of strings:

Compute the central feature of Boolean vectors:

Compute the central feature of a list of date objects:

Compute the central feature of geodetic positions:

## Options(2)

### DistanceFunction(2)

By default, Euclidean distance is used:

The ChessboardDistance only takes into account the dimension with the largest separation:

The DistanceFunction can be given as a symbol:

Or as a pure function:

## Applications(4)

Obtain a robust estimate of multivariate location when outliers are present:

Extreme values have a large influence on the Mean:

Sample points from a convex polygon:

Estimate the center of the polygon by computing the central feature of random points:

Find the central feature of California, based on the location of cities:

Find the central feature of California, based on the location of cities weighted by population:

Draw the cities' locations (gray), unweighted central feature (red) and weighted central feature (black):

The top eight largest cities in Ohio:

The central feature of the eight cities based on TravelDistance:

The sum of distances from the central feature to the other cities, based on TravelDistance:

Draw the cities' locations (gray) and the central feature (red):

## Properties & Relations(5)

CentralFeature is a multivariate location measure:

Mean is also a location measure:

Visualize the data points with central feature and mean:

CentralFeature finds a point belonging to the data that minimizes the sum of distances:

Compute the central feature directly from the definition:

Visualize the sum of distances function together with the data points:

CentralFeature is the same as Median with univariate data when the data length is odd:

CentralFeature finds an element in the data that minimizes the sum of distances to other data points:

SpatialMedian finds a point in the domain that minimizes the sum of distances:

The sum of distances with respect to CentralFeature is greater than or equal to the one with respect to SpatialMedian:

Create a random graph with edge weights sampled uniformly between 0 and 1:

Locate the GraphCenter:

Specify the distance between each pair of vertices using GraphDistance:

Locate the center using CentralFeature:

## Possible Issues(1)

CentralFeature of a non-weighted, two-element list returns the first element:

For weighted two-element lists, it chooses the element with the highest weight, which trivially minimizes :