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CentralFeature
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CentralFeature

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CentralFeature[{x1,x2,…}]

gives the central feature of the elements .

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CentralFeature[{x1v1,x2v2,…}]

gives the vi corresponding to the central feature .

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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,…}Automatictake the vi to be successive integers i
    GeoPosition[…]array of geodetic positions
    WeightedData[…]data with weights
  • The following option can be given:
  • DistanceFunction Automaticthe 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:
  • EuclideanDistancenumeric data
    ImageDistanceimages
    JaccardDissimilarityBoolean data
    EditDistancetext and nominal sequences
    Abs[DateDifference[#1,#2]]&dates and times
    ColorDistancecolors
    GeoDistancegeospatial data
    Boole[SameQ[#1,#2]]&nominal data
    HammingDistancenominal vector data
    WarpingDistancenumerical 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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find the central feature in a list of vectors:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-ei8
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-f2un79
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Same inputs with different output formats:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-0a946l
In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-cg1nsz
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01ytjx49nma-baimh9
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/01ytjx49nma-xk77u
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/01ytjx49nma-jxkqh
Out[5]=5

Central feature works with WeightedData:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-cmk87u
In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-hau2mb
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-87iwb
Out[3]=3

Central feature of a large array:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-nknun
Out[1]=1

Weighted central feature:

In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-ma3v2m
Out[2]=2

Find the central feature of data involving quantities:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-jopin9
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-e8c21s
Out[2]=2

Find the central feature of a list of images:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-86ek1f
Out[1]=1

List of pictures:

In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-5h0a58
Out[2]=2

List of 3D images:

In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-g60h57
Out[3]=3

Compute the central feature of strings:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-jgyc34
Out[1]=1

Compute the central feature of Boolean vectors:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-et6i42
Out[1]=1

Compute the central feature of a list of date objects:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-fo7hey
Out[1]=1

Compute the central feature of geodetic positions:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-h00b0u
In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-zscajd
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-xh3bo
Out[3]=3

Options  (2)Common values & functionality for each option

DistanceFunction  (2)

By default, Euclidean distance is used:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-hgtsmh
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-88aqlu
Out[2]=2

The DistanceFunction can be given as a symbol:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-cse7si
In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-hb61z3
In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-pw5mgr
Out[3]=3

Or as a pure function:

In[4]:=4
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https://wolfram.com/xid/01ytjx49nma-vs537w
Out[4]=4

Applications  (4)Sample problems that can be solved with this function

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

In[4]:=4
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https://wolfram.com/xid/01ytjx49nma-g5ces4
In[5]:=5
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https://wolfram.com/xid/01ytjx49nma-cexxtn
Out[5]=5

Extreme values have a large influence on the Mean:

In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-blrzc0
Out[3]=3

Sample points from a convex polygon:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-os2mws
In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-eu6ef9
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-km5uzj
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/01ytjx49nma-htcbm0
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-ch0km6
In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-fej0tx
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-vaphe
In[4]:=4
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https://wolfram.com/xid/01ytjx49nma-ffpsfp
Out[4]=4

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

In[5]:=5
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https://wolfram.com/xid/01ytjx49nma-35pzkk
In[6]:=6
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https://wolfram.com/xid/01ytjx49nma-bzitte
Out[6]=6

The top eight largest cities in Ohio:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-n79bt
Out[1]=1

The central feature of the eight cities based on TravelDistance:

In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-ief8up
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-oy5ptc
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/01ytjx49nma-p0o21w
Out[4]=4

Properties & Relations  (5)Properties of the function, and connections to other functions

CentralFeature is a multivariate location measure:

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-3l3hs
In[2]:=2
✖
https://wolfram.com/xid/01ytjx49nma-49w8bq
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/01ytjx49nma-d706y
Out[3]=3

Mean is also a location measure:

In[4]:=4
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https://wolfram.com/xid/01ytjx49nma-gqcaip
Out[4]=4

Visualize the data points with central feature and mean:

In[9]:=9
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https://wolfram.com/xid/01ytjx49nma-mgzvm
Out[9]=9

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

In[1]:=1
✖
https://wolfram.com/xid/01ytjx49nma-hsraxu
In[2]:=2
✖
https://wolfram.com/xid/01ytjx49nma-d1hk4a
Out[2]=2

Compute the central feature directly from the definition:

In[3]:=3
✖
https://wolfram.com/xid/01ytjx49nma-p2lkxt
In[4]:=4
✖
https://wolfram.com/xid/01ytjx49nma-h5g0m
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/01ytjx49nma-9fojv
Out[5]=5

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

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-nl9yne
In[2]:=2
✖
https://wolfram.com/xid/01ytjx49nma-fto1z
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/01ytjx49nma-l6968w
In[2]:=2
✖
https://wolfram.com/xid/01ytjx49nma-fpw0ac
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/01ytjx49nma-dtgm5w
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/01ytjx49nma-dq0fzw
In[5]:=5
✖
https://wolfram.com/xid/01ytjx49nma-otjqfs
Out[5]=5

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

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-4hyg3
Out[1]=1

Locate the GraphCenter:

In[2]:=2
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https://wolfram.com/xid/01ytjx49nma-f3adkp
Out[2]=2

Specify the distance between each pair of vertices using GraphDistance:

In[3]:=3
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https://wolfram.com/xid/01ytjx49nma-52owx

Locate the center using CentralFeature:

In[4]:=4
✖
https://wolfram.com/xid/01ytjx49nma-dmt0h
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
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https://wolfram.com/xid/01ytjx49nma-qczayo
In[2]:=2
✖
https://wolfram.com/xid/01ytjx49nma-oxzelz
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/01ytjx49nma-jczf9o
Out[3]=3
Wolfram Research (2017), CentralFeature, Wolfram Language function, https://reference.wolfram.com/language/ref/CentralFeature.html.
✖
Wolfram Research (2017), CentralFeature, Wolfram Language function, https://reference.wolfram.com/language/ref/CentralFeature.html.

Text

Wolfram Research (2017), CentralFeature, Wolfram Language function, https://reference.wolfram.com/language/ref/CentralFeature.html.

✖
Wolfram Research (2017), CentralFeature, Wolfram Language function, https://reference.wolfram.com/language/ref/CentralFeature.html.

CMS

Wolfram Language. 2017. "CentralFeature." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CentralFeature.html.

✖
Wolfram Language. 2017. "CentralFeature." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CentralFeature.html.

APA

Wolfram Language. (2017). CentralFeature. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CentralFeature.html

✖
Wolfram Language. (2017). CentralFeature. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CentralFeature.html

BibTeX

@misc{reference.wolfram_2025_centralfeature, author="Wolfram Research", title="{CentralFeature}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/CentralFeature.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_centralfeature, author="Wolfram Research", title="{CentralFeature}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/CentralFeature.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_centralfeature, organization={Wolfram Research}, title={CentralFeature}, year={2017}, url={https://reference.wolfram.com/language/ref/CentralFeature.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_centralfeature, organization={Wolfram Research}, title={CentralFeature}, year={2017}, url={https://reference.wolfram.com/language/ref/CentralFeature.html}, note=[Accessed: 29-March-2025 ]}