SpatialMedian [{x1,x2,…}]
gives the spatial median of the elements .
SpatialMedian [data]
gives the spatial median for several different forms of data.
SpatialMedian
SpatialMedian [{x1,x2,…}]
gives the spatial median of the elements .
SpatialMedian [data]
gives the spatial median for several different forms of data.
Details and Options
- SpatialMedian is also known as geometric median, median center and 1-median.
- SpatialMedian is a robust location measure. It gives a point with minimum total distance to every point in the data.
- SpatialMedian is equivalent to x^*=ArgMin[sum_(i=1)^nd(x,x_i),x] for the unweighted case and x^*=ArgMin[sum_(i=1)^nw_i d(x,x_i),x] for the weighted case.
- The data data has the following forms and interpretations:
-
{x1,x2,…} univariate numerical data{{x1,y1,…},{x2,y2,…},…} multivariate numerical data
- Under Euclidean distance, SpatialMedian coincides with median in the one-dimensional case. In the multidimensional case, spatial median is unique whenever the points are not collinear.
- The following options can be given:
-
- By default, the following distance functions are used for different types of elements:
-
EuclideanDistance numeric dataGeoDistance geospatial data
- The setting Method {"InitialPoint"{x0,y0,…}} allows for a custom initial point for computing SpatialMedian .
Examples
open all close allBasic Examples (3)
Find the spatial median of a list of vectors:
Find the spatial median of a list of vectors with given weights:
Compute the spatial median of geo locations:
Scope (5)
Same inputs with different precisions:
Spatial median works with WeightedData :
Spatial median of a large array:
Weighted spatial median:
Spatial median of data involving quantities:
Compute the spatial median of geodetic positions:
Options (4)
DistanceFunction (2)
By default, EuclideanDistance is used for numerical data:
The ChessboardDistance only takes into account the dimension with the largest separation:
DistanceFunction can be given as a symbol:
Or as a pure function:
Method (2)
Specify the initial point for the iterative procedure of spatial median:
"NMinimize" and the method options of FindMinimum can be used:
Use with specified DistanceFunction :
Applications (8)
Obtain a robust estimate of a multivariate location when outliers are present:
Extreme values have a large influence on the Mean :
Consider data from a Gaussian mixture distribution:
Estimate the center with Mean :
The sample mean estimator has a large spread for non-Gaussian data. The standard deviation of the estimator is:
Estimate the center with SpatialMedian :
Assess the spread via bootstrapping. The spatial median has a smaller spread compared to the mean:
Consider the stock prices of five companies: GOOG, MSFT, FB, AAPL and INTC in 2015 as five-dimensional data:
Compute the log returns and estimate the center using Mean and SpatialMedian :
Fit the data with MultivariateTDistribution and extract the location parameters:
Spatial median estimator gives a closer estimate to the location parameters of multivariate t distribution than the empirical mean with the given stock data:
With the number of points equal to 3, spatial median is also the Fermat point:
Create equilateral triangles on each side:
Construct the Fermat point geometrically and compare it with the result of SpatialMedian (red):
Sample points from a convex polygon:
Estimate the center of the polygon by computing the spatial median of random points:
Find the spatial median of California based on the locations of cities:
Find the spatial median of California based on the locations of cities, weighted by population:
Draw the city locations (gray), unweighted spatial median (red) and weighted spatial median (black):
For geo locations that are far enough apart on the surface of the Earth, the spatial median depends significantly on the choice of the distance function:
The spatial median under GeoPosition :
The spatial median of the projected coordinates under EuclideanDistance :
Show the locations of the spatial medians and the cities:
Centroids of geographic entities can be approximated by the spatial median of the uniformly sampled geo locations. Obtain the country polygon of Spain:
Sample points from the region and compute the corresponding spatial median:
Find the closest city from the spatial median:
Visualize the results:
Properties & Relations (5)
SpatialMedian is a multivariate location measure:
Compute the spatial median:
Mean is also a location measure:
Visualize the data points with spatial median and mean:
SpatialMedian is the L1 location estimator of spatial points:
Compute SpatialMedian from the definition with FindMinimum :
Visualize the sum of distances function:
Mean (or spatial mean) is the L2 location estimator of spatial points:
Compute Mean from the definition with FindMinimum :
Visualize the sum of distances function:
SpatialMedian is the same as Median for univariate data:
SpatialMedian under ManhattanDistance for multivariate data is the same as Median :
SpatialMedian finds a point in the domain that minimizes the sum of distances:
CentralFeature finds a point that belongs to the data 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 :
History
Text
Wolfram Research (2017), SpatialMedian, Wolfram Language function, https://reference.wolfram.com/language/ref/SpatialMedian.html.
CMS
Wolfram Language. 2017. "SpatialMedian." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SpatialMedian.html.
APA
Wolfram Language. (2017). SpatialMedian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SpatialMedian.html
BibTeX
@misc{reference.wolfram_2025_spatialmedian, author="Wolfram Research", title="{SpatialMedian}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/SpatialMedian.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_spatialmedian, organization={Wolfram Research}, title={SpatialMedian}, year={2017}, url={https://reference.wolfram.com/language/ref/SpatialMedian.html}, note=[Accessed: 17-November-2025]}