--- title: "CentralFeature" language: "en" type: "Symbol" summary: "CentralFeature[{x1, x2, ...}] gives the central feature of the elements xi. CentralFeature[{x1 -> v1, x2 -> v2, ...}] gives the vi corresponding to the central feature xi. CentralFeature[data] gives the central feature for several different forms of data." keywords: - central feature - central tendency - location estimator canonical_url: "https://reference.wolfram.com/language/ref/CentralFeature.html" source: "Wolfram Language Documentation" related_guides: - title: "Descriptive Statistics" link: "https://reference.wolfram.com/language/guide/DescriptiveStatistics.en.md" - title: "Locations, Paths, and Routing" link: "https://reference.wolfram.com/language/guide/LocationsPathsAndRouting.en.md" - title: "Spatial Point Collections" link: "https://reference.wolfram.com/language/guide/SpatialPointCollections.en.md" - title: "Spatial Statistics" link: "https://reference.wolfram.com/language/guide/SpatialStatistics.en.md" - title: "Distance and Similarity Measures" link: "https://reference.wolfram.com/language/guide/DistanceAndSimilarityMeasures.en.md" - title: "Robust Descriptive Statistics" link: "https://reference.wolfram.com/language/guide/RobustDescriptiveStatistics.en.md" related_functions: - title: "Mean" link: "https://reference.wolfram.com/language/ref/Mean.en.md" - title: "Median" link: "https://reference.wolfram.com/language/ref/Median.en.md" - title: "TrimmedMean" link: "https://reference.wolfram.com/language/ref/TrimmedMean.en.md" - title: "WinsorizedMean" link: "https://reference.wolfram.com/language/ref/WinsorizedMean.en.md" - title: "BiweightLocation" link: "https://reference.wolfram.com/language/ref/BiweightLocation.en.md" - title: "SpatialMedian" link: "https://reference.wolfram.com/language/ref/SpatialMedian.en.md" - title: "GraphCenter" link: "https://reference.wolfram.com/language/ref/GraphCenter.en.md" - title: "DistanceMatrix" link: "https://reference.wolfram.com/language/ref/DistanceMatrix.en.md" --- # CentralFeature CentralFeature[{x1, x2, …}] gives the central feature of the elements $x_i$. CentralFeature[{x1 -> v1, x2 -> v2, …}] gives the vi corresponding to the central feature $x_i$. 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 $x^*\in \left\{x_1,x_2,\text{...}\right\}$ that minimizes the sum of distances $\sum _{i=1}^n d\left(x^*,x_i\right)$ for the unweighted case and $\sum _{i=1}^n w_id\left(x^*,x_i\right)$ for the weighted case. [image] * 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`](https://reference.wolfram.com/language/ref/DistanceFunction.en.md) [`Automatic`](https://reference.wolfram.com/language/ref/Automatic.en.md) 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 (23) ### Basic Examples (2) Find the central feature in a list of vectors: ```wl In[1]:= CentralFeature[{{1., 3., 5.}, {7., 1., 2.}, {9., 3., 1.}, {4., 5., 6.}}] Out[1]= {7., 1., 2.} ``` --- Find the central feature in a list of vectors with given weights: ```wl In[1]:= CentralFeature[WeightedData[{{1., 3., 5.}, {-4., 1., 2.}, {3., 3., 1.}, {4., 5., 6.}}, {1., 3., 4., 5.}]] Out[1]= {3., 3., 1.} ``` ### Scope (9) Same inputs with different output formats: ```wl In[1]:= data = {{1, 2}, {3, 4}, {8, 7}, {6, 5}, {9, 4}, {1, 3}}; In[2]:= CentralFeature[data] Out[2]= {3, 4} In[3]:= CentralFeature[data -> Automatic] Out[3]= 2 In[4]:= CentralFeature[data -> Table[f[i], {i, 6}]] Out[4]= f[2] In[5]:= CentralFeature[{{1, 2} -> g[1], {3, 4} -> g[2], {8, 7} -> g[3], {6, 5} -> g[4], {9, 4} -> g[5], {1, 3} -> g[6]}] Out[5]= g[2] ``` --- Central feature works with ``WeightedData`` : ```wl In[1]:= wd = WeightedData[{{1., 3.}, {-4., 2.}, {3., 1.}, {5., 6.}}, {1., 3., 4., 5.}]; In[2]:= CentralFeature[wd] Out[2]= {3., 1.} In[3]:= CentralFeature[wd -> {1, 2, 3, 4}] Out[3]= 3 ``` --- Central feature of a large array: ```wl In[1]:= CentralFeature[RandomReal[1, {10 ^ 5, 2}]] Out[1]= {0.501217, 0.500012} ``` Weighted central feature: ```wl In[2]:= CentralFeature[WeightedData[RandomReal[1, {10 ^ 5, 3}], RandomReal[1, 10 ^ 5]]] Out[2]= {0.492451, 0.492348, 0.502146} ``` --- Find the central feature of data involving quantities: ```wl In[1]:= data = Quantity[RandomReal[1, {6, 2}], "Meters"] Out[1]= {{Quantity[0.11079720282181693, "Meters"], Quantity[0.6515314303412563, "Meters"]}, {Quantity[0.49613268680883627, "Meters"], Quantity[0.8521439683724525, "Meters"]}, {Quantity[0.07466180322204763, "Meters"], Quantity[0.3659169954207231, "Meters"]}, {Quantity[0.29822857534926706, "Meters"], Quantity[0.3699025216424252, "Meters"]}, {Quantity[0.23107340697669554, "Meters"], Quantity[0.8761732437389362, "Meters"]}, {Quantity[0.41766891691505825, "Meters"], Quantity[0.9796850527424898, "Meters"]}} In[2]:= CentralFeature[data] Out[2]= {Quantity[0.11079720282181693, "Meters"], Quantity[0.6515314303412563, "Meters"]} ``` --- Find the central feature of a list of images: ```wl In[1]:= CentralFeature[{[image], [image], [image], [image], [image], [image], [image], [image], [image], [image]}] Out[1]= [image] ``` List of pictures: ```wl In[2]:= CentralFeature[{[image], [image], [image], [image], [image], [image]}] Out[2]= [image] ``` List of 3D images: ```wl In[3]:= CentralFeature[{[image], [image], [image], [image], [image]}] Out[3]= [image] ``` --- Compute the central feature of strings: ```wl In[1]:= CentralFeature[{"abcd", "bcde", "abab", "abcdef", "agi"}] Out[1]= "abcd" ``` --- Compute the central feature of Boolean vectors: ```wl In[1]:= CentralFeature[{{True, False, True}, {True, True, True}, {True, False, False}, {False, False, False}}] Out[1]= {True, False, True} ``` --- Compute the central feature of a list of date objects: ```wl In[1]:= CentralFeature[{DateObject[{1985, 10, 3}], DateObject[{1989, 5, 30}], DateObject[{1999, 11, 15}], DateObject[]}] Out[1]= DateObject[{1989, 5, 30}, "Day", "Gregorian", -6.] ``` --- Compute the central feature of geodetic positions: ```wl In[1]:= cities = {Entity["City", {"Paris", "IleDeFrance", "France"}], Entity["City", {"Sydney", "NewSouthWales", "Australia"}], Entity["City", {"Boston", "Massachusetts", "UnitedStates"}], Entity["City", {"SanFrancisco", "California", "UnitedStates"}], Entity["City", {"Tokyo", "Tokyo", "Japan"}]}; In[2]:= cf = CentralFeature[GeoPosition[cities]] Out[2]= GeoPosition[{37.7599, -122.437}] In[3]:= GeoGraphics[{GeoPath[Thread[{cf, cities}]], GeoMarker[cities, Gray], GeoMarker[cf, Red]}, GeoProjection -> {"AzimuthalEquidistant", "Centering" -> cf}, GeoRange -> "World", GeoGridLines -> Automatic] Out[3]= [image] ``` ### Options (2) #### DistanceFunction (2) By default, Euclidean distance is used: ```wl In[1]:= CentralFeature[RandomVariate[NormalDistribution[], {10, 3}]] Out[1]= {-0.813317, 0.390383, -0.572312} ``` The ``ChessboardDistance`` only takes into account the dimension with the largest separation: ```wl In[2]:= CentralFeature[{{1., 3.}, {-2.6, 4.}, {9.2, 5.}, {7.3, -5.}}, DistanceFunction -> ChessboardDistance] Out[2]= {1., 3.} ``` --- The ``DistanceFunction`` can be given as a symbol: ```wl In[1]:= dist[{u_, v_}, {x_, y_}] := Sqrt[3(u - x) ^ 2 + 2(v - y) ^ 2] In[2]:= data = {{1.5, .6}, {2, 0}, {1.25, 1.25}, {3., 0.3}}; In[3]:= CentralFeature[data, DistanceFunction -> dist] Out[3]= {1.5, 0.6} ``` Or as a pure function: ```wl In[4]:= CentralFeature[data, DistanceFunction -> (Sqrt[{3, 2}.(#1 - #2) ^ 2]&)] Out[4]= {1.5, 0.6} ``` ### Applications (4) Obtain a robust estimate of multivariate location when outliers are present: ```wl In[1]:= data = {{3., -5.}, {2., -5.}, {0., 2.}, {-4., -3.}, {10 ^ 8, -1.}, {8., -20000}}; In[2]:= CentralFeature[data] Out[2]= {2., -5.} ``` Extreme values have a large influence on the ``Mean`` : ```wl In[3]:= Mean[data] Out[3]= {1.6666668166666666*^7, -3335.33} ``` --- Sample points from a convex polygon: ```wl In[1]:= poly = Polygon[{{2., -5.}, {0., 2.}, {-4., -3.}, {-5, -7}}]; pts = RandomPoint[poly, 10 ^ 3]; In[2]:= Graphics[{Lighter[Blue, .9], poly, Black, PointSize[Small], Point[pts]}] Out[2]= [image] ``` Estimate the center of the polygon by computing the central feature of random points: ```wl In[3]:= cf = CentralFeature[pts] Out[3]= {-1.28721, -3.29255} In[4]:= Graphics[{Lighter[Blue, .9], poly, Black, PointSize[Small], Point[pts], PointSize[0.06], Red, Point[cf]}] Out[4]= [image] ``` --- Find the central feature of California, based on the location of cities: ```wl In[1]:= cities = CityData[{All, "California"}]; locations = GeoPosition[CityData[#, "Coordinates"]& /@ cities]; In[2]:= unweighted = cities[[CentralFeature[locations -> Automatic]]] Out[2]= Entity["City", {"McSwain", "California", "UnitedStates"}] ``` Find the central feature of California, based on the location of cities weighted by population: ```wl In[3]:= populations = CityData[#, "Population"]& /@ cities; In[4]:= weighted = cities[[CentralFeature[WeightedData[locations, populations] -> Automatic]]] Out[4]= Entity["City", {"Burbank", "California", "UnitedStates"}] ``` Draw the cities' locations (gray), unweighted central feature (red) and weighted central feature (black): ```wl In[5]:= legend = PointLegend[{Red, Blue}, {"unweighted central feature", "central feature weighted by population"}, LegendMarkerSize -> Large]; In[6]:= Legended[GeoGraphics[{EdgeForm[Black], FaceForm[Orange], Polygon[Entity["AdministrativeDivision", {"California", "UnitedStates"}]], GrayLevel[.4], PointSize[0.0075], Point[locations], PointSize[0.04], Red, Point[unweighted], Blue, Point[weighted]}, GeoRange -> {{32, 43}, {-125, -114}}, ImageSize -> 350], Placed[legend, "Bottom"]] Out[6]= [image] ``` --- The top eight largest cities in Ohio: ```wl In[1]:= cities = CityData[{Large, "Ohio"}] Out[1]= {Entity["City", {"Columbus", "Ohio", "UnitedStates"}], Entity["City", {"Cleveland", "Ohio", "UnitedStates"}], Entity["City", {"Cincinnati", "Ohio", "UnitedStates"}], Entity["City", {"Toledo", "Ohio", "UnitedStates"}], Entity["City", {"Akron", "Ohio", "UnitedStates"}], Entity["City", {"Dayton", "Ohio", "UnitedStates"}]} ``` The central feature of the eight cities based on ``TravelDistance`` : ```wl In[2]:= center = CentralFeature[cities, DistanceFunction -> TravelDistance] Out[2]= Entity["City", {"Columbus", "Ohio", "UnitedStates"}] ``` The sum of distances from the central feature to the other cities, based on ``TravelDistance`` : ```wl In[3]:= Total[Table[TravelDistance[center, city], {city, cities}]] Out[3]= Quantity[3.1049121030183723*^6, "Feet"] ``` Draw the cities' locations (gray) and the central feature (red): ```wl In[4]:= GeoGraphics[{EdgeForm[Black], FaceForm[Orange], Polygon[Entity["AdministrativeDivision", {"Ohio", "UnitedStates"}]], Red, TravelDirections[{center, #}, "TravelPath"]& /@ cities, GrayLevel[.4], PointSize[0.03], Point /@ cities, PointSize[0.03], Red, Point[center]}] Out[4]= [image] ``` ### Properties & Relations (5) ``CentralFeature`` is a multivariate location measure: ```wl In[1]:= pts = {{3, -5}, {2, -5}, {0, 2}, {-4, -3}, {-3, 2}, {-5, -5}, {-1, -5}, {-1, -5}, {5, -1}, {-3, 3}, {10, -1}, {8, -2}}; In[2]:= Graphics[{PointSize[0.03], Orange, Point[pts]}] Out[2]= [image] In[3]:= cf = CentralFeature[pts] Out[3]= {-1, -5} ``` ``Mean`` is also a location measure: ```wl In[4]:= mean = Mean[pts] Out[4]= {(11/12), -(25/12)} ``` Visualize the data points with central feature and mean: ```wl In[5]:= Graphics[{PointSize[0.03], Orange, Point[pts], Locator[cf], Blue, PointSize[.04], Point[mean], Gray, Text["Central Feature", cf - {0, .8}], Text["Mean", mean + {0, .8}]}, ImageSize -> 250] Out[5]= [image] ``` --- ``CentralFeature`` finds a point belonging to the data that minimizes the sum of distances: ```wl In[1]:= data = {{1., 3.}, {4., 6.}, {9., 2.}, {5., 5.}, {7., -2.}}; In[2]:= cf = CentralFeature[data] Out[2]= {5., 5.} ``` Compute the central feature directly from the definition: ```wl In[3]:= sumofdists[x_, y_] := Total[Sqrt[Total[(Transpose[data] - {x, y}) ^ 2]]] In[4]:= {ind} = Ordering[sumofdists@@@data, 1]; data[[ind]] Out[4]= {5., 5.} ``` Visualize the sum of distances function together with the data points: ```wl In[5]:= sumofdists = Total[Sqrt[Total[(Transpose[data] - {#1, #2}) ^ 2]]]&; ContourPlot[sumofdists[x, y], {x, 0, 10}, {y, -3, 8}, Contours -> 10, PlotLegends -> Automatic, Epilog -> {PointSize[Large], Black, Point[data], Red, PointSize[Large], Point[cf], White, Text["Central Feature", cf - {0, 0.7}]}] Out[5]= [image] ``` --- ``CentralFeature`` is the same as ``Median`` with univariate data when the data length is odd: ```wl In[1]:= data = RandomReal[1, 1001]; In[2]:= CentralFeature[data] == Median[data] Out[2]= True ``` --- ``CentralFeature`` finds an element in the data that minimizes the sum of distances to other data points: ```wl In[1]:= data = RandomVariate[NormalDistribution[], {10, 3}]; In[2]:= MemberQ[data, CentralFeature[data]] Out[2]= True ``` ``SpatialMedian`` finds a point in the domain that minimizes the sum of distances: ```wl In[3]:= MemberQ[data, SpatialMedian[data]] Out[3]= False ``` The sum of distances with respect to ``CentralFeature`` is greater than or equal to the one with respect to ``SpatialMedian`` : ```wl In[4]:= sumofdists = Total[Sqrt[Total[(Transpose[data] - #) ^ 2]]]&; In[5]:= sumofdists[CentralFeature[data]] ≥ sumofdists[SpatialMedian[data]] Out[5]= True ``` --- Create a random graph with edge weights sampled uniformly between 0 and 1: ```wl In[1]:= SeedRandom[7]; graph = RandomGraph[{10, 15}, EdgeWeight -> RandomReal[1, 15]] Out[1]= [image] ``` Locate the ``GraphCenter`` : ```wl In[2]:= GraphCenter[graph] Out[2]= {4} ``` Specify the distance between each pair of vertices using ``GraphDistance`` : ```wl In[3]:= distfun = GraphDistance[graph, #1, #2]&; ``` Locate the center using ``CentralFeature`` : ```wl In[4]:= CentralFeature[Range[10], DistanceFunction -> distfun] Out[4]= 4 ``` ### Possible Issues (1) ``CentralFeature`` of a non-weighted, two-element list returns the first element: ```wl In[1]:= data = {GeoPosition[{20, -30}], GeoPosition[{15, 15}]}; In[2]:= CentralFeature[data] Out[2]= GeoPosition[{20, -30}] ``` For weighted two-element lists, it chooses the element with the highest weight, which trivially minimizes $\sum _{i=1}^n w_id\left(x^*,x_i\right)$ : ```wl In[3]:= CentralFeature[WeightedData[data, {1, 2}]] Out[3]= GeoPosition[{15, 15}] ``` ## See Also * [`Mean`](https://reference.wolfram.com/language/ref/Mean.en.md) * [`Median`](https://reference.wolfram.com/language/ref/Median.en.md) * [`TrimmedMean`](https://reference.wolfram.com/language/ref/TrimmedMean.en.md) * [`WinsorizedMean`](https://reference.wolfram.com/language/ref/WinsorizedMean.en.md) * [`BiweightLocation`](https://reference.wolfram.com/language/ref/BiweightLocation.en.md) * [`SpatialMedian`](https://reference.wolfram.com/language/ref/SpatialMedian.en.md) * [`GraphCenter`](https://reference.wolfram.com/language/ref/GraphCenter.en.md) * [`DistanceMatrix`](https://reference.wolfram.com/language/ref/DistanceMatrix.en.md) ## Related Guides * [Descriptive Statistics](https://reference.wolfram.com/language/guide/DescriptiveStatistics.en.md) * [Locations, Paths, and Routing](https://reference.wolfram.com/language/guide/LocationsPathsAndRouting.en.md) * [Spatial Point Collections](https://reference.wolfram.com/language/guide/SpatialPointCollections.en.md) * [Spatial Statistics](https://reference.wolfram.com/language/guide/SpatialStatistics.en.md) * [Distance and Similarity Measures](https://reference.wolfram.com/language/guide/DistanceAndSimilarityMeasures.en.md) * [Robust Descriptive Statistics](https://reference.wolfram.com/language/guide/RobustDescriptiveStatistics.en.md) ## History * [Introduced in 2017 (11.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn111.en.md)