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Symbolic data analysis V. Batagelj Symbolic data analysis Symbolic data analysis Clustering of large data sets of mixed units Clustering and optimization Leaders method Vladimir Batagelj Agglomerative method Examples IMFM Ljubljana

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Symbolic data analysis

V. Batagelj

Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Symbolic data analysis

Clustering of large data sets of mixed units Vladimir Batagelj

IMFM Ljubljana and IAM UP Koper

CRoNoS meeting 16 - 17 April 2016, Colegio de Espa˜ na, Paris

V. Batagelj

Symbolic data analysis

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Symbolic data analysis

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Outline

1 Symbolic data analysis 2 Clustering and optimization 3 Leaders method 4 Agglomerative method 5 Examples 6 References

V. Batagelj

Symbolic data analysis

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Symbolic data analysis

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Symbolic data

Data table · · · variablej · · · · · · · · · · · · · · · uniti · · · valuei,j · · · · · · · · · · · · · · · In classical data analysis valuei,j is a single element (number or label) measured in standard measurement scales (absolute, ratio, interval, ordinal, nominal). In symbolic data table valuei,j can be also a complex data such as: interval, set of values, distribution (in general sense), time series, tree, text, function, etc. Rules linking variables and taxonomies of values can be specified.

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Symbolic data analysis

Symbolic data analysis aims to extend existing data analysis methods to symbolic data and to develop new ones. It was introduced in 1987 by Edwin Diday [Diday, E. (1987)]. Three European projects:

SODAS - Symbolic Official Data Analysis System (1996-99),
ISO-3D - Interpretation of Symbolic Objects with 3D

Representation (1998-01),

ASSO – Analysis System of Symbolic Official Data (2001-03).

resulted in a program for symbolic data analysis SODAS 2. The results were published in many papers in conference proceedings and scientific journals, and three books [Bock, H-H.,Diday, E. (2000), Billard, L., Diday, E. (2006), Diday, E., Noirhomme, M. (2008)]. Additional two books are to appear soon.

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Symbolic data analysis

The SDA group regularly meets at workshops: Wienerwaldhof (2009), Namur (2011), Beijing (2011), Madrid (2012), Taipei (2014), Orl´ eans (2015). Three packages for SDA are available in R:

RSDA,
symbolicDA,
Clamix

Symbolic data analysis group at LinkedIn

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Symbolic data analysis and big data big data

aggregation − − − − − − − − − − →

symbolic data table

Aggregating data into symbolic data preserves much more information than the standard approach using mean values. Let Σ(S, V ) denote a summary – a symbolic value of variable V over the subset of units S. A good summary satisfies the condition: for S1 ∩ S2 = ∅ it holds Σ(S1 ∪ S2, V ) = f (Σ(S1, V ), Σ(S2, V )) With my collaborators (Simona Korenjak-ˇ Cerne and Nataˇ sa Kejˇ zar) we are developing the clustering algorithms for symbolic objects described by modal valued symbolic data [Korenjak-ˇ Cerne, S., Batagelj, V. (1998), Korenjak-ˇ Cerne, S., Batagelj, V.. (2002), Batagelj, V. et al. (2014)].

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Clustering symbolic data

For clustering of SOs we adapted two classical clustering methods:

leaders method (a generalization of k-means method

[Hartigan, J. A. (1975)], dynamic clouds [Diday, E. (1979)]).

Ward’s hierarchical clustering method [Ward, J. H. (1963)].

Both adapted methods are based on the same criterion function – they are solving the same clustering problem. With the leaders method the size of the sets of units is reduced to a manageable number of leaders. The obtained leaders can be further clustered with the compatible agglomerative hierarchical clustering method to reveal relations among them and using the dendrogram also to decide upon the right number of clusters.

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Symbolic data analysis

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Symbolic objects described with distributions

An SO X is described by a list X = [xi] of descriptions of variables Vi. The values NA (not available) are treated as an additional category for each variable. In our model, each variable is described with frequency distribution (bar chart) of its values fxi = [fxi1, fxi2, . . . , fxiki]. With xi = [pxi1, pxi2, . . . , pxiki] we denote the corresponding probability distribution.

ki

pxij = 1, i = 1, . . . , m

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Clustering and optimization

We approach the clustering problem as an optimization problem

ver the set of feasible clusterings Φk – partitions of units into

k clusters. The criterion function has the following form P(C) =

C∈C

p(C). (1) The total error P(C) of the clustering C is a sum of cluster errors p(C).

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

The cluster error

There are many possibilities how to express the cluster error p(C). In this paper we shall assume a model in which the error

f a cluster is a sum of differences of its units from the cluster’s

representative T p(C, T) =

X∈C

d(X, T). (2) Note that in general the representative needs not to be from the same ”space” (set) as units.

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Representatives

The best representative is called a leader TC = argmin

T

p(C, T). (3) Then we define p(C) = p(C, TC) = min

T

X∈C

d(X, T). (4) The SO X is described by a list X = [xi]. Assume that also representatives are described in the same way T = [ti], ti = [ti1, ti2, . . . , tiki].

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Dissimilarity between SOs

We introduce a dissimilarity measure between SOs with d(X, T) =

αid(xi, ti), αi ≥ 0,

αi = 1, (5) where d(xi, ti) =

ki

wxijδ(pxij, tij), wxij ≥ 0. (6) This is a kind of a generalization of the squared Euclidean distance. The weight wxij can be for the same unit X different for each variable Vi (needed in descriptions of ego-centric networks, population pyramids, etc.).

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Leaders method

Leaders method is a generalization of a popular nonhierarchical clustering k-means method. The idea is to get ”optimal” clustering into a pre-specified number of clusters with the following iterative procedure: determine an initial clustering repeat determine leaders of the clusters in the current clustering; assign each unit to the nearest new leader – producing a new clustering until the leaders stabilize.

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Selection of the new leaders

Given a cluster C, the corresponding leader TC is the solution

f the problem

TC = argmin

T

X∈C

d(X, T) =

argmin

ti

X∈C

d(xi, ti) m

i=1

Therefore TC = [t∗

i ] and t∗ i = argminti

X∈C d(xi, ti). To

simplify the notation we omit the index i. t∗ = argmin

t

X∈C

d(x, t) =

argmin

tj∈R

X∈C

wxjδ(pxj, tj) k

j=1

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Leaders method Agglomerative method Examples References

Leaders

Again we omit the index j t∗ = argmin

t∈R

X∈C

wxδ(px, t) This is a standard optimization problem with one real variable. The solution has to satisfy the condition ∂ ∂t

X∈C

wxδ(px, t) = 0

r
X∈C

wx ∂δ(px, t) ∂t = 0 (7)

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Dissimilarities δ

δ(x, t) t∗

ij

1 (px − t)2

Pij Aij

2 ( px−t

t

)2

Qij Pij

3

(px−t)2 t

Aij

4 ( px−t

px )2 Hij Fij

5

(px−t)2 px Aij Hij

6

(px−t)2 pxt

Hij

Aij =

X∈C

wxij Pij =

X∈C

wxijpxij Qij =

X∈C

wxijp2

xij

Hij =

X∈C

wxij pxij Fij =

X∈C

wxij p2

xij

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Leaders method Agglomerative method Examples References

Leaders

For δ1(px, t) = (px − t)2 we get from (8) 0 =

X∈C

wx ∂ ∂t (px − t)2 = −2

X∈C

wx(px − t) Therefore t∗ =

X∈C wxpx
X∈C wx

= P A

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Leaders for δ1

Let wxij = wxi then for each i = 1, . . . , m:

ki

t∗

ij = 1

Ai

ki

j=1
X∈C

wxipxij = 1 The leaders’ components are distributions. Let further wxij = nxi then for each i = 1, . . . , m: t∗

Cij =

X∈C nxipxij
X∈C nxi

=

X∈C fxij
X∈C nxi

= fCij nCi = pCij The leader of a cluster is its distribution.

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Determining the new clustering

Given leaders T the corresponding optimal clustering C∗ is determined from P(C∗) =

X∈U

min

T∈T d(X, T) =

X∈U

d(X, Tc∗(X)) (8) where c∗(X) = argmin

k

d(X, Tk) We assign each unit X to the closest leader Tk ∈ T.

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Hierarchical agglomerative clustering

The hierarchical agglomerative clustering procedure is based on a step-by-step merging of the two closest clusters. each unit forms a cluster: Cn = {{X}: X ∈ U} ; they are at level 0: h({X}) = 0, X ∈ U ; for k = n − 1 to 1 do determine the closest pair of clusters (u, v) = argmini,j : i=j{D(Ci, Cj): Ci, Cj ∈ Ck+1} ; join the closest pair of clusters C(uv) = Cu ∪ Cv Ck = (Ck+1 \ {Cu, Cv}) ∪ {C(uv)} ; h(C(uv)) = D(Cu, Cv) determine the dissimilarities D(C(uv), Cs), Cs ∈ Ck endfor Ck is a partition of the finite set of units U into k clusters. The level h(C) of the cluster C(uv) = Cu ∪ Cv.

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

Dissimilarity between clusters

Therefore the computation of dissimilarities between new (merged) cluster and the rest has to be specified. To obtain the compatibility with the adapted leaders method, we define the dissimilarity between clusters Cu and Cv, Cu ∩ Cv = ∅, as D(Cu, Cv) = p(Cu ∪ Cv) − p(Cu) − p(Cv) =

αi

Auij · Avij Auij + Avij (uij − vij)2 (9) a generalized Ward’s relation. ui and vi are components of the leaders of clusters Cu and Cv.

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Other dissimilarities

Instead of the squared Euclidean distance other dissimilarity measures δ(x, t) can be used (see [Kejˇ zar, N. et al. (2011)]). Relations similar to Ward’s can be derived for them. The proposed approach is implemented in the R-package Clamix. It was successfully applied on different data sets (population pyramids, TIMSS, cars, foods, citation patterns of patents, and

thers).
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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Sheme of analysis

raw data ⇓ ENCODE ⇓ unified data ⇓ MAKE SOs ⇓ SOs - lists of distributions ⇓ ⇓ leaderSO ⇓ clustering and cluster leaders ⇓ hclustSO ⇓ hierarchy and cluster leaders ⇓ ANALYSIS ⇓ dendrogram, reports

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Population pyramids / World 2001

Japan Germany Italy Greece Portugal Spain France Sweden Jersey Luxembourg Liechtenstein Netherlands Denmark Norway Finland Gibraltar Belgium Monaco Isle of Man Guernsey San Marino Austria Switzerland United Kingdom Bulgaria Czech Republic Hungary Romania Moldova Poland Slovakia Russia Bosnia and Herzegovina Georgia Belarus Estonia Latvia Lithuania Malta Serbia Croatia Slovenia Ukraine Canada Aruba Bermuda Cayman Islands Australia United States China Cuba Korea, North Andorra Hong Kong S.A.R. Macau S.A.R. Singapore Barbados Korea, South Saint Helena Taiwan United Arab Emirates Argentina Israel Ireland Cyprus Faroe Islands Iceland Macedonia Montenegro New Zealand Uruguay Armenia Kazakhstan Trinidad and Tobago Kuwait Bahrain Qatar Albania Dominica Saint Kitts and Nevis Chile Greenland Netherlands Antilles Saint Pierre and Miquelon Sri Lanka Thailand Mauritius Anguilla Virgin Islands, British Brazil French Polynesia Bahamas, The New Caledonia Lebanon Burma Guyana Saint Lucia Seychelles Suriname Saint Vincent and the Grenadines Montserrat Tunisia Turkey Indonesia Colombia Brunei Antigua and Barbuda Palau Turks and Caicos Islands Panama Costa Rica Tuvalu Kyrgyzstan Turkmenistan Uzbekistan Philippines Jordan Grenada Libya Morocco Jamaica Ecuador Vanuatu India Egypt Malaysia Peru Dominican Republic Venezuela Bangladesh Iran Mexico Fiji South Africa Algeria Mongolia Azerbaijan Vietnam Oman Liberia Comoros Guinea−Bissau Djibouti Nigeria Gabon Laos Cameroon Senegal Sudan Guatemala Central African Republic Equatorial Guinea Belize Honduras Swaziland Haiti Solomon Islands Togo Mozambique Mayotte Somalia Angola Afghanistan Gambia, The Guinea Sierra Leone Eritrea Ethiopia Madagascar Maldives Congo (Brazzaville) Mauritania Western Sahara Chad Congo (Kinshasa) Niger Mali Burkina Faso Sao Tome and Principe Uganda Yemen Burundi Malawi Benin Tanzania Zambia Saudi Arabia Cote d’Ivoire Iraq Kenya Marshall Islands Rwanda El Salvador Lesotho Bolivia Micronesia, Federated States of Namibia Nepal Papua New Guinea Bhutan Kiribati Paraguay Ghana Pakistan Samoa Cambodia Cape Verde East Timor Nauru Syria Tajikistan Botswana Nicaragua Tonga Zimbabwe

0e+00 1e+06 2e+06 3e+06 4e+06

Males Females Ages 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80+ 0.075 0.15 0.075 0.15 Males Females Ages 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80+ 0.075 0.15 0.075 0.15 Males Females Ages 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80+ 0.075 0.15 0.075 0.15 Males Females Ages 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80+ 0.075 0.15 0.075 0.15

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

World 2001 / 4 clusters on the map

[Korenjak-ˇ Cerne, S. et al. (2015)]

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

Population pyramids / US counties

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ming

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w Jersey w Jersey , Maine , Maine ennsylvania ennsylvania vada exas ennsylvania , Florida , Illinois , Kansas , Indiana w York , Virginia th Carolina th Carolina , Virginia , Virginia , Georgia , Indiana , Indiana , Iowa , Oregon , Colorado , South Dakota , California , Wisconsin , Indiana ermont , Virginia , Virginia , Virginia yland , Virginia th Carolina , Virginia , Florida , Virginia , Virginia , Virginia , Virginia , Connecticut , Massachusetts , Montana , Virginia , Ohio exas , California , Wisconsin w Mexico , Montana , Virginia , Colorado , Idaho Haines Borough, Alaska , Idaho ashington

ming

w Mexico , Ohio , Kansas , Nebraska , Ohio exas , Kansas , South Dakota ashington , Illinois , Kansas , Indiana , Ohio , Illinois , Colorado exas , Texas , Georgia ish, Louisiana ashington , Iowa , Kansas exas , Wisconsin , Iowa , South Dakota , Ohio , Ohio , South Dakota exas , Minnesota , Wisconsin , Illinois , Iowa w York , Wisconsin , Wisconsin , Nebraska , Iowa , Nebraska , Minnesota , Montana , Kansas , South Dakota , South Dakota , Minnesota , Minnesota , Indiana , Minnesota , Missouri , Iowa , Nebraska , Oklahoma , Wisconsin , Minnesota , Wisconsin , Minnesota , Kansas , Minnesota , Minnesota , Minnesota , Iowa , Wisconsin th Dakota , Wisconsin , Minnesota , Montana , Wisconsin , Wisconsin , Wisconsin , Ohio , Ohio , Ohio , South Carolina , Oklahoma ennessee , Missouri w Jersey entucky ennsylvania w York , Indiana , Indiana ermont w York , Maine w York , Ohio , Michigan , Ohio , Georgia , South Carolina th Carolina exas w York w York ennsylvania , Illinois , Illinois , Missouri kansas , Illinois ennsylvania , Indiana , Michigan , Oregon ermont , Virginia , Indiana , Michigan , Ohio , Oklahoma , Michigan , Michigan , Oklahoma exas , Indiana , Michigan , Illinois yland , Michigan , Missouri vada , Oklahoma , Indiana , Michigan , Michigan , Missouri , Indiana est Virginia , Minnesota w York , Missouri exas , Minnesota exas , Kansas , Kansas exas , Nebraska , Texas th Carolina , Wisconsin , California , Idaho , Colorado exas , Missouri , Missouri , Ohio kansas , Oklahoma , Missouri , South Dakota , Michigan , Michigan , Missouri , Michigan , Oklahoma exas , California , Colorado , Colorado , Oregon ashington kansas th Carolina th Carolina entucky , Mississippi exas kansas ashington ashington , Montana

ming

, Idaho , Idaho ashington ashington , Colorado

ming

exas

ming

w York est Virginia , Wisconsin

ming

, Oregon

ming

, Minnesota , Montana , Montana

ming

, Indiana , Ohio , Michigan , Ohio th Carolina , Indiana , Ohio , Indiana , Indiana entucky , Ohio , Ohio kansas , Kansas , Missouri , Ohio , Illinois , Ohio , Indiana , Michigan , Ohio , Indiana , Indiana , Ohio , Indiana , Oklahoma , Ohio ennessee est Virginia th Carolina th Carolina , Georgia , Georgia , Ohio , Michigan , Ohio , Ohio , California ennsylvania entucky est Virginia , Ohio , Illinois entucky entucky est Virginia , Oregon ennsylvania , Ohio , South Carolina , Oregon , Nebraska ashington , Colorado , Illinois , Illinois , Idaho , Montana , Wisconsin w York , Missouri w York ennsylvania ennsylvania ennsylvania , Rhode Island w York w York , Illinois , Iowa , Iowa , Illinois , Illinois , Iowa exas ennsylvania ennsylvania ennsylvania ennsylvania ennsylvania , Wisconsin , Wisconsin , Ohio , Missouri yland , Illinois , Ohio , Maine ennsylvania w York ennsylvania ennsylvania , Illinois ish, Louisiana w York ennsylvania , Wisconsin , Indiana , Ohio vada th Carolina , Ohio , Florida , Oklahoma , Florida kansas entucky , Georgia entucky , Virginia , Michigan entucky entucky , Virginia , Indiana entucky , Michigan entucky kansas ennessee , Wisconsin , Michigan vada w York , Virginia w York w York , Texas w York , Illinois , Wisconsin , Minnesota , Missouri , Missouri , Illinois , Missouri , Iowa , Illinois , Wisconsin , Wisconsin , Illinois , South Dakota , Wisconsin , Mississippi entucky kansas , Mississippi , Ohio , Georgia kansas exas , Alabama , Indiana th Carolina ennessee , Indiana , Indiana , South Carolina , Alabama , Georgia ennessee , Mississippi , Georgia entucky , Virginia , Ohio , Ohio , Georgia entucky kansas kansas , South Carolina ennessee , South Carolina entucky entucky ennessee , Alabama kansas , Georgia ennessee th Carolina , Indiana , South Carolina , Alabama , Georgia entucky entucky entucky , Alabama ennessee , Alabama , Alabama th Carolina , Alabama ennessee entucky ennessee entucky entucky ennessee , Alabama , Alabama ennessee , Georgia ennessee entucky entucky entucky ennessee th Carolina th Carolina , Georgia ennessee , Virginia , Georgia ish, Louisiana th Carolina , Florida , Indiana entucky , Ohio ashington , Alabama entucky entucky entucky entucky , Alabama entucky entucky entucky entucky entucky entucky est Virginia , South Carolina th Carolina th Carolina th Carolina th Carolina , Georgia ennessee ennessee ennessee , Georgia ennessee ennessee ennessee ennessee ennessee entucky ennessee est Virginia , Alabama entucky ennsylvania entucky ennessee ennessee , Alabama ennessee , Alabama kansas ennessee , Alabama , Ohio entucky ennessee , Georgia entucky ennessee est Virginia th Carolina , Virginia entucky , Virginia , Virginia ennessee ennessee ennessee ennessee , Virginia , Virginia , Virginia , Virginia , Georgia est Virginia entucky , Virginia , Virginia , Virginia , Ohio , Ohio ennessee est Virginia est Virginia , Virginia est Virginia entucky est Virginia est Virginia est Virginia w York ennsylvania ennessee , Alabama , Virginia est Virginia entucky entucky entucky entucky entucky est Virginia entucky est Virginia yland , Virginia , Illinois est Virginia , Virginia est Virginia th Carolina est Virginia th Carolina est Virginia est Virginia est Virginia est Virginia est Virginia , Florida izona , Florida exas exas , Florida , Florida , Utah izona , Alabama kansas kansas kansas entucky th Carolina , Oregon , Oregon , Georgia , South Carolina , Indiana kansas yland , Virginia , Mississippi ennessee est Virginia , Virginia est Virginia , Alabama , Georgia est Virginia w Mexico ish, Louisiana kansas , Missouri , Oklahoma , Florida , Oklahoma kansas , Oklahoma exas kansas , Missouri , Missouri , Missouri , Missouri , Oklahoma , Missouri exas kansas entucky , Illinois entucky , Georgia , Illinois , Alabama , Illinois , Illinois , Illinois , Missouri th Carolina th Carolina , Oklahoma , Oklahoma exas kansas , Missouri , Virginia , Alabama kansas entucky , Georgia , Missouri , Alabama , Alabama th Carolina est Virginia est Virginia ennsylvania , Indiana entucky , Ohio , Indiana , Indiana ennsylvania , Indiana , Indiana , Indiana , Alabama , South Carolina , Oklahoma ennessee , Indiana ennessee , Illinois , Alabama entucky , Illinois ennessee , Alabama ennessee , Virginia th Carolina th Carolina , Alabama , Florida , Alabama kansas , Mississippi , Alabama ennessee , Illinois entucky est Virginia ennsylvania ennsylvania , Alabama ennessee ennessee , Georgia ennessee ennessee entucky ennessee entucky , Virginia , Virginia , Virginia est Virginia est Virginia est Virginia ennessee ennessee , Virginia entucky est Virginia , Iowa , Iowa , Kansas , Missouri , Kansas , Nebraska , Kansas , Oklahoma , Illinois , Illinois , Illinois , Missouri , Oklahoma , Illinois , Iowa , Kansas , Minnesota , Nebraska , Missouri , Oklahoma , Iowa , Iowa , Illinois , Iowa , Oklahoma kansas , Illinois kansas kansas ashington , Mississippi kansas , Mississippi entucky kansas ish, Louisiana , Missouri , Oklahoma , Alabama , Georgia , Mississippi , Mississippi , Missouri , Alabama , Georgia , Georgia ish, Louisiana , Oklahoma exas , Missouri , Missouri , Missouri ish, Louisiana kansas kansas exas , Indiana , Missouri exas , Missouri , Oklahoma exas exas , Mississippi exas ish, Louisiana , Mississippi , Mississippi exas , Oklahoma , Oklahoma exas , Illinois , Mississippi , Alabama , Texas ish, Louisiana kansas , Mississippi exas exas w York , Oklahoma , Alabama , Mississippi , Missouri ish, Louisiana , Colorado , Oklahoma exas , Missouri , Oklahoma exas exas , Texas , Idaho exas , Utah , South Dakota , Idaho , Oklahoma kansas , Minnesota

ming

, Indiana , Kansas , Ohio , Iowa entucky , Montana , Ohio ish, Louisiana , Oklahoma , Colorado , Oklahoma exas , Missouri , Missouri , Missouri , Indiana kansas exas exas kansas exas , Indiana , Georgia th Carolina kansas , Georgia , Georgia , Mississippi exas ish, Louisiana kansas kansas , Georgia , Georgia exas , Mississippi , Mississippi , Mississippi , Oklahoma ish, Louisiana ish, Louisiana , Mississippi , Mississippi , Mississippi , Mississippi ennessee , Mississippi kansas , Mississippi , Georgia , Mississippi , Mississippi , Mississippi exas , Mississippi , Mississippi w Mexico w Mexico , South Carolina , South Carolina , Georgia , Idaho , Nebraska , South Dakota , Mississippi exas , Montana , Kansas , Utah , Indiana , Iowa , Ohio ennsylvania yland , Ohio , Ohio ennsylvania , Minnesota w York , Michigan , Texas , Indiana kansas , Mississippi , Ohio ish, Louisiana ish, Louisiana , Illinois , Oklahoma , Georgia entucky entucky , Virginia kansas , South Carolina exas w York , Virginia , Indiana , Minnesota w York w York , Illinois , Virginia yland , Missouri ennsylvania entucky ennessee , Indiana est Virginia , Minnesota , Oregon , Oregon , Iowa , Illinois , Iowa , Kansas , Nebraska th Dakota th Dakota , Nebraska th Dakota , Iowa , Iowa th Dakota , South Dakota , Nebraska , South Dakota , Missouri , Missouri , Nebraska , Wisconsin w Jersey w York , Connecticut w York , Indiana , Ohio , Ohio , South Carolina , Indiana , Indiana ennessee exas , Michigan , Indiana , Missouri , Ohio , Michigan , Michigan , Michigan , Michigan , Indiana , Indiana , Illinois , Wisconsin w York , Massachusetts w York ennsylvania , Michigan , Georgia , Indiana , Georgia entucky , Georgia , Mississippi kansas , Georgia , Missouri , Missouri vada , Georgia , Mississippi w Mexico , Georgia , Illinois , Indiana , Indiana , Iowa , Minnesota ish, Louisiana , Mississippi , Ohio , Georgia , Georgia , Michigan , Indiana , Georgia ish, Louisiana , Ohio , Michigan , Indiana , Oregon , Ohio , Ohio , Ohio exas w Jersey , Illinois , Indiana ennessee , Wisconsin , Iowa , Nebraska , Indiana ish, Louisiana , Ohio , Georgia , Indiana , Missouri , Oklahoma ish, Louisiana exas exas , South Carolina , South Carolina entucky th Carolina , South Carolina , Mississippi , Texas , Georgia , Georgia , South Carolina , Georgia , Georgia , Georgia entucky , South Carolina , South Carolina , Alabama , Michigan , Ohio , Michigan , Georgia , Indiana , Iowa , Georgia , Mississippi , Missouri th Carolina , South Carolina , Michigan , Idaho , Ohio , South Carolina ashington , Indiana , Illinois , Ohio , Ohio , Kansas , Montana , Ohio , Utah

ming

w York , Georgia , Oklahoma , South Carolina , Georgia exas kansas , Oklahoma , Mississippi , Missouri , Indiana ennessee th Carolina exas , Alabama ish, Louisiana , Alabama , Alabama , Mississippi , Oklahoma exas , Georgia , Oregon to Rico , Utah , Utah , Utah , Georgia th Carolina , Georgia , Texas , Virginia , Florida th Carolina , Colorado , Montana , Oregon , Alabama , Illinois , Missouri th Dakota exas , Georgia ish, Louisiana , Mississippi w York , Virginia , Minnesota , Illinois entucky ashington exas , Indiana , Iowa , Illinois , Florida , Idaho est Virginia , Idaho , Virginia , Virginia , Virginia , Virginia , Mississippi , Virginia , Kansas exas , Georgia , Virginia , South Dakota ashington , Missouri , Missouri , Nebraska , Illinois th Carolina , Michigan , Ohio , Indiana , Kansas ennsylvania , Illinois , Mississippi , Oklahoma , South Dakota , Iowa

ming

, California w York , Virginia , Virginia , Massachusetts , Colorado , Colorado exas , Colorado w Jersey th Carolina ennessee ict of Columbia , Virginia , Arizona , Illinois , Indiana w York , Florida , Ohio , California w York , Texas , Michigan , Virginia w York , Michigan , Colorado , Georgia ish, Louisiana entucky entucky , Mississippi , Georgia exas , Kansas th Carolina , Mississippi , Oklahoma , South Carolina exas , California , California , Georgia th Carolina , Virginia , Michigan , California , Indiana , Indiana , Indiana , Kansas , Michigan ennessee , Nevada , California , Georgia th Carolina , Georgia entucky , Georgia entucky , Georgia , Mississippi , Georgia , Georgia th Carolina , Maryland entucky , South Carolina , Colorado , Virginia , Georgia , Georgia , Georgia th Carolina th Carolina , California , Georgia exas th Star Borough, Alaska , Colorado ennessee exas , Colorado , Texas , Oregon , Idaho exas , Utah , Missouri , Mississippi , Florida , Georgia , Oklahoma entucky ish, Louisiana , Georgia , Idaho th Carolina , Georgia , Kansas exas th Carolina ennessee to Rico exas , Florida , Illinois ennessee exas , Illinois exas , Texas exas , Georgia , Mississippi kansas , Ohio exas exas exas Aleutians East Borough, Alaska est Census Area, Alaska , Florida ish, Louisiana , Texas , California , Florida exas , Missouri , Colorado , Virginia w Jersey , Missouri th Carolina , Georgia , Ohio , Ohio , South Carolina , Alabama ennessee , Ohio ennessee ennessee w York , California , Colorado w York , Georgia , Georgia , Georgia , Georgia , Florida entucky , Florida , Florida , Georgia ennessee ish, Louisiana vada exas exas , Georgia entucky , Alabama ish, Louisiana ennsylvania , Indiana , Florida , Indiana ennsylvania , Florida entucky exas , Missouri , Colorado , Illinois , Oklahoma exas w Mexico exas , Iowa , Kansas , Oklahoma exas , Illinois , Illinois , Illinois th Carolina , Virginia entucky , Florida ennessee , Virginia , Virginia , Colorado , South Carolina , Montana , Colorado , Michigan , Virginia th Carolina ennessee , Virginia , Illinois , Illinois , Michigan th Carolina , Oklahoma , Virginia , Colorado , Virginia w York w Jersey , California , Virginia , Massachusetts , Virginia , California , California , Minnesota , Oregon , Massachusetts ashington Denali Borough, Alaska , Colorado , Colorado , California , Colorado , Idaho , Colorado , Colorado

ming

, Florida , Indiana entucky th Carolina , Georgia , Indiana entucky , South Carolina th Carolina th Carolina th Carolina th Carolina , South Carolina entucky , South Carolina , South Carolina , South Carolina th Carolina th Carolina , Ohio entucky , Virginia , Indiana th Carolina th Carolina th Carolina th Carolina th Carolina th Carolina w York , Alabama entucky th Carolina , Alabama , Georgia , Georgia entucky ennessee ennessee est Virginia , Alabama , Florida , Mississippi , Missouri th Carolina entucky th Carolina vada , Georgia entucky , Florida ish, Louisiana , Alabama

ming

, Florida , Missouri w Jersey w Jersey , Iowa , Florida entucky , Nebraska , South Dakota , Wisconsin ware w Jersey kansas w Mexico , Iowa entucky , Oklahoma , Mississippi ish, Louisiana , Missouri , Mississippi , Mississippi , Georgia , Mississippi , Georgia , Georgia , South Carolina , Alabama , Alabama , Georgia , Georgia , Florida ish, Louisiana th Carolina to Rico , Hawaii ennessee th Carolina ennessee yland , Rhode Island , Indiana , Indiana , Missouri th Carolina izona , Illinois , Missouri , Georgia , South Carolina , Illinois , Illinois , Michigan , Florida , Mississippi kansas , Colorado , Alabama , Oklahoma th Carolina entucky ennessee , South Carolina , Georgia , South Carolina , Alabama ish, Louisiana exas exas , Florida , Georgia exas exas exas , Alabama , Ohio , Ohio ennessee ish, Louisiana , Iowa , Florida entucky entucky entucky , South Carolina ennessee entucky ennessee , Georgia , Georgia entucky entucky ennessee , Indiana entucky , Georgia entucky , Missouri exas kansas , Georgia , Georgia , Georgia , Oregon , Alabama , Georgia , Iowa , Nebraska , Oregon , Georgia th Carolina , Georgia , Georgia , Mississippi , Mississippi exas , Georgia kansas , Mississippi , South Carolina exas kansas th Carolina th Carolina kansas , Indiana entucky , Indiana ennessee kansas , Georgia , Georgia , Alabama , Mississippi , Virginia , Georgia , Alabama , Georgia ish, Louisiana , Minnesota , California , Minnesota th Dakota th Carolina , South Carolina , Oklahoma , Georgia , Virginia , Missouri , Wisconsin ennsylvania to Rico , California , Michigan w Hampshire , California , Oregon , Georgia yland , Michigan ennsylvania , Missouri , Virginia est Virginia , Iowa , Iowa w York w York , South Dakota ennsylvania ennsylvania ennsylvania ennsylvania exas , Indiana entucky th Dakota , Nebraska , South Dakota , Missouri , Wisconsin kansas , Mississippi , Missouri , Missouri kansas , Oklahoma , Oklahoma kansas exas , California , Indiana , Missouri , Minnesota ashington w York , Wisconsin , Maine , Maine w Hampshire , Rhode Island , Ohio , Montana , Oklahoma w York ermont w York , Ohio , Oregon th Dakota ashington , California , Connecticut ermont w York w Hampshire , Ohio , Virginia , Indiana w York , Wisconsin entucky , Michigan kansas , Georgia , Nebraska th Dakota , Colorado , Colorado , Montana , Oklahoma , South Carolina , Colorado , Wisconsin , Georgia , Georgia ish, Louisiana ish, Louisiana , Mississippi , Colorado , Minnesota , Mississippi , Alabama , Mississippi w Mexico , Alabama , Oklahoma , Ohio ennessee exas exas , Iowa , Minnesota , Alabama ish, Louisiana w Mexico , Utah izona , Idaho kansas , Georgia , Georgia ish, Louisiana , Alabama entucky , South Carolina ennessee kansas , Michigan ashington entucky , Missouri , Illinois , Michigan th Carolina , California , Mississippi , Kansas , Nebraska exas exas , Massachusetts , Wisconsin , Wisconsin , Minnesota exas , Minnesota w York , Minnesota , Ohio , Wisconsin , Wisconsin , Michigan , Michigan , Oklahoma , Minnesota , Nebraska , Virginia kansas , Kansas , Minnesota , Oklahoma exas , Wisconsin , Indiana ennessee , Kansas exas , Iowa , Virginia entucky th Carolina est Virginia to Rico , California , California , California , California exas , Kansas , Kansas , Georgia , Idaho , Nebraska w Mexico , Oklahoma , Utah ade Hampton Census Area, Alaska , South Dakota th Dakota , South Dakota , South Dakota Dillingham Census Area, Alaska Nome Census Area, Alaska th Slope Borough, Alaska Bethel Census Area, Alaska est Arctic Borough, Alaska w Mexico , South Dakota , South Dakota , Utah izona , Wisconsin exas exas exas exas , California , California exas , Ohio , Idaho , Utah , Indiana exas exas ashington , Utah , Kansas ashington to Rico , Utah to Rico , Utah to Rico to Rico to Rico to Rico to Rico to Rico ish, Louisiana ish, Louisiana , Alabama ish, Louisiana ish, Louisiana ish, Louisiana ish, Louisiana , Mississippi w Mexico , Mississippi , Mississippi , Mississippi , California , Oklahoma , Colorado , Mississippi , Mississippi kansas , Georgia , Mississippi w Mexico exas , California ware ish, Louisiana ish, Louisiana ish, Louisiana ish, Louisiana ish, Louisiana ish, Louisiana ish, Louisiana exas ish, Louisiana , South Dakota , South Dakota exas , California w Mexico , Idaho , Idaho , Kansas , Kansas w Mexico , Oregon , Georgia exas izona , Idaho exas exas ish, Louisiana exas ashington , California , California kansas exas exas , California exas exas exas , California ashington ashington izona , Montana th Dakota , South Dakota , South Dakota , Arizona , Kansas , Nebraska th Dakota , South Dakota , Texas exas exas exas exas exas eninsula Borough, Alaska yukuk Census Area, Alaska , Montana , Montana , South Dakota , Utah , Utah , Utah , Utah exas , Idaho , Idaho , Utah , Utah ish, Louisiana , Mississippi , Mississippi , Mississippi , Mississippi , Mississippi , Idaho , Colorado exas , Utah , Idaho , Utah , Idaho , Utah , Alabama kansas exas , Utah , Florida , Oregon exas exas exas exas izona exas ish, Louisiana exas izona , Utah exas , Georgia exas exas , Georgia , Georgia exas , Idaho , Montana exas exas exas exas exas , Nebraska , South Dakota exas exas vada , Utah exas , Idaho , Idaho , Utah , Georgia , Georgia , Indiana , Alabama , Georgia , Georgia , Mississippi , Colorado , Kansas , Nebraska , Idaho w Mexico exas exas , California , Idaho , Idaho , Mississippi exas , Idaho , Kansas exas , Indiana exas , Kansas , Texas , Alabama , Missouri , Mississippi exas , Alabama , Alabama , Georgia , Georgia ish, Louisiana ish, Louisiana , Alabama kansas , Mississippi , Mississippi , Oklahoma ish, Louisiana w Mexico , Texas exas w Mexico , South Dakota , Idaho exas exas exas , Nebraska , Wisconsin , Florida , Missouri th Carolina , Florida , South Carolina to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico , Mississippi to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico to Rico

5000 10000 15000 20000 25000

Males Females Ages 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−84 85+ 0.075 0.15 0.075 0.15 Males Females Ages 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−84 85+ 0.075 0.15 0.075 0.15 Males Females Ages 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−84 85+ 0.075 0.15 0.075 0.15 Males Females Ages 0−4 5−9 10−14 15−19 20−24 25−29 30−34 35−39 40−44 45−49 50−54 55−59 60−64 65−69 70−74 75−79 80−84 85+ 0.075 0.15 0.075 0.15

V. Batagelj

Symbolic data analysis

SLIDE 27

Symbolic data analysis

V. Batagelj

Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

US counties map

Autauga Baldwin Barbour Bibb Blount Bullock Butler Mississippi Monroe Montgomery Nevada Newton Ouachita Perry Phillips Pike Poinsett Union Upson Walker Walton Ware Warren Washington Wayne Webster Wheeler Grant Graves Grayson Green Greenup Hancock Hardin Harlan Harrison Benton Boone Bradley Calhoun Carroll Chicot Clark Clay Cleburne Cleveland Columbia Conway Craighead Franklin Gadsden Gilchrist Glades Gulf Hamilton Hardee Hendry Hernando Highlands Hillsborough Holmes Indian River Wilcox Wilkes Wilkinson Worth Ada Adams Bannock Bear Lake McLean Menard Mercer Monroe Montgomery Morgan Moultrie Ogle Peoria Perry Piatt Pike Pope Stevens Sumner Thomas Trego Wabaunsee Wallace Washington Wichita Wilson Woodson Wyandotte Adair Clark Clay Clinton Cole Cooper Crawford Dade Dallas Daviess De Kalb Dent Douglas Dunklin Calhoun Chambers Cherokee Chilton Choctaw Clarke Clay Cleburne Coffee Colbert Conecuh Coosa Covington Crenshaw Cullman Dale Dallas De Kalb Elmore Escambia Etowah Fayette Franklin Geneva Greene Hale Henry Houston Jackson Jefferson Lamar Lauderdale Lawrence Lee Limestone Lowndes Macon Madison Marengo Marion Marshall Mobile Monroe Montgomery Morgan Perry Pickens Pike Randolph Russell Saint Clair Shelby Sumter Talladega Tallapoosa Tuscaloosa Walker Washington Wilcox Winston Apache Cochise Coconino Gila Graham Greenlee La Paz Maricopa Mohave Navajo Pima Pinal Santa Cruz Yavapai Yuma Arkansas Ashley Baxter Crawford Crittenden Cross Dallas Desha Drew Faulkner Franklin Fulton Garland Grant Greene Hempstead Hot Spring Howard Independence Izard Jackson Jefferson Johnson Lafayette Lawrence Lee Lincoln Little River Logan Lonoke Madison Marion Miller Polk Pope Prairie Pulaski Randolph Saint Francis Saline Scott Searcy Sebastian Sevier Sharp Stone Union Van Buren Washington White Woodruff Yell Alameda Alpine Amador Butte Calaveras Colusa Contra Costa Del Norte El Dorado Fresno Glenn Humboldt Imperial Inyo Kern Kings Lake Lassen Los Angeles Madera Marin Mariposa Mendocino Merced Modoc Mono Monterey Napa Nevada Orange Placer Plumas Riverside Sacramento San Benito San Bernardino San Diego San Francisco San Joaquin San Luis Obispo San Mateo Santa Barbara Santa Clara Santa Cruz Shasta Sierra Siskiyou Solano Sonoma Stanislaus Sutter Tehama Trinity Tulare Tuolumne Ventura Yolo Yuba Adams Alamosa Arapahoe Archuleta Baca Bent Boulder Broomfield Chaffee Cheyenne Clear Creek Conejos Costilla Crowley Custer Delta Denver Dolores Douglas Eagle El Paso Elbert Fremont Garfield Gilpin Grand Gunnison Hinsdale Huerfano Jackson Jefferson Kiowa Kit Carson La Plata Lake Larimer Las Animas Lincoln Logan Mesa Mineral Moffat Montezuma Montrose Morgan Otero Ouray Park Phillips Pitkin Prowers Pueblo Rio Blanco Rio Grande Routt Saguache San Juan San Miguel Sedgwick Summit Teller Washington Weld Yuma Fairfield Hartford Litchfield Middlesex New Haven New London Tolland Windham Kent New Castle Sussex District of Columbia Alachua Baker Bay Bradford Brevard Broward Calhoun Charlotte Citrus Clay Collier Columbia Desoto Dixie Duval Escambia Flagler Jackson Jefferson Lafayette Lake Lee Leon Levy Liberty Madison Manatee Marion Martin Miami−Dade Monroe Nassau Okaloosa Okeechobee Orange Osceola Palm Beach Pasco Pinellas Polk Putnam Saint Johns Saint Lucie Santa Rosa Sarasota Seminole Sumter Suwannee Taylor Union Volusia Wakulla Walton Washington Appling Atkinson Bacon Baker Baldwin Banks Barrow Bartow Ben Hill Berrien Bibb Bleckley Brantley Brooks Bryan Bulloch Burke Butts Calhoun Camden Candler Carroll Catoosa Charlton Chatham Chattahoochee Chattooga Cherokee Clarke Clay Clayton Clinch Cobb Coffee Colquitt Columbia Cook Coweta Crawford Crisp Dade Dawson Decatur DeKalb Dodge Dooly Dougherty Douglas Early Echols Effingham Elbert Emanuel Evans Fannin Fayette Floyd Forsyth Franklin Fulton Gilmer Glascock Glynn Gordon Grady Greene Gwinnett Habersham Hall Hancock Haralson Harris Hart Heard Henry Houston Irwin Jackson Jasper Jeff Davis Jefferson Jenkins Johnson Jones Lamar Lanier Laurens Lee Liberty Lincoln Long Lowndes Lumpkin Macon Madison Marion McDuffie McIntosh Meriwether Miller Mitchell Monroe Montgomery Morgan Murray Muscogee Newton OconeeOglethorpe Paulding Peach Pickens Pierce Pike Polk Pulaski Putnam Quitman Rabun Randolph Richmond Rockdale Schley Screven Seminole Spalding Stephens Stewart Sumter Talbot Taliaferro Tattnall Taylor Telfair Terrell Thomas Tift Toombs Towns Treutlen Troup Turner Twiggs White Whitfield Benewah Bingham Blaine Boise Bonner Bonneville Boundary Butte Camas Canyon Caribou Cassia Clark Clearwater Custer Elmore Franklin Fremont Gem Gooding Idaho Jefferson Jerome Kootenai Latah Lemhi Lewis Lincoln Madison Minidoka Nez Perce Oneida Owyhee Payette Power Shoshone Teton Twin Falls Valley Washington Adams Alexander Bond Boone Brown Bureau Calhoun Carroll Cass Champaign Christian Clark Clay Clinton Coles Cook Crawford Cumberland De Kalb De Witt Douglas Dupage Edgar Edwards Effingham Fayette Ford Franklin Fulton Gallatin Greene Grundy Hamilton Hancock Hardin Henderson Henry Iroquois Jackson Jasper Jefferson Jersey Jo Daviess Johnson Kane Kankakee Kendall Knox La Salle Lake Lake Michigan Lawrence Lee Livingston Logan Macon Macoupin Madison Marion Marshall Mason Massac McDonough McHenry Pulaski Putnam Randolph Richland Rock Island Saint Clair Saline Sangamon Schuyler Scott Shelby Stark Stephenson Tazewell Union Vermilion Wabash Warren Washington Wayne White Whiteside Will Williamson Winnebago Woodford Adams Allen Bartholomew Benton Blackford Boone Brown Carroll Cass Clark Clay Clinton Crawford Daviess De Kalb Dearborn Decatur Delaware Dubois Elkhart Fayette Floyd Fountain Franklin Fulton Gibson Grant Greene Hamilton Hancock Harrison Hendricks Henry Howard Huntington Jackson Jasper Jay Jefferson Jennings Johnson Knox Kosciusko LaGrange Lake Lake Michigan LaPorte Lawrence Madison Marion Marshall Martin Miami Monroe Montgomery Morgan Newton Noble Ohio Orange Owen Parke Perry Pike Porter Posey Pulaski Putnam Randolph Ripley Rush Saint Joseph Scott Shelby Spencer Starke Steuben Sullivan Switzerland Tippecanoe Tipton Union Vanderburgh Vermillion Vigo Wabash Warren Warrick Washington Wayne Wells White Whitley Adair Adams Allamakee Appanoose Audubon Benton Black Hawk Boone Bremer Buchanan Buena Vista Butler Calhoun Carroll Cass Cedar Cerro Gordo Cherokee Chickasaw Clarke Clay Clayton Clinton Crawford Dallas Davis Decatur Delaware Des Moines Dickinson Dubuque Emmet Fayette Floyd Franklin Fremont Greene Grundy Guthrie Hamilton Hancock Hardin Harrison Henry Howard Humboldt Ida Iowa Jackson Jasper Jefferson Johnson Jones Keokuk Kossuth Lee Linn Louisa Lucas Lyon Madison Mahaska Marion Marshall Mills Mitchell Monona Monroe Montgomery Muscatine O’Brien Osceola Page Palo Alto Plymouth Pocahontas Polk Pottawattamie Poweshiek Ringgold Sac Scott Shelby Sioux Story Tama Taylor Union Van Buren Wapello Warren Washington Wayne Webster Winnebago Winneshiek Woodbury Worth Wright Allen Anderson Atchison Barber Barton Bourbon Brown Butler Chase Chautauqua Cherokee Cheyenne Clark Clay Cloud Coffey Comanche Cowley Crawford Decatur Dickinson Doniphan Douglas Edwards Elk Ellis Ellsworth Finney Ford Franklin Geary Gove Graham Grant Gray Greeley Greenwood Hamilton Harper Harvey Haskell Hodgeman Jackson Jefferson Jewell Johnson Kearny Kingman Kiowa Labette Lane Leavenworth Lincoln Linn Logan Lyon Marion Marshall McPherson Meade Miami Mitchell Montgomery Morris Morton Nemaha Neosho Ness Norton Osage Osborne Ottawa Pawnee Phillips Pottawatomie Pratt Rawlins Reno Republic Rice Riley Rooks Rush Russell Saline Scott Sedgwick Seward Shawnee Sheridan Sherman Smith Stafford Stanton Allen Anderson Ballard Barren Bath Bell Boone Bourbon Boyd Boyle Bracken Breathitt Breckinridge Bullitt Butler Caldwell Calloway Campbell Carlisle Carroll Carter Casey Christian Clark Clay Clinton Crittenden Cumberland Daviess Edmonson Elliott Estill Fayette Fleming Floyd Franklin Fulton Gallatin Garrard Hart Henderson Henry Hickman Hopkins Jackson Jefferson Jessamine Johnson Kenton Knott Knox Larue Laurel Lawrence Lee Leslie Letcher Lewis Lincoln Livingston Logan Lyon Madison Magoffin Marion Marshall Martin Mason McCracken McCreary McLean Meade Menifee Mercer Metcalfe Chisago Clay Clearwater Cook Cottonwood Crow Wing Dakota Dodge Douglas Faribault Clarke Clay Coahoma Copiah Covington Desoto Forrest Franklin George Worth Wright Beaverhead Big Horn Blaine Broadwater Carbon Carter Cascade Chouteau Furnas Gage Garden Garfield Gosper Grant Greeley Hall Hamilton Harlan Hayes Hitchcock Holt Hooker Howard Jefferson Johnson Kearney Keith Keya Paha Brown Burleson Burnet Caldwell Calhoun Callahan Cameron Camp Carson Cass King Kinney Kleberg Knox La Salle Lamar Lamb Lampasas Lavaca Lee Monroe Montgomery Morgan Muhlenberg Nelson Nicholas Ohio Oldham Owen Owsley Pendleton Perry Pike Powell Pulaski Robertson Rockcastle Rowan Russell Scott Shelby Simpson Spencer Taylor Todd Trigg Trimble Union Warren Washington Wayne Webster Whitley Wolfe Woodford Acadia Allen Ascension Assumption Avoyelles Beauregard Bienville Bossier Caddo Calcasieu Caldwell Cameron Catahoula Claiborne Concordia De Soto East Baton Rouge East Carroll East Feliciana Evangeline Franklin Grant Iberia Iberville Jackson Jefferson Jefferson Davis La Salle Lafayette Lafourche Lincoln Livingston Madison Morehouse Natchitoches Orleans Ouachita Plaquemines Pointe Coupee Rapides Red River Richland Sabine Saint Bernard Saint Charles Saint Helena Saint James Saint John the Baptist Saint Landry Saint Martin Saint Mary Saint Tammany Tangipahoa Tensas Terrebonne Union Vermilion Vernon Washington Webster West Baton Rouge West Carroll West Feliciana Winn Androscoggin Aroostook Cumberland Franklin Hancock Kennebec Knox Lincoln Oxford Penobscot Piscataquis Sagadahoc Somerset Waldo Washington York Allegany Anne Arundel Baltimore Calvert Caroline Carroll Cecil Charles Dorchester Frederick Garrett Harford Howard Kent Montgomery Prince George’s Queen Anne’s Saint Mary’s Somerset Talbot Washington Wicomico Worcester Barnstable Berkshire Bristol Dukes Essex Franklin Hampden Hampshire Middlesex Nantucket Norfolk Plymouth Suffolk Worcester Alcona Alger Allegan Alpena Antrim Arenac Baraga Barry Bay Benzie Berrien Branch Calhoun Cass Charlevoix Cheboygan Chippewa Clare Clinton Crawford Delta Dickinson Eaton Emmet Genesee Gladwin Gogebic Grand Traverse Gratiot Hillsdale Houghton Huron Ingham Ionia Iosco Iron Isabella Jackson Kalamazoo Kalkaska Kent Keweenaw Lake Lake Hurron Lake Michigan Lake St. Clair Lake Superior Lapeer Leelanau Lenawee Livingston Luce Mackinac Macomb Manistee Marquette Mason Mecosta Menominee Midland Missaukee Monroe Montcalm Montmorency Muskegon Newaygo Oakland Oceana Ogemaw Ontonagon Osceola Oscoda Otsego Ottawa Presque Isle Roscommon Saginaw Saint Clair Saint Joseph Sanilac Schoolcraft Shiawassee Tuscola Van Buren Washtenaw Wayne Wexford Aitkin Anoka Becker Beltrami Benton Big Stone Blue Earth Brown Carlton Carver Cass Chippewa Fillmore Freeborn Goodhue Grant Hennepin Houston Hubbard Isanti Itasca Jackson Kanabec Kandiyohi Kittson Koochiching Lac qui Parle Lake Lake of the Woods Lake Superior Le Sueur Lincoln Lyon Mahnomen Marshall Martin McLeod Meeker Mille Lacs Morrison Mower Murray Nicollet Nobles Norman Olmsted Otter Tail Pennington Pine Pipestone Polk Pope Ramsey Red Lake Redwood Renville Rice Rock Roseau Saint Louis Scott Sherburne Sibley Stearns Steele Stevens Swift Todd Traverse Wabasha Wadena Waseca Washington Watonwan Wilkin Winona Wright Yellow Medicine Adams Alcorn Amite Attala Benton Bolivar Calhoun Carroll Chickasaw Choctaw Claiborne Greene Grenada Hancock Harrison Hinds Holmes Humphreys Issaquena Itawamba Jackson Jasper Jefferson Jefferson Davis Jones Kemper Lafayette Lamar Lauderdale Lawrence Leake Lee Leflore Lincoln Lowndes Madison Marion Marshall Monroe Montgomery Neshoba Newton Noxubee Oktibbeha Panola Pearl River Perry Pike Pontotoc Prentiss Quitman Rankin Scott Sharkey Simpson Smith Stone Sunflower Tallahatchie Tate Tippah Tishomingo Tunica Union Walthall Warren Washington Wayne Webster Wilkinson Winston Yalobusha Yazoo Adair Andrew Atchison Audrain Barry Barton Bates Benton Bollinger Boone Buchanan Butler Caldwell Callaway Camden Cape Girardeau Carroll Carter Cass Cedar Chariton Christian Franklin Gasconade Gentry Greene Grundy Harrison Henry Hickory Holt Howard Howell Iron Jackson Jasper Jefferson Johnson Knox Laclede Lafayette Lawrence Lewis Lincoln Linn Livingston Macon Madison Maries Marion McDonald Mercer Miller Mississippi Moniteau Monroe Montgomery Morgan New Madrid Newton Nodaway Oregon Osage Ozark Pemiscot Perry Pettis Phelps Pike Platte Polk Pulaski Putnam Ralls Randolph Ray Reynolds Ripley Saint Charles Saint Clair Saint Francois Saint Louis Sainte Genevieve Saline Schuyler Scotland Scott Shannon Shelby Stoddard Stone Sullivan Taney Texas Vernon Warren Washington Wayne Webster Custer Daniels Dawson Deer Lodge Fallon Fergus Flathead Gallatin Garfield Glacier Golden Valley Granite Hill Jefferson Judith Basin Lake Lewis and Clark Liberty Lincoln Madison McCone Meagher Mineral Missoula Musselshell Park Petroleum Phillips Pondera Powder River Powell Prairie Ravalli Richland Roosevelt Rosebud Sanders Sheridan Silver Bow Stillwater Sweet Grass Teton Toole Treasure Valley Wheatland Wibaux Yellowstone Adams Antelope Arthur Banner Blaine Boone Box Butte Boyd Brown Buffalo Burt Butler Cass Cedar Chase Cherry Cheyenne Clay Colfax Cuming Custer Dakota Dawes Dawson Deuel Dixon Dodge Douglas Dundy Fillmore Franklin Frontier Kimball Knox Lancaster Lincoln Logan Loup Madison McPherson Merrick Morrill Nance Nemaha Nuckolls Otoe Pawnee Perkins Phelps Pierce Platte Polk Red Willow Richardson Rock Saline Sarpy Saunders Scotts Bluff Seward Sheridan Sherman Sioux Stanton Thayer Thomas Thurston Valley Washington Wayne Webster Wheeler York Carson City Churchill Clark Douglas Elko Esmeralda Eureka Humboldt Lander Lincoln Lyon Mineral Nye Pershing Storey Washoe White Pine Belknap Carroll Cheshire Coos Grafton Hillsborough Merrimack Rockingham Strafford Sullivan Atlantic Bergen Burlington Camden Cape May Cumberland Essex Gloucester Hudson Hunterdon Mercer Middlesex Monmouth Morris Ocean Passaic Salem Somerset Sussex Union Warren Bernalillo Catron Chaves Cibola Colfax Curry Debaca Dona Ana Eddy Grant Guadalupe Harding Hidalgo Lea Lincoln Los Alamos Luna McKinley Mora Otero Quay Rio Arriba Putnam Queens Rensselaer Richmond Rockland Saint Lawrence Saratoga Schenectady Schoharie Schuyler Seneca Steuben Alexander Alleghany Anson Ashe Avery Beaufort Bertie Bladen Brunswick Buncombe Burke Cabarrus Caldwell Camden Carteret Caswell Catawba Chatham Cherokee Chowan Clay Cleveland Columbus Craven Washington Wheeler Yamhill Adams Allegheny Armstrong Beaver Bedford Berks Blair Bradford Marshall Maury McMinn McNairy Meigs Monroe Montgomery Moore Morgan Obion Overton Perry Gillespie Glasscock Goliad Gonzales Gray Grayson Gregg Grimes Guadalupe Hale Hall Hamilton Hansford Walla Walla Whatcom Whitman Yakima Barbour Berkeley Boone Braxton Brooke Cabell Calhoun Clay Roosevelt San Juan San Miguel Sandoval Santa Fe Sierra Socorro Taos Torrance Union Valencia Albany Allegany Bronx Broome Cattaraugus Cayuga Chautauqua Chemung Chenango Clinton Columbia Cortland Delaware Dutchess Erie Essex Franklin Fulton Genesee Greene Hamilton Herkimer Jefferson Kings Lake Ontario Lewis Livingston Madison Monroe Montgomery Nassau New York Niagara Oneida Onondaga Ontario Orange Orleans Oswego Otsego Suffolk Sullivan Tioga Tompkins Ulster Warren Washington Wayne Westchester Wyoming Yates Alamance Cumberland Currituck Dare Davidson Davie Duplin Durham Edgecombe Forsyth Franklin Gaston Gates Graham Granville Greene Guilford Halifax Harnett Haywood Henderson Hertford Hoke Hyde Iredell Jackson Johnston Jones Lee Lenoir Lincoln Macon Madison Martin McDowell Mecklenburg Mitchell Montgomery Moore Nash New Hanover Northampton Onslow Orange Pamlico Pasquotank Pender Perquimans Person Pitt Polk Randolph Richmond Robeson Rockingham Rowan Rutherford Sampson Scotland Stanly Stokes Surry Swain Transylvania Tyrrell Union Vance Wake Warren Washington Watauga Wayne Wilkes Wilson Yadkin Yancey Adams Barnes Benson Billings Bottineau Bowman Burke Burleigh Cass Cavalier Dickey Divide Dunn Eddy Emmons Foster Golden Valley Grand Forks Grant Griggs Hettinger Kidder Lamoure Logan McHenry McIntosh McKenzie McLean Mercer Morton Mountrail Nelson Oliver Pembina Pierce Ramsey Ransom Renville Richland Rolette Sargent Sheridan Sioux Slope Stark Steele Stutsman Towner Traill Walsh Ward Wells Williams Adams Allen Ashland Ashtabula Athens Auglaize Belmont Brown Butler Carroll Champaign Clark Clermont Clinton Columbiana Coshocton Crawford Cuyahoga Darke Defiance Delaware Erie Fairfield Fayette Franklin Fulton Gallia Geauga Greene Guernsey Hamilton Hancock Hardin Harrison Henry Highland Hocking Holmes Huron Jackson Jefferson Knox Lake Lake Erie Lawrence Licking Logan Lorain Lucas Madison Mahoning Marion Medina Meigs Mercer Miami Monroe Montgomery Morgan Morrow Muskingum Noble Ottawa Paulding Perry Pickaway Pike Portage Preble Putnam Richland Ross Sandusky Scioto Seneca Shelby Stark Summit Trumbull Tuscarawas Union Van Wert Vinton Warren Washington Wayne Williams Wood Wyandot Adair Alfalfa Atoka Beaver Beckham Blaine Bryan Caddo Canadian Carter Cherokee Choctaw Cimarron Cleveland Coal Comanche Cotton Craig Creek Custer Delaware Dewey Ellis Garfield Garvin Grady Grant Greer Harmon Harper Haskell Hughes Jackson Jefferson Johnston Kay Kingfisher Kiowa Latimer Le Flore Lincoln Logan Love Major Marshall Mayes McClain McCurtain McIntosh Murray Muskogee Noble Nowata Okfuskee Oklahoma Okmulgee Osage Ottawa Pawnee Payne Pittsburg Pontotoc Pottawatomie Pushmataha Roger Mills Rogers Seminole Sequoyah Stephens Texas Tillman Tulsa Wagoner Washington Washita Woods Woodward Baker Benton Clackamas Clatsop Columbia Coos Crook Curry Deschutes Douglas Gilliam Grant Harney Hood River Jackson Jefferson Josephine Klamath Lake Lane Lincoln Linn Malheur Marion Morrow Multnomah Polk Sherman Tillamook Umatilla Union Wallowa Wasco Bucks Butler Cambria Cameron Carbon Centre Chester Clarion Clearfield Clinton Columbia Crawford Cumberland Dauphin Delaware Elk Erie Fayette Forest Franklin Fulton Greene Huntingdon Indiana Jefferson Juniata Lackawanna Lancaster Lawrence Lebanon Lehigh Luzerne Lycoming Mc Kean Mercer Mifflin Monroe Montgomery Montour Northampton Northumberland Perry Philadelphia Pike Potter Schuylkill Snyder Somerset Sullivan Susquehanna Tioga Union Venango Warren Washington Wayne Westmoreland Wyoming York Bristol Kent Newport Providence Washington Abbeville Aiken Allendale Anderson Bamberg Barnwell Beaufort Berkeley Calhoun Charleston Cherokee Chester Chesterfield Clarendon Colleton Darlington Dillon Dorchester Edgefield Fairfield Florence Georgetown Greenville Greenwood Hampton Horry Jasper Kershaw Lancaster Laurens Lee Lexington Marion Marlboro McCormick Newberry Oconee Orangeburg Pickens Richland Saluda Spartanburg Sumter Union Williamsburg York Aurora Beadle Bennett Bon Homme Brookings Brown Brule Buffalo Butte Campbell Charles Mix Clark Clay Codington Corson Custer Davison Day Deuel Dewey Douglas Edmunds Fall River Faulk Grant Gregory Haakon Hamlin Hand Hanson Harding Hughes Hutchinson Hyde Jackson Jerauld Jones Kingsbury Lake Lawrence Lincoln Lyman Marshall McCook McPherson Meade Mellette Miner Minnehaha Moody Pennington Perkins Potter Roberts Sanborn Shannon Spink Stanley Sully Todd Tripp Turner Union Walworth Yankton Ziebach Anderson Bedford Benton Bledsoe Blount Bradley Campbell Cannon Carroll Carter Cheatham Chester Claiborne Clay Cocke Coffee Crockett Cumberland Davidson Decatur DeKalb Dickson Dyer Fayette Fentress Franklin Gibson Giles Grainger Greene Grundy Hamblen Hamilton Hancock Hardeman Hardin Hawkins Haywood Henderson Henry Hickman Houston Humphreys Jackson Jefferson Johnson Knox Lake Lauderdale Lawrence Lewis Lincoln Loudon Macon Madison Marion Pickett Polk Putnam Rhea Roane Robertson Rutherford Scott Sequatchie Sevier Shelby Smith Stewart Sullivan Sumner Tipton Trousdale Unicoi Union Van Buren Warren Washington Wayne Weakley White Williamson Wilson Anderson Andrews Angelina Aransas Archer Armstrong Atascosa Austin Bailey Bandera Bastrop Baylor Bee Bell Bexar Blanco Borden Bosque Bowie Brazoria Brazos Brewster Briscoe Brooks Castro Chambers Cherokee Childress Clay Cochran Coke Coleman Collin Collingsworth Colorado Comal Comanche Concho Cooke Coryell Cottle Crane Crockett Crosby Culberson Dallam Dallas Dawson Deaf Smith Delta Denton Dewitt Dickens Dimmit Donley Duval Eastland Ector Edwards El Paso Ellis Erath Falls Fannin Fayette Fisher Floyd Foard Fort Bend Franklin Freestone Frio Gaines Galveston Garza Hardeman Hardin Harris Harrison Hartley Haskell Hays Hemphill Henderson Hidalgo Hill Hockley Hood Hopkins Houston Howard Hudspeth Hunt Hutchinson Irion Jack Jackson Jasper Jeff Davis Jefferson Jim Hogg Jim Wells Johnson Jones Karnes Kaufman Kendall Kenedy Kent Kerr Kimble Leon Liberty Limestone Lipscomb Live Oak Llano Loving Lubbock Lynn Madison Marion Martin Mason Matagorda Maverick McCulloch McLennan McMullen Medina Menard Midland Milam Mills Mitchell Montague Montgomery Moore Morris Motley Nacogdoches Navarro Newton Nolan Nueces Ochiltree Oldham Orange Palo Pinto Panola Parker Parmer Pecos Polk Potter Presidio Rains Randall Reagan Real Red River Reeves Refugio Roberts Robertson Rockwall Runnels Rusk Sabine San Augustine San Jacinto San Patricio San Saba Schleicher Scurry Shackelford Shelby Sherman Smith Somervell Starr Ozaukee Pepin Pierce Polk Portage Price Racine Richland Rock Rusk Stephens Sterling Stonewall Sutton Swisher Tarrant Taylor Terrell Terry Throckmorton Titus Tom Green Travis Trinity Tyler Upshur Upton Uvalde Val Verde Van Zandt Victoria Walker Waller Ward Washington Webb Wharton Wheeler Wichita Wilbarger Willacy Williamson Wilson Winkler Wise Wood Yoakum Young Zapata Zavala Beaver Box Elder Cache Carbon Daggett Davis Duchesne Emery Garfield Grand Iron Juab Kane Millard Morgan Piute Rich Salt Lake San Juan Sanpete Sevier Summit Tooele Uintah Utah Wasatch Washington Wayne Weber Addison Bennington Caledonia Chittenden Essex Franklin Grand Isle Lamoille Orange Orleans Rutland Washington Windham Windsor Accomack Albemarle Alexandria Alleghany Amelia Amherst Appomattox Arlington Augusta Bath Bedford Bland Botetourt Bristol Brunswick Buchanan Buckingham Buena Vista Campbell Caroline Carroll Charles City Charlotte Charlottesville Chesapeake Chesterfield Clarke Clifton Forge City Colonial Heights Covington Craig Culpeper Cumberland Danville Dickenson Dinwiddie Emporia Essex Fairfax Falls Church Fauquier Floyd Fluvanna Franklin Frederick Hanover Harrisonburg Henrico Henry Highland Hopewell Isle of Wight James City King and Queen King George Fredericksburg Galax Giles Gloucester Goochland Grayson Greene Greensville Halifax Hampton King William Lancaster Lee Lexington Loudoun Louisa Lunenburg Lynchburg Madison Manassas Manassas Park Martinsville Mathews Mecklenburg Middlesex Montgomery Nelson New Kent Newport News Norfolk Northampton Northumberland Norton Nottoway Orange Page Patrick Petersburg Pittsylvania Poquoson Portsmouth Powhatan Prince Edward Prince George Prince William Pulaski Radford Rappahannock Richmond Roanoke Rockbridge Rockingham Russell Salem Scott Shenandoah Smyth Southampton Spotsylvania Stafford Staunton Suffolk Surry Sussex Tazewell Virginia Beach Warren Washington Waynesboro Westmoreland Williamsburg Winchester Wise Wythe York Adams Asotin Benton Chelan Clallam Clark Columbia Cowlitz Douglas Ferry Franklin Garfield Grant Grays Harbor Island Jefferson King Kitsap Kittitas Klickitat Lewis Lincoln Mason Okanogan Pacific Pend Oreille Pierce San Juan Skagit Skamania Snohomish Spokane Stevens Thurston Wahkiakum Doddridge Fayette Gilmer Grant Greenbrier Hampshire Hancock Hardy Harrison Jackson Jefferson Kanawha Lewis Lincoln Logan Marion Marshall Mason McDowell Mercer Mineral Mingo Monongalia Monroe Morgan Nicholas Ohio Pendleton Pleasants Pocahontas Preston Putnam Raleigh Randolph Ritchie Roane Summers Taylor Tucker Tyler Upshur Wayne Webster Wetzel Wirt Wood Wyoming Adams Ashland Barron Bayfield Brown Buffalo Burnett Calumet Chippewa Clark Columbia Crawford Dane Dodge Door Douglas Dunn Eau Claire Florence Fond du Lac Forest Grant Green Green Lake Iowa Iron Jackson Jefferson Juneau Kenosha Kewaunee La Crosse Lafayette Lake Michigan Lake Superior Langlade Lincoln Manitowoc Marathon Marinette Marquette Menominee Milwaukee Monroe Oconto Oneida Outagamie Saint Croix Sauk Sawyer Shawano Sheboygan Taylor Trempealeau Vernon Vilas Walworth Washburn Washington Waukesha Waupaca Waushara Winnebago Wood Albany Big Horn Campbell Carbon Converse Crook Fremont Goshen Hot Springs Johnson Laramie Lincoln Natrona Niobrara Park Platte Sheridan Sublette Sweetwater Teton Uinta Washakie Weston

all−American−2 youth all−American−1 student

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Leaders method Agglomerative method Examples References

k-means clusters of patents with patents’ temporal distributions and cluster leaders

# patents: 89 ; Cluster 4

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 66 ; Cluster 18

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

The classical k-means approach based on δ1 gives uninteresting results – the clusters have a single peak. The peak value prevails over other smaller values in the distributions. Using δ3 as the basic dissimilarity we obtained much more interesting results [Kejˇ zar, N. et al. (2011)].

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Leaders method Agglomerative method Examples References

Patents clusters for δ3

# patents: 198 ; Cluster 11

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 100 ; Cluster 18

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 37 ; Cluster 2

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 27 ; Cluster 27

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 201 ; Cluster 14

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 167 ; Cluster 17

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 25 ; Cluster 5

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 99 ; Cluster 15

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 109 ; Cluster 8

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 32 ; Cluster 21

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 195 ; Cluster 19

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 31 ; Cluster 25

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 44 ; Cluster 7

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 74 ; Cluster 22

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 175 ; Cluster 12

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 65 ; Cluster 24

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 239 ; Cluster 16

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 67 ; Cluster 28

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 105 ; Cluster 10

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

# patents: 68 ; Cluster 23

1980 1985 1990 1995 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Fig. 2. Citations 20+, leaders for δ4 - 1st part.
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Leaders method Agglomerative method Examples References

Patents / clustering of leaders

C5 C15 C8 C21 C17 C11 C18 C2 C14 C7 C19 C25 C13 C1 C22 C4 C20 C29 C30 C26 C28 C23 C24 C27 C3 C6 C9 C12 C10 C16

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References I

Anderberg, M. R. (1973). Cluster Analysis for Applications. Academic Press: New York. Batagelj, V. (1988). Generalized Ward and Related Clustering Problems. Classification and Related Methods of Data Analysis. H.H. Bock (editor). North-Holland, Amsterdam, 1988. p. 67-74. Batagelj, V., Kejˇ zar, N., Korenjak-ˇ Cerne, S. (2014) Clustering of modal valued symbolic data. Submitted. arXiv Billard, L., Diday, E. (2006). Symbolic data analysis. Conceptual statistics and data mining. Wiley: New York. Bock, H-H., Diday, E. (eds.) (2000). Analysis of Symbolic Data: Exploratory Methods for Extracting Statistical Information from Complex Data. Springer-Verlag, Berlin. Diday, E. (1979). Optimisation en classification automatique, Tome 1.,2.. INRIA, Rocquencourt (in French). Diday, E. (1987). Introduction ` a l’approche symbolique en analyse des donn´

ees. Premi`

ere Journ´ ees ’Symbolique-Numerique’. CEREMADE, Universit´ e Paris IX - Dauphine, 21-56.

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Symbolic data analysis Clustering and

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Leaders method Agglomerative method Examples References

References II

Diday, E. and Noirhomme, M. (eds.) (2008). Symbolic Data and the SODAS Software. John Wiley & Sons, Ltd, Chichester. Hartigan, J. A. (1975). Clustering algorithms, Wiley-Interscience: New York. Japelj Paveˇ si´ c, B., Korenjak-ˇ Cerne, S. (2004). ”Differences in teaching and learning mathematics in classes over the world : the application of adapted leaders clustering method”. In Proceedings of the IRC - 2004 : IEA International Research Conference. Nicosia: University of Cyprus, Department of Education, 2004, p. 85–101. Kejˇ zar, N., Korenjak-ˇ Cerne, S., Batagelj, V. (2011). “Clustering of discrete distributions: A case of patent citations”. J. classif., vol. 28, no. 2, p. 156-183. Korenjak-ˇ Cerne, S., Kejˇ zar, N., Batagelj, V. (2015). A weighted clustering

f population pyramids for the world’s countries, 1996, 2001, 2006.

Population Studies: A Journal of Demography, 69(1), 105-120. Korenjak-ˇ Cerne, S., Batagelj, V., Japelj Paveˇ si´ c, B. (2011). Clustering large data sets described with discrete distributions and its application on TIMSS data set. Stat. anal. data min., vol. 4, iss. 2, p. 199-215.

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Symbolic data analysis Clustering and

ptimization

Leaders method Agglomerative method Examples References

References III

Korenjak-ˇ Cerne, S., Kogovˇ sek, T., Batagelj, V. (2000). Clustering ego-centered networks. In: Blasius, J¨

rg (ed.). Social science methodology

in the new millennium: proceedings of the Fifth International Conference on Logic and Methodology. Cologne: TT-Publikaties, 4 pages. Korenjak-ˇ Cerne, S., Batagelj, V. (1998). Clustering large datasets of mixed

units. In: Rizzi, A., Vichi, M., Bock, H-H. (eds.). 6th Conference of the

International Federation of Classification Societies (IFCS-98) Universit´ a ”La Sapienza”, Rome, 21-24 July, 1998. Advances in data science and

classification. Berlin: Springer, p. 43-48.

Korenjak-ˇ Cerne, S., Batagelj, V. (2002). Symbolic data analysis approach to clustering large datasets. In: Jajuga, K., Soko lowski, A., Bock, H-H. (eds.). 8th Conference of the International Federation of Classification Societies, July 16-19, 2002, Cracow, Poland. Classification, clustering and data

analysis. Berlin: Springer, p. 319-327.

Ward, J. H. (1963). “Hierarchical grouping to optimize an objective function”, Journal of the American Statistical Association, 58, 236–244.

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