Statistical Archetypal Analysis

24 Pages·2017·1.21 MB·English

Statistical Archetypal Analysis
Chenyue Wua,∗, Esteban G. Tabaka
aDepartment of Mathematics, Courant Institute of Mathematical Sciences, New York
University, 251 Mercer Street, New York, NY 10012, United States
Abstract
Statistical Archetypal Analysis (SAA) is introduced for the dimensional reduc-
tionofacollectionofprobabilitydistributionsknownviasamples. Applications
includemedicaldiagnosisfromclinicaldataintheformofdistributions(suchas
distributionsofbloodpressureorheartratesfromdifferentpatients),theanaly-
sisofclimatedatasuchastemperatureorwindspeedatdifferentlocations,and
the study of bifurcations in stochastic dynamical systems. Distributions can be
embedded into a Hilbert space with a suitable metric, and then analyzed simi-
larly to feature vectors in Euclidean space. However, most dimensional reduc-
tion techniques –such as Principal Component Analysis– are not interpretable
fordistributions,asneitherthecomponentsnorthereconstructionofinputdata
by components are themselves distributions. To obtain an interpretable result,
Archetypal Analysis (AA) is extended to distributions, requiring the compo-
nents to be mixtures of the input distributions and approximating the input
distributions by mixtures of components.
Keywords: archetypal analysis, dimension reduction, energy distance, kernel
embedding, principal component analysis
1. Introduction
1
Finitecollectionsofprobabilitydistributionsappearnaturallyinavarietyof
2
settings, often as conditional distributions ρ(x|z) where z adopts a discrete set
3
of values. For instance, x may represent a collection of clinical variables such
4
as body temperature, blood pressure and cholesterol level, and z may stand for
5
covariatessuchassex, agegroupormedicaltreatment. Inanexamplethatthis
6
paper analyzes in some detail, x is the atmospheric temperature measured at
7
groundlevelandzstandsforthestationwherethemeasurementsareperformed.
8
It is therefore a natural extension of data analysis to use as either labels
9
or features, probability distributions instead of the more conventional discrete-
10
valued variables, continuum scalars or vectors. Thus one might want to predict
11
∗Correspondingauthor
Email addresses: [email protected](ChenyueWu),[email protected](Esteban
G.Tabak)
Preprint submitted to Elsevier February 2, 2017
not the temperature at a particular location and time but its probability distri-
12
bution,orclusterpopulationsformedicalpurposesaccordingtotheprobability
13
distributions of a group of clinical variables.
14
A basic quantity that permeates data analysis is the distance between data
15
points. There are several statistical distances in the literature that measure
16
thedissimilaritybetweentwoprobabilitydistributions. Somearebasedonana-
17
logues of the Euclidean distance, some on information theory, some on optimal
18
transport. Typically, each sheds a different light on what makes two distri-
19
butions different. In this article, we use the energy distance as a measure of
20
dissimilarity among distributions, as it is easy to evaluate efficiently from sam-
21
ple points and can be derived from an inner product, thus rendering accessible
22
many data analysis tools.
23
We study the problem of dimensional reduction of sets of distributions. Af-
24
ter being equipped with a metric and embedded into a Hilbert space, distri-
25
butions can be analyzed similarly to conventional feature vectors. However,
26
there is a gap between the dimensional reduction of distributions and vectors:
27
interpretability. Traditional dimension reduction techniques, such as principal
28
components analysis, lack interpretability when applied to probability distribu-
29
tions, as the projection of each distribution onto the low dimensional subspace
30
found is almost surely not a probability distribution: even though probability
31
distributions can be embedded into a Hilbert space, almost all elements in this
32
spacearenotprobabilitydistributions,sincetheseareconstrainedbypositivity
33
and normalization.
34
To overcome this difficulty in interpretation, we use the tools of archetypal
35
analysis. Archetypal analysis finds a small number of “archetypes” that are
36
convex combinations of the original data points, and approximates the origi-
37
nal data points again via convex combinations of these archetypes. A convex
38
combination can be interpreted as a mixture of probability distributions, so
39
the archetypes found by archetypal analysis are mixtures of the original dis-
40
tributions and the original distributions are approximated within the family of
41
mixtures of the archetypes.
42
Thispaperisarrangedasfollows: Section2givesareviewofarchetypalanal-
43
ysis,ofthealgorithmsforarchetypalanalysisinthegeneralcaseandspecifically
44
for energy distance. Section 3 reviews reproducing kernel Hilbert space, energy
45
distance, describes how distributions equipped with the energy distance can be
46
embeddedintoaHilbertspace, anddescribesalgorithmstoevaluatetheenergy
47
distance from samples. Section 4 introduces statistical archetypal analysis for
48
thedimensionalreductionofprobabilitydistributionsandincludesapplications
49
with numerical experiment.
50
2. Archetypal Analysis
51
Archetypalanalysisapproximatesdatapointsbyconvexcombinationofpro-
52
totypes, where these prototypes, denoted “archetypes”, are themselves convex
53
combinations of the data points.
54
2
Archetypal analysis was introduced in [1] –see also [2]– as a dimensional
55
reduction method alternative to principal components analysis (PCA), yielding
56
more interpretable results. It originated in the study of a dataset consisting of
57
6headdimensionsfor200soldiers,withthegoalofdesigningfacemasksforthe
58
Swiss Army. For this dataset, PCA found principal components that did not
59
resemble a head shape. To have patterns resembling “pure types” in the data,
60
each entry in the dataset was approximated by a mixture of the patterns. To
61
make patterns resemble the data, each pattern itself was a mixture of the data
62
points.
63
For a data matrix X =(x ,x ,··· ,x ) representing n observations, each of
1 2 n
dimension m, Archetypal Analysis seeks k (cid:28)n m-dimensional archetypes Z =
(z ,z ,··· ,z ),suchthateachx canbeapproximatedbyaconvexcombination
1 2 k i
of the z :
k
(cid:88)
x ≈a z +a z +···+a z , a ≥0, a =1,
i 1i 1 2i 2 ki k ji ji
j
where the z themselves are convex combinations of the data:
j
(cid:88)
z =b x +b x +···+b x , b ≥0, b =1.
j 1j 1 2j 2 nj n ij ij
i
After setting a number of archetypes k, the coefficients a and b arise from the
optimization problem
(cid:13) (cid:13)2
n (cid:13) k n (cid:13)
(cid:88)(cid:13) (cid:88) (cid:88) (cid:13)
min (cid:13)xi− aji bljxl(cid:13) , (1)
aji,blj i=1(cid:13)(cid:13) j=1 l=1 (cid:13)(cid:13)
with constraints
(cid:88) (cid:88)
a ≥0, a =1, b ≥0, b =1,
ji ji ij ij
j i
or, in terms of the matrices A=(a ) and B =(b ) ,
ji k×n lj n×k
min(cid:107)X−XBA(cid:107)2 (2)
F
A,B
under the same constraints, with (cid:107)·(cid:107) denoting the Frobenius norm
64 F
(cid:32) (cid:33)1
p q 2
(cid:107)M(cid:107) = (cid:80) (cid:80) |m |2 . Alternatively, this can be written as:
F ij
i=1j=1
(cid:124)
A,B =argmin tr[(I −BA) G(I −BA)], (3)
n n
where G = X(cid:124)X is the Gram matrix of data. This restatement is particularly
65
convenient,asitwillallowustoformulatetheproblemintermsofinnerproducts
66
among the data points instead of the points themselves, which in our problem
67
are distributions. Thus we need a norm for distributions that derive from an
68
inner product, for which we will adopt the energy distance.
69
3
3. Energy Distance
70
The energy distance is a metric defined on probability measures ([3, 4]),
71
which we will use to measure dissimilarity among probability distributions.
72
Definition 1 (Energy Distance). For probability measures µ,ν on Rd, random
vectorsX,X(cid:48) ∼µ(x),Y,Y(cid:48) ∼ν(y),E(cid:107)X(cid:107)<∞,E(cid:107)Y(cid:107)<∞,theenergydistance
between µ and ν, D(µ,ν), is defined by
D2(µ,ν)=2E(cid:107)X−Y(cid:107)−E(cid:107)X−X(cid:48)(cid:107)−E(cid:107)Y −Y(cid:48)(cid:107), (4)
where (cid:107)·(cid:107) is the Euclidean norm on Rd, and X, X(cid:48), Y and Y(cid:48) are pairwise
73
independent.
74
The energy distance as defined above is a metric on distributions ([5], [6]).
It can be viewed as the metric induced by kernel embedding ([7]) with kernel
k(x,y)=(cid:107)x−x (cid:107)+(cid:107)y−y (cid:107)−(cid:107)x−y(cid:107), (5)
0 0
wherex isafixedvalueinRd,whosechoicedoesnotaffecttheinducedmetric.
0
The kernel induces an inner product between distributions P and Q:
(cid:104)P,Q(cid:105)=E k(X,Y) (6)
X,Y
where X ∼P, Y ∼Q, with X and Y independent. The corresponding square-
distance is given by
γ2(P,Q)=(cid:104)P,P(cid:105)+(cid:104)Q,Q(cid:105)−2(cid:104)P,Q(cid:105)
k (7)
=E k(X,X(cid:48))+E k(Y,Y(cid:48))−2E k(X,Y),
XX(cid:48) YY(cid:48) XY
wheretherandomvectorsX,X(cid:48) ∼P(x),Y,Y(cid:48) ∼Q(y)arepairwiseindependent
(conditions for kernels to yield a metric can be found in [5, 8]). In terms of the
kernel in (5),
γ2(µ,ν)=2E(cid:107)X−x (cid:107)−E(cid:107)X−X(cid:48)(cid:107)+2E(cid:107)Y −x (cid:107)−E(cid:107)Y −Y(cid:48)(cid:107)
k 0 0
−2E(cid:107)X−x (cid:107)−2E(cid:107)Y −x (cid:107)+2E(cid:107)X−Y(cid:107)=D2(µ,ν).
0 0
Anumberofdistancesfordistributionsisavailableintheliteratureofstatis-
75
tics,probabilityandinformationtheory,suchastheKullback-Leiblerdivergence
76
([9, 10]) and the p-Wasserstein metric between two probability measures µ(x)
77
and ν(x) on a metric space (M,d) ([11]). We chose the energy distance be-
78
causeitcanbeestimatedefficientlyfromsamplesanditembedstheprobability
79
measures into a Hilbert space, which facilitates further analysis.
80
3.1. Estimating the Energy Distance from Data
81
In calculating the energy distance between two distributions µ and ν given
independent random vectors X ∼µ, Y ∼ν and their i.i.d. copies X(cid:48), Y(cid:48),
(cid:112)
D(µ,ν)= 2E(cid:107)X−Y(cid:107)−E(cid:107)X−X(cid:48)(cid:107)−E(cid:107)Y −Y(cid:48)(cid:107),
4
oneneedstoevaluatethreeexpectations: E(cid:107)X−Y(cid:107),E(cid:107)X−X(cid:48)(cid:107)andE(cid:107)Y −Y(cid:48)(cid:107).
82
If we only have samples of µ and ν, these expectation can be approximated by
83
their empirical means.
84
Specifically, when we have samples {x }nX of µ and {y }nY of ν, we can
i i=1 j j=1
estimate the energy distance between µ and ν by the energy distance between
their corresponding empirical distributions µˆ and νˆ:
(cid:113)
D(µˆ,νˆ)= 2E(cid:107)Xˆ −Yˆ(cid:107)−E(cid:107)Xˆ −Xˆ(cid:48)(cid:107)−E(cid:107)Yˆ −Yˆ(cid:48)(cid:107). (8)
In the equations above,
E(cid:107)Xˆ −Yˆ(cid:107)= 1 i=nX(cid:88),j=nY (cid:107)x −y (cid:107) (9)
n n i j
X Y
i,j=1
is the empirical mean of E(cid:107)X−Y(cid:107). For Xˆ(cid:48), we use the same samples available
for X,
E(cid:107)Xˆ −Xˆ(cid:48)(cid:107)= 1 i=nX(cid:88),i(cid:48)=nX(cid:107)x −x (cid:107). (10)
n n i i(cid:48)
X X
i,i(cid:48)=1
Similarly,
E(cid:107)Yˆ −Yˆ(cid:48)(cid:107)= 1 j=n(cid:88)Y,j=nY (cid:107)y −y (cid:107). (11)
n n j j(cid:48)
Y Y
j,j(cid:48)=1
According to the formulations above for estimating energy distance from
85
samples, if we have n sample points for µ and n sample points for ν, the
86 X Y
time complexity of estimating their energy distance is O(n n +n2 +n2).
87 X Y X Y
Thecorrespondinginnerproductbetweendistributionsµandν, giveninde-
pendent random vectors X ∼µ, Y ∼ν and X(cid:48), Y(cid:48), is
(cid:104)µ,ν(cid:105)=E(cid:107)X−x (cid:107)+E(cid:107)Y −x (cid:107)−E(cid:107)X−Y(cid:107), (12)
0 0
where x is a fixed point. Similarly, calculation of this inner product involves
0
threeexpectations: E(cid:107)X−x (cid:107),E(cid:107)Y−x (cid:107),E(cid:107)X−Y(cid:107). Whenµandν areknown
0 0
via their samples {x }nX and {y }nY , their inner product (µ,ν) is estimated
i i=1 j j=1
by
(cid:104)µˆ,νˆ(cid:105)=E(cid:107)Xˆ −x (cid:107)+E(cid:107)Yˆ −x (cid:107)−E(cid:107)Xˆ −Yˆ(cid:107), (13)
0 0
where
E(cid:107)Xˆ −x (cid:107)= 1 (cid:88)nX (cid:107)x −x (cid:107), (14)
0 n i 0
X
i=1
E(cid:107)Yˆ −x (cid:107)= 1 (cid:88)nY (cid:107)y −x (cid:107), (15)
0 n j 0
Y
j=1
E(cid:107)Xˆ −Yˆ(cid:107)= 1 i=nX(cid:88),j=nY (cid:107)x −y (cid:107), (16)
n n i j
X Y
i,j=1
5
Algorithm 1 Generic algorithm for estimating the energy distance
Input: Samples x ,y of X,Y respectively.
i j
Output: Empirical estimation of E(cid:107)X−Y(cid:107).
1: procedure Energy({xi},{yj})
2: sum=0
3: for all xi do
4: for all yj do
5: sum=sum+(cid:107)xi−yj(cid:107)
6: end for
7: end for
8: return sum
nXnY
9: end procedure
and the time complexity of estimating this inner product is O(n n ). If we
88 X Y
have n sample points for both µ and ν, the time complexity is O(n2).
89
Noticethatestimatingtheenergydistanceandthecorrespondinginnerprod-
90
uct from n sample points of both distributions have time complexities O(n2),
91
whichbecomescomputationallyexpensivewhenusingalargenumberofsample
92
points. In the following section, a fast algorithm for energy distance between
93
one-dimensional distributions is introduced, making the application of energy
94
distance much more efficient.
95
3.2. Fast Algorithm in One Dimension
96
According to (9), (10), (11), (14), (15) and (16), both the data-based com-
putations of energy distance (8) and corresponding inner product (13) have the
same complexity of evaluating
E(cid:107)Xˆ −Yˆ(cid:107)= 1 (cid:88)nX (cid:88)nY (cid:107)x −y (cid:107), (17)
n n i j
X Y
i=1j=1
where the (x , y ) are (n , n ) samples of (X, Y). Generally, the time com-
97 i j X Y
plexity of evaluating (17) is O(n2) via Algorithm 1, which simply takes the
98
arithmetic mean of (cid:107)x −y (cid:107).
99 i j
100
In one-dimensional space, however, thefact that (cid:107)·(cid:107)=|·| enables us to use
6
the identity |x−y|=1 (x−y)−1 (x−y) to obtain
x−y>0 x−y≤0
E(cid:107)Xˆ −Yˆ(cid:107)
1 (cid:88)nX (cid:88)nY
= |x −y |
n n i j
X Y
i=1j=1
1 (cid:88)nX (cid:88)nY
= 1 (x −y )−1 (x −y )
n n {xi−yj>0} i j {xi−yj≤0} i j
X Y
i=1j=1
= 1 (cid:88)nX #{j|yj <xi}−#{j|yj ≥xi}x + 1 (cid:88)nY #{i|xi ≤yj}−#{i|xi >yj}y ,
n n i n n j
X Y Y X
i=1 j=1
where #{···} denote the number of elements in a set.
101
If {x }nX and {y }nY are sorted arrays, the latter expression can be calcu-
102 i i=1 j j=1
lated in the linear time O(n +n ), since each of #{j|y <x }, #{j|y ≥x },
103 X Y j i j i
#{i|x ≤y } and #{i|x >y } can be calculated in linear time by merging
104 i j i j
{x }nX and {y }nY into one sorted array (Algorithm 2.)
105 i i=1 j j=1
If given unsorted samples, we need to sort them before applying Algorithm
106
2. Feasible sorting algorithms are quick sort, which has an O(nlogn) average
107
complexityandanO(n2)worstcasecomplexity,heapsortandmergesort,which
108
have an O(nlogn) worst case complexity. Therefore even for unsorted samples,
109
thecomplexityofestimatingtheenergydistancecanbeboundedbyO(nlogn).
110
4. Statistical Archetypal Analysis
111
4.1. Dimensional Reduction
112
In this section, we study the dimensional reduction of probability distri-
113
butions, mapping a collection of distributions to a low-dimensional space with
114
minimal loss of information. Probability distributions have infinite dimension;
115
when they are known via samples, they can be said to have a dimensionality of
116
the order of the number of samples points. Our dimensional reduction on this
117
high-dimensional dataset consists of two steps: we embed the distributions into
118
an Euclidean space, and then use dimensional reduction methods developed for
119
Euclidean spaces.
120
Probability distributions equipped with the energy distance form a convex
121
subset of a Hilbert space. Therefore a collection of N distributions µ can be
122 i
naturally embedded into an N-dimensional Euclidean space, since every finite
123
dimension subspace of a Hilbert space is isometric to an Euclidean space.
124
Assume that x ∈ RN, i ∈ [1,2,··· ,N], are points in RN such that (cid:107)x −
i i
x (cid:107)=D(µ ,µ )whereD(·)istheenergydistance(Inotherwords, x istheim-
j i j i
ageofµ undertheembeddingintoanEuclideanspace.) PrincipalComponents
i
Analysis (PCA) solves the following optimization problem for centered x :
i
(cid:13) (cid:13)2
N (cid:13) K (cid:13)
(cid:88)(cid:13) (cid:88) (cid:13)
min (cid:13)xi− ajizj(cid:13) (18)
zj,ajii=1(cid:13)(cid:13) j=1 (cid:13)(cid:13)
7
Algorithm 2 Fast algorithm for estimating the energy distance in 1D
Input: Sorted samples x ,y of 1D random variable X,Y respectively
i j
Output: Empirical estimation of E(cid:107)X−Y(cid:107)
1: procedure FastEnergy({xi},{yj})
2: sumX =0,sumY =0,i=1,j =1
3: while i≤nX and j ≤nY do
4: if xi ≤yj then
5: sumX =sumX + (j−1)−[nnYY−(j−1)]xi
6: i=i+1
7: else
8: sumY =sumY + (i−1)−[nnXX−(i−1)]yj
9: j =j+1
10: end if
11: end while
12: if i>nX then
n(cid:80)Y
13: sumY =sumY + yk
k=j
14: else
n(cid:80)X
15: sumX =sumX + xk
k=i
16: end if
17: return sumX/nX +sumY/nY
18: end procedure
under the constraints that the z are orthonormal vectors. Thus PCA maps
125 j
eachdatapointtotheclosestpointinthevectorspacespannedbythez . Here
126 j
K is the dimension of the low dimensional space sought, and (cid:80)K a z is the
127 j=1 ji j
image of x under this dimensional reduction.
128 i
PCA and other mainstream dimensional reduction techniques are not ap-
129
propriate for probability distributions from two perspectives: 1) the z in (18)
130 j
is generally not a probability distribution, neither are almost all points in the
131
spacespannedbythez . 2)thecoefficientsa foreachx in(18)maybenega-
132 j ji i
tive,sotheycannotclearlyexpresshoweachz contributestotherepresentation
133 j
of x . We will use instead archetypal analysis for the dimensional reduction of
134 i
distributions, which does not suffer from this lack of interpretability.
135
4.2. Statistical Archetypal Analysis
136
As seen in Section 2, archetypal analysis has a formulation similar to (18),
except that it requires the optimization of
(cid:13) (cid:13)2
N (cid:13) K (cid:13)
(cid:88)(cid:13) (cid:88) (cid:13)
min (cid:13)xi− ajizj(cid:13) (19)
zj,ajii=1(cid:13)(cid:13) j=1 (cid:13)(cid:13)
8
K
under the constraints that a (cid:62) 0, (cid:80) a = 1, and each z is a convex combi-
137 ji ji j
j=1
N N
nation of the x , i.e. z = (cid:80)b x , with b (cid:62)0, (cid:80)b =1.
138 i j lj l lj lj
l=1 l=1
Switchingfromvectorsx ,z todistributionsµ ,ν andusingenergydistance
i j i j
instead of Euclidean distance, statistical archetypal analysis adopts the form
(cid:13) (cid:13)2
N (cid:13) K (cid:13) N
(cid:88)(cid:13) (cid:88) (cid:13) (cid:88)
min (cid:13)µi− ajiνj(cid:13) , νj = bljµl, (20)
νj,ajii=1(cid:13)(cid:13) j=1 (cid:13)(cid:13) l=1
with the same constraints over the a and b, which now adopt the natural inter-
139
pretationthattheν aremixturesoftheµ andthelatterarewell-approximated
140 j i
by mixtures of the ν . (cid:107)·(cid:107) in (20) is the energy distance, but can naturally be
141 j
extended to any metric induced by kernel embedding as discussed in Section 3.
142
Since the energy distance, that we shall use for the norm in (20), derives
fromaninnerproduct,statisticalarchetypalanalysiscanberewrittenasin(3):
(cid:124)
argmin tr[(I −BA) G(I −BA)], (21)
n n
A,B
where each column of A represents one archetype as a convex combination of
theoriginaldistributions,andeachcolumnofB containsthecoefficientsforthe
approximate reconstruction of each original distribution from the archetypes.
G is the Gram matrix of pairwise inner products among the distributions,
G =Ek(X ,X )
ij i j
for independent X ∼µ and X ∼µ and kernel k.
143 i i j j
144 When each µi is known via samples {ym(i)}Mm=i1 of size Mi, we can replace µi
byitsempiricaldistributionatdatapointsy(i) withweights 1 . Inthissetting,
145 m Mi
ν becomes an empirical distribution concentrated at the union of the y(l) over
146 j m
l=1,2,··· ,N, with weights blj for all m. The resulting number of samples of
147 Ml
ν appears large, since it contains the support of every empirical distribution
148 j
µ . However, since the solution of (19) is sparse, most entries in b are zero, so
149 i lj
we only need to keep those data points y(l) for ν where b is non-zero.
150 m j lj
Statistical archetypal analysis overcomes the two difficulties in interpreta-
151
tionwhenapplyingdimensionreductiononprobabilitydistribution. Archetypes
152
{ν }k found by archetypal analysis, which are mixtures of the {µ }n , are all
153 i i=1 i i=1
probability distributions. The low-dimensional space used to capture informa-
154
tionofthedatasetofdistributionsinthiscaseistheconvexhullofallarchetypes,
155
i.e. the family of mixtures of all archetypes. Each coefficient a in (19) stands
156 ji
for the contribution of the jth archetype ν to µ .
157 j i
4.3. Numerical Examples
158
4.3.1. Synthetic Data
159
Inourfirstexample, wesimulate100probabilitydistributions{µ }100, each
160 i i=1
a Gaussian mixture µ =λ N(−6,2)+(1−λ )N(6,1), where each λ is drawn
161 i i i i
9
independently from the uniform distribution in [0,1].
162
Figure1: Archetypesofsyntheticdatafork=2. Thecurvesarethefoundarchetypesandthe
shadowsarethetwocomponentsN(−6,2)andN(6,1)inthemixturefamilyrespectively.
We set number of archetypes k to 2 and perform archetypal analysis on the
163
syntheticdata. ThetwoarchetypesfoundareshownandcomparedtoN(−6,2)
164
and N(6,1), the two components in the mixture family, in Figure 1. Both of
165
themareclosetothecomponentsexceptatthecenterandtailpart. Thisisdue
166
tothedefinitionofarchetypes,whichisamixtureofinputdistributions. Unless
167
wehaveexactlythesetwocomponentsasinput,thearchetypeswillalwayshave
168
a heavier tail.
169
4.3.2. Temperature Data
170
We work with ground temperature data from United States Climate Refer-
171
ence Network (USCRN) Quality Controlled Datasets ([dataset][12], [13], [14]).
172
Thetemperaturedataaremeasuredhourlyin43citiesacrosstheUnitedStates.
173
Weoperateondatafromwhichthediurnalandseasonalsignalhasbeenremoved
174
usingtheoptimal-transportbasedmethodologyin[15]. Inaddition,thisdataset
175
has missing values, which are filled using a low rank approximation to the data
176
matrix. Figure 2 shows the 43 cities on the map with Table 1 a complete list of
177
the cities.
178
10