COMP9318: Data Warehousing and Data Miningcs9318/20t1/lect/8clst.pdf · COMP9318: Data Warehousing...

COMP9318: Data Warehousing and Data Mining 1

COMP9318: Data Warehousing and Data Mining

— L8: Clustering —


n What is Cluster Analysis?

Cluster Analysisn Motivations

n Arranging objects into groups is a natural and necessary skill that we all share

Human Being’s Approach

Computer’s Approach

sex glasses moust-ache smile hat

1 m y n y n

2 f n n y n

3 m y n n n

4 m n n n n

5 m n n y? n

6 m n y n y

7 m y n y n

8 m n n y n

9 m y y y n

10 f n n n n

11 m n y n n

12 f n n n n

Colour Histograms (Swain & Ballard, 1990)

Find matching model in subsequent video frames

We see a person

Model the target statistically

Find Similarity

Create a target model

Computers see numbers

Object Tracking in Computer Vision

Bucket 1 2 … 121 40 70 … 70

2 36 71 … 69

IRDM WS 20097-10

Example 1: Clustering of Flickr Tags

IRDM WS 20097-11

Example 2: Clustering of Social Tags

Grigory Begelmann et al: Automated Tag Clustering: Improving search and exploration in the tag spacehttp://www.pui.ch/phred/automated_tag_clustering

IRDM WS 20097-12

Example 3: Clustering in Social Networks

Jeffrey Heer, Danah Boyd: Vizster: Visualizing Online Social Networks. INFOVIS 2005

http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/b/Boyd:Danah.html

IRDM WS 20097-13

Example 4: Clustering Search Resultsfor Visualization and Navigation

http://www.grokker.com/

http://www.grokker.com/


What is Cluster Analysis?

n Cluster: a collection of data objectsn Similar to one another within the same clustern Dissimilar to the objects in other clusters

n Cluster analysisn Grouping a set of data objects into clusters

n Clustering belongs to unsupervised classification: no predefined classes

n Typical applicationsn As a stand-alone tool to get insight into data

distribution n As a preprocessing step for other algorithms


General Applications of Clustering

n Pattern Recognitionn Spatial Data Analysis

n create thematic maps in GIS by clustering feature spaces

n detect spatial clusters and explain them in spatial data mining

n Image Processingn Economic Science (especially market research)n WWW

n Document classificationn Cluster Weblog data to discover groups of similar

access patterns


Examples of Clustering Applications

n Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs

n Land use: Identification of areas of similar land use in an earth observation database

n Insurance: Identifying groups of motor insurance policy holders with a high average claim cost

n City-planning: Identifying groups of houses according to their house type, value, and geographical location

n Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults


What Is Good Clustering?

n A good clustering method will produce high quality clusters withn high intra-class similarityn low inter-class similarity

n The quality of a clustering result depends on both the similarity measure used by the method and its implementation.

n The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns.


Requirements of Clustering in Data Mining

n Scalabilityn Ability to deal with different types of attributesn Discovery of clusters with arbitrary shapen Minimal requirements for domain knowledge to

determine input parametersn Able to deal with noise and outliersn Insensitive to order of input recordsn High dimensionalityn Incorporation of user-specified constraintsn Interpretability and usability


Chapter 8. Cluster Analysis

n Preliminaries


Typical Inputs

n Data matrixn N objects, each

represented by a m-dimensional feature vector

n Dissimilarity matrixn A square matrix giving

distances between all pairs of objects.

n If similarity functions are used è similarity matrix

úúúúúúú

û

ù

êêêêêêê

ë

é

nmx...nfx...n1x...............imx...ifx...i1x...............1mx...1fx...11x

úúúúúú

û

ù

êêêêêê

ë

é

0...)2,()1,(:::

)2,3()

...ndnd

0dd(3,10d(2,1)

0n x m

n x n

Key component for clustering: the dissimilarity/similarity metric:d(i, j)


Comments

n The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables.

n Weights should be associated with different variables based on applications and data semantics, or appropriate preprocessing is needed.

n There is a separate “quality” function that measures the “goodness” of a cluster.

n It is hard to define “similar enough” or “good enough” n the answer is typically highly subjective.


Type of data in clustering analysis

n Interval-scaled variables:

n Binary variables:

n Nominal, ordinal, and ratio variables:

n Variables of mixed types:


Interval-valued variables

n Standardize data

n Calculate the mean absolute deviation:

where

n Calculate the standardized measurement (z-score)

n Using mean absolute deviation is more robust than using standard deviation

.... )1 21

f f f nfm (x x xn + += +

1 21(| | | | ... | |)f f f f f nf fs x m x m x mn= - + - + + -

f

fifif s

mx z

-=


Similarity and Dissimilarity Between Objects

n Distances are normally used to measure the similarity or dissimilarity between two data objects

n A popular choice is the Minkowski distance, or the Lpnorm of difference vector

n Special cases: n if p = 1, d is the Manhattan distancen if p = 2, d is the Euclidean distancen


Similarity and Dissimilarity Between Objects (Cont.)

n Other similarity/distance functions:n Mahalanobis distancen Jaccard, Dice, cosine similarity, Pearson correlation

coefficientn Metric distance

n Propertiesn d(i,j) ³ 0n d(i,i) = 0n d(i,j) = d(j,i)n d(i,j) £ d(i,k) + d(k,j)

positivenesssymmetryreflexivity

triangular inequality

common to all distance functions


Areas within a unit distance from q under different Lp distances

q

L1

q

L2 L8

L¥

q

q


Binary Variablesn A contingency table for binary data

n Simple matching coefficient (invariant, if the binary variable is symmetric):

n Jaccard coefficient (noninvariant if the binary variable is asymmetric):

dcbacb jid+++

+=),(

pdbcasumdcdcbaba

sum

++++

01

01

cbacb jid++

+=),(

Object i

Object j

Obj Vector Representationi [0, 1, 0, 1, 0, 0, 1, 0]j [0, 0, 0, 0, 1, 0, 1, 1]


Dissimilarity between Binary Variables

n Example

n gender is a symmetric attributen the remaining attributes are asymmetric binaryn let the values Y and P be set to 1, and the value N be set to 0

Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4Jack M Y N P N N NMary F Y N P N P NJim M Y P N N N N

75.021121),(

67.011111),(

33.010210),(

=++

+=

=++

+=

=++

+=

maryjimd

jimjackd

maryjackd

cbacb jid++

+=),(


Nominal Variables

n A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green

n Method 1: Simple matchingn m: # of matches, p: total # of variables

n Method 2: One-hot encodingn creating a new binary variable for each of the M

nominal states

pmpjid -=),(


Ordinal Variables

n An ordinal variable can be discrete or continuousn Order is important, e.g., rankn Can be treated like interval-scaled

n replace xif by their rank n map the range of each variable onto [0, 1] by replacing

i-th object in the f-th variable by

n compute the dissimilarity using methods for interval-scaled variables

11--

=f

ifif M

rz

},...,1{ fif Mr Î


Ratio-Scaled Variables

n Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale,

such as AeBt or Ae-Bt

n Methods:1. treat them like interval-scaled variables—not a good

choice! (why?—the scale can be distorted), or2. apply logarithmic transformation

yif = log(xif)3. or treat them as continuous ordinal data; treat their

rank as interval-scaled


n A Categorization of Major Clustering Methods


Major Clustering Approaches

n Partitioning algorithms: Construct various partitions and then evaluate them by some criterion

n Hierarchy algorithms: Create a hierarchical decomposition of the set of data (or objects) using some criterion

n Graph-based algorithms: Spectral clustering

n Density-based: based on connectivity and density functionsn Grid-based: based on a multiple-level granularity structuren Model-based: A model is hypothesized for each of the

clusters and the idea is to find the best fit of that model to each other


n Partitioning Methods

35

Partitioning Algorithms: Problem Definition

n Partitioning method: Construct a “good” partition of a database of n objects into a set of k clustersn Input: a n x m data matrix

n How to measure the “goodness” of a given partitioning scheme? n Cost of a cluster, cost(Ci) =

n Note: L2 distance usedn Analogy with binning? n How to choose the center of a cluster?

n Centroid (i.e., Avg) of Xj è Minimizes cost(Ci)

n Cost of k clusters: sum of cost(Ci)

X

xj2Ci

kxj � center(Ci)k22

Example (2D)



Partitioning Algorithms: Basic Concept

n It’s an optimization problem! n Global optimal:

n NP-hard (for a wide range of cost functions) n Requires exhaustively enumerate all partitions

n Stirling numbers of the second kind

n Heuristic methods: n k-means: an instance of the EM (expectation-maximization)

algorithmn Many variants

nn

k

o= ⇥

✓kn

k!

◆


The K-Means Clustering Method

n Lloyds Algorithm:1. Initialize k centers randomly2. While stopping condition is not met

i. Assign each object to the cluster with the nearest center

ii. Compute the new center for each cluster.

n Stopping condition =?

n What are the final clusters?

Cost function: Total squared distance of points to its cluster representative


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reassign

The K-Means Clustering Method

n Example

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K=2Arbitrarily choose K object as initial cluster center

Assign each objects to most similar center

Update the cluster means

Update the cluster means

reassign


Comments on the K-Means Method

n Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n.n Comparing: PAM: O(k(n-k)2), CLARA: O(ks2 + k(n-k))

n Comment:n Often terminates at a local optimum. The global optimum may be

found using techniques such as: deterministic annealing and genetic algorithms

n No guarantee on the quality. Use k-means++.n Weakness

n Applicable only when mean is defined, then what about categorical data?

n Need to specify k, the number of clusters, in advancen Unable to handle noisy data and outliersn Not suitable to discover clusters with non-convex shapes


Variations of the K-Means Method

n A few variants of the k-means which differ in

n Selection of the initial k means

n Dissimilarity calculations

n Strategies to calculate cluster means

n Handling categorical data: k-modes (Huang’98)

n Replacing means of clusters with modes

n Using new dissimilarity measures to deal with categorical objects

n Using a frequency-based method to update modes of clusters

n A mixture of categorical and numerical data: k-prototype method

k-Means++ [Arthur and Vassilvitskii, SODA 2007]

n A simple initialization routine that guarantees to find a solution that is O(log k) competitive to the optimal k-means solution.

n Algorithm:1. Find first center uniformly at random2. For each data point x, compute D(x) as the distance

to its nearest center3. Randomly sample one point as the new center, with

probabilities proportional to D2(x)4. Goto 2 if less than k centers5. Run the normal k-means with the k centers


k-means: Special Matrix Factorizationn Xn x d ≈ Un x k Vk x d

n Loss function: ǁ X – UV ǁF2

n Squared Frobenius normn Constraints:

n Rows of U must be a one-hot encodingn Alternative view

n Xj,* ≈ Uj,* V è Xj,* can be explained as a “special” linear combination of rows in V

43

≈ --- x2 --- 1 0 0--- c1 ------ c2 ------ c3 ---

Expectation Maximization Algorithmn Xn x d ≈ Un x k Vk x d

n Loss function: ǁ X – UV ǁF2

n Finding the best U and V simutaneously is hard, but n Expectation step:

n Given V, find the best U è easyn Maximization step:

n Given U, find the best V è easy n Iterate until converging at a local minima.

44

≈ --- x2 --- 1 0 0--- c1 ------ c2 ------ c3 ---


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What is the problem of k-Means Method?n The k-means algorithm is sensitive to outliers !

n Since an object with an extremely large value may substantially distort the distribution of the data.

n K-Medoids: Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster.

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K-medoids (PAM)

n k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster


The K-Medoids Clustering Method

n Find representative objects, called medoids, in clusters

n PAM (Partitioning Around Medoids, 1987)

n starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering

n PAM works effectively for small data sets, but does not scale well for large data sets

n CLARA (Kaufmann & Rousseeuw, 1990)

n CLARANS (Ng & Han, 1994): Randomized sampling

n Focusing + spatial data structure (Ester et al., 1995)


Typical k-medoids algorithm (PAM)

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Total Cost = 20

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K=2

Arbitrary choose k object as initial medoids

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Assign each remaining object to nearest medoids

Fo each nonmedoid object,Oa

Compute total cost of swapping

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Total Cost = 26

Swapping O and Oa

If quality is improved.

Do loopUntil no change

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PAM (Partitioning Around Medoids) (1987)

n PAM (Kaufman and Rousseeuw, 1987), built in Splusn Use real object to represent the cluster

n Select k representative objects arbitrarilyn For each pair of non-selected object h and selected

object i, calculate the total swapping cost TCih

n For each pair of i and h, n If TCih < 0, i is replaced by hn Then assign each non-selected object to the most

similar representative objectn repeat steps 2-3 until there is no change


What is the problem with PAM?n PAM is more robust than k-means in the presence of

noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean

n PAM works efficiently for small data sets but does not scale well for large data sets.n O(k(n-k)2) for each iteration

where n is # of data,k is # of clustersèSampling based method,

CLARA(Clustering LARge Applications)

Gaussian Mixture Model for Clustering

n k-means can be deemed as a special case of the EM algorithm for GMM

n GMMn allows “soft” cluster assignment:

n model Pr(C | x)n also a good example of

n Generative modeln Latent variable model

n Use the Expectation-Maximization (EM) algorithm to obtain a local optimal solution



n Hierarchical Methods

Hierarchical Clustering n Produces a set of nested clusters organized as a hierarchical treen Can be visualized as a dendrogram

n A tree like diagram that records the sequences of merges or splitsn A clustering of the data objects is obtained by cutting the

dendrogram at the desired level, then each connected component forms a cluster.

1 3 2 5 4 60

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0.15

0.2

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23 4

5

Objects

Distance

Strengths of Hierarchical Clustering

n Do not have to assume any particular number of clustersn Any desired number of clusters can be

obtained by ‘cutting’ the dendogram at the proper level

n They may correspond to meaningful taxonomiesn Example in biological sciences (e.g., animal

kingdom, phylogeny reconstruction, …)

Hierarchical Clusteringn Two main types of hierarchical clustering

n Agglomerative: n Start with the points as individual clustersn At each step, merge the closest pair of clusters until only one

cluster (or k clusters) left

n Divisive: n Start with one, all-inclusive cluster n At each step, split a cluster until each cluster contains a point (or

there are k clusters)

n Traditional hierarchical algorithms use a similarity or distance matrixn Merge or split one cluster at a time

Agglomerative Clustering Algorithm

n More popular hierarchical clustering techniquen Basic algorithm is straightforward

1. Compute the proximity matrix (i.e., matrix of pair-wise distances)2. Let each data point be a cluster3. Repeat4. Merge the two closest clusters5. Update the proximity matrix6. Until only a single cluster remains

n Key operation is the computation of the proximity of two clusters ç different from that of two points

n Different approaches to defining the distance between clusters distinguish the different algorithms

Starting Situation

n Start with clusters of individual points and a proximity matrix

p1

p3

p5p4

p2

p1 p2 p3 p4 p5 . . .

.

.

. Proximity Matrix

...p1 p2 p3 p4 p9 p10 p11 p12

Intermediate Situation

n After some merging steps, we have some clusters

C1

C4

C2 C5

C3

C2C1C1

C3

C5C4

C2

C3 C4 C5

Proximity Matrix

...p1 p2 p3 p4 p9 p10 p11 p12

Intermediate Situation

n We want to merge the two closest clusters (C2 and C5) and update the proximity matrix.

C1

C4

C2 C5

C3

C2C1C1

C3

C5C4

C2

C3 C4 C5

Proximity Matrix

...p1 p2 p3 p4 p9 p10 p11 p12

After Mergingn The question is “How do we update the proximity matrix?”

C1

C4

C2 U C5

C3 ? ? ? ?

?

?

?

C2 U C5C1

C1

C3

C4

C2 U C5

C3 C4

Proximity Matrix

...p1 p2 p3 p4 p9 p10 p11 p12

How to Define Inter-Cluster Distance

p1

p3

p5

p4

p2

p1 p2 p3 p4 p5 . . .

.

.

.

Similarity?

● MIN● MAX● Centroid-based● Group Average● Other methods driven by an objective

function– Ward’s Method uses squared error

Proximity Matrix

How to Define Inter-Cluster Similarity

p1

p3

p5

p4

p2

p1 p2 p3 p4 p5 . . .

.

.

. Proximity Matrix




p1

p3

p5

p4

p2

p1 p2 p3 p4 p5 . . .

.

.

. Proximity Matrix



´ ´


p1

p3

p5

p4

p2

p1 p2 p3 p4 p5 . . .

.

.

. Proximity Matrix



Note: not simple avg distance between the clusters

Cluster Similarity: MIN or Single Link/LINK

n Similarity of two clusters is based on the two most similar (closest) points in the different clustersn i.e., sim(Ci, Cj) = min(dissim(px, py)) // px ∈ Ci, py ∈ Cj

= max(sim(px, py))n Determined by one pair of points, i.e., by one link in the

proximity graph.

1 2 3 4 5

Usu. Called single-link to avoid confusions (min similarity or dissimilarity?)

P1 P2 P3 P4 P5

P1 1.00 0.90 0.10 0.65 0.20

P2 0.90 1.00 0.70 0.60 0.50

P3 0.10 0.70 1.00 0.40 0.30

P4 0.65 0.60 0.40 1.00 0.80

P5 0.20 0.50 0.30 0.80 1.00

Single-Link Example

1 2 3 4 5

P1 P2 P3 P4 P5

P1 1.00 0.90 0.10 0.65 0.20

P2 0.90 1.00 0.70 0.60 0.50

P3 0.10 0.70 1.00 0.40 0.30

P4 0.65 0.60 0.40 1.00 0.80

P5 0.20 0.50 0.30 0.80 1.00

P1 P2 P3 P4 P5

P1 1.00 0.90 0.10 0.65 0.20

P2 1.00 0.70 0.60 0.50

P3 1.00 0.40 0.30

P4 1.00 0.80

P5 1.00

12 P3 P4 P5

12 1.00

P3 1.00 0.40 0.30

P4 1.00 0.80

P5 1.00

0.70 0.65 0.50

12 P3 45

12 1.00 0.70

P3 1.00

45 1.00

0.650.40

sim(Ci, Cj) = max(sim(px, py))

Hierarchical Clustering: MIN

Nested Clusters Dendrogram

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Strength of MIN

Original Points Two Clusters

• Can handle non-elliptical shapes

Limitations of MIN


• Sensitive to noise and outliers

Cluster Similarity: MAX or Complete Link (CLINK)

n Similarity of two clusters is based on the two least similar (most distant) points in the different clustersn i.e., sim(Ci, Cj) = max(dissim(px, py)) // px ∈ Ci, py ∈ Cj

= min(sim(px, py))n Determined by all pairs of points in the two clusters

1 2 3 4 5

P1 P2 P3 P4 P5

P1 1.00 0.90 0.10 0.65 0.20

P2 0.90 1.00 0.70 0.60 0.50

P3 0.10 0.70 1.00 0.40 0.30

P4 0.65 0.60 0.40 1.00 0.80

P5 0.20 0.50 0.30 0.80 1.00

Complete-Link ExampleP1 P2 P3 P4 P5

P1 1.00 0.90 0.10 0.65 0.20

P2 0.90 1.00 0.70 0.60 0.50

P3 0.10 0.70 1.00 0.40 0.30

P4 0.65 0.60 0.40 1.00 0.80

P5 0.20 0.50 0.30 0.80 1.00

P1 P2 P3 P4 P5

P1 1.00 0.90 0.10 0.65 0.20

P2 1.00 0.70 0.60 0.50

P3 1.00 0.40 0.30

P4 1.00 0.80

P5 1.00

12 P3 P4 P5

12 1.00

P3 1.00 0.40 0.30

P4 1.00 0.80

P5 1.00

0.10 0.60 0.20

12 P3 45

12 1.00 0.10

P3 1.00

45 1.00

0.200.30

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sim(Ci, Cj) = min(sim(px, py))

Hierarchical Clustering: MAX

Nested Clusters Dendrogram

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Strength of MAX


• Less susceptible to noise and outliers

Limitations of MAX


•Tends to break large clusters

•Biased towards globular clusters

Cluster Similarity: Group Averagen GAAC (Group Average Agglomerative Clustering)n Similarity of two clusters is the average of pair-wise similarity

between points in the two clusters.

n Why not using simple average distance? This method guarantees that no inversions can occur.

�

similarity(Clusteri,Clusterj) =

similarity(pi,pj)pi ,pj ∈Clusteri∪Clusterjpj ≠pi

∑

(|Clusteri|+|Clusterj|)∗(|Clusteri|+|Clusterj|-1)

(3, 4.5)

(5, 1)(1, 1)

Group Average ExampleP1 P2 P3 P4 P5

P1 1.00 0.90 0.10 0.65 0.20

P2 0.90 1.00 0.70 0.60 0.50

P3 0.10 0.70 1.00 0.40 0.30

P4 0.65 0.60 0.40 1.00 0.80

P5 0.20 0.50 0.30 0.80 1.00

P1 P2 P3 P4 P5

P1 1.00 0.90 0.10 0.65 0.20

P2 1.00 0.70 0.60 0.50

P3 1.00 0.40 0.30

P4 1.00 0.80

P5 1.00

12 P3 P4 P5

12 1.00 0.567 0.717 0.533

P3 1.00 0.40 0.30

P4 1.00 0.80

P5 1.00

12 P3 45

12 1.0 0.567 0.608

P3 1.00 0.5

45 1.00

sim(Ci, Cj) = avg(sim(pi, pj))

Sim(12,3)=2*(0.1+0.7+0.9)/6 = 0.56666661 2 34 5Sim(12,45)=2*(0.9+0.65+0.2+0.6+0.5+0.8)/12 = 0.608

Hierarchical Clustering: Centroid-based and Group Average

n Compromise between Single and Complete Link

n Strengthsn Less susceptible to noise and outliers

n Limitationsn Biased towards globular clusters


More on Hierarchical Clustering Methods

n Major weakness of agglomerative clustering methodsn do not scale well: time complexity of at least O(n2),

where n is the number of total objectsn can never undo what was done previously

n Integration of hierarchical with distance-based clusteringn BIRCH (1996): uses CF-tree and incrementally adjusts

the quality of sub-clustersn CURE (1998): selects well-scattered points from the

cluster and then shrinks them towards the center of the cluster by a specified fraction

n CHAMELEON (1999): hierarchical clustering using dynamic modeling

Spectral Clustering

n See additional slides.


COMP9318: Data Warehousing and Data Miningcs9318/20t1/lect/8clst.pdf · COMP9318: Data Warehousing...

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