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STATS 102B: Introduction to Computation

创建日期：2024-04-12 18:00:28

所属分类：统计学statistics

标签： STATS 102B

Introduction to Computation

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STATS 102B: Introduction to Computation and Optimization for
Statistics

Introduction
I The aim of cluster analysis is to create a grouping of objects
such that objects within a group are similar and objects in
different groups are not similar. Before we look at the method
in more detail, we first motivate cluster analysis with an
example.
I Customer preference: Imagine you run a large online store and
would like to personalise users’ shopping experience. You hope
that by improving their shopping experience, users will buy
more.
I One way to do this is to provide each user with a set of unique
recommendations that they see when they access your site.
You do not directly know each user’s personal preferences and
tastes but you do have lots of data.
I Assuming that we can define a measure of similarity between
customers based on their purchasing history, we could use
cluster analysis to group customers into K groups.
I Within each group, customers have similar shopping patterns.
Differences between customers in the same group could form
the basis of a recommender system.
I A recommender system could also be created by clustering the
items based on the customers they were bought by. If items 1
and 2 were bought by customers A, D, E and G, then they
could be considered similar. Customers could then be
recommended items that were similar to items they had already
bought.
Synthetic dataset for clustering examples
Consider the data shown below. It consists of 200 observations,
x1, . . . , x200, each represented by two atributes, xT =[x1, x2].
0 5 10
−
5
0
5
10
x1
x2
K -Means Clustering
I This method was presented by MacQueen (1967) in the
Proceedings of the 5th Berkeley Symposium on Mathematical
Statistics and Probability.
I MacQueen suggests the term K-means for describing an
algorithm of assigning each observation to the cluster having
the nearest centroid (mean).
I By clustering the data, we have implicitly defined what similar
means — similar observations are those those that are close to
one another in terms of the Euclidean distance.
I There are other ways of defining similarity that may be more
appropriate, for example, the Mahalanobis distance
d(x , y) =
√
(x − y)TS−1(x − y),
K -Means Clustering Procedure
1. Partition the observations into K initial clusters.
2. Scan through the list of n observations, assigning each
observation to the cluster whose centroid (mean) is nearest. If
an observation is reassigned, we recalculate the cluster mean or
centroid for the cluster receiving that item and the cluster
losing that item.
3. Repeat Step 2 over and over again until no more reassignments
are made.
Simple Example
Suppose we measure two variables for each of four observations.
## item X1 X2
## 1 A 3 3
## 2 B 7 9
## 3 C 4 1
## 4 D 3 4
Our objective is to partition the items into two clusters such that
the observations within a cluster are closer to one another than they
are to the observations in different clusters.
Step 1: Compute Centroid
Assume we initially decide to partition the items into two clusters
(A, B) and (C, D). The centroid for cluster (A, B) is
## X1 X2
## 5 6
The centroid for cluster (C, D) is
## X1 X2
## 3.5 2.5
Step 2: Compute the Eclidean Distance
Compute the distance between A and the centroid of cluster (A, B)
## [1] 3.605551
Compute the distance between A and the centroid of cluster (C, D)
## [1] 0.7071068
Here, we see that A is closer to cluster (C, D) than cluster (A, B).
Therefore, item A will be reassigned, resulting in the new clusters
(B) and (A, C, D).
Recompute Centroid
The centroids of the new clusters are calculated as follows:
## [1] 7 9
## X1 X2
## 3.333333 2.666667
We then calculate the distance between the observations and each
of the clusters.
## [,1] [,2] [,3] [,4]
## [1,] 7.211 0.00 8.54 6.40
## [2,] 0.471 7.32 1.80 1.37
K-Means Example
0 5 10
−
5
0
5
10
x1
x2
0 5 10
−
5
0
5
10
x1
x2
0 5 10
−
5
0
5
10
x1
x2
0 5 10
−
5
0
5
10
x1
x2
K-Means Example (Cont.)
0 5 10
−
5
0
5
10
x1
x2

Comments
I If two or more seed points inadvertently lie within a single
cluster, their resulting clusters will be poorly differntiated.
I The existence of an outlier might produce at least one group
with very disperse observations.
I Even if the population is known to consist of K groups, the
sampling method may be such that data from the rarest group
do not appear in the sample. Forceing the data into K groups
would lead to nonsensical clusters.
I It is always a good idea to retun the algorithm for several
choices.

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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