a new platform for categorical variables, ready to be explored - … · 2020. 4. 28. · who uses...
Post on 17-Aug-2021
1 Views
Preview:
TRANSCRIPT
www.SAS.comCopyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
MULTIPLE CORRESPONDENCE
ANALYSIS (MCA)
• A New Platform for Categorical Variables, Ready to be
explored
• Jianfeng Ding
• JMP R&D
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
COVERED TOPICS
• What is CA and MCA
• Who uses MCA
• How to use MCA (Live Demos of JMP)
• Conclusions
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
WHAT IS CORRESPONDENCE
ANALYSIS(CA)
• Scenario: new sales person who has the car sale information from
the last quarter
• Challenge: how to sell cars efficiently
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
WHAT IS CORRESPONDENCE
ANALYSIS(CA)
• Look at two categorical variables each time
• Determine the association between two categorical variables
SVD
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
WHAT IS MULTIPLE CORRESPONDENCE
ANALYSIS (MCA)
• Look at multiple categorical variables at the same time
• Determine the association among all categorical variables
Design matrix
SV
D
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
WHAT IS MULTIPLE CORRESPONDENCE
ANALYSIS (MCA)
Only in JMP12 Available in JMP11
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
ALGORITHM AND NOTATION OF CA
• Two design matrices Z1, Z2
• Scale vectors s1, s2
• Canonical correlation s12=(1/n) st1z
t1z2s2=st
1p12s2
• Find the scale vectors that maximize the correlation
• Do SVD to the matrix to reveal the association
structure
• S= 𝐷−1
2𝑟(𝑃 − 𝑟𝑐𝑇)𝐷
−1
2𝑐
=UΣVT
• Row coordinates and column coordinates
• S1=𝐷−1
2𝑟
U Σ s2=𝐷−1
2𝑐
V Σ
• Trace(sst) is the total inertia that quantifies the total
variances in the cross-table.
Element of
correspondence
matrix pij
Row masses ri
Column masses cj
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
ALGORITHM AND NOTATION OF MCA
• Two ways to perform MCA
• CA of the design matrix Z = [ Z1 Z2,…,ZQ ], do SVD
to
• 𝑛𝑍
𝑄𝑛−1
𝑛11𝑇𝐷 𝐷−1/2
• Ca of the burt matrix c = ztz, do svd to
• 𝐷−1/2𝐶
𝑄2𝑛− 𝐷11𝑇𝐷 𝐷−1/2
• Where D=(1/Q)diag(D1 D2,…,Dq )
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
WHO USES MCA?
• MCA is popular in Europe and Japan
• Social science research
• Health in psychology
• Market data, product preferences, and food preferences
• Elections in political science and credit scoring
• Epidemiology of animal health and microarray studies in genetics
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
STEPS OF ANALYZING DATA WITH MCA
• Preparation of the data for MCA
• Elementary statistical analysis
• Examination of the basic results such as eigenvalues(inertia), coordinates,
contributions, and clouds
• Informal inspection of the clouds in several principal planes
• Interpretation of axes
• Inspecting and “dressing up” the cloud of individuals
• Supplementary elements: individuals and variables
• Deep investigation of the cloud of individuals
• Statistical inference
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
CLOUDS OF CATEGORIES AND INDIVIDUALS
𝑦𝑖 =1
λ
𝑘∈𝐾𝑖
𝑦𝑘 𝑄
𝑦𝑘 =1
λ
𝑖∈𝐼𝑘
𝑦𝑖 𝑛𝑘
Copyr i g ht © 2012, SAS Ins t i tu t e Inc . A l l r ights reser ve d .
CONCLUSIONS
• Support CA and MCA
• Support supplementary elements
• Visualizing the output statistics
• The interactivity makes it easy and fun to analyze and interpret data
top related