Exploratory Analysis of Multivariate Data

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Matt Moores

Exploratory data analysis in R using 'visdat' and 'FactoMineR'

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Exploratory Analysis of Multivariate Data
Matt Moores
Department of Statistics, University of Warwick
WDSI Vacation School 2017
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 1 / 27

1 Multivariate Data
2 Principal Components Analysis (PCA)
3 Clustering
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 2 / 27

Multivariate Data
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 3 / 27

Multivariate Data
High-dimensional (more than 2D)
Complexity increases combinatorially
Correlation between columns
Mixed data types
Missing data
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 4 / 27

Visualisation
vis_dat(typical_data)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 5 / 27

Missing Data
Missing completely at random (MCAR)
“the dog ate my homework”
Missing at random (MAR)
Missing not at random (MNAR)
nonignorable nonresponse
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 6 / 27

Examples
Surveys
Consumer data
Medical records
Morphological measurements
Structured data with millions of dimensions:
*-omics
Images
Text
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 7 / 27

Purple Rock Crab
Figure 1: Leptograpsus variegatus
Image courtesy Wikimedia Commons, user Benjamint444
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 8 / 27

Morphology
library(MASS)
vis_dat(crabs)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 9 / 27

Data Dictionary
sp: species (“B” or “O” for blue or orange)
sex: “F” or “M”
index: 1:50 within each group
FL: frontal lobe size (mm)
RW: rear width (mm)
CL: carapace length (mm)
CW: carapace width (mm)
BD: body depth (mm)
Campbell & Mahon (1974) Aust. J. Zoology 22: 417–425.
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 10 / 27

Exploratory Data Analysis
ggpairs(crabs, ggplot2::aes(colour=sp))
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 11 / 27

Principal Components Analysis (PCA)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 12 / 27

PCA
PCA is a transformation of the original variables:
Useful for high-dimensional, multicollinear data
Principal components are uncorrelated
orthogonal vectors
Decreasing order of variance
An example of unsupervised machine learning
Pearson (1901) Philosophical Magazine 2(11): 559–572.
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 13 / 27

PCA for 2D Gaussian
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 14 / 27

PCA for crabs
crab_pca <- PCA(crabs, quanti.sup=3, quali.sup=1:2,
graph=FALSE)
knitr::kable(crab_pca$eig, digits=4,
col.names = c("eigenvalue","% variance","cumulative %"))
eigenvalue % variance cumulative %
comp 1 4.7888 95.7767 95.7767
comp 2 0.1517 3.0337 98.8104
comp 3 0.0466 0.9327 99.7431
comp 4 0.0111 0.2227 99.9658
comp 5 0.0017 0.0342 100.0000
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 15 / 27

Scree Plot
fviz_screeplot(crab_pca)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 16 / 27

Pairs Plot
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 17 / 27

Species
fviz_pca_ind(crab_pca, axes = c(1,3), habillage=1,
addEllipses=TRUE, ellipse.level=0.95)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 18 / 27

Sex
fviz_pca_ind(crab_pca, axes = c(1,2), habillage=2,
addEllipses=TRUE, ellipse.level=0.95)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 19 / 27

Another example: genetic variation
Novembre et al. (2008) Nature 456(7218): 98–101.
n = 1, 387 European individuals from POPRES project
Aﬀymetrix 500K single nucleotide polymorphism (SNP) chip
n p
Country of origin of each individual’s grandparents
Data publicly available from the database of Genotypes and Phenotypes
(dbGaP), http://www.ncbi.nlm.nih.gov/sites/entrez?Db=gap
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 20 / 27

PCA analysis of 197,146 genetic loci
Figure 2: PCA
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 21 / 27

Clustering
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 22 / 27

Clustering
Another unsupervised approach for ﬁnding patterns in
data:
Grouping observations based on distance
Many diﬀerent distance metrics (Euclidean,
Manhattan, Gower, etc.)
Also many diﬀerent clustering algorithms
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 23 / 27

Heatmap
data("USArrests")
USdf <- scale(USArrests)
US.dist <- get_dist(USdf, method = "euclidean")
fviz_dist(US.dist)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 24 / 27

k-means
res_km <- eclust(USdf, "kmeans", k = 3)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 25 / 27

Chloropleth
library(maps)
UScat <-data.frame(state = tolower(names(res_km$cluster)), clu
states_map <-map_data("state")
ggplot(UScat, aes(map_id = state)) +
geom_map(aes(fill = cluster), map = states_map) + expand_l
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 26 / 27

Dendogram
res.hc <- eclust(USdf, "hclust")
fviz_dend(res.hc, rect = TRUE)
Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 27 / 27

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Exploratory Analysis of Multivariate Data

1. Exploratory Analysis of Multivariate Data Matt Moores Department of Statistics, University of Warwick WDSI Vacation School 2017 Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 1 / 27

2. 1 Multivariate Data 2 Principal Components Analysis (PCA) 3 Clustering Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 2 / 27

3. Multivariate Data Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 3 / 27

4. Multivariate Data High-dimensional (more than 2D) Complexity increases combinatorially Correlation between columns Mixed data types Missing data Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 4 / 27

5. Visualisation vis_dat(typical_data) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 5 / 27

6. Missing Data Missing completely at random (MCAR) “the dog ate my homework” Missing at random (MAR) Missing not at random (MNAR) nonignorable nonresponse Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 6 / 27

7. Examples Surveys Consumer data Medical records Morphological measurements Structured data with millions of dimensions: *-omics Images Text Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 7 / 27

8. Purple Rock Crab Figure 1: Leptograpsus variegatus Image courtesy Wikimedia Commons, user Benjamint444 Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 8 / 27

9. Morphology library(MASS) vis_dat(crabs) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 9 / 27

10. Data Dictionary sp: species (“B” or “O” for blue or orange) sex: “F” or “M” index: 1:50 within each group FL: frontal lobe size (mm) RW: rear width (mm) CL: carapace length (mm) CW: carapace width (mm) BD: body depth (mm) Campbell & Mahon (1974) Aust. J. Zoology 22: 417–425. Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 10 / 27

11. Exploratory Data Analysis ggpairs(crabs, ggplot2::aes(colour=sp)) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 11 / 27

12. Principal Components Analysis (PCA) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 12 / 27

13. PCA PCA is a transformation of the original variables: Useful for high-dimensional, multicollinear data Principal components are uncorrelated orthogonal vectors Decreasing order of variance An example of unsupervised machine learning Pearson (1901) Philosophical Magazine 2(11): 559–572. Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 13 / 27

14. PCA for 2D Gaussian Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 14 / 27

15. PCA for crabs crab_pca <- PCA(crabs, quanti.sup=3, quali.sup=1:2, graph=FALSE) knitr::kable(crab_pca$eig, digits=4, col.names = c("eigenvalue","% variance","cumulative %")) eigenvalue % variance cumulative % comp 1 4.7888 95.7767 95.7767 comp 2 0.1517 3.0337 98.8104 comp 3 0.0466 0.9327 99.7431 comp 4 0.0111 0.2227 99.9658 comp 5 0.0017 0.0342 100.0000 Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 15 / 27

16. Scree Plot fviz_screeplot(crab_pca) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 16 / 27

17. Pairs Plot Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 17 / 27

18. Species fviz_pca_ind(crab_pca, axes = c(1,3), habillage=1, addEllipses=TRUE, ellipse.level=0.95) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 18 / 27

19. Sex fviz_pca_ind(crab_pca, axes = c(1,2), habillage=2, addEllipses=TRUE, ellipse.level=0.95) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 19 / 27

20. Another example: genetic variation Novembre et al. (2008) Nature 456(7218): 98–101. n = 1, 387 European individuals from POPRES project Aﬀymetrix 500K single nucleotide polymorphism (SNP) chip n p Country of origin of each individual’s grandparents Data publicly available from the database of Genotypes and Phenotypes (dbGaP), http://www.ncbi.nlm.nih.gov/sites/entrez?Db=gap Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 20 / 27

21. PCA analysis of 197,146 genetic loci Figure 2: PCA Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 21 / 27

22. Clustering Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 22 / 27

23. Clustering Another unsupervised approach for finding patterns in data: Grouping observations based on distance Many different distance metrics (Euclidean, Manhattan, Gower, etc.) Also many different clustering algorithms Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 23 / 27

24. Heatmap data("USArrests") USdf <- scale(USArrests) US.dist <- get_dist(USdf, method = "euclidean") fviz_dist(US.dist) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 24 / 27

25. k-means res_km <- eclust(USdf, "kmeans", k = 3) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 25 / 27

26. Chloropleth library(maps) UScat <-data.frame(state = tolower(names(res_km$cluster)), clu states_map <-map_data("state") ggplot(UScat, aes(map_id = state)) + geom_map(aes(fill = cluster), map = states_map) + expand_l Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 26 / 27

27. Dendogram res.hc <- eclust(USdf, "hclust") fviz_dend(res.hc, rect = TRUE) Matt Moores (Warwick) Exploratory Analysis of Multivariate Data WDSI Vacation School 2017 27 / 27

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