Multivariate data

“simultaneous observation and analysis of more than one outcome variable”

Multivariate data

   Achimill Agrostol Airaprae Alopgeni Anthodor Bellpere Bromhord Chenalbu
1         1        0        0        0        0        0        0        0
2         3        0        0        2        0        3        4        0
3         0        4        0        7        0        2        0        0
4         0        8        0        2        0        2        3        0
5         2        0        0        0        4        2        2        0
6         2        0        0        0        3        0        0        0
7         2        0        0        0        2        0        2        0
8         0        4        0        5        0        0        0        0
9         0        3        0        3        0        0        0        0
10        4        0        0        0        4        2        4        0
11        0        0        0        0        0        0        0        0
12        0        4        0        8        0        0        0        0
13        0        5        0        5        0        0        0        1
14        0        4        0        0        0        0        0        0
15        0        4        0        0        0        0        0        0
16        0        7        0        4        0        0        0        0
17        2        0        2        0        4        0        0        0
18        0        0        0        0        0        2        0        0
19        0        0        3        0        4        0        0        0
20        0        5        0        0        0        0        0        0
   Cirsarve Comapalu Eleopalu Elymrepe Empenigr Hyporadi Juncarti Juncbufo
1         0        0        0        4        0        0        0        0
2         0        0        0        4        0        0        0        0
3         0        0        0        4        0        0        0        0
4         2        0        0        4        0        0        0        0
5         0        0        0        4        0        0        0        0
6         0        0        0        0        0        0        0        0
7         0        0        0        0        0        0        0        2
8         0        0        4        0        0        0        4        0
9         0        0        0        6        0        0        4        4
10        0        0        0        0        0        0        0        0
11        0        0        0        0        0        2        0        0
12        0        0        0        0        0        0        0        4
13        0        0        0        0        0        0        0        3
14        0        2        4        0        0        0        0        0
15        0        2        5        0        0        0        3        0
16        0        0        8        0        0        0        3        0
17        0        0        0        0        0        2        0        0
18        0        0        0        0        0        0        0        0
19        0        0        0        0        2        5        0        0
20        0        0        4        0        0        0        4        0
   Lolipere Planlanc Poaprat Poatriv Ranuflam Rumeacet Sagiproc Salirepe
1         7        0       4       2        0        0        0        0
2         5        0       4       7        0        0        0        0
3         6        0       5       6        0        0        0        0
4         5        0       4       5        0        0        5        0
5         2        5       2       6        0        5        0        0
6         6        5       3       4        0        6        0        0
7         6        5       4       5        0        3        0        0
8         4        0       4       4        2        0        2        0
9         2        0       4       5        0        2        2        0
10        6        3       4       4        0        0        0        0
11        7        3       4       0        0        0        2        0
12        0        0       0       4        0        2        4        0
13        0        0       2       9        2        0        2        0
14        0        0       0       0        2        0        0        0
15        0        0       0       0        2        0        0        0
16        0        0       0       2        2        0        0        0
17        0        2       1       0        0        0        0        0
18        2        3       3       0        0        0        0        3
19        0        0       0       0        0        0        3        3
20        0        0       0       0        4        0        0        5
   Scorautu Trifprat Trifrepe Vicilath Bracruta Callcusp
1         0        0        0        0        0        0
2         5        0        5        0        0        0
3         2        0        2        0        2        0
4         2        0        1        0        2        0
5         3        2        2        0        2        0
6         3        5        5        0        6        0
7         3        2        2        0        2        0
8         3        0        2        0        2        0
9         2        0        3        0        2        0
10        3        0        6        1        2        0
11        5        0        3        2        4        0
12        2        0        3        0        4        0
13        2        0        2        0        0        0
14        2        0        6        0        0        4
15        2        0        1        0        4        0
16        0        0        0        0        4        3
17        2        0        0        0        0        0
18        5        0        2        1        6        0
19        6        0        2        0        3        0
20        2        0        0        0        4        3

dune dataset from the package vegan shows meadow vegetation of 30 species in 20 sites.

SES data is multidimensional

dune.env has environmenta data for the 20 sites

     A1 Moisture Management      Use Manure
1   2.8        1         SF Haypastu      4
2   3.5        1         BF Haypastu      2
3   4.3        2         SF Haypastu      4
4   4.2        2         SF Haypastu      4
5   6.3        1         HF Hayfield      2
6   4.3        1         HF Haypastu      2
7   2.8        1         HF  Pasture      3
8   4.2        5         HF  Pasture      3
9   3.7        4         HF Hayfield      1
10  3.3        2         BF Hayfield      1
11  3.5        1         BF  Pasture      1
12  5.8        4         SF Haypastu      2
13  6.0        5         SF Haypastu      3
14  9.3        5         NM  Pasture      0
15 11.5        5         NM Haypastu      0
16  5.7        5         SF  Pasture      3
17  4.0        2         NM Hayfield      0
18  4.6        1         NM Hayfield      0
19  3.7        5         NM Hayfield      0
20  3.5        5         NM Hayfield      0

Data types

Type	Features	Notation	Category	Distribution
Count data (N)	Discrete (countable)	Nominal data	Qualitative	Non parametric
Count with 2 groups (dichotomous)	Discrete categorical, non-rankable	Nominal data	Qualitative	Non parametric
Count with > 2 groups	Discrete categorical, non-rankable	Nominal data	Qualitative	Non parametric
Count with rankable groups	Discrete categorical rankable	Ordinal data	Qualitative	Non parametric
Measurements (but not true zero)	Continuous	Interval data	Quantitative	Parametric
Relative measurements (true zero)	Relative continuous	Interval ratio data	Quantitative	Parametric
Time series	Continuous data linked by a time vector	Interval data along a time series	Quantitative	Parametric

From Fetzer et al (2022)

Multivariate analysis

Multivariate analysis of variance (MANOVA)
Clustering & ordination
Principal component analysis (PCA)
Non-metric multidimensional scaling (nMDS)
Redundancy analysis (RDA)
Canonical correspondence analysis (CCA)
Factor Analysis (FA)
Multiple correspondence analysis (MCA)

Clustering

“grouping sets of objects that are more similar to each other than objects in other groups (or clusters)”

Clustering

hierarchical
k-means
diana: divisive hierarchical
pam: partition around mediods
clara: sampling based pam
fanny: fuzzy clustering with partial memberships
som: self organizing maps
model based: mixture of Gaussian distributions to fit data (EM)
sota: self-organizing tree algorithm

Example: k-means

out <- kmeans(as.matrix(dune), centers = 5)

out

K-means clustering with 5 clusters of sizes 4, 4, 4, 4, 4

Cluster means:
  Achimill Agrostol Airaprae Alopgeni Anthodor Bellpere Bromhord Chenalbu
1      0.5        0     1.25     0.00     2.00     0.50     0.00     0.00
2      0.0        5     0.00     1.00     0.00     0.00     0.00     0.00
3      0.0        4     0.00     5.25     0.00     0.00     0.00     0.25
4      1.0        3     0.00     2.75     0.00     1.75     1.75     0.00
5      2.5        0     0.00     0.00     3.25     1.00     2.00     0.00
  Cirsarve Comapalu Eleopalu Elymrepe Empenigr Hyporadi Juncarti Juncbufo
1      0.0        0     0.00      0.0      0.5     2.25      0.0     0.00
2      0.0        1     5.25      0.0      0.0     0.00      2.5     0.00
3      0.0        0     1.00      1.5      0.0     0.00      2.0     2.75
4      0.5        0     0.00      4.0      0.0     0.00      0.0     0.00
5      0.0        0     0.00      1.0      0.0     0.00      0.0     0.50
  Lolipere Planlanc Poaprat Poatriv Ranuflam Rumeacet Sagiproc Salirepe
1     2.25      2.0    2.00    0.00      0.0      0.0     1.25     1.50
2     0.00      0.0    0.00    0.50      2.5      0.0     0.00     1.25
3     1.50      0.0    2.50    5.50      1.0      1.0     2.50     0.00
4     5.75      0.0    4.25    5.00      0.0      0.0     1.25     0.00
5     5.00      4.5    3.25    4.75      0.0      3.5     0.00     0.00
  Scorautu Trifprat Trifrepe Vicilath Bracruta Callcusp
1     4.50     0.00     1.75     0.75     3.25      0.0
2     1.50     0.00     1.75     0.00     3.00      2.5
3     2.25     0.00     2.50     0.00     2.00      0.0
4     2.25     0.00     2.00     0.00     1.00      0.0
5     3.00     2.25     3.75     0.25     3.00      0.0

Clustering vector:
 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
 4  4  4  4  5  5  5  3  3  5  1  3  3  2  2  2  1  1  1  2 

Within cluster sum of squares by cluster:
[1] 144.25 115.25 141.00 164.25 112.50
 (between_SS / total_SS =  57.6 %)

Available components:

[1] "cluster"      "centers"      "totss"        "withinss"     "tot.withinss"
[6] "betweenss"    "size"         "iter"         "ifault"

Example: hierarchical

out <- hclust(dist(dune), method = "average")
plot(out)

Important considerations

Jargon mine: e.g. dissimilarity
Any algorithm is sensitive to the distance metric used.
- Presence / absence data: Jaccard, Sorensen
- Gower, Bray–Curtis, Jaccard and Kulczynski indices are good in detecting underlying ecological gradients
- Morisita, Horn–Morisita, Binomial, Cao and Chao indices should be able to handle different sample sizes
- Mountford and Raup-Crick indices for presence–absence data should be able to handle unknown (and variable) sample sizes
- For guidance see help("vegdist") and vignette of vegan package

Important considerations

Clustering is sensitive to the raw units of the data (distributions), data transformations can help.
- Log-transform heavy distributions (e.g. income)
- Zero-inflated distributions
- Re-scale to zero mean, unit variance

Decisions depend on case-by-case basis, but the idea is to reduce influence of outliers or non-informative observations

Sensitivity and validation

How many clusters to fit?

NbClust package test 10 different methods
Compare 30 different metrics of performance

library(NbClust)
library(clValid)
help("NbClust")

Real world example

Mapping social-ecological systems archetypes

Rocha et al., 2020. ERL

Ostrom’s heritage

Challenge of SDGs: how do we find context dependent solutions?
There is no panaceas!!
Social-ecological systems framework
- 2-tier variables (n=53)
Over 100 case study coded but local in temporal and spatial scales:
- SES Library
- SES Meta Analysis Database

To develop a data driven method to upscale Ostrom’s SES framework

Volta river basin

West African Sahel is vulnerable area due to:

wide-spread of poverty
recurrent droughts and dry spells
political upheaval
growing food demand

Volta basin is 2/3 Ghana and Burkina Faso
- S: wet, rich, urbanization

Ostrom’s framework

FWFig

1st tier	2nd tier	Indicators
Socio-economic and political settings (S)	S2-Demographic trends	Population trend
		Inter regional migration
		Intra regional migration
	S5-Market Incentives	Market access
Resource System (RS)	RS4-Human constructed facilities	Dams
	RS7-Predictability of system’s dynamics	Variance of production (kcals)
Resource Units (RU)	RU5-Number of units	Cattle per
		Small ruminants per capita
Users (U)	U1-Number of users	Population density (persons/)
		Ratio of farmers (%)
	U2-Socioeconomic attributes	Ratio of children (% < 14yr)
		Ratio of woman (%)
		Literacy (%)
Related Ecosystems (ECO)	ECO1-Climate patterns	Aridity
		Mean temperature (C)
	ECO3-Flows	Soil water
		Wet season (months precip. > 60mm)
		Slope 75%
Interactions (I)	I1-Harvesting levels	Kilo calories for diverse crops

How many clusters, which algorithm?

Lessons

Lear more: Rocha et al., 2020 Mapping Social-Ecological System Archetypes. ERL