HellCor: The Hellinger Correlation
In HellCor: The Hellinger Correlation
Description
Usage
Arguments
Details
Value
Author(s)
References
Examples
Description
Empirical value of the Hellinger correlation between two continuous random variables X and Y.
Usage
1
2
3
Arguments
x
Sample of X-values.
y
Sample of Y-values.
Kmax
Maximal number of terms to consider in the expansion of
the square root of the copula density in the basis of Legendre polynomials.
Lmax
Maximal number of terms to consider in the expansion of
the square root of the copula density in the basis of Legendre polynomials.
K
If K and L are not equal to zero, and Kmax = Lmax = 0, the
number of terms is set to K in the expansion of
the square root of the copula density in the basis of Legendre polynomials.
L
If K and L are not equal to zero, and Kmax = Lmax = 0, the
number of terms is set to L in the expansion of
the square root of the copula density in the basis of Legendre polynomials.
alpha
Parameter of the Beta(alpha,alpha) distribution
used to fix boundary issues through transformation when estimating the Hellinger correlation.
pval.comp
Logical. Should we compute the p-value?
conf.level
If not NULL, a number in (0, 1) that sets the
confidence level for the computed confidence interval.
B1
Numeric. Number of Bootstrap replicates for the outer loop
in the double bootstrap procedure used to obtain the confidence
interval. Only used if conf.level is not NULL. For some
examples, one might need to increase this value to 1,000.
B2
Numeric. Number of Bootstrap replicates for the inner loop
in the double bootstrap procedure used to obtain the confidence
interval. Only used if conf.level is not NULL.
C.version
Logical. If FALSE, the R version is used instead
of the C version. Not really useful to change to FALSE unless
one wants to understand the algorithm used.
Details
When Kmax = Lmax = K = L = 0, the value returned in Hcor is the
unnormalized version of the empirical Hellinger correlation.
Value
List with the following components:
Hcor
Value of the empirical Hellinger correlation.
p.value
The p-value associated to the null hypothesis that the Hellinger correlation is
equal to 0 (computed using a Monte-Carlo simulation). Will be
NA if pval.comp is not set to TRUE.
conf.int
Confidence interval for the population Hellinger
correlation (computed using a double Bootstrap). Will be
NA if conf.level is not set to a value in (0,1).
Khat
Value of K selected by cross-validation, if
computed, otherwise NA.
Lhat
Value of L selected by cross-validation, if computed, otherwise NA.
Author(s)
Geenens G., Lafaye de Micheaux P.
References
Geenens G., Lafaye de Micheaux P., (2018). The Hellinger correlation. (), –.
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
# We illustrate the application of our new measure using data extracted
# from a study on the coral reef published in
# [Graham, N.A.J. and Wilson, S.K. and Carr, P. and Hoey, A.S. and
# Jennings, S. and MacNeil, M.A. (2018), Seabirds enhance coral reef
# productivity and functioning in the absence of invasive rats, Nature,
# 559, 250--253.].
# The two variables we consider are the number of fishes and the nitrogen
# input by seabirds per hectare recorded on n = 12 islands in the Chagos
# Archipelago. Nitrogen input is an indirect measure of the abundance of
# seabirds. It is worthwhile to notice that nitrogen is absorbed by algae
# and that herbivorous fishes eat these algae. Fishes and birds living in
# two different worlds, it might seem odd to suspect a dependence between
# these two variables. Since our measure is tailored to capture dependence
# induced by a lurking variable, we can suspect the existence of such a
# hidden variable. This is indeed what researchers in (Graham et al., 2018)
# were able to show by finding that the presence and abundance of rats on
# an island had dramatic effects on the number of seabirds (the rats eating
# their eggs) and consequently on the input of nitrogen in seabirds' guano.
# In turn, this would diminish the abundance of algae and fishes eating
# these algae.
data(Chagos)
n <- nrow(Chagos)
par.save <- par()$mfrow
par(mfrow = c(1, 2))
plot(Chagos$Seabirds_ha, Chagos$Number_of_fishes, main = "Original data",
xlab = "Density of seabirds", ylab = "Density of fishes")
plot(rank(Chagos$Seabirds_ha) / (n + 1), rank(Chagos$Number_of_fishes) /
(n + 1), main = "Rank-Copula transformed data",
xlab = "Density of seabirds", ylab = "Density of fishes")
par(mfrow = par.save)
set.seed(1)
# Empirical Hellinger correlation
HellCor(Chagos$Seabirds_ha, Chagos$Number_of_fishes, pval.comp = TRUE)
# Pearson correlation
cor.test(Chagos$Seabirds_ha, Chagos$Number_of_fishes)
# Distance correlation
dcor.test(Chagos$Seabirds_ha, Chagos$Number_of_fishes, R = 200)
set.seed(1)
# Empirical Hellinger correlation
HellCor(Chagos$kg_N_ha_yr, Chagos$Seabirds_ha, pval.comp = TRUE)
# Pearson correlation
cor.test(Chagos$kg_N_ha_yr, Chagos$Seabirds_ha)
# Distance correlation
dcor.test(Chagos$kg_N_ha_yr, Chagos$Seabirds_ha, R = 200)
set.seed(1)
# Empirical Hellinger correlation
HellCor(Chagos$kg_N_ha_yr, Chagos$Number_of_fishes, pval.comp = TRUE)
# Pearson correlation
cor.test(Chagos$kg_N_ha_yr, Chagos$Number_of_fishes)
# Distance correlation
dcor.test(Chagos$kg_N_ha_yr, Chagos$Number_of_fishes, R = 200)
t.test(Chagos$kg_N_ha_yr ~ Chagos$Treatment)
######################################################################
# Geenens G., Lafaye de Micheaux P., (2020). The Hellinger correlation
######################################################################
# Figure 5.2
## Not run:
n <- 500
set.seed(1)
par(mfrow = c(3, 5))
XX <- .datagenW.corrected(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(W~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenDiamond(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Diamond~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenParabola.corrected(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Parabola~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagen2Parabolas.corrected(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat<-HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Two~~parabolae~~-~~hat(eta)==etahat, list(etahat
= round(etahat, 3))))
XX <- .datagenCircle.corrected(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Circle~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagen4indclouds(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(4~~clouds~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenCubic(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Cubic~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenSine(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Sine~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenWedge(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Wedge~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenCross(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Cross~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenSpiral(n, sigma = 0.05)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Spiral~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagen4Circles(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Circles~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenHeavisine(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Heavisine~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenDoppler(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat<-HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Doppler~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagen5Clouds(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(5~~clouds~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
## End(Not run)
############################
# Birth rate vs Death rate #
############################
data(worlddemographics)
x <- wdemographics$Death.Rate.Pop
y <- wdemographics$Birth.Rate.Pop
plot(x, y, xlab = "DEATHS/1,000 POPULATION", ylab =
"BIRTHS/1,000 POPULATION", main =
"Birth rate vs Death rate for 229 countries and territories (in 2020)", col = "orangered", pch = 20)
text(x, y, labels = wdemographics$Country, pos = 3, cex = 0.4, offset = 0.2)
## Not run:
HellCor(x, y, pval.comp = TRUE)
cor.test(x, y) # Pearson
dcor.test(x, y, R = 200)
## End(Not run)
HellCor documentation built on July 2, 2020, 3:57 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Description Usage Arguments Details Value Author(s) References Examples
Description
Empirical value of the Hellinger correlation between two continuous random variables X and Y.
Usage
1 2 3 |
Arguments
x |
Sample of X-values. |
y |
Sample of Y-values. |
Kmax |
Maximal number of terms to consider in the expansion of the square root of the copula density in the basis of Legendre polynomials. |
Lmax |
Maximal number of terms to consider in the expansion of the square root of the copula density in the basis of Legendre polynomials. |
K |
If K and L are not equal to zero, and Kmax = Lmax = 0, the number of terms is set to K in the expansion of the square root of the copula density in the basis of Legendre polynomials. |
L |
If K and L are not equal to zero, and Kmax = Lmax = 0, the number of terms is set to L in the expansion of the square root of the copula density in the basis of Legendre polynomials. |
alpha |
Parameter of the Beta(alpha,alpha) distribution used to fix boundary issues through transformation when estimating the Hellinger correlation. |
pval.comp |
Logical. Should we compute the p-value? |
conf.level |
If not |
B1 |
Numeric. Number of Bootstrap replicates for the outer loop
in the double bootstrap procedure used to obtain the confidence
interval. Only used if |
B2 |
Numeric. Number of Bootstrap replicates for the inner loop
in the double bootstrap procedure used to obtain the confidence
interval. Only used if |
C.version |
Logical. If |
Details
When Kmax = Lmax = K = L = 0, the value returned in Hcor is the
unnormalized version of the empirical Hellinger correlation.
Value
List with the following components:
Hcor |
Value of the empirical Hellinger correlation. |
p.value |
The p-value associated to the null hypothesis that the Hellinger correlation is
equal to 0 (computed using a Monte-Carlo simulation). Will be
|
conf.int |
Confidence interval for the population Hellinger
correlation (computed using a double Bootstrap). Will be
|
Khat |
Value of K selected by cross-validation, if
computed, otherwise |
Lhat |
Value of L selected by cross-validation, if computed, otherwise |
Author(s)
Geenens G., Lafaye de Micheaux P.
References
Geenens G., Lafaye de Micheaux P., (2018). The Hellinger correlation. (), –.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | # We illustrate the application of our new measure using data extracted
# from a study on the coral reef published in
# [Graham, N.A.J. and Wilson, S.K. and Carr, P. and Hoey, A.S. and
# Jennings, S. and MacNeil, M.A. (2018), Seabirds enhance coral reef
# productivity and functioning in the absence of invasive rats, Nature,
# 559, 250--253.].
# The two variables we consider are the number of fishes and the nitrogen
# input by seabirds per hectare recorded on n = 12 islands in the Chagos
# Archipelago. Nitrogen input is an indirect measure of the abundance of
# seabirds. It is worthwhile to notice that nitrogen is absorbed by algae
# and that herbivorous fishes eat these algae. Fishes and birds living in
# two different worlds, it might seem odd to suspect a dependence between
# these two variables. Since our measure is tailored to capture dependence
# induced by a lurking variable, we can suspect the existence of such a
# hidden variable. This is indeed what researchers in (Graham et al., 2018)
# were able to show by finding that the presence and abundance of rats on
# an island had dramatic effects on the number of seabirds (the rats eating
# their eggs) and consequently on the input of nitrogen in seabirds' guano.
# In turn, this would diminish the abundance of algae and fishes eating
# these algae.
data(Chagos)
n <- nrow(Chagos)
par.save <- par()$mfrow
par(mfrow = c(1, 2))
plot(Chagos$Seabirds_ha, Chagos$Number_of_fishes, main = "Original data",
xlab = "Density of seabirds", ylab = "Density of fishes")
plot(rank(Chagos$Seabirds_ha) / (n + 1), rank(Chagos$Number_of_fishes) /
(n + 1), main = "Rank-Copula transformed data",
xlab = "Density of seabirds", ylab = "Density of fishes")
par(mfrow = par.save)
set.seed(1)
# Empirical Hellinger correlation
HellCor(Chagos$Seabirds_ha, Chagos$Number_of_fishes, pval.comp = TRUE)
# Pearson correlation
cor.test(Chagos$Seabirds_ha, Chagos$Number_of_fishes)
# Distance correlation
dcor.test(Chagos$Seabirds_ha, Chagos$Number_of_fishes, R = 200)
set.seed(1)
# Empirical Hellinger correlation
HellCor(Chagos$kg_N_ha_yr, Chagos$Seabirds_ha, pval.comp = TRUE)
# Pearson correlation
cor.test(Chagos$kg_N_ha_yr, Chagos$Seabirds_ha)
# Distance correlation
dcor.test(Chagos$kg_N_ha_yr, Chagos$Seabirds_ha, R = 200)
set.seed(1)
# Empirical Hellinger correlation
HellCor(Chagos$kg_N_ha_yr, Chagos$Number_of_fishes, pval.comp = TRUE)
# Pearson correlation
cor.test(Chagos$kg_N_ha_yr, Chagos$Number_of_fishes)
# Distance correlation
dcor.test(Chagos$kg_N_ha_yr, Chagos$Number_of_fishes, R = 200)
t.test(Chagos$kg_N_ha_yr ~ Chagos$Treatment)
######################################################################
# Geenens G., Lafaye de Micheaux P., (2020). The Hellinger correlation
######################################################################
# Figure 5.2
## Not run:
n <- 500
set.seed(1)
par(mfrow = c(3, 5))
XX <- .datagenW.corrected(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(W~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenDiamond(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Diamond~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenParabola.corrected(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Parabola~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagen2Parabolas.corrected(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat<-HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Two~~parabolae~~-~~hat(eta)==etahat, list(etahat
= round(etahat, 3))))
XX <- .datagenCircle.corrected(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Circle~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagen4indclouds(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(4~~clouds~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenCubic(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Cubic~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenSine(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Sine~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenWedge(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Wedge~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenCross(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Cross~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenSpiral(n, sigma = 0.05)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Spiral~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagen4Circles(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Circles~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenHeavisine(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Heavisine~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagenDoppler(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat<-HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(Doppler~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
XX <- .datagen5Clouds(n)
plot(XX, xlab = expression(X[1]), ylab = expression(X[2]))
etahat <- HellCor(XX[,1], XX[,2])$Hcor
title(main = substitute(5~~clouds~~-~~hat(eta)==etahat, list(etahat =
round(etahat, 3))))
## End(Not run)
############################
# Birth rate vs Death rate #
############################
data(worlddemographics)
x <- wdemographics$Death.Rate.Pop
y <- wdemographics$Birth.Rate.Pop
plot(x, y, xlab = "DEATHS/1,000 POPULATION", ylab =
"BIRTHS/1,000 POPULATION", main =
"Birth rate vs Death rate for 229 countries and territories (in 2020)", col = "orangered", pch = 20)
text(x, y, labels = wdemographics$Country, pos = 3, cex = 0.4, offset = 0.2)
## Not run:
HellCor(x, y, pval.comp = TRUE)
cor.test(x, y) # Pearson
dcor.test(x, y, R = 200)
## End(Not run)
|
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.