| sdcov {semidist} | R Documentation |
Semi-distance covariance and correlation statistics
Description
Compute the statistics (or sample estimates) of semi-distance covariance and correlation. The semi-distance correlation is a standardized version of semi-distance covariance, and it can measure the dependence between a multivariate continuous variable and a categorical variable. See Details for the definition of semi-distance covariance and semi-distance correlation.
Usage
sdcov(X, y, type = "V", return_mat = FALSE)
sdcor(X, y)
Arguments
X |
Data of multivariate continuous variables, which should be an
|
y |
Data of categorical variables, which should be a factor of length
|
type |
Type of statistic: |
return_mat |
A boolean. If |
Details
For \bm{X} \in \mathbb{R}^{p} and Y \in \{1, 2, \cdots,
R\}, the (population-level) semi-distance covariance is defined as
\mathrm{SDcov}(\bm{X}, Y) =
\mathrm{E}\left[\|\bm{X}-\widetilde{\bm{X}}\|\left(1-\sum_{r=1}^R
I(Y=r,\widetilde{Y}=r)/p_r\right)\right],
where p_r = P(Y = r) and
(\widetilde{\bm{X}}, \widetilde{Y}) is an iid copy of (\bm{X},
Y).
The (population-level) semi-distance correlation is defined as
\mathrm{SDcor}(\bm{X}, Y) = \dfrac{\mathrm{SDcov}(\bm{X},
Y)}{\mathrm{dvar}(\bm{X})\sqrt{R-1}},
where \mathrm{dvar}(\bm{X}) is
the distance variance (Szekely, Rizzo, and Bakirov 2007) of \bm{X}.
With n observations \{(\bm{X}_i, Y_i)\}_{i=1}^{n}, sdcov()
and sdcor() can compute the sample estimates for the semi-distance
covariance and correlation.
If type = "V", the semi-distance covariance statistic is computed as a
V-statistic, which takes a very similar form as the energy-based statistic
with double centering, and is always non-negative. Specifically,
\text{SDcov}_n(\bm{X}, y) = \frac{1}{n^2} \sum_{k=1}^{n}
\sum_{l=1}^{n} A_{kl} B_{kl},
where
A_{kl} = a_{kl} - \bar{a}_{k.} - \bar{a}_{.l} + \bar{a}_{..}
is the double centering (Szekely, Rizzo, and Bakirov 2007) of
a_{kl} = \| \bm{X}_k - \bm{X}_l \|, and
B_{kl} =
1 - \sum_{r=1}^{R} I(Y_k = r) I(Y_l = r) / \hat{p}_r
with \hat{p}_r =
n_r / n = n^{-1}\sum_{i=1}^{n} I(Y_i = r).
The semi-distance correlation statistic is
\text{SDcor}_n(\bm{X}, y)
= \dfrac{\text{SDcov}_n(\bm{X}, y)}{\text{dvar}_n(\bm{X})\sqrt{R - 1}},
where \text{dvar}_n(\bm{X}) is the V-statistic of distance variance
of \bm{X}.
If type = "U", then the semi-distance covariance statistic is computed as
an “estimated U-statistic”, which is utilized in the independence test
statistic and is not necessarily non-negative. Specifically,
\widetilde{\text{SDcov}}_n(\bm{X}, y) = \frac{1}{n(n-1)}
\sum_{i \ne j} \| \bm{X}_i - \bm{X}_j \| \left(1 - \sum_{r=1}^{R}
I(Y_i = r) I(Y_j = r) / \tilde{p}_r\right),
where \tilde{p}_r = (n_r-1) / (n-1) = (n-1)^{-1}(\sum_{i=1}^{n} I(Y_i
= r) - 1). Note that the test statistic of the semi-distance independence
test is
T_n = n \cdot \widetilde{\text{SDcov}}_n(\bm{X}, y).
Value
The value of the corresponding sample statistic.
If the argument return_mat of sdcov() is set as TRUE, a list with
elements
-
sdcov: the semi-distance covariance statistic; -
mat_x, mat_y: the matrices of the distances of X and the divergences of y, respectively;
will be returned.
See Also
-
sd_test()for implementing independence test via semi-distance covariance; -
sd_sis()for implementing groupwise feature screening via semi-distance correlation.
Examples
X <- mtcars[, c("mpg", "disp", "drat", "wt")]
y <- factor(mtcars[, "am"])
print(sdcov(X, y))
print(sdcor(X, y))
# Man-made independent data -------------------------------------------------
n <- 30; R <- 5; p <- 3; prob <- rep(1/R, R)
X <- matrix(rnorm(n*p), n, p)
y <- factor(sample(1:R, size = n, replace = TRUE, prob = prob), levels = 1:R)
print(sdcov(X, y))
print(sdcor(X, y))
# Man-made functionally dependent data --------------------------------------
n <- 30; R <- 3; p <- 3
X <- matrix(0, n, p)
X[1:10, 1] <- 1; X[11:20, 2] <- 1; X[21:30, 3] <- 1
y <- factor(rep(1:3, each = 10))
print(sdcov(X, y))
print(sdcor(X, y))