R: Projection Pursuit Indices based on the bivariate empirical...
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Projection Pursuit Indices based on the bivariate empirical distribution function.
Description
This function can be used to compute the projection pursuit
indices described in Perisic and Posse (2005).
Usage
ecdf.indices(A, sphered = FALSE)
Arguments
A
The projected data.
sphered
Whether the data has already been sphered or not. If
set to FALSE (default), the function will sphere the data before
computing the indices.
Details
The two-dimensional empirical distribution function is defined as,
F_n(x, y) = frac{1}{n} #{(x_j, y_j): x_j ≤q x mbox{ and }
y_j ≤q y}
The indices described in Perisic and Posse (2005) use this function to
construct the following four indices.
Cramer-von-Mises:
∑_i (F_n(x_i, y_i) -
Φ(x_i)Φ(y_i))^2
Kolmogorov-Smirnov:
max_i |F_n(x_i, y_i) - Φ(x_i)Φ(y_i)|
D2:
∑_i (F_n(x_i, y_i) - F_n(y_i, x_i))^2
D-infinity:
max_i |F_n(x_i, y_i) - F_n(y_i, x_i)|
where Φ(.) is the cumulative distribution function of the
standard normal distribution.
When using any of these indices, the original authors recommended
rotating the data projection several times to obtain rotational
invariance. In simulations, the indices performed well even without
rotations.
Value
A named numeric vector with the values of the following indices : the
Cramer-von-Mises index, the Kolmogorov-Smirnov index, the D2 Symmetry
index, and the D-infinity Symmetry index.
Author(s)
Mohit Dayal
References
Perisic, Igor, and Christian Posse. "Projection pursuit indices based on the empirical distribution function." Journal of Computational and Graphical Statistics 14.3 (2005).