Functional PCA, graph convolution and attention, persistent homology, possibilistic clustering, REML components

CurveFpca: functional PCA on a common grid with trapezoid weights (eigenfunctions of W^(1/2) C W^(1/2) rescaled to unit L2 norm, largest value positive). GcnLayer: sigma(D^(-1/2) (A + I) D^(-1/2) H W). SgcRidge: ridge regression on S^K X (intercept unpenalised). GatLayer: one-head graph attention with LeakyReLU scores and softmax over neighbours. RipsPersistence: Vietoris-Rips persistence in dimensions 0 and 1 by boundary-matrix reduction over Z/2 (rows are dimension, birth, death; zero-persistence pairs dropped). PersistenceLandscape: lambda_k(t), the k-th largest tent max(0, min(t - b, d - t)). PossibilisticCmeans: possibilistic c-means with omega from fuzzy c-means (as ppclust::pcm). RemlComponents: REML variance components by average-information Newton steps with step halving and a boundary floor of 1e-10 var(y); CrossedRandomEffects and NestedRandomEffects build the indicator designs.

Usage

CurveFpca(curves, t, n_components = 2)

GcnLayer(A, H, Wt, activation = "relu", self_loops = TRUE)

SgcRidge(A, X, y, k = 2, l2 = 1, self_loops = TRUE)

GatLayer(
  A,
  H,
  Wt,
  a,
  negative_slope = 0.2,
  activation = "linear",
  self_loops = TRUE
)

RipsPersistence(points, max_dim = 1, max_scale = Inf, distance_matrix = FALSE)

PersistenceLandscape(diagram, t, k_max = 3, dimension = NULL)

PossibilisticCmeans(
  X,
  centers,
  m = 2,
  omega = NULL,
  K = 1,
  max_iter = 1000,
  con_val = 1e-09
)

RemlComponents(y, X, Zs, max_iter = 200, tol = 1e-12)

CrossedRandomEffects(y, a, b, X = NULL)

NestedRandomEffects(y, a, b, X = NULL)

Arguments

curves: Matrix of curves (rows).
t: Grid, or evaluation points for the landscape.
n_components: Number of components.
A: Adjacency matrix.
H: Node features.
Wt: Layer weights.
activation: "relu", "tanh" or "linear".
self_loops: Add self loops.
X: Node features, data matrix, or fixed-effects design.
y: Response.
k: Number of propagation steps.
l2: Ridge penalty.
a: Attention vector (length twice the output width), or factor codes.
negative_slope: LeakyReLU slope.
points: Point coordinates (rows) or a distance matrix.
max_dim: Largest homology dimension (0 or 1).
max_scale: Largest filtration value.
distance_matrix: Treat points as distances.
diagram: Persistence diagram rows.
k_max: Number of landscape functions.
dimension: Homology dimension to use, or NULL.
centers: Initial prototypes.
m: Fuzzifier.
omega: Bandwidths, or NULL to derive them from fuzzy c-means.
K: Omega multiplier.
max_iter: Iteration limit.
con_val: Convergence tolerance on the prototypes.
Zs: List of random-effect design matrices.
tol: Relative step tolerance.
b: Second factor codes.

Value

A list, matrix or vector.

References

Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis. Kipf, T. N. and Welling, M. (2017). ICLR 2017. Wu, F. et al. (2019). ICML 2019. Velickovic, P. et al. (2018). ICLR 2018. Edelsbrunner, H., Letscher, D. and Zomorodian, A. (2002). Discrete Comput. Geom. 28, 511-533. Bubenik, P. (2015). JMLR 16, 77-102. Krishnapuram, R. and Keller, J. M. (1993). IEEE Trans. Fuzzy Systems 1, 98-110. Patterson, H. D. and Thompson, R. (1971). Biometrika 58, 545-554. Gilmour, A. R., Thompson, R. and Cullis, B. R. (1995). Biometrics 51, 1440-1450.

Examples

RipsPersistence(rbind(c(0, 0), c(1, 0), c(1, 1), c(0, 1)))$diagram
#>      dimension birth    death
#> [1,]         0     0 1.000000
#> [2,]         0     0 1.000000
#> [3,]         0     0 1.000000
#> [4,]         0     0      Inf
#> [5,]         1     1 1.414214
PersistenceLandscape(rbind(c(0, 2), c(1, 3)), c(0.5, 1, 1.5, 2), k_max = 2)
#>      [,1] [,2] [,3] [,4]
#> [1,]  0.5    1  0.5    1
#> [2,]  0.0    0  0.5    0