-
Notifications
You must be signed in to change notification settings - Fork 1
/
RBF_moments.m
191 lines (183 loc) · 8.05 KB
/
RBF_moments.m
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
181
182
183
184
185
186
187
188
189
190
191
%% RBF_moments
% Author: Jan Glaubitz
% Date: June 22, 2021
%
% Compute the RBF's moments
%
% INPUT:
% a, b : left and right boundary of the domain
% kernel : kernel
% rbf : radial basis function
% ep : shape parameter
% X : data points
%
% OUTPUT:
% m_RBF : moments of the translated RBFs
%%
function m_RBF = RBF_moments( a, b, kernel, rbf, ep, X )
[N,dim] = size(X); % number of data points
m_RBF = zeros(N,1);
%% One dimensional
if dim==1
%% Gaussian
if strcmp(kernel,'G')
m_RBF = (0.5/ep)*sqrt(pi)*( erf( ep*(b-X) ) - erf( ep*(a-X) ) ); % moments
elseif strcmp(kernel,'TPS')
k = 2;
m_RBF = (X-a).^(k+1).*( log(X-a+10^(-14))/(k+1) - 1/(k+1)^2 ) + ...
(b-X).^(k+1).*( log(b-X+10^(-14))/(k+1) - 1/(k+1)^2 ); % moments
elseif strcmp(kernel,'cubic')
k = 3;
m_RBF = ( (a-X).^(k+1) + (b-X).^(k+1) )/(k+1); % moments
elseif strcmp(kernel,'quintic')
k = 3;
m_RBF = ( (a-X).^(k+1) + (b-X).^(k+1) )/(k+1); % moments
elseif strcmp(kernel,'Wendland')
for n=1:N
% Support of kernel with center x_n is [c,d] with
c = max(a,X(n)-1/ep(n)); d = min(X(n)+1/ep(n),b);
int = @(x) rbf(ep(n),abs( x-X(n) )); % integrand
m_RBF(n) = integral( @(x) int(x), c, d ); % nth moment
end
else
rbf_basis = @(x) rbf(ep,abs(x-X')); % RBF basis
syms x
m_RBF = integral( @(x) rbf_basis(x), a, b, 'ArrayValued', true )'; % moments
end
%% Two dimensional
elseif dim==2
if strcmp(kernel,'G')
for n=1:N
mx = (0.5/ep)*sqrt(pi)*( erf( ep*(b-X(n,1)) ) - erf( ep*(a-X(n,1)) ) ); % component in x direction
my = (0.5/ep)*sqrt(pi)*( erf( ep*(b-X(n,2)) ) - erf( ep*(a-X(n,2)) ) ); % component in x direction
m_RBF(n) = mx*my; % moments
end
elseif strcmp(kernel,'TPS')
I_tr = @(u,v) (u/144)*( ...
24*u^3*atan(v/u) + 6*v*(3*u^2+v^2)*log(u^2+v^2) - 33*u^2*v - 7*v^3 ...
); % reference integral
% compute moments
for n=1:N
% shifted edges of the rectangle
c = a; d = b; % we assume the domain [a,b]^2
a_tilde = abs( a - X(n,1) );
b_tilde = abs( b - X(n,1) );
c_tilde = abs( c - X(n,2) );
d_tilde = abs( d - X(n,2) );
% partition rectangle in 8 right triangles and compute the
% corresponding integrals
I(1) = I_tr(b_tilde,d_tilde);
I(2) = I_tr(d_tilde,b_tilde);
I(3) = I_tr(d_tilde,a_tilde);
I(4) = I_tr(a_tilde,d_tilde);
I(5) = I_tr(a_tilde,c_tilde);
I(6) = I_tr(c_tilde,a_tilde);
I(7) = I_tr(c_tilde,b_tilde);
I(8) = I_tr(b_tilde,c_tilde);
I(isnan(I))=0; % set all NaN values to zero;
% sum these up to get the moment
m_RBF(n) = ( b_tilde*d_tilde ~= 0 )*(I(1)+I(2)) + ...
( a_tilde*d_tilde ~= 0 )*(I(3)+I(4)) + ...
( a_tilde*c_tilde ~= 0 )*(I(5)+I(6)) + ...
( b_tilde*c_tilde ~= 0 )*(I(7)+I(8));
end
elseif strcmp(kernel,'cubic')
I_tr = @(u,v) (u/40)*( ...
3*u^4*asinh(v/u) + ...
v*( 5*u^2 + 2*v^2 )*sqrt( u^2 + v^2 ) ...
); % reference integral
% compute moments
for n=1:N
% shifted edges of the rectangle
c = a; d = b; % we assume the domain [a,b]^2
a_tilde = abs( a - X(n,1) );
b_tilde = abs( b - X(n,1) );
c_tilde = abs( c - X(n,2) );
d_tilde = abs( d - X(n,2) );
% partition rectangle in 8 right triangles and compute the
% corresponding integrals
I(1) = I_tr(b_tilde,d_tilde);
I(2) = I_tr(d_tilde,b_tilde);
I(3) = I_tr(d_tilde,a_tilde);
I(4) = I_tr(a_tilde,d_tilde);
I(5) = I_tr(a_tilde,c_tilde);
I(6) = I_tr(c_tilde,a_tilde);
I(7) = I_tr(c_tilde,b_tilde);
I(8) = I_tr(b_tilde,c_tilde);
I(isnan(I))=0; % set all NaN values to zero;
% sum these up to get the moment
m_RBF(n) = ( b_tilde*d_tilde ~= 0 )*(I(1)+I(2)) + ...
( a_tilde*d_tilde ~= 0 )*(I(3)+I(4)) + ...
( a_tilde*c_tilde ~= 0 )*(I(5)+I(6)) + ...
( b_tilde*c_tilde ~= 0 )*(I(7)+I(8));
end
elseif strcmp(kernel,'quintic')
I_tr = @(u,v) (u/336)*( ...
15*u^6*asinh(v/u) + ...
v*( 33*u^4 + 26*u^2*v^2 + 8*v^4 )*sqrt( u^2 + v^2 ) ...
); % reference integral
% compute moments
for n=1:N
% shifted edges of the rectangle
c = a; d = b; % we assume the domain [a,b]^2
a_tilde = abs( a - X(n,1) );
b_tilde = abs( b - X(n,1) );
c_tilde = abs( c - X(n,2) );
d_tilde = abs( d - X(n,2) );
% partition rectangle in 8 right triangles and compute the
% corresponding integrals
I(1) = I_tr(b_tilde,d_tilde);
I(2) = I_tr(d_tilde,b_tilde);
I(3) = I_tr(d_tilde,a_tilde);
I(4) = I_tr(a_tilde,d_tilde);
I(5) = I_tr(a_tilde,c_tilde);
I(6) = I_tr(c_tilde,a_tilde);
I(7) = I_tr(c_tilde,b_tilde);
I(8) = I_tr(b_tilde,c_tilde);
I(isnan(I))=0; % set all NaN values to zero;
% sum these up to get the moment
m_RBF(n) = ( b_tilde*d_tilde ~= 0 )*(I(1)+I(2)) + ...
( a_tilde*d_tilde ~= 0 )*(I(3)+I(4)) + ...
( a_tilde*c_tilde ~= 0 )*(I(5)+I(6)) + ...
( b_tilde*c_tilde ~= 0 )*(I(7)+I(8));
end
elseif strcmp(kernel,'septic')
I_tr = @(u,v) (u/3346)*( ...
105*u^8*asinh(v/u) + ...
v*( 279*u^6 + 326*u^4*v^2 + 200*u^2*v^4 + 48*v^6 )*sqrt( u^2 + v^2 ) ...
); % reference integral
% compute moments
for n=1:N
% shifted edges of the rectangle
c = a; d = b; % we assume the domain [a,b]^2
a_tilde = abs( a - X(n,1) );
b_tilde = abs( b - X(n,1) );
c_tilde = abs( c - X(n,2) );
d_tilde = abs( d - X(n,2) );
% partition rectangle in 8 right triangles and compute the
% corresponding integrals
I(1) = I_tr(b_tilde,d_tilde);
I(2) = I_tr(d_tilde,b_tilde);
I(3) = I_tr(d_tilde,a_tilde);
I(4) = I_tr(a_tilde,d_tilde);
I(5) = I_tr(a_tilde,c_tilde);
I(6) = I_tr(c_tilde,a_tilde);
I(7) = I_tr(c_tilde,b_tilde);
I(8) = I_tr(b_tilde,c_tilde);
I(isnan(I))=0; % set all NaN values to zero;
% sum these up to get the moment
m_RBF(n) = ( b_tilde*d_tilde ~= 0 )*(I(1)+I(2)) + ...
( a_tilde*d_tilde ~= 0 )*(I(3)+I(4)) + ...
( a_tilde*c_tilde ~= 0 )*(I(5)+I(6)) + ...
( b_tilde*c_tilde ~= 0 )*(I(7)+I(8));
end
else
for n=1:N
int = @(x,y) rbf( ep, sqrt( (x-X(n,1)).^2 + (y-X(n,2)).^2 ) ); % integrand
m_RBF(n) = integral2( int, a,b, a,b ); % moment
end
end
%% Higher dimensional
else
error('Desried dimension not yet implemented')
end