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HPFiltering.cpp
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HPFiltering.cpp
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/*
question description
https://web.stanford.edu/~boyd/papers/pdf/l1_trend_filter.pdf
primal-dual interior-point method
http://www.stat.cmu.edu/~ryantibs/convexopt/lectures/primal-dual.pdf
algorithm of solving for banded-matrix
https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm
*/
// [[Rcpp::plugins(cpp11)]]
// [[Rcpp::depends(RcppArmadillo)]]
#include <RcppArmadillo.h>
using namespace Rcpp;
using arma::uword;
// Hyperparameters for search algorithm
static const double ALPHA = 0.7;
static const double BETA = 0.9;
static const double MU = 1.2;
// Parameters
struct Params {
double lambda;
arma::vec filter_d, filter_dd;
arma::vec dd1, dd2, dd3, dy;
Params(uword n, const arma::vec &y, double l): lambda(l) {
dd1.resize(n);
dd2.resize(n);
dd3.resize(n);
dd1.fill(6);
dd2.fill(-4);
dd3.fill(1);
filter_dd = {1, -4, 6, -4, 1};
filter_d = {1, -2, 1};
dy.resize(n);
dy.fill(0);
for (uword i=0; i<n; ++i)
for (uword j=0; j<3; ++j)
dy[i] += y[i+j] * filter_d[j];
}
};
// solve the first row of equation (17)
arma::vec solve_banded_matrix(arma::vec a,
arma::vec b,
arma::vec &c,
arma::vec d,
arma::vec e,
arma::vec &f) {
// ...
uword n = f.n_elem;
arma::vec res(n);
double w;
for (uword i=2; i<n; ++i) {
w = a[i] / b[i-1];
b[i] -= w * c[i-1];
c[i] -= w * d[i-1];
d[i] -= w * e[i-1];
f[i] -= w * f[i-1];
}
for (uword i=1; i<n; ++i) {
w = b[i] / c[i-1];
c[i] -= w * d[i-1];
d[i] -= w * e[i-1];
f[i] -= w * f[i-1];
}
for (uword i=n-2; i>0; --i) {
w = e[i-1] / d[i];
d[i-1] -= w * c[i];
f[i-1] -= w * f[i];
}
res[n-1] = f[n-1] / c[n-1];
for (uword i=n-1; i>0; --i) {
res[i-1] = (f[i-1] - d[i-1] * res[i]) / c[i-1];
}
return res;
}
// compute D D^T v
arma::vec DD_filter(const arma::vec &v) {
uword n = v.n_elem;
arma::vec res(n, arma::fill::zeros);
arma::rowvec filter = {1, -4, 6, -4, 1};
for (uword i=2; i<n-2; ++i)
for (uword j=0; j<5; ++j)
res[i] += filter[j] * v[i-2+j];
uword idx;
for (uword i=0; i<2; ++i)
for (uword j=0; j<5; ++j) {
idx = i-2+j;
if (idx>=0) {
res[i] += filter[j] * v[idx];
res[n-1-i] += filter[4-j] * v[n-1-idx];
}
}
return res;
}
// for memory efficiency
void inline fn_f_inv(arma::vec &f, const double &lambda) {
f.for_each( [&lambda](double x){ x = 1/(x-lambda); } );
}
// initialize s
double init_s(const arma::vec &mu1,
const arma::vec &mu2,
const arma::vec &dmu1,
const arma::vec &dmu2) {
// ...
uword n = mu1.n_elem;
double s = 1.0;
for (uword i=0; i<n; ++i) {
if (dmu1[i]<0)
s = std::min(s, -mu1[i] / dmu1[i]);
if (dmu2[i]<0)
s = std::min(s, -mu2[i] / dmu2[i]);
}
return s * 0.999;
}
double compute_r_norm_plus(const arma::vec &v,
const arma::vec &mu1,
const arma::vec &mu2,
const arma::vec &dv,
const arma::vec &dmu1,
const arma::vec &dmu2,
double s,
double t,
Params *par) {
// ...
arma::vec v_plus = v + s * dv;
arma::vec mu1_plus = mu1 + s * dmu1;
arma::vec mu2_plus = mu2 + s * dmu2;
arma::vec r_v = par->dy;
// r_v -= DD_filter(v_plus);
r_v -= arma::conv(v_plus, par->filter_dd, "same");
r_v -= mu1_plus;
r_v += mu2_plus;
// modify in-place to avoid unnecessary copy
mu1_plus %= (-par->lambda + v_plus);
mu2_plus %= (-par->lambda - v_plus);
mu1_plus += 1/t;
mu2_plus += 1/t;
return arma::dot(r_v.t(), r_v) + arma::dot(mu1_plus.t(), mu1_plus) + arma::dot(mu2_plus.t(), mu2_plus);
}
//' @export
// [[Rcpp::export]]
List hp_filter(const arma::vec &y,
const double lambda) {
// ...
uword n = y.n_elem - 2;
Params *par = new Params(n, y, lambda);
arma::vec v(n, arma::fill::zeros);
arma::vec mu1(n, arma::fill::ones);
arma::vec mu2(n, arma::fill::ones);
double eta = - arma::dot(mu1.t(), (-lambda+v)) - arma::dot(mu2.t(), (-lambda-v));
int k = 1;
double r_norm = 1.0, r_norm_plus = 1.0;
double t, s;
arma::vec J1_inv, J2_inv, f1_inv, f2_inv, v_lhs, v_rhs;
arma::vec r_v, r_mu1, r_mu2;
arma::vec dv, dmu1, dmu2;
while (eta > 1e-6 || r_norm > 1e-6) {
t = MU * n / std::pow(eta, k-1);
// prepare parts for computing dv
f1_inv = v;
f2_inv = -v;
fn_f_inv(f1_inv, lambda);
fn_f_inv(f2_inv, lambda);
J1_inv = f1_inv % mu1;
J2_inv = f2_inv % mu2;
v_rhs = par->dy;
// v_rhs -= DD_filter(v);
v_rhs -= arma::conv(v, par->filter_dd, "same");
// save the temporary result for computing norm of r dual
r_v = v_rhs;
v_rhs += f1_inv / t;
v_rhs -= f1_inv / t;
v_lhs = par->dd1;
v_lhs -= J1_inv;
v_lhs += J2_inv;
dv = solve_banded_matrix(par->dd3, par->dd2, v_lhs, par->dd2, par->dd3, v_rhs);
dmu1 = - mu1 - f1_inv / t;
dmu2 = - mu2 - f2_inv / t;
dmu1 += J1_inv % dv;
dmu2 += J2_inv % dv;
// backtracking line search
// search proper s
// the selection of s ensures that updated mu1, mu2 are positive
s = init_s(mu1, mu2, dmu1, dmu2);
r_v -= mu1;
r_v += mu2;
r_mu1 = mu1 % (-lambda + v) + 1/t;
r_mu2 = mu2 % (-lambda - v) + 1/t;
r_norm = arma::dot(r_v.t(), r_v) + arma::dot(r_mu1.t(), r_mu1), arma::dot(r_mu2.t(), r_mu2);
s /= BETA;
do {
s *= BETA;
r_norm_plus = compute_r_norm_plus(v, mu1, mu2, dv, dmu1, dmu2, s, t, par);
} while (r_norm_plus > (1-ALPHA*s) * r_norm);
// update values
v += s * dv;
mu1 += s * dmu1;
mu2 += s * dmu2;
// compute error terms
eta = - arma::dot(mu1.t(), (-lambda+v)) - arma::dot(mu2.t(), (-lambda-v));
r_norm = std::sqrt(r_norm);
++k;
}
arma::vec x = y;
x -= arma::conv(v, par->filter_d);
return List::create(
_["x"] = x
);
}