map_anfit.h File Reference

Fit a superposition of interrupted Poisson processes to a Hurst parameter. More...

#include <cmath>
#include <cstddef>
#include <vector>
#include "line/api/mam/map_moment.h"
#include "line/api/mam/mmap_lambda.h"
#include "line/api/mam/map_transform.h"
#include "line/num/number.h"
#include "line/util/auglag.h"
#include "line/util/error.h"
#include "line/util/matrix.h"

Include dependency graph for map_anfit.h:

Go to the source code of this file.

Classes
struct	line::mam::MapAnfitResult< T >
	What map_anfit returns: the fitted MAP and the ladder it was built on. More...

Namespaces
namespace	line
namespace	line::mam

Functions
template<class T>
MapAnfitResult< T >	line::mam::map_anfit (const T &ls, const T &rho, const T &H, double n, std::size_t ds)
	Fit a superposition of interrupted Poisson processes to a Hurst parameter.
template<class T>
MapAnfitResult< T >	line::mam::map_anfit_lsq (const T &ls, const T &rho, const T &H, double n, std::size_t ds, const std::vector< T > &SA, const std::vector< unsigned > &SAlags, unsigned iter_max=100)
	The least-squares variant: after the deterministic construction, the per-IPP ratios are tuned so the fitted autocorrelation matches a supplied one.

Detailed Description

Fit a superposition of interrupted Poisson processes to a Hurst parameter.

Templated port of matlab/lib/kpctoolbox/map/map_anfit.m (Andersen-Nielsen). The construction builds d two-state IPPs whose switching rates form a geometric ladder k(2,i) = a^(1-i) k(2,1) with a = 10^(n/(d-1)), and whose weights phi(i) are chosen so that the aggregate variance-time curve follows t^beta with beta = 2 - 2H over n decades. Superposing them with an optional Poisson remainder gives a MAP whose autocorrelation decays like a fractional process over that range, which is what makes it a long-range-dependence surrogate rather than a moment fit.

The phi ladder is built by the reference's backward recursion D = a^(i beta) - sum_{j<i} phi(d-j)^2 exp(1 - a^(i-j)), phi(d-i) = sqrt(D) when D >= 0, else 0 and the depth d is grown by one, which is a greedy fill: a level that cannot carry positive variance is set to zero and the ladder is extended instead of failing.

REFERENCE INCONSISTENCY, resolved in favour of the least-squares branch. map_anfit.m builds the Poisson remainder as map_exponential(lP), and the global map_exponential takes a MEAN, whereas the file's own objfun builds the SAME component as {[-lP], [lP]}, i.e. with lP as a RATE. The two cannot both be right. The rate reading is taken here, for two reasons: objfun is unambiguous, and lP = 0 – which the ls < L branch produces deliberately to mean "no Poisson remainder" – is the null stream under the rate reading and a division by zero under the mean reading.

THE LEAST-SQUARES BRANCH replaces MATLAB's fmincon with line/util/auglag.h, whose header carries the acceptance contract for that substitution. It minimizes the 2-norm between the fitted autocorrelation and the supplied one over the per-IPP ratios r(i), subject to the reference's feasibility constraint ls(i) >= sqrt(k1(i) k2(i) / r(i)), which is what keeps the second IPP rate non-negative.

ARITHMETIC: transcendental.

Definition in file map_anfit.h.