{"paper":{"title":"Limitation of the Least Square Method in the Evaluation of Dimension of Fractal Brownian Motions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"physics.data-an","authors_text":"Benzhong Dai, Bingqiang Qiao, Houdun Zeng, Siming Liu, Xiang Li","submitted_at":"2015-07-12T17:08:52Z","abstract_excerpt":"With the standard deviation for the logarithm of the re-scaled range $\\langle |F(t+\\tau)-F(t)|\\rangle$ of simulated fractal Brownian motions $F(t)$ given in a previous paper \\cite{q14}, the method of least squares is adopted to determine the slope, $S$, and intercept, $I$, of the log$(\\langle |F(t+\\tau)-F(t)|\\rangle)$ vs $\\rm{log}(\\tau)$ plot to investigate the limitation of this procedure. It is found that the reduced $\\chi^2$ of the fitting decreases with the increase of the Hurst index, $H$ (the expectation value of $S$), which may be attributed to the correlation among the re-scaled ranges"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03250","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}