Rational spline histopolation
Kuupäev
2015-05-21
Autorid
Ajakirja pealkiri
Ajakirja ISSN
Köite pealkiri
Kirjastaja
Abstrakt
Väga tihti on andmetöötluseks esitatavad andmed histogrammide kujul, s.t., edasiseks analüüsiks on kasutada mingitel (aja)perioodidel leitud keskväärtused. Enamasti soovitakse, et andmed esituksid pideval kujul ja nende teisendamiseks sellele kujule kasutatakse erinevaid meetodeid. Nendest levinuim on interpoleerimine, vähem kasutatav aga histopoleerimine.
Andmete lähendamisel soovitakse enamasti, et säilitataks andmete geomeetrilisi omadusi nagu näiteks positiivsust (või mittenegatiivsust), monotoonsust, kumerust. Üldiselt on aga teada, et polünomiaalsete splainidega interpoleerimine ja histopoleerimine algandmete geomeetrilisi omadusi ei säilita.
Käesoleva dissertatsiooni põhiprobleemiks on uurida algandmete geomeetriliste omaduste säilitamist ratsionaalsplainidega histopoleerimisel. Töös on uuritud lineaar/lineaar ratsionaalfunktsioonidest ja ruutpolünoomidest kombineeritud splainidega monotoonsuse säilitamist ja ruut/lineaar ratsionaalsplainidega kumeruse säilitamist. Igasuguste algandmete korral on esitatud erinevad strateegiad osalõikude valikuks (kas ratsionaal- või ruutlõik), et säilitada võimalikult suures piirkonnas lähteandmete monotoonsust. Histopolandi tegelikuks leidmiseks tuleb selle parameetrite määramiseks lahendada mittelineaarne süsteem. Selleks sobivad Newtoni meetod ja harilik iteratsioonimeetod. On leitud monotoonsust säilitava meetodi koonduvuskiirus. Kumerust säilitava meetodi korral on raskeks probleemiks lahendi olemasolu küsimus, selleks on leitud piisavad tingimused. Toodud on hulgaliselt arvutuseksperimentide tulemusi ja illustreerivaid jooniseid.
The data needed to use in data analysis is often given as an histogram, i.e., averages of some variable on known (time)interval. For further analysis it is important them to be continuous. To do that several methods can be used. The most used method is the interpolation, the histopolation has not been used so often. Usually the preservation of geometrical properties of given data like positivity (nonnegativity), monotonicity, convexity, etc., is needed. It is generally known that interpolation and histopolation with polynomial splines do not preserve geometrical properties of data. The main research task in this thesis is to study the shape-preserving rational spline histopolation. Combined splines using linear/linear rational functions and quadratic polynomials on corresponding subintervals for preservation of monotonicity and quadratic/linear rational splines to preserve the convexity is studied. For any data, strategies to preserve monotonicity on maximal possible number of intervals are investigated. For actual construction of the histopolant a system of nonlinear equations has to be solved. Newton's method and ordinary iteration method are convenient for that. The convergence rate of such combined comonotone histopolating splines is established. An important and difficult point in research is finding sufficient conditions for the existence of convexity preserving histopolating rational spline. Several numerical examples and illustrative figures are presented.
The data needed to use in data analysis is often given as an histogram, i.e., averages of some variable on known (time)interval. For further analysis it is important them to be continuous. To do that several methods can be used. The most used method is the interpolation, the histopolation has not been used so often. Usually the preservation of geometrical properties of given data like positivity (nonnegativity), monotonicity, convexity, etc., is needed. It is generally known that interpolation and histopolation with polynomial splines do not preserve geometrical properties of data. The main research task in this thesis is to study the shape-preserving rational spline histopolation. Combined splines using linear/linear rational functions and quadratic polynomials on corresponding subintervals for preservation of monotonicity and quadratic/linear rational splines to preserve the convexity is studied. For any data, strategies to preserve monotonicity on maximal possible number of intervals are investigated. For actual construction of the histopolant a system of nonlinear equations has to be solved. Newton's method and ordinary iteration method are convenient for that. The convergence rate of such combined comonotone histopolating splines is established. An important and difficult point in research is finding sufficient conditions for the existence of convexity preserving histopolating rational spline. Several numerical examples and illustrative figures are presented.
Kirjeldus
Märksõnad
funktsioonid (mat.), lähendamine, splainid, functions (math.), approximation, splines