Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Curve Fitting Using Calculus in Normed Spaces
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2011 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

Curve fitting is used in a variety of fields, especially in physics, mathematics and economics.

The method is often used to smooth noisy data and for doing path planning. In this bachelor

thesis calculus of variations will be used to derive a formula for finding an optimal curve to fit a

set of data points. We evaluate a cost function (defined on the set of all curves

f on the interval

[

a; b]) given by F(f) =

R

b

a

(f00(x))2dx +

P

n

i

=1(f(xi) 􀀀 yi)2. The integral term represents the

smoothness of the curve, the interpolation error is given by the summation term and

> 0 is

defined as the interpolation parameter. An ideal curve minimizes the interpolation error and

is relatively smooth. This is problematic since a smooth function generally has a large interpolation

error when doing curve fitting, and therefore the interpolation parameter

is needed

to decide how much consideration should be given to each attribute. For the cost function

F

a larger value of

decreases the interpolation error of the curve. The analytical calculations

performed made it possible to construct a

Matlab program, that could be used to solve the

minimization problem. In the result part some examples are presented for different values of

.

The conclusion is that a larger value of the interpolation parameter

is generally needed when

using more data points and if the points are closely placed on the x-axis. Further on, a method

called Ordinary Cross Validation (OCV) is evaluated to find an optimal value of

. This method

gave good results, except for the case when the points could almost be fitted with a straight line.

Abstract [sv]

Kurvanpassning används inom flera olika ämnesområden, särskilt fysik, matematik och ekonomi.

Metoden används ofta vid anpassning av mätdata och vid banplanering. I denna kandidatexamensuppsats

används variationskalkyl för att ta fram en optimal kurva som passar mätdata. Vi

utvärderar kostnaden

F(f) =

R

b

a

(f00(x))2dx+

P

n

i

=1(f(xi)􀀀yi)2, som är definierad på mängden

av alla kurvor

f på intervallet [a; b]. Intergraltermen representerar kurvans släthet medan interpolationsfelet

ges av summatermen där

> 0 definieras som interpolationsparametern. En ideal

kurva minimerar interpolationsfelet och är relativt slät. En svårighet är att en slät funktion ofta

har en stor felkvadratsumma och därför används konstanten

för att bestämma vilken av de två

egenskaperna som ska väga tyngst. En ökning av värdet på

ger ett mindre interpolationsfel för

kurvan. Våra analytiska beräkningar gav oss en metod för att skriva om problemet, vilket ledde

till att ett

Matlab-program kunde konstrueras för att lösa minimeringsproblemet. I resultatdelen

presenteras exempel med olika värden på konstanten

. Slutsatsen är att det generellt

sett behövs ett högre

-värde när man använder många punkter och när punkterna ligger nära

varandra på x-axeln. Vidare i rapporten utvärderas Ordinary Cross Validation (OCV), för att

hitta ett optimalt värde på

. Denna metod gav bra resultat, förutom då punkterna nästan

kunde anpassas med en rät linje.

Place, publisher, year, edition, pages
2011. , 43 p.
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:kth:diva-105479OAI: oai:DiVA.org:kth-105479DiVA: diva2:571089
Uppsok
Technology
Supervisors
Available from: 2012-11-21 Created: 2012-11-21 Last updated: 2013-02-18Bibliographically approved

Open Access in DiVA

fulltext(831 kB)316 downloads
File information
File name FULLTEXT01.pdfFile size 831 kBChecksum SHA-512
938593c262bd27b233118e6799fecd4a337078f792939e008e6c192e8f638e363c5542ddce6a94ec84e2c178777a974ee70bb02b1326f187417a8f4f3326d7cd
Type fulltextMimetype application/pdf

By organisation
Optimization and Systems Theory
Engineering and Technology

Search outside of DiVA

GoogleGoogle Scholar
Total: 316 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 336 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf