MATHHX A
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4.2 Regression i Analysis Toolpak
Analysis Toolpak er en udvidelse til Excel. Man kan bruge den til at lave almindelig lineær regression, men man kan få flere informationer, end med den metode vi kender fra Mat-B.
Start med at installer Analysis Toolpak, hvis du ikke allerede har den. Du kan se hvordan her: https://support.office.com/en-us/article/load-the-analysis-toolpak-in-excel-6a63e598-cd6d-42e3-9317-6b40ba1a66b4.
Vi vil tage udgangspunkt i datasættet:
\(\begin {array}{ | c | c |} \hline x_i & y_i \\ \hline 1 & 2 \\ 3 & 5 \\ 5 & 4 \\ \hline \end {array}\)
Vi skriver tabellen ind i Excel og finder "Dataanalyse"(Under båndet "Data"):
Nu kommer der en dialogboks frem og alt efter hvilke bokse vi tjekker, kan vi få Analysis Toolpak til at bestemme forskellige relevante størrelser. Vi skal se på hvordan man bestemmer nogle af disse størrelser, men også hvad de
betyder. Vi vil kigge på følgende
Forskrift for regressionslinjen
I dialogboksen vælger vi "Regression"og i den efterfølgende dialogbox vælger vi vores x og y-værdier:
Vi finder nu tabellen:
Vi aflæser resultatet i kolonnen "Koefficienter". Det øverste tal ("Skæring") er \(b\) og det nederste ("X-variabel") er \(a\). I alt får vi
\[y=0{,}5x+2{,}17.\]
Denne forskrift er præcis magen til den vi ville finde hvis vi brugte den sædvanlige metode til at lave regression. Vi finder også vores \(R^2\):
Også her er der ikke noget nyt. Det er bare den goe gamle \(R^2\) vi har fundet. Altså \(R^2=0{,}43\).
Residualer og residualplot
Residualerne viser afvigelsen fra modellen til de faktiske data. Vi får følgende når vi sætter flueben i ”residualer” og ”residualplot”.
I tabellen til venstre ser vi residualerne og til højre er residualerne illustret med et residualplot. Læg mærke til at vi har x-værdierne fra data på x-aksen og residualerne ud ad y-aksen.
Betydningen af residualerne ses her:
De sorte punkter er vores data og de lodrette streger viser forskellen mellem model og data. Sammenligner vi tallene på tegningen med dem i tabellen ser vi at to af dem i tabellen har et minus foran. Det er fordi de ligger under
grafen og derfor er der en negativ afvigelse.
Vi skal se nærmere på residualerne i afsnit 4.4 om modelkontrol.
Konfidensintervaller
Vi skal nu se hvordan man bestemmer et konfidensinterval for \(\alpha \) i regressionsmodellen
\[y=\alpha x+\beta +\varepsilon \]
Helt konkret vil vi bestemme et 90%-konfidensinterval for \(\alpha \). Vi er desværre nødt til at starte forfra, da vi skal bruge andre indstillinger.
Vi finder grænserne for vores konfidensinterval her:
Vi får vores 90%-konfidensinterval til at være [-3,14;4,15]. Men hvad har vi fundet? Jo, vi har fundet ud af, at \(\alpha \) med 90% sandsynlighed ligger i intervallet [-3,14;4,15]. Som vi lærte i afsnit 4.1, er det jo ikke den rigtige hældning \(\alpha \) vi finder, når vi laver lineær regression, men bare vores bedste bud \(a\).
Vi bemærker at vores konfidensinterval indeholder 0. Det betyder, at selv om vi har bestemt \(a\) til at være positiv, og vi dermed har en voksende funktion, så kan vi ikke være sikre på, det forholder sig sådan i virkeligheden. For
hvis \(\alpha =0\), så er \(y\)-værdierne slet ikke afhængige af \(x\)-værdierne, og så er der jo ingen sammenhæng mellem \(x\) og \(y\). Så når vi har et konfidensinterval som indeholder nul, skal vi være forsigtige med at
konkludere noget ud fra modellen.
I eksamensopgaverne på hhx optræder tit formuleringen ”konfidensinterval for \(a\)”. Med det menes konfidensinterval for \(\alpha \).
Øvelse 4.2.1
Betragt data:
\(\begin {array}{ | c | c |} \hline x_i & y_i \\ \hline 1 & 2 \\ 3 & 5 \\ 5 & 4 \\ 6 & 7 \\ \hline \end {array}\)
Bestem følgende ved hjælp af Analysis Toolpak
-
a) En lineær model for data
-
b) \(R^2\)
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c) Residualer og residualplot
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d) Et 99%-konfidensinterval for hældningen i den lineære model
Løsning 4.2.1
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a) \(y=0{,}78x+1{,}58\)
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b) \(R^2=0{,}69\)
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c)
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d) \([-2{,}89;4{,}45]\)