MATHHX A

MATHHX A

\(\newcommand{\footnotename}{footnote}\) \(\def \LWRfootnote {1}\) \(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\let \LWRorighspace \hspace \) \(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\newcommand {\mathnormal }[1]{{#1}}\) \(\newcommand \ensuremath [1]{#1}\) \(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \) \(\newcommand {\setlength }[2]{}\) \(\newcommand {\addtolength }[2]{}\) \(\newcommand {\setcounter }[2]{}\) \(\newcommand {\addtocounter }[2]{}\) \(\newcommand {\arabic }[1]{}\) \(\newcommand {\number }[1]{}\) \(\newcommand {\noalign }[1]{\text {#1}\notag \\}\) \(\newcommand {\cline }[1]{}\) \(\newcommand {\directlua }[1]{\text {(directlua)}}\) \(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\) \(\newcommand {\protect }{}\) \(\def \LWRabsorbnumber #1 {}\) \(\def \LWRabsorbquotenumber "#1 {}\) \(\newcommand 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#1.#2.#3\LWRsiunitxEND {\LWRsiunitxprintdecimalsubtwo #1,,\LWRsiunitxENDTWO \ifblank {#2}{}{{\LWRsiunitxdecimal }\LWRsiunitxprintdecimalsubtwo #2,,\LWRsiunitxENDTWO }}\) \(\newcommand {\LWRsiunitxprintdecimal }[1]{\LWRsiunitxprintdecimalsub #1...\LWRsiunitxEND }\) \(\def \LWRsiunitxnumplus #1+#2+#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxprintdecimal {#1}}{\ifblank {#1}{\LWRsiunitxprintdecimal {#2}}{\LWRsiunitxprintdecimal {#1}\unicode {x02B}\LWRsiunitxprintdecimal {#2}}}\LWRsiunitxdistribunit }\) \(\def \LWRsiunitxnumminus #1-#2-#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumplus #1+++\LWRsiunitxEND }{\ifblank {#1}{}{\LWRsiunitxprintdecimal {#1}}\unicode {x02212}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\def \LWRsiunitxnumpmmacro #1\pm #2\pm #3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumminus #1---\LWRsiunitxEND }{\LWRsiunitxprintdecimal {#1}\unicode {x0B1}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\def \LWRsiunitxnumpm #1+-#2+-#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumpmmacro #1\pm \pm \pm \LWRsiunitxEND }{\LWRsiunitxprintdecimal {#1}\unicode {x0B1}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\newcommand {\LWRsiunitxnumscientific }[2]{\ifblank {#1}{}{\ifstrequal {#1}{-}{-}{\LWRsiunitxprintdecimal {#1}\times }}10^{\LWRsiunitxprintdecimal {#2}}\LWRsiunitxdistribunit }\) \(\def \LWRsiunitxnumD #1D#2D#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumpm #1+-+-\LWRsiunitxEND }{\mathrm {\LWRsiunitxnumscientific {#1}{#2}}}}\) \(\def \LWRsiunitxnumd #1d#2d#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumD #1DDD\LWRsiunitxEND }{\mathrm {\LWRsiunitxnumscientific {#1}{#2}}}}\) \(\def \LWRsiunitxnumE #1E#2E#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumd #1ddd\LWRsiunitxEND }{\mathrm {\LWRsiunitxnumscientific {#1}{#2}}}}\) \(\def \LWRsiunitxnume #1e#2e#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumE #1EEE\LWRsiunitxEND }{\mathrm {\LWRsiunitxnumscientific {#1}{#2}}}}\) \(\def \LWRsiunitxnumx #1x#2x#3x#4\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnume #1eee\LWRsiunitxEND }{\ifblank {#3}{\LWRsiunitxnume #1eee\LWRsiunitxEND \times \LWRsiunitxnume #2eee\LWRsiunitxEND }{\LWRsiunitxnume #1eee\LWRsiunitxEND \times \LWRsiunitxnume #2eee\LWRsiunitxEND \times \LWRsiunitxnume #3eee\LWRsiunitxEND }}}\) \(\newcommand {\num }[2][]{\LWRsiunitxnumx #2xxxxx\LWRsiunitxEND }\) \(\newcommand {\si }[2][]{\mathrm {\gsubstitute {#2}{~}{\,}}}\) \(\def \LWRsiunitxSIopt #1[#2]#3{\def \LWRsiunitxdistribunit {\,\si {#3}}{#2}\num {#1}\def \LWRsiunitxdistribunit {}}\) \(\newcommand {\LWRsiunitxSI }[2]{\def \LWRsiunitxdistribunit {\,\si {#2}}\num {#1}\def \LWRsiunitxdistribunit {}}\) \(\newcommand {\SI }[2][]{\ifnextchar [{\LWRsiunitxSIopt {#2}}{\LWRsiunitxSI {#2}}}\) \(\newcommand {\numlist }[2][]{\text {#2}}\) \(\newcommand {\numrange }[3][]{\num {#2}\ \LWRsiunitxrangephrase \ \num {#3}}\) \(\newcommand {\SIlist }[3][]{\text {#2}\,\si {#3}}\) \(\newcommand {\SIrange }[4][]{\num {#2}\,#4\ \LWRsiunitxrangephrase \ \num {#3}\,#4}\) \(\newcommand {\tablenum }[2][]{\mathrm {#2}}\) \(\newcommand {\ampere }{\mathrm {A}}\) \(\newcommand {\candela }{\mathrm {cd}}\) \(\newcommand {\kelvin }{\mathrm {K}}\) \(\newcommand {\kilogram }{\mathrm {kg}}\) \(\newcommand {\metre }{\mathrm {m}}\) \(\newcommand {\mole }{\mathrm {mol}}\) \(\newcommand {\second }{\mathrm {s}}\) \(\newcommand {\becquerel }{\mathrm {Bq}}\) \(\newcommand {\degreeCelsius }{\unicode {x2103}}\) \(\newcommand {\coulomb }{\mathrm {C}}\) \(\newcommand {\farad }{\mathrm {F}}\) \(\newcommand {\gray }{\mathrm {Gy}}\) \(\newcommand {\hertz }{\mathrm {Hz}}\) \(\newcommand {\henry }{\mathrm {H}}\) \(\newcommand {\joule }{\mathrm {J}}\) \(\newcommand {\katal }{\mathrm {kat}}\) \(\newcommand {\lumen }{\mathrm {lm}}\) \(\newcommand {\lux }{\mathrm {lx}}\) \(\newcommand {\newton }{\mathrm {N}}\) \(\newcommand {\ohm }{\mathrm {\Omega }}\) \(\newcommand {\pascal }{\mathrm {Pa}}\) \(\newcommand {\radian }{\mathrm {rad}}\) \(\newcommand {\siemens }{\mathrm {S}}\) 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{\morecmidrules }{}\) \(\newcommand {\specialrule }[3]{\hline }\) \(\newcommand {\addlinespace }[1][]{}\) \(\def \LWRsiunitxrangephrase { \protect \mbox {to (numerical range)} }\) \(\def \LWRsiunitxdecimal {.}\)

4.5 Guide til fancy lineær regression

Vi slutter af med en oversigt over hvordan man laver lineær regression på et højere niveau end vi kender fra B-niveau. Det bemærkes at det er en lidt omstændelig proces, så i forbindelse med skriftlige prøver og eksamen anbefaler jeg, at man dropper modelkontrollen (eller i hvert fald begrænser den) medmindre opgaven ligger op til at man laver modelkontrol.

Simpel lineær regression
  • 1. Lav regression med Analysis Toolpak. Sørg for at du har sat flueben ved residualer og residualplot.

  • 2. Lav modelkontrol. Dvs. undersøg om residualerne er normalfordelte med middelværdi nul, har konstant varians og er uafhængige. I praksis vil modelkontrollen ofte give et lidt mudret billede, da det tit ser ud til at kravene ikke helt er opfyldt. Det betyder dog ikke at modellen ikke kan bruges, men det gør den mindre pålidelig - så du skal tage forbehold. Husk også at overveje om du vil fjerne outliers (hvorefter du så må lave en ny model).

  • 3. Opskriv den endelige model, skriv hvad de forskellige variable står for, og kommenter på modellens kvalitet ud fra din modelkontrol (inkl. \(R^2)\).

Multipel lineær regression
  • 1. Lav først en regression med alle variable med Analysis Toolpak.

  • 2. Du kan nu vælge om du opstille den fulde model eller en korrigeret model. Du skal altså beslutte om du vil vælge ikke signifikante variable fra. Der er ikke nogen fast regel der siger om du skal gøre det eller ej. Vi husker at en høj p-værdi ikke betyder, at variablen er irrelevant - det betyder, at vi ikke kan være sikre på den er relevant ud fra de data vi har. Måske har vi på forhånd en formodning om at den er relevant? Summa summarum, man skal tænke sig om inden man fjerner variable fra modellen. Så tænker I nu... åhhh hvad gør jeg til eksamen? Ja vi må håbe det fremgår af opgaven om det forventes at I fjerner dem. Hvis ikke, kan I evt. opstille både en fuld og en reduceret model. I den virkelige verden vil man typisk bruge en mere raffineret metode til variabelselektion end den som er præsenteret her. Hvis du beslutter, at du vil fjerne ikke-signifikante variable, så se om der er nogle variable som har en p-værdi på over \(5\%\) (eller hvad du nu har valgt som signifikansniveau). Hvis ja, så fjern den variabel som har den største p-værdi og lav en ny model. Dette gentager du, indtil alle p-værdierne er under signifikansniveauet. Overvej også om du vil fjerne outliers.

  • 3. Lav modelkontrol som ved simpel lineær regression. Men her skal du huske også at tjekke for korrelation mellem de forklarende variable med en korrelationsmatrix. Hvis der er korrelation, så overvej at fjerne variable. Hvis en variabel både er korreleret og ikke-signifikant, er der endnu mere grund til at fjerne den.

  • 4. Opskriv den endelige model, skriv hvad de forskellige variable står for, og kommenter på modellens kvalitet ud fra din modelkontrol (inkl. justeret \(R^2)\).

Øvelse 4.5.1

Du skal nu tage udgangspunkt i det samme datasæt som jeg har brugt som eksempel igennem sidste afsnit. Det er vedlagt her

  • a) Gennemføre grundig multipel regression efter opskriften ovenover.

Løsning 4.5.1

  • a) Vi har følgende model:

    \[\hat {y}=2{,}0 + 0{,}35x_1-14x_2,\]

    hvor \(\hat {y}\) er den forventede karakter, \(x_1\) er tiden brugt på lektier og \(x_2\) er fraværet. Pga. antallet af observationer er det svært at sige om residualerne er normalfordelte samt om variansen er konstant, men det ser ok ud. Vi kan se at residualerne har en middelværdi på 0. Da der ikke er tale om en tidsserie undersøger vi ikke for uafhængighed mellem residualer. Der er ikke korrelation mellem de to forklarende variable. Da vi yderligere har en høj determinationskoefficient konkluderer vi, at vi har fundet en model som beskriver data godt.