MATHHX A

MATHHX A

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#1.#2.#3\LWRsiunitxEND {\LWRsiunitxprintdecimalsubtwo #1,,\LWRsiunitxENDTWO \ifblank {#2}{}{{\LWRsiunitxdecimal }\LWRsiunitxprintdecimalsubtwo #2,,\LWRsiunitxENDTWO }}\) \(\newcommand {\LWRsiunitxprintdecimal }[1]{\LWRsiunitxprintdecimalsub #1...\LWRsiunitxEND }\) \(\def \LWRsiunitxnumplus #1+#2+#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxprintdecimal {#1}}{\ifblank {#1}{\LWRsiunitxprintdecimal {#2}}{\LWRsiunitxprintdecimal {#1}\unicode {x02B}\LWRsiunitxprintdecimal {#2}}}\LWRsiunitxdistribunit }\) \(\def \LWRsiunitxnumminus #1-#2-#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumplus #1+++\LWRsiunitxEND }{\ifblank {#1}{}{\LWRsiunitxprintdecimal {#1}}\unicode {x02212}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\def \LWRsiunitxnumpmmacro #1\pm #2\pm #3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumminus #1---\LWRsiunitxEND }{\LWRsiunitxprintdecimal {#1}\unicode {x0B1}\LWRsiunitxprintdecimal {#2}\LWRsiunitxdistribunit }}\) \(\def \LWRsiunitxnumpm #1+-#2+-#3\LWRsiunitxEND {\ifblank 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{#2}{\LWRsiunitxnume #1eee\LWRsiunitxEND }{\ifblank {#3}{\LWRsiunitxnume #1eee\LWRsiunitxEND \times \LWRsiunitxnume #2eee\LWRsiunitxEND }{\LWRsiunitxnume #1eee\LWRsiunitxEND \times \LWRsiunitxnume #2eee\LWRsiunitxEND \times \LWRsiunitxnume #3eee\LWRsiunitxEND }}}\) \(\newcommand {\num }[2][]{\LWRsiunitxnumx #2xxxxx\LWRsiunitxEND }\) \(\newcommand {\si }[2][]{\mathrm {\gsubstitute {#2}{~}{\,}}}\) \(\def \LWRsiunitxSIopt #1[#2]#3{\def \LWRsiunitxdistribunit {\,\si {#3}}{#2}\num {#1}\def \LWRsiunitxdistribunit {}}\) \(\newcommand {\LWRsiunitxSI }[2]{\def \LWRsiunitxdistribunit {\,\si {#2}}\num {#1}\def \LWRsiunitxdistribunit {}}\) \(\newcommand {\SI }[2][]{\ifnextchar [{\LWRsiunitxSIopt {#2}}{\LWRsiunitxSI {#2}}}\) \(\newcommand {\numlist }[2][]{\text {#2}}\) \(\newcommand {\numrange }[3][]{\num {#2}\ \LWRsiunitxrangephrase \ \num {#3}}\) \(\newcommand {\SIlist }[3][]{\text {#2}\,\si {#3}}\) \(\newcommand {\SIrange }[4][]{\num {#2}\,#4\ \LWRsiunitxrangephrase \ \num {#3}\,#4}\) 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{\morecmidrules }{}\) \(\newcommand {\specialrule }[3]{\hline }\) \(\newcommand {\addlinespace }[1][]{}\) \(\def \LWRsiunitxrangephrase { \protect \mbox {to (numerical range)} }\) \(\def \LWRsiunitxdecimal {.}\)

4.1 Lineære modeller

Lad os sige at vi vil bestemme sammenhængen mellem penge brugt på markedsføring og omsætning for virksomheder. Vi spørger nogle virksomheder og laver et xy-plot:

(image)

Det ser ud til at punkterne følger en lineær funktion, men med tilfældige afvigelser (op og ned) fra linjen:

(image)

I afsnittet om lineære regression (mat-b) ignorerede vi afvigelserne, og beskæftiget os kun med den lineære funktion. Nu skal vi se på en model, som også beskriver afvigelserne. Den ser således ud:

\[y=\alpha x +\beta +\varepsilon \]

Bogstaverne \(\alpha , \beta \) og \(\varepsilon \), er de græske bogstaver alfa, beta og epsilon. Vi kan se at \(y\) har form som en lineær funktion bortset fra \(\varepsilon \). Størrelsen \(\varepsilon \) er en normalfordelt stokastisk variabel med middelværdi 0. Denne stokastiske variabel giver afvigelserne fra linjen og derfor kaldes den også fejlleddet.

Estimeret lineær model

Når vi laver lineær regression, så antager vi først at data kan beskrives med den lineære model

\[y=\alpha x +\beta +\varepsilon \]

hvorefter vi estimere \(\alpha \) og \(\beta \). Estimaterne for \(\alpha \) og \(\beta \) kalder vi for \(a\) og \(b\), og ud fra dem kan vi opskrive den estimerede model:

\[y=ax+b+\varepsilon \]

Det er vigtigt at forstå, at vi egentligt er interesseret i \(\alpha \) og \(\beta \), men at det er umuligt at bestemme dem eksakt, fordi vores data er ”forurenet” med de tilfældige \(\varepsilon \)’er (afvigelser). Dvs. laver vi en ny undersøgelse af sammenhæng mellem markedsføringsbudget og omsætning, hvor vi spørger andre virksomheder, vil sandsynligvis finde en anden model, fordi de tilfældige afvigelser \(\varepsilon \) falder anderledes ud. Selvom vi ikke kan sige præcis, hvad \(\alpha \) og \(\beta \) er, kan vi dog bestemme konfidensintervaller for dem, hvilket vi kommer til at gøre i næste afsnit. Når vi skal bruge den estimerede model i praksis, dropper vi \(\varepsilon \) og skriver blot:

\[y=ax+b\]

... og så er vi tilbage til den goe gamle regressionsmodel fra mat-b. Grunden til at vi dropper \(\varepsilon \) er, at det er en stokastisk variabel som har middelværdi \(0\). Dvs. at vores bedste bud på dens værdi (for et konkret punkt på grafen) er \(0\). Vil vil i nogle sammenhænge skrive modellen som

\[\hat {y}=ax+b\]

Her skriver vi \(\hat {y}\), for at understrege, at der er tale om vores bedste bud på \(y\), og ikke den virkelige \(y\)-værdi (som svinger tilfældigt omkring linjen).