Question $4/$Vraag $4$

You have a dataset containing $4$ measured

parameters of wine, namely acidity, sugar,

sulphates and alcohol content. Each wine also

has a quality score between $0$ and $10,$ based on

a professional tasting You wish to use muitiple

linear regression to predict the score of a wine,

for a given set of these parameters.

$($a$)$ The first wine in the dataset has

$\{7\mathrm{.}4,1\mathrm{.}9,0\mathrm{.}56,9\mathrm{.}4\}$ for the four pa$-$

rameters listed above. It has a quality

score of $5.$ The second wine in the dataset

has $\{11\mathrm{.}2,1\mathrm{.}9,0\mathrm{.}58,9\mathrm{.}8\}$ for the same

parameters, and has a quality score of $6.$

Write down the design matrix $x($includ$-$

ing a bias term$)$ and the target vector $y$ for

just these two data points.

$($b$)$ Next, you read $4000$ wines' data into the

design matrix. In your Python program

this is stored as a matrix variable X$.$ You

have also read all the target values into

the vector variable $y.$ Write Python$-$like

pseudocode to show how you would use

the normal equation to solve for the co$-$

efficients of the linear regression, w$.$ It is

not necessary to write syntactically cor$-$

rect Python code, but it should be clear

from your solution what you are trying to

do$.$

$($c$)$ Assume that, after performing multiple

linear regression, you find that hat$(w)=$

$\{16\mathrm{.}7,0\mathrm{.}284,-0\mathrm{.}518,-3\mathrm{.}63,-0\mathrm{.}727\}.$ You

are given a new wine that has parameters

$\{6,1\mathrm{.}9,0\mathrm{.}57,9\mathrm{.}5\}.$ What would the pre$-$

dicted quality score for this wine be$?$

U het $\text{'}$n datastel wat $4$ gemete parameters vas

wyn beval, naamlik suurvlak, suiker, sulfate en

alkeholinhoud. Elke wyn het ook $\text{'}$n kwaliteits $-$

punt tussen $0$ en $10,$ gebaseet op $\text{'}$is professio$-$

nele proesessie. U wil veelvuldige line$$re regres$-$

sie gebrulk om die kwaliteitspunt van $\text{'}$n wyn te

voorspel, vir $\text{'}$n gegewe stel van hierdie parame$-$

ters.

$($a$)$ Die cerste wyn in die datastel het

$(7\mathrm{.}4,1\mathrm{.}9,0\mathrm{.}56,9\mathrm{.}4\}$ vir die vier para$-$

meters wat hierbo gelys is$.$ Dit het $\text{'}n$

kwaliteitspunt van $5.$ Die tweede wyn in

die datastel het $\{11\mathrm{.}2,1\mathrm{.}9,0\mathrm{.}58,9\mathrm{.}8\}$ vir

dieselfde parameters, en het $\text{'}n$ kwaliteits$-$

punt van $6.$ Skryf die ontwerpsmatriks X

$($Instuitende die afsetterm$)$ en die teiken$-$

vektor $y$ vir net hierdie twee datapunte

nees.

$($b$)$ Volgende lees $44000$ wyne se data in die

ontwerpsmatriks in$.$ In u Python$-$program

word dit gestoor as $\text{'}n$ matriksverander$-$

like $x.$ U het ook al die teikenwaardes in$-$

gelees in die vektorveranderlike $y.$ Skryf

Python$-$styl$-$pseudokode om te wys hoe u

die normaalvergelyking sou gebruik om

op te los vir die ko$$ffisiente van die line$-$

$$re regressie, hat$(w).$ Dit is nie nodig om sin$-$

takties korrekte Python$-$kode te skryf nie,

maar die moet duidelik wees vanuit $u$ op$-$

lossing wat u probeer regkry.

$($c$)$ Aanvaar dat i na afloop van die

line$$re regressie vind dat hat$(w{)}^{*}=$

$16\mathrm{.}7,0\mathrm{.}284,-0\mathrm{.}518,-3\mathrm{.}63,-0\mathrm{.}727.$

$U$ word $\text{'}$n nuwe wyn gegee met para$-$

meters $\{6,1\mathrm{.}9,0\mathrm{.}57,9\mathrm{.}5\}.$ Wat sou die

voorspelde kwaliteitspunt vir hierdie wyn

wees?