def my$\_$pca$($X$,$ n$\_$components$)$:

$\text{'}\text{'}\text{'}$

Performs PCA on X to reduce it to $`$n$\_$components$`,$ using the method

described in lecture but with the 'centering' of X before SVD is done.

Parameters

$----------$

X: np$.$darray

An $`$m $*$ n$`$ matrix where $`$m$`$ is the number of samples and $`$n$`$ is the

number of features which each sample has. In other words, the $`$i$`$th sample

is given by $`$X$[$i$]`.$

n$\_$components: int

No$.$ of components that the reduced space has.

Returns

$-------$

The tuple $`($components$,$ singular$\_$values$)`.$ Here, $`$components$`$ is an

$`$n$\_$components $*$ n$`$ matrix such that $`$components$[$i$]`$ returns the $`$i$`$th

principal axis that has the $`$i$`$th largest singular value. In addition,

$`$singular$\_$values$`$ is a $1$D numpy array with $`$n$\_$components$`$ entries such that

$`$singular$\_$values$[$i$]`$ returns the $`$i$`$th singular value.

Note

$----$

'centering' here refers to subtracting the mean from X such that the resulting

X$\text{'}$ has a mean of $0$ for each feature.

$\text{'}\text{'}\text{'}$

# TODO: add your solution here and remove $`$raise NotImplementedError$`$

# no loop allowed

X$\_$centered $=$ X $-$ np$.$mean$($X$,$ axis$=0)$

# Performing SVD

U$,$ S$,$ Vt $=$ np$.$linalg.svd$($X$\_$centered.T$)$

# Extracting the first n$\_$components principal components

components $=$ Vt$[$:n$\_$components, :$]$

# Extracting the corresponding singular values

singular$\_$values $=$ S$[$:n$\_$components$]$

return components, singular$\_$values

X$\_$test$\_1=$ np$.$array$([[0\mathrm{.}16550775,-16\mathrm{.}57982642,-50\mathrm{.}13603715,-20\mathrm{.}68348877,8\mathrm{.}80989065],[27\mathrm{.}74489674,17\mathrm{.}10507978,18\mathrm{.}71626655,32\mathrm{.}02624683,20\mathrm{.}99010911],[18\mathrm{.}05469443,-43\mathrm{.}28464778,31\mathrm{.}50703811,-18\mathrm{.}97769479,0\mathrm{.}44841138],[-45\mathrm{.}96509892,42\mathrm{.}75939443,-0\mathrm{.}08726752,7\mathrm{.}63493673,-30\mathrm{.}24841115]])$

expected$\_$pca $=$ my$\_$pca$($X$\_$test$\_1,2)$

# Public test case $1$

assert$($not np$.$all$($expected$\_$pca$[1]==$ np$.$array$([7282\mathrm{.}76644566,4817\mathrm{.}22025846])))$

# Public test case $2$

assert$($not np$.$all$($expected$\_$pca$[1]==$ np$.$array$([2427\mathrm{.}58881522,1605\mathrm{.}74008615])))$

# Public test case $3$

diff$\_$singular$\_$values $=$ np$.$abs$($expected$\_$pca$[1]-$ np$.$array$([1820\mathrm{.}69161142,1204\mathrm{.}30506462]))$

print$("$Difference in singular values:", diff$\_$singular$\_$values$)$

assert$($np$.$all$($diff$\_$singular$\_$values $<0\mathrm{.}000001))$

# Public test case $4$

X$\_$test$\_4=$ np$.$array$([[63\mathrm{.}69616873,26\mathrm{.}97867138,4\mathrm{.}09735239,1\mathrm{.}65276355,81\mathrm{.}32702392],[91\mathrm{.}27555773,60\mathrm{.}66357758,72\mathrm{.}9496561,54\mathrm{.}36249915,93\mathrm{.}50724238],[81\mathrm{.}58535541,0\mathrm{.}27385002,85\mathrm{.}74042766,3\mathrm{.}35855753,72\mathrm{.}96554464],[17\mathrm{.}56556206,86\mathrm{.}31789223,54\mathrm{.}14612202,29\mathrm{.}97118905,42\mathrm{.}26872212]])$

diff$\_$singular$\_$values$\_4=$ np$.$abs$($my$\_$pca$($X$\_$test$\_4,2)[1]-$ np$.$array$([1820\mathrm{.}69161142,1204\mathrm{.}30506462]))$

assert$($np$.$all$($diff$\_$singular$\_$values$\_4<0\mathrm{.}000001))$

# Public test case $5$

X$\_$test$\_5=$ np$.$array$([[0\mathrm{.}22733602,0\mathrm{.}31675834,0\mathrm{.}79736546,0\mathrm{.}67625467,0\mathrm{.}39110955,0\mathrm{.}33281393,0\mathrm{.}59830875],[0\mathrm{.}18673419,0\mathrm{.}67275604,0\mathrm{.}94180287,0\mathrm{.}24824571,0\mathrm{.}94888115,0\mathrm{.}66723745,0\mathrm{.}09589794],[0\mathrm{.}44183967,0\mathrm{.}88647992,0\mathrm{.}6974535,0\mathrm{.}32647286,0\mathrm{.}73392816,0\mathrm{.}22013496,0\mathrm{.}08159457],[0\mathrm{.}1598956,0\mathrm{.}34010018,0\mathrm{.}46519315,0\mathrm{.}26642103,0\mathrm{.}8157764,0\mathrm{.}19329439,0\mathrm{.}12946908]])$

diff$\_$singular$\_$values$\_5=$ np$.$abs$($my$\_$pca$($X$\_$test$\_5,3)[1]-$ np$.$array$([0\mathrm{.}14205335,0\mathrm{.}06453807,0\mathrm{.}0472502]))$

assert$($np$.$all$($diff$\_$singular$\_$values$\_5<0\mathrm{.}000001))$

# Public test case $6$

X$\_$test$\_6=$ np$.$array