$($a$)$ Please identify the appropriate data transformation methods for the following situations.

Give a brief description about your answers:

$[4]$ Consider a dataset containing information about student performance in two

subjects: Math and English. The Math scores range from $35$ and $200($mean $=88,$

standard deviation $=18),$ while the English scores range from $84$ to $112($mean $=$

$98\mathrm{.}6,$ standard deviation $=0\mathrm{.}95).$

For each feature, apply normalization $($transformed data has: $x^{\text{'}}in[0,1])$ and

calculate the new mean and new standard deviation of the normalized feature.

Compare their means and standard deviations. And

for each feature, apply standardization to it and show the range of transformed

data and compare their ranges.

$[4]$ During the design of an artificial neural network, we sometimes need to transform

a variable $x$ that has a range of $(-,)$ to an open set zin$(-1,1).$ Note that

$z$ monotonically increases as $x$ increases in this transformation. Please specify a

proper function for such transformation.

$($b$)$ In natural language processing $($NLP$),$ there are diverse ways to represent words such

as one$-$hot encoding, bag of words, $T{F}^{*}$IDF, and distributed word representations. In

one hot encoding, a bit vector whose length is the size of the vocabulary of words is

created, where only the associated word bit is on $($i$.$e$.,1)$ while all other bits are off $($i$.$e$.,$

$0).$ Here is a toy example: suppose there is a $5-$dimensional feature vector to represent

a vocabulary of five words: $[$king$,$ queen, man, woman, power$].$ In this case, 'king' is

encoded into $1,0,0,0,0,$ 'queen' is encoded into $0,1,0,0,0,$ etc. Due to the nature of this

representation, the feature vector encodes the vocabulary of a sentence where all words

are equally distant. On the other hand, in distributed word vectors, a real$-$valued

vector whose length is defined by some common properties of words is created, then

each word can be represented as a linear combination of the defined properties. Using

the toy example above, given a $3-$dimensional feature vector of $[$man$,$ woman, power$]$ as

the common properties, then words such as 'king', 'queen', 'man', and 'woman' could be

encoded into $0\mathrm{.}98,0\mathrm{.}1,0\mathrm{.}8$

$0,0\mathrm{.}99,0\mathrm{.}85$

$0\mathrm{.}9,0,0\mathrm{.}5,$ and $0,0\mathrm{.}97,0\mathrm{.}5,$ respectively.

In this case, if you subtract a vector of 'man' from a vector of 'king', and add a vector

of 'woman', then you will get a vector close to a vector of 'queen'.

$[4]$ What is a major advantage$/$disadvantage of one hot encoding as compared to

distributed word vectors. Briefly justify your answer.

$[4]$ What is a major advantage$/$disadvantage of distributed word vectors as com$-$

pared to one hot encoding. Briefly justify your answer.