1.What's the difference between PMI and PPMI, and why is PPMI preferred for NLP problems?

2.Complete the code to calculate the frequency of words that co-occur within a symmetric window (\(k =1 \)) for the given word.

3.When \(k = 2 \), what will be target word and context words in the \(k\)-window for the sentence "The Spanish galleon disappeared into the night"?

4.Explain the intuition behind TF-IDF.

5.Calculate TF-IDF for the documents in Figure A.

6.Complete the code to calculate the Euclidean distance. Do not use NumPy.

8.Complete the code to calculate the 2-norm (aka Euclidean norm) of a vector. Do not use NumPy.

9.Complete the code to normalize a vector to unit length using the 2-norm (aka Euclidean norm). Do not use NumPy for this.

10.What is meant if the cosine of two words is 1?

11.What's the cosine similarity between [-3, 3, 4] and [2, -5, -1]?

12.Cosine similarity

Give the formula for cosine similarity. Describe what it measures.

13.What's the cosine similarity between [-68, -92, 21, -19, 54] and [28, 32, 51, -78, -10]?

14.What's the cosine similarity between [86, -36, -34, -51] and [51, 67, 27, 52]?

15.What's the cosine similarity between [-53, 81, 96, 3] and [3, 49, 93, -1]?

16.What's the cosine similarity between [-76.81, 5.51] and [44.62, 36.53]?

17.Euclidean distance

Implement Euclidean distance using numpy:

$$
d_{2}(\mathbf{a}, \mathbf{b}) = \sqrt{\sum_{i=0}^{\vert \mathbf{a} \vert} (a_{i} - b_{i})^{2}}
$$

where \(d_{2}\) represents the Euclidean distance of vectors \(\mathbf{a}\) and \(\mathbf{b}\).

18.Dot product

Implement the function dot_product using numpy. Your function should take two np.ndarrays representing \(\mathbf{x}\) and \(\mathbf{y}\) and return a single number.

19.What is \([3, 9, 1] \cdot [0, 3, -4]\)?

20.Provide symmetric \(k\)-windows of size \(k=1\), \(k=2\), \(k=3\), and \(k=4\) for Shakespeare in the following sentence:

>the greatest Shakespeare play is Macbeth

21.What does a vector's 2-norm represent?

22.If \(\sum_{i=0}^{ \vert \mathbf{x} \vert } x_{i}^{2} = 1\), what can you say about \(\mathbf{x}\)?

23.When does \(\text{cos}(\mathbf{a}, \mathbf{a}) = \mathbf{a} \cdot \mathbf{b}\)?

24.Given \(\mathbf{a} = [7, 2]\), what vector has the same direction, but 3 times the magnitude of \(\mathbf{a}\)?

25.Find the centroid


Implement the function find_centroid using numpy. Your function should take one np.ndarray representing a multirow matrix X and return a vector representing the centroid.

26.Create a term-term matrix for the sentences in Figure A.

27.Create a term-term matrix for the sentences in Figure A.