data analysis

Question 1

What is an orthogonal matrix?

Answer 1

In linear algebra, an orthogonal matrix or real orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors) Its transpose is equal to the inverse.

Question 2

How do you find the covarience matrix when given a set of vectors?

Answer 2

At the end you divide by N-1

Question 3

After finding a covarience matrix C, you might apply eigenvector decomposition so that C=UDU^T. What infomation can you find from U and D?

Answer 3

Columns of U are a new basis {u1, · · · , u_N } for R^N . Basis vectors u_n point in directions of maximum variance of X The eigenvector (element of D) d_n is the variance of the data in the direction u_n

Question 4

Suppose that each vector x ∈ X is replaced by its 1-dimensional projection x’ onto the direction defined in part (5). What is that average squared error between x and x’

Answer 4

Each point can be projected on the vector which gives the greatest varience.

There will be a certain varience between the orginal points and the projected ones.

The varience is given by the magnitude of the varience in the orthogonal direction, which is the other eigenvalue.

Question 5

Is translation a linear transformation?

Answer 5

No

Question 6

How would you use vector augmentation to perform a transformation?

Answer 6

Question 7

If you wanted to rotate a vector through an angle about point A, what would you do?

Answer 7

You would then do Ax where x is the vector you wish to rotate.

If it was a column vector like [1;2] you would add a 1 to make [1;2;1] so that it could be multiplied.

Question 8

What are all the DFT basis vectors in relation to each other?

Answer 8

Orthogonal. However they all have magnitude N^0.5 so it doesn’t act as a basis.

Question 9

If you had a 4*1 vector f, how would you find the DFT , F, of f?

Answer 9

Find the DFT basis vector for N=4

F=Df

Question 10

For a given n, how do you find the corresponding DFT basis vector D?

Answer 10

Question 11

What are some of the hints that make finding a DFT basis vector easier?

Answer 11

b₀ only consists of 1s

b_0.5*n goes 1,-1,1,-1,1…

Question 12

What is the frequency resolution?

Answer 12

sample rate / frame length

The frame length is the number of sample points, N

Question 13

What is the frame rate?

Answer 13

the reciprocal of the time betweent the sampling bands.

Question 14

How do you find the number of frames processed?

Answer 14

Length of the sample / frame rate

Question 15

How do you perform a reflection about a line at an ange to the horizontal?

Answer 15

Question 16

What does a markov process consist of?

Answer 16

A matrix whose rows consist of positive numbers that sum to 1 is called a stochastic matrix

Question 17

In a markov process, how can you find the probabilty of being in certain places at a time t?

Answer 17

p^k=A^tP^t-1

p^k=(A^t)^kp⁰

Question 18

On a markov transition matrix what does the a_ij entry denote?

Answer 18

The probabilty of going from i to j

Question 19

When you have a set of data, and centroids, how do you find some better centroids?

Answer 19

Question 20

How do you use the different distortion matrics e.g. d₁(u,v) d₂(u,v) and d∞ (u,v)

Answer 20

Question 21

How do you find the distortion from some central centroids ?

Answer 21

Question 22

Let X be a set of data points and C⁰ a set of K centroids in D-dimensional space. Questions: 1. Let C¹ be the set of K centroids after 1 iteration of K-means clustering applied to C⁰ , X and the Euclidean metric d₂. Is the centroid set C¹ globally optimal? In other words, is it true that for any set of K centroidsC, Dist(C¹, X) ≥ Dist(C¹ , X)?

Answer 22

. No. The centroid set that you obtain from one iteration of k-means clustering depends on the initial estimates of the centroids. Different initial estimates will give different solutions.

Question 23

Let X be a set of data points and C⁰ a set of K centroids in D-dimensional space:

Suppose t hat after n iterations the K-means clustering algorithm converges to a set C (in other words, applying K-means clustering to C with the data set X and the Euclidean metric d₂ does not change C). Is the centroid set C globally optimal (as defined above)?

Answer 23

. No. As in the previous part of this question, the solution that the k-means algorithm converges to will also depend on the initial estimates of the centroids. Remember the MatLab demonstration that I showed in the lecture (and which you can run yourself using the code on Canvas). The final set of centroids after 20 iterations of k-means clustering (the blue circles) depends on the initial estimates (black circles).

Question 24

How do you find the dot product with complex vectors?

Answer 24

Question 25

How do you find the magnitude of a complex vector?

Answer 25

Question 26

What is the Inverse Document Frequency?

Answer 26

N is the total number of documents

Question 27

What is the term frequency?

Answer 27

The Term Frequency (TF) f_t.d of word t relative to document d is the number of times t occurs in d

Question 28

How do you find the TF-IDF weight?

Answer 28

Question 29

What is the TF-IDF similarity between two documents?

Answer 29

Question 30

Answer 30

The 6th and 7th column are vec(d₁) and vec(d₂)

Csim(d₁,d₂) is the cross product of vec(d₁) and vec(d₂)

Question 31

How does the linear dependancy of vectors work?

Answer 31

You detmine if they are dependent if you put them in a matix and are able to gaussioan eleminate to just get numbers in the diagonal. Then they are independent

If they are dependent, then one of the vectors can be written as a combination of the others.

Question 32

What defines a basis?

Answer 32

All the vectors are orthogonal with each other and have length 1.

Question 33

How do you apply the Gram-Schmidt algorithm?

Answer 33

Question 34

How is the ampltiude spectrum related to F?

Answer 34

Question 35

How do you find the inverse DFT?

Answer 35

N^{-1 *}D’*F

The D’ is the complex concugate.

Question 36

Answer 36

Question 37

What would make a transfomation t linear?

Answer 37

Question 38

What is the definition of a basis?

Answer 38