Data Fundamentals (H)

2. Vectorised operations:

Do many operations to the same data items at once

Do operations without a defined order

Do the same operation on many data items at once

Do one operation on one number at a time

Do not work on numbers

3. Arrays have fixed:

shape and dtype

legs

variable names

elements

sums

4. An (8,4) array has:

8 rows, 1 column, 1 depth

8 rows, 4 columns

4 rows, 8 columns

4 legs, 8 arms

2 frames, 4 rows, 4 columns

5. x[::2, 1:5] is a slice which indexes:

Every row, all columns

Every column, rows 1-6

Every row, columns 1-5

Every other row, columns 1-5

Every row, columns 0-6

6. np.arange(1,5) produces the array:

[[1], [1,2], [1,2,3], [1,2,3,4], [1,2,3,4,5]]

[1,2,3,4]

[0,1,2,3,4]

[0,1,2,3,4,5]

[1,1,1,1,1]

7. x is (15,4). y is (15,). How would I add the values in y to each row of x?

x[:,0] * y

Impossible, shape mismatch

x + y[:,:]

x + y

(x.T + y).T

8. x is shape (3,3). What does the operation x[:, [1,0,2]] do?

Invokes the NumPy self-destruct sequence

Impossible, undefined operation

Returns a new array with the first two columns of x exchanged

Exchanges the exponent and mantissa of x

Returns a new array with the columns of x replaced with [1,0,2]

9. A strided array data structure means that:

transpose has time complexity O(N)

transpose cannot be performed without making a copy.

array legs are very long

values in arrays are in undefined order

transpose has time complexity O(1)

10. Not a number (NaN) is represented in IEEE 754 as:

a mantissa equal to the exponent

a sign of -3

an exponent of all 0s

an exponent of all 1s

a mantissa that evaluates to 666

11. Which of these is not a IEEE 754 floating point exception?

underflow

incomplete operation

invalid operation

overflow

inexact operation

12. x is (10,4) and y is (4,4). Which of these operations results in x and y joined into a (14,4) array?

np.join(x,y)

np.stack([x,y])

np.concatentate([x,y], axis=0)

np.concatentate([x,y], axis=1)

np.stack([x,y], axis=-1)

13. A line is a:

coord

layer

stat

facet

geom

14. Faceted and layered

faceted: set of small scatterplots; layered: a 3D plot

layered: multiple coords with the same geom; faceted: creates many geoms for same coord

faceted: stacks multiple plots on top of each other; layered: many separate views of one dataset

faceted: uses human faces to represent data; layered: uses the lasagne model of visualisation

faceted: many separate views of one dataset; layered: stacks multiple plots on top of each other

15.
Which of these things is missing from this plot?

[Figure: The heart mass of cats, plotted against the cats' body weight]

Axis labels

A title

Colours

A caption

A scale

16.
I want to compute

\( z = sum_i i^2 x_i^3 y_i \)

\(x\) and \(y\) are 1D vectors of the same shape. z will be a scalar. Which of these does that?

i = np.linspace(0, x.shape[0], 100) z = np.sum(x**3 * y * i**2)

i = np.ones_like(x) z = np.cumsum(x**3 * y * i**2)

i = np.arange(x.shape[0]) z = np.sum(x**3 * y * i**2)

z = np.sum(x**3 * y * x.shape[0]**2)

i = np.arange(x.shape[1]) z = np.sum(x**3) * np.sum(y) * np.sum(i**2)

17. x is shape (100,5,5,8). What does np.einsum('ijkl->likj', x) result in?

confusion

A (1, 100, 5, 5, 8) tensor

A (100, 100, 5, 5) tensor

A (8, 100, 5, 5) tensor

A (100, 8, 5, 5) tensor

18. The memory layout of numerical arrays is stored using:

A linked list data structure, with a hierarchy of pointers

Tiny snakes and ladders.

An array of strides specifying byte offsets to move along each dimension

Sentinel NaN values at the end of each row

An array map, that gives stores memory locations in an array of the same size

19. \(\|\vec{x}\|_\infty\) could be computed by which operation?

np.cumsum(x*x[::-1])

np.min(np.abs(x))

np.sqrt(np.sum(x**2))

np.max(np.abs(x))

np.sum(np.abs(x))

20. Which of these operations is not defined over vectors in a topological vector space equipped with an inner product?

inner product

length measurement

scalar multiplication

addition

square root

21. \(\vec{x}\bullet \vec{y}\) is zero if and only if:

\(x\) is orthogonal to \(y\)

\(x\) is nonzero

\(x\) is not equal to \(y\)

\(x\) is a scaled version of \(y\)

\(x\) is equal to \(y\)

22. Distances in high-dimension can be counter intuitive because:

Every vector will have a very similar distance to every other vector

Distances will span a huge range of possible values

Distances cannot be computed

Only the \(L_\infty\) norm can be applied.

There are so many different kinds of distances

23. The covariance matrix represents:

the colour of the dataset

the size of the largest element of a dataset

the cross product of the mean vector with itself

the spread of the dataset around its mean

the number of non-zero elements in a dataset

24. When rendering a graph with unsigned scalar values mapped to colours, what property should the colour map have?

Perceptually uniform, monotonically increasing brightness

Perceptually cuniform, moronically unceasing colour

Monotonic red-blue separation.

A diverging hue around zero.

Monotonic, perceptually nonuniform hue-saturation separation

25. Applying the linear map defined by the matrix \(A\) to the column vector \(\vec{x}\) should be written:

\(\vec{x}A\)

\(\vec{x}^TA\vec{x}\)

\(\vec{x}A\vec{x}^T\)

\(A\vec{x}^T\)

\(A\vec{x}\)

26. Repeatedly applying a matrix \(A\) to a random initial vector \(\vec{x}_0\), normalising after each step, will lead to:

infinity

the minor eigenvector

zero

the leading eigenvector

the cross eigenvector

27. If \(A\) is orthogonal, then:

\(A^A=T^A\)

\(A = AA^T\)

\(A^T=A\)

\(A^{-1} = A^T\)

\(A = A^{-1}\)

28. The adjacency matrix of an undirected graph is:

Self-similar

Non-singular

Symmetric

Circulant

Adiabatic

29. In a stochastic matrix:

the sum of all elements is -1

the sum of all elements is 1

all elements are either 0 or 1

the sum of each row is 1

the sum of each row 0

30. The determinant of a matrix is equal to:

The Frobenious norm of the matrix

The product of the eigenvalues

The sum of the eigenvalues

The product of the rows

The sum of the diagonal

31. I want to find the shape of an object, with constant surface area, that holds the least water. What is the objective function?

The colour of the surface.

The shape of the object.

The surface area of the object.

None of the above.

The amount of water the object holds.

32. A convex constraint is equivalent to a restriction to a portion of the parameter space:

where the parameter vector has a fixed \(L_\infty\) norm.

defined by a collection of planes.

within a torus of fixed radius.

inside an axis-aligned box.

where the minima are.

33. An objective function is nonconvex, iff:

It is incomputable.

It is partially differentiable.

It more than one minimum.

It is discontinuous.

It has two maxima.

34. The feasible set in an optimisation problem is:

the most distant configurations in the parameter space

a kind of metaheuristic

the possible configurations of the parameters

the possible values of the objective function

the best solutions to the problem

35. In an approximation problem, we'd often have a loss function of the form:

\(L(\theta) = \theta \vec{x}\)

\(L(\theta) = \|f(\vec{x};\theta)-y\|\)

\(L(\theta) = \|\theta - \vec{x}\|\)

\(L(\theta) = \frac{\theta}{f(\vec{x}-\vec{\theta})}\)

\(L(\theta) = \frac{1}{\theta}\)

36. The definition of an eigenvector is:

\(A\vec{x} = \lambda x\)

\(A\lambda = \vec{x}A\)

\(A\vec{x} = x\)

\(A^{-1}\vec{x} = A^{+}\lambda\)

\(\lambda = \|\vec{x}\|_2\)

37. Simulated annealing uses what metaheuristic to help avoid getting trapped in local minima?

A population of solutions.

Randomised restart.

Crossover rules.

Hill climbing.

A temperature schedule.

38. A hyperparameter of an optimisation algorithm is:

A value that is used to impose constraints on the solution.

A direction in hyperspace.

A measure of how good a solution is.

The determinant of the Hessian.

A value that affects how a solution is searched for.

39. First-order optimisation requires that objective functions be:

one-dimensional

\(C^1\) continuous

invertible

monotonic

disconcerting

40. The gradient vector \(\nabla L(\theta)\) is a vector which, at any given point \(\theta\) will:

be equal to \(\theta\)

point towards the global minimum of \(L(\theta)\)

point in the direction of steepest descent

have \(L_2\) norm 1

be zero

41. Finite differences is not an effective approach to apply first-order optimisation because:

the curse of dimensionality

the effect of measurement noise

of numerical roundoff issues.

none of the above

all of the above

42. Ant colony optimisation applies which two metaheuristics to improve random local search?

random restart and hyperdynamics

memory and population

thants

gradient descent and crossover

temperature and memory

43. For a multi-objective optimisation, Pareto optimality means that:

Every combination of the sub-objective functions has been searched.

Gradient descent is invalid.

All sub-objective functions are zero.

There is no possible improvement to any sub-objective function.

Any improvement in any sub-objective functions makes at least one other worse.

44. What property of a probability distribution always holds true?

The sum of all probabilities is 0

The determinant of probabilities is \(\infty\)

The product of all probabilities is 1

Probabilities are equally divided among outcomes

The sum of all probabilities is 1

45. Bayesians use probability as:

a calculus of belief

a prayer book

a representation of the long-term average of frequencies of outcomes

a calculus of truth

complex angles

46. The conditional probability P(A|B) is defined to be: (\(\land\) means "and" and \(\lor\) means "or")

\(P(A \lor B) + P(B \lor A)\)

\(P(A)P(B)\)

\(P(A \land B) P(B)\)

\(P(A||B) - B(A||P)\)

\(P(A \land B) / P(B)\)

47. If I have a joint distribution over two random variables \(A\) and \(B\), \(P(A,B)\), how can I compute \(P(A)\)?

\(P(A,B)âˆ’P(A|B)\)

Sum/integrate \(P(A,B)\) for every value of \(A\) and \(B\).

Divide \(P(A,B)\) by \(P(B)\)

Sum/integrate over \(P(A,B)\) for every value of \(B\)

Sum/integrate over \(P(A,B)\) for every value of \(A\)

48. In an optimisation problem, a penalty function can be used to:

implement genetic algorithms

accelerate gradient descent

issue red cards

implement soft constraints

reduce the need for random search

49. The entropy of a random variable \(H(X) = \sum_i -\log(P(x_i))P(x_i)\) is:

A measure of validity

A measure of surprise

A measure of dimension

A measure of likelihood

A measure of expectation

50. Bayes' Rule is:

P(A|B) = P(A) - P(A ^ B)

P(A|B) = P(A)P(B)

P(A|B) = P(B|A)P(A) / P(B)

P(A|B) = P(B|B)P(A|A)

P(A|B) = P(B|A)P(A)P(B)

51. The name for P(B|A) and P(A) in Bayes Rule are:

likelihood and prior

posterior and evidence

evidence and prior

likelihood and posterior

priory and little red riding hood

52. The expectation \(\mathbb{E}[X+1]\) for a discrete random variable \(X\) would be computed as:

\(\sum_x P(X=x+1)(x+1)\)

\(\int_x P(X=x)P(X=1)x\)

\(\sum_x P(X=x+1)\)

\(\sum_x P(X=x)(x+1)\)

\(\sum_x P(X+1=x)(x)\)

53. Which of these is a statistic which is an estimator of a population parameter for a normal distribution?

The probability

The sample mean

The entropy

The sample minimum

The bootstrap

54. Which of these is a nonparametric statistic?

mean

standard deviation

gradient

expectation

median

55. The Nyquist limit \(f_n\) is equal to:

1.0Hz

half the amplitude quantization levels

twice the sampling rate \(f_s\)

half the sampling rate \(f_s\)

twice the amplitude quantization levels

56. Decreasing the number of levels of amplitude quantization will have what affect on the sampled representation of a signal?

No effect

Decreased SNR

Decreased Nyquist rate

Frequency shift

Increased SNR

57. The exponential smooth is often used instead of a moving average because:

it is probabilistic

it is more numerically stable

it is super-quadratic

it is nonlinear

it requires storing/computing less data

58. Aliasing is caused by sampling signals with:

frequencies less than the Nyquist limit

undefined values present

noise levels greater than the maximum SNR

frequencies greater than the Nyquist limit

noise levels less than the maximum SNR

59. Along with a way to evaluate the likelihood and prior at any parameter setting \(\theta\), what else does Metropolis-Hastings need to sample from the posterior distribution?

A proposal distribution \(q(\theta^\prime|\theta)\)

An integration function \(V(\theta|D)\)

A way to evaluate the evidence \(P(\theta)[/theta]

A maximum likelihood estimation procedure.

The square root of 2.

60. In medical device, if you had an initial heart/pulse rate p0 and a 1D vector of changes in pulse rates captured at evenly spaced intervals, delta_p, how would you compute p, the pulse rate at each of these times?

p = np.prod(delta_p) * p0

p = p0 * delta_p

p = np.sum(delta_p) + p0

p = p0 + np.cumsum(delta_p)

p = p0 + delta_p[:]