1 (c) (ii) Is automatic differentation relevant to this problem? Justify your answer.

1 (d) (ii) There is a rule that if bubbles are present in the water, a diver will see approximately of them. What is the expected number of bubbles that the diver will see when a wild dolphin appears?

Write a Python expression that will compute this, again choosing any sensible variable names following the style in the question.

1 (c) (iii) Write down a NumPy function expression to penalise off-diagonal elements of a matrix , taking a parameter controlling the strength of the penalty.

Remember that you do not have to write fully working code to get full marks; marks are not deducted for minor errors.

1 (a) (iii) A new, high speed satellite receives the position data from a dolphin sensor every millisecond and has to to record dolphin position and speed. The dolphin sensor can report the current location and/or the change in location in the last millisecond, where and are reported in degrees. Speed is computed using the norm of the change in position. Dolphins typically move around 1mm in 1 millisecond. There are 11720000mm in one degree.

• Team A suggest sending just the position changes then running the following algorithm:

# assume p is initialised to an initial position 
# and record is an empty list
  ...
  while running:
    d = read_comms() # returns float32
    p = p + d  
    v = np.linalg.norm(d)
    record.append((p, v))

• Team B suggest sending both the position and change , and using:

  ...
  while running:
    p, d = read_comms() # returns two float32
    v = np.linalg.norm(d)
    record.append((p, v))

• Team C suggest sending just the position changes and using:

  last_p = p
  ...
  while running:
    d = read_comms() # returns float32
    p = p + d
    df = p - last_p
    last_p = p
    v = np.linalg.norm(df)
    record.append((p, v))

Rank these in order of numerical stability assuming computations are done in float32, and very briefly justify your ranking.

1 (d) (i) An member of the team goes diving. A wild dolphin appears! Bubbles are seen.

Write down a Python expression that will compute the probability that this was a common dolphin.

Use any sensible variable names of the style given above.

1 (c) (i) Team member 3 says that she doesn't trust np.cov and suggests "fitting an MV Gaussian with MLE and penalising off-diagonal elements". Explain what is being proposed, describe the form of the objective function and the effect of the penalty.

1(b)(i) A junior member of the team is excited about using the SVD. He proposes to compute of to and then to do a second SVD on to get to gain additional insights. Describe , justifying your explanation, and comment on the usefulness of this approach.

1 (b) (ii) Another member of the team thinks it would be interesting to compute the covariance matrix of this data. What is the shape of the covariance matrix? Would you expect the covariance matrix to be diagonal in this case? Why or why not?

1 (a) (ii) The team want to rearrange the hour of data into two minute chunks, with all of the dolphins across all pods combined in one axis, keeping just the longitude. For computational reasons they want to promote this to a five dimensional tensor with singletons in the first axes.

Write down a NumPy expression to do that on the shape you described above; assume the array is called d.

1 (a) (i) What would a suitable array shape be for a block of an hours' worth of dolphin data?