JANUARY 2023 ASSESSMENT PERIOD

School of Engineering and Materials Science

Module code:

Module name:

Rubric:

I)   ASSESSMENT INSTRUCTIONS

●  Answers should be written legibly by hand or word-processed. All text, sketches and mathematics should be integrated into a single document, and submitted as a pdf.

●   Include a cover page (first page), detailing your student number, and the module code and name.

●   If you have a DDS (Disability & Dyslexia Service) cover sheet, include a scanned copy of that as the second page of your submission

During the assessment

●  Your submission must be your own work, and you must not break any of the rules in theAcademic Misconduct Policy. Do not collude with others. Do not ask anyone else to answer the questions for you. All such activities are expressively forbidden. You must answer questions entirely independently.

●   Please be aware that:

1.  All submissions will be subject to inspection for plagiarism and use of external tutoring support. Detection of such activity will constitute an assessment offence

2.  We will viva a percentage of students after each assessment to discuss the answers they have provided as confirmation this is your own work

●  Any assessment offences will be referred to the Academic Registrar for consideration by the Chair of the Academic Misconduct Panel, and may subsequently lead to a severe penalties, including expulsion.

●   Please ensure that you read the question rubric (above) carefully, and answer all the questions that you are expected to. Make sure it is clear which answer refers to which question.

●   If you answer more questions than specified in the rubric, only the first answers up to the number

required will be marked. Ensure you delete anything you do not wish to be marked before submission.

●  Any question regarding the assessment should be directed to the QMPlus module forum. We will staff these during UK working hours, so you can always expect an answer within 24 hours but may already   find the answer to your question within the forum post. We will only help with queries around clarity of the question paper and will not provide any guidance on the answers.

Submitting the assessment

●  Submit your answer as a single pdf document using the submission point on the QMPlus module page. Your filename should be the module code plus your student number. Leave yourself plenty of time for  the upload.

●   In case of difficulties with the upload, please email your submission to:

[email protected]c.uk  before the submission deadline, putting the module code and your student number in the email subject line.

Question 1

A supervised learning approach based on a stochastict learning algorithm is proposed to predict the taxiing time of arrival aircraft for an international hub airport. Due to the stochastic nature of the learning algorithm, the proposed supervised learning approach has been trained 20 times indepently on the same training data and tested on the testing data set in order to evaluate the performance of the proposed approach. The prediction accuracy measured by the Mean Absolute Error (MAE) for the whole testing data set, in minutes, are as follows:

0.8616    2.1604    4.3546    1.9278    3.9610     3.1240    4.4367    1.0391     2.8023

1.7922    5.9080    3.8252    4.3790     1.9418    2.5314    2.7275    4.0984    2.7221

3.7015    0.9482

Assuming that the MAEs are normally distributed with unknown mean, μ , and unkown variance σ2 ,

(a) Estimate μ and σ 2 ;          [6 marks]

(b) Find a two-sided 95% confidence interval for the mean, μ;             [8 marks]

(c) Find a two-sided 95% confidence interval for the variance , σ 2 ;             [9 marks]

(d) Find a two-sided 95% confidence interval for the standard deviation, σ .            [2 marks]

Question 2

Consider we have 5 sets of observed data (red points on the coordinate system)

(x, y) = [(0.0, 1.2);  (0.5, 2.2); (1.0, 3.5); (1.5,7.3); (2.0, 10.8)]

Evaluate the regression model as a second-order polynomial function:

y = 1 + a1x + a2x 2 by

(a) Formulate the regression model into a matrix form. and show the least Squares (LS) representation of a1  and a2 ;    [8 marks]

(b) Estimating the values of a1, a2  from the observed data by using the LS method;       [8 marks]

(c) Drawing  the  regression  model  on  the  coordinate  system  (Show  the  applied coordinates);         [6 marks]

(d) Validating the regression results by calculating the Mean Squared Error (MSE);          [3 marks]

Question 3

Consider we have 2 sets of observed data

(x, y ) = [(1.0, 0.6); (2.0, 0.7)]

Use gradient decent method to determine the one-dimensional neural network model:

with  by solving the following problems:

(a) Calculate the feed-forward outputs and the cost function value with w1  and w2   being initialized as w1  = 0.5 and w2  = 0.2 ;            [4 marks]

(b) Calculate the gradient of the cost function with respect to the weight  w2   for the first back-propagation step; [8 marks]

(c) Calculate the gradient of the cost function with respect to the weight  w1   for the first back-propagation step;  [9 marks]

(d) Calculate the updated weights w1  and w2  for the first back-propagation step under the learning rate λ = 1;  [4 marks]

where exp (0.1) = 1.105 ; The required cost function is

with y (k) being the model prediction.

Question 4

(1)  Condier three finite fuzzy sets of X defined by three linguistic qualifiers: positive big (PB),  positive  medium  (PM)  and  positive  small  (PS)  on  the  finite  universe  of  real numbers [0, 6]:

Find the membership function of

(a) PB or PM, PM and PS, Not PM, Not (PB or PM)   [4 marks]

(b) Consider the rule:

R: IF (error is PB) THEN (regulator position is reduced to PS).

For PB, PS defined above, find the relational matrix R corresponding to this rule.   [5 marks]

(2)  Condider the following fuzzy  partitioning of the variables ‘x’ and ‘y’  in a  normalised universe of discourse ‘x’ and ‘y’ respectively:

Let us assume the following set of 4 Takagi Sugeno-Kang (TSK) fuzzy rules:

Rule 1:   IF ‘x’ is NB AND ‘y’ is N THEN Z = a1x + b1  + c1y  

Rule 2:   IF ‘x’ is NM AND ‘y’ is N THEN Z = a2x + b2  + c2y 

Rule 3:   IF ‘x’ is Z     AND ‘y’ is Z THEN Z = a3x + b3  + c3y 

Rule 4:   IF ‘x’ is PM AND ‘y’ is P THEN Z = a4x + b4  + c4y  

(a) Is the above rule-base complete? Why?  [4 marks]

(b) Find the defuzzified values of the output ‘Z’ if the recorded crips values of ‘x’ and ‘y’ are:

●   x = −0.75 ANDY = −0.5

●   x = −0.25 ANDY = −0.75                                                 [12 marks]