Autocorrelation Function

Version 1 Page 1 of 3 MA4 022 All Candidates Semester 2 Examinations 201 9 DO NOT OPEN THE QUESTION PAPER UNTIL INS TRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Mathematics Module Code MA4022 Module Title Data mining and neural networks Exam Duration (in words) Two Hours CHECK YOU HAVE THE CORRECT QUESTION PAPE R Number of Pages 3 Number of Questions 3 Instructions to Candidates Answer all questions. All answers will be taken into account. The questions 1- 3 carry almost equal weights (33, 33 and 34 marks). FOR THIS EXAM YOU ARE ALLOWED TO USE THE FOLLOWING: Calculators Permitted calculators are the Casio FX83 and FX85 models Books/Statutes provided by the University No Are students permitted to bring their own Books/Statu tes/Notes? No Additional Stationery No Version 1 Page 2 of 3 MA4 022 All Candidates 1. Time series [33 marks] 1a) [ 8 marks ] Give the definition of a stationary random process (in the strong and in the weak sense). Are the following random processes stationary: white noise, random walk, MA(1) process? Find the autocorrelation function for the MA(1) process. 1b) [ 8 marks ] Define the ARMA process in the general form. Give the conditions of stationarity and invertibility of suc h a process. Consider the model ���� (����) = ���� Version 1 Page 3 of 3 MA4 022 All Candidates 3d) [ 10 marks ] A PhD student analysed data about efficiency of video lectures. He has analysed performance of 300 students in Cambridge and 300 students from another university. The data are summarised in the table below. Do the video courses increase performance? According to the table for Cambridge, the answer is ‘No’. Similarly, the answer is negative for another university. However, ignoring the information about the university, and collating the data into one combined table, we find that video helps! Should we recommend the video courses? Create a Bayes net for this problem. How is it possible to avoid the ‘paradox’ by sampling? Cambridge 1st class Not 1st class % of 1 st class Video 120 80 60% Standard 75 25 75 % Another Uni 1st class Not 1st class % of 1 st class Video 10 90 10% Standard 30 170 15% Combined 1st class Not 1st class % of 1 st class Video 130 170 43.3% Standard 105 195 35% Table for Paradox END OF PAPER ����( ���� |���� ) = 0.4 ���� ( ���� | ~ ����) = 0.4 ���� ( ���� |���� ∧����) = 0.3 ���� ( ���� | ~ ���� ∧����) = 0.6 ���� ( ���� |���� ∧~����) = 0.1 ���� ( ���� | ~ ���� ∧~����) = 0.9 ���� ( ���� |���� ) = 0.1 ���� ( ���� | ~ ����) = 0.3 ���� ( ���� ) = 0.4 ���� ( ���� ) = 0.3 A B C D ����( ���� |���� ) = 0.8 ���� ( ���� | ~ ����) = 0.4 E F