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Sma2110:Real Analysis Ii Question Paper

Sma2110:Real Analysis Ii

Course:Bachelor Of Computer Science

Institution: Meru University Of Science And Technology question papers

Exam Year:2012

QUESTION ONE - (30 MARKS)
a) Briefly explain the objective of time series analysis. (4 Marks) b) Distinguish the following terms as used in time series analysis i. Weak and strong stationarity. (2 Marks) ii. Time domain and frequency domain. (2 Marks) c) Consider the following data containing 12 observations taken over time Time 1 2 3 4 5 6 7 8 9 10 11 12 Yt 71 70 69 68 64 65 72 78 75 75 75 70
Fit the trend using exponential smoothing technique with parameter = 0.1and compute the MSE. (6 Marks) d) Determine whether the following process is invertible and if so find the autocorrelation function. ???? = ?? + 0.7??-1 - 0.2??-2 (7 Marks) e) Let ???? = ?? + ??-1 + ??-2. Find the spectral density function of ???? and sketch the spectrogram. (7 Marks) f) Suppose ???? = 5 + 2?? + ???? where ???? is a zero when stationary process with autocovariance function ??. Find the autocovariance function of ????. (2 Marks)
2
QUESTION TWO (20 MARKS)
a) The number of distinguished guests visiting a certain hotel from 1994 to 2000 is given below: Time 1994 1995 1996 1997 1998 1999 2000 2001 Value 3 5 10 18 29 43 60 -
Determine which trend curve is more appropriate for this data, fit the data to the trend curve and forecast the trend value for 2001. (10 Marks) b) Given that ??3 = 14.8,??7 = 26.8 and ??11 = 218.8 and that ???? follows a modified exponential curve. Find ??12. (10 Marks)
QUESTION THREE (20 MARKS)
a) Find the autocorrelation function of the following function and check for stationarity. (10 Marks) ???? = 1 4 ????-1 + 3 64 ????-2 + 9?? b) If ???? = Asin ?? + where A is a random variable with mean zero and unit variance and is a random variable with uniform distribution on the interval ??,??independent of A. Show that ???? is weakly stationary. (10 Marks)
QUESTION FOUR (20 MARKS)
a) Suppose ???? = ei?? = cos?? sin??. Find the effect of a linear filter on ????. (11 Marks) b) Show that the spectral density function of (1) process ???? = ????-1 + ???? is given by ? ? 22 cos212)( ?? ? ? ??? hf . (9 Marks)
QUESTION FIVE (20 MARKS)
a) Consider the following AR(2) process ???? = 1 3 ????-1 + 2 9 ????-2 + ?? where ?? is a white noise process. Investigate whether the process is stationary and hence or otherwise find the autocorrelation function. (2 Marks)
b) The data below gives the average quarterly price of a commodity for 4 years. Year Q1 Q2 Q3 Q4 2007 40.3 44.8 46.0 48.0 2008 50.1 55.1 55.3 59.5 2009 47.2 50.1 52.1 55.2 2010 55.4 59.0 61.6 65.3 Totals 193 209 215 228 Average 48.25 52.25 53.75 57 Calculate the seasonal variation indices and adjust them if necessary. (8 Marks)

Sma2110:Real Analysis Ii Question Paper

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