SIO 221B, Data Analysis
Professor Sarah Gille

Homework #5, problem 3

I found the data set for pressure in Darwin, Australia from 1980 to 2002.  It is frequently 
used in the Southern Oscillation Index, pertaining to the El Nino cycle.  The data clearly 
has an annual trend, and appears to have some slower oscillation, as well.  First I 
removed the linear trend from the data by fitting a best-fit sinusoidal solution to the data.  
For my model, I used: 

M1: average trend	9.8830
M2: cos(2?t)		-3.3265
M3: sin(2?t)		-0.5529
   
The fit (figure 2) appears very good!  However, the L2 norm of the model misfit is 
colossal: 1,685.  (what are units here?)

Then I wanted to find what the longer frequency oscillation was.  The remaining signal 
did not offer many clues, so I made a massive model, using sin and cos with a period of 2 
to 20 years.  

The model parameters are:  for Cosine: 
 Period (y)	M
    2.0000    0.0671
    3.0000   -0.0967
    4.0000   -0.0156
    5.0000   -0.1597
    6.0000    0.0193
    7.0000    0.0441
    8.0000    0.0370
    9.0000   -0.0730
   10.0000    0.0207
   11.0000   -0.0101
   12.0000   -1.4716
   13.0000   -6.1314
   14.0000  -15.0349
   15.0000  -35.8864
   16.0000   95.2218
   17.0000 -112.3644
   18.0000 -104.2515
   19.0000  -51.9291
   20.0000    5.6438



model parameters for Sine: 
  Period (y)	M
    2.0000   -0.0967
    3.0000    0.0622
    4.0000    0.0250
    5.0000   -0.0167
    6.0000   -0.0806
    7.0000    0.1052
    8.0000    0.0606
    9.0000   -0.0985
   10.0000    0.1068
   11.0000   -0.2496
   12.0000    0.0171
   13.0000   -1.6673
   14.0000   14.6509
   15.0000  -39.2123
   16.0000   27.8723
   17.0000   68.7997
   18.0000  -52.8989
   19.0000   34.2019
   20.0000  -13.8460


It looks like the largest variation is in the period range of 16-18 years.  The fit of the 
model clearly doesn't represent the large fluctuations in pressure difference.   And the 
leftover data doesn't appear to have much periodicity remaining. 

The model misfit for this second model is now 1,510, about the same.