LEAST SQUARES APPLICATION: EL NINO
Fourier series deal with functions on an interval. Once the coefficients have been estimated by the trapezoid rule the series only relias on a dicrete set of data. The truncated sum tn(x) with coefficients computed by the 2n+1 point trapezoid rule interpolates the original data. In some ways it is more useful than the "ordinary" series and thinking of it as an approximation should be avoided but rather as an entity in its own right.
El Nino is a typical application. Scientist have observed that in the southern Pasific the prevailing winds are from East to West. This causes the surface water to move in the same direction, resulting in an upwelling of lower water, colder water on the west coast of South America. The colder water is richer in nutrients and allows large numbers of marine organisms to thrive on the continental shell of these coasts. The coastal weather is also affected. The effect is not constant but varies during the year and weather and from one year to year in a more or less regular cycle called el Nino. As food production and weather prediction are of great economic importance scientists try to analyze these cycles. One way is by use of the "Southern Oscillation Index", the difference in atmospheric pressure between the Easter Island and Darwin Australia, measured at the same moment. There is one data point per month representing an average of a number of values, and it is thought of as occuring at the middle of the month. The below figure shows this index for the 14 year period 1962-1975. There are 168 points plotted at the half integers (representing mid-months) between 0 and 168.
Figure of the southern oscillation between
AUTOCORRELATION OF EL NINO
In numerical analysis, if f(x) and g(x) are functions defined on , the cross covariance of f and g, written f o g, is the function
and, are the mean values of f and g. If f=g, z/z(0) is called the autocorrelation function and often written . The division by z(0) amounts to a normalization so that -1<<1.
By the means of some numerical analysis, one may compute the discrete autocorrelation of
the El Nino data.