**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 t_{n}(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 di****rection, 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 NINOIn numerical analysis, if

f(x)andg(x)are functions defined on , thecross covarianceoffandg, writtenfog,is the functionwhere

and, are the mean values of f and g.

Iff=g,z/z(0)is called theautocorrelation functionand 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.