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Example - Wheather Analysis

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I. Introduction

This program performs a very simple statistical analysis of some meteorological data, with a view to constructing a formula that could be used to estimate the daily temperature considered as a function of time of year.

II. Mathematics Involved

The most basic approximation in this example is that of the arithmetic mean, calculated in the normal way,

where n is the number of data points. The further two approximations use an approach which is essentially that of finding terms in a finite Fourier series. That is achieved by scaling the days in the year into a range of 0 to 2π, so that each data point x_i relates to some time value t_i in that range. The coefficients in the series can then be calculated using the formula:

where it is assumed that the value of a₀ is . Once a series of coefficients has been calculated it is then possible to calculate an estimate of a value from the population from which the x_i values were drawn. The approximation y_i is given by the formula:

The final stage is to compare the approximations with the real data. This is done by calculating the mean square of the difference between the approximation and real values, i.e.:

III. How the Program Works

The program uses data that has been read from a database file and is stored in two arrays, t% and d%.

   OPEN FILE "WEATHER"
   
   n%% = RECCOUNT ( FILE ):' number of data points
   DIM d%(n%%):'             dates 
   DIM t%(n%%):'             temperatures
     
   SET FORM USING 2
   SET FORM FG 1,1
   Superbase.StatusText = "Loading data..."
   
   i%% = 0
   SELECT FIRST 
   WHILE NOT EOF ( FILE )
     d%(i%%) = Date.WEATHER
     t%(i%%) = TempAtNoon.WEATHER
     i%% = i%% + 1
     SELECT NEXT 
   WEND 

Then as it goes through the stages of the analysis the program informs the user of its progress. The formulae above are all based on summing values over a range. This is simplified by using the Superbase.Summation() method, which sums the value of some expression over a specified range of values for some input variable.

   a0% = Summation("t%(i%%)","i%%",0,n%% - 1) / n%%

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   a1% = Summation("(t%(i%%)-a0%) * cos (2 * pi * ((d%(i%%)-days(s$+str$\
   (year(d%(i%%)),f$)))/366))","i%%",0,n%% - 1) / (n%% / 2)

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   a2% = Summation("(t%(i%%)-a0%) * cos (4 * pi * ((d%(i%%)-days(s$+str$\
   (year(d%(i%%)),f$)))/366))","i%%",0,n%% - 1) / (n%% /