30 CLIMATIC CYCLES AND TREE GROWTH 



along the diagonals. Three factors are really needed to give the result: 

 the ellipticity of the dotted area, the semimajor axis, and the direction of 

 the major axis. (See page 91 for discussion of an example.) 



Periodogram — The periodogram is a summary of cycles found in a series 

 of data: the cycle lengths are commonly expressed along the axis of x between 

 limits that are evident. The ordinates decide the type of periodogram. In 

 Schuster's original form they were derived from amplitudes of the sine curves 

 involved in the analysis. Dr. Alter has used the name correlation periodo- 

 gram as applied to the curve of correlation coefficient values obtained by 

 comparing a curve with itself at lags of 1 year (or unit), 2 years, and so forth, 

 up to about half of the total number of terms. 1 With the large number of 

 cycles obtained in many curves of considerable length, we find it very con- 

 venient to use a diagram showing the number of occurrences of the various 

 cycle lengths: this is called a frequency periodogram. When this frequency 

 is determined separately for successive intervals of time as for successive 

 centuries, we are calling it a "progressive" or chrono-periodogram. (See 

 figs. 21 and 55.) 



GRAPHIC EXPRESSION OF CYCLES 



Cycle Integration or Summation — This has been the fundamental method 

 of testing for cycles in any succession of data. It consists in placing suc- 

 cessive blocks or sets of data one under the other, each just containing the 

 cycle length. To illustrate — suppose we have a series of values of sunspot 

 data (smoothed annual mean numbers, neglecting the fractions) as follows: 



We can test for a 10-year cycle by adding and averaging the columns as 

 they stand, and as a result we find no cycle of any consequence (see fig. 14a). 

 But we see in table 1 that the data have a somewhat suggestive distribution 

 with the larger numbers moving to the right in each successive set, and to 

 see it better we plot figure 14b in a crude quantitative distribution and pass 

 an estimated line through the maxima. If the best solution were 11 years, 

 the line of maxima would slant to the right 1 year for each decade. It does 

 more than that, for it changes to the right about 3 years for each 2 decades. 

 That gives us a double value in 23 years, or 11.5 years as a rough solution. 



Now if we wish to integrate these data on an 11.5-year period, we do it 

 by the following approximation, which is accurate enough for our purpose, 

 namely, to plot a fairly good average of the sun-spot cycle from 1833 to 

 1924. 



With this table before us we realize how much extra trouble it is to re- 

 arrange our numbers for testing each proposed cycle and especially to test a 

 fractional cycle, or one that does not consist of an even number of units. 

 This fractional test is especially irksome because the values really should be 

 interpolated. We could do all this very easily if we could make the additions 

 at any desired diagonal direction, including those giving fractional periods, 



1 Dr. Alter is now using a simpler coefficient in forming a periodogram, see Bibli- 

 ography, 1933 (3). 



