DISCONTINUOUS PERIOD IN CYCLOGRAM ANALYSIS 59 



prominent in them. It was not at first called by that name but simply- 

 described as a lack of those short cycles which are so characteristic of scram- 

 bled or random data. It was used in 1922 and especially in 1932 to dis- 

 tinguish between random and natural sequences. It had even started the idea 

 of a natural cyclic unit in climatic change which might include several ter- 

 restrial units such as the year. Each climatic cycle (even though tempo- 

 rary) could possibly be called a natural unit of which several could exist at 

 the same time. In 1932 Dr. Alter had applied his lag correlations to certain 

 tree-ring data and reported strong conservation of 1 to 3 years. When I 

 proposed a natural unit of this sort to Dr. Bartels he referred to his <r, de- 

 scribed below, as his idea of a natural unit. 1 This a would become a test of 

 the presence of conservation and a measure of its effect on the data. 



From the viewpoint of probabilities (and following Dr. Bartels' discus- 

 sion) this conservation is the tying together of several successive values so 

 that they are not independent. It affects the reliability of periods suspected 

 in the data, which should be evaluated on the basis of the number of inde- 

 pendent terms. So we desire to find how many of our original terms must be 

 united into one term in order to make the values independent of each other. 



Bartels calls the conservation factor a and describes it (and therefore the 

 natural unit) as the equivalent length of the quasi-persistence. The Bartels 

 ratio, which we will call B, may be stated thus : 



M _B 



m 



in which M is the standard deviation of the mean; m is the average standard 

 deviation of the sets as before; N is the number of sets and B is usually more 

 than unity. In random data B is unity. Therefore to make our formula 

 produce unity on the right-hand side of the equation — as in random data — 

 we must divide each side of the equation by B and we get (placing B under 

 the radical) : 



M 



m 



= 1 



• 



N 

 B 2 



This B 2 is Bartels' a or the number of sets or unit values, whole or frac- 

 tional, that must be grouped together in order to get independent terms, for 

 it divides the number of sets N by a quantity over unity and reduces the 

 number of independent sets. Of course, we understand that all this becomes 

 more strictly true as the number of data or sets increases very greatly. 



For example, in an apparent 11-year cycle in Eberswalde trees (giving a 

 correlation coefficient of 0.51 ± 0.07 with the smoothed annual means of 

 the sunspot numbers) we find 7 sets with an average standard deviation of 



1 See more recent paper in Bibliography, Bartels, 1935. 2 



