58 CLIMATIC CYCLES AND TREE GROWTH 



A difficult case, sometimes occurring in natural data, is well brought out by 

 this method. When the 180°-phase reversals occur at regular intervals, as in 

 figure 27f, the pattern obviously contains an interference between two 

 periods, of which one takes the definite direction AB and the other may be 

 either CD or EF with "infection," as Bartels has called it, from AB. A 

 pattern of this type was encountered in a Southern California rainfall cycle 

 near 2 years in length, that could only express itself in winter and thus clings 

 a bit to the annual value. In the cyclogram obtained, the interfering cycle 

 could be read at once as 1.8 years or 2.2 years and the 1-2-year cycle could be 

 stated as a 2-year cycle with regular phase reversals or a 1-year cycle with 

 alternating intervals of appearance and suppression. 



Evaluation of Quasi-Persistence — Bartels has done more than recognize 

 QP's; he has arranged a criterion 1 which may be applied to express their 

 weight. This is derived through a principle of probability. In setting up a 

 correct average under normal distribution, the probable error of the average 

 is inversely proportional to the square root of the number of terms; the 

 standard deviation is the square root of the mean square of the residuals. 

 This may be applied to separate random values which are to be averaged into 

 a mean but it may be applied equally to separate random sets of values such 

 as a cycle length, commonly used in the summation process in deriving the 

 form of a cycle that exists in a series of data. A "set" is a row of data cover- 

 ing one period of the period length considered. Bartels compares the stand- 

 ard deviation of the mean of the sets with a theoretical value derived from the 



average standard deviation of a single set, and uses the —^ relation stated 



vN 



above for estimating the random part of the final values. Thus, if we have 

 N random sets of values in which the first set shows a standard deviation of 

 mi and the second of m 2 , etc., the average standard deviation m of the sets is 

 the square root of the fraction 2m 2 n /N. The mean of the N random sets of 

 values theoretically would have an average standard deviation of m/VN. If, 

 therefore, we divide the actual standard deviation of the mean by this theoret- 

 ical value for random data, we get unity as a ratio in case the data are 

 really random values and a sufficiently large number are used; if, however, 

 the values are not random but tend to show the same cycle in each set, the 

 standard deviation of the mean is larger than this quantity for random values 

 and on being divided by it gives a ratio over unity, which for the moment we 

 are calling the Bartels A.D. ratio or the A.D. ratio. Such a ratio, therefore, 

 becomes a measure of the reality of a cycle when compared with unity on one 

 side and with VN on the other. This latter is the maximum value it can 

 reach in the case of a cycle with no random values. 



The Conservation Factor 2 — Conservation in natural sequences was recog- 

 nized by the writer very early in the study of tree-ring records because it is 



1 C. F. Marvin published in 1921 his "periodocrite" test to which Bartels' criterion 

 bears a relation. 



2 In statistical studies conservation refers to correlations between successive values 

 without regard to the physical cause. 



