CYCLOGRAM ANALYSIS 29 



presence of abundant and lasting water supply; high sensitivity results from 

 its scarcity. It is evident that we have here a character figure for tree-growth 

 curves containing important information regarding environment. 



Correlation Coefficient — The popular method of expressing relationship 

 between two curves is the correlation coefficient. This is in essence the 

 ratio between the algebraic sum of the products of corresponding residuals 

 (departures from a mean) and the square root of the product of the sum of 

 their respective squares. It may be written: 



r = Sxy 



\/2x 2 -y 2 



Thus it may be stated as the average product of residuals divided by the 

 square root of the product of their average squares. 



In the correlation coefficient the values are taken as departures from a 

 mean or as "residuals." It makes no difference where the zero point is; for 

 instance, one set of residuals may average three times as large as the other. 

 One set of increments from each term to the next may be three times as 

 large as the other, and no change is produced in the coefficient. In tree- 

 ring work, this overlooks important features, for cross-identification depends 

 on large increments as compared to ring size and fails with small increments. 



The similarity between two trees curves then is only partly expressed by 

 a correlation coefficient. The reliability of a bit of cross-identification may 

 be estimated at over 90 per cent when the correlation coefficient is below 50 

 per cent. Complacent curves may show high correlation coefficient with 

 doubtful cross-identity. In general, however, they are not far apart. 



We have made various attempts to hit upon an index of similarity that 

 is more satisfactory than the correlation coefficient. Dr. Glock has used a 

 "trend coefficient" reached by dividing the sum of the positive products of 

 corresponding increments from two curves by the sum total of positive and 

 negative products disregarding signs. This is easier to compute than the 

 correlation coefficient, but here again one set of increments could be increased 

 or diminished in a definite ratio without affecting the result. 



A simple trend comparison, occasionally made use of by ourselves and 

 others, is the percentage of cases of agreement in rising or falling value of the 

 increment. It gives a fair idea of similarity. 



One of the best devices is the scatter diagram using original values from 

 zero and arranging the vertical and horizontal scales so that the mean values 

 of each curve come near the center of the diagram and the zeros come at the 

 lower left corner. This arrangement appears automatically if each curve 

 has been standardized; in fact the comparison is usually made between 

 standardized curves. Departures from a mean may be used with equalized 

 average residuals. 



The diagram resulting from such plotting of standardized curves gives a 

 ready and vivid picture of the points of similarity between two curves by 

 the shape of the cloud of dots, especially its departure from the circular form 



