Validation of the correlation between two variables by means of transition to a new coordinate system in the case when the correlation and regression coefficients are both small

A problem of estimating the significance of revealed trends appears in the analysis of long-term variations in the parameters of different processes that describe the state of geophysical phenomena and similar processes. If the time interval is limited, the range of variations in the studied values appears to be small. This causes small values of the correlation and regression coefficients. Such a situation is usually interpreted as a low significance level of the observed processes, which is not always true and a detailed analysis is needed.