Identification and quantification of trends

Descriptive statistics and basement levels assesment

The p-value is the most important concept of statistical significance and therefore an applicability of the results of a (trend) test. The p-value represents the probability of an error when considering the real value of estimated parameter differs from the computed (or static) one, e.g., that the zero hypothesis holds although we considered the alternative one. Usually, if the p-value is under 5%, we accept the alternative hypothesis, because the risk of its invalidity is relatively low.

The very last step of the whole time series analysis consist of an assessment of its progress in time. There are two essential steps of the trend analysis - a test of a randomness of the trend (identification) and an estimation of its magnitude (quantification), if the trend is present and significant. Therefore, the statistical significance of these tests is of the highest importance, showing, whether the results can be used as tools for description of the processes in the area of monitoring.

There are three types of the test: parametric for normal and lognormal distribution (Pearson trend test and least squares regression) and nonparametric (other) for a universal use.

Together with the vizualisations provided in the first step, seasonality description computed in the fifth step, optional annual aggregation and descriptive statistics from the previous step, these tests cover an exhaustive characterization of POPs concentrations and their development. There are naturally more methods which could be applied on the data, especially for future predictions, but their usefulness is lower in comparison with the methods mentioned above, which should be obvious when evaluating POPs burden at any site.

Trend identification and quantification

There are two essential steps in the trend analysis. In the first one, it is necessary to estimate a probability of a randomness of a trend, in the second, the magnitude of the change in time is estimated.

Trend identification:

Pearson correlation trend test and its p-value

Parametric test of time trend based on the size of Pearson correlation coefficient computed as a moment statistic. The p-value represents a probability of the error when expecting, that the trend differs from zero (i.e. probability, that there is no time change and the value is based on random fluctuations only).

Daniels trend test and its p-value

Nonparametric test of time trend based on the size of Spearman correlation coefficient computed as a rank statistic. The p-value represents a probability of the error when expecting, that the trend differs from zero (i.e. probability, that there is no time change and the value is based on random fluctuations only).

Mann-Kendall trend test and its p-value

Parametric test of time trend based on the size of Kendall correlation coefficient computed as a rank statistic, which provides similar values as the previous one. The p-value represents a probability of the error when expecting, that the trend differs from zero (i.e. probability, that there is no time change and the value is based on random fluctuations only).

Trend quantification:

Least squares linear regression

Interlacing of the time series points by a straight line with the lowest sum of squared distances (in direction of y-axis) from all the points.

where xi denotes the time points of measurements as in the previous equations.

Theil-Sen regression

Nonparametric variant of the previous trend statistic computed as a median slope from all of the slopes of pairs of points in the time series (such that, the first point is predecessor of the second one in the pair).

Delta

Basic difference of the final and initial time point of the time series:

.

Previous step.

References

Sen, P. K., Estimates of the Regression Coefficient Based on Kendall's Tau. Journal of the American Statistical Association 1968, 63 (324), 1379-1389.
8: Trend analysis
8: Trend analysis