Mann-Kendall Trend Test and Sen's Slope Estimation

The Mann-Kendall trend test and Sen's slope estimation are two classical non-parametric statistical methods widely used for trend analysis and change rate quantification in time series data. These methods require no strict assumptions about data distribution, effectively handle outlier interference, and are applicable in fields such as meteorology, hydrology, and environmental monitoring.

Mann-Kendall Trend Test This method determines trend significance by comparing the relative magnitude relationships between data points in a sequence. The core principle states that if a series shows an upward trend, later data points should generally be larger than earlier ones. The test statistic is constructed based on sign tests, with trend significance determined by a standardized Z-value (e.g., Z>1.96 indicates significant upward trend at 5% significance level). In MATLAB implementation, the algorithm typically involves pairwise comparison loops using nested for-loops to count concordant/discordant pairs, with tie-handling adjustments using functions like tiedrank.

Sen's Slope Estimation Used to quantify the rate of trend change, this method calculates the median of slopes between all data point pairs, producing robust results resistant to extreme values. A positive Sen's slope indicates an upward trend, while negative values signify downward trends. Code implementation often uses vectorized operations with meshgrid or nchoosek functions to efficiently compute all possible pairwise slopes, followed by median calculation while ignoring NaN values through nanmedian.

Standard MATLAB implementation typically follows these logical steps: Data preprocessing: Handle missing values using interpolation or deletion methods like isnan detection to ensure time series continuity. Mann-Kendall statistic calculation: Iterate through all data pairs using combinatorial approaches, count instances where later values exceed earlier values, and apply tie correction factors. Significance testing: Compute variance based on series length using variance formulas incorporating tie adjustments, derive Z-values, and compare against critical values from standard normal distribution. Sen's slope calculation: Compute slopes for all valid data pairs through (y_j-y_i)/(x_j-x_i) operations, then determine median value as final estimate using robust statistical functions.

These methods are frequently combined—Mann-Kendall test determines trend existence while Sen's slope quantitatively describes trend magnitude. Practical applications require attention to serial autocorrelation effects, which may necessitate modified approaches like pre-whitening procedures using autocorrelation function (acf) analysis before applying standard tests.