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3. Trend magnitude estimation

Source: vignettes/d-slope-estimation.Rmd
d-slope-estimation.Rmd

Why this matters

Trend tests and slope estimators answer different questions: the former assess statistical evidence for change, while the latter quantify its magnitude and rate over time.

Having established whether a trend exists in the previous vignette, this one turns to the second question: how large is it? slope_estimator() answers that question directly, independently of whether the trend reached statistical significance.

What slope_estimator() does

Trend magnitude describes how rapidly a variable changes over time. In sptrends, it is quantified through slope estimation and expressed in the units of the input variable per unit of time. slope_estimator() estimates one temporal slope for each valid raster cell and returns a raster representing the rate of change across the study area. Theil-Sen (TS) (Theil, 1950; Sen, 1968) is the default and generally recommended estimator because it provides a robust balance between computational efficiency and resistance to outliers.

Basic workflow

The example below applies the recommended Theil-Sen estimator to the complete annual NDVI raster series at its original spatial resolution. The map shows raw slope estimates without filtering by statistical significance.

years <- 1982:2023
ts <- slope_estimator(
  r, method = "TS", t = years,
  report = FALSE, verbose = FALSE
)
plot(ts)

Global map of raw Theil-Sen NDVI slopes

Understanding the results

Positive values indicate increasing trends, whereas negative values indicate decreasing trends. A slope of 0.02 represents a rate of change of 0.02 input units per unit of time. Unless the original variable is expressed as a percentage, slope values should not be interpreted as percentages. A slope represents a rate of change, not the total change over the complete study period.

Slope estimation is particularly useful as a complement to CMK: CMK evaluates the statistical evidence for a spatially contextual trend, whereas the slope quantifies its magnitude. However, CMK does not correct serial correlation internally, so temporal dependence should be diagnosed and treated when necessary before inference. The raw slope map should not be interpreted as a map of statistically significant trends; significance and multiple testing must be evaluated separately.

Choosing the main options

Method Robustness Relative computational cost Guidance
OLS Low Low Use when assumptions and efficiency justify it
TS (Theil, 1950; Sen, 1968) High Moderate Recommended general-purpose choice
RM (Siegel, 1982) Very high High Use when extreme contamination is plausible

OLS is fastest but sensitive to outliers. Theil-Sen (TS) usually provides the best balance between robustness and computation. Siegel’s repeated median (RM) is more resistant but substantially slower.

Common mistakes

  • Do not assume that OLS, TS and RM produce identical estimates; they can diverge substantially in the presence of outliers.
  • Do not interpret a slope estimate as statistically significant by itself; combine it with a trend test and multiple-testing correction for inferential interpretation (see trend-test vignette and multiple-testing vignette).
  • Do not ignore serial correlation when combining slope estimates with significance results; diagnose and treat temporal dependence when necessary (see prewhitening vignette).
  • Do not compare slopes expressed in different units without appropriate standardisation; supply actual observation times when measurements are irregularly spaced.

Next steps

Continue to vignette("e-fdr-correction") to decide which trend-test results remain reliable after testing many cells.

Further details

See ?slope_estimator for formulas, assumptions, computational costs, robustness, optional smoothing, validation and complete references.

References