Workflow diagram

Example

• Given 40 years of monthly ocean temperature data, calculate the linear trend line fit for each pixel to see where and how temperature changed over time.
• Given daily precipitation data collected over 10 years, use the harmonic trend line option and examine the R-squared goodness of fit statistic to see whether a seasonal trend is present in the data.

Usage notes

This tool can be used to fit data along a linear, harmonic, or polynomial trend line, or it can be used to perform trend detection using the Mann-Kendall or Seasonal-Kendall test.

The output trend raster generated with this tool is used as the input to the Predict Using Trend Raster tool.

The Mann-Kendall and Seasonal-Kendall tests are used to determine whether there is a monotonic trend in the data. They are nonparametric, meaning they do not assume a specific distribution of data The Mann-Kendall test does not consider serial correlation or seasonal effects. If the data is seasonal, the Seasonal-Kendall test is more appropriate.

There are three trend line options for fitting a trend to the variable values along a dimension: linear, harmonic, and polynomial.

• Linear—The linear trend line is a best-fit straight line that is used to estimate simple linear relationships. A linear trend highlights a rate of change that is increasing or decreasing at a steady rate. The formula for the linear trend line is as follows:

• Where:
  • y = the pixel's variable value
  • x = the dimension value
  • ß0 = the y-intercept
  • ß1 = the linear slope or rate of change
    • ß1 > 0 indicates an increasing trend
    • ß1 < 0 indicates a decreasing trend

• Harmonic—The harmonic trend line is a periodically repeating curved line that is best used to describe data that follows a cyclical pattern, such as seasonal temperature changes. The formula for the harmonic trend line is as follows:

• Where:
  • y = the pixel's variable value
  • t = the Julian date
  • ß0 = the y-intercept
  • ß1 = the rate of change
  • α, γ = coefficients of inter-annual or intra-annual changes
  • ω = 1 / i
  • f = the harmonic frequency

• Polynomial—The polynomial trend line is a curved line that is useful for data that fluctuates. In this case, a polynomial order value is used to indicate the maximum number of fluctuations that occur. The formula for the polynomial trend line is as follows:

• Where:
  • y = the pixel's variable value
  • x = the dimension value
  • ß0, ß1, ß2, ß3, ..., ßn = constant coefficients

If you perform linear, harmonic, or polynomial trend fitting, the output is a multidimensional imagery layer in which each slice is a multiband raster containing information about the trend line. If you are analyzing the trend for a single variable for a dataset containing a single dimension (for example, time), there will be a single slice in the output dataset. If you are analyzing a single variable for a dataset containing multiple dimensions (for example, time and depth), each slice will contain trend information for each dimension value along the dimension that was not included in the analysis.

For linear trend analysis, the output contains three-band rasters, where:

• Band 1 = Slope
• Band 2 = Intercept
• Band 3 = Root Mean Square Error (RMSE) or the error around the line of best fit

For harmonic trend analysis, the number of bands in the output depends on the harmonic frequency. When the frequency is set to 1, the output is a five-band raster, where:

• Band 1 = Slope
• Band 2 = Intercept
• Band 3 = Harmonic_sin1
• Band 4 = Harmonic_cos1
• Band 5 = RMSE

When the frequency is set to 2, the output is a seven-band raster, where:

• Band 1 = Slope
• Band 2 = Intercept
• Band 3 = Harmonic_sin1
• Band 4 = Harmonic_cos1
• Band 5 = Harmonic_sin2
• Band 6 = Harmonic_cos2
• Band 7 = RMSE

For polynomial trend analysis, the number of bands in the output depends on the polynomial order. Second-order polynomial fitting produces a four-band raster, where:

• Band 1 = Polynomial_2
• Band 2 = Polynomial_1
• Band 3 = Polynomial_0
• Band 4 = RMSE

Third-order polynomial fitting produces a five-band raster, where the following occurs:

• Band 1 = Polynomial_3
• Band 2 = Polynomial_2
• Band 3 = Polynomial_1
• Band 4 = Polynomial_0
• Band 5 = RMSE

If the tool is used to perform either the Mann-Kendall test or the Seasonal-Kendall test, the output is a five-band raster as follows:

• Band 1 = Sen's slope
• Band 2 = p-value
• Band 3 = Mann-Kendall score (S)
• Band 4 = S Variance
• Band 5 = Z-score

The outputs from the Mann-Kendall or Seasonal-Kendall test can be used to determine which pixels in your multidimensional time series have a statistically significant trend. You can use this information in conjunction with the linear, harmonic, or polynomial trend analysis to extract significant trends in your time series. You can generate a mask including pixels with significant p-values, apply the mask to the multidimensional raster, and use this masked multidimensional raster as the input to the tool to perform linear, harmonic, or polynomial trend analysis.

Model goodness-of-fit statistics can be generated as optional outputs for linear, harmonic, and polynomial trend rasters. The root mean square error (RMSE), R-squared, and the trend slope p-value can be calculated and displayed in the output raster using the RGB renderer and specifying the statistics as the red, green, and blue channels.

For information on how to publish multidimensional imagery layers, see Publish multidimensional imagery layers.

If Use current map extent is checked, only the pixels that are visible in the current map extent will be analyzed. If unchecked, the entire input imagery layer will be analyzed.

The parameters for this tool are listed in the following table:

Environments

Analysis environment settings are additional parameters that affect a tool's results. You can access the tool's analysis environment settings by clicking the gear icon at the top of the tool pane.

This tool honors the following Analysis Environments:

• Output coordinate system—Specifies the coordinate system of the output layer.
• Extent—Specifies the area to be used for analysis.
• Snap Raster—Adjusts the extent of the output so it matches the cell alignment of the specified snap raster layer.
• Cell size—The cell size to use in the output layer.
• Resampling method—The method to use to interpolate pixel values.
• Parallel processing factor—Controls the raster processing CPU or GPU instances.

Similar tools and raster functions

Use Generate Multidimensional Anomaly to calculate anomaly values for variables over time. Other tools may be useful in solving similar problems.