Creating maps using local polynomial interpolation

Available with Geostatistical Analyst license.

Local polynomial interpolation is not an exact interpolator (that is, the surface is a best fit to the data, but does not pass through all the data points). It produces a smooth surface and is best suited to data that exhibits short-range (local) variation.

1. Click a point layer in the ArcMap table of contents that contains the attributes in which you are interested.

   Alternatively, go directly to Step 2 and browse to the dataset you are interested in on the first page of the Geostatistical Wizard.

2. Start the Geostatistical Wizard
3. Under the Methods section, choose Local Polynomial Interpolation, which is located under Deterministic Methods.

   The lower portion of the Geostatistical Wizard shows information about Local Polynomial Interpolation.

4. Under the Input Data section, you will see that Source Dataset has been set to the layer you clicked on in the ArcMap table of contents. Under Data Field, select the attribute that you want to interpolate.

   In addition, you can specify a Weight field. This will weigh the data values and alter the interpolated surface. Including a Weight can be a useful option when you want to include a measure of confidence in the data (for example, GPS locations taken inside a forest may be less reliable than those taken in clear areas, so you may choose to assign them less weight in the interpolation).

5. Click Next.
6. LPI offers two ways to create a surface: one uses a reduced set of options while the second offers full control over all the parameters. By default, the advanced options are turned off and are inaccessible until the Advanced Mode option is changed from False to True.
   • In Default mode, Advanced mode = False:
     • Exploratory Trend Surface Analysis is a slider bar that controls the Major semiaxis, Minor semiaxis and Bandwidth parameters simultaneously. For more on the search neighborhood parameters, see Altering the search neighborhood by changing its size and shape and Altering the search neighborhood by changing the number of neighbors. The smaller the value of the Exploratory Trend Surface Analysis (by typing in a small value or moving the slider to the left), the larger the values of these three parameters. At the extreme, when the Exploratory Trend Surface Analysis value is zero, LPI produces the same output as Global Polynomial Interpolation because all the data in the dataset is used to fit a surface constructed from one (global) polynomial. On the other extreme, when the Exploratory Trend Surface Analysis value is 100, a very small (local) subset of the data will be used to fit a short-range polynomial, which generates a surface that honors much of the local variation in the data values.
     • The Order of Polynomial can be varied between 1 and 10, although values above 3 are not recommended for most situations.
     • A Kernel Function can be chosen from: exponential, Gaussian, quartic, Epanechnikov, PolynomialOrder5 and Constant.
     • Result Type is limited to Prediction only in Default mode.
     • Goodness of Fit reports a goodness of fit statistic that indicates how well or poorly the model performs. The lower the value, the better the model fit to the data. This value should be complemented with the cross validation results shown on the next dialog of the wizard.
     • The Optimize button will find parameter values that result in the model with the lowest Root Mean Square error. Note that options only available in Advanced mode (Bandwidth, Spatial Condition Number Threshold and the Search neighborhood options) will be modified during the optimization process.
   • Advanced mode set to True:
     • The Exploratory Trend Surface Analysis is turned off.
     • Order of Polynomial and Kernel Functionare available, as in the Default mode.
     • Result Type can be set to Prediction (to preview and generate a map of predicted values), Prediction Standard Error (to assess the uncertainty associated with the predicted values) or Condition Number (to preview and generate a map of the Spatial Condition Number, which is a measure of the stability of the predictions).
     • Search neighborhood options can be adjusted (refer to Altering the search neighborhood by changing its size and shape and Altering the search neighborhood by changing the number of neighbors to see how to modify the number of neighbors and the shape of the search neighborhood). Anisotropy (directional influences present in the phenomenon that the data represents) can be accounted for at this stage by modifying the Major semiaxis, Minor semiaxis and Angle parameters.
     • Neighborhood type can be changed from Standard to Smooth. In this case, Maximum neighbors, Minimum neighbors and Sector type are replaced by a Smoothing factor, and the method will produce a smoother surface. Refer to smooth interpolation for more details.
     • Bandwidth can be manually set. There is also an option to find an optimal Bandwidth value by clicking the Optimize button . This will only optimize the Bandwidth, no other parameter values will be changed.
     • Use Spatial Condition Number Threshold can be changed from False to True to limit the output to areas where the predictions are reliable. The output surfaces will be clipped to exclude areas where the Spatial Condition Number exceeds the Spatial Condition Number Threshold.
     • With Use Spatial Condition Number Threshold set to True, the Spatial Condition Number Threshold becomes accessible. Values can be entered manually, or an optimal value can be found by clicking the Optimize button . This will only optimize the Spatial Condition Number Threshold, no other parameter values will be changed. Rule of thumb values for this parameter are shown in the table below, and are rendered in yellow in the Condition Number map.

       Order of Polynomial	Critical Spatial Condition Number Threshold value
       1	10
       2	100
       3	1000
       Greater than 3	Not recommended for most situations

7. Click Next.
8. Assess the goodness of fit of the model using the Predicted, Error, Standardized Error and Normal QQ Plot graphs, the summary information on Prediction Errors, and by examining particular pairs of measured and predicted values in the table on the left hand side.

   For more information on how to assess the goodness of fit of a model, refer to Performing cross validation and validation.

9. Once you are satisfied with the model, click Finish.
10. A Method Report window will appear. Click OK to produce the surface.

    The Method Report window contains a summary showing the dataset, attribute, interpolation method and parameter values used to create the surface. This information can be retrieved for any geostatistical layer by right-clicking on it in the ArcMap table of contents, choosing Properties from the menu and then clicking on the Method Summary tab.

The result is a surface generated by interpolating attribute values using Local Polynomial Interpolation. The surface is added directly to the ArcMap table of contents, and is displayed using a default color scheme and class breaks, which can be changed by accessing the layer's properties.