Root Mean Square Error RMSE in GIS

Root Mean Square Error

What is Root Mean Square Error RMSE?

Root Mean Square Error (RMSE) (also known as Root Mean Square Deviation) is one of the most widely used statistics in GIS.

RMSE can be used for a variety of geostatistical applications such as comparing elevation data sets.

RMSE measures how much error there is between two datasets comparing a predicted value and an observed or known value.

Let’s examine RMSE with a bit more detail.

Root Mean Square Error Example

For example, a LiDAR elevation point (predicted value) might be compared with a surveyed ground measurement (observed value).

  • Predicted value: LiDAR elevation value
  • Observed value: Surveyed elevation value



Root mean square error takes the difference for each LiDAR value and surveyed value.

You can swap the order of subtraction because the next step is to take the square of the difference. (The square of a negative or positive value will always be a positive value).

But just make sure that you keep tha order through out.

After that, divide the sum of all values by the number of observations. This is how RMSE is calculated.

RMSE Formula:

RMSE = {\sqrt {\frac{1} {N}{\sum\limits_{i = 1}^N {(x_{i} - \hat{x}_{i} } })^{2} } }

How to Calculate RMSE in Excel

Here is a quick and easy guide to calculate RMSE in Excel. You will need a set of observed and predicted values:

1. In cell A1, type “observed value” as a title. In B1, type “predicted value”. In C2, type “difference”.

2. If you have 10 observations, place observed elevation values in A2 to A11. Place predicted values in B2 to B11.




3. In column C2, subtract observed value and predicted value: =A2-B2. Repeat for all rows below where predicted and observed values exist.

4. In cell D2, use the following formula to calculate RMSE: =SQRT(SUMSQ(C2:C11)/COUNTA(C2:C11))

Cell D2 is the root mean square error value.

What’s Next?

RMSE quantifies how different a set of values are.

The smaller an RMSE value, the closer predicted and observed values are.

Give this quick RMSE guide a try and master one of the most widely used statistics in GIS…

Root Mean Square Error

7 Comments on Root Mean Square Error RMSE in GIS

  1. There is no need to create the C column, this Excel formula can calculate the RMSE from the A and B columns only.

    =SQRT(SUMXMY2(A2:A11,B2:B11)/COUNTA(A2:A11))

  2. Can we use RMSE to compare land surface temperature from Landsat (predicted value) with surveyed measurment (observed value) of land surface temperature?

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