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Function to calculate common numerical model evaluation statistics with flexible conditioning


  mod = "mod",
  obs = "obs",
  statistic = c("n", "FAC2", "MB", "MGE", "NMB", "NMGE", "RMSE", "r", "COE", "IOA"),
  type = "default", = NULL,



A data frame.


Name of a variable in mydata that respresents modelled values.


Name of a variable in mydata that respresents measured values.


The statistic to be calculated. See details below for a description of each.


type determines how the data are split i.e. conditioned, and then plotted. The default is will produce statistics using the entire data. type can be one of the built-in types as detailed in cutData e.g. “season”, “year”, “weekday” and so on. For example, type = "season" will produce four sets of statistics --- one for each season.

It is also possible to choose type as another variable in the data frame. If that variable is numeric, then the data will be split into four quantiles (if possible) and labelled accordingly. If type is an existing character or factor variable, then those categories/levels will be used directly. This offers great flexibility for understanding the variation of different variables and how they depend on one another.

More than one type can be considered e.g. type = c("season", "weekday") will produce statistics split by season and day of the week.

Simple model ranking can be carried out if is supplied. will generally refer to a column representing a model name, which is to ranked. The ranking is based the COE performance, as that indicator is arguably the best single model performance indicator available.


Other aruments to be passed to cutData e.g. hemisphere = "southern"


Returns a data frame with model evaluation statistics.


This function is under development and currently provides some common model evaluation statistics. These include (to be mathematically defined later):

  • \(n\), the number of complete pairs of data.

  • \(FAC2\), fraction of predictions within a factor of two.

  • \(MB\), the mean bias.

  • \(MGE\), the mean gross error.

  • \(NMB\), the normalised mean bias.

  • \(NMGE\), the normalised mean gross error.

  • \(RMSE\), the root mean squared error.

  • \(r\), the Pearson correlation coefficient. Note, can also supply and aurument method e.g. method = "spearman"

  • \(COE\), the Coefficient of Efficiency based on Legates and McCabe (1999, 2012). There have been many suggestions for measuring model performance over the years, but the COE is a simple formulation which is easy to interpret.

    A perfect model has a COE = 1. As noted by Legates and McCabe although the COE has no lower bound, a value of COE = 0.0 has a fundamental meaning. It implies that the model is no more able to predict the observed values than does the observed mean. Therefore, since the model can explain no more of the variation in the observed values than can the observed mean, such a model can have no predictive advantage.

    For negative values of COE, the model is less effective than the observed mean in predicting the variation in the observations.

  • \(IOA\), the Index of Agreement based on Willmott et al. (2011), which spans between -1 and +1 with values approaching +1 representing better model performance.

    An IOA of 0.5, for example, indicates that the sum of the error-magnitudes is one half of the sum of the observed-deviation magnitudes. When IOA = 0.0, it signifies that the sum of the magnitudes of the errors and the sum of the observed-deviation magnitudes are equivalent. When IOA = -0.5, it indicates that the sum of the error-magnitudes is twice the sum of the perfect model-deviation and observed-deviation magnitudes. Values of IOA near -1.0 can mean that the model-estimated deviations about O are poor estimates of the observed deviations; but, they also can mean that there simply is little observed variability - so some caution is needed when the IOA approaches -1.

All statistics are based on complete pairs of mod and obs.

Conditioning is possible through setting type, which can be a vector e.g. type = c("weekday", "season").

Details of the formulas are given in the openair manual.


Legates DR, McCabe GJ. (1999). Evaluating the use of goodness-of-fit measures in hydrologic and hydroclimatic model validation. Water Resources Research 35(1): 233-241.

Legates DR, McCabe GJ. (2012). A refined index of model performance: a rejoinder, International Journal of Climatology.

Willmott, C.J., Robeson, S.M., Matsuura, K., 2011. A refined index of model performance. International Journal of Climatology.


David Carslaw


## the example below is somewhat artificial --- assuming the observed
## values are given by NOx and the predicted values by NO2.

modStats(mydata, mod = "no2", obs = "nox")
#>    default     n      FAC2        MB      MGE        NMB     NMGE     RMSE
#> 1 all data 63095 0.1703868 -129.6739 129.7277 -0.7252306 0.725532 166.6203
#>           r        COE       IOA
#> 1 0.7874487 -0.3374671 0.3312664

## evaluation stats by season

modStats(mydata, mod = "no2", obs = "nox", type = "season")
#>         season     n       FAC2        MB      MGE        NMB      NMGE
#> 2 spring (MAM) 17343 0.25763083 -107.5466 107.5466 -0.6846951 0.6846951
#> 3 summer (JJA) 14658 0.17140113 -116.2004 116.2004 -0.7053470 0.7053470
#> 1 autumn (SON) 14775 0.09825963 -154.1534 154.1534 -0.7526324 0.7526324
#> 4 winter (DJF) 16319 0.14201438 -143.1283 143.3366 -0.7494609 0.7505519
#>       RMSE         r        COE       IOA
#> 2 139.9177 0.8196724 -0.2497589 0.3751205
#> 3 144.7206 0.7568891 -0.4197370 0.2901315
#> 1 191.9351 0.7881365 -0.4370783 0.2814608
#> 4 185.3849 0.8260512 -0.3207156 0.3396422