### Description

This function computes the mean-absolute deviation (MAD) – "the average of the magnitudes of the errors or deviations."

### Usage

 1 madstat(observed, na.rm = FALSE) 

### Arguments

 observed numeric vector, matrix, data.frame, or data.table that contains the observed data points. na.rm logical vector that determines whether the missing values should be removed or not.

### Details

n^{-1} ∑ \limits_{i=1}^n{ ≤ft| O_i - \bar{O} \right|}

n

the number of observations

O

the "pairwise-matched observations that are judged to be reliable"

\bar{O}

the "true" mean of the observations

Reference 1 fully discusses MAD, while Reference 2 provides the formula used to calculate the MAD.

### Value

mean-absolute deviation (MAD) as a numeric vector or a named numeric vector if using a named object (matrix, data.frame, or data.table). MAD has the same units as the observed values. The default choice is that any NA values will be kept (na.rm = FALSE). This can be changed by specifying na.rm = TRUE, such as madstat(obs, na.rm = TRUE).

### Source

kurtosis for use of na.rm for numeric vector, matrix, and data.frame objects

### References

1. Cort J. Willmott, Kenji Matsuura, and Scott M. Robeson, "Ambiguities inherent in sums-of-squares-based error statistics", Atmospheric Environment, vol. 43, no. 3, pp. 749-752, 2009, http://www.sciencedirect.com/science/article/pii/S1352231008009564.

2. Cort J. Willmott, Scott M. Robeson, and Kenji Matsuura, "Short Communication: A refined index of model performance", International Journal of Climatology, Volume 32, Issue 13, pages 2088-2094, 15 November 2012, http://onlinelibrary.wiley.com/doi/10.1002/joc.2419/pdf.

3. Nathabandu T. Kottegoda and Renzo Rosso, Statistics, Probability, and Reliability for Civil and Environmental Engineers, New York City, New York: The McGraw-Hill Companies, Inc., 1997, page 15.

mad for median absolute deviation (MAD)
mape for mean absolute percent error (MAPE), mae for mean-absolute error (MAE), dr for "index of agreement (dr)", vnse for Nash-Sutcliffe model efficiency (NSE), and rmse for root mean square error (RMSE).
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 library(ie2misc) # Example 1.18 from Kottegoda (page 15) obs <- c(50, 56, 42, 53, 49) # annual rainfall in cm madstat(obs) require(stats) set.seed(100) # makes the example reproducible obs1 <- rnorm(100) # observed # using the numeric vector obs1 madstat(obs1) # using a matrix of the numeric vector obs1 mat1 <- matrix(data = obs1, nrow = length(obs1), ncol = 1, byrow = FALSE, dimnames = list(c(rep("", length(obs1))), "Observed")) madstat(mat1) # using a data.frame of the numeric vector obs1 df1 <- data.frame(obs1) madstat(df1) # using a data.table of the numeric vector obs1 df2 <- data.table(obs1) madstat(df2)