Language Reference
MAD Function
MAD (x <, method> ) ;
This function is supported by the IML procedure and the iml action.
The MAD function computes the univariate (scaled) median absolute deviation of each column of the input matrix.
The arguments to the MAD function are as follows:
- x
is an
input data matrix. - method
-
is an optional string argument with the following values:
- "MAD"
for computing the median absolute deviation (MAD); this is the default.
- "NMAD"
for computing the normalized version of MAD
- "SN"
for computing
- "QN"
for computing
For simplicity, the following descriptions assume that the input argument x is a column vector. The notation 
means the ith element of the column vector x.
The MAD function can be used for computing one of the following three robust scale estimates:
-
median absolute deviation (MAD) or normalized form of MAD,
where
is the unscaled default and 
is used for the scaled version (consistency with the Gaussian distribution). -
, which is a more efficient alternative to MAD,
where the outer median is a low median (order statistic of rank
) and the inner median is a high median (order statistic of rank 
), and where 
is a scalar that depends on sample size n. -
is another efficient alternative to MAD. It is based on the kth-order statistic of the
inter-point distances, 
where
is a scalar similar to but different from . See Rousseeuw and Croux (1993) for more details.
The scalars and are defined as follows:
Example
The following example uses the univariate data set of Barnett and Lewis (1978). The data set is used in Chapter 14 to illustrate the univariate LMS and LTS estimates.
b = {3, 4, 7, 8, 10, 949, 951};
rmad1 = mad(b);
rmad2 = mad(b,"mad");
rmad3 = mad(b,"nmad");
rmad4 = mad(b,"sn");
rmad5 = mad(b,"qn");
print "Default MAD=" rmad1,
"Common MAD =" rmad2,
"MAD*1.4826 =" rmad3,
"Robust S_n =" rmad4,
"Robust Q_n =" rmad5;
Figure 240: Median Absolute Deviations
| rmad1 | |
|---|---|
| Default MAD= | 4 |
| rmad2 | |
|---|---|
| Common MAD = | 4 |
| rmad3 | |
|---|---|
| MAD*1.4826 = | 5.9304089 |
| rmad4 | |
|---|---|
| Robust S_n = | 7.143674 |
| rmad5 | |
|---|---|
| Robust Q_n = | 5.7125049 |