compute Rousseeuw and Croux' Sn
Data Set Options
Random Variable Options
data set or Matrix data set
algebraic; random variable or distribution
(optional) equation(s) of the form option=value where option is one of correction, ignore, or weights; specify options for computing Rousseeuw and Croux' Sn statistic of a data set
(optional) equation of the form numeric=value; specifies options for computing Rousseeuw and Croux' Sn statistic of a random variable
The RousseeuwCrouxSn function computes a robust measure of the dispersion of the specified data set or random variable, as introduced by Rousseeuw and Croux in .
This statistic, referred to as Sn in the remainder of this help page, is defined for a data set A1,A2,...,An as:
where the LowMedian of n values is its n2+12th OrderStatistic and the HighMedian is its n2+12th OrderStatistic. (HighMedian and LowMedian are not Maple functions - they are only used here to define Sn.)
Sn is a robust statistic: it has a high breakdown point (the proportion of arbitrarily large observations it can handle before giving an arbitrarily large result). The breakdown point of Sn is the maximum possible value, 12.
Sn is a measure of dispersion, also called a measure of scale: if Sn⁡X=a, then for all real constants α and β, we have Sn⁡α⁢X+β=α⁢a.
The first parameter can be a data set, a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]). For a data set A, RousseeuwCrouxSn computes Sn as defined above. For a distribution or random variable X, RousseeuwCrouxSn computes the asymptotic equivalent - the value that Sn converges to for ever larger samples of X.
By default, all computations involving random variables are performed symbolically (see option numeric below).
All computations involving data are performed in floating-point; therefore, all data provided must have type/realcons and all returned solutions are floating-point, even if the problem is specified with exact values.
For more information about computation in the Statistics package, see the Statistics[Computation] help page.
The ds_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.
ignore=truefalse -- This option controls how missing data is handled by the RousseeuwCrouxSn command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the RousseeuwCrouxSn command may return undefined. If ignore=true all missing items in A will be ignored. The default value is false.
weights=Vector -- Data weights. The number of elements in the weights array must be equal to the number of elements in the original data sample. By default all elements in A are assigned weight 1.
correction=samplesize or correction=none -- In , Rousseeuw and Croux define a correction factor cn for finite sample size as:
If the option correction = samplesize is given, then this correction factor is applied before the result is returned. The default is correction = none, that is, no correction factor is applied.
The rv_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.
numeric=truefalse -- By default, Sn is computed using exact arithmetic. To compute Sn numerically, specify the numeric or numeric = true option.
Compute Sn for a data sample.
s ≔ 1,5,2,2,7,4,1,6
Employ Rousseeuw and Croux's finite sample size correction.
Let's replace three of the values with very large values.
t ≔ copy⁡s:
t1..3 ≔ 10100:
The value of Sn stays bounded, because it has a high breakdown point.
Compute Sn for a normal distribution.
The symbolic result is a rather complicated expression. It evaluates to the same floating-point number.
Generate a random sample of size 1000000 from the same distribution and compute the sample's Sn.
A ≔ Sample⁡'Normal'⁡3,5,1000000:
Consider the following Matrix data set.
M ≔ Matrix⁡3,1130,114694,4,1527,127368,3,907,88464,2,878,96484,4,995,128007
We compute Sn for each of the columns.
 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.
 Rousseeuw, Peter J., and Croux, Christophe. Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association 88(424), 1993, pp.1273-1283.
The Statistics[RousseeuwCrouxSn] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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