Qn

Qn robust estimator of scale (first quartile of interpoint distances $|x_i-x_j|$)

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Syntax

Description

example

y =Qn(X, dim) Qn with all default otpions.

Examples

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  • Qn with all default otpions.

    • X = [1 2 4 4  7;
3 4 6 6  8;
5 6 8 8  10;
5 7 10 12 1500];
y1=Qn(X)
y2=Qn(X,2)
    y1 =

    2.2752    2.2752    2.2752    4.5505    3.4128


y2 =

    3.7506
    3.7506
    3.7506
    5.6259

    Input Arguments

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    X — Input array. Vector | matrix | 3D array.

    Input array, specified as a vector, matrix, or 3D array.

    For vectors, Qn(X) is the scale estimator of the elements in X. For matrices, Qn(X) is a row vector containing the scale estimator value of each column. For 3D arrays, Qn(X) is the robust scale estimator of the elements along the first non-singleton dimension of X.

    Data Types: ingle | double | int8 | int16 | int32 | int64 |uint8 | uint16 | uint32 | uint64

    dim — Dimension to operate along. Positive integer scalar.

    Dimension to operate along, specified as a positive integer scalar. If no value is specified, then the default is the first array dimension whose size does not equal 1.

    Data Types: ingle | double | int8 | int16 | int32 | int64 |uint8 | uint16 | uint32 | uint64

    Output Arguments

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    y —robust estimator of scale. Scalar | Vector or 3D array

    Qn(X,dim) takes the robust estimator of scale along the dimension dim of X.

    More About

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    Additional Details

    $Q_n$ is the first quartile of the distances { $|x_i-x_j|$; $i <j$} Note that $Q_n$ does not need any location estimate. More in detail, let $d_{(1)} \leq d_{(2)} \leq ... \leq d_{(m)}$ the ordered values of the $m$ differences $|x_i-x_j|$ with $i>j$ and $m = {n \choose 2}$. $Q_n=d_{(k)}$ where $k= {[n/2]+1 \choose 2}$. Since $k$ is approximately $m/4$, $Q_n$ is approximately the first quartile of the ordered distances $d_{(1)} \leq d_{(2)} \leq ... \leq d_{(m)}$. $Q_n$ is multiplyed by $c$ and $c_n$.

    $c$ is the so called asymptotic consistency factor and is equal to 2.2219 while $c_n$ is a finite sample correction factor to make the estimator unbiased.

    References

    Rousseeuw P.J. and Croux C., (1993), Alternatives to the median absolute deviation, "Journal of American Statistical Association", Vol. 88, pp. 1273-1283

    Croux C. and Rousseeuw P.J.(1992) Time-efficient algorithms for two highly robust estimators of scale, in "Computational Statistics", Vol. 1, eds. Y . Dodge and J. Whittaker, Heidelberg: Physika-Verlag, 41 1-428.

    See Also

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