Tau <: RobustScale

Tau(; c1::Float64=4.5, c2::Float64=3.0, consistency_correction::Bool=true, can_handle_nan::Bool=false)

Robust Tau-Estimate of Scale

Computes the robust τ-estimate of univariate scale, as proposed by Maronna and Zamar (2002); improved by a consistency factor

Keywords

`c1::Float64=4.5`, optional: constant for the weight function, defaults to 4.5
`c2::Float64=3.0`, optional: constant for the rho function, defaults to 3.0
`consistency_correction::Bool`, optional: boolean indicating if consistency for normality should be applied. Defaults to True.
`can_handle_nan::Bool`, optional: boolean indicating whether NANs can be handled, defaults to false.

References

Robust Estimates of Location and Dispersion for High-Dimensional Datasets,
Ricarco A Maronna and Ruben H Zamar (2002)

Examples

julia> x = hbk[:,1];
julia> tau = Tau();
julia> fit!(tau, x);
julia> location(tau)
1.5554543865072337

julia> scale(tau)
2.012461881814477

Qn <: RobustScale

Qn(;location_func::Function=median, consistency_correction::Bool=true, finite_correction::Bool=true, can_handle_nan::Bool = false)

Robust Location-Free Scale Estimate More Efficient than MAD

Keywords

`location_func::LocationOrScaleEstimator`, optional: as the Qn estimator does not
    estimate location, a location function should be explicitly passed.
`consistency_correction::Bool`, optional: boolean indicating if consistency for normality should be applied. Defaults to true.
`finite_correction::Bool`, optional: boolean indicating if finite sample correction should be applied. Defaults to true.

References

Christophe Croux and Peter J. Rousseeuw (1992). Time-efficient algorithms for two highly robust estimators of scale.

Donald B. Johnson and Tetsuo Mizoguchi (1978). Selecting the k^th element in X+Y and X1+...+Xm

Examples

julia> x = hbk[:,1];

julia> qn = Qn();
julia> fit!(qn, x);

julia> location(qn)         # the median
1.8

julia> scale(qn)
1.7427832460732984

location(rs::RobustScale)

Univariate location

location(model::CovarianceEstimator)::Vector{Float64}

Location vector

scale(rs::RobustScale)

Univariate scale

CovClassic <: CovarianceEstimator <: Any

CovClassic(;assume_centered=false)

Classical Location and Scatter Estimation

Compute the classical estimetas of the location vector and covariance matrix of a data matrix.

Examples

julia> cc=CovClassic();
julia> fit!(cc, hbk[:,1:3])
-> Method:  Classical Estimator. 

Estimate of location:
[3.20667, 5.59733, 7.23067]

Estimate of covariance:
3×3 Matrix{Float64}:
 13.3417  28.4692   41.244
 28.4692  67.883    94.6656
 41.244   94.6656  137.835

covariance(model::CovarianceEstimator)::Matrix{Float64}

Covariance matrix

correlation(model::CovarianceEstimator)::Matrix{Float64}

Correlation matrix

CovMcd <: RobustCovariance <: CovarianceEstimator 

CovMcd(;assume_centered=false, alpha=nothing, n_initial_subsets=500, n_initial_c_steps=2,
    n_best_subsets=10, n_partitions=nothing, tolerance=1e-8, reweighting=true, verbosity=Logging.Warn)

Robust Location and Scatter Estimation via MCD

Compute the Minimum Covariance Determinant (MCD) estimator, a robust multivariate location and scale estimate with a high breakdown point, via the 'Fast MCD' algorithm proposed in Rousseeuw and Van Driessen (1999).

Keywords:

`alpha::Float64 | Int | Nothing`, optional:
    size of the h subset.
    If an integer between n/2 and n is passed, it is interpreted as an absolute value.
    If a float between 0.5 and 1 is passed, it is interpreted as a proportation
    of n (the training set size).
    If None, it is set to (n+p+1) / 2.
    Defaults to Nothing.
`n_initial_subsets::Int`, optional: number of initial random subsets of size p+1
`n_initial_c_steps::Int`, optional: number of initial c steps to perform on all initial subsets
`n_best_subsets::Int`, optional: number of best subsets to keep and perform c steps on until convergence
`n_partitions::Int` optional: Number of partitions to split the data into.
    This can speed up the algorithm for large datasets (n > 600 suggested in paper)
    If None, 5 partitions are used if n > 600, otherwise 1 partition is used.
`tolerance::Float64`, optional: Minimum difference in determinant between two iterations to stop the C-step
`reweighting:Bool`, optional: Whether to apply reweighting to the raw covariance estimate

References:

Rousseeuw and Van Driessen, A Fast Algorithm for the Minimum Covariance Determinant
Estimator, 1999, American Statistical Association and
the American Society for Quality, TECHNOMETRICS

Examples

julia> mcd=CovMcd();
julia> fit!(mcd, hbk[:,1:3])
-> Method:  Fast MCD Estimator: (alpha=nothing ==> h=39)

Robust estimate of location:
[1.55833, 1.80333, 1.66]

Robust estimate of covariance:
3×3 Matrix{Float64}:
 1.21312    0.0239154  0.165793
 0.0239154  1.22836    0.195735
 0.165793   0.195735   1.12535

juilia> dd_plot(mcd)

DetMcd(; assume_centered=false,  alpha=nothing, n_maxcsteps=200, tolerance=1e-8,
    reweighting=true, verbosity=Logging.Warn)

Deterministic MCD estimator (DetMCD) based on the algorithm proposed in Hubert, Rousseeuw and Verdonck (2012)

Keywords:

`alpha::Float64 | Int | Nothing`, optional: size of the h subset.
    If an integer between n/2 and n is passed, it is interpreted as an absolute value.
    If a float between 0.5 and 1 is passed, it is interpreted as a proportation
    of n (the training set size).
    If None, it is set to (n+p+1) / 2.
    Defaults to None.
`n_maxcsteps::Int=200`, optional: Maximum number of C-step iterations
`tolerance::Float64`, optional: Minimum difference in determinant between two iterations to stop the C-step
`reweighting::Bool`, optional: Whether to apply reweighting to the raw covariance estimate

References:

Hubert, Rousseeuw and Verdonck, A deterministic algorithm for robust location
and scatter, 2012, Journal of Computational and Graphical Statistics

Examples

julia> mcd=DetMcd();
julia> fit!(mcd, hbk[:,1:3])
-> Method:  Deterministic MCD: (alpha=nothing ==> h=39)

Robust estimate of location:
[1.5377, 1.78033, 1.68689]

Robust estimate of covariance:
3×3 Matrix{Float64}:
 1.2209     0.0547372  0.126544
 0.0547372  1.2427     0.151783
 0.126544   0.151783   1.15414

julia> dd_plot(mcd);

CovOgk(;store_precision::Bool=true, assume_centered::Bool=false, location_estimator::Function=median,
    scale_estimator::Function=MAD_scale, n_iterations::Int=2, reweighting::Bool=false,
    reweighting_beta::Float64=0.9)

Implementation of the Orthogonalized Gnanadesikan-Kettenring estimator for location dispersion proposed in Maronna, R. A., & Zamar, R. H. (2002)

Keywords:

store_precision (boolean, optional): whether to store the precision matrix
assume_centered (boolean, optional): whether the data is already centered
location_estimator (LocationOrScaleEstimator, optional): function to estimate the
    location of the data, should accept an array like input as first value and a named
    argument axis
scale_estimator (LocationOrScaleEstimator, optional): function to estimate the scale
    of the data, should accept an array like input as first value and a named argument
    axis
n_iterations (int, optional): number of iteration for orthogonalization step
reweighting (boolean, optional): whether to apply reweighting at the end
    (i.e. calculating regular location and covariance after filtering outliers based on
    Mahalanobis distance using OGK estimates)
reweighting_beta (float, optional): quantile of chi2 distribution to use as cutoff for
    reweighting

References:

Maronna, R. A., & Zamar, R. H. (2002).
Robust Estimates of Location and Dispersion for High-Dimensional Datasets.
Technometrics, 44(4), 307–317. http://www.jstor.org/stable/1271538

Examples

julia> ogk=CovOgk();
julia> fit!(ogk, hbk[:,1:3])
-> Method:  Orthogonalized Gnanadesikan-Kettenring Estimator

Robust estimate of location:
[1.56005, 2.22345, 2.12035]

Robust estimate of covariance:
3×3 Matrix{Float64}:
 3.3575    0.587449  0.699388
 0.587449  2.09268   0.285757
 0.699388  0.285757  2.77527

julia> dd_plot(ogk);

Hawkins & Bradu & Kass data

Components

• x1::Float64: first independent variable.
• x2::Float64: second independent variable.
• x3::Float64: third independent variable.
• y::Float64: dependent (response) variable.

Reference

Hawkins, D.M., Bradu, D., and Kass, G.V. (1984) Location of several outliers in multiple regression data using elemental sets. Technometrics 26, 197–208.

Animals data

Components

• names::AbstractString: names of animals.
• body::Float64: body weight in kg.
• brain::Float64: brain weight in g.

References

 Venables, W. N. and Ripley, B. D. (1999) _Modern Applied
 Statistics with S-PLUS._ Third Edition. Springer.

 P. J. Rousseeuw and A. M. Leroy (1987) _Robust Regression and
 Outlier Detection._ Wiley, p. 57.

Stack loss data

Components

• airflow::Float64: flow of cooling air (independent variable).
• watertemp::Float64: cooling water inlet temperature (independent variable).
• acidcond::Float64: concentration of acid (independent variable).
• stackloss::Float64: stack loss (dependent variable).

Outliers

Observations 1, 3, 4, and 21 are outliers.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S Language_.  Wadsworth & Brooks/Cole.

Dodge, Y. (1996) The guinea pig of multiple regression. In: _Robust Statistics, Data Analysis, and Computer Intensive Methods;
In Honor of Peter Huber's 60th Birthday_, 1996, _Lecture Notes in Statistics_ *109*, Springer-Verlag, New York.

Modified Wood Gravity Data

Components

• x1::Float64: Random values.
• x2::Float64: Random values.
• x3::Float64: Random values.
• x4::Float64: Random values.
• x5::Float64: Random values.
• y::Float64: Random values (independent variable).

References

P. J. Rousseeuw and A. M. Leroy (1987) Robust Regression and Outlier Detection. Wiley, p.243, table 8.