BiweightLocation

BiweightLocation[list]

gives the value of the biweight location estimator of the elements in list.

BiweightLocation[list,c]

gives the value of the biweight location estimator with scaling parameter c.

Details and Options

  • BiweightLocation is a robust location estimator.
  • BiweightLocation is given by a weighted mean of the elements. Elements farther from the center have lower weights.
  • The width scale of the weight function is controlled by a parameter c. Larger c indicates more data values are included in the computation of the statistic, and vice versa. »
  • For the list {x1,x2,,xn}, the value of the biweight location estimator is given by , where and is Median[{x1-x*,x2-x*,,xn-x*}]. The value x* of the estimator is computed iteratively, with the initial value chosen automatically by default.
  • BiweightLocation[list] is equivalent to BiweightLocation[list,6].
  • BiweightLocation[{{x1,y1,},{x2,y2,},}] gives {BiweightLocation[{x1,x2,}],BiweightLocation[{y1,y2,}],}.
  • BiweightLocation allows c to be any positive real number.
  • The following options can be given:
  • AccuracyGoalAutomaticthe accuracy sought
    MaxIterationsAutomaticmaximum number of iterations to use
    MethodAutomaticmethod to use
    PrecisionGoalAutomaticthe precision sought
    WorkingPrecisionMachinePrecisionthe precision used in internal computations
  • The setting Method{"InitialPoint"x0} allows for a custom initial value .

Examples

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Basic Examples  (3)

BiweightLocation of a list:

BiweightLocation of columns of a matrix:

BiweightLocation of a list with scaling parameter 7:

Scope  (6)

Same inputs with different precisions:

Biweight location with different scaling parameters:

Biweight location for a matrix gives columnwise estimate:

Biweight location of a large array:

Find a biweight location of a TimeSeries:

The biweight location depends only on the values:

Biweight location works with data involving quantities:

Options  (2)

MaxIterations  (1)

The value of BiweightLocation is computed iteratively. Limit the number of iterations attempted in the computation:

Method  (1)

Adjust the starting value in the computation of BiweightLocation:

Limit the number of iterations with a better starting value:

Applications  (3)

Obtain a robust estimate of location when outliers are present:

Extreme values have a large influence on the Mean:

Consider data from a Gaussian mixture distribution:

Estimate the center with Mean:

The sample mean estimator has a large spread for non-Gaussian data. The standard deviation of the estimator is:

Estimate the center with BiweightLocation:

Use bootstrapping to assess the spread of the biweight location estimator:

Simulate a trajectory with heavy-tailed measurement noise:

The underlying signal and simulated path with noise:

Smooth the trajectory using a moving BiweightLocation:

Increasing the block size gives a smoother trajectory:

Properties & Relations  (3)

Compute the biweight location of a sample:

Values outside of the interval have no effect on the statistic. Here is the value of biweight location and is the median absolute deviation with respect to . is a scaling parameter with default value equal to 6:

The shape of the weight function w(x) being used in computing biweight location:

Multiply the smallest and the largest values in the sample by 2 and compute the biweight location again:

For normally distributed samples, BiweightLocation and Mean are nearly the same:

For non-normally distributed samples such as data from CauchyDistribution, BiweightLocation gives a better estimate of the center location than Mean:

BiweightLocation approaches Mean for large values of c:

Neat Examples  (2)

Variation of biweight location around the mean of univariate data depending on the scaling factor c:

Variation of biweight location around the mean of bivariate data depending on the scaling factor c:

Introduced in 2017
 (11.1)