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https://github.com/daleroberts/hdmedians

High-dimensional medians (medoid, geometric median, etc.). Fast implementations in Python.
https://github.com/daleroberts/hdmedians

high-dimensional-data machine-learning median python statistics

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High-dimensional medians (medoid, geometric median, etc.). Fast implementations in Python.

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README 

        
# Hdmedians



Did you know there is no unique way to mathematically extend the concept of a

[median](https://en.wikipedia.org/wiki/Median) to higher dimensions?



Various definitions for a **high-dimensional median** exist and this Python

package provides a number of fast implementations of these definitions.

Medians are extremely useful due to their high breakdown point (up to 50%

contamination) and have a number of nice applications in machine learning,

computer vision, and high-dimensional statistics.










This package currently has implementations of [medoid](#medoid) and [geometric

median](#geometric-median) with support for missing data using `NaN`. 



### Installation



The latest version of the package is always available on [pypi](https://pypi.python.org/pypi/hdmedians), 

so can be easily installed by typing:

```{sh}

pip3 install hdmedians

```



## Medoid



Given a finite set  of -dimensional observation vectors , 

the [medoid](https://en.wikipedia.org/wiki/Medoid)  of these observations is given by




The current implementation of `medoid` is in vectorized Python and can handle

any data type supported by

[ndarray](https://docs.scipy.org/doc/numpy/reference/generated/numpy.ndarray.html).

If you would like the algorithm to take care of missing values encoded as `nan`

then you can use the `nanmedoid` function.



### Examples



Create an 6 x 10 array of random integer observations.

```{python}

>>> import numpy as np

>>> X = np.random.randint(100, size=(6, 10))

array([[12,  9, 61, 76,  2, 17, 12, 11, 26,  0],

       [65, 72,  7, 64, 21, 92, 51, 48,  9, 65],

       [39,  7, 50, 56, 29, 79, 47, 45, 10, 52],

       [70, 12, 23, 97, 86, 14, 42, 90, 15, 16],

       [13,  7,  2, 47, 80, 53, 23, 59,  7, 15],

       [83,  2, 40, 12, 22, 75, 69, 61, 28, 53]])

```



Find the medoid, taking the last axis as the number of observations.

```{python}

>>> import hdmedians as hd

>>> hd.medoid(X)

array([12, 51, 47, 42, 23, 69])

```



Take the first axis as the number of observations.

```{python}

>>> hd.medoid(X, axis=0)

array([39,  7, 50, 56, 29, 79, 47, 45, 10, 52])

```



Since the medoid is one of the observations, the `medoid` function has the ability to only return the index if required.

```{python}

>>> hd.medoid(X, indexonly=True)

6

>>> X[:,6]

array([12, 51, 47, 42, 23, 69])

```



## Geometric Median



The [geometric median](https://en.wikipedia.org/wiki/Geometric_median) is also known as the 1-median, spatial median,

Euclidean minisum, or Torricelli point. Given a finite set  of -dimensional observation vectors , 

the geometric median  of these observations is given by



Note there is a subtle difference between the definition of the geometric median and the medoid: the search space 

for the solution differs and has the effect that the medoid returns one of the true observations whereas the geometric median can be described 

as a synthetic (not physically observed) observation.



The current implementation of `geomedian` uses Cython and can handle `float64`

or `float32`. If you would like the algorithm to take care of missing values

encoded as `nan` then you can use the `nangeomedian` function.



### Examples



Create an 6 x 10 array of random `float64` observations.

```{python}

>>> import numpy as np

>>> np.set_printoptions(precision=4, linewidth=200)

>>> X = np.random.normal(1, size=(6, 10))

array([[ 1.1079,  0.5763,  0.3072,  1.2205,  0.8596, -1.5082,  2.5955,  2.8251,  1.5908,  0.4575],

       [ 1.555 ,  1.7903,  1.213 ,  1.1285,  0.0461, -0.4929, -0.1158,  0.5879,  1.5807,  0.5828],

       [ 2.1583,  3.4429,  0.4166,  1.0192,  0.8308, -0.1468,  2.6329,  2.2239,  0.2168,  0.8783],

       [ 0.7382,  1.9453,  0.567 ,  0.6797,  1.1654, -0.1556,  0.9934,  0.1857,  1.369 ,  2.1855],

       [ 0.1727,  0.0835,  0.5416,  1.4416,  1.6921,  1.6636,  1.6421,  1.0687,  0.6075, -0.0301],

       [ 2.6654,  1.6741,  1.1568,  1.3092,  1.6944,  0.2574,  2.8604,  1.6102,  0.4301, -0.3876]])

>>> X.dtype

dtype('float64')

```



Find the geometric median, taking the last axis as the number of observations.

```{python}

>>> import hdmedians as hd

>>> hd.geomedian(X)

array([ 1.0733,  0.8974,  1.1935,  0.9122,  0.9975,  1.3422])

```



Take the first axis as the number of observations.

```{python}

>>> hd.geomedian(X, axis=0)

array([ 1.4581,  1.6377,  0.7147,  1.1257,  1.0493, -0.091 ,  1.7907,  1.4168,  0.9587,  0.6195])

```



Convert to `float32` and compute the geometric median.

```{python}

>>> X = X.astype(np.float32)

>>> m = hd.geomedian(X)

```



## References



  * Small, C. G. (1990). [A survey of multidimensional medians](http://www.jstor.org/stable/1403809). *International Statistical Review/Revue Internationale de Statistique*, 263-277.