| Name | PyDistances JSON |
| Version |
0.0.18
JSON |
| download |
| home_page | https://github.com/FabioScielzoOrtiz/Distances_Package |
| Summary | This is a package for computing distances among observations of statistical variables, such as: Euclidean, Minkowski, Canberra, Pearson, Mahalanobis, Robust Mahalanobis, Gower, Generalized Gower and Related Metric Scaling (RelMS). A total of 41 statistical distances can be computed. |
| upload_time | 2023-06-06 17:48:47 |
| maintainer | |
| docs_url | None |
| author | Fabio Scielzo Ortiz |
| requires_python | >=3.7 |
| license | |
| keywords |
|
| VCS |
 |
| bugtrack_url |
|
| requirements |
No requirements were recorded.
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| Travis-CI |
No Travis.
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| coveralls test coverage |
No coveralls.
|
# PyDistances: A Statistical Distances Python Package
This is a package for computing distances among observations of statistical variables, such as: Euclidean, Minkowski, Canberra, Pearson, Mahalanobis, Robust Mahalanobis, Gower, Generalized Gower and Related Metric Scaling (RelMS). A total of 41 statistical distances can be computed.
## Installation
```python
pip install PyDistances
```
## Example of use
```python
import PyDistances
```
```python
from PyDistances import Euclidean_Dist, Euclidean_Dist_Matrix, Minkowski_Dist, Minkowski_Dist_Matrix, Canberra_Dist, Canberra_Dist_Matrix, Pearson_Dist, Pearson_Dist_Matrix, Mahalanobis_Dist, Mahalanobis_Dist_Matrix, a_b_c_d_Matrix, Sokal_Similarity, Sokal_Dist, Sokal_Dist_Matrix, Jaccard_Similarity, Jaccard_Dist, Jaccard_Dist_Matrix, alpha, Matching_Similarity, Matching_Dist, Matching_Dist_Matrix, Gower_Similarity_Matrix, Gower_Dist_Matrix, Robust_Mahalanobis_Dist, Robust_Mahalanobis_Dist_Matrix, GeneralizedGowerDistance
```
### Getting data
We load the data we are going to work with throughout this tutorial. This data-set is available in the following link: https://github.com/FabioScielzoOrtiz/Distances_Package/blob/master/Tests/House_Price.csv
```python
Data = pd.read_csv('House_Price.csv')
```
```python
Data = Data.loc[0:150, ['latitude', 'longitude', 'price', 'size_in_m_2', 'balcony_recode', 'private_garden_recode', 'private_gym_recode', 'quality_recode', 'no_of_bathrooms', 'no_of_bedrooms']]
```
```python
Data_quant = Data.loc[:,['latitude', 'longitude', 'price', 'size_in_m_2']]
Data_binary = Data.loc[:,['balcony_recode', 'private_garden_recode', 'private_gym_recode']]
Data_multiclass = Data.loc[:,['quality_recode', 'no_of_bathrooms', 'no_of_bedrooms']]
```
```python
Data.head() # p1=4, p2=3, p3=3
```
| latitude | longitude | price | size_in_m_2 | balcony | private_garden | private_gym | quality | no_of_bathrooms | no_of_bedrooms |
|:----------:|:-----------:|:----------:|:-------------:|:----------------:|:-----------------------:|:--------------------:|:----------------:|:-----------------:|:----------------:|
| 25.1132 | 55.1389 | 2.7e+06 | 100.242 | 1 | 0 | 0 | 2 | 2 | 1 |
| 25.1068 | 55.1512 | 2.85e+06 | 146.973 | 1 | 0 | 0 | 2 | 2 | 2 |
| 25.0633 | 55.1377 | 1.15e+06 | 181.254 | 1 | 0 | 0 | 2 | 5 | 3 |
| 25.2273 | 55.3418 | 2.85e+06 | 187.664 | 1 | 0 | 0 | 1 | 3 | 2 |
| 25.1143 | 55.1398 | 1.7292e+06 | 47.1018 | 0 | 0 | 0 | 2 | 1 | 0 |
<br>
## Computing Euclidean distance
We compute the Euclidean distance between observation of index 0 and itself.
```python
Euclidean_Dist(Data_quant.iloc[0,:], Data_quant.iloc[0,:])
```
0.0
We compute the Euclidean distance between observation of index 0 and the one of index 2.
```python
Euclidean_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:])
```
1550000.002117049
We compute the Euclidean distances matrix for the data-set `Data_quant`.
```python
Euclidean_Dist_Matrix(Data_quant)
```
```
array([[ 0. , 150000.00727904, 1550000.00211705, ...,
1500000.00009635, 2700000.01899102, 12100000.00553371],
[ 150000.00727904, 0. , 1700000.00034565, ...,
1650000.00026782, 2550000.0146678 , 11950000.00426352],
[ 1550000.00211705, 1700000.00034565, 0. , ...,
50000.040973 , 4250000.00673279, 13650000.00297389],
...,
[ 1500000.00009635, 1650000.00026782, 50000.040973 , ...,
0. , 4200000.01094663, 13600000.00447653],
[ 2700000.01899102, 2550000.0146678 , 4250000.00673279, ...,
4200000.01094663, 0. , 9400000.00011113],
[12100000.00553371, 11950000.00426352, 13650000.00297389, ...,
13600000.00447653, 9400000.00011113, 0. ]])
```
<br>
Now, we are going to repeat the same procedure with other available distances in `PyDistances`.
<br>
## Computing Minkowski distance
```python
Minkowski_Dist(Data_quant.iloc[0,:], Data_quant.iloc[0,:], q=1)
```
0.0
```python
Minkowski_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:], q=1)
```
1550081.062526
```python
Minkowski_Dist_Matrix(Data_quant, q=1)
```
```
array([[ 0. , 150046.748877, 1550081.062526, ...,
1500017.050769, 2700320.266531, 12100365.997115],
[ 150046.748877, 0. , 1700034.338187, ...,
1650029.78435 , 2550273.554024, 11950319.272776],
[ 1550081.062526, 1700034.338187, 0. , ...,
50064.027555, 4250239.302851, 13650284.955165],
...,
[ 1500017.050769, 1650029.78435 , 50064.027555, ...,
0. , 4200303.29563 , 13600348.947944],
[ 2700320.266531, 2550273.554024, 4250239.302851, ...,
4200303.29563 , 0. , 9400045.764238],
[12100365.997115, 11950319.272776, 13650284.955165, ...,
13600348.947944, 9400045.764238, 0. ]])
```
<br>
## Computing Canberra distance
```python
Canberra_Dist(Data_quant.iloc[0,:], Data_quant.iloc[0,:])
```
0.0
```python
Canberra_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:])
```
0.6913917083019879
```python
Canberra_Dist_Matrix(Data_quant)
```
```
array([[0. , 0.21629237, 0.69139171, ..., 0.463675 , 0.9485963 ,
1.33838751],
[0.21629237, 0. , 0.53043317, ..., 0.52079671, 0.79157752,
1.19854721],
[0.69139171, 0.53043317, 0. , ..., 0.23597883, 1.04765637,
1.29619958],
...,
[0.463675 , 0.52079671, 0.23597883, ..., 0. , 1.20126891,
1.44813664],
[0.9485963 , 0.79157752, 1.04765637, ..., 1.20126891, 0. ,
0.51782969],
[1.33838751, 1.19854721, 1.29619958, ..., 1.44813664, 0.51782969,
0. ]])
```
<br>
## Computing Pearson distance
```python
Pearson_Dist(Data_quant.iloc[0,:], Data_quant.iloc[0,:], variance=Data.var())
```
0.0
```python
Pearson_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:], variance=Data.var())
```
1.5393297661160206
```python
Pearson_Dist_Matrix(Data_quant)
```
```
array([[0. , 0.63961801, 1.53932977, ..., 1.03084131, 4.32943281,
7.47171915],
[0.63961801, 0. , 1.20505141, ..., 1.09780711, 3.76643257,
7.04893716],
[1.53932977, 1.20505141, 0. , ..., 0.84617436, 3.79891055,
7.4670243 ],
...,
[1.03084131, 1.09780711, 0.84617436, ..., 0. , 4.44143053,
7.87905955],
[4.32943281, 3.76643257, 3.79891055, ..., 4.44143053, 0. ,
4.57460318],
[7.47171915, 7.04893716, 7.4670243 , ..., 7.87905955, 4.57460318,
0. ]])
```
<br>
## Computing Mahalanobis distance
```python
Mahalanobis_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:], S_inv=np.linalg.inv( np.cov(Data_quant , rowvar=False) ))
```
0.0
```python
Mahalanobis_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:], S_inv=np.linalg.inv( np.cov(Data_quant , rowvar=False) ))
```
2.7671855371187757
```python
Mahalanobis_Dist_Matrix(Data_quant)
```
```
array([[0. , 0.92801614, 2.76718554, ..., 1.52541554, 5.21105193,
6.45997793],
[0.92801614, 0. , 1.96135599, ..., 0.98693199, 4.43479282,
6.2920865 ],
[2.76718554, 1.96135599, 0. , ..., 1.3592188 , 3.4307313 ,
7.27986558],
...,
[1.52541554, 0.98693199, 1.3592188 , ..., 0. , 4.41360406,
7.01503103],
[5.21105193, 4.43479282, 3.4307313 , ..., 4.41360406, 0. ,
7.4691448 ],
[6.45997793, 6.2920865 , 7.27986558, ..., 7.01503103, 7.4691448 ,
0. ]])
```
<br>
## Computing Sokal similarity
```python
a,b,c,d,p = a_b_c_d_Matrix(Data_binary)
```
```python
Sokal_Similarity(i=0, r=2, a=a, d=d, p=p)
```
1.0
```python
Sokal_Dist(i=0, r=2, a=a, d=d, p=p)
```
0.0
```python
Sokal_Dist_Matrix(Data_binary)
```
```
array([[0. , 0. , 0. , ..., 0. , 0. ,
0.81649658],
[0. , 0. , 0. , ..., 0. , 0. ,
0.81649658],
[0. , 0. , 0. , ..., 0. , 0. ,
0.81649658],
...,
[0. , 0. , 0. , ..., 0. , 0. ,
0.81649658],
[0. , 0. , 0. , ..., 0. , 0. ,
0.81649658],
[0.81649658, 0.81649658, 0.81649658, ..., 0.81649658, 0.81649658,
0. ]])
```
<br>
## Computing Jaccard similarity
```python
Jaccard_Similarity(i=0, r=2, a=a, d=d, p=p)
```
1.0
```python
Jaccard_Dist(i=0, r=2, a=a, d=d, p=p)
```
0.0
```python
Jaccard_Dist_Matrix(Data_binary)
```
```
array([[0., 0., 0., ..., 0., 0., 1.],
[0., 0., 0., ..., 0., 0., 1.],
[0., 0., 0., ..., 0., 0., 1.],
...,
[0., 0., 0., ..., 0., 0., 1.],
[0., 0., 0., ..., 0., 0., 1.],
[1., 1., 1., ..., 1., 1., 0.]])
```
<br>
## Computing Matching similarity
```python
Matching_Similarity(x_i=Data_multiclass.iloc[0,:], x_r=Data_multiclass.iloc[2,:], Data=Data_multiclass)
```
0.3333333333333333
```python
Matching_Dist(x_i=Data_multiclass.iloc[0,:], x_r=Data_multiclass.iloc[2,:], Data=Data_multiclass)
```
1.1547005383792517
```python
Matching_Dist_Matrix(Data_multiclass)
```
```
array([[0. , 0.81649658, 1.15470054, ..., 0.81649658, 1.15470054,
1.41421356],
[0.81649658, 0. , 1.15470054, ..., 0. , 1.15470054,
1.41421356],
[1.15470054, 1.15470054, 0. , ..., 1.15470054, 0.81649658,
1.15470054],
...,
[0.81649658, 0. , 1.15470054, ..., 0. , 1.15470054,
1.41421356],
[1.15470054, 1.15470054, 0.81649658, ..., 1.15470054, 0. ,
1.15470054],
[1.41421356, 1.41421356, 1.15470054, ..., 1.41421356, 1.15470054,
0. ]])
```
<br>
## Computing Gower distance
From a theoretical perspective Gower (1971) has been followed.
```python
Gower_Similarity_Matrix(Data, p1=4, p2=3, p3=3)
```
```
array([[1. , 0.85175283, 0.68485131, ..., 0.83008431, 0.62482353,
0.34709882],
[0.85175283, 1. , 0.69489168, ..., 0.94863663, 0.63064768,
0.35833279],
[0.68485131, 0.69489168, 1. , ..., 0.72293677, 0.73120218,
0.48172501],
...,
[0.83008431, 0.94863663, 0.72293677, ..., 1. , 0.59776459,
0.36311382],
[0.62482353, 0.63064768, 0.73120218, ..., 0.59776459, 1. ,
0.55654437],
[0.34709882, 0.35833279, 0.48172501, ..., 0.36311382, 0.55654437,
1. ]])
```
```python
Gower_Dist_Matrix(Data, p1=4, p2=3, p3=3)
```
```
array([[0. , 0.38502879, 0.56138105, ..., 0.41220831, 0.61251651,
0.808023 ],
[0.38502879, 0. , 0.55236611, ..., 0.22663488, 0.60774363,
0.80104133],
[0.56138105, 0.55236611, 0. , ..., 0.52636796, 0.51845716,
0.71991318],
...,
[0.41220831, 0.22663488, 0.52636796, ..., 0. , 0.63422032,
0.79805149],
[0.61251651, 0.60774363, 0.51845716, ..., 0.63422032, 0. ,
0.66592464],
[0.808023 , 0.80104133, 0.71991318, ..., 0.79805149, 0.66592464,
0. ]])
```
<br>
## Computing Robust Mahalanobis distance
From a theoretical perspective Gnanadesikan (1997) and Delvin et al. (1975) have been followed.
```python
Robust_Mahalanobis_Dist(x_i=Data_quant.iloc[0,:], x_r=Data_quant.iloc[2,:], Data=Data_quant, Method='MAD', epsilon=0.05, n_iters=20)
```
2.1448247626892223
```python
Robust_Mahalanobis_Dist(x_i=Data_quant.iloc[0,:], x_r=Data_quant.iloc[2,:], Data=Data_quant, Method='trimmed', alpha=0.1, epsilon=0.05, n_iters=20)
```
2.7434709885399884
```python
Robust_Mahalanobis_Dist(x_i=Data_quant.iloc[0,:], x_r=Data_quant.iloc[2,:], Data=Data_quant, Method='winsorized', alpha=0.1, epsilon=0.05, n_iters=20)
```
2.8446274140577943
```python
Robust_Mahalanobis_Dist_Matrix(Data=Data_quant, Method='trimmed', alpha=0.1, epsilon=0.05, n_iters=20)
```
```
array([[ 0. , 0.89250845, 2.74347099, ..., 1.48503889,
5.95276234, 8.49453068],
[ 0.89250845, 0. , 1.99959936, ..., 0.96839524,
5.33355737, 8.32070442],
[ 2.74347099, 1.99959936, 0. , ..., 1.36336733,
4.12306341, 9.38094479],
...,
[ 1.48503889, 0.96839524, 1.36336733, ..., 0. ,
5.1322854 , 9.00337923],
[ 5.95276234, 5.33355737, 4.12306341, ..., 5.1322854 ,
0. , 11.06785954],
[ 8.49453068, 8.32070442, 9.38094479, ..., 9.00337923,
11.06785954, 0. ]])
```
<br>
## Computing Generalized Gower distance and Releted Metric Scaling
To end this tutorial we are going to compute both the Gower distance matrix and the Related Metric Scaling matrix for the mixed data-set `Data`. And we are going to do that considering all the possible combinations of the quantitative, binary and multiclass distances. Then, we will save all the resulting matrix in a Python dictionary.
From a theoretical perspective we have followed Cuadras and Fortiana (1998), Albarrán et al. (2015) and Grané et al. (2021).
```python
D_GG_list_maha_robust = []
D_RelMS_list_maha_robust = []
D_GG_list_not_maha_robust = []
D_RelMS_list_not_maha_robust = []
d1_list = ['Euclidean', 'Minkowski', 'Canberra', 'Pearson', 'Mahalanobis']
d2_list = ['Sokal', 'Jaccard']
d3_list = ['Matching']
```
```python
for d in itertools.product(d1_list, d2_list, d3_list) :
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1=d[0], d2=d[1], d3=d[2], q=1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)
D_GG_list_not_maha_robust.append(D)
```
```python
for d in itertools.product(['Robust_Mahalanobis'], d2_list, d3_list, ['trimmed', 'winsorized', 'MAD']) :
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1=d[0], d2=d[1], d3=d[2], epsilon=0.05, Method=d[3], alpha=0.1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)
D_GG_list_maha_robust.append(D)
```
```python
for d in itertools.product(d1_list, d2_list, d3_list) :
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1=d[0], d2=d[1], d3=d[2], q=1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True, tol=0.009, d=2)
D_RelMS_list_not_maha_robust.append(D)
```
```python
for d in itertools.product(['Robust_Mahalanobis'], d2_list, d3_list, ['trimmed', 'winsorized', 'MAD']) :
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1=d[0], d2=d[1], d3=d[2], epsilon=0.05, Method=d[3], alpha=0.1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True, tol=0.009, d=2)
D_RelMS_list_maha_robust.append(D)
```
```python
D_GG_list = D_GG_list_not_maha_robust + D_GG_list_maha_robust
D_RelMS_list = D_RelMS_list_not_maha_robust + D_RelMS_list_maha_robust
```
```python
search_space = [x for x in D_GG_list] + [x for x in D_RelMS_list]
distance_names = ['GG_'+x[0]+'_'+x[1]+'_'+x[2] for x in itertools.product(d1_list, d2_list, d3_list)] + ['GG_'+x[0]+'_'+x[1]+'_'+x[2]+'_'+x[3] for x in itertools.product(['Robust_Mahalanobis'], d2_list, d3_list, ['trimmed', 'winsorized', 'MAD'])] + ['RelMS_'+x[0]+'_'+x[1]+'_'+x[2] for x in itertools.product(d1_list, d2_list, d3_list)] + ['RelMS_'+x[0]+'_'+x[1]+'_'+x[2]+'_'+x[3] for x in itertools.product(['Robust_Mahalanobis'], d2_list, d3_list, ['trimmed', 'winsorized', 'MAD'])]
dic_distance_matrix = dict(zip(distance_names, search_space))
```
```python
dic_distance_matrix
```
```
{'GG_Euclidean_Sokal_Matching': array([[0. , 1.01161446, 1.60800698, ..., 1.23798333, 1.92432848,
6.35838514],
[1.01161446, 0. , 1.64229596, ..., 0.7889253 , 1.87696727,
6.29319748],
[1.60800698, 1.64229596, 0. , ..., 1.42723912, 2.26882579,
6.96673669],
...,
[1.23798333, 0.7889253 , 1.42723912, ..., 0. , 2.4635748 ,
7.01727531],
[1.92432848, 1.87696727, 2.26882579, ..., 2.4635748 , 0. ,
5.11270638],
[6.35838514, 6.29319748, 6.96673669, ..., 7.01727531, 5.11270638,
0. ]]),
'GG_Euclidean_Jaccard_Matching': array([[0. , 1.01161446, 1.60800698, ..., 1.23798333, 1.92432848,
6.21923207],
[1.01161446, 0. , 1.64229596, ..., 0.7889253 , 1.87696727,
6.15257024],
[1.60800698, 1.64229596, 0. , ..., 1.42723912, 2.26882579,
6.83997121],
...,
[1.23798333, 0.7889253 , 1.42723912, ..., 0. , 2.4635748 ,
6.89143953],
[1.92432848, 1.87696727, 2.26882579, ..., 2.4635748 , 0. ,
4.93857798],
[6.21923207, 6.15257024, 6.83997121, ..., 6.89143953, 4.93857798,
0. ]]),
'GG_Minkowski_Sokal_Matching': array([[0. , 1.01161589, 1.60801451, ..., 1.23797549, 1.92440501,
6.35838512],
[1.01161589, 0. , 1.64229192, ..., 0.78891568, 1.87702827,
6.29317915],
[1.60801451, 1.64229192, 0. , ..., 1.42723962, 2.2688732 ,
6.96667937],
...,
[1.23797549, 0.78891568, 1.42723962, ..., 0. , 2.46364348,
7.01724763],
[1.92440501, 1.87702827, 2.2688732 , ..., 2.46364348, 0. ,
5.11260609],
[6.35838512, 6.29317915, 6.96667937, ..., 7.01724763, 5.11260609,
0. ]]),
'GG_Minkowski_Jaccard_Matching': array([[0. , 1.01161589, 1.60801451, ..., 1.23797549, 1.92440501,
6.21923205],
[1.01161589, 0. , 1.64229192, ..., 0.78891568, 1.87702827,
6.15255149],
[1.60801451, 1.64229192, 0. , ..., 1.42723962, 2.2688732 ,
6.83991282],
...,
[1.23797549, 0.78891568, 1.42723962, ..., 0. , 2.46364348,
6.89141134],
[1.92440501, 1.87702827, 2.2688732 , ..., 2.46364348, 0. ,
4.93847416],
[6.21923205, 6.15255149, 6.83991282, ..., 6.89141134, 4.93847416,
0. ]]),
'GG_Canberra_Sokal_Matching': array([[0. , 1.1089173 , 2.04873576, ..., 1.41070641, 2.47064802,
3.88007815],
[1.1089173 , 0. , 1.81887649, ..., 1.10728448, 2.20656591,
3.66760203],
[2.04873576, 1.81887649, 0. , ..., 1.51266848, 2.44536222,
3.67890583],
...,
[1.41070641, 1.10728448, 1.51266848, ..., 0. , 2.92569072,
4.05431191],
[2.47064802, 2.20656591, 2.44536222, ..., 2.92569072, 0. ,
2.67423498],
[3.88007815, 3.66760203, 3.67890583, ..., 4.05431191, 2.67423498,
0. ]]),
'GG_Canberra_Jaccard_Matching': array([[0. , 1.1089173 , 2.04873576, ..., 1.41070641, 2.47064802,
3.64757349],
[1.1089173 , 0. , 1.81887649, ..., 1.10728448, 2.20656591,
3.42068569],
[2.04873576, 1.81887649, 0. , ..., 1.51266848, 2.44536222,
3.43280265],
...,
[1.41070641, 1.10728448, 1.51266848, ..., 0. , 2.92569072,
3.83239234],
[2.47064802, 2.20656591, 2.44536222, ..., 2.92569072, 0. ,
2.32407372],
[3.64757349, 3.42068569, 3.43280265, ..., 3.83239234, 2.32407372,
0. ]]),
'GG_Pearson_Sokal_Matching': array([[0. , 1.0588577 , 1.62258227, ..., 1.13386485, 2.59878376,
4.5833716 ],
[1.0588577 , 0. , 1.54980561, ..., 0.55073019, 2.36782324,
4.41160916],
[1.62258227, 1.54980561, 0. , ..., 1.48883715, 2.15643298,
4.46893998],
...,
[1.13386485, 0.55073019, 1.48883715, ..., 0. , 2.64592015,
4.75194328],
[2.59878376, 2.36782324, 2.15643298, ..., 2.64592015, 0. ,
3.34753806],
[4.5833716 , 4.41160916, 4.46893998, ..., 4.75194328, 3.34753806,
0. ]]),
'GG_Pearson_Jaccard_Matching': array([[0. , 1.0588577 , 1.62258227, ..., 1.13386485, 2.59878376,
4.38828909],
[1.0588577 , 0. , 1.54980561, ..., 0.55073019, 2.36782324,
4.20857237],
[1.62258227, 1.54980561, 0. , ..., 1.48883715, 2.15643298,
4.26863098],
...,
[1.13386485, 0.55073019, 1.48883715, ..., 0. , 2.64592015,
4.56407174],
[2.59878376, 2.36782324, 2.15643298, ..., 2.64592015, 0. ,
3.07502796],
[4.38828909, 4.20857237, 4.26863098, ..., 4.56407174, 3.07502796,
0. ]]),
'GG_Mahalanobis_Sokal_Matching': array([[0. , 1.11128701, 1.9908619 , ..., 1.26642065, 2.97833241,
4.17851469],
[1.11128701, 0. , 1.73337267, ..., 0.49510815, 2.64311668,
4.11353573],
[1.9908619 , 1.73337267, 0. , ..., 1.5815777 , 1.99507289,
4.39053781],
...,
[1.26642065, 0.49510815, 1.5815777 , ..., 0. , 2.63417571,
4.3979867 ],
[2.97833241, 2.64311668, 1.99507289, ..., 2.63417571, 0. ,
4.4698317 ],
[4.17851469, 4.11353573, 4.39053781, ..., 4.3979867 , 4.4698317 ,
0. ]]),
'GG_Mahalanobis_Jaccard_Matching': array([[0. , 1.11128701, 1.9908619 , ..., 1.26642065, 2.97833241,
3.96355535],
[1.11128701, 0. , 1.73337267, ..., 0.49510815, 2.64311668,
3.89499193],
[1.9908619 , 1.73337267, 0. , ..., 1.5815777 , 1.99507289,
4.18647921],
...,
[1.26642065, 0.49510815, 1.5815777 , ..., 0. , 2.63417571,
4.19429052],
[2.97833241, 2.64311668, 1.99507289, ..., 2.63417571, 0. ,
4.26956454],
[3.96355535, 3.89499193, 4.18647921, ..., 4.19429052, 4.26956454,
0. ]]),
'GG_Robust_Mahalanobis_Sokal_Matching_trimmed': array([[0. , 1.0738818 , 1.81990287, ..., 1.17982158, 2.83584093,
4.38026385],
[1.0738818 , 0. , 1.64744788, ..., 0.39866732, 2.61869851,
4.3233478 ],
[1.81990287, 1.64744788, 0. , ..., 1.53344794, 1.97466567,
4.56660697],
...,
[1.17982158, 0.39866732, 1.53344794, ..., 0. , 2.54962302,
4.5492545 ],
[2.83584093, 2.61869851, 1.97466567, ..., 2.54962302, 0. ,
5.16721825],
[4.38026385, 4.3233478 , 4.56660697, ..., 4.5492545 , 5.16721825,
0. ]]),
'GG_Robust_Mahalanobis_Sokal_Matching_winsorized': array([[0. , 1.10035027, 1.96521318, ..., 1.24876507, 3.02193061,
4.2158267 ],
[1.10035027, 0. , 1.72244788, ..., 0.45786845, 2.71169847,
4.170886 ],
[1.96521318, 1.72244788, 0. , ..., 1.57396145, 2.01907767,
4.45138733],
...,
[1.24876507, 0.45786845, 1.57396145, ..., 0. , 2.6589383 ,
4.42575055],
[3.02193061, 2.71169847, 2.01907767, ..., 2.6589383 , 0. ,
4.74960743],
[4.2158267 , 4.170886 , 4.45138733, ..., 4.42575055, 4.74960743,
0. ]]),
'GG_Robust_Mahalanobis_Sokal_Matching_MAD': array([[0. , 1.09006233, 1.80375514, ..., 1.18201607, 2.67497233,
4.55678538],
[1.09006233, 0. , 1.62058379, ..., 0.44488228, 2.40606721,
4.40232615],
[1.80375514, 1.62058379, 0. , ..., 1.53278692, 1.93813141,
4.46679441],
...,
[1.18201607, 0.44488228, 1.53278692, ..., 0. , 2.48916367,
4.64371521],
[2.67497233, 2.40606721, 1.93813141, ..., 2.48916367, 0. ,
4.16671594],
[4.55678538, 4.40232615, 4.46679441, ..., 4.64371521, 4.16671594,
0. ]]),
'GG_Robust_Mahalanobis_Jaccard_Matching_trimmed': array([[0. , 1.0738818 , 1.81990287, ..., 1.17982158, 2.83584093,
4.17570322],
[1.0738818 , 0. , 1.64744788, ..., 0.39866732, 2.61869851,
4.11595944],
[1.81990287, 1.64744788, 0. , ..., 1.53344794, 1.97466567,
4.37077626],
...,
[1.17982158, 0.39866732, 1.53344794, ..., 0. , 2.54962302,
4.35264315],
[2.83584093, 2.61869851, 1.97466567, ..., 2.54962302, 0. ,
4.99499053],
[4.17570322, 4.11595944, 4.37077626, ..., 4.35264315, 4.99499053,
0. ]]),
'GG_Robust_Mahalanobis_Jaccard_Matching_winsorized': array([[0. , 1.10035027, 1.96521318, ..., 1.24876507, 3.02193061,
4.00287155],
[1.10035027, 0. , 1.72244788, ..., 0.45786845, 2.71169847,
3.95551209],
[1.96521318, 1.72244788, 0. , ..., 1.57396145, 2.01907767,
4.25025118],
...,
[1.24876507, 0.45786845, 1.57396145, ..., 0. , 2.6589383 ,
4.22339365],
[3.02193061, 2.71169847, 2.01907767, ..., 2.6589383 , 0. ,
4.5616397 ],
[4.00287155, 3.95551209, 4.25025118, ..., 4.22339365, 4.5616397 ,
0. ]]),
'GG_Robust_Mahalanobis_Jaccard_Matching_MAD': array([[0. , 1.09006233, 1.80375514, ..., 1.18201607, 2.67497233,
4.36051361],
[1.09006233, 0. , 1.62058379, ..., 0.44488228, 2.40606721,
4.19884049],
[1.80375514, 1.62058379, 0. , ..., 1.53278692, 1.93813141,
4.26638468],
...,
[1.18201607, 0.44488228, 1.53278692, ..., 0. , 2.48916367,
4.45127812],
[2.67497233, 2.40606721, 1.93813141, ..., 2.48916367, 0. ,
3.95111474],
[4.36051361, 4.19884049, 4.26638468, ..., 4.45127812, 3.95111474,
0. ]]),
'RelMS_Euclidean_Sokal_Matching': array([[0. , 1.01092438, 1.68587263, ..., 1.2435966 , 1.75479379,
5.76354972],
[1.01092436, 0. , 1.72123768, ..., 0.78892531, 1.71977376,
5.69924943],
[1.68587264, 1.7212377 , 0. , ..., 1.42997022, 2.20660915,
6.5504967 ],
...,
[1.24359658, 0.78892532, 1.42997021, ..., 0. , 2.26671431,
6.42377887],
[1.7547938 , 1.71977375, 2.20660914, ..., 2.26671431, 0. ,
4.781135 ],
[5.76354972, 5.69924943, 6.55049671, ..., 6.42377887, 4.78113499,
0. ]]),
'RelMS_Euclidean_Jaccard_Matching': array([[0. , 1.01092435, 1.68587263, ..., 1.24359659, 1.75479381,
5.73873464],
[1.01092437, 0. , 1.72123769, ..., 0.78892532, 1.71977378,
5.67208311],
[1.68587264, 1.72123769, 0. , ..., 1.42997021, 2.20660914,
6.53309456],
...,
[1.24359658, 0.78892529, 1.42997021, ..., 0. , 2.26671431,
6.41402297],
[1.7547938 , 1.71977375, 2.20660914, ..., 2.2667143 , 0. ,
4.6957284 ],
[5.73873463, 5.67208312, 6.53309457, ..., 6.41402297, 4.69572838,
0. ]]),
'RelMS_Minkowski_Sokal_Matching': array([[0. , 1.0104344 , 1.68473307, ..., 1.24302039, 1.75451827,
5.7636572 ],
[1.01043437, 0. , 1.72039524, ..., 0.78891568, 1.71978231,
5.69946617],
[1.68473308, 1.72039525, 0. , ..., 1.42922921, 2.20651554,
6.55109162],
...,
[1.24302037, 0.7889157 , 1.4292292 , ..., 0. , 2.2667207 ,
6.42402052],
[1.75451827, 1.71978229, 2.20651553, ..., 2.2667207 , 0. ,
4.78235997],
[5.7636572 , 5.69946616, 6.55109161, ..., 6.42402052, 4.78235997,
0. ]]),
'RelMS_Minkowski_Jaccard_Matching': array([[0. , 1.01043437, 1.68473307, ..., 1.24302038, 1.75451828,
5.73875343],
[1.01043439, 0. , 1.72039525, ..., 0.78891569, 1.71978232,
5.67221733],
[1.68473307, 1.72039524, 0. , ..., 1.4292292 , 2.20651553,
6.5336026 ],
...,
[1.24302038, 0.78891568, 1.4292292 , ..., 0. , 2.2667207 ,
6.41417732],
[1.75451828, 1.7197823 , 2.20651553, ..., 2.2667207 , 0. ,
4.6969009 ],
[5.73875342, 5.67221732, 6.5336026 , ..., 6.41417732, 4.6969009 ,
0. ]]),
'RelMS_Canberra_Sokal_Matching': array([[0. , 3.29475825, 3.63767326, ..., 3.42002989, 3.78234978,
4.28387746],
[3.29475817, 0. , 3.54627477, ..., 3.36365755, 3.64707779,
4.11290306],
[3.63767327, 3.5462748 , 0. , ..., 3.36371231, 3.88636668,
4.26421609],
...,
[3.42002989, 3.36365756, 3.36371231, ..., 0. , 4.08835735,
4.43146723],
[3.78234979, 3.64707779, 3.88636667, ..., 4.08835736, 0. ,
3.55682862],
[4.28387745, 4.11290305, 4.26421607, ..., 4.43146723, 3.55682862,
0. ]]),
'RelMS_Canberra_Jaccard_Matching': array([[0. , 3.29475816, 3.63767325, ..., 3.42002988, 3.7823498 ,
4.18398249],
[3.29475818, 0. , 3.54627479, ..., 3.36365756, 3.64707782,
4.00084943],
[3.63767326, 3.54627478, 0. , ..., 3.36371229, 3.88636666,
4.15092751],
...,
[3.42002988, 3.36365755, 3.36371228, ..., 0. , 4.08835736,
4.3378168 ],
[3.78234979, 3.64707778, 3.88636666, ..., 4.08835735, 0. ,
3.36218137],
[4.18398248, 4.00084941, 4.15092752, ..., 4.3378168 , 3.36218137,
0. ]]),
'RelMS_Pearson_Sokal_Matching': array([[0. , 1.04250916, 1.57029271, ..., 1.11835441, 2.35030151,
3.99961285],
[1.04250913, 0. , 1.55642417, ..., 0.55073019, 2.17276224,
3.83629275],
[1.5702927 , 1.55642418, 0. , ..., 1.44481248, 2.11094744,
4.05200057],
...,
[1.11835439, 0.55073021, 1.44481248, ..., 0. , 2.43447697,
4.16544183],
[2.35030151, 2.17276223, 2.11094745, ..., 2.43447697, 0. ,
3.00502738],
[3.99961283, 3.83629274, 4.05200056, ..., 4.16544183, 3.00502738,
0. ]]),
'RelMS_Pearson_Jaccard_Matching': array([[0. , 1.04250913, 1.57029271, ..., 1.11835441, 2.35030152,
3.89789603],
[1.04250915, 0. , 1.55642418, ..., 0.55073023, 2.17276226,
3.72479069],
[1.5702927 , 1.55642415, 0. , ..., 1.44481247, 2.11094744,
3.94329467],
...,
[1.11835439, 0.55073016, 1.44481248, ..., 0. , 2.43447698,
4.07654071],
[2.35030152, 2.17276223, 2.11094745, ..., 2.43447697, 0. ,
2.77842982],
[3.89789601, 3.72479067, 3.94329467, ..., 4.0765407 , 2.77842982,
0. ]]),
'RelMS_Mahalanobis_Sokal_Matching': array([[0. , 1.0872495 , 1.91566724, ..., 1.23718333, 2.78694322,
3.59368169],
[1.08724948, 0. , 1.72190382, ..., 0.49510814, 2.51013925,
3.52430362],
[1.91566725, 1.72190383, 0. , ..., 1.53860587, 1.97114821,
3.91897956],
...,
[1.23718333, 0.49510818, 1.53860586, ..., 0. , 2.47401146,
3.7944967 ],
[2.78694323, 2.51013924, 1.97114821, ..., 2.47401146, 0. ,
4.10401609],
[3.59368167, 3.52430361, 3.91897955, ..., 3.7944967 , 4.10401609,
0. ]]),
'RelMS_Mahalanobis_Jaccard_Matching': array([[0. , 1.08724947, 1.91566724, ..., 1.23718333, 2.78694323,
3.46907215],
[1.0872495 , 0. , 1.72190383, ..., 0.49510817, 2.51013926,
3.39550188],
[1.91566724, 1.72190381, 0. , ..., 1.53860586, 1.97114821,
3.80535063],
...,
[1.23718333, 0.49510812, 1.53860586, ..., 0. , 2.47401147,
3.68911387],
[2.78694323, 2.51013924, 1.97114821, ..., 2.47401147, 0. ,
3.96214705],
[3.46907213, 3.39550187, 3.80535063, ..., 3.68911387, 3.96214705,
0. ]]),
'RelMS_Robust_Mahalanobis_Sokal_Matching_trimmed': array([[0. , 1.05396495, 1.74951184, ..., 1.15390312, 2.67058462,
3.82780883],
[1.05396493, 0. , 1.63479812, ..., 0.39866731, 2.51224528,
3.76362714],
[1.74951185, 1.63479814, 0. , ..., 1.49657109, 1.961588 ,
4.09825745],
...,
[1.15390311, 0.39866735, 1.49657109, ..., 0. , 2.41854434,
3.97375586],
[2.67058463, 2.51224527, 1.961588 , ..., 2.41854434, 0. ,
4.81269468],
[3.82780882, 3.76362713, 4.09825744, ..., 3.97375586, 4.81269468,
0. ]]),
'RelMS_Robust_Mahalanobis_Sokal_Matching_winsorized': array([[0. , 1.07688717, 1.88851059, ..., 1.21940102, 2.83800382,
3.64003684],
[1.07688713, 0. , 1.70819251, ..., 0.45786842, 2.58662722,
3.59029333],
[1.8885106 , 1.70819253, 0. , ..., 1.53220354, 1.99808026,
3.97860895],
...,
[1.21940101, 0.45786849, 1.53220353, ..., 0. , 2.50787408,
3.829693 ],
[2.83800382, 2.58662721, 1.99808026, ..., 2.50787408, 0. ,
4.38739858],
[3.64003683, 3.59029333, 3.97860894, ..., 3.829693 , 4.38739858,
0. ]]),
'RelMS_Robust_Mahalanobis_Sokal_Matching_MAD': array([[0. , 1.06915308, 1.73228661, ..., 1.15789936, 2.45834684,
3.97049139],
[1.06915305, 0. , 1.61195487, ..., 0.44488227, 2.24973009,
3.81621214],
[1.73228661, 1.61195488, 0. , ..., 1.4894837 , 1.90536576,
4.00431571],
...,
[1.15789934, 0.44488231, 1.4894837 , ..., 0. , 2.30824179,
4.04102682],
[2.45834685, 2.24973009, 1.90536577, ..., 2.30824178, 0. ,
3.79967402],
[3.97049139, 3.81621213, 4.0043157 , ..., 4.04102682, 3.79967402,
0. ]]),
'RelMS_Robust_Mahalanobis_Jaccard_Matching_trimmed': array([[0. , 1.05396492, 1.74951184, ..., 1.15390312, 2.67058463,
3.7103996 ],
[1.05396495, 0. , 1.63479813, ..., 0.39866734, 2.51224529,
3.64245313],
[1.74951185, 1.63479812, 0. , ..., 1.49657109, 1.961588 ,
3.98729219],
...,
[1.15390311, 0.39866728, 1.49657109, ..., 0. , 2.41854435,
3.87035377],
[2.67058464, 2.51224527, 1.961588 , ..., 2.41854434, 0. ,
4.69932707],
[3.71039959, 3.64245311, 3.9872922 , ..., 3.87035377, 4.69932707,
0. ]]),
'RelMS_Robust_Mahalanobis_Jaccard_Matching_winsorized': array([[0. , 1.07688714, 1.88851059, ..., 1.21940102, 2.83800383,
3.51619033],
[1.07688715, 0. , 1.70819252, ..., 0.45786846, 2.58662723,
3.46347473],
[1.88851059, 1.70819251, 0. , ..., 1.53220354, 1.99808026,
3.86606614],
...,
[1.219401 , 0.45786843, 1.53220353, ..., 0. , 2.50787409,
3.72394257],
[2.83800382, 2.58662721, 1.99808026, ..., 2.50787408, 0. ,
4.25828147],
[3.51619032, 3.46347472, 3.86606614, ..., 3.72394256, 4.25828147,
0. ]]),
'RelMS_Robust_Mahalanobis_Jaccard_Matching_MAD': array([[0. , 1.06915304, 1.73228661, ..., 1.15789935, 2.45834686,
3.86694579],
[1.06915307, 0. , 1.61195488, ..., 0.4448823 , 2.24973011,
3.7045599 ],
[1.7322866 , 1.61195486, 0. , ..., 1.48948369, 1.90536575,
3.89571711],
...,
[1.15789934, 0.44488225, 1.48948369, ..., 0. , 2.30824179,
3.9478467 ],
[2.45834686, 2.24973009, 1.90536576, ..., 2.30824179, 0. ,
3.64285626],
[3.86694578, 3.70455988, 3.8957171 , ..., 3.9478467 , 3.64285626,
0. ]])}
```
## Computational Cost Testing
In this case, we are going to use the entire `House_Price.csv` dataset, which has 1905 rows, to perform a computational cost test (in terms of time) of the new distance metrics included in `PyDistances`.
```python
Data = pd.read_csv('House_Price.csv')
Data = Data.loc[:, ['latitude', 'longitude', 'price', 'size_in_m_2', 'balcony_recode', 'private_garden_recode', 'private_gym_recode', 'quality_recode', 'no_of_bathrooms', 'no_of_bedrooms']]
```
```python
Data.shape
```
```
(1905, 10)
```
```python
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='trimmed', alpha=0.1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)
# Time: 1.11 minutes.
```
```python
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='winsorized', alpha=0.1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)
# Time: 1.15 minutes.
```
```python
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='MAD', alpha=0.1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)
# Time: 1.12 minutes.
```
```python
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='trimmed', alpha=0.1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True)
# Time: 1.58 minutes.
```
```python
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='winsorized', alpha=0.1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True)
# Time: 1.53 minutes.
```
```python
Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='MAD', alpha=0.1)
D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True)
# Time: 1.55 minutes.
```
We can compare these times with the one obtained by (simple) Gower distance.
```python
Gower_Dist_Matrix(Data, p1=4, p2=3, p3=3)
# Time: 38 seconds.
```
# Bibliography
Albarrán, I., P. Alonso, and A. Grané “Profile Identification via Weighted Related Metric Scaling: An Application to Dependent Spanish Children.” Journal of the Royal Statistical Society. Series A, Statistics in Society 178, no. 3 (2015): 593–618. https://doi.org/10.1111/rssa.12084stex:B88856BB540BB0134A72028E02D7B00CBED08217.
Cuadras, C. M., and J. Fortiana. “Chapter 25 - Visualizing Categorical Data with Related Metric Scaling.” In Visualization of Categorical Data, 365–76. Academic Press, 1998. https://doi.org/10.1016/B978-012299045-8/50028-0.
Devlin, S. J., R. Gnanadesikan, and J. R. Kettenring. “Robust Estimation and Outlier Detection with Correlation Coefficients.” Biometrika 62, no. 3 (1975): 531–45. https://doi.org/10.1093/biomet/62.3.531.
Grané, A., Manzi G. and S. Salini. "Smart Visualization of Mixed Data". Stats n.º 4 (2021): 472–485. https://doi.org/10.3390/stats4020029
Gower, J. C. “A General Coefficient of Similarity and Some of Its Properties.” Biometrics 27, no. 4 (1971): 857–71. https://doi.org/10.2307/2528823.
Gnanadesikan, R. Methods for Statistical Data Analysis of Multivariate Observations. 2nd ed. New York etc.: : John Wiley and Sons, 1997.
Raw data
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"description": "# PyDistances: A Statistical Distances Python Package\r\n\r\nThis is a package for computing distances among observations of statistical variables, such as: Euclidean, Minkowski, Canberra, Pearson, Mahalanobis, Robust Mahalanobis, Gower, Generalized Gower and Related Metric Scaling (RelMS). A total of 41 statistical distances can be computed.\r\n\r\n\r\n## Installation\r\n\r\n```python\r\npip install PyDistances\r\n```\r\n\r\n## Example of use\r\n\r\n\r\n```python\r\nimport PyDistances\r\n```\r\n\r\n```python\r\nfrom PyDistances import Euclidean_Dist, Euclidean_Dist_Matrix, Minkowski_Dist, Minkowski_Dist_Matrix, Canberra_Dist, Canberra_Dist_Matrix, Pearson_Dist, Pearson_Dist_Matrix, Mahalanobis_Dist, Mahalanobis_Dist_Matrix, a_b_c_d_Matrix, Sokal_Similarity, Sokal_Dist, Sokal_Dist_Matrix, Jaccard_Similarity, Jaccard_Dist, Jaccard_Dist_Matrix, alpha, Matching_Similarity, Matching_Dist, Matching_Dist_Matrix, Gower_Similarity_Matrix, Gower_Dist_Matrix, Robust_Mahalanobis_Dist, Robust_Mahalanobis_Dist_Matrix, GeneralizedGowerDistance\r\n```\r\n\r\n### Getting data\r\n\r\nWe load the data we are going to work with throughout this tutorial. This data-set is available in the following link: https://github.com/FabioScielzoOrtiz/Distances_Package/blob/master/Tests/House_Price.csv\r\n```python\r\nData = pd.read_csv('House_Price.csv')\r\n```\r\n\r\n\r\n```python\r\nData = Data.loc[0:150, ['latitude', 'longitude', 'price', 'size_in_m_2', 'balcony_recode', 'private_garden_recode', 'private_gym_recode', 'quality_recode', 'no_of_bathrooms', 'no_of_bedrooms']]\r\n```\r\n\r\n\r\n```python\r\nData_quant = Data.loc[:,['latitude', 'longitude', 'price', 'size_in_m_2']]\r\nData_binary = Data.loc[:,['balcony_recode', 'private_garden_recode', 'private_gym_recode']]\r\nData_multiclass = Data.loc[:,['quality_recode', 'no_of_bathrooms', 'no_of_bedrooms']]\r\n```\r\n\r\n\r\n\r\n```python\r\nData.head() # p1=4, p2=3, p3=3\r\n```\r\n| latitude | longitude | price | size_in_m_2 | balcony | private_garden | private_gym | quality | no_of_bathrooms | no_of_bedrooms |\r\n|:----------:|:-----------:|:----------:|:-------------:|:----------------:|:-----------------------:|:--------------------:|:----------------:|:-----------------:|:----------------:|\r\n| 25.1132 | 55.1389 | 2.7e+06 | 100.242 | 1 | 0 | 0 | 2 | 2 | 1 |\r\n| 25.1068 | 55.1512 | 2.85e+06 | 146.973 | 1 | 0 | 0 | 2 | 2 | 2 |\r\n| 25.0633 | 55.1377 | 1.15e+06 | 181.254 | 1 | 0 | 0 | 2 | 5 | 3 |\r\n| 25.2273 | 55.3418 | 2.85e+06 | 187.664 | 1 | 0 | 0 | 1 | 3 | 2 |\r\n| 25.1143 | 55.1398 | 1.7292e+06 | 47.1018 | 0 | 0 | 0 | 2 | 1 | 0 |\r\n\r\n\r\n<br>\r\n\r\n## Computing Euclidean distance\r\n\r\nWe compute the Euclidean distance between observation of index 0 and itself.\r\n```python\r\nEuclidean_Dist(Data_quant.iloc[0,:], Data_quant.iloc[0,:])\r\n```\r\n \r\n 0.0\r\n\r\n\r\nWe compute the Euclidean distance between observation of index 0 and the one of index 2.\r\n\r\n```python\r\nEuclidean_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:])\r\n```\r\n\r\n 1550000.002117049\r\n\r\n\r\nWe compute the Euclidean distances matrix for the data-set `Data_quant`.\r\n```python\r\nEuclidean_Dist_Matrix(Data_quant)\r\n```\r\n```\r\narray([[ 0. , 150000.00727904, 1550000.00211705, ...,\r\n 1500000.00009635, 2700000.01899102, 12100000.00553371],\r\n [ 150000.00727904, 0. , 1700000.00034565, ...,\r\n 1650000.00026782, 2550000.0146678 , 11950000.00426352],\r\n [ 1550000.00211705, 1700000.00034565, 0. , ...,\r\n 50000.040973 , 4250000.00673279, 13650000.00297389],\r\n ...,\r\n [ 1500000.00009635, 1650000.00026782, 50000.040973 , ...,\r\n 0. , 4200000.01094663, 13600000.00447653],\r\n [ 2700000.01899102, 2550000.0146678 , 4250000.00673279, ...,\r\n 4200000.01094663, 0. , 9400000.00011113],\r\n [12100000.00553371, 11950000.00426352, 13650000.00297389, ...,\r\n 13600000.00447653, 9400000.00011113, 0. ]])\r\n```\r\n\r\n<br>\r\n\r\nNow, we are going to repeat the same procedure with other available distances in `PyDistances`.\r\n\r\n<br>\r\n\r\n## Computing Minkowski distance\r\n\r\n\r\n```python\r\nMinkowski_Dist(Data_quant.iloc[0,:], Data_quant.iloc[0,:], q=1)\r\n```\r\n\r\n 0.0\r\n\r\n\r\n```python\r\nMinkowski_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:], q=1)\r\n```\r\n 1550081.062526\r\n\r\n\r\n\r\n```python\r\nMinkowski_Dist_Matrix(Data_quant, q=1)\r\n```\r\n```\r\narray([[ 0. , 150046.748877, 1550081.062526, ...,\r\n 1500017.050769, 2700320.266531, 12100365.997115],\r\n [ 150046.748877, 0. , 1700034.338187, ...,\r\n 1650029.78435 , 2550273.554024, 11950319.272776],\r\n [ 1550081.062526, 1700034.338187, 0. , ...,\r\n 50064.027555, 4250239.302851, 13650284.955165],\r\n ...,\r\n [ 1500017.050769, 1650029.78435 , 50064.027555, ...,\r\n 0. , 4200303.29563 , 13600348.947944],\r\n [ 2700320.266531, 2550273.554024, 4250239.302851, ...,\r\n 4200303.29563 , 0. , 9400045.764238],\r\n [12100365.997115, 11950319.272776, 13650284.955165, ...,\r\n 13600348.947944, 9400045.764238, 0. ]])\r\n```\r\n\r\n<br>\r\n\r\n## Computing Canberra distance\r\n\r\n```python\r\nCanberra_Dist(Data_quant.iloc[0,:], Data_quant.iloc[0,:])\r\n```\r\n\r\n 0.0\r\n\r\n```python\r\nCanberra_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:])\r\n``` \r\n\r\n 0.6913917083019879\r\n\r\n```python\r\nCanberra_Dist_Matrix(Data_quant)\r\n```\r\n\r\n```\r\narray([[0. , 0.21629237, 0.69139171, ..., 0.463675 , 0.9485963 ,\r\n 1.33838751],\r\n [0.21629237, 0. , 0.53043317, ..., 0.52079671, 0.79157752,\r\n 1.19854721],\r\n [0.69139171, 0.53043317, 0. , ..., 0.23597883, 1.04765637,\r\n 1.29619958],\r\n ...,\r\n [0.463675 , 0.52079671, 0.23597883, ..., 0. , 1.20126891,\r\n 1.44813664],\r\n [0.9485963 , 0.79157752, 1.04765637, ..., 1.20126891, 0. ,\r\n 0.51782969],\r\n [1.33838751, 1.19854721, 1.29619958, ..., 1.44813664, 0.51782969,\r\n 0. ]])\r\n```\r\n\r\n<br>\r\n\r\n## Computing Pearson distance\r\n\r\n```python\r\nPearson_Dist(Data_quant.iloc[0,:], Data_quant.iloc[0,:], variance=Data.var())\r\n```\r\n\r\n 0.0\r\n\r\n```python\r\nPearson_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:], variance=Data.var())\r\n```\r\n\r\n 1.5393297661160206\r\n\r\n\r\n```python\r\nPearson_Dist_Matrix(Data_quant)\r\n```\r\n```\r\narray([[0. , 0.63961801, 1.53932977, ..., 1.03084131, 4.32943281,\r\n 7.47171915],\r\n [0.63961801, 0. , 1.20505141, ..., 1.09780711, 3.76643257,\r\n 7.04893716],\r\n [1.53932977, 1.20505141, 0. , ..., 0.84617436, 3.79891055,\r\n 7.4670243 ],\r\n ...,\r\n [1.03084131, 1.09780711, 0.84617436, ..., 0. , 4.44143053,\r\n 7.87905955],\r\n [4.32943281, 3.76643257, 3.79891055, ..., 4.44143053, 0. ,\r\n 4.57460318],\r\n [7.47171915, 7.04893716, 7.4670243 , ..., 7.87905955, 4.57460318,\r\n 0. ]])\r\n```\r\n\r\n\r\n<br>\r\n\r\n## Computing Mahalanobis distance\r\n\r\n```python\r\nMahalanobis_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:], S_inv=np.linalg.inv( np.cov(Data_quant , rowvar=False) ))\r\n```\r\n\r\n 0.0\r\n\r\n\r\n```python\r\nMahalanobis_Dist(Data_quant.iloc[0,:], Data_quant.iloc[2,:], S_inv=np.linalg.inv( np.cov(Data_quant , rowvar=False) ))\r\n```\r\n\r\n 2.7671855371187757\r\n\r\n```python\r\nMahalanobis_Dist_Matrix(Data_quant)\r\n```\r\n\r\n```\r\narray([[0. , 0.92801614, 2.76718554, ..., 1.52541554, 5.21105193,\r\n 6.45997793],\r\n [0.92801614, 0. , 1.96135599, ..., 0.98693199, 4.43479282,\r\n 6.2920865 ],\r\n [2.76718554, 1.96135599, 0. , ..., 1.3592188 , 3.4307313 ,\r\n 7.27986558],\r\n ...,\r\n [1.52541554, 0.98693199, 1.3592188 , ..., 0. , 4.41360406,\r\n 7.01503103],\r\n [5.21105193, 4.43479282, 3.4307313 , ..., 4.41360406, 0. ,\r\n 7.4691448 ],\r\n [6.45997793, 6.2920865 , 7.27986558, ..., 7.01503103, 7.4691448 ,\r\n 0. ]])\r\n```\r\n\r\n\r\n<br>\r\n\r\n## Computing Sokal similarity\r\n\r\n```python\r\na,b,c,d,p = a_b_c_d_Matrix(Data_binary)\r\n```\r\n\r\n\r\n```python\r\nSokal_Similarity(i=0, r=2, a=a, d=d, p=p)\r\n```\r\n\r\n 1.0\r\n\r\n```python\r\nSokal_Dist(i=0, r=2, a=a, d=d, p=p)\r\n```\r\n 0.0\r\n\r\n\r\n```python\r\nSokal_Dist_Matrix(Data_binary)\r\n```\r\n```\r\narray([[0. , 0. , 0. , ..., 0. , 0. ,\r\n 0.81649658],\r\n [0. , 0. , 0. , ..., 0. , 0. ,\r\n 0.81649658],\r\n [0. , 0. , 0. , ..., 0. , 0. ,\r\n 0.81649658],\r\n ...,\r\n [0. , 0. , 0. , ..., 0. , 0. ,\r\n 0.81649658],\r\n [0. , 0. , 0. , ..., 0. , 0. ,\r\n 0.81649658],\r\n [0.81649658, 0.81649658, 0.81649658, ..., 0.81649658, 0.81649658,\r\n 0. ]])\r\n```\r\n\r\n\r\n<br>\r\n\r\n## Computing Jaccard similarity\r\n\r\n```python\r\nJaccard_Similarity(i=0, r=2, a=a, d=d, p=p)\r\n```\r\n 1.0\r\n\r\n\r\n```python\r\nJaccard_Dist(i=0, r=2, a=a, d=d, p=p)\r\n```\r\n 0.0\r\n\r\n```python\r\nJaccard_Dist_Matrix(Data_binary)\r\n```\r\n```\r\narray([[0., 0., 0., ..., 0., 0., 1.],\r\n [0., 0., 0., ..., 0., 0., 1.],\r\n [0., 0., 0., ..., 0., 0., 1.],\r\n ...,\r\n [0., 0., 0., ..., 0., 0., 1.],\r\n [0., 0., 0., ..., 0., 0., 1.],\r\n [1., 1., 1., ..., 1., 1., 0.]])\r\n```\r\n\r\n\r\n<br>\r\n\r\n## Computing Matching similarity\r\n\r\n```python\r\nMatching_Similarity(x_i=Data_multiclass.iloc[0,:], x_r=Data_multiclass.iloc[2,:], Data=Data_multiclass)\r\n```\r\n\r\n 0.3333333333333333\r\n\r\n\r\n```python\r\nMatching_Dist(x_i=Data_multiclass.iloc[0,:], x_r=Data_multiclass.iloc[2,:], Data=Data_multiclass)\r\n```\r\n\r\n 1.1547005383792517\r\n\r\n\r\n```python\r\nMatching_Dist_Matrix(Data_multiclass)\r\n```\r\n```\r\narray([[0. , 0.81649658, 1.15470054, ..., 0.81649658, 1.15470054,\r\n 1.41421356],\r\n [0.81649658, 0. , 1.15470054, ..., 0. , 1.15470054,\r\n 1.41421356],\r\n [1.15470054, 1.15470054, 0. , ..., 1.15470054, 0.81649658,\r\n 1.15470054],\r\n ...,\r\n [0.81649658, 0. , 1.15470054, ..., 0. , 1.15470054,\r\n 1.41421356],\r\n [1.15470054, 1.15470054, 0.81649658, ..., 1.15470054, 0. ,\r\n 1.15470054],\r\n [1.41421356, 1.41421356, 1.15470054, ..., 1.41421356, 1.15470054,\r\n 0. ]])\r\n```\r\n\r\n<br>\r\n\r\n## Computing Gower distance\r\n\r\nFrom a theoretical perspective Gower (1971) has been followed.\r\n\r\n```python\r\nGower_Similarity_Matrix(Data, p1=4, p2=3, p3=3)\r\n```\r\n\r\n```\r\narray([[1. , 0.85175283, 0.68485131, ..., 0.83008431, 0.62482353,\r\n 0.34709882],\r\n [0.85175283, 1. , 0.69489168, ..., 0.94863663, 0.63064768,\r\n 0.35833279],\r\n [0.68485131, 0.69489168, 1. , ..., 0.72293677, 0.73120218,\r\n 0.48172501],\r\n ...,\r\n [0.83008431, 0.94863663, 0.72293677, ..., 1. , 0.59776459,\r\n 0.36311382],\r\n [0.62482353, 0.63064768, 0.73120218, ..., 0.59776459, 1. ,\r\n 0.55654437],\r\n [0.34709882, 0.35833279, 0.48172501, ..., 0.36311382, 0.55654437,\r\n 1. ]])\r\n```\r\n\r\n```python\r\nGower_Dist_Matrix(Data, p1=4, p2=3, p3=3)\r\n```\r\n\r\n```\r\narray([[0. , 0.38502879, 0.56138105, ..., 0.41220831, 0.61251651,\r\n 0.808023 ],\r\n [0.38502879, 0. , 0.55236611, ..., 0.22663488, 0.60774363,\r\n 0.80104133],\r\n [0.56138105, 0.55236611, 0. , ..., 0.52636796, 0.51845716,\r\n 0.71991318],\r\n ...,\r\n [0.41220831, 0.22663488, 0.52636796, ..., 0. , 0.63422032,\r\n 0.79805149],\r\n [0.61251651, 0.60774363, 0.51845716, ..., 0.63422032, 0. ,\r\n 0.66592464],\r\n [0.808023 , 0.80104133, 0.71991318, ..., 0.79805149, 0.66592464,\r\n 0. ]])\r\n```\r\n\r\n\r\n<br>\r\n\r\n## Computing Robust Mahalanobis distance\r\n\r\nFrom a theoretical perspective Gnanadesikan (1997) and Delvin et al. (1975) have been followed.\r\n\r\n```python\r\nRobust_Mahalanobis_Dist(x_i=Data_quant.iloc[0,:], x_r=Data_quant.iloc[2,:], Data=Data_quant, Method='MAD', epsilon=0.05, n_iters=20)\r\n```\r\n 2.1448247626892223\r\n\r\n```python\r\nRobust_Mahalanobis_Dist(x_i=Data_quant.iloc[0,:], x_r=Data_quant.iloc[2,:], Data=Data_quant, Method='trimmed', alpha=0.1, epsilon=0.05, n_iters=20)\r\n```\r\n 2.7434709885399884\r\n\r\n\r\n```python\r\nRobust_Mahalanobis_Dist(x_i=Data_quant.iloc[0,:], x_r=Data_quant.iloc[2,:], Data=Data_quant, Method='winsorized', alpha=0.1, epsilon=0.05, n_iters=20)\r\n```\r\n 2.8446274140577943\r\n\r\n```python\r\nRobust_Mahalanobis_Dist_Matrix(Data=Data_quant, Method='trimmed', alpha=0.1, epsilon=0.05, n_iters=20)\r\n```\r\n\r\n```\r\narray([[ 0. , 0.89250845, 2.74347099, ..., 1.48503889,\r\n 5.95276234, 8.49453068],\r\n [ 0.89250845, 0. , 1.99959936, ..., 0.96839524,\r\n 5.33355737, 8.32070442],\r\n [ 2.74347099, 1.99959936, 0. , ..., 1.36336733,\r\n 4.12306341, 9.38094479],\r\n ...,\r\n [ 1.48503889, 0.96839524, 1.36336733, ..., 0. ,\r\n 5.1322854 , 9.00337923],\r\n [ 5.95276234, 5.33355737, 4.12306341, ..., 5.1322854 ,\r\n 0. , 11.06785954],\r\n [ 8.49453068, 8.32070442, 9.38094479, ..., 9.00337923,\r\n 11.06785954, 0. ]])\r\n```\r\n\r\n<br>\r\n\r\n## Computing Generalized Gower distance and Releted Metric Scaling\r\n\r\nTo end this tutorial we are going to compute both the Gower distance matrix and the Related Metric Scaling matrix for the mixed data-set `Data`. And we are going to do that considering all the possible combinations of the quantitative, binary and multiclass distances. Then, we will save all the resulting matrix in a Python dictionary.\r\n\r\nFrom a theoretical perspective we have followed Cuadras and Fortiana (1998), Albarr\u00e1n et al. (2015) and Gran\u00e9 et al. (2021).\r\n\r\n```python\r\nD_GG_list_maha_robust = []\r\nD_RelMS_list_maha_robust = []\r\nD_GG_list_not_maha_robust = []\r\nD_RelMS_list_not_maha_robust = []\r\n\r\nd1_list = ['Euclidean', 'Minkowski', 'Canberra', 'Pearson', 'Mahalanobis']\r\nd2_list = ['Sokal', 'Jaccard']\r\nd3_list = ['Matching']\r\n```\r\n\r\n```python\r\nfor d in itertools.product(d1_list, d2_list, d3_list) :\r\n Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1=d[0], d2=d[1], d3=d[2], q=1)\r\n D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)\r\n D_GG_list_not_maha_robust.append(D)\r\n```\r\n\r\n```python\r\nfor d in itertools.product(['Robust_Mahalanobis'], d2_list, d3_list, ['trimmed', 'winsorized', 'MAD']) :\r\n Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1=d[0], d2=d[1], d3=d[2], epsilon=0.05, Method=d[3], alpha=0.1)\r\n D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)\r\n D_GG_list_maha_robust.append(D)\r\n```\r\n\r\n\r\n```python\r\nfor d in itertools.product(d1_list, d2_list, d3_list) :\r\n Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1=d[0], d2=d[1], d3=d[2], q=1)\r\n D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True, tol=0.009, d=2)\r\n D_RelMS_list_not_maha_robust.append(D)\r\n```\r\n \r\n\r\n```python\r\nfor d in itertools.product(['Robust_Mahalanobis'], d2_list, d3_list, ['trimmed', 'winsorized', 'MAD']) :\r\n Generalized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1=d[0], d2=d[1], d3=d[2], epsilon=0.05, Method=d[3], alpha=0.1)\r\n D, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True, tol=0.009, d=2)\r\n D_RelMS_list_maha_robust.append(D)\r\n```\r\n\r\n```python\r\nD_GG_list = D_GG_list_not_maha_robust + D_GG_list_maha_robust\r\nD_RelMS_list = D_RelMS_list_not_maha_robust + D_RelMS_list_maha_robust\r\n```\r\n\r\n\r\n```python\r\nsearch_space = [x for x in D_GG_list] + [x for x in D_RelMS_list]\r\ndistance_names = ['GG_'+x[0]+'_'+x[1]+'_'+x[2] for x in itertools.product(d1_list, d2_list, d3_list)] + ['GG_'+x[0]+'_'+x[1]+'_'+x[2]+'_'+x[3] for x in itertools.product(['Robust_Mahalanobis'], d2_list, d3_list, ['trimmed', 'winsorized', 'MAD'])] + ['RelMS_'+x[0]+'_'+x[1]+'_'+x[2] for x in itertools.product(d1_list, d2_list, d3_list)] + ['RelMS_'+x[0]+'_'+x[1]+'_'+x[2]+'_'+x[3] for x in itertools.product(['Robust_Mahalanobis'], d2_list, d3_list, ['trimmed', 'winsorized', 'MAD'])]\r\ndic_distance_matrix = dict(zip(distance_names, search_space))\r\n```\r\n\r\n```python\r\ndic_distance_matrix\r\n```\r\n\r\n\r\n```\r\n{'GG_Euclidean_Sokal_Matching': array([[0. , 1.01161446, 1.60800698, ..., 1.23798333, 1.92432848,\r\n 6.35838514],\r\n [1.01161446, 0. , 1.64229596, ..., 0.7889253 , 1.87696727,\r\n 6.29319748],\r\n [1.60800698, 1.64229596, 0. , ..., 1.42723912, 2.26882579,\r\n 6.96673669],\r\n ...,\r\n [1.23798333, 0.7889253 , 1.42723912, ..., 0. , 2.4635748 ,\r\n 7.01727531],\r\n [1.92432848, 1.87696727, 2.26882579, ..., 2.4635748 , 0. ,\r\n 5.11270638],\r\n [6.35838514, 6.29319748, 6.96673669, ..., 7.01727531, 5.11270638,\r\n 0. ]]),\r\n 'GG_Euclidean_Jaccard_Matching': array([[0. , 1.01161446, 1.60800698, ..., 1.23798333, 1.92432848,\r\n 6.21923207],\r\n [1.01161446, 0. , 1.64229596, ..., 0.7889253 , 1.87696727,\r\n 6.15257024],\r\n [1.60800698, 1.64229596, 0. , ..., 1.42723912, 2.26882579,\r\n 6.83997121],\r\n ...,\r\n [1.23798333, 0.7889253 , 1.42723912, ..., 0. , 2.4635748 ,\r\n 6.89143953],\r\n [1.92432848, 1.87696727, 2.26882579, ..., 2.4635748 , 0. ,\r\n 4.93857798],\r\n [6.21923207, 6.15257024, 6.83997121, ..., 6.89143953, 4.93857798,\r\n 0. ]]),\r\n 'GG_Minkowski_Sokal_Matching': array([[0. , 1.01161589, 1.60801451, ..., 1.23797549, 1.92440501,\r\n 6.35838512],\r\n [1.01161589, 0. , 1.64229192, ..., 0.78891568, 1.87702827,\r\n 6.29317915],\r\n [1.60801451, 1.64229192, 0. , ..., 1.42723962, 2.2688732 ,\r\n 6.96667937],\r\n ...,\r\n [1.23797549, 0.78891568, 1.42723962, ..., 0. , 2.46364348,\r\n 7.01724763],\r\n [1.92440501, 1.87702827, 2.2688732 , ..., 2.46364348, 0. ,\r\n 5.11260609],\r\n [6.35838512, 6.29317915, 6.96667937, ..., 7.01724763, 5.11260609,\r\n 0. ]]),\r\n 'GG_Minkowski_Jaccard_Matching': array([[0. , 1.01161589, 1.60801451, ..., 1.23797549, 1.92440501,\r\n 6.21923205],\r\n [1.01161589, 0. , 1.64229192, ..., 0.78891568, 1.87702827,\r\n 6.15255149],\r\n [1.60801451, 1.64229192, 0. , ..., 1.42723962, 2.2688732 ,\r\n 6.83991282],\r\n ...,\r\n [1.23797549, 0.78891568, 1.42723962, ..., 0. , 2.46364348,\r\n 6.89141134],\r\n [1.92440501, 1.87702827, 2.2688732 , ..., 2.46364348, 0. ,\r\n 4.93847416],\r\n [6.21923205, 6.15255149, 6.83991282, ..., 6.89141134, 4.93847416,\r\n 0. ]]),\r\n 'GG_Canberra_Sokal_Matching': array([[0. , 1.1089173 , 2.04873576, ..., 1.41070641, 2.47064802,\r\n 3.88007815],\r\n [1.1089173 , 0. , 1.81887649, ..., 1.10728448, 2.20656591,\r\n 3.66760203],\r\n [2.04873576, 1.81887649, 0. , ..., 1.51266848, 2.44536222,\r\n 3.67890583],\r\n ...,\r\n [1.41070641, 1.10728448, 1.51266848, ..., 0. , 2.92569072,\r\n 4.05431191],\r\n [2.47064802, 2.20656591, 2.44536222, ..., 2.92569072, 0. ,\r\n 2.67423498],\r\n [3.88007815, 3.66760203, 3.67890583, ..., 4.05431191, 2.67423498,\r\n 0. ]]),\r\n 'GG_Canberra_Jaccard_Matching': array([[0. , 1.1089173 , 2.04873576, ..., 1.41070641, 2.47064802,\r\n 3.64757349],\r\n [1.1089173 , 0. , 1.81887649, ..., 1.10728448, 2.20656591,\r\n 3.42068569],\r\n [2.04873576, 1.81887649, 0. , ..., 1.51266848, 2.44536222,\r\n 3.43280265],\r\n ...,\r\n [1.41070641, 1.10728448, 1.51266848, ..., 0. , 2.92569072,\r\n 3.83239234],\r\n [2.47064802, 2.20656591, 2.44536222, ..., 2.92569072, 0. ,\r\n 2.32407372],\r\n [3.64757349, 3.42068569, 3.43280265, ..., 3.83239234, 2.32407372,\r\n 0. ]]),\r\n 'GG_Pearson_Sokal_Matching': array([[0. , 1.0588577 , 1.62258227, ..., 1.13386485, 2.59878376,\r\n 4.5833716 ],\r\n [1.0588577 , 0. , 1.54980561, ..., 0.55073019, 2.36782324,\r\n 4.41160916],\r\n [1.62258227, 1.54980561, 0. , ..., 1.48883715, 2.15643298,\r\n 4.46893998],\r\n ...,\r\n [1.13386485, 0.55073019, 1.48883715, ..., 0. , 2.64592015,\r\n 4.75194328],\r\n [2.59878376, 2.36782324, 2.15643298, ..., 2.64592015, 0. ,\r\n 3.34753806],\r\n [4.5833716 , 4.41160916, 4.46893998, ..., 4.75194328, 3.34753806,\r\n 0. ]]),\r\n 'GG_Pearson_Jaccard_Matching': array([[0. , 1.0588577 , 1.62258227, ..., 1.13386485, 2.59878376,\r\n 4.38828909],\r\n [1.0588577 , 0. , 1.54980561, ..., 0.55073019, 2.36782324,\r\n 4.20857237],\r\n [1.62258227, 1.54980561, 0. , ..., 1.48883715, 2.15643298,\r\n 4.26863098],\r\n ...,\r\n [1.13386485, 0.55073019, 1.48883715, ..., 0. , 2.64592015,\r\n 4.56407174],\r\n [2.59878376, 2.36782324, 2.15643298, ..., 2.64592015, 0. ,\r\n 3.07502796],\r\n [4.38828909, 4.20857237, 4.26863098, ..., 4.56407174, 3.07502796,\r\n 0. ]]),\r\n 'GG_Mahalanobis_Sokal_Matching': array([[0. , 1.11128701, 1.9908619 , ..., 1.26642065, 2.97833241,\r\n 4.17851469],\r\n [1.11128701, 0. , 1.73337267, ..., 0.49510815, 2.64311668,\r\n 4.11353573],\r\n [1.9908619 , 1.73337267, 0. , ..., 1.5815777 , 1.99507289,\r\n 4.39053781],\r\n ...,\r\n [1.26642065, 0.49510815, 1.5815777 , ..., 0. , 2.63417571,\r\n 4.3979867 ],\r\n [2.97833241, 2.64311668, 1.99507289, ..., 2.63417571, 0. ,\r\n 4.4698317 ],\r\n [4.17851469, 4.11353573, 4.39053781, ..., 4.3979867 , 4.4698317 ,\r\n 0. ]]),\r\n 'GG_Mahalanobis_Jaccard_Matching': array([[0. , 1.11128701, 1.9908619 , ..., 1.26642065, 2.97833241,\r\n 3.96355535],\r\n [1.11128701, 0. , 1.73337267, ..., 0.49510815, 2.64311668,\r\n 3.89499193],\r\n [1.9908619 , 1.73337267, 0. , ..., 1.5815777 , 1.99507289,\r\n 4.18647921],\r\n ...,\r\n [1.26642065, 0.49510815, 1.5815777 , ..., 0. , 2.63417571,\r\n 4.19429052],\r\n [2.97833241, 2.64311668, 1.99507289, ..., 2.63417571, 0. ,\r\n 4.26956454],\r\n [3.96355535, 3.89499193, 4.18647921, ..., 4.19429052, 4.26956454,\r\n 0. ]]),\r\n 'GG_Robust_Mahalanobis_Sokal_Matching_trimmed': array([[0. , 1.0738818 , 1.81990287, ..., 1.17982158, 2.83584093,\r\n 4.38026385],\r\n [1.0738818 , 0. , 1.64744788, ..., 0.39866732, 2.61869851,\r\n 4.3233478 ],\r\n [1.81990287, 1.64744788, 0. , ..., 1.53344794, 1.97466567,\r\n 4.56660697],\r\n ...,\r\n [1.17982158, 0.39866732, 1.53344794, ..., 0. , 2.54962302,\r\n 4.5492545 ],\r\n [2.83584093, 2.61869851, 1.97466567, ..., 2.54962302, 0. ,\r\n 5.16721825],\r\n [4.38026385, 4.3233478 , 4.56660697, ..., 4.5492545 , 5.16721825,\r\n 0. ]]),\r\n 'GG_Robust_Mahalanobis_Sokal_Matching_winsorized': array([[0. , 1.10035027, 1.96521318, ..., 1.24876507, 3.02193061,\r\n 4.2158267 ],\r\n [1.10035027, 0. , 1.72244788, ..., 0.45786845, 2.71169847,\r\n 4.170886 ],\r\n [1.96521318, 1.72244788, 0. , ..., 1.57396145, 2.01907767,\r\n 4.45138733],\r\n ...,\r\n [1.24876507, 0.45786845, 1.57396145, ..., 0. , 2.6589383 ,\r\n 4.42575055],\r\n [3.02193061, 2.71169847, 2.01907767, ..., 2.6589383 , 0. ,\r\n 4.74960743],\r\n [4.2158267 , 4.170886 , 4.45138733, ..., 4.42575055, 4.74960743,\r\n 0. ]]),\r\n 'GG_Robust_Mahalanobis_Sokal_Matching_MAD': array([[0. , 1.09006233, 1.80375514, ..., 1.18201607, 2.67497233,\r\n 4.55678538],\r\n [1.09006233, 0. , 1.62058379, ..., 0.44488228, 2.40606721,\r\n 4.40232615],\r\n [1.80375514, 1.62058379, 0. , ..., 1.53278692, 1.93813141,\r\n 4.46679441],\r\n ...,\r\n [1.18201607, 0.44488228, 1.53278692, ..., 0. , 2.48916367,\r\n 4.64371521],\r\n [2.67497233, 2.40606721, 1.93813141, ..., 2.48916367, 0. ,\r\n 4.16671594],\r\n [4.55678538, 4.40232615, 4.46679441, ..., 4.64371521, 4.16671594,\r\n 0. ]]),\r\n 'GG_Robust_Mahalanobis_Jaccard_Matching_trimmed': array([[0. , 1.0738818 , 1.81990287, ..., 1.17982158, 2.83584093,\r\n 4.17570322],\r\n [1.0738818 , 0. , 1.64744788, ..., 0.39866732, 2.61869851,\r\n 4.11595944],\r\n [1.81990287, 1.64744788, 0. , ..., 1.53344794, 1.97466567,\r\n 4.37077626],\r\n ...,\r\n [1.17982158, 0.39866732, 1.53344794, ..., 0. , 2.54962302,\r\n 4.35264315],\r\n [2.83584093, 2.61869851, 1.97466567, ..., 2.54962302, 0. ,\r\n 4.99499053],\r\n [4.17570322, 4.11595944, 4.37077626, ..., 4.35264315, 4.99499053,\r\n 0. ]]),\r\n 'GG_Robust_Mahalanobis_Jaccard_Matching_winsorized': array([[0. , 1.10035027, 1.96521318, ..., 1.24876507, 3.02193061,\r\n 4.00287155],\r\n [1.10035027, 0. , 1.72244788, ..., 0.45786845, 2.71169847,\r\n 3.95551209],\r\n [1.96521318, 1.72244788, 0. , ..., 1.57396145, 2.01907767,\r\n 4.25025118],\r\n ...,\r\n [1.24876507, 0.45786845, 1.57396145, ..., 0. , 2.6589383 ,\r\n 4.22339365],\r\n [3.02193061, 2.71169847, 2.01907767, ..., 2.6589383 , 0. ,\r\n 4.5616397 ],\r\n [4.00287155, 3.95551209, 4.25025118, ..., 4.22339365, 4.5616397 ,\r\n 0. ]]),\r\n 'GG_Robust_Mahalanobis_Jaccard_Matching_MAD': array([[0. , 1.09006233, 1.80375514, ..., 1.18201607, 2.67497233,\r\n 4.36051361],\r\n [1.09006233, 0. , 1.62058379, ..., 0.44488228, 2.40606721,\r\n 4.19884049],\r\n [1.80375514, 1.62058379, 0. , ..., 1.53278692, 1.93813141,\r\n 4.26638468],\r\n ...,\r\n [1.18201607, 0.44488228, 1.53278692, ..., 0. , 2.48916367,\r\n 4.45127812],\r\n [2.67497233, 2.40606721, 1.93813141, ..., 2.48916367, 0. ,\r\n 3.95111474],\r\n [4.36051361, 4.19884049, 4.26638468, ..., 4.45127812, 3.95111474,\r\n 0. ]]),\r\n 'RelMS_Euclidean_Sokal_Matching': array([[0. , 1.01092438, 1.68587263, ..., 1.2435966 , 1.75479379,\r\n 5.76354972],\r\n [1.01092436, 0. , 1.72123768, ..., 0.78892531, 1.71977376,\r\n 5.69924943],\r\n [1.68587264, 1.7212377 , 0. , ..., 1.42997022, 2.20660915,\r\n 6.5504967 ],\r\n ...,\r\n [1.24359658, 0.78892532, 1.42997021, ..., 0. , 2.26671431,\r\n 6.42377887],\r\n [1.7547938 , 1.71977375, 2.20660914, ..., 2.26671431, 0. ,\r\n 4.781135 ],\r\n [5.76354972, 5.69924943, 6.55049671, ..., 6.42377887, 4.78113499,\r\n 0. ]]),\r\n 'RelMS_Euclidean_Jaccard_Matching': array([[0. , 1.01092435, 1.68587263, ..., 1.24359659, 1.75479381,\r\n 5.73873464],\r\n [1.01092437, 0. , 1.72123769, ..., 0.78892532, 1.71977378,\r\n 5.67208311],\r\n [1.68587264, 1.72123769, 0. , ..., 1.42997021, 2.20660914,\r\n 6.53309456],\r\n ...,\r\n [1.24359658, 0.78892529, 1.42997021, ..., 0. , 2.26671431,\r\n 6.41402297],\r\n [1.7547938 , 1.71977375, 2.20660914, ..., 2.2667143 , 0. ,\r\n 4.6957284 ],\r\n [5.73873463, 5.67208312, 6.53309457, ..., 6.41402297, 4.69572838,\r\n 0. ]]),\r\n 'RelMS_Minkowski_Sokal_Matching': array([[0. , 1.0104344 , 1.68473307, ..., 1.24302039, 1.75451827,\r\n 5.7636572 ],\r\n [1.01043437, 0. , 1.72039524, ..., 0.78891568, 1.71978231,\r\n 5.69946617],\r\n [1.68473308, 1.72039525, 0. , ..., 1.42922921, 2.20651554,\r\n 6.55109162],\r\n ...,\r\n [1.24302037, 0.7889157 , 1.4292292 , ..., 0. , 2.2667207 ,\r\n 6.42402052],\r\n [1.75451827, 1.71978229, 2.20651553, ..., 2.2667207 , 0. ,\r\n 4.78235997],\r\n [5.7636572 , 5.69946616, 6.55109161, ..., 6.42402052, 4.78235997,\r\n 0. ]]),\r\n 'RelMS_Minkowski_Jaccard_Matching': array([[0. , 1.01043437, 1.68473307, ..., 1.24302038, 1.75451828,\r\n 5.73875343],\r\n [1.01043439, 0. , 1.72039525, ..., 0.78891569, 1.71978232,\r\n 5.67221733],\r\n [1.68473307, 1.72039524, 0. , ..., 1.4292292 , 2.20651553,\r\n 6.5336026 ],\r\n ...,\r\n [1.24302038, 0.78891568, 1.4292292 , ..., 0. , 2.2667207 ,\r\n 6.41417732],\r\n [1.75451828, 1.7197823 , 2.20651553, ..., 2.2667207 , 0. ,\r\n 4.6969009 ],\r\n [5.73875342, 5.67221732, 6.5336026 , ..., 6.41417732, 4.6969009 ,\r\n 0. ]]),\r\n 'RelMS_Canberra_Sokal_Matching': array([[0. , 3.29475825, 3.63767326, ..., 3.42002989, 3.78234978,\r\n 4.28387746],\r\n [3.29475817, 0. , 3.54627477, ..., 3.36365755, 3.64707779,\r\n 4.11290306],\r\n [3.63767327, 3.5462748 , 0. , ..., 3.36371231, 3.88636668,\r\n 4.26421609],\r\n ...,\r\n [3.42002989, 3.36365756, 3.36371231, ..., 0. , 4.08835735,\r\n 4.43146723],\r\n [3.78234979, 3.64707779, 3.88636667, ..., 4.08835736, 0. ,\r\n 3.55682862],\r\n [4.28387745, 4.11290305, 4.26421607, ..., 4.43146723, 3.55682862,\r\n 0. ]]),\r\n 'RelMS_Canberra_Jaccard_Matching': array([[0. , 3.29475816, 3.63767325, ..., 3.42002988, 3.7823498 ,\r\n 4.18398249],\r\n [3.29475818, 0. , 3.54627479, ..., 3.36365756, 3.64707782,\r\n 4.00084943],\r\n [3.63767326, 3.54627478, 0. , ..., 3.36371229, 3.88636666,\r\n 4.15092751],\r\n ...,\r\n [3.42002988, 3.36365755, 3.36371228, ..., 0. , 4.08835736,\r\n 4.3378168 ],\r\n [3.78234979, 3.64707778, 3.88636666, ..., 4.08835735, 0. ,\r\n 3.36218137],\r\n [4.18398248, 4.00084941, 4.15092752, ..., 4.3378168 , 3.36218137,\r\n 0. ]]),\r\n 'RelMS_Pearson_Sokal_Matching': array([[0. , 1.04250916, 1.57029271, ..., 1.11835441, 2.35030151,\r\n 3.99961285],\r\n [1.04250913, 0. , 1.55642417, ..., 0.55073019, 2.17276224,\r\n 3.83629275],\r\n [1.5702927 , 1.55642418, 0. , ..., 1.44481248, 2.11094744,\r\n 4.05200057],\r\n ...,\r\n [1.11835439, 0.55073021, 1.44481248, ..., 0. , 2.43447697,\r\n 4.16544183],\r\n [2.35030151, 2.17276223, 2.11094745, ..., 2.43447697, 0. ,\r\n 3.00502738],\r\n [3.99961283, 3.83629274, 4.05200056, ..., 4.16544183, 3.00502738,\r\n 0. ]]),\r\n 'RelMS_Pearson_Jaccard_Matching': array([[0. , 1.04250913, 1.57029271, ..., 1.11835441, 2.35030152,\r\n 3.89789603],\r\n [1.04250915, 0. , 1.55642418, ..., 0.55073023, 2.17276226,\r\n 3.72479069],\r\n [1.5702927 , 1.55642415, 0. , ..., 1.44481247, 2.11094744,\r\n 3.94329467],\r\n ...,\r\n [1.11835439, 0.55073016, 1.44481248, ..., 0. , 2.43447698,\r\n 4.07654071],\r\n [2.35030152, 2.17276223, 2.11094745, ..., 2.43447697, 0. ,\r\n 2.77842982],\r\n [3.89789601, 3.72479067, 3.94329467, ..., 4.0765407 , 2.77842982,\r\n 0. ]]),\r\n 'RelMS_Mahalanobis_Sokal_Matching': array([[0. , 1.0872495 , 1.91566724, ..., 1.23718333, 2.78694322,\r\n 3.59368169],\r\n [1.08724948, 0. , 1.72190382, ..., 0.49510814, 2.51013925,\r\n 3.52430362],\r\n [1.91566725, 1.72190383, 0. , ..., 1.53860587, 1.97114821,\r\n 3.91897956],\r\n ...,\r\n [1.23718333, 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'RelMS_Robust_Mahalanobis_Jaccard_Matching_winsorized': array([[0. , 1.07688714, 1.88851059, ..., 1.21940102, 2.83800383,\r\n 3.51619033],\r\n [1.07688715, 0. , 1.70819252, ..., 0.45786846, 2.58662723,\r\n 3.46347473],\r\n [1.88851059, 1.70819251, 0. , ..., 1.53220354, 1.99808026,\r\n 3.86606614],\r\n ...,\r\n [1.219401 , 0.45786843, 1.53220353, ..., 0. , 2.50787409,\r\n 3.72394257],\r\n [2.83800382, 2.58662721, 1.99808026, ..., 2.50787408, 0. ,\r\n 4.25828147],\r\n [3.51619032, 3.46347472, 3.86606614, ..., 3.72394256, 4.25828147,\r\n 0. ]]),\r\n 'RelMS_Robust_Mahalanobis_Jaccard_Matching_MAD': array([[0. , 1.06915304, 1.73228661, ..., 1.15789935, 2.45834686,\r\n 3.86694579],\r\n [1.06915307, 0. , 1.61195488, ..., 0.4448823 , 2.24973011,\r\n 3.7045599 ],\r\n [1.7322866 , 1.61195486, 0. , ..., 1.48948369, 1.90536575,\r\n 3.89571711],\r\n ...,\r\n [1.15789934, 0.44488225, 1.48948369, ..., 0. , 2.30824179,\r\n 3.9478467 ],\r\n [2.45834686, 2.24973009, 1.90536576, ..., 2.30824179, 0. ,\r\n 3.64285626],\r\n [3.86694578, 3.70455988, 3.8957171 , ..., 3.9478467 , 3.64285626,\r\n 0. ]])}\r\n```\r\n\r\n\r\n## Computational Cost Testing\r\n\r\nIn this case, we are going to use the entire `House_Price.csv` dataset, which has 1905 rows, to perform a computational cost test (in terms of time) of the new distance metrics included in `PyDistances`.\r\n\r\n```python\r\nData = pd.read_csv('House_Price.csv')\r\nData = Data.loc[:, ['latitude', 'longitude', 'price', 'size_in_m_2', 'balcony_recode', 'private_garden_recode', 'private_gym_recode', 'quality_recode', 'no_of_bathrooms', 'no_of_bedrooms']]\r\n```\r\n\r\n```python\r\nData.shape\r\n```\r\n``` \r\n(1905, 10)\r\n```\r\n\r\n```python\r\nGeneralized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='trimmed', alpha=0.1)\r\nD, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)\r\n\r\n# Time: 1.11 minutes.\r\n```\r\n\r\n```python\r\nGeneralized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='winsorized', alpha=0.1)\r\nD, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)\r\n\r\n# Time: 1.15 minutes.\r\n```\r\n\r\n\r\n```python\r\nGeneralized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='MAD', alpha=0.1)\r\nD, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=False)\r\n\r\n# Time: 1.12 minutes.\r\n```\r\n\r\n```python\r\nGeneralized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='trimmed', alpha=0.1)\r\nD, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True)\r\n\r\n# Time: 1.58 minutes.\r\n```\r\n\r\n```python\r\nGeneralized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='winsorized', alpha=0.1)\r\nD, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True)\r\n\r\n# Time: 1.53 minutes.\r\n```\r\n\r\n```python\r\nGeneralized_Gower_Distance_init = GeneralizedGowerDistance(Data=Data, p1=4, p2=3, p3=3, d1='Robust_Mahalanobis', d2='Jaccard', d3='Matching', epsilon=0.05, Method='MAD', alpha=0.1)\r\nD, D_2 = Generalized_Gower_Distance_init.compute(Related_Metric_Scaling=True)\r\n\r\n# Time: 1.55 minutes.\r\n```\r\n\r\nWe can compare these times with the one obtained by (simple) Gower distance.\r\n\r\n```python\r\nGower_Dist_Matrix(Data, p1=4, p2=3, p3=3)\r\n\r\n# Time: 38 seconds.\r\n```\r\n\r\n\r\n\r\n# Bibliography\r\n\r\nAlbarr\u00e1n, I., P. Alonso, and A. Gran\u00e9 \u201cProfile Identification via Weighted Related Metric Scaling: An Application to Dependent Spanish Children.\u201d Journal of the Royal Statistical Society. Series A, Statistics in Society 178, no. 3 (2015): 593\u2013618. https://doi.org/10.1111/rssa.12084stex:B88856BB540BB0134A72028E02D7B00CBED08217.\r\n\r\nCuadras, C. M., and J. Fortiana. \u201cChapter 25 - Visualizing Categorical Data with Related Metric Scaling.\u201d In Visualization of Categorical Data, 365\u201376. Academic Press, 1998. https://doi.org/10.1016/B978-012299045-8/50028-0.\r\n\r\nDevlin, S. J., R. Gnanadesikan, and J. R. Kettenring. \u201cRobust Estimation and Outlier Detection with Correlation Coefficients.\u201d Biometrika 62, no. 3 (1975): 531\u201345. https://doi.org/10.1093/biomet/62.3.531.\r\n\r\nGran\u00e9, A., Manzi G. and S. Salini. \"Smart Visualization of Mixed Data\". Stats n.\u00ba 4 (2021): 472\u2013485. https://doi.org/10.3390/stats4020029\r\n\r\n\r\nGower, J. C. \u201cA General Coefficient of Similarity and Some of Its Properties.\u201d Biometrics 27, no. 4 (1971): 857\u201371. https://doi.org/10.2307/2528823.\r\n\r\nGnanadesikan, R. Methods for Statistical Data Analysis of Multivariate Observations. 2nd ed. New York etc.: : John Wiley and Sons, 1997.\r\n\r\n",
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