In [1]:

from google.colab import drive
import os
drive.mount('/content/gdrive')
# Establecer ruta de acceso en dr
import os
print(os.getcwd())
os.chdir("/content/gdrive/My Drive")

Mounted at /content/gdrive
/content

Analisis Factorial (FA)¶

El análisis factorial es un enfoque útil para encontrar variables latentes que no se miden directamente en una sola variable, sino que se infieren de otras variables en el conjunto de datos. Estas variables latentes se denominan factores. Entonces, el análisis factorial es un modelo de medición de variables latentes. Por ejemplo, si encontramos dos variables latentes en nuestro modelo, se llama modelo de dos factores. La suposición principal del FA es que existen tales variables latentes en nuestros datos.

En la actualidad, realizamos el análisis factorial utilizando el método de componentes principales, que es muy similar al análisis de componentes principales. Utilizaremos datos de registros nacionales de mujeres que representan a 55 países en siete eventos diferentes.

Las variables son:

X1: 100m (s)
X2: 200m (s)
X3: 400m (s)
X4: 800m (min)
X5: 1500m (min)
X6: 3000m (min)
X7: Marathon (min)

In [2]:

import pandas as pd
df = pd.read_csv("women_track_records.csv")
df.head()

Out[2]:

	COUNTRY	X1	X2	X3	X4	X5	X6	X7
0	Argentina	11.61	22.94	54.50	2.15	4.43	9.79	178.52
1	Australia	11.20	22.35	51.80	1.98	4.13	9.08	152.37
2	Austria	11.43	23.09	50.62	1.99	4.22	9.34	159.37
3	Belgium	11.41	23.04	52.00	2.00	4.14	8.88	157.85
4	Bermuda	11.46	23.05	53.30	2.16	4.58	9.81	169.98

In [5]:

X = df.iloc[:, 1:8]
X.head()

Out[5]:

	X1	X2	X3	X4	X5	X6	X7
0	11.61	22.94	54.50	2.15	4.43	9.79	178.52
1	11.20	22.35	51.80	1.98	4.13	9.08	152.37
2	11.43	23.09	50.62	1.99	4.22	9.34	159.37
3	11.41	23.04	52.00	2.00	4.14	8.88	157.85
4	11.46	23.05	53.30	2.16	4.58	9.81	169.98

In [7]:

from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_scaled = sc.fit_transform(X)
X_scaled

Out[7]:

array([[-2.24428486e-02, -5.62237236e-01,  3.00209295e-01,
         6.86677436e-01,  8.06867242e-02,  3.93983736e-01,
         1.74496205e-01],
       [-9.42599641e-01, -1.08128697e+00, -6.11096516e-01,
        -8.98614916e-01, -6.80073818e-01, -2.51633758e-01,
        -6.91870707e-01],
       [-4.26414124e-01, -4.30275438e-01, -1.00937091e+00,
        -8.05362425e-01, -4.51845656e-01, -1.52104506e-02,
        -4.59956046e-01],
       [-4.71299821e-01, -4.74262704e-01, -5.43592382e-01,
        -7.12109934e-01, -6.54715134e-01, -4.33497841e-01,
        -5.10314658e-01],
       [-3.59085578e-01, -4.65465251e-01, -1.04815510e-01,
         7.79929927e-01,  4.61066996e-01,  4.12170144e-01,
        -1.08439681e-01],
       [-6.95728307e-01, -3.59895812e-01, -2.73575845e-01,
         2.20414979e-01,  2.32838833e-01,  3.75797327e-01,
        -1.49190400e-01],
       [ 1.16702813e+00,  7.83773102e-01,  4.68969630e-01,
         9.66434910e-01,  1.31404094e-01,  1.39374020e-01,
         5.88629529e-01],
       [-1.39145661e+00, -1.16926151e+00, -1.01612132e+00,
        -7.12109934e-01, -8.57584612e-01, -4.97150270e-01,
        -7.88612251e-01],
       [ 8.52828247e-01, -1.81147559e+00,  4.35217563e-01,
        -2.45847477e-01, -4.26486971e-01,  1.20691619e-02,
        -6.20567490e-02],
       [ 7.40614004e-01,  7.30988383e-01,  4.58844010e-01,
         3.39099968e-02, -1.72900123e-01, -4.24900630e-02,
        -1.58135680e-01],
       [-4.48856972e-02,  3.70292802e-01, -1.18316337e-01,
         3.13667471e-01, -1.22182754e-01,  9.39079991e-02,
        -2.59515518e-01],
       [ 2.87268462e+00,  3.09750329e+00,  2.29158125e+00,
         2.08546481e+00,  1.12039280e+00,  1.58519348e+00,
         1.98674363e+00],
       [ 7.63056853e-01,  8.98139994e-01,  1.56591181e+00,
         1.24619238e+00,  7.14653843e-01,  9.75948801e-01,
        -4.81418694e-02],
       [-1.18947098e+00, -1.41559019e+00, -1.89705027e+00,
        -1.73788734e+00, -6.54715134e-01, -4.94372709e+00,
        -4.77183992e-01],
       [-4.48856972e-01, -5.19849507e-02,  3.37164740e+00,
        -4.32352460e-01, -5.53280395e-01, -5.88082311e-01,
        -7.12411720e-01],
       [ 3.81528426e-01,  4.14280068e-01,  8.23366334e-01,
         1.52594986e+00,  8.66805952e-01,  4.84915777e-01,
         1.01468989e+00],
       [-1.09969958e+00, -1.04609716e+00, -1.17138083e+00,
        -4.32352460e-01, -7.56149873e-01, -3.97125024e-01,
        -6.30247669e-01],
       [-1.05481388e+00, -1.22204622e+00, -6.34722963e-01,
        -7.12109934e-01, -6.54715134e-01, -3.42565800e-01,
        -6.28922442e-01],
       [-1.63832795e+00, -1.64432398e+00, -1.83967176e+00,
        -1.36487737e+00, -1.11117146e+00, -5.51709495e-01,
        -5.15946871e-01],
       [-1.36901376e+00, -1.04609716e+00, -1.30301389e+00,
        -1.17837239e+00, -9.33660666e-01, -6.97200761e-01,
        -8.19092464e-01],
       [-1.39145661e+00, -1.27483094e+00, -1.06337421e+00,
        -8.98614916e-01, -9.33660666e-01, -6.69921149e-01,
        -7.79666972e-01],
       [ 3.81528426e-01,  4.40672428e-01,  4.45343183e-01,
        -5.93424945e-02, -1.22182754e-01,  4.66729369e-01,
         2.96417056e-01],
       [ 4.93742669e-01,  8.45355275e-01,  8.36867160e-01,
         1.89895982e+00,  1.17111017e+00,  1.07597405e+00,
         1.40960743e+00],
       [-3.81528426e-01, -4.56667797e-01, -7.12352717e-01,
        -6.18857442e-01, -6.54715134e-01, -3.42565800e-01,
        -5.59348044e-01],
       [ 7.40614004e-01,  6.16621492e-01, -3.55930889e-03,
         2.20414979e-01, -1.98258808e-01,  5.66754614e-01,
         4.98514117e-01],
       [ 5.16185518e-01,  5.81431679e-01,  5.83726658e-01,
         1.33944488e+00,  5.37143050e-01,  6.03127431e-01,
         9.28550160e-01],
       [-4.26414124e-01, -6.07824039e-02, -1.25066750e-01,
        -2.45847477e-01, -7.30791188e-01, -4.24404637e-01,
        -7.90931398e-01],
       [-3.81528426e-01, -7.99768472e-03,  4.35217563e-01,
         2.20414979e-01, -3.75769601e-01,  1.20691619e-02,
        -4.23181007e-01],
       [-7.40614004e-01, -5.09452517e-01, -5.40217175e-01,
        -1.08511990e+00, -1.06045409e+00, -6.60827944e-01,
        -7.10092573e-01],
       [ 2.46871335e-01,  3.70292802e-01,  4.03183783e-02,
         1.27162488e-01,  2.41368572e+00, -1.42515309e-01,
        -7.53825052e-01],
       [ 2.46871335e-01,  2.64723364e-01, -3.07327912e-01,
        -7.12109934e-01,  1.90651203e+00, -1.42515309e-01,
         2.58316790e-01],
       [ 7.63056853e-01,  8.01368009e-01,  7.05234099e-01,
         6.86677436e-01,  5.53280395e-02,  2.39399265e-01,
        -2.85026130e-01],
       [ 1.41389946e+00,  1.93623947e+00, -8.13608918e-01,
        -9.91867408e-01, -3.75769601e-01, -6.11724642e-03,
         1.96031138e-01],
       [ 9.20156793e-01,  1.21484831e+00,  8.40242367e-01,
        -5.93424945e-02, -4.61066996e-02,  2.57585674e-01,
         4.72744483e-02],
       [ 1.36901376e+00,  5.55039319e-01,  4.99346490e-01,
         1.05968740e+00,  7.40012528e-01,  1.00322841e+00,
         2.95423136e-01],
       [ 3.14199880e-01,  1.32041775e+00,  1.51528371e+00,
         1.80570733e+00,  9.93599375e-01,  1.40332940e+00,
         2.91142051e+00],
       [ 6.05956912e-01,  3.59895812e-02,  5.04439984e-02,
        -3.39099968e-01, -3.75769601e-01,  2.12119653e-01,
        -4.87785805e-01],
       [-8.30385398e-01, -6.76604127e-01, -4.15334527e-01,
        -8.05362425e-01,  1.67828386e+00, -3.15286187e-01,
        -6.88226334e-01],
       [-1.57099940e-01, -3.95085625e-01, -6.78600650e-01,
        -5.25604951e-01,  1.98258808e+00, -5.42616291e-01,
        -9.20140995e-01],
       [-8.97713944e-02, -2.36731468e-01, -1.65569231e-01,
        -4.32352460e-01, -9.84378036e-01, -7.51759986e-01,
        -9.20140995e-01],
       [ 1.41389946e+00,  1.27643048e+00,  1.13051014e+00,
         1.52594986e+00,  1.12039280e+00,  1.21237211e+00,
         1.97945488e+00],
       [ 3.14199880e-01, -3.43900443e-02,  3.33961362e-01,
         1.05968740e+00,  5.11784365e-01,  7.30432289e-01,
         8.98401254e-01],
       [-1.09969958e+00, -1.20445132e+00, -1.45827340e+00,
        -1.17837239e+00, -1.03509541e+00, -3.51659004e-01,
        -4.11916581e-01],
       [ 4.26414124e-01,  5.63836773e-01,  2.32705160e-01,
         1.27162488e-01, -6.03997764e-01, -4.69870658e-01,
        -7.30633586e-01],
       [-4.03971275e-01, -1.04769670e-01, -8.13608918e-01,
        -1.45812986e+00, -1.11117146e+00, -7.51759986e-01,
        -2.58521598e-01],
       [ 1.52611371e+00,  1.25003812e+00,  4.95971283e-01,
         4.06919962e-01,  3.08914887e-01,  5.30381798e-01,
         3.15301535e-01],
       [ 4.03971275e-01,  3.52697896e-01, -6.93451559e-03,
        -2.45847477e-01, -6.54715134e-01, -3.06192983e-01,
        -3.52943995e-01],
       [-1.03237104e+00, -6.67806674e-01, -6.14471722e-01,
        -5.25604951e-01, -7.05432503e-01, -4.69870658e-01,
        -6.21965002e-01],
       [-3.81528426e-01, -2.36731468e-01, -1.68944437e-01,
        -5.25604951e-01, -8.32225927e-01, -5.33523087e-01,
        -6.57083508e-01],
       [-8.97713944e-01, -8.43755738e-01, -3.74832046e-01,
         2.20414979e-01, -4.61066996e-02,  2.48492469e-01,
         1.52961272e-01],
       [ 2.91757032e-01,  7.74975649e-01,  7.38986166e-01,
         1.15293989e+00,  8.16088582e-01,  8.39550739e-01,
        -1.59129600e-01],
       [ 8.07942550e-01,  7.57380743e-01,  9.58374602e-01,
         6.86677436e-01, -7.14653843e-02,  2.11623660e-02,
         9.21924027e-01],
       [-1.86275643e+00, -1.53875454e+00, -1.00937091e+00,
        -1.08511990e+00, -1.13653014e+00, -7.79039598e-01,
        -1.01158163e+00],
       [-1.25679952e+00, -1.22204622e+00, -1.49202547e+00,
        -1.73788734e+00, -1.33939962e+00, -8.24505619e-01,
        -7.29970973e-01],
       [ 2.51359904e+00,  1.99782164e+00,  1.72792173e+00,
         2.36522228e+00,  3.58018522e+00,  3.34927508e+00,
         4.39799349e+00]])

In [8]:

!pip install factor_analyzer

Collecting factor_analyzer
  Downloading factor_analyzer-0.4.0.tar.gz (41 kB)
     |████████████████████████████████| 41 kB 486 kB/s 
Requirement already satisfied: pandas in /usr/local/lib/python3.7/dist-packages (from factor_analyzer) (1.3.5)
Requirement already satisfied: scipy in /usr/local/lib/python3.7/dist-packages (from factor_analyzer) (1.4.1)
Requirement already satisfied: numpy in /usr/local/lib/python3.7/dist-packages (from factor_analyzer) (1.21.6)
Requirement already satisfied: scikit-learn in /usr/local/lib/python3.7/dist-packages (from factor_analyzer) (1.0.2)
Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas->factor_analyzer) (2.8.2)
Requirement already satisfied: pytz>=2017.3 in /usr/local/lib/python3.7/dist-packages (from pandas->factor_analyzer) (2022.1)
Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas->factor_analyzer) (1.15.0)
Requirement already satisfied: threadpoolctl>=2.0.0 in /usr/local/lib/python3.7/dist-packages (from scikit-learn->factor_analyzer) (3.1.0)
Requirement already satisfied: joblib>=0.11 in /usr/local/lib/python3.7/dist-packages (from scikit-learn->factor_analyzer) (1.1.0)
Building wheels for collected packages: factor-analyzer
  Building wheel for factor-analyzer (setup.py) ... done
  Created wheel for factor-analyzer: filename=factor_analyzer-0.4.0-py3-none-any.whl size=41455 sha256=3d1d93c5107dbc5248504614853c4452dfb77e7b27e85ae0941bc08f1d68247e
  Stored in directory: /root/.cache/pip/wheels/ac/00/37/1f0e8a5039f9e9f207c4405bbce0796f07701eb377bfc6cc76
Successfully built factor-analyzer
Installing collected packages: factor-analyzer
Successfully installed factor-analyzer-0.4.0

Realicemos FA nuevamente con nfactors = 2 (anteriormente, nfactors = 4). Esto se debe a que hemos decidido mantener solo dos factores para nuestros datos

In [9]:

from factor_analyzer import FactorAnalyzer
fa = FactorAnalyzer(n_factors=2, rotation="varimax", method="principal", 
                    is_corr_matrix=False)
fa.fit(X_scaled)

/usr/local/lib/python3.7/dist-packages/sklearn/utils/extmath.py:376: FutureWarning: If 'random_state' is not supplied, the current default is to use 0 as a fixed seed. This will change to  None in version 1.2 leading to non-deterministic results that better reflect nature of the randomized_svd solver. If you want to silence this warning, set 'random_state' to an integer seed or to None explicitly depending if you want your code to be deterministic or not.
  FutureWarning,

Out[9]:

FactorAnalyzer(method='principal', n_factors=2, rotation='varimax',
               rotation_kwargs={})

In [10]:

print("Valores propios:")
print(fa.get_eigenvalues()[0])
print()
print("Communalities:")
print(fa.get_communalities())
print()
print("Varianzas especificas:")
print(fa.get_uniquenesses())
print()
print("Cargas de los factores:")
print(fa.loadings_)

Valores propios:
[5.06759677 0.6020256  0.44429295 0.36590389 0.26931274 0.13929091
 0.11157713]

Communalities:
[0.8632044  0.86854473 0.77623794 0.84827979 0.79619776 0.73826138
 0.77889637]

Varianzas especificas:
[0.1367956  0.13145527 0.22376206 0.15172021 0.20380224 0.26173862
 0.22110363]

Cargas de los factores:
[[0.8399412  0.3971186 ]
 [0.86109019 0.35646657]
 [0.81415209 0.33674071]
 [0.61543129 0.6852183 ]
 [0.22614824 0.86316553]
 [0.48965453 0.7060452 ]
 [0.46668107 0.74906952]]

Comunalidades

X1: Alrededor del 86% de la variabilidad de X1 se explica por los dos factores que seleccionamos.
X2: Alrededor del 87% de la variabilidad de X2 se explica por los dos factores que seleccionamos.
X3: Alrededor del 78% de la variabilidad de X3 se explica por los dos factores que seleccionamos.
X4: Alrededor del 85% de la variabilidad de X4 se explica por los dos factores que seleccionamos.
X5: Alrededor del 80% de la variabilidad de X5 se explica por los dos factores que seleccionamos.
X6: Alrededor del 74% de la variabilidad de X6 se explica por los dos factores que seleccionamos.
X7: Alrededor del 78% de la variabilidad de X7 se explica por los dos factores que seleccionamos.

variaciones específicas

El efecto del factor específico sobre X1 es de alrededor del 14%.
El efecto del factor específico sobre X2 es de alrededor del 13%.
El efecto del factor específico sobre X3 es de alrededor del 22%.
El efecto del factor específico sobre X4 es de alrededor del 15%.
El efecto del factor específico sobre X5 es de alrededor del 20%.
El efecto del factor específico sobre X6 es de alrededor del 26%.
El efecto del factor específico en X7 es de alrededor del 22%

In [11]:

import matplotlib.pyplot as plt
plt.style.use("ggplot")

plt.plot(fa.get_eigenvalues()[0], marker='o')
plt.xlabel("Eigenvalue ")
plt.ylabel("Eigenvalue tamaño")
plt.title("Scree Plot")

Out[11]:

Text(0.5, 1.0, 'Scree Plot')

Está claro que las variables X1, X2, X3 definen el factor 1 (cargas altas en el factor 1, cargas relativamente pequeñas en el factor 2) mientras que las variables X4, X5, X6 y X7 definen el factor 2 (cargas altas en el factor 2, cargas relativamente pequeñas en el factor 2). factor 1). Pero la variable X4 tiene aspectos de atributos representados por ambos factores (cargas aproximadamente iguales en ambos factores).

Para dar nombres a los dos factores, centrémonos en el dominio del conocimiento del campo.

El conjunto de datos proporcionado representa los registros nacionales de mujeres que representan a 55 países en siete eventos diferentes. Generalmente, en carreras de corta distancia (por ejemplo, 100 m, 200 m, 400 m), los atletas deben centrarse principalmente en la velocidad. En carreras de larga distancia (por ejemplo, 1500 m, 3000 m, maratón), los atletas deben centrarse principalmente en la tolerancia o la resistencia. En nuestro análisis, el factor 1 representa los antecedentes de corta distancia (ya que X1, X2 y X3) y el factor 2 representa los antecedentes de larga distancia (ya que X4, X5, X6 y X7). Por lo tanto, podemos dar nombres relevantes para los dos factores de la siguiente manera.

Factor 1 → factor de velocidad
Factor 2 → tolerancia o factor de resistencia

In [12]:

transformed_df = pd.DataFrame(fa.transform(X_scaled), columns=['RF1', 'RF2'])
transformed_df

Out[12]:

	RF1	RF2
0	-0.213747	0.492749
1	-0.864122	-0.351732
2	-0.645847	-0.195162
3	-0.335540	-0.573830
4	-0.628536	0.822508
5	-0.719778	0.603115
6	1.014360	-0.016134
7	-1.178504	-0.320728
8	-0.129174	-0.167418
9	1.007324	-0.676723
10	0.200318	-0.146080
11	3.068299	0.444342
12	1.177901	0.243695
13	-1.407983	-1.528352
14	1.721239	-1.727718
15	0.262662	1.080618
16	-1.123549	-0.150988
17	-0.999261	-0.227916
18	-1.856963	-0.146468
19	-1.205515	-0.520131
20	-1.213444	-0.427182
21	0.565193	-0.147662
22	0.309477	1.578275
23	-0.363157	-0.522204
24	0.609653	-0.034210
25	0.430094	0.805991
26	0.173307	-0.864563
27	0.257576	-0.397847
28	-0.263473	-1.020930
29	-0.609446	1.162281
30	-0.777327	1.148980
31	1.087221	-0.419528
32	1.304007	-1.043131
33	1.392007	-0.704095
34	0.712123	0.605590
35	0.618732	1.868953
36	0.509618	-0.612468
37	-1.532721	1.070597
38	-1.285372	1.013866
39	0.365952	-1.256078
40	0.986941	1.328728
41	-0.087738	1.039127
42	-1.278473	-0.287339
43	0.984739	-1.144108
44	-0.088259	-1.106852
45	1.322910	-0.213729
46	0.699542	-0.931063
47	-0.666742	-0.425431
48	0.109772	-0.929756
49	-1.006417	0.695615
50	0.500099	0.559547
51	1.110912	-0.166870
52	-1.406942	-0.571562
53	-1.178561	-0.856376
54	0.564617	4.267786

Analisis Discriminante¶

In [13]:

from sklearn.datasets import load_iris
from sklearn.decomposition import PCA
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis

iris = load_iris()
X = iris.data
y = iris.target
X

Out[13]:

array([[5.1, 3.5, 1.4, 0.2],
       [4.9, 3. , 1.4, 0.2],
       [4.7, 3.2, 1.3, 0.2],
       [4.6, 3.1, 1.5, 0.2],
       [5. , 3.6, 1.4, 0.2],
       [5.4, 3.9, 1.7, 0.4],
       [4.6, 3.4, 1.4, 0.3],
       [5. , 3.4, 1.5, 0.2],
       [4.4, 2.9, 1.4, 0.2],
       [4.9, 3.1, 1.5, 0.1],
       [5.4, 3.7, 1.5, 0.2],
       [4.8, 3.4, 1.6, 0.2],
       [4.8, 3. , 1.4, 0.1],
       [4.3, 3. , 1.1, 0.1],
       [5.8, 4. , 1.2, 0.2],
       [5.7, 4.4, 1.5, 0.4],
       [5.4, 3.9, 1.3, 0.4],
       [5.1, 3.5, 1.4, 0.3],
       [5.7, 3.8, 1.7, 0.3],
       [5.1, 3.8, 1.5, 0.3],
       [5.4, 3.4, 1.7, 0.2],
       [5.1, 3.7, 1.5, 0.4],
       [4.6, 3.6, 1. , 0.2],
       [5.1, 3.3, 1.7, 0.5],
       [4.8, 3.4, 1.9, 0.2],
       [5. , 3. , 1.6, 0.2],
       [5. , 3.4, 1.6, 0.4],
       [5.2, 3.5, 1.5, 0.2],
       [5.2, 3.4, 1.4, 0.2],
       [4.7, 3.2, 1.6, 0.2],
       [4.8, 3.1, 1.6, 0.2],
       [5.4, 3.4, 1.5, 0.4],
       [5.2, 4.1, 1.5, 0.1],
       [5.5, 4.2, 1.4, 0.2],
       [4.9, 3.1, 1.5, 0.2],
       [5. , 3.2, 1.2, 0.2],
       [5.5, 3.5, 1.3, 0.2],
       [4.9, 3.6, 1.4, 0.1],
       [4.4, 3. , 1.3, 0.2],
       [5.1, 3.4, 1.5, 0.2],
       [5. , 3.5, 1.3, 0.3],
       [4.5, 2.3, 1.3, 0.3],
       [4.4, 3.2, 1.3, 0.2],
       [5. , 3.5, 1.6, 0.6],
       [5.1, 3.8, 1.9, 0.4],
       [4.8, 3. , 1.4, 0.3],
       [5.1, 3.8, 1.6, 0.2],
       [4.6, 3.2, 1.4, 0.2],
       [5.3, 3.7, 1.5, 0.2],
       [5. , 3.3, 1.4, 0.2],
       [7. , 3.2, 4.7, 1.4],
       [6.4, 3.2, 4.5, 1.5],
       [6.9, 3.1, 4.9, 1.5],
       [5.5, 2.3, 4. , 1.3],
       [6.5, 2.8, 4.6, 1.5],
       [5.7, 2.8, 4.5, 1.3],
       [6.3, 3.3, 4.7, 1.6],
       [4.9, 2.4, 3.3, 1. ],
       [6.6, 2.9, 4.6, 1.3],
       [5.2, 2.7, 3.9, 1.4],
       [5. , 2. , 3.5, 1. ],
       [5.9, 3. , 4.2, 1.5],
       [6. , 2.2, 4. , 1. ],
       [6.1, 2.9, 4.7, 1.4],
       [5.6, 2.9, 3.6, 1.3],
       [6.7, 3.1, 4.4, 1.4],
       [5.6, 3. , 4.5, 1.5],
       [5.8, 2.7, 4.1, 1. ],
       [6.2, 2.2, 4.5, 1.5],
       [5.6, 2.5, 3.9, 1.1],
       [5.9, 3.2, 4.8, 1.8],
       [6.1, 2.8, 4. , 1.3],
       [6.3, 2.5, 4.9, 1.5],
       [6.1, 2.8, 4.7, 1.2],
       [6.4, 2.9, 4.3, 1.3],
       [6.6, 3. , 4.4, 1.4],
       [6.8, 2.8, 4.8, 1.4],
       [6.7, 3. , 5. , 1.7],
       [6. , 2.9, 4.5, 1.5],
       [5.7, 2.6, 3.5, 1. ],
       [5.5, 2.4, 3.8, 1.1],
       [5.5, 2.4, 3.7, 1. ],
       [5.8, 2.7, 3.9, 1.2],
       [6. , 2.7, 5.1, 1.6],
       [5.4, 3. , 4.5, 1.5],
       [6. , 3.4, 4.5, 1.6],
       [6.7, 3.1, 4.7, 1.5],
       [6.3, 2.3, 4.4, 1.3],
       [5.6, 3. , 4.1, 1.3],
       [5.5, 2.5, 4. , 1.3],
       [5.5, 2.6, 4.4, 1.2],
       [6.1, 3. , 4.6, 1.4],
       [5.8, 2.6, 4. , 1.2],
       [5. , 2.3, 3.3, 1. ],
       [5.6, 2.7, 4.2, 1.3],
       [5.7, 3. , 4.2, 1.2],
       [5.7, 2.9, 4.2, 1.3],
       [6.2, 2.9, 4.3, 1.3],
       [5.1, 2.5, 3. , 1.1],
       [5.7, 2.8, 4.1, 1.3],
       [6.3, 3.3, 6. , 2.5],
       [5.8, 2.7, 5.1, 1.9],
       [7.1, 3. , 5.9, 2.1],
       [6.3, 2.9, 5.6, 1.8],
       [6.5, 3. , 5.8, 2.2],
       [7.6, 3. , 6.6, 2.1],
       [4.9, 2.5, 4.5, 1.7],
       [7.3, 2.9, 6.3, 1.8],
       [6.7, 2.5, 5.8, 1.8],
       [7.2, 3.6, 6.1, 2.5],
       [6.5, 3.2, 5.1, 2. ],
       [6.4, 2.7, 5.3, 1.9],
       [6.8, 3. , 5.5, 2.1],
       [5.7, 2.5, 5. , 2. ],
       [5.8, 2.8, 5.1, 2.4],
       [6.4, 3.2, 5.3, 2.3],
       [6.5, 3. , 5.5, 1.8],
       [7.7, 3.8, 6.7, 2.2],
       [7.7, 2.6, 6.9, 2.3],
       [6. , 2.2, 5. , 1.5],
       [6.9, 3.2, 5.7, 2.3],
       [5.6, 2.8, 4.9, 2. ],
       [7.7, 2.8, 6.7, 2. ],
       [6.3, 2.7, 4.9, 1.8],
       [6.7, 3.3, 5.7, 2.1],
       [7.2, 3.2, 6. , 1.8],
       [6.2, 2.8, 4.8, 1.8],
       [6.1, 3. , 4.9, 1.8],
       [6.4, 2.8, 5.6, 2.1],
       [7.2, 3. , 5.8, 1.6],
       [7.4, 2.8, 6.1, 1.9],
       [7.9, 3.8, 6.4, 2. ],
       [6.4, 2.8, 5.6, 2.2],
       [6.3, 2.8, 5.1, 1.5],
       [6.1, 2.6, 5.6, 1.4],
       [7.7, 3. , 6.1, 2.3],
       [6.3, 3.4, 5.6, 2.4],
       [6.4, 3.1, 5.5, 1.8],
       [6. , 3. , 4.8, 1.8],
       [6.9, 3.1, 5.4, 2.1],
       [6.7, 3.1, 5.6, 2.4],
       [6.9, 3.1, 5.1, 2.3],
       [5.8, 2.7, 5.1, 1.9],
       [6.8, 3.2, 5.9, 2.3],
       [6.7, 3.3, 5.7, 2.5],
       [6.7, 3. , 5.2, 2.3],
       [6.3, 2.5, 5. , 1.9],
       [6.5, 3. , 5.2, 2. ],
       [6.2, 3.4, 5.4, 2.3],
       [5.9, 3. , 5.1, 1.8]])

In [14]:

Out[14]:

array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2])

In [16]:

from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_scaled = sc.fit_transform(X)
X_scaled

Out[16]:

array([[-9.00681170e-01,  1.01900435e+00, -1.34022653e+00,
        -1.31544430e+00],
       [-1.14301691e+00, -1.31979479e-01, -1.34022653e+00,
        -1.31544430e+00],
       [-1.38535265e+00,  3.28414053e-01, -1.39706395e+00,
        -1.31544430e+00],
       [-1.50652052e+00,  9.82172869e-02, -1.28338910e+00,
        -1.31544430e+00],
       [-1.02184904e+00,  1.24920112e+00, -1.34022653e+00,
        -1.31544430e+00],
       [-5.37177559e-01,  1.93979142e+00, -1.16971425e+00,
        -1.05217993e+00],
       [-1.50652052e+00,  7.88807586e-01, -1.34022653e+00,
        -1.18381211e+00],
       [-1.02184904e+00,  7.88807586e-01, -1.28338910e+00,
        -1.31544430e+00],
       [-1.74885626e+00, -3.62176246e-01, -1.34022653e+00,
        -1.31544430e+00],
       [-1.14301691e+00,  9.82172869e-02, -1.28338910e+00,
        -1.44707648e+00],
       [-5.37177559e-01,  1.47939788e+00, -1.28338910e+00,
        -1.31544430e+00],
       [-1.26418478e+00,  7.88807586e-01, -1.22655167e+00,
        -1.31544430e+00],
       [-1.26418478e+00, -1.31979479e-01, -1.34022653e+00,
        -1.44707648e+00],
       [-1.87002413e+00, -1.31979479e-01, -1.51073881e+00,
        -1.44707648e+00],
       [-5.25060772e-02,  2.16998818e+00, -1.45390138e+00,
        -1.31544430e+00],
       [-1.73673948e-01,  3.09077525e+00, -1.28338910e+00,
        -1.05217993e+00],
       [-5.37177559e-01,  1.93979142e+00, -1.39706395e+00,
        -1.05217993e+00],
       [-9.00681170e-01,  1.01900435e+00, -1.34022653e+00,
        -1.18381211e+00],
       [-1.73673948e-01,  1.70959465e+00, -1.16971425e+00,
        -1.18381211e+00],
       [-9.00681170e-01,  1.70959465e+00, -1.28338910e+00,
        -1.18381211e+00],
       [-5.37177559e-01,  7.88807586e-01, -1.16971425e+00,
        -1.31544430e+00],
       [-9.00681170e-01,  1.47939788e+00, -1.28338910e+00,
        -1.05217993e+00],
       [-1.50652052e+00,  1.24920112e+00, -1.56757623e+00,
        -1.31544430e+00],
       [-9.00681170e-01,  5.58610819e-01, -1.16971425e+00,
        -9.20547742e-01],
       [-1.26418478e+00,  7.88807586e-01, -1.05603939e+00,
        -1.31544430e+00],
       [-1.02184904e+00, -1.31979479e-01, -1.22655167e+00,
        -1.31544430e+00],
       [-1.02184904e+00,  7.88807586e-01, -1.22655167e+00,
        -1.05217993e+00],
       [-7.79513300e-01,  1.01900435e+00, -1.28338910e+00,
        -1.31544430e+00],
       [-7.79513300e-01,  7.88807586e-01, -1.34022653e+00,
        -1.31544430e+00],
       [-1.38535265e+00,  3.28414053e-01, -1.22655167e+00,
        -1.31544430e+00],
       [-1.26418478e+00,  9.82172869e-02, -1.22655167e+00,
        -1.31544430e+00],
       [-5.37177559e-01,  7.88807586e-01, -1.28338910e+00,
        -1.05217993e+00],
       [-7.79513300e-01,  2.40018495e+00, -1.28338910e+00,
        -1.44707648e+00],
       [-4.16009689e-01,  2.63038172e+00, -1.34022653e+00,
        -1.31544430e+00],
       [-1.14301691e+00,  9.82172869e-02, -1.28338910e+00,
        -1.31544430e+00],
       [-1.02184904e+00,  3.28414053e-01, -1.45390138e+00,
        -1.31544430e+00],
       [-4.16009689e-01,  1.01900435e+00, -1.39706395e+00,
        -1.31544430e+00],
       [-1.14301691e+00,  1.24920112e+00, -1.34022653e+00,
        -1.44707648e+00],
       [-1.74885626e+00, -1.31979479e-01, -1.39706395e+00,
        -1.31544430e+00],
       [-9.00681170e-01,  7.88807586e-01, -1.28338910e+00,
        -1.31544430e+00],
       [-1.02184904e+00,  1.01900435e+00, -1.39706395e+00,
        -1.18381211e+00],
       [-1.62768839e+00, -1.74335684e+00, -1.39706395e+00,
        -1.18381211e+00],
       [-1.74885626e+00,  3.28414053e-01, -1.39706395e+00,
        -1.31544430e+00],
       [-1.02184904e+00,  1.01900435e+00, -1.22655167e+00,
        -7.88915558e-01],
       [-9.00681170e-01,  1.70959465e+00, -1.05603939e+00,
        -1.05217993e+00],
       [-1.26418478e+00, -1.31979479e-01, -1.34022653e+00,
        -1.18381211e+00],
       [-9.00681170e-01,  1.70959465e+00, -1.22655167e+00,
        -1.31544430e+00],
       [-1.50652052e+00,  3.28414053e-01, -1.34022653e+00,
        -1.31544430e+00],
       [-6.58345429e-01,  1.47939788e+00, -1.28338910e+00,
        -1.31544430e+00],
       [-1.02184904e+00,  5.58610819e-01, -1.34022653e+00,
        -1.31544430e+00],
       [ 1.40150837e+00,  3.28414053e-01,  5.35408562e-01,
         2.64141916e-01],
       [ 6.74501145e-01,  3.28414053e-01,  4.21733708e-01,
         3.95774101e-01],
       [ 1.28034050e+00,  9.82172869e-02,  6.49083415e-01,
         3.95774101e-01],
       [-4.16009689e-01, -1.74335684e+00,  1.37546573e-01,
         1.32509732e-01],
       [ 7.95669016e-01, -5.92373012e-01,  4.78571135e-01,
         3.95774101e-01],
       [-1.73673948e-01, -5.92373012e-01,  4.21733708e-01,
         1.32509732e-01],
       [ 5.53333275e-01,  5.58610819e-01,  5.35408562e-01,
         5.27406285e-01],
       [-1.14301691e+00, -1.51316008e+00, -2.60315415e-01,
        -2.62386821e-01],
       [ 9.16836886e-01, -3.62176246e-01,  4.78571135e-01,
         1.32509732e-01],
       [-7.79513300e-01, -8.22569778e-01,  8.07091462e-02,
         2.64141916e-01],
       [-1.02184904e+00, -2.43394714e+00, -1.46640561e-01,
        -2.62386821e-01],
       [ 6.86617933e-02, -1.31979479e-01,  2.51221427e-01,
         3.95774101e-01],
       [ 1.89829664e-01, -1.97355361e+00,  1.37546573e-01,
        -2.62386821e-01],
       [ 3.10997534e-01, -3.62176246e-01,  5.35408562e-01,
         2.64141916e-01],
       [-2.94841818e-01, -3.62176246e-01, -8.98031345e-02,
         1.32509732e-01],
       [ 1.03800476e+00,  9.82172869e-02,  3.64896281e-01,
         2.64141916e-01],
       [-2.94841818e-01, -1.31979479e-01,  4.21733708e-01,
         3.95774101e-01],
       [-5.25060772e-02, -8.22569778e-01,  1.94384000e-01,
        -2.62386821e-01],
       [ 4.32165405e-01, -1.97355361e+00,  4.21733708e-01,
         3.95774101e-01],
       [-2.94841818e-01, -1.28296331e+00,  8.07091462e-02,
        -1.30754636e-01],
       [ 6.86617933e-02,  3.28414053e-01,  5.92245988e-01,
         7.90670654e-01],
       [ 3.10997534e-01, -5.92373012e-01,  1.37546573e-01,
         1.32509732e-01],
       [ 5.53333275e-01, -1.28296331e+00,  6.49083415e-01,
         3.95774101e-01],
       [ 3.10997534e-01, -5.92373012e-01,  5.35408562e-01,
         8.77547895e-04],
       [ 6.74501145e-01, -3.62176246e-01,  3.08058854e-01,
         1.32509732e-01],
       [ 9.16836886e-01, -1.31979479e-01,  3.64896281e-01,
         2.64141916e-01],
       [ 1.15917263e+00, -5.92373012e-01,  5.92245988e-01,
         2.64141916e-01],
       [ 1.03800476e+00, -1.31979479e-01,  7.05920842e-01,
         6.59038469e-01],
       [ 1.89829664e-01, -3.62176246e-01,  4.21733708e-01,
         3.95774101e-01],
       [-1.73673948e-01, -1.05276654e+00, -1.46640561e-01,
        -2.62386821e-01],
       [-4.16009689e-01, -1.51316008e+00,  2.38717193e-02,
        -1.30754636e-01],
       [-4.16009689e-01, -1.51316008e+00, -3.29657076e-02,
        -2.62386821e-01],
       [-5.25060772e-02, -8.22569778e-01,  8.07091462e-02,
         8.77547895e-04],
       [ 1.89829664e-01, -8.22569778e-01,  7.62758269e-01,
         5.27406285e-01],
       [-5.37177559e-01, -1.31979479e-01,  4.21733708e-01,
         3.95774101e-01],
       [ 1.89829664e-01,  7.88807586e-01,  4.21733708e-01,
         5.27406285e-01],
       [ 1.03800476e+00,  9.82172869e-02,  5.35408562e-01,
         3.95774101e-01],
       [ 5.53333275e-01, -1.74335684e+00,  3.64896281e-01,
         1.32509732e-01],
       [-2.94841818e-01, -1.31979479e-01,  1.94384000e-01,
         1.32509732e-01],
       [-4.16009689e-01, -1.28296331e+00,  1.37546573e-01,
         1.32509732e-01],
       [-4.16009689e-01, -1.05276654e+00,  3.64896281e-01,
         8.77547895e-04],
       [ 3.10997534e-01, -1.31979479e-01,  4.78571135e-01,
         2.64141916e-01],
       [-5.25060772e-02, -1.05276654e+00,  1.37546573e-01,
         8.77547895e-04],
       [-1.02184904e+00, -1.74335684e+00, -2.60315415e-01,
        -2.62386821e-01],
       [-2.94841818e-01, -8.22569778e-01,  2.51221427e-01,
         1.32509732e-01],
       [-1.73673948e-01, -1.31979479e-01,  2.51221427e-01,
         8.77547895e-04],
       [-1.73673948e-01, -3.62176246e-01,  2.51221427e-01,
         1.32509732e-01],
       [ 4.32165405e-01, -3.62176246e-01,  3.08058854e-01,
         1.32509732e-01],
       [-9.00681170e-01, -1.28296331e+00, -4.30827696e-01,
        -1.30754636e-01],
       [-1.73673948e-01, -5.92373012e-01,  1.94384000e-01,
         1.32509732e-01],
       [ 5.53333275e-01,  5.58610819e-01,  1.27429511e+00,
         1.71209594e+00],
       [-5.25060772e-02, -8.22569778e-01,  7.62758269e-01,
         9.22302838e-01],
       [ 1.52267624e+00, -1.31979479e-01,  1.21745768e+00,
         1.18556721e+00],
       [ 5.53333275e-01, -3.62176246e-01,  1.04694540e+00,
         7.90670654e-01],
       [ 7.95669016e-01, -1.31979479e-01,  1.16062026e+00,
         1.31719939e+00],
       [ 2.12851559e+00, -1.31979479e-01,  1.61531967e+00,
         1.18556721e+00],
       [-1.14301691e+00, -1.28296331e+00,  4.21733708e-01,
         6.59038469e-01],
       [ 1.76501198e+00, -3.62176246e-01,  1.44480739e+00,
         7.90670654e-01],
       [ 1.03800476e+00, -1.28296331e+00,  1.16062026e+00,
         7.90670654e-01],
       [ 1.64384411e+00,  1.24920112e+00,  1.33113254e+00,
         1.71209594e+00],
       [ 7.95669016e-01,  3.28414053e-01,  7.62758269e-01,
         1.05393502e+00],
       [ 6.74501145e-01, -8.22569778e-01,  8.76433123e-01,
         9.22302838e-01],
       [ 1.15917263e+00, -1.31979479e-01,  9.90107977e-01,
         1.18556721e+00],
       [-1.73673948e-01, -1.28296331e+00,  7.05920842e-01,
         1.05393502e+00],
       [-5.25060772e-02, -5.92373012e-01,  7.62758269e-01,
         1.58046376e+00],
       [ 6.74501145e-01,  3.28414053e-01,  8.76433123e-01,
         1.44883158e+00],
       [ 7.95669016e-01, -1.31979479e-01,  9.90107977e-01,
         7.90670654e-01],
       [ 2.24968346e+00,  1.70959465e+00,  1.67215710e+00,
         1.31719939e+00],
       [ 2.24968346e+00, -1.05276654e+00,  1.78583195e+00,
         1.44883158e+00],
       [ 1.89829664e-01, -1.97355361e+00,  7.05920842e-01,
         3.95774101e-01],
       [ 1.28034050e+00,  3.28414053e-01,  1.10378283e+00,
         1.44883158e+00],
       [-2.94841818e-01, -5.92373012e-01,  6.49083415e-01,
         1.05393502e+00],
       [ 2.24968346e+00, -5.92373012e-01,  1.67215710e+00,
         1.05393502e+00],
       [ 5.53333275e-01, -8.22569778e-01,  6.49083415e-01,
         7.90670654e-01],
       [ 1.03800476e+00,  5.58610819e-01,  1.10378283e+00,
         1.18556721e+00],
       [ 1.64384411e+00,  3.28414053e-01,  1.27429511e+00,
         7.90670654e-01],
       [ 4.32165405e-01, -5.92373012e-01,  5.92245988e-01,
         7.90670654e-01],
       [ 3.10997534e-01, -1.31979479e-01,  6.49083415e-01,
         7.90670654e-01],
       [ 6.74501145e-01, -5.92373012e-01,  1.04694540e+00,
         1.18556721e+00],
       [ 1.64384411e+00, -1.31979479e-01,  1.16062026e+00,
         5.27406285e-01],
       [ 1.88617985e+00, -5.92373012e-01,  1.33113254e+00,
         9.22302838e-01],
       [ 2.49201920e+00,  1.70959465e+00,  1.50164482e+00,
         1.05393502e+00],
       [ 6.74501145e-01, -5.92373012e-01,  1.04694540e+00,
         1.31719939e+00],
       [ 5.53333275e-01, -5.92373012e-01,  7.62758269e-01,
         3.95774101e-01],
       [ 3.10997534e-01, -1.05276654e+00,  1.04694540e+00,
         2.64141916e-01],
       [ 2.24968346e+00, -1.31979479e-01,  1.33113254e+00,
         1.44883158e+00],
       [ 5.53333275e-01,  7.88807586e-01,  1.04694540e+00,
         1.58046376e+00],
       [ 6.74501145e-01,  9.82172869e-02,  9.90107977e-01,
         7.90670654e-01],
       [ 1.89829664e-01, -1.31979479e-01,  5.92245988e-01,
         7.90670654e-01],
       [ 1.28034050e+00,  9.82172869e-02,  9.33270550e-01,
         1.18556721e+00],
       [ 1.03800476e+00,  9.82172869e-02,  1.04694540e+00,
         1.58046376e+00],
       [ 1.28034050e+00,  9.82172869e-02,  7.62758269e-01,
         1.44883158e+00],
       [-5.25060772e-02, -8.22569778e-01,  7.62758269e-01,
         9.22302838e-01],
       [ 1.15917263e+00,  3.28414053e-01,  1.21745768e+00,
         1.44883158e+00],
       [ 1.03800476e+00,  5.58610819e-01,  1.10378283e+00,
         1.71209594e+00],
       [ 1.03800476e+00, -1.31979479e-01,  8.19595696e-01,
         1.44883158e+00],
       [ 5.53333275e-01, -1.28296331e+00,  7.05920842e-01,
         9.22302838e-01],
       [ 7.95669016e-01, -1.31979479e-01,  8.19595696e-01,
         1.05393502e+00],
       [ 4.32165405e-01,  7.88807586e-01,  9.33270550e-01,
         1.44883158e+00],
       [ 6.86617933e-02, -1.31979479e-01,  7.62758269e-01,
         7.90670654e-01]])

In [17]:

pca = PCA(n_components=2)
X_pca = pca.fit_transform(X_scaled)

lda = LinearDiscriminantAnalysis(n_components=2, solver='svd')
X_lda = lda.fit_transform(X, y)

In [18]:

fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(13.5 ,4))
sns.scatterplot(X_pca[:,0], X_pca[:,1], hue=y, palette='Set1', ax=ax[0])
sns.scatterplot(X_lda[:,0], X_lda[:,1], hue=y, palette='Set1', ax=ax[1])
ax[0].set_title("PCA de IRIS", fontsize=15, pad=15)
ax[1].set_title("LDA de IRIS", fontsize=15, pad=15)
ax[0].set_xlabel("PC1", fontsize=12)
ax[0].set_ylabel("PC2", fontsize=12)
ax[1].set_xlabel("LD1", fontsize=12)
ax[1].set_ylabel("LD2", fontsize=12)

/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  FutureWarning
/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  FutureWarning

Out[18]:

Text(0, 0.5, 'LD2')

Kernel PCA¶

In [19]:

import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.datasets import make_moons

# Creamos los datos
X, y = make_moons(n_samples = 500, random_state=42)
sns.scatterplot(X[:, 0], X[:, 1], hue=y, palette='Set1')

/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  FutureWarning

Out[19]:

<matplotlib.axes._subplots.AxesSubplot at 0x7f367c4450d0>

Implementemos PCA y Kernel PCA y veamos las diferencias

In [20]:

from sklearn.decomposition import PCA
from sklearn.decomposition import KernelPCA

pca = PCA(n_components=2)
X_pca = pca.fit_transform(X)

kpca = KernelPCA(n_components=2, kernel='rbf', gamma=15, random_state=42)
X_kpca = kpca.fit_transform(X)

In [21]:

fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(13.5 ,4))
sns.scatterplot(X_pca[:, 0], X_pca[:, 1], hue=y, palette='Set1', ax=ax[0])
sns.scatterplot(X_kpca[:, 0], X_kpca[:, 1], hue=y, palette='Set1', ax=ax[1])
ax[0].set_title("PCA", fontsize=15, pad=15)
ax[1].set_title("RBF Kernel PCA", fontsize=15, pad=15)
ax[0].set_xlabel("Componente 1", fontsize=12)
ax[0].set_ylabel("Componente 2", fontsize=12)
ax[1].set_xlabel("Componente 1", fontsize=12)
ax[1].set_ylabel("Componente 2", fontsize=12)

/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  FutureWarning
/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  FutureWarning

Out[21]:

Text(0, 0.5, 'Componente 2')

El PCA normal no puede transformar datos no lineales en una forma lineal. Después de aplicar Kernel PCA a los mismos datos, las dos clases están linealmente bien separadas (ahora, las clases se pueden dividir dibujando una línea recta vertical).

Aquí, los datos originales tienen una dimensión de 2 y los datos trazados también tienen una dimensión de 2. Entonces, el Kernel PCA realmente redujo la dimensionalidad de los datos? La respuesta es 'Sí' porque la función kernel RBF proyecta temporalmente los datos bidimensionales en un nuevo espacio de características de dimensiones superiores donde las clases se vuelven linealmente separables y luego el algoritmo proyecta esos datos de dimensiones superiores nuevamente en los datos bidimensionales que se puede trazar en un gráfico 2D.

Una limitación del uso de Kernel PCA para la reducción de dimensionalidad es que tenemos que especificar un valor para el hiperparámetro gamma antes de ejecutar el algoritmo y para esto e puede usar una técnica de ajuste de hiperparámetros como Grid Search para encontrar un valor óptimo para la gamma.

t-SNE¶

In [24]:

from sklearn.decomposition import PCA
from sklearn.manifold import TSNE
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_scaled = sc.fit_transform(X)
pca = PCA()
X_pca = pca.fit_transform(X_scaled)
tsne = TSNE()
X_tsne = tsne.fit_transform(X_pca)

/usr/local/lib/python3.7/dist-packages/sklearn/manifold/_t_sne.py:783: FutureWarning: The default initialization in TSNE will change from 'random' to 'pca' in 1.2.
  FutureWarning,
/usr/local/lib/python3.7/dist-packages/sklearn/manifold/_t_sne.py:793: FutureWarning: The default learning rate in TSNE will change from 200.0 to 'auto' in 1.2.
  FutureWarning,

Ahora creamos un pipeline

In [26]:

from sklearn.pipeline import Pipeline
from sklearn.decomposition import PCA
from sklearn.manifold import TSNE
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
pca = PCA()
tsne = TSNE()
tsne_after_pca = Pipeline([
    ('std_scaler', sc),
    ('pca', pca),
    ('tsne', tsne)
])
X_tsne = tsne_after_pca.fit_transform(X)

/usr/local/lib/python3.7/dist-packages/sklearn/manifold/_t_sne.py:783: FutureWarning: The default initialization in TSNE will change from 'random' to 'pca' in 1.2.
  FutureWarning,
/usr/local/lib/python3.7/dist-packages/sklearn/manifold/_t_sne.py:793: FutureWarning: The default learning rate in TSNE will change from 200.0 to 'auto' in 1.2.
  FutureWarning,

Ahora, aplicamos t-SNE al conjunto de datos Iris. Tiene solo 4 características. Por lo tanto, no necesitamos ejecutar PCA antes de t-SNE

In [27]:

from sklearn.datasets import load_iris
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.manifold import TSNE
from sklearn.preprocessing import StandardScaler
sns.set_style('darkgrid')

iris = load_iris()
X = iris.data
y = iris.target

sc = StandardScaler()
X_scaled = sc.fit_transform(X)

tsne = TSNE(n_components=2, random_state=1)
X_tsne = tsne.fit_transform(X_scaled)

sns.scatterplot(X_tsne[:,0], X_tsne[:,1], hue=y, palette='Set1')
plt.title("t-SNE de IRIS", fontsize=15, pad=15)

/usr/local/lib/python3.7/dist-packages/sklearn/manifold/_t_sne.py:783: FutureWarning: The default initialization in TSNE will change from 'random' to 'pca' in 1.2.
  FutureWarning,
/usr/local/lib/python3.7/dist-packages/sklearn/manifold/_t_sne.py:793: FutureWarning: The default learning rate in TSNE will change from 200.0 to 'auto' in 1.2.
  FutureWarning,
/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  FutureWarning

Out[27]:

Text(0.5, 1.0, 't-SNE de IRIS')

Multidimensional Scaling¶

In [28]:

from sklearn.datasets import load_iris
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.manifold import MDS
sns.set_style('darkgrid')

iris = load_iris()
X = iris.data
y = iris.target

In [30]:

mds = MDS(n_components=2, metric=True, random_state=2)
X_mds = mds.fit_transform(X)

sns.scatterplot(X_mds[:,0], X_mds[:,1], hue=y, palette='Set1')
plt.title("MDS de IRIS dataset", fontsize=15, pad=15)

/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  FutureWarning

Out[30]:

Text(0.5, 1.0, 'MDS de IRIS dataset')

Isomap¶

In [31]:

from sklearn.datasets import load_iris
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.manifold import Isomap
sns.set_style('darkgrid')

iris = load_iris()
X = iris.data
y = iris.target

In [32]:

isomap = Isomap(n_neighbors=5, n_components=2, 
                eigen_solver='auto')
X_isomap = isomap.fit_transform(X)

sns.scatterplot(X_isomap[:,0], X_isomap[:,1], hue=y, palette='Set1')
plt.title("Isomap de IRIS", fontsize=15, pad=15)

/usr/local/lib/python3.7/dist-packages/sklearn/manifold/_isomap.py:324: UserWarning: The number of connected components of the neighbors graph is 2 > 1. Completing the graph to fit Isomap might be slow. Increase the number of neighbors to avoid this issue.
  self._fit_transform(X)
/usr/local/lib/python3.7/dist-packages/scipy/sparse/_index.py:84: SparseEfficiencyWarning: Changing the sparsity structure of a csr_matrix is expensive. lil_matrix is more efficient.
  self._set_intXint(row, col, x.flat[0])
/usr/local/lib/python3.7/dist-packages/seaborn/_decorators.py:43: FutureWarning: Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.
  FutureWarning

Out[32]:

Text(0.5, 1.0, 'Isomap de IRIS')

Created in Deepnote