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import numpy as np
import seaborn as sns
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib import cm
from sklearn import datasets
from sklearn.datasets import make_blobs
from mpl_toolkits.mplot3d import Axes3D
import scipy.ndimage
import warnings
warnings.filterwarnings("ignore", category=FutureWarning)
pd.set_option("display.max_rows", None)
Motivación: visualizar datos multidimensionales#
Clasificación y clustering#
Datos bidimensionales con etiquetas
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X, y = make_blobs(n_samples=100, centers=3, n_features=2,random_state=0)
#np.unique(y)
df1 = pd.DataFrame(X,columns=['x1','x2'])
df1['label'] = y
df1.head(10)
| x1 | x2 | label | |
|---|---|---|---|
| 0 | 2.631858 | 0.689365 | 1 |
| 1 | 0.080804 | 4.690690 | 0 |
| 2 | 3.002519 | 0.742654 | 1 |
| 3 | -0.637628 | 4.091047 | 0 |
| 4 | -0.072283 | 2.883769 | 0 |
| 5 | 0.628358 | 4.460136 | 0 |
| 6 | -2.674373 | 2.480062 | 2 |
| 7 | -0.577483 | 3.005434 | 2 |
| 8 | 2.727562 | 1.305125 | 1 |
| 9 | 0.341948 | 3.941046 | 0 |
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f,axes = plt.subplots(nrows=1,ncols=2,figsize=(10,5))
ax = axes.ravel()
ax[0].plot(X[:,0],X[:,1],'.')
colors = ["#4EACC5", "#FF9C34", "#4E9A06"]
for yy in np.unique(y):
dataSelection = y ==yy
data = X[dataSelection,:]
ax[1].plot(data[:,0],data[:,1],'.',c=colors[yy],label=yy)
ax[1].legend()
for a in ax:
a.set_xlabel(r'$X_1$')
a.set_ylabel(r'$X_2$')
Datos tridimensionales con etiquetas
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X, y = make_blobs(n_samples=200, centers=4, n_features=3,random_state=4)
#np.unique(y)
df1 = pd.DataFrame(X,columns=['x1','x2','x3'])
df1['label'] = y
df1.head()
| x1 | x2 | x3 | label | |
|---|---|---|---|---|
| 0 | 9.863844 | 0.448826 | 9.282223 | 0 |
| 1 | 10.596115 | -11.280679 | -5.449909 | 2 |
| 2 | -2.083092 | 7.565793 | -5.662472 | 3 |
| 3 | 5.578687 | 5.149093 | -6.176931 | 1 |
| 4 | 4.101746 | 3.373981 | -6.774126 | 1 |
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colorsDict = {0:'orange',
1:'red',
2:'green',
3:'blue'}
fig = plt.figure(figsize=(20,10))
ax = fig.add_subplot(121, projection='3d')
for ii in df1.index:
ax.scatter(
df1.loc[ii,'x1'],
df1.loc[ii,'x2'],
df1.loc[ii,'x3'],
marker='.',
s=100,
color='blue'
#color = colorsDict[
# df1.loc[ii,'label']
#]
)
ax = fig.add_subplot(122, projection='3d')
for ii in df1.index:
ax.scatter(
df1.loc[ii,'x1'],
df1.loc[ii,'x2'],
df1.loc[ii,'x3'],
marker='.',
s=100,
color = colorsDict[
df1.loc[ii,'label']
]
)
plt.show()
Regresión#
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x = np.linspace(-10,10,100)
y = 4*np.sin(x)*np.cos(2*x)+np.sin(3*x)/3+11/6*np.cos(5*x)
f,ax = plt.subplots(nrows=1,ncols=1,figsize=(7,7))
ax.plot(x,y,'-',color='red')
ydata = y+np.random.uniform(-0.5,0.5,y.shape[0])
ax.plot(x,ydata,'.',color='blue',label='datos')
ax.legend()
<matplotlib.legend.Legend at 0x13ee1fbed10>
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fig, ax = plt.subplots(figsize=(10,10),subplot_kw={"projection": "3d"})
# Make data.
X = np.arange(-5, 5, 0.25)
Y = np.arange(-5, 5, 0.25)
X, Y = np.meshgrid(X, Y)
#R = np.sqrt(X**2 + Y**2)
#Z = np.sin(R)
Z=np.sin(X/3)*np.cos(Y/2)+2
Z2 = np.sin(X/3)*np.cos(Y/2)+2+np.random.uniform(-0.5,0.5,(Z.shape[0],Z.shape[1]))
# Plot the surface.
surf = ax.plot_surface(X, Y, Z, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
ax.scatter(X,Y,Z2,s=2,label='datos')
ax.legend()
plt.show()
Más de tres dimensiones#
Cuando tenemos más dimensiones no podemos visualizar los datos.
Datos iris
Usamos la scatter_matrix
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# iris
iris = datasets.load_iris()
iris_data = iris.data
iris_data.shape
irisDf = pd.DataFrame(iris_data,columns=iris.feature_names)
irisDf['target'] = iris.target
irisDf.head()
| sepal length (cm) | sepal width (cm) | petal length (cm) | petal width (cm) | target | |
|---|---|---|---|---|---|
| 0 | 5.1 | 3.5 | 1.4 | 0.2 | 0 |
| 1 | 4.9 | 3.0 | 1.4 | 0.2 | 0 |
| 2 | 4.7 | 3.2 | 1.3 | 0.2 | 0 |
| 3 | 4.6 | 3.1 | 1.5 | 0.2 | 0 |
| 4 | 5.0 | 3.6 | 1.4 | 0.2 | 0 |
df = pd.DataFrame(iris.data, columns=iris.feature_names)
colors=np.array(50*['r']+50*['g']+50*['b'])
pd.plotting.scatter_matrix(df,
alpha=0.6,
figsize=(10,10),
#color=colors,
hist_kwds={'bins':30})
plt.show()
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df = pd.DataFrame(iris.data, columns=iris.feature_names)
colors=np.array(50*['r']+50*['g']+50*['b'])
pd.plotting.scatter_matrix(df,
alpha=0.6,
figsize=(10,10),
color=colors,
hist_kwds={'bins':30})
plt.show()
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import seaborn as sns
sns.set_theme(style="ticks")
df = sns.load_dataset("penguins")
df.head(10)
| species | island | bill_length_mm | bill_depth_mm | flipper_length_mm | body_mass_g | sex | |
|---|---|---|---|---|---|---|---|
| 0 | Adelie | Torgersen | 39.1 | 18.7 | 181.0 | 3750.0 | Male |
| 1 | Adelie | Torgersen | 39.5 | 17.4 | 186.0 | 3800.0 | Female |
| 2 | Adelie | Torgersen | 40.3 | 18.0 | 195.0 | 3250.0 | Female |
| 3 | Adelie | Torgersen | NaN | NaN | NaN | NaN | NaN |
| 4 | Adelie | Torgersen | 36.7 | 19.3 | 193.0 | 3450.0 | Female |
| 5 | Adelie | Torgersen | 39.3 | 20.6 | 190.0 | 3650.0 | Male |
| 6 | Adelie | Torgersen | 38.9 | 17.8 | 181.0 | 3625.0 | Female |
| 7 | Adelie | Torgersen | 39.2 | 19.6 | 195.0 | 4675.0 | Male |
| 8 | Adelie | Torgersen | 34.1 | 18.1 | 193.0 | 3475.0 | NaN |
| 9 | Adelie | Torgersen | 42.0 | 20.2 | 190.0 | 4250.0 | NaN |
Método pairplot del paquete seaborn
https://seaborn.pydata.org/generated/seaborn.pairplot.html
sns.pairplot(df)
<seaborn.axisgrid.PairGrid at 0x13ee40c4410>
sns.pairplot(df, hue="species")
<seaborn.axisgrid.PairGrid at 0x13ee2244410>
Puede haber datos sin etiquetas
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import numpy as np
import pandas as pd
np.random.seed(134)
N = 1000
x1 = np.random.normal(0, 1, N)
x2 = x1 + np.random.normal(0, 3, N)
x3 = 2 * x1 - x2 + np.random.normal(0, 2, N)
x4 = x3 * x1 -4*x2 + np.random.normal(0, 1, N)
x5 = x2-x4*x1 + np.random.normal(0, 2, N)
df = pd.DataFrame({'x1':x1,
'x2':x2,
'x3':x3,
'x4':x4,
'x5':x5
})
df.head()
| x1 | x2 | x3 | x4 | x5 | |
|---|---|---|---|---|---|
| 0 | -0.224315 | -8.840152 | 10.145993 | 33.286302 | -1.376902 |
| 1 | 1.337257 | 2.383882 | -1.854636 | -11.590022 | 18.471552 |
| 2 | 0.882366 | 3.544989 | -1.117054 | -14.303068 | 14.009670 |
| 3 | 0.295153 | -3.844863 | 3.634823 | 15.538617 | -4.391063 |
| 4 | 0.780587 | -0.465342 | 2.121288 | 2.874332 | 1.209348 |
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import matplotlib.pyplot as plt
pd.plotting.scatter_matrix(df,figsize=(10,10))
plt.show()
Proyecciones#
Las gráficas anteriores son proyecciones ortogonales de los datos sobre los diferentes planos formados eligiendo coordenadas de los diferentes atributos, por parejas
Ejemplo ad hoc en tres dimensiones
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clusters=[1,2,3,4,5,6]
Npuntos = 40*len(clusters)
dictClusters={1:(1,0,0),
2:(0,1,0),
3:(0,0,1),
4:(-1,0,0),
5:(0,-1,0),
6:(0,0,-1)}
dictDesvests={1:0.15,
2:0.15,
3:0.15,
4:0.15,
5:0.15,
6:0.15}
label = []
df = pd.DataFrame(columns = ['x1','x2','x3'])
for _ in range(Npuntos):
cluN = np.random.choice(clusters)
clu=dictClusters[cluN]
clx,cly,clz = clu
desvest = dictDesvests[cluN]
clx += np.random.normal(0,desvest)
cly += np.random.normal(0,desvest)
clz += np.random.normal(0,desvest)
label.append(cluN)
df.loc[df.shape[0]] = [clx,cly,clz]
df['label'] = label
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df.head()
| x1 | x2 | x3 | label | |
|---|---|---|---|---|
| 0 | -1.017333 | 0.098265 | 0.038147 | 4 |
| 1 | -1.266728 | -0.147091 | 0.011565 | 4 |
| 2 | -1.010595 | -0.059136 | -0.128969 | 4 |
| 3 | 0.018001 | -1.076903 | 0.076536 | 5 |
| 4 | -0.088302 | -0.908741 | -0.319811 | 5 |
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colorsDict = {6:'orange',
1:'red',
2:'green',
3:'blue',
4:'black',
5:'purple'}
#maxLabel = np.max(df.label.unique())
fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(111, projection='3d')
u = np.linspace(0, 2 * np.pi, 39)
v = np.linspace(0,np.pi, 21)
x = np.outer(np.cos(u), np.sin(v))
y = np.outer(np.sin(u), np.sin(v))
z = np.outer(np.ones(np.size(u)), np.cos(v))
## Use 3x the stride, no scipy zoom
#ax = fig.gca(projection='3d')
##ax.plot_surface(x, y, z, rstride=3, cstride=3, color='black', shade=0)
#ax.plot_wireframe(x, y, z,rstride=2,cstride=2)
#plt.show()
# Normalize to [0,1]
norm = plt.Normalize(z.min(), z.max())
colors = cm.viridis(norm(z))
rcount, ccount, _ = colors.shape
#fig = plt.figure()
ax.plot_wireframe(x, y, z,rstride=2,cstride=2)
#surf = ax.plot_surface(x, y, z, rcount=rcount, ccount=ccount,
# facecolors=colors, shade=False)
for ii in df.index:
ax.scatter(
df.loc[ii,'x1'],
df.loc[ii,'x2'],
df.loc[ii,'x3'],
marker='.',
s=100,
color=colorsDict[df.loc[ii,'label']]
)
ax.set_xlim((-1.5,1.5))
ax.set_ylim((-1.5,1.5))
ax.set_zlim((-1.5,1.5))
plt.show()
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sns.pairplot(df, hue="label")
<seaborn.axisgrid.PairGrid at 0x13ee5a7cfd0>
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colorsDict = {6:'orange',
1:'red',
2:'green',
3:'blue',
4:'black',
5:'purple'}
#maxLabel = np.max(df.label.unique())
fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(111, projection='3d')
u = np.linspace(0, 2 * np.pi, 39)
v = np.linspace(0,np.pi, 21)
x = np.outer(np.cos(u), np.sin(v))
y = np.outer(np.sin(u), np.sin(v))
z = np.outer(np.ones(np.size(u)), np.cos(v))
## Use 3x the stride, no scipy zoom
#ax = fig.gca(projection='3d')
##ax.plot_surface(x, y, z, rstride=3, cstride=3, color='black', shade=0)
#ax.plot_wireframe(x, y, z,rstride=2,cstride=2)
#plt.show()
# Normalize to [0,1]
norm = plt.Normalize(z.min(), z.max())
colors = cm.viridis(norm(z))
rcount, ccount, _ = colors.shape
#fig = plt.figure()
ax.plot_wireframe(x, y, z,rstride=2,cstride=2)
#surf = ax.plot_surface(x, y, z, rcount=rcount, ccount=ccount,
# facecolors=colors, shade=False)
for ii in df.index:
ax.scatter(
df.loc[ii,'x1'],
df.loc[ii,'x2'],
df.loc[ii,'x3'],
marker='.',
s=100,
color=colorsDict[df.loc[ii,'label']]
)
# proyeccion 1
for ii in df.index:
ax.scatter(
-3,
df.loc[ii,'x2'],
df.loc[ii,'x3'],
marker='.',
s=100,
color=colorsDict[df.loc[ii,'label']]
)
# proyeccion 2
for ii in df.index:
ax.scatter(
df.loc[ii,'x1'],
3,
df.loc[ii,'x3'],
marker='.',
s=100,
color=colorsDict[df.loc[ii,'label']]
)
# proyeccion 2
for ii in df.index:
ax.scatter(
df.loc[ii,'x1'],
df.loc[ii,'x2'],
-3,
marker='.',
s=100,
color=colorsDict[df.loc[ii,'label']]
)
plt.show()
Ninguna proyección es completamente satisfactoria: problema con la distancia.
Nota: estas no son las únicas proyecciones posibles.Existen más proyecciones posibles
Recordatorio: proyección ortogonal en una recta
Ejemplo ad hoc en el plano
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c1 = np.array((1,1))
c2 = np.array((2,1))
c3 = np.array((1,2))
c4 = np.array((2,2))
centers = [c1,c2,c3,c4]
Npuntos = 20
df = pd.DataFrame(columns=['x1','x2'])
labels=[]
for lb,c in enumerate(centers):
for _ in range(Npuntos):
df.loc[df.shape[0]] = c+np.random.uniform(-0.25,0.25,2)
labels.append(lb)
df['label'] = labels
df.head()
| x1 | x2 | label | |
|---|---|---|---|
| 0 | 1.056981 | 0.935569 | 0 |
| 1 | 0.781720 | 0.935267 | 0 |
| 2 | 0.810779 | 0.867002 | 0 |
| 3 | 0.906029 | 1.030259 | 0 |
| 4 | 1.103518 | 1.027526 | 0 |
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sns.pairplot(df, hue="label")
<seaborn.axisgrid.PairGrid at 0x13ee1c661d0>
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# proyecciones
p1 = []
vR1 = np.array((1,1))
iR1 = np.inner(vR1,vR1)
p2 = []
vR2 = np.array((1,-1))
iR2 = np.inner(vR2,vR2)
for ii in df.index:
v = np.array((df.loc[ii,'x1'],df.loc[ii,'x2']))
p1.append(np.inner(v,vR1)/iR1)
p2.append(np.inner(v,vR2)/iR2)
df['p1'] = p1
df['p2'] = p2
df.head()
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| x1 | x2 | label | p1 | p2 | |
|---|---|---|---|---|---|
| 0 | 1.056981 | 0.935569 | 0 | 0.996275 | 0.060706 |
| 1 | 0.781720 | 0.935267 | 0 | 0.858494 | -0.076773 |
| 2 | 0.810779 | 0.867002 | 0 | 0.838890 | -0.028112 |
| 3 | 0.906029 | 1.030259 | 0 | 0.968144 | -0.062115 |
| 4 | 1.103518 | 1.027526 | 0 | 1.065522 | 0.037996 |
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colorsDict = {0:'orange',
1:'red',
2:'green',
3:'blue'}
colores = [colorsDict[df.label.loc[i]] for i in df.index]
f,ax=plt.subplots(nrows=1,ncols=2,figsize=(20,10))
ax[0].scatter(df.x1,df.x2,c=colores,alpha=0.5)
ax[0].scatter(np.zeros(df.shape[0]),df.x2,marker='<',c=colores,alpha=0.5)
ax[0].scatter(df.x1,np.zeros(df.shape[0]),marker='v',c=colores,alpha=0.5)
ax[0].set_xlim((-0.5,3))
ax[0].set_ylim((-0.5,3))
ax[0].grid()
ax[1].scatter(df.x1,df.x2,c=colores,alpha=0.5)
ax[1].plot(np.linspace(-0,3,100),np.linspace(-0,3,100))
ax[1].plot(np.linspace(-3,3,100),-np.linspace(-3,3,100))
ax[1].scatter(df.p1,df.p1,marker = '+',c=colores,alpha=0.5)
ax[1].scatter(df.p2,-df.p2,marker = '+',c=colores,alpha=0.5)
ax[1].set_xlim((-1,3))
ax[1].set_ylim((-1,3))
ax[1].grid()
Nota: existen direcciones para las cuales las proyecciones son más adecuadas
Mapa#
Fig. 26 Mapa de Madrid#
Un mapa es una proyección cartográfica en la que puntos geográficos cercanos están cerca en el mapa.