Question about applying PCA with one Component

Question

I have a set of data that I've been assigned to apply PCA and retain one component and then visualize the distribution in a scatter plot which indicates the class of each data point.

For context: The data we're working with has three columns. X is column 1 and 2 and y is column 3 which contains the class of each data point.

It was implied that the resulting visualization should be a horizontal line, but I'm not seeing that. The resulting visualization is a scatter plot that looks like a positive linear distribution.

import pandas as pd
import sklearn
from sklearn.model_selection import train_test_split
import numpy as np
from sklearn.preprocessing import StandardScaler
from sklearn.decomposition import PCA
import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap


df = pd.read_csv("data.csv", header=None)
X = df.iloc[:, 0:2].values
y = df.iloc[:,-1].values
X_train, X_test, y_train, y_test = train_test_split(X,y, test_size=0.3,random_state=np.random)
sc_X = StandardScaler()
X_train = sc_X.fit_transform(X_train)
X_test = sc_X.transform(X_test)

pcaObj1 = PCA(n_components=1)
X_train_PCA = pcaObj1.fit_transform(X_train)
X_test_PCA = pcaObj1.transform(X_test)
X_set, y_set = X_test_PCA, y_test
X3 = np.meshgrid(np.arange(start = X_set[:, 0].min() - 1, stop = X_set[:, 0].max() + 1, step = 0.01))
X3 = np.array(X3)

plt.xlim(X3.min(), X3.max())
plt.ylim(X3.min(), X3.max())
for i, j in enumerate(np.unique(y_set)):
    plt.scatter(X_set[y_set == j, 0], X_set[y_set == j, 0],
                c = ListedColormap(('purple', 'yellow'))(i), label = j)

Answer 1

This example should give some clarity. Make sure you read all the comments so you can follow what's going on.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib as mpl
import urllib.request
import random
# seaborn is a layer on top of matplotlib which has additional visualizations -
# just importing it changes the look of the standard matplotlib plots.
# the current version also shows some warnings which we'll disable.
import seaborn as sns
sns.set(style="white", color_codes=True)
import warnings
warnings.filterwarnings("ignore")


from sklearn import datasets
iris = datasets.load_iris()
X = iris.data[:, 0:4]  # we take the first four features.
y = iris.target

print(X.sample(5))
print(y.sample(5))

# see how many samples we have of each species 
data["species"].value_counts()


from sklearn import preprocessing

scaler = preprocessing.StandardScaler()

scaler.fit(X)
X_scaled_array = scaler.transform(X)
X_scaled = pd.DataFrame(X_scaled_array, columns = X.columns)

X_scaled.sample(5)


# try clustering on the 4d data and see if can reproduce the actual clusters.

# ie imagine we don't have the species labels on this data and wanted to
# divide the flowers into species. could set an arbitrary number of clusters
# and try dividing them up into similar clusters.

# we happen to know there are 3 species, so let's find 3 species and see
# if the predictions for each point matches the label in y.

from sklearn.cluster import KMeans

nclusters = 3 # this is the k in kmeans
seed = 0

km = KMeans(n_clusters=nclusters, random_state=seed)
km.fit(X_scaled)

# predict the cluster for each data point
y_cluster_kmeans = km.predict(X_scaled)
y_cluster_kmeans


# use seaborn to make scatter plot showing species for each sample
sns.FacetGrid(data, hue="species", size=4) \
   .map(plt.scatter, "sepal_length", "sepal_width") \
   .add_legend();

# do same for petals
sns.FacetGrid(data, hue="species", size=4) \
   .map(plt.scatter, "petal_width", "sepal_width") \
   .add_legend();

# if you have a lot of features it can be helpful to do some feature reduction
# to avoid the curse of dimensionality (i.e. needing exponentially more data
# to do accurate predictions as the number of features grows).

# you can do this with Principal Component Analysis (PCA), which remaps the data
# to a new (smaller) coordinate system which tries to account for the
# most information possible.

# you can *also* use PCA to visualize the data by reducing the 
# features to 2 dimensions and making a scatterplot. 
# it kind of mashes the data down into 2d, so can lose 
# information - but in this case it's just going from 4d to 2d, 
# so not losing too much info. 

# so let's just use it to visualize the data...

# mash the data down into 2 dimensions

from sklearn.decomposition import PCA

ndimensions = 2

seed = 10 
pca = PCA(n_components=ndimensions, random_state=seed)
pca.fit(X_scaled)
X_pca_array = pca.transform(X_scaled)
X_pca = pd.DataFrame(X_pca_array, columns=['PC1','PC2']) # PC=principal component
X_pca.sample(5)



# so that gives us new 2d coordinates for each data point.

# at this point, if you don't have labelled data,
# you can add the k-means cluster ids to this table and make a
# colored scatterplot. 

# we do actually have labels for the data points, but let's imagine
# we don't, and use the predicted labels to see what the predictions look like.

# first, convert species to an arbitrary number
y_id_array = pd.Categorical(data['species']).codes

df_plot = X_pca.copy()
df_plot['ClusterKmeans'] = y_cluster_kmeans
df_plot['SpeciesId'] = y_id_array # also add actual labels so we can use it in later plots
df_plot.sample(5)


# so now we can make a 2d scatterplot of the clusters
# first define a plot fn

def plotData(df, groupby):
    "make a scatterplot of the first two principal components of the data, colored by the groupby field"

    # make a figure with just one subplot.
    # you can specify multiple subplots in a figure, 
    # in which case ax would be an array of axes,
    # but in this case it'll just be a single axis object.
    fig, ax = plt.subplots(figsize = (7,7))

    # color map
    cmap = mpl.cm.get_cmap('prism')

    # we can use pandas to plot each cluster on the same graph.
    # see http://pandas.pydata.org/pandas-docs/stable/generated/pandas.DataFrame.plot.html
    for i, cluster in df.groupby(groupby):
        cluster.plot(ax = ax, # need to pass this so all scatterplots are on same graph
                     kind = 'scatter', 
                     x = 'PC1', y = 'PC2',
                     color = cmap(i/(nclusters-1)), # cmap maps a number to a color
                     label = "%s %i" % (groupby, i), 
                     s=30) # dot size
    ax.grid()
    ax.axhline(0, color='black')
    ax.axvline(0, color='black')
    ax.set_title("Principal Components Analysis (PCA) of Iris Dataset");

# plot the clusters each datapoint was assigned to
plotData(df_plot, 'ClusterKmeans')

Answer 2

I see that you have a test set in addition to a training set, however this not the usual setup for PCA. PCA has multiple applications, but one of the main ones is dimensionality reduction. Dimensionality reduction is about removing variables, and PCA serves this purpose by changing the basis of your data and ordering them by the amount (or relative amount) of the total variation that they linearly explain. Since this does not require test data, we can think of this as unsupervised machine learning, although many would also prefer to call this feature engineering as it is often used to preprocess data to improve the performance of models trained on that preprocessed data.

Let me generate a random dataset with 10 variables and 1000 entries for the sake of example. Fitting the PCA transform for 1 component, you're selecting a new variable (feature) that is a linear combination of the original variables that attempts to linearly explain the most variance in the data. As you say, it is a number line; just as a quick-and-easy plot let's just use the x-axis as the index of the new variable array and the y-axis as the value of the variable.

import matplotlib.pyplot as plt
import numpy as np
from sklearn.decomposition import PCA

X_train = np.random.random((1000, 10))
y_labels = np.array([0] * 500 + [1] * 500)

pcaObj1 = PCA(n_components=1)
X_PCA = pcaObj1.fit_transform(X_train)

plt.scatter(range(len(y_labels)), X_PCA, c=['red' if i==0 else 'green' for i in y_labels])
plt.show()

You can see this produces a 1000 x 1 array representing your new variable.

>>> X_PCA.shape
(1000, 1)

If you had selected n_components=2 instead, you'd have a 1000 x 2 array with two such variables. Let's see that as example. This time I'll plot the two principal components against each other instead of using a single principal component against its index.

import matplotlib.pyplot as plt
import numpy as np
from sklearn.decomposition import PCA

X_train = np.random.random((1000, 10))
y_labels = np.array([0] * 500 + [1] * 500)

pcaObj1 = PCA(n_components=2)
X_PCA = pcaObj1.fit_transform(X_train)

plt.scatter(X_PCA[:,0], X_PCA[:,1], c=['red' if i==0 else 'green' for i in y_labels])
plt.show()

Now, my randomly-generated data may not have the same properties as your data set. If you really expect the output to be a line, then I'd say certainly not as my example generates a very eratic trace. You'll see even in the 2D case that the data doesn't seem structured by class, but that's what you would expect from random data.