Factor analysis techniques in Python

Factor analysis is a statistical method of unsupervised learning that is used to analyze the relationship between multiple variables and find out the potential factors that affect these variables. There are a variety of factor analysis techniques and libraries available in Python, and this article will introduce several of them.

1. Principal Component Analysis (PCA)

Principal Component Analysis (PCA) is a method of factor analysis, which can transform a high-dimensional data set into a low-dimensional subspace. PCA can be used to reduce the impact of noisy or redundant variables while retaining the most important information in the data set. In Python, PCA can be easily implemented using the sklearn library.

The following is a sample code that shows how to use PCA to reduce the dimensionality of a handwritten digits dataset:

import numpy as np
from sklearn.decomposition import PCA
from sklearn.datasets import load_digits

digits = load_digits()
X = digits.data
y = digits.target

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

print(f"Original shape: {X.shape}, reduced shape: {X_reduced.shape}")

The code first loads the handwritten digits dataset and then uses PCA to reduce the dimensionality of the dataset to 2 dimensions, and finally output the data shape before and after dimensionality reduction.

2. Independent Component Analysis (ICA)

Independent Component Analysis (ICA) is a statistical method for finding multiple signal sources. In Python, ICA can be implemented using scikit-learn’s FastICA class. The FastICA algorithm assumes that each signal source is independent of each other and has a non-Gaussian distribution.

Here is a sample code showing how to use FastICA to separate signals:

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

np.random.seed(0)
n_samples = 2000
time = np.linspace(0, 8, n_samples)

s1 = np.sin(2 * time)  # 信号1
s2 = np.sign(np.sin(3 * time))  # 信号2
S = np.c_[s1, s2]
S += 0.2 * np.random.normal(size=S.shape)

# 混合信号
A = np.array([[1, 1], [0.5, 2]])
X = np.dot(S, A.T)

# ICA潜在成分分离
ica = FastICA(n_components=2)
S_ = ica.fit_transform(X)
A_ = ica.mixing_

# 打印结果
fig, ax = plt.subplots(3, figsize=(10, 10))
ax[0].plot(time, S)
ax[0].set_title('True Sources')
ax[1].plot(time, X)
ax[1].set_title('Mixed Signals')
ax[2].plot(time, S_)
ax[2].set_title('ICA Recovered Signals')
fig.tight_layout()
plt.show()

The code first generates two random signals and mixes them into two new signals. The signals are then separated using FastICA and finally the results are plotted.

3. Factor rotation

When performing factor analysis, factor rotation is an important step. Factor rotation can make the correlations between factors clearer and allow for better identification of underlying factors. There are several factor rotation techniques to choose from in Python, such as varimax and promax rotation.

Here is a sample code showing how to use varimax rotation to analyze the Iris dataset:

import numpy as np
from factor_analyzer import FactorAnalyzer
from sklearn.datasets import load_iris

iris = load_iris()
X = iris.data

# 因子分析
fa = FactorAnalyzer(rotation="varimax", n_factors=2)
fa.fit(X)

# 输出结果
print(fa.loadings_)

The code first loads the Iris dataset and then uses factor analysis and varimax rotation to extract latent factors. Finally, the factor loadings are output.

Summary:

Python provides a variety of factor analysis techniques and libraries to choose from. Principal component analysis can be used to reduce the dimensionality of data, independent component analysis can be used to separate signals, and factor rotation can help us better understand the relationship between underlying factors. Different methods have different advantages and application scenarios, and it is necessary to choose a suitable method based on the characteristics of the data set.

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