Robust Principal Component Analysis Overview and Implementation Examples

Robust Principal Component Analysis(RPCA)

Robust Principal Component Analysis (RPCA) is a method for finding a basis in data and is characterized by its robustness to data containing outliers and noise.

On the other hand, ordinary principal component analysis (PCA), described in “About Principle Component Analysis (PCA)” finds principal components by decomposing the covariance matrix of the data into singular values decomposition as described in “Overview of Singular Value Decomposition (SVD) and examples of algorithms and implementations“, and is very sensitive to outliers and noise, and if they are included If they are included, accurate results cannot be obtained.

Robust Principal Component Analysis Implementation Flow

The general flow of performing a robust principal component analysis is as follows

1. Data Preparation: Before performing RPCA, the dataset to which the principal component analysis will be applied is prepared. The data is represented as a matrix, where each row corresponds to an observed value and each column corresponds to a variable or feature.
2. Outlier detection: RPCA is a robust method against outliers and noise. The first step is to detect outliers in the data set. This is done using an outlier detection technique (e.g., MCD).
3. Perform Principal Component Analysis: Once outliers have been detected, a principal component analysis is performed. Principal component analysis is used to reduce the dimensionality of data and extract features, extracting principal components from the data set and projecting the data into the space of principal components.
4. Evaluate Results: Evaluate the results of the Principal Component Analysis. This includes calculating the contribution and cumulative contribution of each principal component, interpreting the principal components, and plotting the results. Evaluating the results will allow you to understand the characteristics and patterns of the data set.
5. Applications: There are many ways to apply the results of RPCA. As an example, principal component scores can be used for data clustering and anomaly detection, or for visualization and feature extraction through dimensionality reduction.

Robust Principal Component Analysis Application Examples

The following are some examples of applications of Robust Principal Component Analysis.

• Image Processing: Robust Principal Component Analysis is used to separate background and foreground in images and to remove noise and chorus in images. This can, for example, be used to robustly process background noise and motion when detecting and tracking people in video surveillance footage.
• Data cleansing: Robust Principal Component Analysis is used to identify and remove outliers and anomalies in a data set. This is used when anomaly detection and data cleansing are critical tasks, such as in the analysis of financial or sensor data.
• Acoustic Signal Processing: Robust Principal Component Analysis is also used for audio data and acoustic signals. It is used, for example, to remove noise and ambient noise effects from musical data, to separate musical components, and as a preprocessor for speech recognition and speech analysis.
• Big Data Analysis: Robust Principal Component Analysis can be an effective method for large data sets. It is used in big data analysis when noise and outliers in the data set need to be handled accurately, for example, in the analysis of customer data and web log data.
• Financial Data Analysis: Robust Principal Component Analysis (RPCA) is also widely applied to the analysis of financial data.
• Sensor data processing: Robust Principal Component Analysis (RPCA) can also be a useful technique in processing sensor data. Sensor data typically contain noise and outliers, and RPCA provides a robust analysis method for these elements.

Robust Principal Component Analysis is used in a variety of application areas where robustness is required for the data or where noise and outliers are expected to be present.

Libraries and algorithms used in robust principal component analysis

Robust Principal Component Analysis (RPCA) is a method of principal component analysis that is robust against outliers and noise in a data set. Some common libraries and algorithms are listed below.

• RRCM (Robust Recursive Component Analysis): RRCM is a robust principal component analysis algorithm for detecting outliers in data. In this algorithm, the data is updated sequentially and the principal components are recalculated.
• ROBPCA (Robust Principal Component Analysis): ROBPCA is an R language package that provides robust principal component analysis methods. This package includes features such as outlier detection and data plotting.
• PCA with Outliers (PCAO): PCAO is a method provided by the Python scikit-learn library. This method takes into account outliers in the data when applying Principal Component Analysis.
• MCD (Minimum Covariance Determinant): MCD is a method for estimating the covariance matrix taking into account outliers. This method may be applied to robust principal component analysis.

Robust Principal Component Analysis (RPCA) Implementation Example

As an example of robust principal component analysis (RPCA) implementation, a method using Python’s scikit-learn library is shown below.

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import numpy as np

def robust_pca(X):
    # Data Standardization
    scaler = StandardScaler()
    X_scaled = scaler.fit_transform(X)

    # Running a Principal Component Analysis
    pca = PCA()
    pca.fit(X_scaled)

    # Removal of noise components
    X_noise_removed = X_scaled - np.dot(pca.transform(X_scaled), pca.components_)

    # Extraction of sparse components
    sparse_pca = PCA()
    sparse_pca.fit(X_noise_removed)

    # Return of results
    return pca.components_, sparse_pca.components_

# Data matrix for testing
X = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12]])

# Perform robust principal component analysis
components, sparse_components = robust_pca(X)

# Display Results
print("Components:")
print(components)
print("Sparse Components:")
print(sparse_components)

The code performs a robust principal component analysis of a given data matrix X. The implementation uses scikit-learn’s PCA class to perform the principal component analysis.

The specific steps here are as follows

1. Standardize data: Standardize data using StandardScaler.
2. Execution of Principal Component Analysis: Principal component analysis is performed on the standardized data using the PCA class.
3. Remove noise components: Remove noise components from the original data using the obtained principal components.
4. Extract sparse components: Run Principal Component Analysis again on the noise-eliminated data to extract sparse components.

Finally, the obtained principal components and sparse components are displayed.

Foreground image extraction using robust principal component analysis

Here is a specific example of foreground image extraction using robust principal component analysis. The input is an image series of length 200, with each frame being a 176 × 144 image. The 25344 × 200 matrix Y obtained by vectorizing each frame into a column vector of 25344 dimensions is the input matrix. The length of the original image series proposed by Li et al. is 3584, but here it is downsized to 200 for ease of computation. In this experiment, the regularization parameter λ=10-6,𝜃=25344‾‾‾‾‾‾√=159.2. The reason for taking θ in this way is minimized in Candes et al.

By decomposing Y into a low-rank matrix L and a sparse matrix S, we expect to separate backgrounds that change slowly over time into L and objects that are only represented in some frames, such as passersby, into the sparse matrix S. In general, image processing involves taking the median (median) for each pixel to achieve such a separation. Since the median is obtained by minimizing the sum of the L1 errors, such a process can be viewed as estimation based on a more constrained model where all L columns are equal (the median does not change from frame to frame).

The figure below shows the input data Y and some corresponding frames of the matrices L and S obtained from the experiment.

It can be seen that many passers-by are extracted into the sparse matrix S and successfully separated from the low-rank matrix L. On the other hand, we can see that objects that remain motionless in the camera’s field of view for long periods of time, such as the person with the suitcase in frames 2054 and 3080, cannot be distinguished from the background and are included in the low-rank matrix L. The background image is unlikely to change from this figure, but its overall brightness changes slowly over time, indicating that the low-rank assumption is appropriate.

Financial Data Analysis with Robust Principal Component Analysis

<Overview>

The following are examples of the use of RPCA in financial analysis.

• Portfolio optimization: RPCA can be used to optimize a portfolio of financial assets. By applying principal component analysis and extracting the characteristics of the covariance matrix, it is possible to optimize the balance between risk and return of the portfolio.
• Anomaly detection: RPCA can help detect outliers and noise within financial data. In financial markets, price fluctuations and abnormal behavior of trades can occur, and RPCA makes it possible to detect these abnormal patterns and take appropriate action.
• Feature extraction of time-series data: Time-series data in financial markets contain complex patterns and trends, and RPCA can be applied to extract important features buried in the time-series data. This helps in predicting future price fluctuations and understanding market trends.
• Market Risk Assessment: RPCA can be used to assess financial market risk. Through principal component analysis, characteristic risk factors can be extracted from the covariance matrix, enabling risk assessment of portfolios and investment strategies.

These are only a few examples. RPCA can be applied to financial analysis in a variety of ways, and can be used to extract characteristics of financial data to improve risk management and investment strategies.

<Implementation in Python>

Below is an example of a Python implementation of RPCA for financial data analysis.

1. Example of RPCA implementation using Scikit-learn:

from sklearn.decomposition import PCA
from sklearn.preprocessing import RobustScaler

# Load financial data (load data as appropriate)
data = load_financial_data()

# Data scaling (using robust scaler)
scaler = RobustScaler()
scaled_data = scaler.fit_transform(data)

# Execution of RPCA
pca = PCA(n_components=2)  # Specify the number of principal components
principal_components = pca.fit_transform(scaled_data)

# Subsequent processing, including evaluation and visualization of results

In this example, RPCA is implemented using the PCA class in the scikit-learn library; RobustScaler is used to robustly scale the data, and the fit_transform method in the PCA class is used to perform principal component analysis.

2. Example implementation of RPCA using StatsModels:

import statsmodels.api as sm
import numpy as np

# Load financial data (load data as appropriate)
data = load_financial_data()

# Execution of RPCA
results = sm.RobustPCA(data)

# Acquisition of noise and signal components
noise = results.resid
signal = results.fittedvalues

# Subsequent processing, including evaluation and visualization of results

In this example, RPCA is implemented using the RobustPCA function from the StatsModels library.The RobustPCA function decomposes data into noise and signal components. The noise component can be obtained from the resid attribute, and the signal component can be obtained from the fittedvalues attribute.

Sensor data processing with robust principal component analysis

<Overview>

The following is a general procedure for processing sensor data using RPCA.

1. Data Preparation: Collect sensor data and prepare a data set in an appropriate format. Data typically contains multiple sensor measurements observed along a time step.
2. Noise Removal: Sensor data may contain noise generated during measurements; RPCA can be used to remove noise, and principal component analysis is applied to identify and remove noise components in the data set.
3. Anomaly Detection: Sensor data may contain anomalous patterns associated with system anomalies or failures; RPCA can be used to detect anomalous data points and anomalous patterns, and principal component analysis is used to capture anomalous data features.
4. Feature Extraction: Sensor data may contain multiple measurements from multiple sensors, and RPCA can be used to extract characteristic patterns and correlations in the data set. This allows for dimensionality reduction and feature extraction of the data.
5. Data visualization: Visualization of RPCA results can facilitate understanding and interpretation of sensor data, plotting the results of principal component analysis, and marking anomalous data.

<Implementation in Python>

Below is an example of RPCA implementation in Python.

1. Example implementation of RPCA using Scikit-learn:

from sklearn.decomposition import PCA
from sklearn.preprocessing import RobustScaler

# Load sensor data (load data as appropriate)
data = load_sensor_data()

# Data scaling (using robust scaler)
scaler = RobustScaler()
scaled_data = scaler.fit_transform(data)

# Execution of RPCA
pca = PCA(n_components=2)  # Specify the number of principal components
principal_components = pca.fit_transform(scaled_data)

# Perform subsequent processing, including visualization of results

In this example, RPCA is implemented using the PCA class in the scikit-learn library; RobustScaler is used to robustly scale the data, and the fit_transform method in the PCA class is used to perform principal component analysis.

2. Example implementation of RPCA using StatsModels:

import statsmodels.api as sm
import numpy as np

# Load sensor data (load data as appropriate)
data = load_sensor_data()

# Execution of RPCA
results = sm.RobustPCA(data)

# Acquisition of noise and signal components
noise = results.resid
signal = results.fittedvalues

# Perform subsequent processing, including visualization of results

In this example, RPCA is implemented using the RobustPCA function from the StatsModels library, which decomposes data into noise and signal components, with the noise component available from the resid attribute and the signal component available from the fittedvalues attribute.

Reference Information and Reference Books

For reference information and implementations of robust principal component analysis, see”PCA and Robust PCA for Modern Datasets“、”New Robust Principal Component Analysis for Joint Image Alignment and Recovery via Affine Transformations, Frobenius and  Norms“、”Robust PCA“.

Reference book are “Constrained Principal Component Analysis and Related Techniques“,

“Advances in Principal Component Analysis: Research and Development“.