Cross decomposition

Cross decomposition is an algorithm used for solving complex problems where the data is divided into multiple sets that are related to each other. The goal is to find a common structure or relationship among these sets.

Imagine you have three sets of data: A, B, and C. These sets are interconnected, meaning that there is some kind of relationship between them. However, the relationship might not be immediately obvious or easy to analyze.

Cross decomposition aims to uncover this hidden relationship by decomposing the original problem into smaller, more manageable subproblems. It does this by creating a model that connects the sets together.

Here's how cross decomposition works:

1. First, the algorithm takes the original data sets A, B, and C.

2. It decomposes the problem by creating subproblems. Each subproblem involves one of the data sets and tries to find a relationship with the other sets.

3. The algorithm starts by analyzing the relationship between data set A and the other sets (B and C). It creates a model that describes how A is related to B and C.

4. Once it has learned the relationship between A and the other sets, it moves on to the next subproblem. It analyzes the relationship between B and the remaining sets (A and C) and creates another model.

5. Finally, it analyzes the relationship between C and the remaining sets (A and B) and creates a third model.

6. After creating these models, cross decomposition combines them to find a common structure or relationship among the original sets A, B, and C. It uses these models to predict or infer information about the sets that might not have been apparent before.

The main idea behind cross decomposition is that by breaking down a complex problem into smaller, more manageable subproblems and finding the relationships between these subproblems, we can gain a deeper understanding of the underlying structure and uncover hidden patterns or connections in the data.

Python Example

from sklearn.cross_decomposition import PLSRegression

# Let's say we have three sets of data: A, B, and C
A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]  # Features in set A
B = [[0.1, 0.2], [0.3, 0.4], [0.5, 0.6]]  # Features in set B
C = [10, 20, 30]  # Target variable in set C

# Create a PLSR model and fit it to the data
model = PLSRegression(n_components=1)
model.fit(A, C)

# Predict the target variable C based on the features in set A
predicted_C = model.predict(A)

# Print the predicted values
print("Predicted C values:", predicted_C)

Results:

Predicted C values: [[10.]
[20.]
[30.]]

We use the PLSRegression class from the sklearn.cross_decomposition module in scikit-learn to create a PLSR model. We set the number of components to 1, indicating that we want to find a one-dimensional relationship between the sets.

Next, we fit the model using the features in set A and the target variable in set C. The algorithm analyzes the relationship between these sets and creates a model that captures this relationship.

Once the model is trained, we can use it to predict the target variable C based on the features in set A. We call the predict method on the model and pass in the features in set A. The algorithm uses the learned relationship to make predictions.

Finally, we print the predicted values of C. These values represent the model's predictions for the target variable based on the features in set A.

Note that this is a simplified example to demonstrate the usage of cross decomposition. In practice, you would typically work with larger and more complex datasets and consider more sophisticated variations of cross decomposition algorithms depending on your specific problem.