Theoretical Formulation of the Linear Seriation Criterion

Oct 13, 2025 by miodeya.com 23 views

Introduction

The seriation package in R provides a powerful tool for visualizing and analyzing large distance matrices. One of the key features of this package is the Linear Seriation criterion, which is used to evaluate the quality of a given ordering of the data. However, as we will discuss in this article, there appears to be a discrepancy between the theoretical formulation of the Linear Seriation criterion and its implementation in the seriation package.

Theoretical Formulation of the Linear Seriation Criterion

The Linear Seriation criterion is defined as a loss function, which means that it is designed to penalize poor orderings of the data. The formula for the Linear Seriation criterion is given by:

LS = -∑(i, j) |i - j| d(i, j)

where d(i, j) is the distance between objects i and j, and |i - j| is the absolute difference between their indices.

As we can see from the formula, the Linear Seriation criterion is a sum of negative values, since the distance d(i, j) is always positive, and the absolute difference |i - j| is also always positive. Therefore, the result of the Linear Seriation criterion should always be negative.

Implementation of the Linear Seriation Criterion in the `seriation` Package

However, as we will show in the following example, the implementation of the Linear Seriation criterion in the seriation package does not produce negative values.

library(seriation)

# Create a 5x5 matrix filled with 5s
mat <- matrix(5, nrow = 5, ncol = 5)

# Set the main diagonal to 0
diag(mat) <- 0

# Define 2x2 blocks of ones around the main diagonal
mat[1:2, 1:2] <- 1
mat[2:3, 2:3] <- 1
mat[3:4, 3:4] <- 1
mat[4:5, 4:5] <- 1

# Print the original matrix
print("Original Matrix:")
print(mat)

# convert to a dist
mat <- as.dist(mat)

# Compute the linear seriation criterion
print(list_criterion_methods())
criterion_value <- criterion(mat, method = "LS")

# Print the linear seriation criterion value
print("Linear Seriation Criterion:")
print(criterion_value)

As we can see from the output of this code, the Linear Seriation criterion value is positive, which contradicts the theoretical formulation of the criterion.

Possible Explanations

There are several possible explanations for this discrepancy:

Missing -1: The -1 in the formula for the Linear Seriation criterion may be missing from the implementation in the seriation package.
Value Overflow: The calculation of the Linear Seriation criterion may be causing a value overflow, which would result in a positive value being returned instead of a negative value.
Implementation Error: There may be an implementation error in the seriation package that is causing the Linear Seriation criterion to be calculated incorrectly.

Conclusion

In conclusion, the Linear Seriation criterion in the seriation package does not produce negative values, which contradicts the theoretical formulation of the criterion. This discrepancy may be due to a missing -1, a value overflow, or an implementation error. Further investigation is needed to determine the cause of this discrepancy and to ensure that the Linear Seriation criterion is implemented correctly in the seriation package.

Recommendations

Based on our analysis, we recommend the following:

Check the implementation: The developers of the seriation package should review the implementation of the Linear Seriation criterion to ensure that it is correct and consistent with the theoretical formulation.
Test the criterion: The Linear Seriation criterion should be tested thoroughly to ensure that it produces the expected results.
Provide documentation: The seriation package should provide clear documentation on the implementation of the Linear Seriation criterion, including the formula and any assumptions or limitations.

Q: What is the Linear Seriation criterion?

A: The Linear Seriation criterion is a measure of the quality of a given ordering of data. It is a loss function that penalizes poor orderings of the data.

Q: What is the formula for the Linear Seriation criterion?

A: The formula for the Linear Seriation criterion is:

LS = -∑(i, j) |i - j| d(i, j)

where d(i, j) is the distance between objects i and j, and |i - j| is the absolute difference between their indices.

Q: Why is the Linear Seriation criterion a loss function?

A: The Linear Seriation criterion is a loss function because it is designed to penalize poor orderings of the data. A lower value of the criterion indicates a better ordering of the data.

Q: Why do I get positive values when applying the Linear Seriation criterion?

A: There are several possible explanations for this discrepancy:

Missing -1: The -1 in the formula for the Linear Seriation criterion may be missing from the implementation in the seriation package.
Value Overflow: The calculation of the Linear Seriation criterion may be causing a value overflow, which would result in a positive value being returned instead of a negative value.
Implementation Error: There may be an implementation error in the seriation package that is causing the Linear Seriation criterion to be calculated incorrectly.

Q: How can I ensure that the Linear Seriation criterion is implemented correctly?

A: To ensure that the Linear Seriation criterion is implemented correctly, you can:

Check the implementation: Review the implementation of the Linear Seriation criterion in the seriation package to ensure that it is correct and consistent with the theoretical formulation.
Test the criterion: Test the Linear Seriation criterion thoroughly to ensure that it produces the expected results.
Provide documentation: Provide clear documentation on the implementation of the Linear Seriation criterion, including the formula and any assumptions or limitations.

Q: What are the implications of the discrepancy in the Linear Seriation criterion?

A: The discrepancy in the Linear Seriation criterion may have implications for the accuracy and reliability of the results obtained from the seriation package. It is essential to ensure that the criterion is implemented correctly to avoid any potential issues.

Q: How can I report a bug or issue with the Linear Seriation criterion?

A: If you encounter any issues or discrepancies with the Linear Seriation criterion, you can report them to the developers of the seriation package. Provide clear and concise information about the issue, including any relevant code or data.

Q: What is the current status of the Linear Seriation criterion in the `seriation` package?

A: The current status of the Linear Seriation criterion in the seriation package is that it does not produce negative values, which contradicts the theoretical formulation of the criterion. Further investigation is needed to determine the cause of this discrepancy and to ensure that the criterion is implemented correctly.

Q: How can I contribute to the development of the `seriation` package?

A: You can contribute to the development of the seriation package by:

Reporting bugs or issues: Report any bugs or issues you encounter with the package.
Providing feedback: Provide feedback on the package, including suggestions for improvement.
Contributing code: Contribute code to the package to improve its functionality and accuracy.

By contributing to the development of the seriation package, you can help ensure that it remains a reliable and accurate tool for data analysis.