Calculating the conditional cumulative residual entropy (CRE) for two continuous variables involves several steps. CRE measures the uncertainty of one variable given the knowledge of another variable. Here's a high-level overview of the process:

Collect Data: Gather a dataset containing observations of the two continuous variables, which we'll call X and Y.

Binning or Density Estimation: Since the variables are continuous, you may need to discretize them using binning or estimate their probability density functions (PDFs) using techniques like kernel density estimation.

Conditional Probability: Calculate the conditional probability distribution of X given Y. This can be done using:

Conditional PDF: If you estimated PDFs in step 2, you can find the conditional PDF of X|Y by dividing the joint PDF of X and Y by the marginal PDF of Y.

Histogram: If you used binning, calculate the conditional probabilities by dividing the joint histogram of X and Y by the histogram of Y.

Residual Entropy: Calculate the residual entropy for each conditional distribution of X|Y. This is done using the formula for entropy:

H(X|Y) = -∫[p(x|y) * log2(p(x|y))] dx

Where p(x|y) is the conditional probability of X given Y.

Cumulative Residual Entropy: Integrate the residual entropy values obtained in step 4 over the range of Y values. This will give you the cumulative residual entropy for X given Y. The formula for this integration depends on the specific mathematical expressions you obtained in step 4.

Normalization (Optional): If needed, you can normalize the cumulative residual entropy to make it comparable across different datasets or conditions.

Keep in mind that this process may require some computational tools or programming skills, especially when dealing with continuous variables. Additionally, the choice of binning method or PDF estimation technique can affect the results, so consider the appropriateness of these choices for your specific dataset and research question.

Ensure that you have a solid understanding of probability and information theory concepts to perform these calculations effectively

Please refer to the following website, in which the Python code is provided.

Link: https://dit.readthedocs.io/en/latest/measures/other/cumulative_residual_entropy.html#:~:text=The%20conditional%20cumulative%20residual%20entropy,probabilities%20conditioned%20on%20Y%3Dy%20.