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Applying Gaussian Processes to Compositional Data
Based on your question about applying Gaussian Processes (GPs) to compositional data, I can provide some guidance on your approach and implementation considerations in PyMC.
Your Proposed Approach
Your idea of using a hierarchical model with a Dirichlet prior on a multinomial response is a reasonable approach for compositional data. The structure you’ve outlined:
y_i ~ Mult(M_i, p_i)
Where you want to use a GP to model the mean of the Dirichlet vector at a given time t:
(mu_alpha) ~ GP(0, K)
This approach makes sense conceptually for modeling compositional time series data [1].
Implementation Considerations
1. Hierarchical Structure
Your hierarchical structure is similar to what’s shown in Document 1, where you’d use a GP to model the parameters of a Dirichlet distribution. This is a valid approach for compositional data that preserves the simplex constraint (proportions sum to 1) [1].
2. GP Approximation Methods
For practical implementation, you might want to consider using the Hilbert Space Gaussian Process (HSGP) approximation mentioned in Document 2. This would be particularly useful if:
- Your dataset is large (many time points)
- You need faster computation than a full GP would allow
- Your input dimension is low (1D for time or 2D for space-time) [2]
The HSGP approximation reduces computational complexity from O(n³) to O(mn + m), where n is the number of data points and m is the number of basis vectors [2].
3. Link Function
For the link function between the GP and the Dirichlet parameters, you’ll need to ensure the parameters remain positive. A common approach is to use an exponential or softmax transformation of the GP outputs to maintain the constraints of the Dirichlet distribution [1].
4. Multiple GPs for Multiple Dimensions
Since your compositional data has d dimensions, you could model this using multiple GPs with a shared covariance structure. Document 3 mentions that you can have “multiple GPs in the same problem” and use ARD (Automatic Relevance Determination) for multidimensional inputs [3].
Implementation in PyMC
Here’s a sketch of how you might implement this in PyMC:
import pymc as pm
import numpy as np
# Assuming X is your input (time points) and Y is your compositional data
with pm.Model() as model:
# GP hyperparameters
length_scale = pm.InverseGamma("length_scale", alpha=2.0, beta=1.0)
eta = pm.HalfNormal("eta", sigma=1.0)
# Define covariance function
cov_func = eta**2 * pm.gp.cov.Matern52(1, ls=length_scale)
# For each dimension of your compositional data (except one)
# We model d-1 dimensions since the last is determined by the constraint
gps = []
for i in range(d-1):
gp = pm.gp.Latent(cov_func=cov_func)
# Place GP prior
f = gp.prior(f"f_{i}", X=X)
# Transform GP output to ensure positivity
alpha_i = pm.Deterministic(f"alpha_{i}", pm.math.exp(f))
gps.append(alpha_i)
# The last dimension is determined by constraint
# Or you could model all dimensions and normalize
# Combine to form Dirichlet parameters
alpha = pm.math.stack(gps, axis=1)
# Dirichlet-Multinomial likelihood
for i in range(len(X)):
pm.Multinomial(f"y_{i}", n=M[i], p=alpha[i]/alpha[i].sum(), observed=Y[i])
Practical Considerations
-
Separability: Your assumption about separable space and time kernels is valid and commonly used. This can be implemented by taking the product of a spatial kernel and a temporal kernel [2].
-
Approximation Methods: If computational efficiency is a concern, the HSGP approach from Document 2 would be beneficial. For very large datasets, you might also consider variational inference methods [2].
-
Data Containers: Be aware that there have been issues using Data Containers with GPs in older versions of PyMC, as mentioned in Document 4. However, this may be resolved in newer versions [4].
Conclusion
Your approach is theoretically sound. The main challenges will be in the practical implementation, particularly in ensuring computational efficiency and proper transformation of the GP outputs to maintain the constraints of the Dirichlet distribution.
For implementation, I would recommend starting with a simplified version (perhaps with synthetic data) to ensure the model structure works as expected before scaling to your full dataset.