2-Cats

2d Copula Approximating Transforms (2-Cats). 2-Cats is a neural network approach to learn 2-D Copulas. It relies on monotonic transforms of the input data from the I ([0, 1]) domain to the Real ([-∞, ∞]) domain. Input data consists of the Cumulative Distribution Functions (CDFs) values for 2-D marginals (e.g., x and y).

After this data is mapped to a vector of two numbers each on the real domain, the marginals are copulated using any Bivariate Cumulative Probability Distribution.

Using the Model

See the notebooks Example and Example on Boston Dataset. Also, the code snippet below.

from copulae.input import generate_copula_net_input

from copulae.training.cflax.mono_aux import EluPOne

from copulae.training.cflax.two_cats import FlexibleBi
from copulae.training.cflax.two_cats import NormalBi
from copulae.training.cflax.two_cats import PositiveLayer
from copulae.training.cflax.two_cats import TransformLayer
from copulae.training.cflax.two_cats import TwoCats

from copulae.training import setup_training

from copulae.training.loss import copula_likelihood
from copulae.training.loss import sq_error
from copulae.training.loss import sq_error_partial
     
from tdqm import tdqm

import jax
import jax.numpy as jnp
import numpy as np
import optax

D = # some 2d dataset with dimensions on columns (2, n) shape. Transpose a (n, 2) data.
TrainingTensors = generate_copula_net_input(
    D=D,  # 2 cats receives transposed data for historic reasons
    bootstrap=False
)

# Losses to consider when training
losses = [
    (0.01, sq_error),
    (0.5, sq_error_partial),
    (0.1, copula_likelihood),
]

# The Model
layer_widths = [128, 64, 32, 16]
model = TwoCats(         # the 2-Cats models
    [                    # Will perform a sequence of monotonic transforms, usually one
        TransformLayer(
            PositiveLayer(layer_widths, EluPOne, EluPOne, EluPOne)  # Using a NN that outputs Positive Numbers
        )
    ],
    NormalBi()           # and copulate it with a Bivariate CDF. Change to FlexibleBi for the Bivariate Logistic.
)


# Symbolic differentitation
# nn_C is the Copula
# nn_dC is the first derivative
# nn_c  is the second derivative, pseudo-likelihood
# cop_state is a training state for Flax
# forward is the forward function to perform gradient descent
# grad is the derivative to perform gradient descent
nn_C, nn_dC, nn_c, cop_state, forward, grad = setup_training(
    model, TrainingTensors, losses, rescale=True
)

# Initialize Model
key, subkey = jax.random.split(key)
init_params = model.init(subkey, TrainingTensors.UV_batches[0])
del subkey

lr = 2e-3
n_iter = 2000

params = init_params
optimizer = optax.adam(lr)
opt_state = optimizer.init(params)

# Training Loop
for i in tqdm(range(n_iter)):
    grads, cop_state = grad(params, cop_state)
    updates, opt_state = optimizer.update(grads, opt_state)
    params = optax.apply_updates(params, updates)

flaviovdf/copulae

2-Cats

Using the Model