Quantile and CDF Regression Example

Quantile Regression Objective

$$ J(\tau) = E\left(\rho(\tau, Y - u(\tau, X)|X\right)$$

CDF Regression Objective

$$ J(y_c) = E\left(1_{Y < y_c} \log v(y_c, X) + (1 - 1_{Y < y_c}) \log(1 - v(y_x, X)) | X\right)$$

The functions $u$, $v$ must be monotonic in $\tau$ and $y_c$ respectively.

Unconditional Distribution of $Y$

Quantile Regression

TensorFlow	JAX

CDF Estimation via Logistic Regression with Monotone Network

TensorFlow	JAX (logistic)	JAX (CRPS)

Conditional Distribution of $Y|X$

Quantile Regression

TensorFlow	JAX

CDF Estimation via Logistic Regression with Monotone Network

TensorFlow	JAX (logistic)	JAX (CRPS)

TODO

• do more quantitative error plots etc.
• normalizing flows ... i.e. round trip cdf and quantile as consistency constraint ... where is this done, is it actually doing anything to include this constraint.
• randomize $\tau$ sampling during training in jax instead of grid ...
• regularization/calibration/conformal prediction
• review the translation to jax and make sure arch is actually the same as tf
• make sure data is pure and deterministic across envs