Cross-Validation

The fundamental goal of cross-validation is to estimate a model's prediction error on unseen signals. A Volterra model with low training error but high validation error has memorized the training data's noise. This manifests as poor Adjacent Channel Leakage Ratio (ACLR) correction when the DPD encounters a new modulation scheme or power level not present in the training set. Cross-validation provides a statistically sound estimate of this out-of-sample performance.

The most common approach for time-series data involves dividing the captured waveform into K equal-sized folds. The process iterates K times:

• Train the Volterra model on K-1 folds.
• Validate on the held-out fold.
• The final performance metric is the average Normalized Mean Squared Error (NMSE) across all K trials. For PA measurements, a typical choice is K=5 or K=10, balancing computational cost against the variance of the error estimate.

Standard random shuffling is dangerous for PA behavioral modeling due to memory effects. If future samples leak into the training set, the model appears to predict the past from the future, yielding unrealistically optimistic results. The correct approach is blocked time-series cross-validation, where contiguous blocks of samples are held out. This preserves the causal structure of thermal trapping and bias network dynamics.

Cross-validation is the standard mechanism for tuning hyperparameters in sparse regression. When applying LASSO regression to prune a full Volterra series, a sequence of regularization parameters (λ) is tested. For each λ, cross-validation computes the validation error. The optimal λ is selected via the one-standard-error rule: choosing the most regularized model whose error is within one standard deviation of the minimum, maximizing sparsity without sacrificing accuracy.

A critical distinction exists between validation and test data. Cross-validation guides model selection (e.g., choosing memory depth). After the final model is locked, a completely untouched hold-out test set—often captured under different conditions (e.g., a different center frequency or temperature)—provides the final unbiased performance report. Reusing the validation folds to report final performance constitutes a statistical error.

For highly non-stationary signals like burst-mode traffic, repeated random sub-sampling offers advantages. Monte Carlo cross-validation randomly splits the data into training and validation sets many times (e.g., 100 iterations). The resulting distribution of NMSE values reveals the model's sensitivity to specific signal statistics. A wide variance indicates the training data lacks sufficient excitation of all nonlinear orders.