Nonlinear Quadrature ADF / Power-EP Suite

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These rows are deterministic reference-free baselines: they use only the known transition, known observation model, and observed x,y, then project the local tilted distribution back to a strict Gaussian or Gaussian mixture.

x pattern	Model	components	power	state NLL	ref state NLL	cov 90	var ratio	pred-y NLL	ref pred NLL
sinusoidal	reference-free quadrature ADF Gaussian	1	1.00	6.054099	2.691023	0.817546	1.098277	0.687358	0.457118
sinusoidal	reference-free quadrature ADF K2	2	1.00	20.978921	2.691023	0.762207	0.524024	0.502116	0.457118
sinusoidal	reference-free quadrature ADF K4 spread 2pi	4	1.00	6.839985	2.691023	0.809408	0.877449	0.489497	0.457118
sinusoidal	reference-free quadrature Power-EP K4 alpha 0.5 spread 2pi alpha=0.50	4	0.50	2.866602	2.691023	0.852620	0.876197	0.506078	0.457118
weak_sinusoidal	reference-free quadrature ADF Gaussian	1	1.00	2.749367	2.692361	0.905273	0.945751	0.332878	0.301245
weak_sinusoidal	reference-free quadrature ADF K2	2	1.00	2.958450	2.692361	0.884115	0.938871	0.312388	0.301245
weak_sinusoidal	reference-free quadrature ADF K4 spread 2pi	4	1.00	2.648053	2.692361	0.919352	1.290849	0.314055	0.301245
weak_sinusoidal	reference-free quadrature Power-EP K4 alpha 0.5 spread 2pi alpha=0.50	4	0.50	2.637562	2.692361	0.928060	1.215569	0.320576	0.301245
intermittent_sinusoidal	reference-free quadrature ADF Gaussian	1	1.00	2.768658	2.695039	0.886312	0.908330	0.390607	0.351468
intermittent_sinusoidal	reference-free quadrature ADF K2	2	1.00	3.733279	2.695039	0.854736	0.791023	0.358784	0.351468
intermittent_sinusoidal	reference-free quadrature ADF K4 spread 2pi	4	1.00	2.674221	2.695039	0.903402	1.225335	0.356788	0.351468
intermittent_sinusoidal	reference-free quadrature Power-EP K4 alpha 0.5 spread 2pi alpha=0.50	4	0.50	2.528341	2.695039	0.919922	1.162230	0.362396	0.351468
zero	reference-free quadrature ADF Gaussian	1	1.00	2.693344	2.693317	0.913900	1.000501	0.259663	0.259663
zero	reference-free quadrature ADF K2	2	1.00	2.693476	2.693317	0.913900	1.000501	0.259663	0.259663
zero	reference-free quadrature ADF K4 spread 2pi	4	1.00	2.736383	2.693317	0.951904	1.369919	0.259663	0.259663
zero	reference-free quadrature Power-EP K4 alpha 0.5 spread 2pi alpha=0.50	4	0.50	2.736383	2.693317	0.951904	1.369919	0.259663	0.259663
random_normal	reference-free quadrature ADF Gaussian	1	1.00	8.833866	2.697773	0.740153	3.651619	0.803609	0.529506
random_normal	reference-free quadrature ADF K2	2	1.00	38.979881	2.697773	0.687988	0.407669	0.557264	0.529506
random_normal	reference-free quadrature ADF K4 spread 2pi	4	1.00	8.975737	2.697773	0.808838	0.755969	0.546472	0.529506
random_normal	reference-free quadrature Power-EP K4 alpha 0.5 spread 2pi alpha=0.50	4	0.50	5.002086	2.697773	0.815348	0.863937	0.561492	0.529506