Kennen-o Output-Only Methods

• Kennen-o (Output-Only) is a comprehensive suite of methodologies for system identification using only output data, integrating classical statistics, deep learning, and Bayesian inference.
• It effectively extracts latent information such as input forces, modal responses, and governing equations from measurements like vibration, acceleration, or quantum observables.
• Kennen-o methods enable robust, real-time estimation and physics discovery in scenarios with unmeasured inputs, proving useful across quantum mechanics, structural dynamics, and nonlinear systems.

Kennen-o (Output-Only) is a suite of methodologies and algorithmic paradigms for parameter, state, and governing physics identification in physical, dynamical, and engineered systems relying exclusively on output (response) data. It enables extraction of latent system information—such as input forces, parameters, or modal properties—using measurements like vibration, acceleration, or quantum observables, without knowledge or measurement of the input. Approaches span classical statistical inference, machine learning, stochastic dynamics, and modern Bayesian frameworks, targeting robust output-only analysis for system identification, structural health monitoring, and physics discovery.

1. Mathematical and Statistical Foundations

Kennen-o approaches are unified by the treatment of unknown or unmeasured inputs as either disturbances to be marginalized out, stochastic processes to be modelled, or nuisance parameters in joint estimation. Foundational output-only identification problems include (a) system thermalization in quantum mechanics; (b) modal decomposition and parameter identification in structural dynamics; (c) learning governing equations in continuously forced nonlinear systems.

For quantum systems, the task is statistical inference: from output-only observable means, determine whether sample states thermalize onto a low-dimensional Gibbs manifold and (if so) estimate the underlying Hamiltonian. This approach leverages factor/PCA-inspired statistical frameworks and maximum entropy projections, producing model evidence metrics via log-likelihood cost functions (Rau, 2011).

In dynamical systems, classic Kennen-o methods exploit second-order statistics or transfer function analysis, such as the frequency-domain decomposition (FDD) of output response. Recent machine learning variants use deep unsupervised architectures for blind-source separation of modal coordinates (Liu et al., 2020).

In physics discovery, output-only frameworks recast unknown-forced dynamics as stochastic differential equations (SDEs), enabling sparse regression and Bayesian model selection for governing law identification (Tripura et al., 2022).

2. Core Output-Only Methodologies

2.1 Classical Statistical and Subspace Methods

• Gibbs Manifold Inference: Construct data covariance matrices from measured outputs; determine fit to theoretical Gibbs manifolds via relative entropy minimization, yielding evidence for thermalization and, in energy-unique cases, maximum-likelihood Hamiltonian estimates (Rau, 2011).
• Time/Frequency-Domain Modal Analysis: Employ output power spectral densities (PSD), cross-power spectral densities (CPSD), and singular value decompositions to identify modal frequencies and shapes, under assumptions of broad-band or white noise excitation (Jahangiri et al., 2020).

2.2 Machine Learning-based Output-Only Approaches

• Self-Coding Deep Neural Networks: Architectures enforce statistical independence and non-Gaussianity among internal codes (interpreted as modal responses) via composite loss functions, simultaneously recovering mode shapes and dynamics from ambient vibration data (Liu et al., 2020).
• Unsupervised Optimization: No external supervisory labels are required—reconstruction and independence losses suffice for modal separation and output-only identification.

2.3 Bayesian Stochastic Physics Discovery

• Stochastic SDE Formulation: Replace unknown deterministic inputs with white noise; model state increments as SDEs via Euler–Maruyama discretization.
• Spike-and-Slab Priors and Sparse Learning: Use Bayesian inference over candidate basis libraries for drift and diffusion functions, employing Gibbs sampling for parameter, inclusion, and noise variable updates; select models based on posterior-inclusion probabilities (Tripura et al., 2022).

2.4 Real-time Joint Input-Parameter-State Estimation

• Unscented Kalman Filter with Two-Stage Input Estimation: Augment the state vector to include dynamic states and parameters; at each time step, estimate unknown input first using state predictions and again using corrected state/parameter estimates post-update. The filter does not include the input as a state but iteratively reconstructs it from the estimated dynamics (Impraimakis et al., 4 Nov 2025).

3. Theoretical Guarantees and Identifiability

Output-only identification is inherently ill-posed if all inputs are unmeasured. Identifiability can be restored if at least one input channel is known (zero or nonzero), as uniqueness is achieved through constraints in the system equations. This is formalized in MDOF identifiability results: in an $n$-DOF linear system with one known input, all parameters and remaining unknown inputs are locally identifiable, provided standard observability assumptions hold (Impraimakis et al., 4 Nov 2025).

In quantum inference, evidence-based likelihood comparisons (model selection) test whether output states genuinely thermalize and identify the dimensionality and generators (constants of motion) of the equilibrium manifold (Rau, 2011).

4. Algorithmic Workflows and Implementation

4.1 Quantum Inference from Outputs

• Collect multiple samples; obtain empirical means for a set of observables.
• Construct data covariance; propose candidate thermodynamic manifolds.
• Project measurements onto the manifold using maximum-entropy principles.
• Evaluate asymptotic log-likelihood function; assess model evidence; estimate conserved quantities (e.g., Hamiltonian via maximum-likelihood optimization).

4.2 Structural Modal Identification

• Acquire raw acceleration or vibration time series; generate synthetic noise as needed to control SNR.
• Choose appropriate identification method: PP (Peak Picking), FDD, or data-driven stochastic subspace (SSI).
• Compute spectral densities and/or block-Hankel matrices, then recover modal dynamics through eigenanalysis or SVD.
• For deep-learning methods, train autoencoder models to enforce whitened, independent, and non-Gaussian hidden codes; extract modal coordinates and shape matrices from model parameters (Liu et al., 2020).

4.3 Bayesian Output-only Physics Inference

• Formulate system as SDE with unknown forcing replaced by noise; discretize with Euler–Maruyama.
• Construct regression targets $Y^f, Y^g$ for drift and diffusion.
• Define candidate basis functions; assign spike-and-slab priors.
• Employ Gibbs sampling for iterative parameter/posterior updates; retain terms with high posterior-inclusion probability, yielding parsimonious models with uncertainty quantification (Tripura et al., 2022).

4.4 Real-time Output-only Kalman Filtering

• Initialize augmented state covariance and state estimates; propagate sigma points.
• In each iteration, predict state/parameter evolution, estimate input from state dynamics, and update with new measurements.
• Apply input estimation steps both before and after measurement assimilation.
• Repeat in real time for joint input, parameter, and state recovery (Impraimakis et al., 4 Nov 2025).

5. Robustness, Practicality, and Limitations

• Noise Robustness: Identifiability of modes and model parameters degrades with diminishing SNR. FDD has higher robustness than SSI or PP for modal analysis, identifying higher modes down to SNR ≈ –6 dB (Jahangiri et al., 2020). Deep learning and sparse Bayesian approaches tolerate moderate noise and partially missing state data.
• Computational Considerations: Gibbs sampling over modest-sized libraries is computationally efficient; UKF-based algorithms are real-time feasible for moderate state dimensionalities.
• Sensor Requirements: For dynamic systems, the recommendation is to measure as many state components (e.g., displacement, velocity, acceleration) as possible, since derivative operations amplify noise and cause drift if states are omitted (Impraimakis et al., 4 Nov 2025).
• Ill-Posedness: Pure output-only identification remains non-unique absent at least one known external constraint/channel (structural, dynamical, or measurement).

6. Case Studies and Empirical Performance

Representative demonstrations include:

System	Methodology	Identifiable Quantity	Accuracy/Robustness
Qubit ensemble	Output-only Gibbs inference	Hamiltonian, thermalization	ML solution, evidence test (Rau, 2011)
Euler–Bernoulli beam	FDD, SSI, Deep DNN	Modes, frequencies, shapes	FDD robust to SNR down to 6 dB (Jahangiri et al., 2020), DNN MAC >0.99 (Liu et al., 2020)
Duffing–Van der Pol, Black-Scholes, Shear Building	SDE+spike/slab Bayesian	Drift/diffusion structure, parameters	Posterior means within 2–3% of true values (Tripura et al., 2022)
Linear/nonlinear MDOF system	IPS-UKF	States, parameters, input	Real-time recovery, identifiability if ≥1 known input (Impraimakis et al., 4 Nov 2025)

In output-only settings, accuracy is critically dependent on measurement noise, completeness of observable states, and the underlying structure of the excitation input.

7. Extensions and Outlook

Contemporary Kennen-o advances aim to:

• Extend Bayesian and deep-learning approaches to partially observed, hybrid, or multi-fidelity data settings.
• Develop adaptive and online variants of deep architectures for time-varying systems.
• Improve identifiability by fusing partial input knowledge, structural constraints, or additional sensor modalities.
• Formalize theoretical limits for uniqueness and statistical efficiency under various noise and excitation conditions.

The output-only paradigm offers a quantitative framework for system identification and monitoring in scenarios where input data are sparse, inaccessible, or prohibitively costly to measure. Its methodologies are under active development across physics, engineering, and data-driven scientific discovery.