Dynamic Probability Density Decomposition (DPDD)

• DPDD is a nonparametric framework that decomposes probability densities into spectral modes, enabling forecast of stochastic dynamics via Koopman operators.
• It employs extended dynamic mode decomposition (EDMD) on an orthonormal basis to provide closed-form, multi-step density forecasts with optimal convergence rates in Wasserstein metrics.
• The method is validated on synthetic and real data, offering computational efficiency and improved accuracy compared to traditional diffusion forecasts.

Dynamic Probability Density Decomposition (DPDD) is a data-driven, nonparametric framework for representing and forecasting the temporal evolution of probability densities by expanding them in spectral bases adapted to the underlying stochastic dynamics. In the original formulation, the method is based on the stochastic Koopman operator and extended dynamic mode decomposition (EDMD), and it is explicitly introduced as a method for the probability density evolution of stochastic dynamical systems (Zhao et al., 2022). In a later Koopman–Wasserstein formulation, DPDD is described as a modal expansion of time-evolving probability densities driven by an underlying stochastic dynamical system, expressed in the eigenbasis of the Koopman generator and implemented via a finite-dimensional approximation of the Koopman operator (Wang et al., 10 Jul 2025). A closely related precursor, Dynamic Distribution Decomposition (DDD), was introduced as a variation on Dynamic Mode Decomposition that fits continuous-time Markov chains over basis functions and continuously maps between distributions (Taylor-King et al., 2020).

1. Historical emergence and terminological scope

DPDD enters the literature as the name of a spectral density-forecast method for stochastic dynamical systems. The original paper presents a data-driven nonparametric approach for forecasting the probability density evolution of stochastic dynamical systems, based on the stochastic Koopman operator and EDMD, and states: “We call the proposed method as dynamic probability density decomposition (DPDD)” (Zhao et al., 2022). In that setting, DPDD denotes a decomposition of the probability density function into modes associated with the infinitesimal generator and the stochastic Koopman semigroup.

A later line of work extends DPDD to functional time series forecasting of distributions. There, DPDD is framed as a modal expansion of time-evolving probability densities driven by an underlying stochastic dynamical system, with density-ratio dynamics represented in the eigenbasis of the Koopman generator and forecast errors evaluated in $2$-Wasserstein distance (Wang et al., 10 Jul 2025). This extension places DPDD within an overview of functional data analysis on non-Euclidean spaces, Koopman operator theory, and optimal transport.

DDD provides an important precursor and near-neighbor concept. It was introduced as a variation on Dynamic Mode Decomposition that uses basis functions over a continuous state space, allows for the fitting of continuous-time Markov chains over these basis functions, and continuously maps between distributions (Taylor-King et al., 2020). A plausible implication is that DPDD and DDD are best understood as members of a broader operator-theoretic family of dynamic density representations rather than as entirely separate methodological species.

2. Operator-theoretic formulation

The foundational DPDD setting is an Itô diffusion

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$

with density $p_t$ and stationary density $p_s$. In the Koopman–Wasserstein formulation, the key coordinate is the density ratio

$q_t := \frac{p_t}{p_s},$

and DPDD expands $q_t$ in an orthonormal eigenbasis $\{\varphi_i\}_{i\ge 0}$ of the generator $\mathcal{L}$ in the weighted Hilbert space $L^2(p_s)$: $$q_t(x) = 1 + \sum_{i=1}^{\infty} c_i(0)\, e^{\lambda_i t}\, \varphi_i(x), \qquad c_i(0)=\langle q_0,\varphi_i\rangle_{p_s}.$$ This decomposition separates a stationary baseline from time-evolving modes, each governed by an exponential factor $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$0 (Wang et al., 10 Jul 2025).

The original DPDD formulation uses the stochastic Koopman operator

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$1

and its generator

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$2

The Fokker–Planck equation evolves the density under $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$3, and a significant connection between $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$4 and $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$5 is used to construct an orthonormal basis in a weighted Hilbert space adapted to the invariant measure (Zhao et al., 2022).

In the original spectral density expansion, the probability density is written as

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$6

with coefficient dynamics

$dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$7

The Koopman semigroup relation $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$8 makes the temporal evolution diagonal in the modal basis. This is the central mathematical economy of DPDD: the PDE-level density evolution is converted into linear evolution of coefficients in a fixed basis (Zhao et al., 2022).

3. Estimation by EDMD and closed-form density forecasting

In the original construction, EDMD is applied to training data sampled from the stationary distribution of the underlying stochastic dynamical system. Given a dictionary $dX_t = b(X_t)\,dt + \sigma(X_t)\,dW_t,$9, the finite-dimensional Koopman approximation is built from Gram and cross-covariance matrices and yields eigenpairs $p_t$0. The approximate Koopman eigenfunctions are

$p_t$1

and the generator eigenvalues are obtained by

$p_t$2

The initial density is projected onto this basis, and the modal coefficients are propagated exponentially in time before the density is reconstructed (Zhao et al., 2022).

The Koopman–Wasserstein extension introduces an importance-weighted EDMD variant. The stationary density is estimated by kernel density estimation,

$p_t$3

and normalized importance weights are defined by

$p_t$4

The weighted EDMD matrices are then

$p_t$5

with Koopman estimate

$p_t$6

After projecting the last observed density onto the estimated Koopman basis, the coefficients are advanced by

$p_t$7

and the density forecast is reconstructed as

$p_t$8

Because the evolution is exponential in $p_t$9, the method provides closed-form multi-step forecasts for any horizon $p_s$0 (Wang et al., 10 Jul 2025).

For locally stationary time series, a sliding-window variant recalculates local stationary densities, local weighted EDMD matrices, local eigenpairs, and local coefficients on moving windows. This produces SW-DPDD, a localized version intended for slowly time-varying dynamics (Wang et al., 10 Jul 2025).

4. Theoretical guarantees and empirical behavior

The original DPDD paper states that, in the limit of the large number of snapshots and observables, the data-driven probability density approximation converges to the Galerkin projection of the semigroup solution of the Fokker–Planck equation on a basis adapted to an invariant measure. It further reports that the proposed method shares the similar idea to diffusion forecast, but renders more accurate probability density than the diffusion forecast does (Zhao et al., 2022).

The Koopman–Wasserstein extension strengthens the statistical theory. Its weighted EDMD estimator satisfies spectral convergence: under the stated assumptions, the leading Koopman eigenvalues and eigenvectors converge at rate $p_s$1, and the paper further states that $p_s$2 and $p_s$3 in $p_s$4. For fixed forecast horizon $p_s$5, the finite-sample $p_s$6 prediction risk satisfies

$p_s$7

which the paper summarizes as optimal convergence rates under $p_s$8 loss with closed-form density evolution (Wang et al., 10 Jul 2025).

The reported simulation results compare DPDD with Wasserstein Autoregression (WAR) using empirical mean-squared $p_s$9-Wasserstein error:

Model	DPDD $q_t := \frac{p_t}{p_s},$0	WAR $q_t := \frac{p_t}{p_s},$1
AR(1)	0.009	0.019
AR(2)	0.014	0.033
OU	0.004	0.015
2D OU	0.004	0.012
Mixing AR(1)+OU	0.015	0.022

On U.S. housing price distributions, the reported average $q_t := \frac{p_t}{p_s},$2 MSE is $q_t := \frac{p_t}{p_s},$3 for DPDD and $q_t := \frac{p_t}{p_s},$4 for WAR. In a locally stationary experiment, the reported values are $q_t := \frac{p_t}{p_s},$5 for global DPDD, $q_t := \frac{p_t}{p_s},$6 for WAR, and $q_t := \frac{p_t}{p_s},$7 for SW-DPDD (Wang et al., 10 Jul 2025).

The original DPDD paper also reports strong performance relative to diffusion forecast in benchmark stochastic systems. For a $q_t := \frac{p_t}{p_s},$8D quadratic turbulence example, the approximate computational cost is reported as $q_t := \frac{p_t}{p_s},$9 s for DPDD with $q_t$0 basis functions, versus approximately $q_t$1 s for diffusion forecast with $q_t$2 basis functions, while both methods approximate the reference distribution closely (Zhao et al., 2022).

5. Relation to DDD, Wasserstein geometry, and other dynamic density methods

DDD provides a finite-basis representation of density evolution by projecting the Perron–Frobenius operator onto basis functions. With nonnegative basis functions $q_t$3, a density is approximated by

$q_t$4

and the coefficient vector obeys

$q_t$5

where $q_t$6 is constrained to be a continuous-time Markov chain generator. Sparse DDD restricts the method to compact basis functions, reformulates the estimation problem in terms of sparse matrices, and integrates both trajectory time series and snapshot time series data (Taylor-King et al., 2020). In conceptual terms, DDD and DPDD share the idea that evolving distributions can be represented by linear dynamics on coefficients over a chosen basis.

The Koopman–Wasserstein version of DPDD departs from tangent-space methods such as Wasserstein Autoregression by not linearizing around a Fréchet mean through Wasserstein logarithm and exponential maps. Instead, it uses density ratios $q_t$7 in $q_t$8 and evaluates forecast quality in $q_t$9. The paper states that density ratio dynamics “define a coordinate chart on the Wasserstein manifold, embedding distributional evolution within the intrinsic geometry of optimal transport” (Wang et al., 10 Jul 2025).

A distinct but related direction appears in Parametric Density Path Optimization (PDPO). That work does not use the term DPDD, but it represents a target probability path as the pushforward of a reference density through a parametric map, reducing an infinite-dimensional optimization over densities to a finite-dimensional one over parameters of the map, and uses cubic spline interpolation in parameter space (Hernandez et al., 24 May 2025). This suggests a different dynamic-density paradigm centered on transport maps, action minimization, and path optimization rather than spectral Koopman evolution.

Another non-equivalent use of dynamic probability decomposition appears in RNA designability. A recent RNA paper introduces a theory of ensemble approximation and a probability decomposition framework for bounding the folding probabilities of RNA structures in an explainable way, together with a linear-time dynamic programming algorithm that searches over exponentially many decompositions and identifies the optimal one yielding the tightest probabilistic bound for a given structure (Zhou et al., 14 Feb 2026). A plausible implication is that the phrase “dynamic decomposition” now spans both continuous density evolution and discrete ensemble probability analysis, and therefore requires contextual specification.

6. Applications, interpretive value, and limitations

DPDD has been applied to benchmark stochastic systems, moment forecasting, and real data. The original paper presents examples including a one-dimensional double-well-type SDE, a one-dimensional Ornstein–Uhlenbeck process, a two-dimensional quadratic turbulence system, a noisy Lorenz-63 system, and sea-surface temperature forecasting using NOAA ERSST v5 data from 1971–2020 for training and 2020–2022 for testing (Zhao et al., 2022). The Koopman–Wasserstein extension emphasizes applications in behavioral science, public health, finance, and neuroimaging, and uses U.S. housing price distributions over $\{\varphi_i\}_{i\ge 0}$0 MSAs and $\{\varphi_i\}_{i\ge 0}$1 months as a real-data case study (Wang et al., 10 Jul 2025). DDD is presented as particularly relevant for biomedical data because it can integrate both trajectory time series and snapshot time series, a setting common when studies observe populations at fixed time points while also recording repeated follow-ups for individuals (Taylor-King et al., 2020).

Interpretively, the appeal of DPDD lies in the separation of spatial and temporal structure. The modes $\{\varphi_i\}_{i\ge 0}$2 or $\{\varphi_i\}_{i\ge 0}$3 encode recurrent patterns of density variation, while the coefficients evolve by explicit semigroup factors. In the Koopman–Wasserstein formulation, this yields closed-form multi-step forecasts; in DDD, it yields continuous-time Markov-chain dynamics over basis-function indices; and in both settings it supports continuous-time interpolation between observed distributions (Wang et al., 10 Jul 2025, Taylor-King et al., 2020).

The limitations are likewise structural. The original DPDD formulation assumes an ergodic, time-homogeneous, reversible stochastic dynamical system with a stationary density and a discrete generator spectrum in the weighted space; it also relies on snapshot pairs and on a dictionary choice that captures the dominant eigenspace (Zhao et al., 2022). The Koopman–Wasserstein theory requires assumptions B1–B5, simple leading Koopman eigenvalues, and compact-support conditions in the finite-horizon $\{\varphi_i\}_{i\ge 0}$4 analysis (Wang et al., 10 Jul 2025). In the original DPDD construction, the expansion is in an $\{\varphi_i\}_{i\ge 0}$5 sense and the paper does not do an explicit positivity-enforcement step; it works in the Hilbert-space sense and validates approximation quality empirically (Zhao et al., 2022). For DDD, the sparse formulation reduces the parameter count but depends on compact, local basis functions and on constrained nonlinear optimization over generator parameters (Taylor-King et al., 2020).

Taken together, these developments place DPDD at the intersection of stochastic analysis, semigroup methods, spectral approximation, and distributional forecasting. Its core identity remains stable across variants: a probability density is decomposed into dynamically meaningful modes, and the time evolution of those modes is made explicit through operator-theoretic structure.