Conditionally Identifiable Latent Representation for Multivariate Time Series with Structural Dynamics

Minkey Chang; Jae-Young Kim

cs.LGMar 24, 2026

Conditionally Identifiable Latent Representation for Multivariate Time Series with Structural Dynamics

Authors: Minkey Chang, Jae-Young Kim

Abstract

We propose the Identifiable Variational Dynamic Factor Model (iVDFM) for multivariate time series. The model combines variational inference with partial identifiability guarantees up to a known ambiguity class. Innovations follow a conditional exponential-family prior over observed auxiliary context and regime embeddings. Linear diagonal dynamics map innovations to factors and preserve that class, so factors are partially identifiable up to permutation and component-wise affine maps. We train by maximizing the ELBO and estimate uncertainty in both latent trajectories and forecasts. Under explicit assumptions, we prove partial identifiability up to the stated class and provide the full proof in the appendix. We evaluate factor recovery on synthetic DGPs, intervention behavior on synthetic SCMs, and forecasting on real benchmarks with CRPS and MSE.

Figures & tables

Figure 1: iVDFM in three steps. (a) Extraction. Observations $\bm{y}_{1:T}$ and context $(\mathbf{u}_{t},\mathbf{e}_{t})$ are mapped by the encoder to a variational distribution $q_{\phi}(\bm{\eta}_{t}\mid\cdot)$ over innovations, while the prior $p_{\theta}(\bm{\eta}_{t}\mid\mathbf{u}_{t},\mathbf{e}_{t})$ conditions on the same context. (b) Propagation. Innovations $\bm{\eta}_{t}$ drive factors $\bm{f}_{t}$ through diagonal dynamics $\bar{A}_{t}$ , $\bar{B}_{t}$ . (c) Forecasting and generation. Factors are decoded by $g$ to reconstructed or forecast observations $\widehat{\bm{y}}_{t}$ .

Model	MCC $\uparrow$	Subspace $\downarrow$	Smoothness $\downarrow$	Trace $R^{2}$ $\uparrow$
iVDFM	0.6548 $\pm$ 0.0549	0.4345 $\pm$ 0.1022	1.7392 $\pm$ 0.1118	0.8202 $\pm$ 0.0603
DDFM	0.5202 $\pm$ 0.0360	0.8441 $\pm$ 0.0564	2.0060 $\pm$ 0.1068	0.4998 $\pm$ 0.0435
iVAE	0.6146 $\pm$ 0.0560	0.5149 $\pm$ 0.1264	1.5864 $\pm$ 0.0710	0.7664 $\pm$ 0.0871
VAE	0.6065 $\pm$ 0.0587	0.5418 $\pm$ 0.1286	1.5844 $\pm$ 0.0741	0.7526 $\pm$ 0.0838
DFM	0.4681 $\pm$ 0.0310	0.9565 $\pm$ 0.0414	0.3149 $\pm$ 0.0566	0.3783 $\pm$ 0.0324

Table 1: Factor recovery on the dynamic DGP (

T{=}200

N{=}20

r{=}5

). Metrics are MCC, subspace distance, smoothness, and Trace

R^{2}

after MCC-based matching. Higher values are better for MCC and Trace

R^{2}

, while lower values are better for subspace distance and smoothness. Reported values are mean

\pm

std over simulations.

Model	MCC $\uparrow$	Subspace $\downarrow$	Smoothness $\downarrow$	Trace $R^{2}$ $\uparrow$
iVDFM	0.5683 $\pm$ 0.0661	0.6949 $\pm$ 0.0544	1.7930 $\pm$ 0.1281	0.6178 $\pm$ 0.0460
DDFM	0.5989 $\pm$ 0.0673	0.6147 $\pm$ 0.0540	2.3296 $\pm$ 0.0911	0.6745 $\pm$ 0.0502
iVAE	0.5678 $\pm$ 0.0751	0.5055 $\pm$ 0.0953	2.2504 $\pm$ 0.1186	0.7766 $\pm$ 0.0589
VAE	0.6405 $\pm$ 0.0945	0.4520 $\pm$ 0.0675	2.3255 $\pm$ 0.1438	0.8078 $\pm$ 0.0472
DFM	0.3374 $\pm$ 0.0397	1.1192 $\pm$ 0.0448	0.0994 $\pm$ 0.0173	0.2476 $\pm$ 0.0446

Table 2: Factor recovery, static DGP (

T{=}200

N{=}20

r{=}5

). Metrics are computed after MCC-based matching; we report MCC, subspace distance, smoothness, and Trace

R^{2}

(mean

\pm

std over simulations).

SCM	IRF-MSE $\downarrow$	IRF-MAE $\downarrow$	Sign Acc $\uparrow$	IRF Corr $\uparrow$
Base (linear)	$0.89\pm 1.01$	$0.45\pm 0.23$	$0.58\pm 0.25$	$0.21\pm 0.75$
Regime	$1.28\pm 0.92$	$0.77\pm 0.27$	$0.60\pm 0.27$	$0.13\pm 0.62$
Chain	$2.18\pm 2.40$	$0.97\pm 0.40$	$0.58\pm 0.33$	$0.20\pm 0.62$

Table 3: Causal intervention on synthetic SCMs (iVDFM). IRF-MSE, IRF-MAE: mean squared / absolute error between ground-truth and model-implied impulse response; Sign Acc: fraction of (time step, variable) pairs with correct sign; IRF Corr: Pearson correlation with ground truth. Mean

\pm

std over five simulations.

Dataset	iVDFM		iTrans.		TimeMix		TimeXer		DDFM
	CRPS	MSE	CRPS	MSE	CRPS	MSE	CRPS	MSE	CRPS	MSE
ETTh1	0.282	1.258	0.693	1.296	0.276	0.801	0.542	1.189	0.334	1.342
ETTh2	0.268	1.016	0.501	1.018	0.230	0.401	0.452	0.990	0.278	1.013
ETTm1	0.315	1.221	0.706	1.288	0.229	0.547	0.598	1.250	0.356	1.277
ETTm2	0.340	1.098	0.644	1.330	0.232	0.434	0.531	1.395	0.460	2.308
Weather	0.271	1.184	0.316	0.703	0.163	0.341	0.281	0.707	0.164	0.439

Table 4: CRPS and MSE (standardized), averaged over horizons. Rows: dataset; columns: per-model CRPS and MSE (std.).

Figure 2: Recovered iVDFM factors for the weekly illness application ( $r{=}2$ ), with standardized outpatient visits overlay.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Model	MCC $\uparrow$	Subspace $\downarrow$	Smoothness $\downarrow$	Trace $R^{2}$ $\uparrow$
VAE	$0.7160\pm 0.0609$	$0.2665\pm 0.1176$	$2.2715\pm 0.0286$	$0.8681\pm 0.0638$
iVAE	$0.7265\pm 0.0828$	$0.2305\pm 0.0915$	$2.2710\pm 0.0635$	$0.8869\pm 0.0374$

Appendix

Table 5: iVAE static diagnostic (static synthetic DGP, iVAE-aligned). Mean

\pm

std over 5 simulations.

$T$	MCC $\uparrow$	Subspace $\downarrow$	Smoothness $\downarrow$	Trace $R^{2}$ $\uparrow$
100	$0.608\pm 0.088$	$0.597\pm 0.112$	$2.050\pm 0.037$	$0.731\pm 0.074$
200	$0.666\pm 0.036$	$0.377\pm 0.065$	$1.723\pm 0.099$	$0.857\pm 0.043$
500	$0.660\pm 0.039$	$0.307\pm 0.021$	$1.665\pm 0.040$	$0.888\pm 0.017$
1000	$0.703\pm 0.092$	$0.337\pm 0.044$	$1.661\pm 0.025$	$0.857\pm 0.041$

Appendix

Table 6: Dynamic factor recovery finite-sample sweep for iVDFM (Laplace innovations), training window equal to

T

. Mean

\pm

std over 5 simulations.

Setting	MCC $\uparrow$	Subspace $\downarrow$	Smoothness $\downarrow$	Trace $R^{2}$ $\uparrow$
DGP Laplace; iVDFM Laplace	$0.6696\pm 0.0696$	$0.4455\pm 0.0842$	$1.6905\pm 0.0810$	$0.8103\pm 0.0471$
DGP Gaussian; iVDFM Laplace	$0.6879\pm 0.0769$	$0.3315\pm 0.0387$	$1.7547\pm 0.0495$	$0.8742\pm 0.0279$
DGP Laplace; iVDFM Gaussian	$0.6805\pm 0.0779$	$0.3531\pm 0.0489$	$1.7221\pm 0.0767$	$0.8720\pm 0.0319$

Appendix

Table 7: Innovation-distribution diagnostics for dynamic factor recovery (iVDFM). Mean

\pm

std over 5 simulations.

SCM comparison ( $T=200$ , $r=3$ )
Setting	IRF-MSE $\downarrow$	IRF-MAE $\downarrow$	Sign Acc $\uparrow$	IRF Corr $\uparrow$
Base (linear)	$0.89\pm 1.01$	$0.45\pm 0.23$	$0.58\pm 0.25$	$0.21\pm 0.75$
Regime	$1.28\pm 0.92$	$0.77\pm 0.27$	$0.60\pm 0.27$	$0.13\pm 0.62$
Chain	$2.18\pm 2.40$	$0.97\pm 0.40$	$0.58\pm 0.33$	$0.20\pm 0.62$
Varying $T$ (base SCM, $r=3$ )
$T=100$	$0.75\pm 0.54$	$0.44\pm 0.16$	$0.02\pm 0.01$	$-0.08\pm 0.52$

Appendix

Table 8: iVDFM causal intervention stress tests: varying

T

and misspecified

r

(base, regime, chain SCMs). IRF-MSE, IRF-MAE: mean squared / absolute error; Sign Acc: fraction of (time step, variable) pairs with correct sign; IRF Corr: Pearson correlation with ground truth. Mean

\pm

std over five simulations.

$K$	sMSE $\downarrow$	CRPS $\downarrow$	MAE $\downarrow$
1	$1.061\pm 0.023$	$0.265\pm 0.005$	$0.739\pm 0.015$
3	$\mathbf{1.032\pm 0.071}$	$\mathbf{0.256\pm 0.013}$	$0.745\pm 0.032$
5	$1.061\pm 0.124$	$0.267\pm 0.028$	$0.759\pm 0.070$
7	$1.113\pm 0.083$	$0.273\pm 0.016$	$0.789\pm 0.045$
9	$1.114\pm 0.042$	$0.274\pm 0.010$	$0.791\pm 0.027$

Appendix

Table 10: iVDFM

K

-ablation on ETTh1 at horizon 96. Mean

\pm

std over 3 seeds (2026, 1337, 42). Other hyperparameters fixed per config/experiment/forecasting.yaml .

chain_strength	IRF-MSE	IRF-MAE	Sign Acc	IRF-Corr
0.0	$0.89\pm 0.91$	$0.45\pm 0.20$	$0.58\pm 0.23$	$0.21\pm 0.67$
0.1	$1.31\pm 1.39$	$0.66\pm 0.28$	$0.70\pm 0.29$	$0.27\pm 0.59$
0.2	$2.18\pm 2.15$	$0.97\pm 0.36$	$0.58\pm 0.30$	$0.20\pm 0.56$
0.3	$4.02\pm 2.94$	$1.44\pm 0.40$	$0.61\pm 0.30$	$0.15\pm 0.49$
0.5	$15.26\pm 7.15$	$2.99\pm 0.73$	$0.56\pm 0.31$	$0.15\pm 0.34$

Appendix

Table 11: Chain-SCM sweep. Each row is mean

\pm

std over 5 simulations; IRF-MSE

\downarrow

, IRF-MAE

\downarrow

, Sign Acc

\uparrow

, IRF-Corr

\uparrow

. Row chain_strength=0.2 is the main-text chain setting.

SCM dim	align corr	ES (model vs true)	ES (null vs do, true)	effect cosine
0	$0.66$	$4.80$	$3.82$	$0.32$
1	$0.52$	$4.35$	$3.44$	$0.21$
2	$0.01$	$5.01$	$3.37$	$-0.04$

Appendix

Table 12: Counterfactual trajectory comparison under the shared-history protocol. Columns: IRF-alignment correlation between model and true IRF for the selected coordinate; mean energy score of the model vs. true counterfactual ensemble (lower is better; the last column gives the energy score between the true no-intervention and true intervened ensembles as a reference scale); cosine between model and true expected causal effects

\mathbb{E}[Y^{do}]-\mathbb{E}[Y^{\mathrm{null}}]

averaged over horizon.

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