---
type: paper
slug: suhyeon-kim-2025-tmf-gnn
title: >-
  TMF-GNN: Temporal matrix factorization-based graph neural network for
  multivariate time series forecasting with missing values
authors:
  - '[[suhyeon-kim]]'
  - '[[taek-ho-lee]]'
  - '[[junghyelee]]'
year: 2025
venue: '[[expert-systems-with-applications]]'
volume: '275'
pages: '127001'
doi: 10.1016/j.eswa.2025.127001
categories:
  - '[[time-series-methods]]'
  - '[[machine-learning]]'
methods:
  - '[[graph-neural-network]]'
  - '[[temporal-matrix-factorization]]'
  - '[[gctc-regularization]]'
  - '[[temporal-convolution-network]]'
  - '[[graph-convolutional-network]]'
  - '[[matrix-factorization]]'
themes:
  - '[[multivariate-time-series]]'
  - '[[missing-values]]'
  - '[[time-series-forecasting]]'
  - '[[mts-imputation]]'
  - '[[adaptive-graph-learning]]'
aliases:
  - TMF-GNN
last_updated: '2026-05-20T02:01:15+09:00'
---
# TMF-GNN: Temporal matrix factorization-based graph neural network for multivariate time series forecasting with missing values

*[[suhyeon-kim]], [[taek-ho-lee]], [[junghyelee]] (2025)* · [[expert-systems-with-applications]] 275:127001 · [DOI ↗](https://doi.org/10.1016/j.eswa.2025.127001)

> **TMF-GNN** — *결측 (missing values) 있는 다변량 시계열 (MTS) 예측* 의 *end-to-end* 방법으로, *imputation + forecasting 의 2-stage* 분리 한계를 해소. 핵심 통찰: 기존 TMF (Yu et al 2016 TRMF, Chen et al 2022 NoTMF) 가 *temporal regularization 만* 사용 → variable-wise inter-correlation 무시. TMF-GNN 은 **(i) Matrix Factorization module** (factor matrix $U \in R^{N \times R}$, $V \in R^{R \times T}$ 의 partial observation reconstruction) + **(ii) GCTC regularization** (Graph Convolution + Temporal Convolution): Adaptive adjacency $A = f_{sc}(\text{ReLU}(L \cdot ZZ^T))$ + top-$k$ neighbor sparsification + GCN 으로 variable interaction + TCN 으로 temporal pattern + concat → fc → forecast. 4 dataset (GFT 3.24% missing, AQ 14.4%, PM2.5 50.5%, Traffic) × 3 baseline (TMF, TRMF, NoTMF) × 5 metric (MAE, RMSE, MAPE, ND, NRMSE) × 3 horizon (1, 3, 6) 평가. TMF-GNN 이 모든 baseline outperform. [[junghyelee]] 의 *실타래 5 (Graph-based methods) + 실타래 4 (Representation Learning)* 의 *MTS forecasting + missing values* 응용. [[suhyeon-kim]] 의 *4-paper series* 의 *graph + sequential ML* 정점.

- **RQ**: (1) MTS 의 *결측값* 을 *separate imputation stage 없이* 어떻게 forecasting 정확도 향상에 활용하는가? (2) 기존 TMF (TRMF, NoTMF) 의 *temporal-only regularization* 한계 (variable interaction 무시) 를 어떻게 해소하는가? (3) Graph structure 와 temporal pattern 을 *end-to-end* 로 동시 학습하는 *deep learning-based* regularization 은?
- **방법론**: **MF module** (Eq 5): $\mathcal{L}_m = \lambda_1 \|\mathcal{P}_{\Omega}(X - UV)\|_F^2$, 관찰된 cell 만 reconstruction. **GCTC regularization** (Eq 8): (i) Adaptive adjacency $A$ 생성 + softmax + top-$k$ neighbor sparsification, (ii) [[graph-convolutional-network]] (Kipf-Welling 2016) 의 $H^{(l+1)} = f_g(A H^{(l)} W^{(l)} + b^{(l)})$, (iii) [[temporal-convolution-network]] (Wan et al 2019) 의 1D FCN + causal convolution + dilated convolution + residual connection, (iv) Sliding window $\delta$ 의 sequence input $V_t$, (v) Row-mean aggregator $h_g$ + concat $h_t^v$ + fc layer → forecast $H_t$, (vi) Graph regularization (smoothness + sparsity). **Total loss** (Eq 9): $\mathcal{L} = \mathcal{L}_m + \mathcal{L}_r$
- **데이터**: 4 dataset (Table 1) — (i) **GFT (Google Flu Trends)** 159 variables × 620 weekly samples (2003-2015), 3.24% missing. (ii) **AQ (Air Quality)** 13 vars × 8,760 hourly (2004-2005, Italy), 14.4% missing. (iii) **PM2.5** 7 vars × 8,760 hourly (Beijing 2013), 50.5% missing. (iv) **Traffic** 962 vars × 17,544 hourly (California Bay Area 2015-2016), no missing. 평가: GFT next 12 weeks, AQ/PM2.5/Traffic next 7 days (24 timesteps × 7 windows). 3 horizon (1, 3, 6), 3 trial avg
- **주요 발견**: (i) **TMF-GNN > TMF (Xiong et al 2010) > TRMF (Yu et al 2016) > NoTMF (Chen et al 2022)** in 모든 5 metric. (ii) **Higher missing ratio (PM2.5 50.5%) 에서 우위 더 큼** — adaptive graph 가 *missing 영향* mitigation. (iii) **Adaptive adjacency** 가 *prior graph structure* 없이도 효과적 — variable inter-correlation 자동 학습. (iv) **TCN > AR/VAR** for temporal regularization — sequential deep learning 우위. (v) Hyperparameter sensitivity (Appendix A): $\lambda_1, \lambda_2, \lambda_3$, hidden layers, $k$, learning rate, epoch
- **시사점**: (a) **End-to-end approach** 의 *2-stage error propagation* 회피 — imputation 정확도가 forecasting 정확도를 보장하지 않는다는 *system identification* 문제 해결. (b) **Graph-based MTS forecasting** 의 *missing values* 확장 — traffic flow 외 *air quality, healthcare, IoT sensor* 의 *high-missing* domain 적용. (c) **Adaptive graph structure 의 자동 학습** — domain knowledge 부재 환경에서 graph prior 불필요. (d) **Deep learning regularization 의 TMF 통합** — naive AR/VAR 의 *naive time-series model* 한계 극복. (e) 환경 모니터링, 의료 시계열, IoT 등 *결측 빈번* 도메인 의 *new baseline*

*(원본 PDF figure 미추출 — 본문 architecture 식 + Algorithm 1 로 구조 surface)*

## 요약

본 paper 는 [[junghyelee]] 의 *3 기 SNU TEMEP 시기* 의 *graph + sequential ML 결합* 작업. [[suhyeon-kim]] (W2V-LSA, LBC, RSAE-BA 의 동일 제 1 저자, Pyeongchang National University Graduate School of Data Science) + [[taek-ho-lee]] (SeoulTech Industrial Engineering + TEMEP) + [[junghyelee]] (corresponding, SNU Institute of Engineering Research + Graduate School of Engineering Practice) 의 공저. 동기는 *MTS missing values 의 pervasive 문제* — sensor 고장, network failure, data transmission 문제, storage 한계가 IoT/health/environmental MTS 의 chronic challenge. 기존 imputation approach 의 한계:

1. **Statistical imputation** (mean/median, AR/MA, KNN, random forest) — *separate stage* 의 inefficiency
2. **Deep learning imputation** (RNN, GAN) — *imputation 정확도 ≠ forecasting 정확도*
3. **2-stage approach 의 systemic 문제** — error propagation, dual hyperparameter tuning, imputation 이 raw distribution 보존 안 함

기존 TMF approach (Xiong et al 2010, Yu et al 2016 TRMF, Chen et al 2022 NoTMF) 가 *imputation 제거* 의 alternative — partial observation 만으로 low-rank $U \cdot V$ 분해 + temporal regularization (AR/VAR). 그러나 두 한계: (a) *variable interaction 무시* — temporal-only, (b) *naive AR/VAR* 의 한계. 본 paper 는 *graph-based variable interaction* + *deep learning sequential regularization* 으로 동시 해소.

**MF module — Reconstruction with missing data.** 학습 가능 $U \in R^{N \times R}$ (variable-wise) + $V \in R^{R \times T}$ (temporal). Reconstruction loss (Eq 5):

$$
\mathcal{L}_m = \lambda_1 \|\mathcal{P}_{\Omega}(X - UV)\|_F^2
$$

여기서 $\mathcal{P}_{\Omega}$ = partial observation mask (관찰된 cell 만), $\|\cdot\|_F$ = Frobenius norm. *SVD/NMF 같은 complete-data MF* 는 missing 불가 — 학습 가능 factor matrix 가 대안.

**GCTC regularization — MTS forecasting.** 4 sub-module.

**(i) Adaptive adjacency $A$** (Eq 6, 7):

$$
A = f_{sc}(\text{ReLU}(L \cdot Z \cdot Z^T)), \quad Z = M W^T
$$

여기서 $M \in R^{N \times K}$ = randomly initialized node embedding, $W \in R^{K \times K}$ = model parameter, $f_{sc}$ = softmax. Top-$k$ neighbor sparsification 으로 noise + computational cost 감소.

**(ii) GCN module**: $A$ 위에 graph convolution — $H_g = \text{GCN}(A, U)$. 각 layer:

$$
H^{(l+1)} = f_g(A H^{(l)} W^{(l)} + b^{(l)})
$$

Initial $H^{(0)} = U$. *Variable-wise factor matrix 가 node feature*.

**(iii) TCN module**: Sliding window $\delta$ 의 sequence $V_t = [v_{t-\delta+1}, \ldots, v_t]^T$. TCN = 1D FCN + causal convolution + dilated convolution + residual connection. Dilated convolution (Eq 3): $F(s) = \sum_{i=0}^{k-1} f(i) \cdot x_{s - d \cdot i}$.

**(iv) Aggregation + forecast**: Row-mean aggregator $h_g$ + concat $h_t^v$ + fc layer $f_h$:

$$
h_t^c = f_h(\text{CONCAT}(h_g, h_t^v))
$$

$H_t$ 가 forecast.

**Graph regularization** (Chen-Wu-Zaki 2020): $\frac{1}{N^2} \sum_{ij} A_{ij} \|u_i - u_j\|^2 + \beta \|A\|_F^2$ — smoothness + sparsity.

**Total loss** (Eq 9):

$$
\mathcal{L} = \lambda_1 \|\mathcal{P}_{\Omega}(X - UV)\|_F^2 + \lambda_2 \sum_{t=\delta+1}^{T} \|x_t - H_t\|_F^2 + \lambda_3 \left( \frac{1}{N^2} \sum_{ij} A_{ij} \|u_i - u_j\|^2 + \beta \|A\|_F^2 \right)
$$

**Experiments** — 4 dataset (GFT, AQ, PM2.5, Traffic) × 3 baseline (TMF, TRMF, NoTMF) × 5 metric × 3 horizon. TMF-GNN 이 모든 dataset + metric + horizon 에서 baseline outperform. 한계: (i) hyperparameter ($\lambda_1, \lambda_2, \lambda_3$, $k$, hidden layer 수, learning rate, epoch) 가 dataset-specific — grid search 필요, (ii) *random missingness* (MCAR/MAR) 가정 — *MNAR* (missing not at random) 미검증, (iii) 4 dataset 한정 — 다른 도메인 generalizability, (iv) computational cost (GCN + TCN + MF 동시) 가 큰 dataset 에서 burden. [[junghyelee]] author page 의 *실타래 5 (Graph-based methods) 의 발전 + 실타래 4 (Representation Learning) 의 MTS 응용* 의 anchor. [[suhyeon-kim]] series 의 *4 번째 작품* — text embedding → multistage data mining → autoencoder embedding → *graph + sequential* 의 progression.

## 핵심 결과

**4 dataset (Table 1)**

| Dataset | # vars | # samples | Sample rate | Missing ratio |
|---|---:|---:|---|---:|
| **GFT** (Google Flu Trends) | 159 | 620 | 1 week | 3.24% |
| **AQ** (Air Quality, Italy) | 13 | 8,760 | 1 hour | 14.4% |
| **PM2.5** (Beijing 2013) | 7 | 8,760 | 1 hour | **50.5%** |
| **Traffic** (California Bay Area) | 962 | 17,544 | 1 hour | 0% |

**5 evaluation metrics** (Eqs 10-14)

| Metric | 식 |
|---|---|
| MAE | $\frac{1}{|\Omega_{test}|}\sum |X - \hat{X}|$ |
| RMSE | $\sqrt{\frac{1}{|\Omega_{test}|}\sum (X - \hat{X})^2}$ |
| MAPE | mean absolute percentage error |
| ND | normalized deviation |
| NRMSE | normalized RMSE |

**Baseline 비교** (TMF, TRMF, NoTMF)

| Method | Variable interaction | Temporal regularization |
|---|---|---|
| TMF (Xiong et al 2010) | ❌ | AR |
| TRMF (Yu et al 2016) | ❌ | regularized AR (Markov) |
| NoTMF (Chen et al 2022) | ❌ | non-temporal MF |
| **TMF-GNN** | ✅ **Adaptive GCN** | ✅ **TCN** |

> **정량 결론.** TMF-GNN 이 4 dataset × 5 metric × 3 horizon 모두에서 baseline outperform. *Adaptive graph learning* + *TCN deep sequential regularization* 의 combined effect. PM2.5 (50.5% missing) 에서 가장 큰 우위 — high missing 환경에서 *graph-based variable interaction* 의 mitigation 효과 명확.

## 방법론 노트

**MF module reconstruction loss** (Eq 5):

$$
\mathcal{L}_m = \lambda_1 \|\mathcal{P}_{\Omega}(X - UV)\|_F^2
$$

여기서 $U \in R^{N \times R}$ = variable-wise factor matrix, $V \in R^{R \times T}$ = temporal factor matrix, $R$ = low-rank dimension, $\mathcal{P}_{\Omega}$ = observation mask (관찰된 cell 만 selection).

**Adaptive adjacency** (Eq 6, 7):

$$
A = f_{sc}(\text{ReLU}(L \cdot Z \cdot Z^T)), \quad Z = M W^T
$$

$$
A_{kj} = \begin{cases} 1 & A_{kj} \in \text{top-}k(A_k, k_n) \\ 0 & \text{otherwise} \end{cases}
$$

End-to-end 학습 + top-$k$ neighbor sparsification.

**GCN layer** (Kipf-Welling 2016):

$$
H^{(l+1)} = f_g(A H^{(l)} W^{(l)} + b^{(l)})
$$

**TCN dilated convolution** (Eq 3):

$$
F(s) = \sum_{i=0}^{k-1} f(i) \cdot x_{s - d \cdot i}
$$

Residual block output (Eq 4): $o = f_g(x + F(x))$.

**Total loss** (Eq 9):

$$
\mathcal{L} = \lambda_1 \|\mathcal{P}_{\Omega}(X - UV)\|_F^2 + \lambda_2 \sum_{t=\delta+1}^{T} \|x_t - H_t\|_F^2 + \lambda_3 \left( \frac{1}{N^2} \sum_{ij} A_{ij} \|u_i - u_j\|^2 + \beta \|A\|_F^2 \right)
$$

식별은 (i) low-rank assumption 의 MTS adequacy, (ii) variable interaction graph 가 *adaptive learning* 으로 식별 가능, (iii) end-to-end joint optimization 의 *single loss surface*, (iv) sparse adjacency (top-$k$) 의 computational tractability. *Limitation*: hyperparameter dataset-specific + MCAR/MAR 가정 + computational cost.

## 연구 계보

본 paper 의 *MTS forecasting with missing* lineage: (i) **Imputation approaches** — statistical (mean/median, Furnkranz-Falcone 2011), RNN-based (Cao et al 2018, Choi et al 2016 doctorAI, Lipton et al 2016, Yoon et al 2017), GAN-based (Luo et al 2018, 2019; Yoon-Jordon-Schaar 2018 GAIN; Du-Côté-Liu 2023 self-attention), spatiotemporal graph (Marisca-Cini-Alippi 2022). (ii) **TMF-based**: Xiong et al (2010) original TMF, Yu et al (2016 NeurIPS) TRMF, Chen et al (2022) NoTMF — temporal-only regularization. (iii) **MTS forecasting deep learning** — RNN variants, TCN (Wan-Mei-Wang-Liu-Yang 2019), Seq2Seq (Sutskever-Vinyals-Le 2014), attention + memory (Fan et al 2019, Yu et al 2020). (iv) **Graph-based MTS** — Bai-Yao-Li-Wang-Wang (2020), Wu-Pan-Long-Jiang-Zhang (2019, 2020 Graph WaveNet), Bian et al (2022). (v) **GCN/GNN**: Kipf-Welling (2016 ICLR), Cao-Zhu-Wang-Zhu-Niu (2022), Chami-Abu-El-Haija-Perozzi-Re-Murphy (2020). (vi) **TCN**: Long-Shelhamer-Darrell (2015 1D FCN), Lea-Flynn-Vidal-Reiter-Hager (2017). (vii) **Graph regularization**: Chen-Wu-Zaki (2020).

TEMEP 내 sibling: [[suhyeon-kim]] *4-paper series* 의 *4 번째 (최종)* — [[suhyeon-kim-2020-w2v-lsa-blockchain]] (text embedding, topic modeling) → [[suhyeon-kim-2021-lbc-data-mining]] (text + manifold + LSTM, port logistics) → [[suhyeon-kim-2022-rsae-biological-age]] (autoencoder embedding, medical AI) → 본 paper (graph + sequential + MF, environmental/health MTS). *방법론적 진화*: word embedding → manifold + LSTM → autoencoder → GNN + TCN + MF. [[junghyelee]] author page 의 *3 기 SNU TEMEP* 의 *기술경영 + ML methodology* 라인의 *최근 정점*. [[taek-ho-lee]] (SeoulTech IE + TEMEP affiliation) 의 *공저* — TEMEP 내 *graph + sequential ML* 협업 라인. [[wonho-sohn]] 의 *항만 무역 + LSTM* 라인 ([[suhyeon-kim-2021-lbc-data-mining]] 의 sibling) 과 함께 [[junghyelee]] *3 기* 의 *MTS forecasting* 라인 anchor.

## See also

- [[graph-neural-network]]
- [[temporal-matrix-factorization]]
- [[gctc-regularization]]
- [[temporal-convolution-network]]
- [[graph-convolutional-network]]
- [[matrix-factorization]]
- [[multivariate-time-series]]
- [[missing-values]]
- [[time-series-forecasting]]
- [[adaptive-graph-learning]]
- [[junghyelee]]
- [[suhyeon-kim]]
- [[taek-ho-lee]]
- [[suhyeon-kim-2020-w2v-lsa-blockchain]]
- [[suhyeon-kim-2021-lbc-data-mining]]
- [[suhyeon-kim-2022-rsae-biological-age]]
- [[expert-systems-with-applications]]