--- abstract: | Recently, diffusion probabilistic models have attracted attention in generative time series forecasting due to their remarkable capacity to generate high-fidelity samples. However, the effective utilization of their strong modeling ability in the probabilistic time series forecasting task remains an open question, partially due to the challenge of instability arising from their stochastic nature. To address this challenge, we introduce a novel **M**ulti-**G**ranularity **T**ime **S**eries **D**iffusion (`\mgtsdnospace`{=latex}) model, which achieves state-of-the-art predictive performance by leveraging the inherent granularity levels within the data as given targets at intermediate diffusion steps to guide the learning process of diffusion models. The way to construct the targets is motivated by the observation that the forward process of the diffusion model, which sequentially corrupts the data distribution to a standard normal distribution, intuitively aligns with the process of smoothing fine-grained data into a coarse-grained representation, both of which result in a gradual loss of fine distribution features. In the study, we derive a novel multi-granularity guidance diffusion loss function and propose a concise implementation method to effectively utilize coarse-grained data across various granularity levels. More importantly, our approach does not rely on additional external data, making it versatile and applicable across various domains. Extensive experiments conducted on real-world datasets demonstrate that our `\mgtsd `{=latex}model outperforms existing time series prediction methods. Our code is available at . author: - "**Xinyao Fan**$^{1*}$, **Yueying Wu**$^{2*}$, **Chang Xu**$^{4\\dagger}$, **Yuhao Huang**$^{3}$, **Weiqing Liu**$^{4}$, **Jiang Bian**$^{4}$" bibliography: - iclr2024_conference.bib title: "MG-TSD: Multi-Granularity Time Series Diffusion Models with Guided Learning Process" --- \newcommand{\figleft}{{\em (Left)}} \newcommand{\figcenter}{{\em (Center)}} \newcommand{\figright}{{\em (Right)}} \newcommand{\figtop}{{\em (Top)}} \newcommand{\figbottom}{{\em (Bottom)}} \newcommand{\captiona}{{\em (a)}} \newcommand{\captionb}{{\em (b)}} \newcommand{\captionc}{{\em (c)}} \newcommand{\captiond}{{\em (d)}} \newcommand{\newterm}[1]{{\bf #1}} \def\figref#1{figure~\ref{#1}} \def\appref#1{Appendix~\ref{#1}} \def\Figref#1{Figure~\ref{#1}} \def\twofigref#1#2{figures \ref{#1} and \ref{#2}} \def\quadfigref#1#2#3#4{figures \ref{#1}, \ref{#2}, \ref{#3} and \ref{#4}} \def\secref#1{Section~\ref{#1}} \def\Secref#1{Section~\ref{#1}} \def\twosecrefs#1#2{sections \ref{#1} 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\newcommand{\pencode}{p_{\rm{encoder}}} \newcommand{\pdecode}{p_{\rm{decoder}}} \newcommand{\precons}{p_{\rm{reconstruct}}} \newcommand{\laplace}{\mathrm{Laplace}} \newcommand{\E}{\mathbb{E}} \newcommand{\Ls}{\mathcal{L}} \newcommand{\R}{\mathbb{R}} \newcommand{\emp}{\tilde{p}} \newcommand{\lr}{\alpha} \newcommand{\reg}{\lambda} \newcommand{\rect}{\mathrm{rectifier}} \newcommand{\softmax}{\mathrm{softmax}} \newcommand{\sigmoid}{\sigma} \newcommand{\softplus}{\zeta} \newcommand{\KL}{D_{\mathrm{KL}}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\standarderror}{\mathrm{SE}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\normlzero}{L^0} \newcommand{\normlone}{L^1} \newcommand{\normltwo}{L^2} \newcommand{\normlp}{L^p} \newcommand{\normmax}{L^\infty} \newcommand{\parents}{Pa} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Tr}{Tr} \let\ab\allowbreak \def \blue #1{\textcolor{blue}{#1}} \def\Nn{N_{*}} \def\baralp{\bar{\alpha}} \def\dd {\mathrm{d}} \def\epsb{{\bm{\epsilon}}} \renewcommand\footnotemark{} \def\UrlAlphabet{% \do\a\do\b\do\c\do\d\do\e\do\f\do\g\do\h\do\i\do\j% \do\k\do\l\do\m\do\n\do\o\do\p\do\q\do\r\do\s\do\t% \do\u\do\v\do\w\do\x\do\y\do\z\do\A\do\B\do\C\do\D% \do\E\do\F\do\G\do\H\do\I\do\J\do\K\do\L\do\M\do\N% \do\O\do\P\do\Q\do\R\do\S\do\T\do\U\do\V\do\W\do\X% \do\Y\do\Z} \def\UrlDigits{\do\1\do\2\do\3\do\4\do\5\do\6\do\7\do\8\do\9\do\0} \newcommand{\fix}{\marginpar{FIX}} \newcommand{\new}{\marginpar{NEW}} \newcommand{\crps}{$\text{CRPS}_{\text{sum}}$ } \newcommand{\crpsnospace}{$\text{CRPS}_{\text{sum}}$} \newcommand{\mgtsd}{\textbf{MG-TSD} } \newcommand{\mgtsdnospace}{\textbf{MG-TSD}} \maketitle `\noindent `{=latex}University of British Columbia$^{1}$, Peking University$^{2}$, Nanjing University$^{3}$, Microsoft Research$^{4}$ `\noindent `{=latex}`xinyao.fan@stat.ubc.ca, wuyueying@stu.pku.edu.cn, huangyh@smail.nju.edu.cn,`\ `{chanx, weiqing.liu, jiang.bian}@microsoft.com ` # Introduction Time series prediction is a critical task with applications in various domains such as finance forecasting [@hou2021stock; @chen2018incorporating], energy planning [@koprinska2018convolutional; @wu2021autoformer], climate modeling [@wu2023interpretable; @wu2021autoformer], and biological sciences [@luo2020hitanet; @rajpurkar2022ai]. Considering that time series forecasting problems can be effectively addressed as a conditional generation task, many works leverage generative models for predictive purposes. For instance, @salinas2019high utilizes a low-rank plus diagonal covariance Gaussian copula; @rasul2021autoregressive models the predictive distribution using normalizing flows. Recent advancements in diffusion probabilistic models [@ho2020denoising] have sparked interest in utilizing them into probabilistic time series prediction. For example, @rasul2020multivariate auto-regressively generates data through iterative denoising diffusion models. @tashiro2021csdi uses a conditional score-based diffusion model explicitly trained for probabilistic time series imputation and prediction. These methods relying on diffusion models have exhibited remarkable predictive capabilities. However, there is still considerable scope for improvement. One challenge that diffusion models face in time series forecasting tasks is the instability due to their stochastic nature when compared to deterministic models like RNNs and variants like LSTMs [@hochreiter1997lstm; @laiModelingLongShortTerm2018], GRUs [@ballakur2020Empirical; @yamak2019comparison], and Transformers that rely on self-attention mechanisms [@Vaswani2017attention; @zhouInformerEfficientTransformer2021; @zhou2022fedformer; @wu2021autoformer]. More specifically, the diffusion models yield diverse samples from the conditional distributions, including possible low-fidelity samples from the low-density regions within the data manifold [@sehwag2022generating]. In the context of time series forecasting, where fixed observations exclusively serve as objectives, such variability would result in forecasting instability and inferior prediction performance. \begin{wrapfigure}{r}{0.45\textwidth} \includegraphics[width=0.45\textwidth]{fig_MG-TSD.pdf} \caption{The process of smoothing data from finest-grained to coarsest-grained naturally aligns with the diffusion process.} \label{fig-concept} \end{wrapfigure} To stabilize the output of a diffusion model in time series prediction, one straightforward method is to constrain the intermediate states during the sampling process. Prior research in the realm of diffusion models has introduced the idea of classifier-guidance [@nichol2021glide] and classifier-free guidance [@ho2022classifier], where the predicted posterior mean is shifted with the gradient of either explicit or implicit classifier. However, these methods require labels as the source of guidance while sampling, which are unavailable during out-of-sample inference. We observe that the forward process of the diffusion model, which sequentially corrupts the data distribution to a standard normal distribution, intuitively aligns with the process of smoothing fine-grained data into a coarser-grained representation, both of which result in a gradual loss of finer distribution features. This provides the insights that intrinsic features within data granularities may also serve as a source of guidance. In this paper, we propose a novel **M**ulti-**G**ranularity **T**ime **S**eries **D**iffusion (`\mgtsdnospace`{=latex}) model that leverages multiple granularity levels within data to guide the learning process of diffusion models. The coarse-grained data at different granularity levels are utilized as targets to guide the learning of the denoising process. These targets serve as constraints for the intermediate latent states, ensuring a regularized sampling path that preserves the trends and patterns within the coarse-grained data. They introduce inductive bias which promotes the generation of coarser features during intermediate steps and facilitates the recovery of finer features in subsequent diffusion steps. Consequently, this design reduces variability and results in high-quality predictions. Our key contributions can be summarized as below: 1. We introduce a novel `\mgtsd `{=latex}model with an innovatively designed multi-granularity guidance loss function that efficiently guides the diffusion learning process, resulting in reliable sampling paths and more precise forecasting results. 2. We provide a concise implementation that leverages coarse-grained data instances at various granularity levels. Furthermore, we explore the optimal configuration for different granularity levels and propose a practical rule of thumb. 3. Extensive experiments conducted on real-world datasets demonstrate the superiority of the proposed model, achieving the best performance compared to the state-of-the-art methods. # Background ## Denoising diffusion probabilistic models {#sec-ddpm} Suppose $\vx_{0}\sim q_{\mathcal{X}}(\vx_{0})$ is a multivariate vector from space $\mathcal{X}=\mathbb{R}^{D}$. Denoising diffusion probabilistic models aim to learn a model distribution $p_\theta(\vx_0)$ that approximates the data distribution $q(\vx_0)$. Briefly, they are latent variable models of the form $p_{\theta}(\vx_{0})=\int p_{\theta}(\vx_{0:N})\dd \vx_{1:N}$, where $\vx_n$ for $n = 1,\ldots,N$ is a sequence of latent variables in the same sample space as $\vx_0$. The denoising diffusion models are composed of two processes: the forward process and the reverse process. During the forward process, a small amount of Gaussian noise is added gradually in $N$ steps to samples. It is characterized by the following Markov chain: ${q(\vx_{1:N}|\vx_0) = \prod^N_{n=1} q(\vx_n | \vx_{n-1})}, \ \text{where} \ q (\vx_{n}|\vx_{n-1}):= \mathcal{N} (\sqrt{1-\beta_n}\vx_{n-1}, \beta_n \mI)$. The step sizes are controlled by a variance schedule $\{\beta_{n}\in (0,1)\}_{n=1}^{N}$, where $n$ represents a diffusion step. A nice property of the above process is that one can sample at any arbitrary diffusion step in a closed form, let $\alpha_n := 1 - \beta_n$ and $\baralp_n =\prod^n_{i=1} \alpha_i$. It has been shown that $\vx_{n}=\sqrt{\baralp_n}\vx_{0}+ \sqrt{1-\baralp_{n}}\epsb$. The reverse diffusion process is to recreate the real samples from a Gaussian noise input. It is defined as a Markov chain with learned Gaussian transitions starting with $p(\vx_N)=\mathcal{N}(\vx_N;\textbf{0},\mI)$. The reverse process is characterized as $p_\theta(\vx_{0:N}) := p(\vx_{N})\prod^1_{n=N} p_\theta(\vx_{n-1} {|} \vx_{n}),\ \text{where}\ p_\theta(\vx_{n-1} {|} \vx_{n}) := \mathcal{N}(\vx_{n-1}; \mu_\theta(\vx_{n}, n), \Sigma_\theta(\vx_n, n)\mI)$; $\mu_{\theta}:\R^{D}\times \mathbb{N} \to \R^{D}$ and $\Sigma_{\theta}: \R^{D}\times \mathbb{N} \to \R^{+}$ take the variable $\vx_{n}\in \R^{D}$ and the diffusion step $n\in \mathbb{N}$ as inputs, and share the parameters $\theta$. The parameters in the model are optimized to minimize the negative log-likelihood $\min_{\theta} \E_{\vx_{0}\sim q(\vx_0)}[-\log p_{\theta}(\vx_{0})]$ via a variational bound. According to denoising diffusion probabilistic models (DDPM) in @ho2020denoising, the parameterization of $p_\theta(\vx_{n-1}| \vx_n)$ is chosen as: $$\begin{equation} \label{eq-mu} \mu_\theta(\vx_n, n) = {\frac{1}{\sqrt{\alpha_n}}} \bigg( \vx_n- \frac{1-\alpha_n}{\sqrt{1-\baralp_n}} \epsb_\theta( \sqrt{\baralp_n}\vx_{0}+\sqrt{1-\baralp_{n}}\epsb,n)\bigg), \end{equation}$$ where $\epsb_{\theta}$ is a network which predicts $\epsb\sim \mathcal{N}(\textbf{0},\textbf{I})$ from $\vx_{n}$. We simplify the objective function into $$\begin{equation} \label{eq-diffusion-loss} L_{n}^{\text{simple}}= \E_{n,\epsb_n,\vx_0}\bigg[\|\epsb_{n}-\epsb_{\theta}(\sqrt{\baralp_n}\vx_0+\sqrt{1-\baralp_n}\epsb_n,n)\|^2\bigg]. \end{equation}$$ Once trained, we can iteratively sample from the reverse process $p_{\theta}(\vx_{n-1}|\vx_{n})$ to reconstruct $\vx_0$. ## TimeGrad Model {#sec-timegrad} We treat the time series forecasting task as a conditional generation task and utilize the diffusion models presented in `\secref{sec-ddpm}`{=latex} as the backbone generative model. TimeGrad model is a related work by @rasul2020multivariate which first explored the use of diffusion models for forecasting multivariate time series. Consider a contiguous time series sampled from the complete history training data, indexed from 1 to $T$. This time series is partitioned into a context window of interval $[1,t_0)$ and a prediction interval $[t_0,T]$. TimeGrad utilizes diffusion models from @ho2020denoising to learn the conditional distribution of the future timesteps of the multivariate time series given their past. An RNN is employed to capture the temporal dependencies, and the time series sequence up to timestep $t$ is encoded in the updated hidden state $\rvh_{t}$. Mathematically, TimeGrad models $q_{\mathcal{X}}(\vx_{t_0:T}|\vx_{1:t_0-1})=\prod_{t=t_0}^{T}q_{\mathcal{X}}(\vx_t|\vx_{1:t-1})\approx \prod_{t=t_0}^{T}q_{\mathcal{X}}(\vx_{t}|\rvh_{t-1})$, where $\vx_{t}\in\R^{D}$ denotes the time series at timestep $t$ and $\rvh_{t}=\text{RNN}_{\psi}(\vx_t,\rvh_{t-1})$. Each factor is learned via a shared conditional denoising diffusion model. In contrast to @ho2020denoising, the hidden states $\rvh_{t-1}$ are taken as an additional input in the denoising network $\epsb_{\theta}(\vx_t^{n},\rvh_{t-1},n)$, and the loss function for timestep $t$ and diffusion step $n$ is given by: $$\begin{equation} \E_{\epsb,\vx_{0,t},n}[\|\epsb-\epsb_{\theta}(\sqrt{\baralp_n}\vx_{0,t}+\sqrt{1-\baralp_n}\epsb,n,\rvh_{t-1})\|^2], \end{equation}$$ where the first subscript in $\vx_{0,t}$ represents the index of the diffusion step, while $t$ denotes the timestep within the time series. ## Problem formulation In the time series prediction task, let $\mX^{(1)}$ represent the original observed data. The time series data is denoted as $\mX^{(1)} = [\vx_{1}^1,\ldots,\vx_{t}^1,\ldots, \vx_{T}^1]$, where $t$ represents the timestep $t\in [1,T]$ and $\vx_{t}\in \R^{D}$. Specifically, our task is to model the conditional distribution of future timesteps of the time series $[\vx_{t_0}^1,\ldots,\vx_{T}^1]$ given the fixed window of history context. Mathematically, the problem we consider can be formulated as follows: $$\begin{equation} q_{\mathcal{X}}\bigg(\vx_{t_0:T}^1|\big\{\vx_{1:t_0-1}^1\big\}\bigg)=\prod_{t=t_0}^{T}q_{\mathcal{X}}\bigg(\vx_{t}^1|\big\{\vx_{1:t-1}^1\big\}\bigg). \end{equation}$$ # Method In this section, we provide an overview of the `\mgtsd `{=latex}model architecture in Section `\ref{sec-overview}`{=latex}, followed by a detailed discussion of the novel guided diffusion process module in Section `\ref{sec-guide-module}`{=latex}, including the derivation of the heuristic loss function and its implementation across various granularity levels. ## MG-TSD model architecture {#sec-overview} The proposed methodology consists of three key modules, as depicted in Figure `\ref{fig-archi}`{=latex}. **Multi-granularity Data Generator** is responsible for generating multi-granularity data from observations. In this module, various coarse-grained time series are obtained by smoothing out the fine-grained data using historical sliding windows with different sizes. Suppose $f$ is a pre-defined smoothing (for example, average) function, and $s^g$ is the pre-defined sliding window size for granularity level $g$. Then $\mX^{(g)}=f(\mX^{(1)},s^g)$. The sliding windows are non-overlapping and the obtained coarse-grained data for granularity $g$ are replicated $s^g$ times to align over the timeline $[1,T]$. **Temporal Process Module** is designed to capture the temporal dynamics of the multi-granularity time series data. We utilize RNN architecture on each granularity level $g$ separately to encode the time series sequence up to a specific timestep $t$ and the encoded hidden states are denoted as $\rvh_t^{g}$. The RNN cell type is implemented as GRU in @chung2014empirical. **Guided Diffusion Process Module** is designed to generate stable time series predictions at each timestep $t$. We utilize multi-granularity data as given targets to guide the diffusion learning process. A detailed discussion of the module can be found in `\secref{sec-guide-module}`{=latex}.
Overview of the Multi-Granularity Time Series Diffusion () model, consisting of three key modules: Multi-granularity Data Generator, Temporal Process Module (TPM), and Guided Diffusion Process Module for time series forecasting at a specific granularity level.
## Multi-granularity Guided Diffusion {#sec-guide-module} In this section, we delve into the details of the **Guided Diffusion Process Module**, a key component in our model. `\secref{sec-2-gran-guidance}`{=latex} presents the derivation of a heuristic guidance loss for the two-granularity case. In `\secref{sec-multi-gran-loss}`{=latex}, we generalize the loss to the multi-granularity case and provide a concise implementation to effectively utilize coarse-grained data across various granularity levels. Briefly, the optimization of the heuristic loss function can be simply achieved by training denoising diffusion models on the multi-granularity data with shared denoising network parameters and partially shared variance schedule. ### Coarse-grained guidance {#sec-2-gran-guidance} Without loss of generality, consider two granularities: finest-grained data $\vx_{t}^{g_1}$ ($g_1=1$) from $\mX^{(g_1)}$ and coarse-grained data $\vx_{t}^{g}$ from $\mX^{(g)}$ at a fixed timestep $t$, where $1