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+ +# Contents + +0. Abstract +1. Introduction +2. Related Works +3. CycleNet + 1. Residual Cycle Foreacsting + 2. Backbone + +4. Experiments +5. Discussion + +
+ +# 0. Abstract + +Stable periodic patterns + +$$\rightarrow$$ Foundation for conducting **long-horizon** forecasts + +
+ +### Residual Cycle Forecasting (RCF) + +- Learnable recurrent cycles + - To model the inherent periodic patterns +- Predictions on the residual components of the modeled cycles + +
+ +RCF + Linear (MLP) + +- Simple yet powerful method + +=> ***CycleNet*** + +
+ +https://github.com/ACAT-SCUT/CycleNet + +
+ +# 1. Introduction + +Long-horizon prediction + +- Understanding the ***inherent periodicity*** + +- Cannot rely solely on ***recent*** temporal information + +- Long-term dependencies + + = Underlying ***stable*** periodicity within the data + + = Practical foundation for conducting long-term predictions + +
+ +Transformer-based + +- Informer [59], Autoformer [51], and PatchTST [40] +- Transformer’s ability for long-distance modeling to address LTSF + +
+ +ModernTCN [38] + +- Large convolutional kernels to enhance TCNs’ ability to capture long-range dependencies + +
+ +SegRNN [31] + +- Segment-wise iterations to improve RNN methods’ handling of long sequences. + +
+ +![figure2](/assets/img/ts2/img179.png) + +
+ +Explicit modeling of periodic patterns in the data + +$$\rightarrow$$ to enhance the model's performance on LTSF tasks! + +
+ +### Residual Cycle Forecasting (RCF) + +- Step 1) **Learnable recurrent cycles** to explicitly model the inherent periodic patterns within TS data +- Step 2) Followed by predicting the **residual components** of the modeled cycles. + +
+ +CycleNet = RCF + Linear/MLP + +
+ +# 2. Related Works + +### RCF technique + +- Type of Seasonal-Trend Decomposition (STD) method. + +- Key difference from existing techniques : + - ***Explicit modeling*** of global periodic patterns within independent sequences using learnable recurrent cycles. + +- Simple, computationally efficient, and yields significant improvements in prediction accuracy + +
+ +Notation + +- Data) TS $$X$$ with $$D$$ channels + +- Objective) $$f: x_{t-L+1: t} \in \mathbb{R}^{L \times D} \rightarrow \bar{x}_{t+1: t+H} \in \mathbb{R}^{H \times D}$$ + - Predict future horizons $$H$$ steps ahead + - Based on past $$L$$ observations + +
+ +![figure2](/assets/img/ts2/img180.png) + +![figure2](/assets/img/ts2/img181.png) + +
+ +# 3. CycleNet + +## (1) Residual Cycle Forecasting + +Two steps + +- Step 1) Modeling the periodic patterns of sequences + - Via learnable recurrent cycles within independent channels, +- Step 2) Predicting the residual components + +
+ +### Step 1) Periodic patterns modeling + +( Notation: $$D$$ channels & Priori cycle length $$W$$ ) + +Generate learnable ***recurrent cycles*** $$Q \in \mathbb{R}^{W \times D}$$ + +- All initialized to zeros +- Globally shared within channels + +
+ +By performing cyclic replications, + +$$\rightarrow$$ obtain ***cyclic components*** $$C$$ of the sequence $$X$$ of the same length. + +
+ +Details of Cycle length $$W$$ + +- (1) Depends on the a priori characteristics of the dataset +- (2) Set to the maximum stable cycle within the dataset. +- (3) Easily availble + - Considering that scenes requiring long-term predictions usually exhibit prominent, explicit cycles (e.g., electrical consumption and traffic data exhibit clear daily and weekly cycles), determining the specific cycle length is available and straightforward. + - Cycles can be further examined through autocorrelation functions (ACF) + +
+ +### Step 2) Residual Forecasting + +Predictions made on the residual components of the modeled cycles, + +
+ +Procedure + +- Step 2-1) Remove the cyclic components $$c_{t-L+1: t}$$ + - From the original input $$x_{t-L+1: t}$$ + - Obtain residual components $$x_{t-L+1: t}^{\prime}$$. +- Step 2-2) Predict residual components $$\bar{x}_{t+1: t+H}^{\prime}$$ + - Using $$x_{t-L+1: t}^{\prime}$$ +- Step 2-3) Add back the predicted residual components $$\bar{x}_{t+1: t+H}^{\prime}$$ + - To the cyclic components $$c_{t+1: t+H}$$ + - Obtain $$\bar{x}_{t+1: t+H}$$. + +
+ +Cyclic components $$C$$ are **virtual sequences** + +- derived from the cyclic replications of $$Q$$, + +$$\rightarrow$$ Cannot directly obtain the aforementioned sub-sequences $$c_{t-L+1: t}$$ and $$c_{t+1: t+H}$$. + +$$\therefore$$ Appropriate ***alignments and repetitions*** of the recurrent cycles $$Q$$ are needed! + +
+ +How? + +![figure2](/assets/img/ts2/img182.png) + +$$\begin{aligned} +& c_{t-L+1: t}=[\underbrace{Q^{(t)}, \cdots, Q^{(t)}}_{\lfloor L / W\rfloor}, Q_{0: L \bmod W}^{(t)}] \\ +& c_{t+1: t+H}=[\underbrace{Q^{(t+L)}, \cdots, Q^{(t+L)}}_{\lfloor H / W\rfloor}, Q_{0: H \bmod W}^{(t+L)} +\end{aligned}$$. + +
+ +## (2) Backbone + +Original prediction task is transformed into... + +***Cyclic residual component modeling*** + +$$\rightarrow$$ Any TS model can be employed! + +
+ +Opt for the most basic backbone = Linear & MLP + +- CycleNet/Linear +- CycleNet/MLP + +
+ +Others + +- CI strategy +- RevIN +- Loss: MSE + +
+ +# 4. Experiments + +## (1) Settings + +![figure2](/assets/img/ts2/img183.png) + +
+ +## (2) LTSF + +![figure2](/assets/img/ts2/img184.png) + +
+ +## (3) Analysis + +### Efficiency analyiss + +- Plug-and-play module +- Requires minimal overhead + - Needing only additional W × D learnable parameters + +![figure2](/assets/img/ts2/img185.png) + +
+ +### Ablation Study + +![figure2](/assets/img/ts2/img186.png) + +
+ +### Comparison of different STD techniques + +![figure2](/assets/img/ts2/img187.png) + +
+ +### Vizualization + +![figure2](/assets/img/ts2/img188.png) + +
+ +# 5. Discussion + +Potential Limitation + +1. **Unstable cycle length** + - Unsuitable for datasets where the cycle length (or frequency) varies over time, + - ex) Electrocardiogram (ECG) data +2. **Varying cycle lengths across channels** + - When different channels within a dataset exhibit cycles of varying lengths ( due to CI ) + - Potential solution: + - pre-process the dataset by splitting it based on cycle lengths + - independently model each channel as a separate dataset. +3. Impact of outliers +4. Long-range cycle modeling: T diff --git a/_posts/2024-10-14-(paper)GasNorm.md b/_posts/2024-10-14-(paper)GasNorm.md new file mode 100644 index 000000000000..14300438a089 --- /dev/null +++ b/_posts/2024-10-14-(paper)GasNorm.md @@ -0,0 +1,361 @@ +--- +title: GAS-Norm; Score-Driven Adaptive Normalization forNon-Stationary Time Series Forecasting in Deep Learning +categories: [TS] +tags: [] +excerpt: arxiv 2024 +--- + + + +# GAS-Norm: Score-Driven Adaptive Normalization for Non-Stationary Time Series Forecasting in Deep Learning + +
+ +# Contents + +0. Abstract +1. Related Works +2. Proposed Method + 1. Why Adaptive Normalization for a DNN? + 2. Parameter Filtering for Non-stationary Time Series + 3. GAS-Norm + +3. Experiments + +
+ +# 0. Abstract + +Suboptimal performance of DL in TS = By ***Non-stationarity*** + +This paper: + +- (1) Show how **DNN** fail in simple **non-stationary settings** +- (2) **GAS-Norm** + - For **Adaptive TS normalization** + +
+ +### Gas-Norm + +Based on the combination of + +- **Generalized Autoregressive Score (GAS)** + - Encompasses a score-driven family of models + - Estimate the mean and variance at each new observation + - Provide updated statistics to normalize the input data of the DNN +- **DNN** + - Output of the DNN is eventually denormalized + - Using the statistics forecasted by the GAS model + +
+ +Leverages the strengths of both statistical modeling and deep + +learning. + +
+ +# 1. Related Works + +## (1) Statistical models for non-stationary data + +Generalized Autoregressive Score models [8] + +- Used for filtering time-varying parameters + - that update the estimation with the observed score + +
+ +GAS model is the GARCH model + +- Widely used to filter time-varying variance in TS + +
+ +## (2) Recent works + +Address challenges caused by non-stationary TS + +$$\rightarrow$$ Dynamic normalization + +
+ +Examples + +Input normalization [27]. + +- [23] Batch normalization method for domain adaptation [23] +- [29] DAIN learns the normalization with a nonlinear network +- [10] normalizes the input both in time and frequency + +$$\rightarrow$$ Ignore non-stationarity over time within the input TS + +
+ +RevIN [19] + +- Denormalization step to restore the statistics removed during the normalization step + +
+ +[11] + +- Adopted a normalization methodology combined with a denormalization step. + +- Also consier intra-space shift, + + = Non-stationarity between input & output + +- Adopts fixed statistics for the forecast + +
+ +SAN [24] + +- Proposes a dynamic normalization approach + +- Splits both input and output into shorter temporal slices, in which non-stationarity can be less impactful + + $$\rightarrow$$ Uses them to estimate means and variances + +
+ +GasNorm (proposed) + +- Adapts the statistics ***online*** +- Avoids possible problems caused by slices that are... + - Too long, like non-stationarity + - Too short, like overly noisy estimations. + +
+ +# 2. Proposed Method + +## (1) Why Adaptive Normalization for a DNN? + +DNNs = Composition of many nonlinear functions + +$$\rightarrow$$ Importance of the ***stability of the input distribution*** of each layer has been extensively explored in the literature [18, 21] + +
+ +### GAS-Norm + +Improving the generalization of the nonlinear forecasting model in ***non-stationary settings*** + +
+ +Example) + +- Complex nonlinear model can be less robust to changes in the input distribution than a linear model when applied to TS data + +Toy dataset + +![figure2](/assets/img/ts2/img193.png) + +
+ +TS: Changes in the mean and variance: Can be either + +- Predictable (like a simple linear trend) +- Unpredictable (like random regime changes) + +
+ +Adaptive and flexible way to filter the location and scale parameters of our data **online** + +$$\rightarrow$$ Allow us to normalize the data even in the presence of **deterministic or unpredictable non-stationarity** + +
+ +## (2) Parameter Filtering for Non-stationary Time Series + +Forecasting problem + +- $$E\left[Y_{t+h} \mid X_t, \ldots, X_{t-l} ; w\right]=f_w\left(X_t, \ldots, X_{t-l}\right)$$. +- Conditional expectation + - Random variable that depends on the realizations of $$\left\{X_t, \ldots, X_{t-l}\right\}$$. + +
+ +Input distribution at time $$t$$ of a $$k$$-dim input vector + +- Unknown joint distribution of all input features +- $$P_{X_t}\left(x_t\right)=P_{X_{1, t}, X_{2, t}, \ldots X_{k, t}}\left(x_{1, t}, x_{2, t}, \ldots x_{k, t}\right)$$, + +
+ +Stationary case + +- (mean, var) of the input distn = constant + +Non-stationary case + +- (mean, var) of the input distn = change in time + +
+ +Assume to know the type of **parametrized density function of the marginal distributions** of each input feature, **conditional to the past observations of that feature itself**. + +- ex) Gaussian distribution with time-varying mean and variance + - $$P_{X_{i, t} \mid \mathcal{F}_{i, t}}\left(x_{i, t}\right)=\mathcal{N}\left(\mu_{i, t}, \sigma_{i, t}^2\right)$$. + +
+ +***Observation-driven state-space representation*** + +Realizations of our input feature $$x_{i, t}$$ are given by: + +$$\begin{aligned} +& x_{i, t}=\mu_{i, t}+\sigma_{i, t} \epsilon_{i, t} \\ +& \mu_{i, t}=g\left(\mu_{i, t-1}, x_{i, t-1}\right) \\ +& \sigma_{i, t}^2=g^{\prime}\left(\sigma_{i, t-1}^2, x_{i, t-1}\right) +\end{aligned}$$. + +
+ +Notation + +- $$\theta_t=\left[\mu_t, \sigma_t^2\right]$$ = Time-varying parameter vector + + ( Drop the feature index $$i$$. ) + +
+ +Modify the GAS formulation in [22] + +By adding a new hyperparameter $$\gamma \in[0,1)$$ + +- Control how much importance is given to + - (1) **maximizing the likelihood** + - (2) **keeping the parameter stable** +- Controls the update speed of the normalization parameters +- **Low** values = **Slow** adaptation + - Normalized input more similar to the original one + - (Extreme) $$\gamma=0$$ : Equivalent to a static normalization. + +- $$\max _\theta \gamma \log p\left(x_t \mid \theta\right)-\frac{1-\gamma}{2} \mid \mid \theta-\theta_{t \mid t-1} \mid \mid _{P_t}^2$$. + - where $$P_t$$ is a penalization matrix + +
+ +![figure2](/assets/img/ts2/img194.png) + +
+ +First-order Taylor expansion: + +- $$\log p\left(x_t \mid \theta\right)=\log p\left(x_t \mid \theta_{t \mid t-1}\right)+\left(\theta-\theta_{t \mid t-1}\right) \nabla_\theta\left(x_t \mid \theta_{t \mid t-1}\right)$$. +- First-order condition + - $$\theta_{t \mid t}=\theta_{t \mid t-1}+\frac{\gamma}{1-\gamma} P_t^{-1} \nabla_\theta\left(x_t \mid \theta_{t \mid t-1}\right)$$. + +
+ +Summary + +- (Update) $$\theta_{t \mid t}=\theta_{t \mid t-1}+\frac{\gamma}{1-\gamma} \alpha \tilde{\nabla}_\theta\left(x_t \mid \theta_{t \mid t-1}\right)$$. +- (Linear prediction) $$\theta_{t+1 \mid t}=\omega+\beta \theta_{t \mid t}$$ +- Parameters = $$(\alpha, \beta, \omega)$$ + +
+ +To optimize these static parameters... + +- Prediction error decomposition + + $$p\left(x_1, x_2, \ldots, x_T\right)=p\left(x_T \mid x_{T-1}\right) \ldots p\left(x_2 \mid x_1\right) p\left(x_1\right)$$. + +
+ +New optimization problem + +$$\begin{aligned} +\max _{\alpha, \omega, \beta} & \gamma \log p\left(x_1\right)-\frac{1-\gamma}{2} \mid \mid \theta_1-\theta_0 \mid \mid _{P_t}^2+ \\ +& \sum_{t=2}^T \gamma \log p\left(x_t \mid x_{t-1}\right)-\frac{1-\gamma}{2} \mid \mid \theta_t-\theta_{t \mid t-1} \mid \mid _{P_t}^2 +\end{aligned}$$, + +
+ +Find optimal $$\alpha, \beta, \gamma$$ + +- Use them to filter the mean and variance +- Normalize the marginal distribution of each feature at each time step + +
+ +Joint input distribution: + +$$\begin{aligned} +& P_{\tilde{X}_{1, t}, \tilde{X}_{2, t}, \ldots \tilde{X}_{k, t}}\left(\tilde{x}_{1, t}, \tilde{x}_{2, t}, \ldots \tilde{x}_{k, t}\right)= \\ +& p\left([0,0, \ldots 0],\left[\begin{array}{ccc} +1 & \operatorname{Corr}\left(x_{1, t}, x_{2, t}\right) & \ldots \\ +\operatorname{Corr}\left(x_{1, t}, x_{2, t}\right) & 1 & \ldots \\ +\ldots & \ldots & 1 +\end{array}\right]\right) +\end{aligned}$$. + +- mean zero and a covariance matrix equal to the correlation matrix of the original joint distribution +- Normalized features can be used as input for a DNN + +
+ +## (3) GAS-Norm + +![figure2](/assets/img/ts2/img195.png) + +Different blocks + +- (1) Normalization +- (2) Forecasting +- (3) Denormalization + +
+ +$$y_{i, t+h} =\mu_{i, t+h}+\sigma_{i, t+h} e_{i, t+h} $$. + +- $$\mu_{i, t+h} =g\left(\mu_{i, t+h-1}\right)$$. +- $$\sigma_{i, t+h}^2 =g^{\prime}\left(\sigma_{i, t+h-1}^2\right)$$. +- $$e_{i, t+h} =f_w\left(\tilde{X}_t, \ldots, \tilde{X}_{t-l}\right)+\epsilon_{i, t+h}$$. + - DNN = ***Residual Learning*** + +
+ +### a) Normalization + +$$\begin{aligned} +& \mu_{t+1}=\omega_\mu+\beta_\mu\left[\frac{\gamma}{1-\gamma} \alpha_\mu \frac{y_t-\mu_t}{1+\frac{\left(y_t-\mu_t\right)^2}{v \sigma_t^2}}+\mu_t\right] \\ +& \sigma_{t+1}^2=\omega_\sigma+\beta_\mu\left[\frac{\gamma}{1-\gamma} \alpha_\sigma\left(\frac{(v+1)\left(y_t-\mu_t\right)^2}{v+\frac{\left(y_t-\mu_t\right)^2}{\sigma_t^2}}-\sigma_t^2\right)+\sigma_t^2\right] +\end{aligned}$$. + +
+ +### b) Forecasting & Denormalization + +$$\begin{aligned} +\mu_{t+1} & =\omega_\mu+\beta_\mu \mu_t \\ +\sigma_{t+1}^2 & =\omega_\sigma+\beta_\sigma \sigma_t^2 +\end{aligned}$$. + +
+ +### Summary + +Autoregressive approach + +- Prediction is assumed as real observation to predict the next step. + +- Forecast by DNN is combined with the means and variances predicted by the GAS + + $$\rightarrow$$ Re-introducing the information removed during the normalization procedure + +
+ +# 3. Experiments + +![figure2](/assets/img/ts2/img196.png) + +![figure2](/assets/img/ts2/img197.png) + +![figure2](/assets/img/ts2/img198.png) + diff --git a/_posts/2024-10-14-(paper)TiVaT.md b/_posts/2024-10-14-(paper)TiVaT.md new file mode 100644 index 000000000000..dce1146d9453 --- /dev/null +++ b/_posts/2024-10-14-(paper)TiVaT.md @@ -0,0 +1,212 @@ +--- +title: TiVaT; Jonit-Axis Attention for TIme Series Forecasting with Lead-Lag Dynamics +categories: [TS] +tags: [] +excerpt: arxiv 2024 +--- + + + +# TiVaT: Jonit-Axis Attention for TIme Series Forecasting with Lead-Lag Dynamics + +
+ +# Contents + +0. Abstract +1. Introduction +2. Methodology + +
+ +# 0. Abstract + +Simultaneously capturing both TD & CD (VD) remains a challenging + +
+ +Previous works + +- CD models: handle these dependencies separately +- Limitation in lead-lag dynamics. + +
+ +### TiVaT (Time-Variable Transformer) + +(1) Integrates TD& VD via **Joint-Axis (JA) attention** + +- Capture variate-temporal dependencies + +(2) Further enhanced by **Distance-aware Time-Variable (DTV) Sampling** + +- Reduces noise and improves accuracy through a learned 2D map that focuses on key interactions + +
+ +# 1. Introduction + +CD models = Handle temporal and inter-variable dependencies ***separately*** + +$$\rightarrow$$ Limiting their ability to capture more complex interactions between variables and temporal dynamics + +( i.e. **lead-lag relationships** ) + +
+ +Why not do both? + +$$\rightarrow$$ Significant increase in computational cost and model ***complexity*** + +
+ +### TiVaT (Time-Variable Transformer) + +Capture both **temporal and variate dependencies** simultaneously + +Joint-Axis (JA) attention mechanism + +![figure2](/assets/img/ts2/img189.png) + +
+ +A key feature of TiVaT + += Integration of offsets inspired by + +- deformable attention (Zhu et al., 2020) +- Distance-aware Time-Variable (DTV) Sampling + +
+ +### DTV Sampling technique + +Constructs a learned 2D map to capture both spatial and temporal distances between variables and time steps + +- Not only reduces computational overhead +- But also mitigates noise! + +$$\rightarrow$$ Scale efficiently to high-dimensional datasets without sacrificing performance. + +
+ +Incorporates TS decomposition + +- To capture long-term trends and cyclical patterns + +
+ +# 2. Methodology + +### Notation + +- $$\mathbf{X}=\left\{\mathbf{x}_{T-L_H+1}, \ldots, \mathbf{x}_T\right\} \in \mathbb{R}^{L_H \times V}$$ . + - $$\mathbf{X}_{(t,:)} \in \mathbb{R}^V$$, + - $$\mathbf{X}_{(:, v)} \in \mathbb{R}^{L_H}$$, +- $$\mathbf{Y}=\left\{\mathbf{x}_{T+1}, \ldots, \mathbf{x}_{T+L_F}\right\} \in \mathbb{R}^{L_F \times V}$$. + + + +MTS forecasting is challenging because it requires capturing complex relationships along ***both the variate and temporal axes*** + + + +## (1) Overview + +![figure2](/assets/img/ts2/img190.png) + +$$\begin{aligned} +\mathbf{X}^{\text {Trend }} & =M A(\mathbf{X}) \\ +\mathbf{X}^{\text {Seasonality }} & =\mathbf{X}-\mathbf{X}^{\text {Trend }} \\ +\mathbf{X}^{\text {Trend }} & =\mathbf{X}^{\text {Trend }}+\text { Linear }\left(\mathbf{X}^{\text {Trend }}\right) \\ +\mathbf{X}^{\text {Seasonality }} & =\mathbf{X}^{\text {Seasonality }}+\text { Linear }\left(\mathbf{X}^{\text {Seasonality }}\right), +\end{aligned}$$. + +
+ +Decomposed components + +- $$\mathbf{X}^{\text {Trend }}$$ and $$\mathbf{X}^{\text {Seasonality }}$$ +- Processed through separate sibling architectures + +
+ +Architectures + +- (1) Patching + Embedding + - 1-1) Patch: $$X_P \in \mathbb{R}^{L_N} \times V \times L_P$$ + - 1-2) Token: $$Z \in \mathbb{R}^{L_N \times V \times D}$$ + - Via embedding layer $$E: \mathbb{R}^{L_N \times V \times L_P} \rightarrow \mathbb{R}^{L_N \times V \times D}$$ +- (2) JA attention blocks +- (3) Projector + - 3-1) Trend prediction + - 3-2) Seasonality prediction + - Common predictor + - $$\operatorname{Proj}: \mathbb{R}^{L_N \times V \times D} \rightarrow$$ $$\mathbb{R}^{L_F \times V}$$ + +
+ +Summary + +- $$\hat{\mathbf{Y}}=\operatorname{Proj}\left(\operatorname{Enc}\left(E\left(X_P^{\text {Trend }}\right)+P E\right)\right)+\operatorname{Proj}\left(\operatorname{Enc}\left(E\left(X_P^{\text {Seasonality }}\right)+P E\right)\right)$$. + +
+ +## (2) Joint-Axis Attention Module + +![figure2](/assets/img/ts2/img192.png) + +
+ +Joint-Axis Attention block + +- Transformer encoder block + - Replacing self-attention with the JA attention module + +- Inspired by ***Deformable attention*** module (Zhu et al., 2020), + + = Capture relationships between a feature vector $$Z_{(t, v)}$$ and other feature vectors $$Z_{\left(t^{\prime}, v^{\prime}\right)}$$, where $$t \neq t^{\prime}, v \neq v^{\prime}$$, or both. + +- Unlike the deformable attention, the JA attention module uses **offset points** as guidelines to minimize information loss +- Uses the **DTV sampling** to capture relationships with other points that are highly relevant + +- Efficient compared to full attention + - As it avoids processing less relevant points + +
+ +### Deformable Attention + +Introduced to tackle the inefficiencies of self-attention operations in CV + +
+ +How does it work? + +- Extracts **offset points** + + - based on the query feature $$q_{(t, v)}$$ at the reference point $$(t, v)$$ + +- Attention operation is performed accordingly + +- Efficiently consider **all axes**, + + while also offering **computational efficiency** compared to self-attention at every location. + +
+ +[ Fig. 3 ] + +- $$Z \in \mathbb{R}^{L_N \times V \times D}$$. + + - [TS] Temporal & variable axes + - [CV] Width & height axes + +- Offset points can indentify relationships between + + - the reference point + - the other data points + + across both axes. + +
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