# NFCL: Simply interpretable neural networks for a short-term multivariate
forecasting

Wonkeun Jo<EMAIL_ADDRESS>0000-0001-7374-2900 Department of Computer
Science and Engineering, Chungnam National University99
Daehak–roYuseong–guDaejeonRepublic of Korea34134 and  Dongil Kim
<EMAIL_ADDRESS>0000-0001-7425-7579 Department of Data Science,
Ewha Womans University52 Ewhayeodae–gilSeodaemun–guSeoulRepublic of Korea

( )

###### Abstract.

Multivariate time-series forecasting (MTSF) stands as a compelling field
within the machine learning community. Diverse neural network based
methodologies deployed in MTSF applications have demonstrated commendable
efficacy. Despite the advancements in model performance, comprehending the
rationale behind the model’s behavior remains an enigma. Our proposed model,
the Neural ForeCasting Layer (NFCL), employs a straightforward amalgamation of
neural networks. This uncomplicated integration ensures that each neural
network contributes inputs and predictions independently, devoid of
interference from other inputs. Consequently, our model facilitates a
transparent explication of forecast results. This paper introduces NFCL along
with its diverse extensions. Empirical findings underscore NFCL’s superior
performance compared to nine benchmark models across 15 available open
datasets. Notably, NFCL not only surpasses competitors but also provides
elucidation for its predictions. In addition, Rigorous experimentation
involving diverse model structures bolsters the justification of NFCL’s unique
configuration.

Deep learning; Multivariate time-series; Forecasting; Interpretable;

††copyright: acmlicensed††journalyear:  ††doi: XXXXXXX.XXXXXXX††conference: ;
; ††isbn:  ††ccs: Computing methodologies Artificial intelligence††ccs:
Computing methodologies Modeling methodologies

##  1\. Introduction

Multivariate Time-Series Forecasting (MTSF) plays a crucial role in the
machine learning society, significantly impacting various aspects of our
lives. For instance, machine learning algorithms control the quantity of
electricity generated (Sun and You, 2021). Additionally, atmospheric
conditions are forecasted using neural network (NN) based on past time-series
data (Venkatachalam et al., 2023; Kim et al., 2022; Zhang et al., 2024; Chen,
2019). Given the multitude of applications beyond these examples, the MTSF
problem attracts numerous researchers striving for more precise solutions
(Chen et al., 2023).

Figure 1. Forecasting the fourth and seventh next steps’ HUFL based on the
historical multivariate time-series. Below, contribution maps illustrate the
reasons why the model predicts the target to be closer to
$\hat{Y}^{\mathrm{HUFL}}=10$. The pattern of the contribution map predicted by
the model changes, similar to moving three steps forward in the fourth future
step while the target time point is moved to three step forward.

NNs are the preferred method for addressing encompassing classification,
regression, anomaly detection, and more (Sarker, 2021; Alzubaidi et al.,
2021). And, recent approaches aim to solve time-series problems, including
forecasting (Zhou et al., 2020; Wu et al., 2023; Zeng et al., 2023). However,
our focus aligns with two trends in the machine learning society: 1)
contemporary research emphasizes performance and architectural complexity (Tan
and Le, 2021; Dehghani et al., 2023), and 2) there is a growing need for
short-term future forecasting using the latest NNs (Zhang et al., 2024; Mounir
et al., 2023; Melo et al., 2022; Ke et al., 2017).

Interpretability is another critical aspect in the machine learning community,
addressing the need to understand why a method made specific predictions
(Rojat et al., 2021). In real-world decision-making, users seek information on
how black-box ML models generate results, and providing this insight can aid
individuals in making informed decisions (Theissler et al., 2022; Veerappa et
al., 2022; Zhang et al., 2024). A simple method, such as an additive model
applying independent functions to each input, represents a solution in machine
learning to interpret the reasons for forecasting time-series (Zeng et al.,
2023).

We propose Neural ForeCasting Layer (NFCL) for forecasting the near future
while elucidating the reasons behind predictions. Figure 1 illustrates the
concept of NFCL. The model receives the input series, depicted in green, and
subsequently forecasts the future, represented in red, using contribution
maps. In these maps, the red color signifies positive weights, while the blue
color denotes negative weights. It is important to note that the grey color,
closer to zero, is negligible in the forecasting process. A simple example
demonstrates that when forecasting the High UseFul Load (HUFL) in the next
fourth and seventh steps future from the present (because the future time step
starts from zero index), NFCL assigns higher positive weights from Middle
UseFul Load (MUFL) and mixed weights from the HUFL. According to the research,
Note that the HUFL must be higher than MUFL, and they are correlated (Zhou et
al., 2020).

We also tested expansion of NFCL, which treats each variable (representing one
step of the time-series) independently, by applying various preprocessing
methods, such as statistical normalization, input re-evaluation, time-series
decomposition and etc. We illustrate that NFCL achieves the best scores in
real-world datasets and elucidate the reasons behind the model’s forecasts.
Our main contributions are as follows:

  * •

NFCL is constructed based on simple weight combinations, such as linear
regression. We observe that capturing cross-correlation between NFCL and other
time-series contributes more to forecasting than handling only its own time-
series independently.

  * •

The advanced expansions are designed; 1) input re-evaluation and 2)
decomposition. NFCL can be enhanced by employing the input re-evaluation
function $h$ (Agarwal et al., 2021). To justify the structure of the input
mapping function $h$, we conduct empirical analyses using real-world datasets,
comparing scenarios with and without the utilization of $h$. Additionally, we
perform experiments applying the average pooling method to decompose the time-
series.

  * •

Three versions of NFCLs are evaluated on 15 datasets collected from four
diverse fields using nine forecasting benchmarks. According to experiments,
NFCL-C (NFCL-Complex) achieved the most competitive performance across four
metrics.

  * •

The input-output directly connected network structure aids users seeking
evidence for forecasting one of the time-series variables at a specific time.
Independent computation for each input variable clarifies which series’ time
step has a more significant impact on the result.

##  2\. Related work

Over the past decades, statistical and algorithmic machine learning methods
have been instrumental in solving MTSF problems (Ho and Xie, 1998; Liu et al.,
2018). The evolution of NN has subsequently transformed conventional
approaches to MTSF, as NNs circumvent the structural limitations inherent in
traditional machine learning methods (Tu, 1996). Convolutional NNs (CNNs) and
transformers have demonstrated superior performance compared to simple
connected networks as NN backbone techniques in various fields where machine
learning has faced recent challenges (Lea et al., 2016; Dosovitskiy et al.,
2021).

Informer (Zhou et al., 2020), based on the transformer (Vaswani et al., 2017),
has introduced its own benchmark datasets (electricity transfer temperature,
ETT) to the community, establishing baselines and serving as the foundation
for various methods (e.g., Autoformer (Wu et al., 2021), PatchTST (Nie et al.,
2023), etc. (Zhou et al., 2022; Liu et al., 2022, 2024)) to address MTSF
problems. Transformer-derived models not only excelled at predicting long-term
time-series, where traditional recurrent NN families struggle, but also
performed well at predicting short-term time-series, an area where
conventional recurrent NN models dominate. However, beyond attention
mechanisms, forecasting NNs often handle cross-correlation of time-series
inadequately (Zeng et al., 2023; Nie et al., 2023). Even without attention,
SCINet, based on CNN, has achieved competitive performance in benchmarks for
MTSF problems (LIU et al., 2022; Zhou et al., 2020). Additionally, built on
simple linear weight combinations, decomposed- and normalized-linear (D-,
N-Linear) models have demonstrated the best performance and interpretability
simultaneously after training (Zeng et al., 2023). TimesNet (Wu et al., 2023)
and TiDE (Das et al., 2023) leverage CNNs and linear combinations in the
model, respectively, as they pass through the input to output, achieving
improved performance.

Another significant focus of our manuscript is interpretability. One of the
reasons Attention mechanisms have garnered interest is their ability to show
evidence of a model’s predictions, indicating how much each point contributes
during the forwarding of inputs to outputs (Zhou et al., 2020). However, while
the attention mechanism clearly indicates how much each time-series point
contributes to the prediction of other time-series points, the contribution
between different time-series corresponding to the input channels is not fully
understood. TimeSHAP (Bento et al., 2021), a time-series explaining method
utilizing SHAP (Lundberg and Lee, 2017), summarizes contributions to simple
weights for any method but requires a post-hoc process to fit the explaining
model after training the recurrent NN-based forecaster. Nevertheless, some
papers have proposed applying a simple additive structure, which can also
provide contributions to model decisions but have predominantly focused on
tabular datasets (Agarwal et al., 2021; Radenovic et al., 2022).

##  3\. Preliminaries

###  3.1. Notations

In dataset $\mathcal{D}$, each time-series is separated to two pairwise
continuous series; historical and forecasting target series as
$\\{\mathcal{X},\mathcal{Y}\\}$, respectively. The sample $X$ of historical
series $\mathcal{X}$ is as,

$X=\\{x^{1,t},x^{2,t},\cdots,x^{K,t}\\}_{t=1}^{L}\in\mathbb{R}^{K,L},$

where $L$ is look-back window length and $x^{i,j}$ denote the $i$-th series at
the $j$-th time step. Also, $K$ denotes the number of the series. As
similarly, the sample $Y$ of forecasting series $\mathcal{Y}$ is as,

$Y=\\{x^{1,t},x^{2,t},\cdots,x^{K,t}\\}_{t=L+1}^{L+T}\in\mathbb{R}^{K,T},$

where the $T$ is forecasting time length. Then, we supposed to hold conditions
that the $L>T>1$ and $K>1$, short-term forecasting with the MTSF in the
manuscript.

###  3.2. Problem statement

We defined the problem that to solve time-series forecasting is finding the
optimal parameters of forecasters as,
$\mathop{\mathrm{argmin}}_{\theta}\mathcal{L}(f_{\theta}(\mathcal{X}),\mathcal{Y})$,
where $\theta$ is the parameter of forecasting model
$f_{\theta}:\mathbb{R}^{K,L}\rightarrow\mathbb{R}^{K,T}$ and $\mathcal{L}$ is
loss function to optimize the parameter. Moreover, we have two kinds of
datasets $\mathcal{D}$ and $\mathcal{D}^{\prime}$. $\mathcal{D}^{\prime}$
denotes the unseen dataset that we can observe in near futures and has the
same shape as $L$, $T$, and $K$. We design the problem that the forecasting
model should train and validate using only $\mathcal{D}$, and evaluate the
model performance with $\mathcal{D}^{\prime}$ that is excepted in the learning
and validation process.

##  4\. NFCLs

###  4.1. Simple Expansion

Figure 2. The simple expansion for forecasting. The black arrow denotes a
fully-connected operation between each input sequence in X and each target
sequence in Y.

Figure 3. The complex expansion for forecasting.

D- and N-Linear, proposed from LSTF-Linear (Zeng et al., 2023), offer an
embarrassingly simple baseline for solving MTSF. Two linears handle the time-
series as univariate to forecast the future with only one weight $w$ to
compute $Xw+b$. For multivariate forecasting, they repeat the process of
univariate as much as the number of time-series variables. So, LSTF-Linear
advocated methods that utilize capacity up to $K\cdot L\cdot T$, where
($\cdot$) represents a product between two more numbers. The critique suggests
that LSTF-Linear neglected their approach, which improved upon using not only
a univariate solution but also cross-correlated multivariate series.
Therefore, we conducted experiments to expand the weight of the linear
combinations, referred to as NFCL-V (NFCL-Vanilla).

We expanded the weight size from LSTF-Linear to $(K\cdot L)\cdot(K\cdot
T)=K^{2}\cdot L\cdot T$, as NFCL-V forecasts each input point (denoted as all
$x^{k,t}$ in $X$) to each forecasting point (denoted as all $y^{k,t}$ in the
paired $Y$). We summarized NFCL-V in Equation 1:

(1)
$\hat{y}^{k,t}=\sum_{i=1}^{K}{\sum_{j=1}^{L}{x^{i,j}w^{i,j,k,t}}}+b^{k,t},$

where $w\in\mathbb{R}^{K,L,K,T}$ and $b\in\mathbb{R}^{K,T}$ represent the
weight and bias, respectively.

Figure 2 depicts the process of NFCL-V. Concerning the weight $w$, the left
side represents the historical input $X$, and the right side represents the
forecasting target $Y$. Each circle and colored row indicates each point of
each series and the time-series variable. The number of circles for each block
indicates the length of $X$ or $Y$, and the arrow indicates a complete
connection between the circle in the colored $X$ block and the circle in the
opposite $Y$ block. The upper red box provides more details; all arrow
connections between blocks in the lower box denote the complete connection
between all points in blocks.

To implement the weight combination, we transformed the shape of the two
matrices, weight, and bias, as $w\in\mathbb{R}^{K\cdot L,K\cdot T}$ and
$b\in\mathbb{R}^{K\cdot T}$ to compute $Xw+b$. To maintain the setup, we also
transformed $X$ and $Y$ into the flatten shape, as $X\in\mathbb{R}^{K\cdot L}$
and $Y\in\mathbb{R}^{K\cdot T}$.

This section introduces a method to enhance the model by adding a mapping
function $h$ to the input variables. Several papers have explored the
complication of these additive operations (Agarwal et al., 2021; Radenovic et
al., 2022). To make the model more complex, we utilize mapping methods $h$ on
inputs to increase the complexity of simple connections, denoted as NFCL-C. In
Equation 2, $h:\mathbb{R}^{K\cdot L}\rightarrow\mathbb{R}^{K\cdot L}$ is an
arbitrary non-linear function that maps each input $x^{i,j}$ as:

###  4.2. Complex Expansion

(2)
$\hat{y}^{k,t}=\sum_{i=1}^{K}{\sum_{j=1}^{L}{h^{i,j}\left(x^{i,j}\right)w^{i,j,k,t}}}+b^{k,t}.$

Figure 3 depicts the process of NFCL-C. Most components in Figure 3 share the
same semantics as their counterparts in Figure 2. However, the diagonal inner
patterned circle in the small red box denotes the re-evaluated value based on
the input $x^{i,j}$. This indicates that we need $K\cdot L$ different
$h^{i,j}$ to dedicate each $x^{i,j}$. To accelerate loop computation, we use
group convolution that supports point-wise convolution, requiring the group
size to be set to $K\cdot L$ (Chollet, 2016; Radenovic et al., 2022). Appendix
A illustrates the algorithm for implementing Equation 2 and the corresponding
algorithm utilizing group convolution.

We explored the structure of $h^{i,j}$ in experiments. We prepared more
complex structures using not only weight combinations and non-linear
activation but also dropout or batch normalization, known as NN performance-
improving methods. However, the drop-out and normalization layer hinders the
production of the re-evalu-ation value from $X$ according to someones works;
hence, we conducted a simple mapping function $h$. Despite the uncomplicated
shape of the function, the simple $h$ form of the network outperformed in most
cases. We discuss our experimental results in Section 5.7 to choose the proper
hyperparameters for $h$.

Figure 4. The decomposition for forecasting.

Figure 5. The NFCLs overall pipeline for forecasting.

###  4.3. Decomposed Expansion

On one hand, we consider pre-processing, such as time-series decomposition.
Several papers have focused on forecasting improvement based on average
pooling and frequency analysis (Wu et al., 2021; Zeng et al., 2023). Thus, we
prepared the separation for time-series with two kinds of methods: 1)
decomposition and 2) statistical normalizing.

The first is moving average pooling with the kernel size $S$ that has three
over diverse. We conduct the pooling sequence from the largest $s_{1}$ to the
smallest $s_{N}$. Before applying the pooling sequence, the time axis of
$\tilde{X}$ is padded to the left side with zero to keep the time-series
length as $L$. At the end of each pooling sequence, $\tilde{X}$ is separated
into two over decomposed series
$\tilde{X}_{dec}=[\tilde{X}_{s_{1}},\cdots,\tilde{X}_{s_{N}}]$, where
$S=\left\\{s_{1},\cdots,s_{N};s_{n}>s_{n+1}\right\\}$. Note that the smallest
$s_{N}$ must be $1$ to capture the residual used to recover the raw time-
series, while the largest $s_{1}$ shows the trend of time-series. We built
independent models to dedicate each $\tilde{X}_{s_{n}}$ respectively and
summed all outputs of each $h_{s}$. In the manuscript, the decomposed NFCL is
denoted as NFCL-D (NFCL-Decomposition), and details are followed in appendix
B. Figure 4 illustrates the time-series decomposition flow. Within the red
area in Figure 4, the time-series $\tilde{X}$ is decomposed into Decomp.
blocks. Each decomposition block comprises time-series left padding and
average pooling. Please refer to Appendix B for more details.

Next is statistical normalizing $X$ according to each time-series variable. In
this case, the normalized input $\tilde{X}$ is delivered to the arbitrary
function $h$. After extraction from $h$, the mapping value $h(\tilde{X})$ is
recovered as much as the shifted values from applying normalization. This
separation makes the module work easy: the first module’s role is only
normalization, and the second module’s role is to capture the forecasting
pattern. We summarized the process with Equation 3 as follows:

(3)
$\displaystyle\begin{split}\tilde{x}^{k,:}&=\alpha\left({\frac{x^{k,:}-\mathrm{mean}(x^{k,:})}{\mathrm{std}(x^{k,:})}}\right)+\beta,\\\
\hat{y}^{k,:}&=\frac{\tilde{y}^{k,:}-\beta}{\alpha}\cdot\mathrm{std}(x^{k,:})+\mathrm{mean}(x^{k,:}),\end{split}$

where $(:)$, $\mathrm{mean}$s, and $\mathrm{std}$s indicate all times of $k$
time-series, the functions of average and standard deviation, respectively.
Moreover, $\alpha\in\mathbb{R}^{k}$ and $\beta\in\mathbb{R}^{k}$ are the
learnable parameters that apply normalization effectively (Xu et al., 2019).
In short, input $X$ is normalized to $\tilde{X}$, $h$ extracts the
contribution based on the center-adjusted time-series $\tilde{X}$, and
normalized target $\tilde{Y}=h(\tilde{X})w+b$ is invert-shifted to $\hat{Y}$
with the original $X$ statistics. Figure 5 illustrates the overall pipeline of
NFCLs for the forecasting process. Each time-series that is colored unique in
a historical time-series $X$ is subjected to time-series specific
normalization as $\tilde{X}$ shown in Equation 3. $\tilde{X}$ are concatenated
along the first dimension and passed to the NFCLs. NFCLs predict the
$\tilde{Y}$s from the normalized input $\tilde{X}$. It then applies
denormalization to each time-series, reusing the statistical values for each
time-series used in the normalization. The denormalization is inversely
applied to the $\tilde{Y}$s, resulting in $Y$.

##  5\. experiments

Metric | Category | NFCL | DLinear | NLinear | TiDE | Informer | Autoformer | PatchTST | iTransformer | SCINet | TimesNet
---|---|---|---|---|---|---|---|---|---|---|---
| Patient | 1.212 | 1.140 | 1.072 | 1.016 | 1.362 | 1.142 | 0.939 | 0.781 | 1.145 | 0.864
| Currency | 0.095 | 0.090 | 0.090 | 0.090 | 0.761 | 0.121 | 0.090 | 0.094 | 0.222 | 0.102
| Electricity | 0.299 | 0.308 | 0.311 | 0.311 | 0.375 | 0.349 | 0.299 | 0.311 | 0.305 | 0.300
MAE | Climate | 0.310 | 0.361 | 0.348 | 0.346 | 0.383 | 0.391 | 0.334 | 0.340 | 0.322 | 0.328
| Patient | 6.349 | 3.080 | 2.843 | 2.698 | 4.312 | 3.018 | 2.972 | 2.326 | 3.238 | 3.413
| Currency | 0.023 | 0.021 | 0.021 | 0.021 | 0.975 | 0.031 | 0.021 | 0.022 | 0.110 | 0.025
| Electricity | 0.245 | 0.249 | 0.266 | 0.265 | 0.304 | 0.292 | 0.245 | 0.257 | 0.231 | 0.241
MSE | Climate | 0.314 | 0.354 | 0.373 | 0.370 | 0.394 | 0.400 | 0.355 | 0.355 | 0.314 | 0.336
| Patient | 38.149 | 43.689 | 41.671 | 40.446 | 52.267 | 43.967 | 34.421 | 30.707 | 43.916 | 32.280
| Currency | 7.066 | 6.820 | 6.816 | 6.800 | 34.877 | 8.835 | 6.857 | 7.047 | 12.000 | 7.499
| Electricity | 27.546 | 28.366 | 28.140 | 28.113 | 34.892 | 32.096 | 27.532 | 28.471 | 29.137 | 28.100
SMAPE | Climate | 29.043 | 33.533 | 31.415 | 31.222 | 35.814 | 36.123 | 30.494 | 31.039 | 31.460 | 30.419
| Patient | -0.484 | -0.179 | 0.269 | 0.274 | -0.580 | 0.244 | 0.074 | 0.339 | -0.050 | 0.144
| Currency | 0.931 | 0.928 | 0.929 | 0.929 | -2.475 | 0.912 | 0.936 | 0.932 | 0.603 | 0.929
| Electricity | 0.650 | 0.636 | 0.636 | 0.635 | 0.350 | 0.565 | 0.644 | 0.623 | 0.617 | 0.644
$R^{2}$ | Climate | 0.659 | 0.614 | 0.595 | 0.600 | 0.366 | 0.464 | 0.616 | 0.611 | 0.639 | 0.633

Table 1. The average score for each model by metric per dataset category. Each
score is an average of the scores across datasets in each category and
forecast length $L$.

###  5.1. Datasets

We prepared 15 datasets collected from various sources, including Patient,
Currency, Electricity, and Climate. For testing the MTSF problem, all $K$
time-series variables were utilized during both training and evaluation. We
adopted the evaluation concept of Informer, dividing all datasets into
training, validation, and test sets in a ratio of 6:2:2 for all datasets (Zhou
et al., 2020). Meanwhile, the Informer employed a 7:1:2 splitting ratio
exclusively for ETT datasets, while others utilized a 6:2:2 splitting ratio.
We adjusted Informer’s evaluation procedure 7:1:2 split to 6:2:2 to maintain
consistency in the splitting approach. Experiments were conducted with
different length settings than those of Informer, setting the historical
length $L$ to $24$ and the predicted length $T\in\\{6,12,18,24\\}$. It’s
important to note that we strictly separated $\mathcal{D}$ and
$\mathcal{D}^{\prime}$ during both the training and testing phases. Additional
details, including dataset abbreviations, are provided in Appendix C.

###  5.2. Metrics

The employed metrics in our experiment include four widely recognized
measures: Mean Absolute Error (MAE), Mean Squared Error (MSE), Symmetric Mean
Absolute Percentage Error (SMAPE), and $R^{2}$. Through our experiments, we
observed that the popular non-scaled metric tended to overestimate the
baselines. While MAE and MSE portray non-scaled errors, SMAPE and $R^{2}$,
which are bounded between 0 and 1 or percentage, facilitate the quantification
of errors across diverse scales. Additionally, SMAPE and $R^{2}$ are related
to MAE and $R^{2}$, respectively. $R^{2}$ is another widely used metric to
evaluate regression results, as it expresses the explained variances in $Y$ by
a model related to the entire variance of $Y$. Notably, SMAPE proves to be a
superior metric compared to Mean Absolute Percentage Error (MAPE). SMAPE
addresses the issue of MAPE diverging to $\infty$ when $Y\simeq 0$ and
$0<Y\ll\hat{Y}$ (Chen et al., 2017). For comprehensive details on the metrics,
please refer to Appendix D.

###  5.3. Baseline

The nine baselines are categorized into three types of forecasters: 1) fully
connected, 2) CNN-based, and 3) attention-based models. So, we employed models
as; D-Lienar, N-Linear, TiDE, Informer, Autoformer, PatchTST, iTransformer,
SCINet, and TimesNet. Note that we did not derive our experimental results
from another paper. Instead, we independently re-implemented each paper’s
methodology and hyperparameter setting based on their descriptions and GitHub
repositories. Subsequently, we rigorously re-tested these implementations on
our dataset split. Although we considered testing additional baselines, a few
models did not work as expected, possibly due to issues with the official
repositories on GitHub. Moreover, some others could work for forecasting while
the models perform too much poorly. A brief abstraction of each baseline’s
properties is attached on Appendix E.

Metric | Type | NFCL | DLinear | NLinear | TiDE | Informer | Autoformer | PatchTST | iTransformer | SCINet | TimesNet
---|---|---|---|---|---|---|---|---|---|---|---
| Rank | 2.33 | 6.98 | 6.05 | 5.33 | 9.13 | 9.25 | 3.58 | 4.95 | 3.93 | 3.45
MAE | Win | 43 | 3 | 0 | 4 | 0 | 0 | 1 | 4 | 5 | 0
| Rank | 2.85 | 5.08 | 6.78 | 6.10 | 8.52 | 9.03 | 4.45 | 5.08 | 3.15 | 3.95
MSE | Win | 23 | 3 | 0 | 0 | 0 | 0 | 5 | 4 | 25 | 0
| Rank | 2.00 | 6.95 | 5.70 | 4.82 | 9.28 | 9.30 | 3.08 | 4.72 | 5.68 | 3.47
SMAPE | Win | 43 | 0 | 3 | 4 | 0 | 0 | 3 | 4 | 0 | 3
| Rank | 2.30 | 5.32 | 6.40 | 5.87 | 9.05 | 8.98 | 4.40 | 4.98 | 4.07 | 3.63
$R^{2}$ | Win | 37 | 0 | 4 | 0 | 0 | 2 | 5 | 2 | 10 | 0

Table 2. The average rank and number of wins of performance on experiments
versus benchmarks.

###  5.4. Experiments Setting

Our experiments were implemented using PyTorch 2.X (Paszke et al., 2019).
While most forecasting papers have followed the experimental code of
Autoformer, we structured our experiments based on PyTorch-Lightning for
improved reusability among researchers. Our GitHub will provide the source
codes to develop forecasting applications for handling time-series datasets.

We conducted multiple runs of our experiments, with each model learning and
testing each dataset up to five times using different seeds and a batch size
of 128. To optimize the models, we employed the AdamW optimizer (Loshchilov
and Hutter, 2019) with a weight decay $\lambda$, and the learning rate
$\mathrm{lr}$ was set to 0.001. We aimed for sufficient convergence, utilizing
100 early-stopping rounds with MSE as the criterion. The model’s trainable
weights were reset to their best-performing state at the end of the training
process. The recovered model was then employed in the testing phase using the
dataset $\mathcal{D}^{\prime}$.

###  5.5. NFCL Setting

All three versions of NFCL underwent validation on the test bed with uniform
normalization applied across all variants. The results indicated that NFCL-C
with a single layer employing 32 hidden nodes demonstrated the best
performance. Consequently, NFCL-C was used in Subsection 5.6 to represent
NFCL. In addition, NFCL-C employed the leaky-ReLU activation. There is no
special reason why we choose the leaky-ReLU that was made for its ability to
express positive and negative values (not close to 0) non-linearly.
Furthermore, we have not considered the task of forecasting univariate time-
series, as NFCL has a similar concept to LSTF-linear when dealing with
univariate time-series (Zeng et al., 2023).

We note that each version of NFCL was designated as follows: Simple Forward
(NFCL-V), Using Mapping Function $h$ (NFCL-C), and Applied Decomposition
(NFCL-D). In the investigation, NFCL-C and -D were examined with diversity,
incorporating four different numbers of hidden nodes, three distinct layer
numbers, and variations in the application of time-series decomposition. The
term NFCL, without any suffix, refers to the best-performing NFCL-C, as we
avoided comparing NFCLs across experiments to prevent overestimation. Thus,
NFCL-C was compared to other versions in experiments, using the full names of
each NFCL in additional experiments.

###  5.6. NFCL’s Performances

Here are Table 1 and 2 displaying the performance of NFCL and the baselines.
Table 1 and 2 summarizes the results based on the information provided in
appendix F, that includes the full benchmark results according to each metric.
In the scores of the tables, blued-bold and underlined scores indicate the
best and next best scores, respectively. In addition, Table 1 has some similar
scores not highlighted because they have differences under the four decimal
points. Note that NFCL in the column refers to NFCL-C with $h=32$ nodes and
one layer.

Table 1 averaged the scores depending on the category where the dataset is
included and the forecast length $L$. The first and second columns in Table 1
show the metric and the category, respectively (Appendix C notice you to which
category included each dataset). Based on the aggregate results, our NFCL
performed best on Electricity and Climate categories when utilizing metrics
other than MSE. While NFCL performed worse than the best-performing reference
in terms of MSE, NFCL excelled in $R^{2}$, the normalized score of MSE.
Whereas MSE indicates that NFCL is performing worse than the other references
for the squared deviation of $Y$ from $\hat{Y}$, $R^{2}$ shows that NFCL is
performing better than the other benchmarks for the predicted deviation from
the variance of $Y$. Therefore, NFCL demonstrates its ability to capture the
variability in the dataset better than other benchmarks when the real-world
data has a large variance. Additionally, even on datasets in the Currency,
NFCL did not show significant degradation compared to other benchmarks.
However, NFCL performed poorly in the Patient, which we attribute to
overfitting of NFCL due to the small number of data samples (under 600
training points, and 200 validation and test points).

Table 2 summarizes the results based on the scores provided in appendix F,
which includes the full benchmark results according to each metric. The first
and second columns in each table show the metrics and the counter type that
evaluated the performance between real $Y$ and forecasted $\hat{Y}$, while the
third column onwards points to the respective average ranks or the number of
wins depend on the model utilized. In Table 2, concerning MAE, NFCL
demonstrates an average ranking of 2.33, and it outperforms competitors in 43
out of 60 total testbeds. Regarding MSE, NFCL achieves the best performance 23
times, which is two instances fewer than SCINet; however, NFCL maintains a
superior average ranking. Additionally, NFCL excels in scaled metrics, such as
SMAPE and $R^{2}$. As indicated in Table 2, the proposed model attains an
average ranking of 2.00 and 2.30 for SMAPE and $R^{2}$, respectively, securing
the top rank in 43 and 37 instances.

There is a discernible contrast in the performance of the models with and
without scaling, with the most recently studied baselines exhibiting
relatively superior results on scaled metrics. Informer and Autoformer
performed inadequately for short time-series forecasting in all cases. SCINet
exhibited a significant decline in performance on scaled metrics compared to
its performance before scaling, whereas PatchTST and TimesNet demonstrated the
opposite behavior. TiDE, a fully connection-based approach, outperformed D-
and N-Linear in most cases but did not surpass the performance of the other
baselines.

We also investigated the learnable parameters for NFCL-V and NFCL-C with
benchmarks, which are presented in Appendix I. According to Table 4, the
number of multivariate time-series in the dataset we tested is
$K\in\left\\{21,10,8,7\right\\}$. The number of learning parameters as a
function of the number of time-series and forecast length variation was larger
for the transformer series benchmark than for the other references. The
smallest models are LSTF-Linear, with parameters $L\cdot T$. The model with
the next smallest number of learning parameters was SCINet. The next model
that consumes more capacity than SCINet is NFCL-V, followed by NFCL-C.

###  5.7. Comparison of NFCL’s module

In this section, we provide a summary of the hyperparameter search process for
NFCL, as detailed in the preceding section. For the purpose of experiment
comparison, we computed the average ranking for each of the four metrics, as
presented in the summary of the complete experiment table among only the
variation of NFCLs, not including the references. Table 3 illustrates the best
performer election process of NFCL across the experiment pipeline. The first
row of Table 3 delineates the categories for the subsequent rows. Accordingly,
the first and second columns represent the variations of NFCL and the
structure of $h$. If you observe (-) in $h$ in Table 3, it corresponds to
NFCL-V (in case there is no $h$). Within the list in $h$, the numbers denote
the count of hidden nodes in each layer, with multiple numbers indicating
stacked layers. Notably, we employed $s_{n}\in\\{10,4,1\\}$ as the
decomposition kernel size for NFCL-D. For each decomposed time-series, NFCL-C
was configured, and the dimension of $h$ utilized in NFCL-D matches the
description in NFCL-C. Similarly, the best scores in Table 3 are presented in
bold, and the next best scores are underlined. The full tables of the NFCL’s
hyperparameter comparison are recorded in Appendix G.

The number of possible combinations of $h$ and decomposition structures in
NFCL’s experiments is vast. The experiments to optimize the structure for 15
datasets were conducted in the following order:

  1. (1)

We investigated whether NFCL-C is more effective than NFCL-V. We initially
tried $h\in\\{[64],[64,64]\\}$ and found that NFCL-C outperforms NFCL-V on all
metrics except SMAPE.

  2. (2)

We assessed whether to utilize decomposition or not to NFCL. Based on
experiments with one or two layers, we found that attempting decomposed MTSF
resulted in worse performance in all cases.

  3. (3)

We explored the depth structure of $h$ that performed best on NFCL-C. We went
as deep as possible, utilizing combinations of 64 and 128 hidden nodes.
Experiments showed that performance degraded as the number of layers
increased.

  4. (4)

We investigated the number of hidden nodes that worked best when only one
layer was utilized in NFCL-C. For one layer, we found that performance dropped
again when hidden nodes were set to 16 or less.

Table 3, on the other hand, is just an average ranking based on the tables
recorded in Appendix G. While the average ranking changes dramatically, we can
see that there is not much difference between the actual scores. According to
Appendix G, NFCL-V performs well on the ETTh1, and there are datasets and $T$
for which increasing the layer of $h$ is significant. In addition to the
average rankings in Table 3, we also looked at the number of times the model
performed best as we went through the four-step hyperparameter exploration
described above. Therefore, we use the NFCL-C as NFCL utilized in this paper
was set to utilize $h$ with one layer and the number of hidden nodes set to
32. We conducted an analysis to identify the factors contributing to the
underperformance of NFCL-D. It was observed that NFCL-D exhibited a more
pronounced overfitting phenomenon compared to other NFCL variants, likely
attributed to the application of decomposition. For further insights, please
consult Appendix J.

Variation | $h$ | MAE | MSE | SMAPE | $R^{2}$ | Avg. rank
---|---|---|---|---|---|---
NFCL-V | - | 8.42 | 9.60 | 5.27 | 9.37 | 8.16
NFCL-C | 16 | 4.57 | 5.08 | 4.45 | 5.37 | 4.87
32 | 3.93 | 4.42 | 4.27 | 4.65 | 4.32
64 | 4.13 | 4.48 | 4.28 | 4.68 | 4.40
128 | 5.15 | 5.25 | 5.62 | 5.52 | 5.38
16, 16 | 5.13 | 4.68 | 5.83 | 4.78 | 5.11
32, 32 | 5.72 | 4.90 | 5.87 | 5.18 | 5.42
64, 64 | 6.47 | 6.23 | 6.80 | 6.80 | 6.58
128, 128 | 7.38 | 6.78 | 8.12 | 7.28 | 7.39
128, 128, 64 | 7.88 | 7.05 | 9.10 | 7.50 | 7.88
NFCL-D | 64 | 10.45 | 10.85 | 10.28 | 10.12 | 10.43
128 | 11.08 | 11.35 | 10.95 | 10.73 | 11.03
64, 64 | 12.12 | 12.08 | 11.82 | 11.32 | 11.83
128, 128 | 12.57 | 12.23 | 12.35 | 11.70 | 12.21

Table 3. Hyper-parameter analysis of the number of layers and nodes according
to NFCL variations.

###  5.8. NFCL interpretation

Figure 6. Illustration of NFCL Interpretations.

After training, NFCL calculates the direct contribution for each point in $X$.
Figure 6 depicts the process of contribution calculating. The data used
originates from one of Dongsi’s test samples, with only a few portions
selected as a visualization example for the illustration. The Dongsi dataset
recorded the atmospheric sensor’s value of two inhalable Particulate Matter
(PM2.5 & PM10, where a number is a micron radius of matter) in the air,
including various climate values. Figure 6 displays the data utilized in sub-
plots a), b), and c).

In sub-plot a) of figure 6, the $X$ of the green area is used to forecast $Y$
of the red area. The continuous line with the circle marker and dotted line
with the x-marker of sub-plot a) denotes the real and predicted series. Sub-
plot b) shows the outcome of computing $w$, and $h(X)$ based on $X$, while
sub-plot c) displays the contribution attributed to $h(X)\odot w$ in
predicting the result of $y_{k,t}$. $X$ is input into the mapping function $h$
in the sub-plot b), where it undergoes re-evaluation. This result is then
combined with $w$, assigning weights to contributions from all variables,
producing the final contribution map. We execute $h(X)\odot w$ as a matrix
multiplication, equivalent to the Hadamard product between the two matrices,
denoted by the $\odot$ operator in the sub-plot b), followed by summation.
Thus, the map in the sub-plot c) depicts a scenario where only $\odot$ is
performed. By reshaping the section of $w$ highlighted by the blue or green
box in the sub-plot b), conducting $h(X)$ and $\odot$, we obtain the
corresponding figure in the same color in the sub-plot c). This NFCL
interpretation process can be applied to all input variables to forecast a
variable at a future point in time. Specifically, the weight map for the
output layer $w$ is excessively large and is not detailed in main context but
is available in Appendix H. The width-broad, and height-narrow blue box region
in Figure 8, as illustrated in Appendix H, represents
$w\left[:,:,3,\mathrm{NO2}\right]$.

Examining the forecast for NO2 three hours into the future, although a more
diverse pattern than PM2.5 and PM10 are not identified, it is observed that
NO2 is autocorrelated and contributed to by factors such as TEMP, DEWP, and
WSPM. Notably, the input consists of 24 points, representing a day in hourly
data. Consequently, the initial part of the historical time-series aligns with
the same time as the first point in $Y$, indicating a robust day-to-day
periodicity, particularly in climate-related data.

###  5.9. Analysis contributions

Figure 7 presents the results of executing the two examples in Figure 8 for
all $w$ in Appendix H. For each subplot, a left-to-right shift on the X-axis
indicates an increase in the historical $L$, and a top-to-bottom shift on the
Y-axis indicates an increase in the forecasting $T$. The low contribution
$h(x)\odot w$ is shown in grayscale, and we applied a similar approach to the
other plots (we used grayscale because using a separate color depth scale for
each plot would not allow us to accurately perceive the global contribution to
a particular prediction).

The upper green-edged subplot illustrates the association of particulate
matter with SO2 and NO2. Examining the summation of contributions, PM2.5
receives a larger contribution from SO2 than other variables and a smaller
contribution from NO2. Notably, In China, research results have shown a strong
correlation between PM2.5 and SO2, while NO2 exhibited inconsistent patterns,
with correlations depending on the region (Li and Bai, 2019). The
autocorrelation and cross-correlation for O3, temperature, and barometric
pressure forecasts are visible in the red-outlined contribution map in the
lower red-edged subplot. Indeed, O3 shows a strong correlation with surface
air temperature (Porter and Heald, 2019), and depending on the terrain, wind
can further enhance the O3–temperature correlation (Yoshikado, 2023).
Visualizing the contributions across time-series can be an important element
in laying the groundwork for future applications and use of NFCLs by ML Non-
Experts in other fields, not just the one to which this study was applied.

Figure 7. Interpretation example.

##  6\. Conclusion

We discuss various variants of NFCL in our paper. Essentially, NFCL provides a
method to leverage auto-correlation and cross-correlation for MTSF. Our
findings indicate that re-evaluating the input time-series with a mapping
function $h$ can lead to a slight improvement in model performance.
Simultaneously, the structural exploration of NFCL-C reveals that a simple
model is adequate for reevaluating historical time-series. We also explored
ensembles of NFCL with decomposition to enhance the performance fundamentally.
However, the performance of NFCL-D is worse than simple NFCL connection.
Moreover, NFCL demonstrates the possibility of interpreting the input-output
structure. It shows that NFCL prevents that even if one comprehends the
reasoning behind a prediction, it may become meaningless on real-world if the
rationale is inaccurate.

NFCL remains imperfect. The learnable parameter of $h$ for independently
processing all values of $w$ and $X$ is directly proportional to the size of
the time-series (precisely proportional to the length $L$ of the time-series
and the number of time-series $K$). The complexity of $w$ also increases
depending on not only $K$ and $L$, but also forecasting length $T$. To enlarge
the output layer weight $w$ for huge sizes of time-series can lead to
misunderstandings or hinder comprehension of the forecasting reasons.

On the other hand, we point to the iterative structure of the algorithm to
ensure an independent $h$ as one of the reasons for NFCL’s poor computational
speedup relative to its parameters. Aside from parallelizing the iterative
algorithm, there is no solution to replace iterative algorithms yet. In the
future, our focus will be on strategies to reduce the computational complexity
of NFCL. Additionally, we aim to transform NFCL into a simple web application,
ensuring accessibility to a broad audience, including those who are not
experts in artificial intelligence.

## References

  * (1)
  * Agarwal et al. (2021) Rishabh Agarwal, Levi Melnick, Nicholas Frosst, Xuezhou Zhang, Ben Lengerich, Rich Caruana, and Geoffrey E Hinton. 2021.  Neural additive models: Interpretable machine learning with neural nets.  _Advances in Neural Information Processing Systems_ 34 (2021).
  * Alzubaidi et al. (2021) Laith Alzubaidi, Jinglan Zhang, Amjad J. Humaidi, Ayad Al-Dujaili, Ye Duan, Omran Al-Shamma, J. Santamaría, Mohammed A. Fadhel, Muthana Al-Amidie, and Laith Farhan. 2021.  Review of deep learning: concepts, CNN architectures, challenges, applications, future directions.  _Journal of Big Data_ 8, 1 (31 Mar 2021), 53.  https://doi.org/10.1186/s40537-021-00444-8
  * Bento et al. (2021) João Bento, Pedro Saleiro, André F. Cruz, Mário A.T. Figueiredo, and Pedro Bizarro. 2021.  TimeSHAP: Explaining Recurrent Models through Sequence Perturbations. In _Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining_ (Virtual Event, Singapore) _(KDD ’21)_. Association for Computing Machinery, New York, NY, USA, 2565–2573.  https://doi.org/10.1145/3447548.3467166
  * Chen et al. (2017) Chao Chen, Jamie Twycross, and Jonathan M Garibaldi. 2017.  A new accuracy measure based on bounded relative error for time series forecasting.  _PLoS One_ 12, 3 (March 2017), e0174202.
  * Chen (2019) Song Chen. 2019.  Beijing Multi-Site Air-Quality Data.  UCI Machine Learning Repository.  DOI: https://doi.org/10.24432/C5RK5G.
  * Chen et al. (2023) Zonglei Chen, Minbo Ma, Tianrui Li, Hongjun Wang, and Chongshou Li. 2023.  Long sequence time-series forecasting with deep learning: A survey.  _Information Fusion_ 97 (2023), 101819.  https://doi.org/10.1016/j.inffus.2023.101819
  * Chollet (2016) François Chollet. 2016.  Xception: Deep Learning with Depthwise Separable Convolutions.  _2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ (2016), 1800–1807.
  * Cooley and Tukey (1965) James W. Cooley and John W. Tukey. 1965.  An Algorithm for the Machine Calculation of Complex Fourier Series.  _Math. Comp._ 19, 90 (1965), 297–301.
  * Das et al. (2023) Abhimanyu Das, Weihao Kong, Andrew Leach, Shaan K Mathur, Rajat Sen, and Rose Yu. 2023.  Long-term Forecasting with TiDE: Time-series Dense Encoder.  _Transactions on Machine Learning Research_ (2023).
  * Dehghani et al. (2023) Mostafa Dehghani, Josip Djolonga, Basil Mustafa, Piotr Padlewski, Jonathan Heek, Justin Gilmer, Andreas Peter Steiner, Mathilde Caron, Robert Geirhos, Ibrahim Alabdulmohsin, Rodolphe Jenatton, Lucas Beyer, Michael Tschannen, Anurag Arnab, Xiao Wang, Carlos Riquelme Ruiz, Matthias Minderer, Joan Puigcerver, Utku Evci, Manoj Kumar, Sjoerd Van Steenkiste, Gamaleldin Fathy Elsayed, Aravindh Mahendran, Fisher Yu, Avital Oliver, Fantine Huot, Jasmijn Bastings, Mark Collier, Alexey A. Gritsenko, Vighnesh Birodkar, Cristina Nader Vasconcelos, Yi Tay, Thomas Mensink, Alexander Kolesnikov, Filip Pavetic, Dustin Tran, Thomas Kipf, Mario Lucic, Xiaohua Zhai, Daniel Keysers, Jeremiah J. Harmsen, and Neil Houlsby. 2023.  Scaling Vision Transformers to 22 Billion Parameters. In _Proceedings of the 40th International Conference on Machine Learning_ _(Proceedings of Machine Learning Research, Vol. 202)_ , Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (Eds.). PMLR, 7480–7512.
  * Dosovitskiy et al. (2021) Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. 2021.  An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale. In _International Conference on Learning Representations_.
  * Ho and Xie (1998) S.L. Ho and M. Xie. 1998.  The use of ARIMA models for reliability forecasting and analysis.  _Computers & Industrial Engineering_ 35, 1 (1998), 213–216.  https://doi.org/10.1016/S0360-8352(98)00066-7
  * Ke et al. (2017) Jintao Ke, Hongyu Zheng, Hai Yang, and Xiqun (Michael) Chen. 2017.  Short-term forecasting of passenger demand under on-demand ride services: A spatio-temporal deep learning approach.  _Transportation Research Part C: Emerging Technologies_ 85 (2017), 591–608.  https://doi.org/10.1016/j.trc.2017.10.016
  * Kim et al. (2022) Bu-Yo Kim, Yun-Kyu Lim, and Joo Wan Cha. 2022.  Short-term prediction of particulate matter (PM10 and PM2.5) in Seoul, South Korea using tree-based machine learning algorithms.  _Atmospheric Pollution Research_ 13, 10 (2022), 101547.  https://doi.org/10.1016/j.apr.2022.101547
  * Lai et al. (2018) Guokun Lai, Wei-Cheng Chang, Yiming Yang, and Hanxiao Liu. 2018.  Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks. In _The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval_ (Ann Arbor, MI, USA) _(SIGIR ’18)_. Association for Computing Machinery, New York, NY, USA, 95–104.  https://doi.org/10.1145/3209978.3210006
  * Lea et al. (2016) Colin Lea, René Vidal, Austin Reiter, and Gregory D. Hager. 2016.  Temporal Convolutional Networks: A Unified Approach to Action Segmentation. In _Computer Vision – ECCV 2016 Workshops_ , Gang Hua and Hervé Jégou (Eds.). Springer International Publishing, Cham, 47–54.
  * Li and Bai (2019) Ke Li and Kaixu Bai. 2019.  Spatiotemporal Associations between PM(2.5) and SO(2) as well as NO(2) in China from 2015 to 2018.  _Int J Environ Res Public Health_ 16, 13 (July 2019).
  * LIU et al. (2022) Minhao LIU, Ailing Zeng, Muxi Chen, Zhijian Xu, Qiuxia LAI, Lingna Ma, and Qiang Xu. 2022.  SCINet: Time Series Modeling and Forecasting with Sample Convolution and Interaction. In _Advances in Neural Information Processing Systems_ , S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Eds.), Vol. 35. Curran Associates, Inc., 5816–5828.
  * Liu et al. (2018) Song Liu, Yaming Cui, Yaze Ma, and Peng Liu. 2018.  Short-term Load Forecasting Based on GBDT Combinatorial Optimization. In _2018 2nd IEEE Conference on Energy Internet and Energy System Integration (EI2)_. 1–5.  https://doi.org/10.1109/EI2.2018.8582108
  * Liu et al. (2022) Shizhan Liu, Hang Yu, Cong Liao, Jianguo Li, Weiyao Lin, Alex X Liu, and Schahram Dustdar. 2022.  Pyraformer: Low-Complexity Pyramidal Attention for Long-Range Time Series Modeling and Forecasting. In _International Conference on Learning Representations_.
  * Liu et al. (2024) Yong Liu, Tengge Hu, Haoran Zhang, Haixu Wu, Shiyu Wang, Lintao Ma, and Mingsheng Long. 2024.  iTransformer: Inverted Transformers Are Effective for Time Series Forecasting. In _The Twelfth International Conference on Learning Representations_.
  * Loshchilov and Hutter (2019) Ilya Loshchilov and Frank Hutter. 2019.  Decoupled Weight Decay Regularization. In _International Conference on Learning Representations_.
  * Lundberg and Lee (2017) Scott M Lundberg and Su-In Lee. 2017.  A Unified Approach to Interpreting Model Predictions.  In _Advances in Neural Information Processing Systems 30_ , I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (Eds.). Curran Associates, Inc., 4765–4774.
  * Melo et al. (2022) João Victor Jales Melo, George Rossany Soares Lira, Edson Guedes Costa, Antonio F. Leite Neto, and Iago B. Oliveira. 2022.  Short-Term Load Forecasting on Individual Consumers.  _Energies_ 15, 16 (2022).  https://doi.org/10.3390/en15165856
  * Mounir et al. (2023) Nada Mounir, Hamid Ouadi, and Ismael Jrhilifa. 2023.  Short-term electric load forecasting using an EMD-BI-LSTM approach for smart grid energy management system.  _Energy and Buildings_ 288 (2023), 113022.  https://doi.org/10.1016/j.enbuild.2023.113022
  * Nie et al. (2023) Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. 2023.  A Time Series is Worth 64 Words: Long-term Forecasting with Transformers. In _International Conference on Learning Representations_.
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. 2019.  _PyTorch: an imperative style, high-performance deep learning library_.  Curran Associates Inc., Red Hook, NY, USA.
  * Porter and Heald (2019) W. C. Porter and C. L. Heald. 2019.  The mechanisms and meteorological drivers of the summertime ozone–temperature relationship.  _Atmospheric Chemistry and Physics_ 19, 21 (2019), 13367–13381.  https://doi.org/10.5194/acp-19-13367-2019
  * Radenovic et al. (2022) Filip Radenovic, Abhimanyu Dubey, and Dhruv Mahajan. 2022.  Neural Basis Models for Interpretability.  _arXiv:2205.14120_ (2022).
  * Rojat et al. (2021) Thomas Rojat, Raphaël Puget, David Filliat, Javier Del Ser, Rodolphe Gelin, and Natalia Díaz Rodríguez. 2021.  Explainable Artificial Intelligence (XAI) on TimeSeries Data: A Survey.  _CoRR_ abs/2104.00950 (2021).  arXiv:2104.00950
  * Sarker (2021) Iqbal H. Sarker. 2021.  Deep Learning: A Comprehensive Overview on Techniques, Taxonomy, Applications and Research Directions.  _SN Computer Science_ 2, 6 (18 Aug 2021), 420.  https://doi.org/10.1007/s42979-021-00815-1
  * Sun and You (2021) Li Sun and Fengqi You. 2021.  Machine Learning and Data-Driven Techniques for the Control of Smart Power Generation Systems: An Uncertainty Handling Perspective.  _Engineering_ 7, 9 (2021), 1239–1247.  https://doi.org/10.1016/j.eng.2021.04.020
  * Tan and Le (2021) Mingxing Tan and Quoc Le. 2021.  EfficientNetV2: Smaller Models and Faster Training. In _Proceedings of the 38th International Conference on Machine Learning_ _(Proceedings of Machine Learning Research, Vol. 139)_ , Marina Meila and Tong Zhang (Eds.). PMLR, 10096–10106.
  * Theissler et al. (2022) Andreas Theissler, Francesco Spinnato, Udo Schlegel, and Riccardo Guidotti. 2022.  Explainable AI for Time Series Classification: A Review, Taxonomy and Research Directions.  _IEEE Access_ 10 (2022), 100700–100724.  https://doi.org/10.1109/ACCESS.2022.3207765
  * Tu (1996) J V Tu. 1996.  Advantages and disadvantages of using artificial neural networks versus logistic regression for predicting medical outcomes.  _J Clin Epidemiol_ 49, 11 (Nov. 1996), 1225–1231.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is All You Need. In _Proceedings of the 31st International Conference on Neural Information Processing Systems_ (Long Beach, California, USA) _(NIPS’17)_. Curran Associates Inc., Red Hook, NY, USA, 6000–6010.
  * Veerappa et al. (2022) Manjunatha Veerappa, Mathias Anneken, Nadia Burkart, and Marco F. Huber. 2022.  Validation of XAI explanations for multivariate time series classification in the maritime domain.  _Journal of Computational Science_ 58 (2022), 101539.  https://doi.org/10.1016/j.jocs.2021.101539
  * Venkatachalam et al. (2023) K. Venkatachalam, Pavel Trojovský, Dragan Pamucar, Nebojsa Bacanin, and Vladimir Simic. 2023.  DWFH: An improved data-driven deep weather forecasting hybrid model using Transductive Long Short Term Memory (T-LSTM).  _Expert Systems with Applications_ 213 (2023), 119270.  https://doi.org/10.1016/j.eswa.2022.119270
  * Wu et al. (2023) Haixu Wu, Tengge Hu, Yong Liu, Hang Zhou, Jianmin Wang, and Mingsheng Long. 2023.  TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis. In _International Conference on Learning Representations_.
  * Wu et al. (2021) Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long. 2021.  Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting. In _Advances in Neural Information Processing Systems_.
  * Xu et al. (2019) Jingjing Xu, Xu Sun, Zhiyuan Zhang, Guangxiang Zhao, and Junyang Lin. 2019.  Understanding and Improving Layer Normalization. In _Advances in Neural Information Processing Systems_ , H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett (Eds.), Vol. 32. Curran Associates, Inc.
  * Yoshikado (2023) Hiroshi Yoshikado. 2023.  Correlation between air temperature and surface ozone in their extreme ranges in the greater Tokyo region.  _Asian Journal of Atmospheric Environment_ 17, 1 (Aug. 2023), 9.
  * Zeng et al. (2023) Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu. 2023.  Are Transformers Effective for Time Series Forecasting?  _Proceedings of the AAAI Conference on Artificial Intelligence_.
  * Zhang et al. (2024) Chu Zhang, Yuhan Wang, Yongyan Fu, Xiujie Qiao, Muhammad Shahzad Nazir, and Tian Peng. 2024.  A novel DWTimesNet-based short-term multi-step wind power forecasting model using feature selection and auto-tuning methods.  _Energy Conversion and Management_ 301 (2024), 118045.  https://doi.org/10.1016/j.enconman.2023.118045
  * Zhou et al. (2020) Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wan Zhang. 2020.  Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting. In _AAAI Conference on Artificial Intelligence_.
  * Zhou et al. (2022) Tian Zhou, Ziqing Ma, Qingsong Wen, Xue Wang, Liang Sun, and Rong Jin. 2022.  FEDformer: Frequency enhanced decomposed transformer for long-term series forecasting. In _Proc. 39th International Conference on Machine Learning (ICML 2022)_ (Baltimore, Maryland).

## Appendix

##  Appendix A NFCL-C implementation

Algorithm 1 Iterative re-evaluation for complex NFCL

$X\in\mathbb{R}^{K,L}$, $H=[[h^{1,1},h^{1,2},\cdots h^{1,L}]\times K]$

$X_{h}\in\mathbb{R}^{K,L}$

$X_{h}\leftarrow$ copy$(X)$.zero()$\triangleright$ the zero array has same
shape as $X$.

for $k$ in $1,2,\cdots,K$ do

for $l$ in $1,2,\cdots,L$ do

$h^{k,l}\leftarrow H[k][l]$$\triangleright$ $h^{k,l}$ is the non-linear
mapping function.

$X_{h}[k][l]\leftarrow h^{k,l}(X[k][l])$

end for

end for

Algorithm 2 Group convolution for complex NFCL

$X\in\mathbb{R}^{K,L}$, $H=[[h^{1,1},h^{1,2},\cdots h^{1,L}]\times K]$

$C=[c_{1},\cdots,c_{N}]$, $\gamma=K\cdot L$ $\triangleright$ argument $\gamma$
is group size.

$X_{h}\in\mathbb{R}^{K,L}$

$X_{h}\leftarrow\textrm{Flatten}(X)\in\mathbb{R}^{K\cdot L,1}$

$C_{in}\leftarrow[1]+[c_{1},\cdots,c_{N-1}]$ ;
$C_{out}\leftarrow[c_{1},\cdots,c_{N}]$

for $c_{in},c_{out}\ \textbf{in}\ C_{in},C_{out}$ do

$X_{h}\leftarrow Conv(X_{h},c_{i}\cdot\gamma,c_{o}\cdot\gamma,\gamma)$

$X_{h}\leftarrow\mathrm{activation}(X_{h})$

end for

$X_{h}\leftarrow Conv(X_{h},c_{N}\cdot\gamma,1\cdot\gamma,\gamma)$

NFCL-V computes predictions for all future time-series points by multiplying
weights from all past time-series points. NFCL-C, on the other hand, involves
a simple process: before multiplying the weights of all past time-series
points, each individual time-series point undergoes re-evaluation by a
trainable NNs. To uphold our proposed concept, it’s crucial that the NN
responsible for each time-series point remains unaffected by other points.
Consequently, the Equation 2 is satisfied by the below Algorithm 1 for NFCL-C.
In the Algorithm 1, $H$ denotes the $K\cdot L$ number of mapping function $h$,
and $x_{h}$ is re-evaluated value by each $h$. As a result of the Algorithm,
we obtain $X_{h}$ a re-evaluation of the input, which can substitute $X$ in
Equation 1. If we substitute $h^{i,j}\left(x^{i,j}\right)$ in Equation 2 with
$X_{h}$, we observe that Equations 1 and 2 are fully compatible.

To enhance the efficiency of the Algorithm during double iterations, we hired
the group convolution. Widely employed in image processing, the group
convolution partitions the input channel into multiple groups, establishing
independent CNN to execute convolutional operations exclusively on these
groups. Specifically, when the size of the image region matches the number of
groups, a CNN is independently created for each pixel in every image to carry
out the operation, and this is called the point-wise convolution. We leverage
this approach to construct a re-evaluation module $H$ that independently
operates at each time point for every time-series variable. In the Algorithm
2, the parameter of the function
$Conv(X_{h},c_{i}\cdot\gamma,c_{o}\cdot\gamma,\gamma)$ sequentially denotes
the input tensor, the number of input tensor’s channel, the number of output
tensor’s channels, and the group size, respectively. Since we set the same
group size $\gamma$ to the number of the first input tensor’s channel, we can
easily implement the point-wise convolution.

##  Appendix B Decomposition algorithm

Algorithm 3 Decomposition for the time-series

$X\in\mathbb{R}^{K,L},S={s_{1},\cdots,s_{N}}$

$X_{dec}\in\mathbb{R}^{K,L}$

$X_{dec}\leftarrow[]$

while $S\neq\varnothing$ do

$s_{n}\leftarrow S\mathrm{.pop()}$

$X_{pad}\leftarrow\mathrm{LeftPad}(X,s_{n}-1)$

$X_{s_{n}}\leftarrow\mathrm{AveragePool}(X_{pad},s_{n})$

$X_{dec}\mathrm{.append}(X_{s_{n}})$

$X\leftarrow X-X_{s_{n}}$

end while

Algorithm 4 Workflow of NFCL-D

$X_{dec}\in\mathbb{R}^{K,L}$,
$M=[\mathrm{NFCL-C}_{s_{1}},\cdots,\mathrm{NFCL-C}_{s_{N}}]$

$\tilde{Y}\in\mathbb{R}^{K,T}$

$\hat{Y}_{dec}\leftarrow[]$

while $M\neq\varnothing\mathrm{\ and\ }X_{dec}\neq\varnothing$ do

$m_{s_{n}}\leftarrow M\mathrm{.pop()}$

$X_{s_{n}}\in\mathbb{R}^{K,L}\leftarrow X_{dec}\mathrm{.pop()}$

$\hat{Y}_{s_{n}}\leftarrow m_{s_{n}}\left(X_{s_{n}}\right)$

$\hat{Y}_{dec}\mathrm{.append}(\hat{Y}_{s_{n}})$

end while

$\tilde{Y}\leftarrow\mathrm{Sum}\left(\mathrm{Concat}\left(\hat{Y}_{dec}\right)\right)$

We describe the process of NFCL-D. Algorithms 3 and 4 provide the sequences of
the pipeline. Algorithm 3 depicts the decomposition process. And Algorithm 4
displays how to forecast the future using NFCL-D. Since we employed
$\left|S\right|$ times NFCL-Cs, the number of NFCL-D’s parameters depends on
$\left|S\right|$.

In the Algorithm 3, the LeftPad function copies the first time value in the
time-series and pads it to the left of the given $X$ by the length specified
as the second argument $s_{n}-1$. Similarly, the AveragePool function rolls
and averages the $X$ using the kernel size $s_{n}$. In the Algorithm 4, the
Sum and Concat functions perform as the Python general function way, such as
NumPy. Two functions focused to sum between the $\hat{Y}_{dec}$ that used
different kernel sizes. Consequently, we can forecast the $\hat{Y}$ to recover
the scale of $\tilde{Y}$.

Category | Dataset | Abbreviation | Freq. | # of var. | # of Train | # of Val. | # of Test.
---|---|---|---|---|---|---|---
Patient | National illness | ILL | 7 days | 7 | 532 | 171 | 170
Currency exchange | Exchange | Exchange | 1 day | 8 | 4505 | 1496 | 1494
Electricity | ETTh1 | ETTh1 | 1 hour | 7 | 10405 | 3461 | 3461
ETTh2 | ETTh2
ETTm1 | ETTm1 | 15 min. | 7 | 41761 | 13913 | 13913
ETTm2 | ETTm2
Climate | Aotizhongxin | Aotiz. | 1 hour | 10 | 20991 | 6991 | 6989
Changping | Chang.
Dingling | Ding.
Dongsi | Dongsi
Guanyuan | Guan.
Gucheng | Guch.
Huairou | Huai.
Nongzhanguan | Nong.
Weather | Weather | 10 min. | 21 | 31570 | 10517 | 10516
Table 4. Details of datasets.

##  Appendix C Detail of datasets

We used the online-available open datasets from the UCI-repository, and
GitHubs. Autoformer111https://github.com/thuml/Autoformer/tree/main aligned
some datasets on their repo: Weather, National illness, and Currency Exchange.
In particular, three datasets that Autoformer provided applied underwent
minimal preprocessing. We got the others, electricity transfer temperature and
Beijing air quality, from its
GitHub222https://github.com/zhouhaoyi/ETDataset/tree/main/ETT-small or UCI-
repository333https://archive.ics.uci.edu/dataset/501/beijing+multi+site+air+quality+data.
The details of datasets are Table 4, and briefly summaries are as follow:

  * •

Weather444https://www.bgc-jena.mpg.de/wetter/ dataset has 21 climate records,
such as temperature, rainfall, and etc. This dataset had collected every 10
minute on 2020 year.

  * •

National illness555https://gis.cdc.gov/grasp/fluview/fluportaldashboard.html
dataset was collected by Centers for Disease Control and Prevention of the
United States between 2002 to 2020. This dataset targeted to the influenza-
like illness patients. Note that we use the abbreviation as ILL.

  * •

Exchange dataset records the daily currency exchange rates of eight nations
(Australia, British, Canada, Switzerland, China, Japan, NewZealand, and
Singapore) from 1990 to 2010 (Lai et al., 2018).

  * •

Note that we use the abbreviation of the electricity transfer temperature as
ETT. ETT recorded the oil temperature and the electricity’s loads, it was
recorded every 15 minutes or hourly between 2016 to 2018. Also, ETT has four
sub-dataset, becuase ETT was collected from two different provinces and has
different acquisition frequency. A number of ETT indicates the province as ”1”
or ”2”. Thus, ”m” and ”h” denotes the frequency of ETT collected.

  * •

Beijing air quality is the record of subdistrict’s climates between 2013–2017.
In contrast to the weather dataset, it also forcused on the two different
sizes’ inhalable particulate matters. In preprocessing, we drop the time-
series variables; the wind direction and rainfall.

##  Appendix D Metrics

We held the manner of Informer experiments for MTSF (Zhou et al., 2020). There
are two manners; one is the sampling the dataset with the one step sliding and
the other is the method how to evaluate the MTSF performance. The evaluation
process performed on zero-mean scaled time-series variables. Kept the
notation, we compute the each metrics as,

  1. (1)

MAE

$MAE(Y^{\prime},\hat{Y}^{\prime})=\frac{1}{|D^{\prime}|\cdot
K}\sum_{n=1}^{|D^{\prime}|}\sum_{k=1}^{K}\left|y_{n}^{k,:}-\hat{y}_{n}^{k,:}\right|$

  2. (2)

MSE

$MAE(Y^{\prime},\hat{Y}^{\prime})=\frac{1}{|D^{\prime}|\cdot
K}\sum_{n=1}^{|D^{\prime}|}\sum_{k=1}^{K}\left(y_{n}^{k,:}-\hat{y}_{n}^{k,:}\right)^{2}$

  3. (3)

SMAPE

$SMAPE(Y^{\prime},\hat{Y}^{\prime})=\frac{100\%}{|D^{\prime}|\cdot
K}\sum_{n=1}^{|D^{\prime}|}\sum_{k=1}^{K}\frac{\left|y_{n}^{k,:}-\hat{y}_{n}^{k,:}\right|}{\left|y_{n}^{k,:}\right|+\left|\hat{y}_{n}^{k,:}\right|}$

  4. (4)

$R^{2}$

$R^{2}(Y^{\prime},\hat{Y}^{\prime})=1-\frac{\sum_{n=1}^{|D^{\prime}|}{\sum_{k=1}^{K}{(y_{n}^{k,:}-\hat{y}_{n}^{k,:})^{2}}}}{\sum_{n=1}^{|D^{\prime}|}{\sum_{k=1}^{K}{(y_{n}^{k,:}-\bar{y}^{k,:})^{2}}}}$

where, $\bar{Y}^{\prime}$ and $|D^{\prime}|$ denote the average of test
$Y^{\prime}$ and the number of samples in $|D^{\prime}|$.

##  Appendix E hyperparameter

Basically, we applied to the baseline’s hyperparameter based on the PatchTST
experiments666https://github.com/yuqinie98/PatchTST/tree/main/PatchTST_supervised/models.
Since their Github provides seven baselines (DLinear, NLinear, Informer,
Autoformer, PatchTST, Linear, and Transformer), we conducted the experiments
to use their baselines. However, the Transformer and simple-Linear layer
showed too much degraded performance, so we excluded the summary of the
scores. Moreover, these baselines didn’t employ the CNN-based model, we
referred to the official GitHub of SCINet777https://github.com/cure-
lab/SCINet, TimesNet888https://github.com/thuml/Time-Series-Library, and
iTransformer999https://github.com/thuml/iTransformer. We also tried to employ
the MICN, and TiDE from the GitHub that also shared the TimesNet baseline.
While the TiDE shows competitive performance, MICN implementation didn’t work.
So, we included only the TiDE in our comparisons. In addition, we double-check
the hyperparameter setting from their papers. The table 5 records the full
setting. And, a brief abstraction of each baseline’s properties is provided
below:

  * •

LSTF-Linear (Zeng et al., 2023) proposed two baselines; where letters D and N
denote decomposition and normalization, respectively, with fully-connected
weights for univariate time-series.

  * •

TiDE (Das et al., 2023) introduced an encoder-decoder forecasting approach
based on linear combination stacking.

  * •

Informer (Zhou et al., 2020) serves as a fundamental method utilizing the
transformer architecture. Informer focuses on reducing computational costs and
provides an open dataset.

  * •

Autoformer (Wu et al., 2021) is composed of three attention blocks for
encoding, trend decoding, and seasonal decoding. To effectively capture auto-
correlation, it employs Fourier transformation and aggregation modules in each
block.

  * •

PatchTST (Nie et al., 2023) is a transformer-based method forecasting channels
independently. Additionally, it incorporates a patching process to enhance
forecasting performance.

  * •

SCINet (LIU et al., 2022) remains a state-of-the-art model in electricity
temperature transfer datasets (Zhou et al., 2020). It forecasts the future
based on sample convolution and interaction blocks that function similarly to
the Fourier transform (Cooley and Tukey, 1965).

  * •

TimesNet (Wu et al., 2023) utilizes CNN and the Fourier transform to extract
information from the frequency-amplitude map.

  * •

iTransformer (Liu et al., 2024) simply applies the attention and feed-forward
network on the transposed dimensions of time and variables. The attention
mechanism captured multivariate correlations instead of the time-step
correlations.

Model | Params. | Setting | Model | Params. | Setting
---|---|---|---|---|---
TimesNet | top_k | 5 | SCINet | hiddens | 1
kernel_size | 6 | kernel_size | 5
d_ff | 16 | dropout | 0.5
d_ff | 64 | group | 1
Former | moving_avg | 25 | PatchTST | fc_dropout | 0.05
factor | 1 | head_dropout | 0
dropout | 0.05 | patch_len | 16
embed | timeF | stride | 8
activation | gelu | padding_patch | end
embed_type | 0 | subtract_last | 0
d_model | 512 | decomposition | 0
n_heads | 8 | kernel_size | 25
e_layers | 2 | individual | 0
d_layers | 1 |  |
d_ff | 2048 |  |

Table 5. hyperparameter setting.

##  Appendix F NFCL Inter-comparison

We only included the rankings between nine benchmarks and NFCL-C with $h=32$
in the main content of this section. To present our complete comparisons, we
have added an additional section with long tables. The title of each table
indicates the evaluation metric. These tables are summarized in Table 2 in
Section 5.6.

Dataset | $T$ | NFCL | DLinear | NLinear | TiDE | Informer | Autoformer | PatchTST | iTransformer | SCINet | TimesNet
---|---|---|---|---|---|---|---|---|---|---|---
| 6 | 0.070 | 0.099 | 0.080 | 0.084 | 0.187 | 0.146 | 0.072 | 0.105 | 0.089 | 0.082
| 12 | 0.090 | 0.131 | 0.106 | 0.108 | 0.236 | 0.175 | 0.096 | 0.128 | 0.115 | 0.102
| 18 | 0.108 | 0.160 | 0.129 | 0.128 | 0.296 | 0.200 | 0.116 | 0.148 | 0.143 | 0.126
Weather | 24 | 0.125 | 0.186 | 0.148 | 0.146 | 0.412 | 0.223 | 0.136 | 0.162 | 0.169 | 0.146
| 6 | 0.899 | 0.720 | 0.647 | 0.634 | 1.212 | 1.028 | 0.630 | 0.542 | 0.840 | 0.615
| 12 | 1.181 | 1.101 | 1.016 | 0.973 | 1.299 | 1.078 | 0.932 | 0.726 | 1.089 | 0.805
| 18 | 1.322 | 1.322 | 1.253 | 1.176 | 1.437 | 1.223 | 1.053 | 0.883 | 1.266 | 0.927
ILL | 24 | 1.444 | 1.418 | 1.371 | 1.282 | 1.502 | 1.239 | 1.140 | 0.975 | 1.383 | 1.109
| 6 | 0.064 | 0.060 | 0.060 | 0.060 | 0.633 | 0.097 | 0.061 | 0.063 | 0.161 | 0.072
| 12 | 0.088 | 0.083 | 0.083 | 0.083 | 0.714 | 0.113 | 0.084 | 0.087 | 0.202 | 0.096
| 18 | 0.106 | 0.101 | 0.101 | 0.101 | 0.783 | 0.132 | 0.101 | 0.104 | 0.258 | 0.114
Exchange | 24 | 0.122 | 0.116 | 0.115 | 0.115 | 0.914 | 0.145 | 0.115 | 0.120 | 0.265 | 0.128
| 6 | 0.368 | 0.357 | 0.358 | 0.358 | 0.439 | 0.420 | 0.348 | 0.354 | 0.371 | 0.372
| 12 | 0.418 | 0.385 | 0.387 | 0.388 | 0.513 | 0.460 | 0.392 | 0.398 | 0.422 | 0.421
| 18 | 0.438 | 0.394 | 0.396 | 0.397 | 0.520 | 0.470 | 0.401 | 0.411 | 0.437 | 0.442
ETTh1 | 24 | 0.438 | 0.397 | 0.400 | 0.401 | 0.530 | 0.487 | 0.407 | 0.416 | 0.444 | 0.447
| 6 | 0.212 | 0.226 | 0.225 | 0.226 | 0.309 | 0.249 | 0.218 | 0.226 | 0.224 | 0.218
| 12 | 0.239 | 0.254 | 0.251 | 0.253 | 0.363 | 0.270 | 0.249 | 0.258 | 0.262 | 0.246
| 18 | 0.253 | 0.266 | 0.262 | 0.264 | 0.356 | 0.281 | 0.262 | 0.271 | 0.281 | 0.258
ETTh2 | 24 | 0.261 | 0.271 | 0.268 | 0.269 | 0.369 | 0.289 | 0.268 | 0.281 | 0.308 | 0.269
| 6 | 0.239 | 0.256 | 0.256 | 0.257 | 0.284 | 0.342 | 0.244 | 0.254 | 0.234 | 0.241
| 12 | 0.325 | 0.356 | 0.361 | 0.361 | 0.364 | 0.416 | 0.331 | 0.351 | 0.307 | 0.319
| 18 | 0.397 | 0.439 | 0.453 | 0.452 | 0.419 | 0.476 | 0.413 | 0.437 | 0.369 | 0.379
ETTm1 | 24 | 0.459 | 0.507 | 0.532 | 0.527 | 0.477 | 0.556 | 0.478 | 0.506 | 0.413 | 0.435
| 6 | 0.148 | 0.159 | 0.160 | 0.159 | 0.221 | 0.184 | 0.153 | 0.158 | 0.161 | 0.153
| 12 | 0.176 | 0.195 | 0.196 | 0.194 | 0.260 | 0.209 | 0.186 | 0.191 | 0.191 | 0.181
| 18 | 0.198 | 0.222 | 0.223 | 0.222 | 0.268 | 0.231 | 0.208 | 0.217 | 0.216 | 0.201
ETTm2 | 24 | 0.216 | 0.244 | 0.245 | 0.244 | 0.302 | 0.242 | 0.228 | 0.246 | 0.233 | 0.221
| 6 | 0.260 | 0.301 | 0.293 | 0.292 | 0.313 | 0.343 | 0.281 | 0.284 | 0.268 | 0.279
| 12 | 0.327 | 0.383 | 0.373 | 0.371 | 0.393 | 0.404 | 0.357 | 0.359 | 0.347 | 0.345
| 18 | 0.369 | 0.430 | 0.416 | 0.414 | 0.445 | 0.451 | 0.400 | 0.403 | 0.388 | 0.385
Aotiz. | 24 | 0.399 | 0.457 | 0.442 | 0.440 | 0.468 | 0.477 | 0.430 | 0.433 | 0.419 | 0.415
| 6 | 0.261 | 0.302 | 0.295 | 0.294 | 0.299 | 0.337 | 0.282 | 0.281 | 0.261 | 0.277
| 12 | 0.320 | 0.373 | 0.363 | 0.360 | 0.363 | 0.395 | 0.346 | 0.347 | 0.326 | 0.336
| 18 | 0.357 | 0.412 | 0.401 | 0.398 | 0.412 | 0.437 | 0.384 | 0.386 | 0.364 | 0.378
Chang. | 24 | 0.385 | 0.438 | 0.424 | 0.421 | 0.422 | 0.456 | 0.410 | 0.412 | 0.391 | 0.403
| 6 | 0.238 | 0.276 | 0.266 | 0.265 | 0.300 | 0.336 | 0.253 | 0.256 | 0.248 | 0.259
| 12 | 0.305 | 0.360 | 0.345 | 0.343 | 0.371 | 0.398 | 0.327 | 0.332 | 0.315 | 0.327
| 18 | 0.351 | 0.407 | 0.392 | 0.390 | 0.427 | 0.437 | 0.375 | 0.380 | 0.361 | 0.374
Ding. | 24 | 0.384 | 0.440 | 0.421 | 0.419 | 0.440 | 0.461 | 0.406 | 0.410 | 0.389 | 0.406
| 6 | 0.258 | 0.295 | 0.287 | 0.287 | 0.308 | 0.354 | 0.278 | 0.280 | 0.260 | 0.278
| 12 | 0.327 | 0.379 | 0.369 | 0.367 | 0.384 | 0.398 | 0.356 | 0.360 | 0.337 | 0.345
| 18 | 0.370 | 0.426 | 0.413 | 0.411 | 0.430 | 0.448 | 0.401 | 0.406 | 0.384 | 0.387
Dongsi | 24 | 0.403 | 0.454 | 0.440 | 0.438 | 0.456 | 0.477 | 0.429 | 0.436 | 0.414 | 0.416
| 6 | 0.264 | 0.301 | 0.295 | 0.294 | 0.312 | 0.356 | 0.282 | 0.285 | 0.271 | 0.284
| 12 | 0.332 | 0.384 | 0.375 | 0.374 | 0.409 | 0.411 | 0.361 | 0.361 | 0.345 | 0.352
| 18 | 0.375 | 0.429 | 0.419 | 0.417 | 0.437 | 0.454 | 0.404 | 0.407 | 0.391 | 0.394
Guan. | 24 | 0.406 | 0.457 | 0.445 | 0.443 | 0.458 | 0.478 | 0.432 | 0.436 | 0.419 | 0.422
| 6 | 0.291 | 0.331 | 0.326 | 0.325 | 0.352 | 0.370 | 0.313 | 0.310 | 0.296 | 0.307
| 12 | 0.358 | 0.411 | 0.406 | 0.404 | 0.429 | 0.436 | 0.389 | 0.387 | 0.365 | 0.373
| 18 | 0.400 | 0.455 | 0.450 | 0.447 | 0.472 | 0.479 | 0.433 | 0.433 | 0.404 | 0.416
Guch. | 24 | 0.430 | 0.481 | 0.475 | 0.473 | 0.489 | 0.507 | 0.461 | 0.461 | 0.431 | 0.447
| 6 | 0.254 | 0.297 | 0.289 | 0.287 | 0.301 | 0.338 | 0.272 | 0.275 | 0.257 | 0.269
| 12 | 0.314 | 0.371 | 0.361 | 0.358 | 0.384 | 0.397 | 0.342 | 0.344 | 0.326 | 0.330
| 18 | 0.355 | 0.415 | 0.402 | 0.398 | 0.431 | 0.432 | 0.383 | 0.387 | 0.367 | 0.371
Huai. | 24 | 0.383 | 0.442 | 0.427 | 0.424 | 0.429 | 0.457 | 0.410 | 0.415 | 0.392 | 0.398
| 6 | 0.247 | 0.286 | 0.279 | 0.279 | 0.295 | 0.339 | 0.268 | 0.268 | 0.252 | 0.267
| 12 | 0.312 | 0.367 | 0.358 | 0.356 | 0.372 | 0.386 | 0.344 | 0.344 | 0.325 | 0.333
| 18 | 0.353 | 0.411 | 0.399 | 0.397 | 0.416 | 0.434 | 0.385 | 0.388 | 0.369 | 0.373
Nong. | 24 | 0.382 | 0.438 | 0.424 | 0.422 | 0.439 | 0.458 | 0.412 | 0.416 | 0.396 | 0.405
| Avg. rank | 2.33 | 6.98 | 6.05 | 5.33 | 9.13 | 9.25 | 3.58 | 4.95 | 3.93 | 3.45
Sum | # of wins | 43 | 3 | 0 | 4 | 0 | 0 | 1 | 4 | 5 | 0

Table 6. MAE ($\downarrow$) comparison among NFCL and nine benchmarks.

Dataset | $T$ | NFCL | DLinear | NLinear | TiDE | Informer | Autoformer | PatchTST | iTransformer | SCINet | TimesNet
---|---|---|---|---|---|---|---|---|---|---|---
| 6 | 0.058 | 0.064 | 0.063 | 0.063 | 0.111 | 0.090 | 0.060 | 0.075 | 0.056 | 0.066
| 12 | 0.075 | 0.087 | 0.087 | 0.085 | 0.174 | 0.115 | 0.080 | 0.098 | 0.074 | 0.082
| 18 | 0.089 | 0.108 | 0.106 | 0.105 | 0.248 | 0.135 | 0.097 | 0.117 | 0.092 | 0.106
Weather | 24 | 0.102 | 0.127 | 0.123 | 0.121 | 0.458 | 0.155 | 0.113 | 0.131 | 0.110 | 0.122
| 6 | 3.229 | 1.394 | 1.272 | 1.275 | 3.519 | 2.417 | 1.442 | 0.945 | 1.856 | 1.276
| 12 | 5.764 | 2.855 | 2.614 | 2.520 | 3.964 | 2.760 | 3.049 | 1.991 | 2.951 | 2.778
| 18 | 7.569 | 3.843 | 3.535 | 3.319 | 4.730 | 3.405 | 3.812 | 2.994 | 3.801 | 4.065
ILL | 24 | 8.835 | 4.226 | 3.948 | 3.678 | 5.033 | 3.491 | 3.586 | 3.376 | 4.346 | 5.531
| 6 | 0.011 | 0.010 | 0.010 | 0.010 | 0.680 | 0.020 | 0.010 | 0.011 | 0.061 | 0.013
| 12 | 0.019 | 0.018 | 0.018 | 0.018 | 0.870 | 0.027 | 0.018 | 0.019 | 0.087 | 0.021
| 18 | 0.027 | 0.025 | 0.025 | 0.025 | 0.991 | 0.035 | 0.025 | 0.025 | 0.148 | 0.029
Exchange | 24 | 0.034 | 0.031 | 0.031 | 0.031 | 1.359 | 0.042 | 0.031 | 0.032 | 0.145 | 0.036
| 6 | 0.319 | 0.304 | 0.305 | 0.305 | 0.410 | 0.370 | 0.293 | 0.297 | 0.319 | 0.330
| 12 | 0.406 | 0.353 | 0.356 | 0.357 | 0.529 | 0.441 | 0.363 | 0.372 | 0.394 | 0.416
| 18 | 0.437 | 0.370 | 0.374 | 0.376 | 0.545 | 0.455 | 0.382 | 0.392 | 0.419 | 0.446
ETTh1 | 24 | 0.437 | 0.377 | 0.383 | 0.384 | 0.547 | 0.484 | 0.387 | 0.398 | 0.425 | 0.444
| 6 | 0.105 | 0.116 | 0.114 | 0.116 | 0.181 | 0.136 | 0.111 | 0.116 | 0.114 | 0.110
| 12 | 0.131 | 0.143 | 0.141 | 0.143 | 0.250 | 0.160 | 0.141 | 0.149 | 0.148 | 0.142
| 18 | 0.146 | 0.157 | 0.155 | 0.157 | 0.241 | 0.171 | 0.156 | 0.163 | 0.168 | 0.155
ETTh2 | 24 | 0.155 | 0.164 | 0.163 | 0.164 | 0.260 | 0.180 | 0.164 | 0.174 | 0.201 | 0.167
| 6 | 0.159 | 0.175 | 0.186 | 0.187 | 0.177 | 0.273 | 0.169 | 0.175 | 0.140 | 0.157
| 12 | 0.285 | 0.323 | 0.362 | 0.362 | 0.281 | 0.397 | 0.301 | 0.329 | 0.231 | 0.273
| 18 | 0.426 | 0.477 | 0.560 | 0.554 | 0.380 | 0.518 | 0.460 | 0.494 | 0.330 | 0.375
ETTm1 | 24 | 0.571 | 0.626 | 0.754 | 0.736 | 0.486 | 0.670 | 0.611 | 0.659 | 0.420 | 0.488
| 6 | 0.059 | 0.064 | 0.064 | 0.064 | 0.101 | 0.076 | 0.062 | 0.064 | 0.067 | 0.061
| 12 | 0.080 | 0.091 | 0.091 | 0.091 | 0.140 | 0.096 | 0.086 | 0.088 | 0.088 | 0.082
| 18 | 0.097 | 0.114 | 0.115 | 0.115 | 0.147 | 0.117 | 0.105 | 0.110 | 0.106 | 0.100
ETTm2 | 24 | 0.113 | 0.134 | 0.135 | 0.135 | 0.185 | 0.128 | 0.123 | 0.135 | 0.122 | 0.117
| 6 | 0.213 | 0.250 | 0.256 | 0.255 | 0.256 | 0.292 | 0.244 | 0.239 | 0.212 | 0.226
| 12 | 0.314 | 0.368 | 0.385 | 0.382 | 0.387 | 0.397 | 0.365 | 0.358 | 0.317 | 0.329
| 18 | 0.382 | 0.435 | 0.462 | 0.458 | 0.472 | 0.484 | 0.439 | 0.437 | 0.382 | 0.399
Aotiz. | 24 | 0.434 | 0.476 | 0.511 | 0.507 | 0.514 | 0.533 | 0.490 | 0.493 | 0.433 | 0.451
| 6 | 0.214 | 0.255 | 0.261 | 0.258 | 0.239 | 0.288 | 0.246 | 0.239 | 0.204 | 0.225
| 12 | 0.303 | 0.352 | 0.368 | 0.363 | 0.341 | 0.383 | 0.344 | 0.338 | 0.294 | 0.318
| 18 | 0.362 | 0.409 | 0.434 | 0.428 | 0.404 | 0.451 | 0.406 | 0.405 | 0.355 | 0.387
Chang. | 24 | 0.407 | 0.445 | 0.477 | 0.469 | 0.439 | 0.487 | 0.451 | 0.450 | 0.394 | 0.437
| 6 | 0.196 | 0.227 | 0.232 | 0.231 | 0.264 | 0.299 | 0.219 | 0.219 | 0.202 | 0.214
| 12 | 0.299 | 0.342 | 0.358 | 0.354 | 0.378 | 0.407 | 0.335 | 0.337 | 0.304 | 0.325
| 18 | 0.377 | 0.415 | 0.442 | 0.438 | 0.448 | 0.477 | 0.418 | 0.424 | 0.369 | 0.415
Ding. | 24 | 0.433 | 0.463 | 0.499 | 0.493 | 0.482 | 0.524 | 0.475 | 0.478 | 0.408 | 0.474
| 6 | 0.213 | 0.242 | 0.248 | 0.247 | 0.259 | 0.310 | 0.241 | 0.238 | 0.210 | 0.230
| 12 | 0.315 | 0.361 | 0.378 | 0.375 | 0.390 | 0.395 | 0.364 | 0.362 | 0.332 | 0.336
| 18 | 0.388 | 0.429 | 0.458 | 0.453 | 0.465 | 0.483 | 0.440 | 0.443 | 0.394 | 0.413
Dongsi | 24 | 0.446 | 0.474 | 0.511 | 0.506 | 0.513 | 0.533 | 0.493 | 0.500 | 0.439 | 0.462
| 6 | 0.219 | 0.255 | 0.261 | 0.259 | 0.267 | 0.313 | 0.246 | 0.242 | 0.228 | 0.236
| 12 | 0.320 | 0.370 | 0.388 | 0.385 | 0.410 | 0.410 | 0.368 | 0.361 | 0.334 | 0.344
| 18 | 0.392 | 0.436 | 0.465 | 0.460 | 0.480 | 0.485 | 0.440 | 0.437 | 0.398 | 0.414
Guan. | 24 | 0.448 | 0.478 | 0.515 | 0.510 | 0.527 | 0.534 | 0.491 | 0.492 | 0.446 | 0.469
| 6 | 0.296 | 0.339 | 0.347 | 0.345 | 0.370 | 0.387 | 0.335 | 0.321 | 0.307 | 0.313
| 12 | 0.415 | 0.473 | 0.498 | 0.493 | 0.559 | 0.508 | 0.474 | 0.461 | 0.437 | 0.436
| 18 | 0.494 | 0.549 | 0.588 | 0.582 | 0.588 | 0.594 | 0.560 | 0.554 | 0.496 | 0.521
Guch. | 24 | 0.552 | 0.596 | 0.645 | 0.637 | 0.635 | 0.659 | 0.617 | 0.612 | 0.537 | 0.587
| 6 | 0.217 | 0.256 | 0.262 | 0.261 | 0.266 | 0.304 | 0.246 | 0.243 | 0.213 | 0.232
| 12 | 0.310 | 0.364 | 0.383 | 0.379 | 0.404 | 0.407 | 0.359 | 0.357 | 0.329 | 0.333
| 18 | 0.379 | 0.428 | 0.459 | 0.453 | 0.493 | 0.479 | 0.431 | 0.431 | 0.385 | 0.405
Huai. | 24 | 0.428 | 0.469 | 0.508 | 0.502 | 0.500 | 0.525 | 0.482 | 0.482 | 0.417 | 0.458
| 6 | 0.192 | 0.224 | 0.229 | 0.228 | 0.224 | 0.277 | 0.218 | 0.211 | 0.188 | 0.207
| 12 | 0.285 | 0.334 | 0.350 | 0.347 | 0.351 | 0.359 | 0.333 | 0.327 | 0.282 | 0.311
| 18 | 0.348 | 0.395 | 0.420 | 0.416 | 0.417 | 0.436 | 0.400 | 0.400 | 0.344 | 0.382
Nong. | 24 | 0.398 | 0.434 | 0.464 | 0.460 | 0.453 | 0.481 | 0.448 | 0.450 | 0.383 | 0.448
| Avg. rank | 2.85 | 5.08 | 6.78 | 6.10 | 8.52 | 9.03 | 4.45 | 5.08 | 3.15 | 3.95
Sum | # of wins | 23 | 3 | 0 | 0 | 0 | 0 | 5 | 4 | 25 | 0

Table 7. MSE ($\downarrow$) comparison among NFCL and nine benchmarks.

Dataset | $T$ | NFCL | DLinear | NLinear | TiDE | Informer | Autoformer | PatchTST | iTransformer | SCINet | TimesNet
---|---|---|---|---|---|---|---|---|---|---|---
| 6 | 12.280 | 15.050 | 13.406 | 13.857 | 29.537 | 25.124 | 12.264 | 16.422 | 17.493 | 13.970
| 12 | 14.712 | 18.482 | 16.264 | 16.538 | 33.287 | 28.120 | 15.165 | 18.863 | 21.764 | 16.322
| 18 | 16.706 | 21.657 | 18.778 | 18.619 | 38.851 | 31.693 | 17.373 | 20.830 | 25.141 | 18.581
Weather | 24 | 18.467 | 24.498 | 20.573 | 20.329 | 45.846 | 33.693 | 19.458 | 22.083 | 28.143 | 20.469
| 6 | 33.436 | 29.769 | 30.668 | 28.678 | 46.836 | 42.205 | 27.313 | 26.136 | 33.631 | 28.070
| 12 | 38.362 | 42.532 | 41.030 | 39.603 | 50.608 | 42.724 | 33.980 | 29.496 | 41.860 | 31.261
| 18 | 39.817 | 49.588 | 46.220 | 45.271 | 55.238 | 45.591 | 36.790 | 32.536 | 48.143 | 33.467
ILL | 24 | 40.979 | 52.867 | 48.765 | 48.233 | 56.387 | 45.350 | 39.603 | 34.661 | 52.032 | 36.320
| 6 | 5.312 | 5.047 | 5.041 | 5.035 | 29.496 | 7.440 | 5.101 | 5.286 | 9.469 | 5.785
| 12 | 6.672 | 6.451 | 6.452 | 6.440 | 32.426 | 8.380 | 6.495 | 6.702 | 11.454 | 7.114
| 18 | 7.719 | 7.496 | 7.495 | 7.477 | 36.478 | 9.431 | 7.529 | 7.697 | 13.334 | 8.161
Exchange | 24 | 8.563 | 8.285 | 8.276 | 8.248 | 41.106 | 10.090 | 8.303 | 8.503 | 13.742 | 8.938
| 6 | 33.694 | 32.968 | 33.034 | 32.867 | 39.128 | 38.254 | 32.576 | 33.026 | 34.624 | 34.329
| 12 | 36.836 | 34.845 | 34.759 | 34.861 | 44.059 | 40.784 | 35.155 | 35.439 | 38.269 | 37.249
| 18 | 38.101 | 35.360 | 35.316 | 35.366 | 44.545 | 41.538 | 35.511 | 36.184 | 39.227 | 38.786
ETTh1 | 24 | 38.065 | 35.543 | 35.536 | 35.622 | 45.786 | 42.635 | 35.879 | 36.635 | 39.993 | 39.200
| 6 | 21.834 | 22.255 | 22.326 | 22.181 | 30.706 | 24.970 | 21.719 | 22.627 | 22.800 | 22.392
| 12 | 23.742 | 24.703 | 24.506 | 24.557 | 34.494 | 26.668 | 24.298 | 25.103 | 26.073 | 24.328
| 18 | 24.566 | 25.655 | 25.298 | 25.391 | 33.086 | 27.271 | 25.164 | 26.190 | 27.576 | 25.119
ETTh2 | 24 | 25.052 | 25.882 | 25.591 | 25.655 | 34.168 | 27.704 | 25.531 | 26.495 | 29.628 | 25.696
| 6 | 24.919 | 25.921 | 25.619 | 25.792 | 29.178 | 32.077 | 25.373 | 26.066 | 25.177 | 25.193
| 12 | 30.357 | 32.562 | 31.977 | 32.092 | 35.142 | 36.562 | 30.844 | 31.892 | 30.596 | 30.188
| 18 | 34.331 | 37.869 | 36.808 | 36.881 | 38.019 | 40.151 | 35.581 | 36.589 | 34.878 | 33.617
ETTm1 | 24 | 37.598 | 42.081 | 40.642 | 40.605 | 42.014 | 44.818 | 39.018 | 40.217 | 37.597 | 36.982
| 6 | 14.332 | 15.460 | 15.607 | 15.380 | 23.777 | 19.816 | 14.742 | 15.568 | 15.847 | 15.894
| 12 | 17.136 | 18.698 | 18.860 | 18.617 | 26.658 | 21.883 | 17.831 | 18.784 | 19.266 | 18.436
| 18 | 19.247 | 21.064 | 21.253 | 21.043 | 27.328 | 23.685 | 19.773 | 21.160 | 21.573 | 20.189
ETTm2 | 24 | 20.930 | 22.993 | 23.103 | 22.895 | 30.187 | 24.721 | 21.512 | 23.566 | 23.072 | 21.997
| 6 | 25.255 | 28.279 | 27.329 | 27.259 | 28.826 | 31.619 | 26.634 | 26.881 | 26.173 | 26.828
| 12 | 29.692 | 34.184 | 32.351 | 32.210 | 34.733 | 35.433 | 31.479 | 31.675 | 32.374 | 31.037
| 18 | 32.372 | 37.628 | 34.904 | 34.722 | 37.606 | 38.139 | 34.089 | 34.327 | 35.186 | 33.453
Aotiz. | 24 | 34.156 | 39.638 | 36.463 | 36.222 | 40.678 | 39.505 | 35.811 | 35.930 | 37.468 | 35.423
| 6 | 26.410 | 29.507 | 28.693 | 28.461 | 29.796 | 32.226 | 27.837 | 27.849 | 26.637 | 27.696
| 12 | 30.522 | 34.903 | 33.294 | 32.916 | 33.960 | 36.546 | 32.264 | 32.345 | 31.716 | 31.666
| 18 | 33.102 | 37.921 | 35.677 | 35.298 | 37.936 | 39.207 | 34.762 | 34.727 | 34.528 | 34.469
Chang. | 24 | 34.877 | 39.917 | 37.124 | 36.661 | 38.786 | 40.253 | 36.243 | 36.341 | 36.664 | 36.080
| 6 | 24.380 | 27.449 | 26.442 | 26.340 | 28.913 | 32.140 | 25.532 | 25.760 | 25.443 | 26.022
| 12 | 29.148 | 33.908 | 31.836 | 31.595 | 33.698 | 36.650 | 30.649 | 30.931 | 30.381 | 30.768
| 18 | 32.151 | 37.495 | 34.729 | 34.475 | 38.641 | 39.389 | 33.700 | 33.956 | 34.150 | 33.753
Ding. | 24 | 34.212 | 40.004 | 36.489 | 36.169 | 40.650 | 40.639 | 35.580 | 35.777 | 36.549 | 35.774
| 6 | 25.611 | 28.460 | 27.583 | 27.514 | 29.166 | 32.797 | 26.863 | 27.001 | 26.120 | 27.224
| 12 | 30.417 | 34.890 | 33.011 | 32.737 | 35.245 | 36.076 | 32.038 | 32.310 | 31.777 | 31.785
| 18 | 33.216 | 38.378 | 35.699 | 35.419 | 38.438 | 39.230 | 34.892 | 35.095 | 35.418 | 34.382
Dongsi | 24 | 35.129 | 40.488 | 37.363 | 37.007 | 40.476 | 40.790 | 36.518 | 36.943 | 37.797 | 36.269
| 6 | 26.164 | 28.965 | 28.147 | 28.083 | 29.632 | 33.126 | 27.366 | 27.608 | 26.736 | 27.754
| 12 | 30.847 | 35.118 | 33.422 | 33.295 | 36.438 | 36.769 | 32.623 | 32.663 | 32.483 | 32.221
| 18 | 33.670 | 38.639 | 36.114 | 35.948 | 38.559 | 39.666 | 35.392 | 35.573 | 35.969 | 34.880
Guan. | 24 | 35.639 | 40.673 | 37.729 | 37.530 | 40.236 | 40.865 | 37.066 | 37.364 | 38.088 | 36.801
| 6 | 26.206 | 29.312 | 28.559 | 28.492 | 31.079 | 32.224 | 27.673 | 27.565 | 27.020 | 27.343
| 12 | 30.526 | 34.861 | 33.405 | 33.331 | 35.791 | 36.114 | 32.539 | 32.386 | 31.834 | 31.610
| 18 | 33.139 | 38.016 | 35.914 | 35.811 | 39.314 | 39.143 | 35.132 | 35.164 | 34.385 | 34.155
Guch. | 24 | 34.871 | 39.862 | 37.336 | 37.255 | 39.852 | 40.392 | 36.797 | 36.741 | 36.501 | 35.993
| 6 | 25.187 | 28.489 | 27.545 | 27.324 | 29.242 | 32.162 | 26.351 | 26.764 | 26.076 | 26.386
| 12 | 29.335 | 34.032 | 32.199 | 31.944 | 35.024 | 35.795 | 31.030 | 31.336 | 31.291 | 30.496
| 18 | 31.965 | 37.464 | 34.794 | 34.412 | 38.599 | 38.085 | 33.625 | 33.957 | 34.656 | 33.154
Huai. | 24 | 33.809 | 39.527 | 36.348 | 35.906 | 38.810 | 39.576 | 35.284 | 35.678 | 36.830 | 34.871
| 6 | 25.097 | 28.056 | 27.270 | 27.188 | 29.226 | 32.472 | 26.436 | 26.567 | 25.906 | 26.703
| 12 | 29.701 | 34.194 | 32.513 | 32.218 | 34.748 | 35.444 | 31.559 | 31.674 | 31.738 | 31.210
| 18 | 32.376 | 37.567 | 35.075 | 34.738 | 38.065 | 39.043 | 34.087 | 34.337 | 34.891 | 33.843
Nong. | 24 | 34.201 | 39.673 | 36.550 | 36.185 | 39.632 | 40.290 | 35.690 | 35.981 | 37.229 | 35.702
| Avg. rank | 2.00 | 6.95 | 5.70 | 4.82 | 9.28 | 9.30 | 3.08 | 4.72 | 5.68 | 3.47
Sum | # of wins | 43 | 0 | 3 | 4 | 0 | 0 | 3 | 4 | 0 | 3

Table 8. SMAPE ($\downarrow$) comparison among NFCL and nine benchmarks.

Dataset | $T$ | NFCL | DLinear | NLinear | TiDE | Informer | Autoformer | PatchTST | iTransformer | SCINet | TimesNet
---|---|---|---|---|---|---|---|---|---|---|---
| 6 | 0.822 | 0.807 | 0.810 | 0.809 | -0.172 | -0.059 | 0.826 | 0.763 | 0.763 | 0.797
| 12 | 0.772 | 0.729 | 0.730 | 0.732 | -0.656 | -0.068 | 0.756 | 0.684 | 0.680 | 0.735
| 18 | 0.717 | 0.652 | 0.655 | 0.661 | -1.109 | -0.074 | 0.688 | 0.610 | 0.577 | 0.645
Weather | 24 | 0.659 | 0.580 | 0.585 | 0.595 | -2.933 | -0.316 | 0.619 | 0.555 | 0.453 | 0.560
| 6 | 0.208 | 0.523 | 0.659 | 0.655 | -0.295 | 0.353 | 0.596 | 0.727 | 0.456 | 0.646
| 12 | -0.391 | -0.072 | 0.314 | 0.317 | -0.453 | 0.294 | 0.068 | 0.419 | 0.081 | 0.275
| 18 | -0.760 | -0.505 | 0.094 | 0.107 | -0.730 | 0.165 | -0.224 | 0.139 | -0.266 | -0.028
ILL | 24 | -0.991 | -0.661 | 0.007 | 0.018 | -0.844 | 0.166 | -0.146 | 0.070 | -0.470 | -0.315
| 6 | 0.958 | 0.952 | 0.952 | 0.952 | -1.529 | 0.933 | 0.962 | 0.957 | 0.768 | 0.954
| 12 | 0.938 | 0.935 | 0.936 | 0.935 | -1.992 | 0.922 | 0.944 | 0.938 | 0.669 | 0.935
| 18 | 0.920 | 0.920 | 0.921 | 0.920 | -2.600 | 0.904 | 0.926 | 0.924 | 0.501 | 0.921
Exchange | 24 | 0.909 | 0.906 | 0.907 | 0.907 | -3.779 | 0.890 | 0.913 | 0.906 | 0.474 | 0.904
| 6 | 0.658 | 0.684 | 0.687 | 0.685 | 0.504 | 0.606 | 0.686 | 0.685 | 0.641 | 0.647
| 12 | 0.565 | 0.636 | 0.640 | 0.639 | 0.381 | 0.522 | 0.617 | 0.607 | 0.555 | 0.559
| 18 | 0.527 | 0.618 | 0.620 | 0.619 | 0.349 | 0.503 | 0.597 | 0.585 | 0.519 | 0.521
ETTh1 | 24 | 0.531 | 0.608 | 0.610 | 0.609 | 0.344 | 0.481 | 0.589 | 0.578 | 0.505 | 0.518
| 6 | 0.694 | 0.673 | 0.680 | 0.675 | 0.227 | 0.567 | 0.676 | 0.662 | 0.670 | 0.683
| 12 | 0.628 | 0.601 | 0.610 | 0.604 | -0.236 | 0.501 | 0.607 | 0.574 | 0.531 | 0.607
| 18 | 0.593 | 0.566 | 0.578 | 0.571 | -0.178 | 0.466 | 0.566 | 0.540 | 0.406 | 0.575
ETTh2 | 24 | 0.569 | 0.548 | 0.560 | 0.554 | -0.288 | 0.455 | 0.552 | 0.515 | 0.337 | 0.544
| 6 | 0.828 | 0.805 | 0.802 | 0.803 | 0.780 | 0.727 | 0.815 | 0.807 | 0.837 | 0.829
| 12 | 0.718 | 0.662 | 0.656 | 0.656 | 0.665 | 0.619 | 0.695 | 0.664 | 0.744 | 0.723
| 18 | 0.609 | 0.517 | 0.506 | 0.507 | 0.557 | 0.515 | 0.565 | 0.521 | 0.653 | 0.631
ETTm1 | 24 | 0.503 | 0.393 | 0.368 | 0.371 | 0.465 | 0.396 | 0.448 | 0.398 | 0.577 | 0.542
| 6 | 0.832 | 0.821 | 0.820 | 0.821 | 0.641 | 0.752 | 0.822 | 0.815 | 0.817 | 0.821
| 12 | 0.767 | 0.745 | 0.743 | 0.744 | 0.485 | 0.697 | 0.747 | 0.740 | 0.743 | 0.752
| 18 | 0.712 | 0.678 | 0.675 | 0.676 | 0.482 | 0.639 | 0.686 | 0.672 | 0.687 | 0.702
ETTm2 | 24 | 0.664 | 0.619 | 0.617 | 0.617 | 0.422 | 0.601 | 0.630 | 0.606 | 0.641 | 0.648
| 6 | 0.781 | 0.744 | 0.738 | 0.740 | 0.730 | 0.688 | 0.749 | 0.755 | 0.781 | 0.769
| 12 | 0.682 | 0.629 | 0.611 | 0.615 | 0.610 | 0.587 | 0.630 | 0.638 | 0.675 | 0.667
| 18 | 0.616 | 0.564 | 0.536 | 0.541 | 0.483 | 0.501 | 0.558 | 0.560 | 0.609 | 0.600
Aotiz. | 24 | 0.566 | 0.523 | 0.489 | 0.494 | 0.481 | 0.455 | 0.509 | 0.507 | 0.559 | 0.551
| 6 | 0.751 | 0.706 | 0.699 | 0.702 | 0.715 | 0.646 | 0.717 | 0.726 | 0.758 | 0.741
| 12 | 0.658 | 0.601 | 0.584 | 0.590 | 0.600 | 0.545 | 0.613 | 0.619 | 0.656 | 0.642
| 18 | 0.595 | 0.542 | 0.515 | 0.523 | 0.528 | 0.472 | 0.547 | 0.549 | 0.590 | 0.569
Chang. | 24 | 0.549 | 0.504 | 0.471 | 0.481 | 0.494 | 0.432 | 0.501 | 0.502 | 0.544 | 0.518
| 6 | 0.793 | 0.759 | 0.752 | 0.755 | 0.718 | 0.675 | 0.765 | 0.767 | 0.785 | 0.776
| 12 | 0.692 | 0.647 | 0.630 | 0.635 | 0.602 | 0.559 | 0.652 | 0.653 | 0.678 | 0.670
| 18 | 0.617 | 0.576 | 0.547 | 0.554 | 0.496 | 0.496 | 0.572 | 0.568 | 0.610 | 0.581
Ding. | 24 | 0.562 | 0.530 | 0.492 | 0.500 | 0.493 | 0.452 | 0.516 | 0.515 | 0.570 | 0.524
| 6 | 0.771 | 0.739 | 0.732 | 0.735 | 0.724 | 0.664 | 0.741 | 0.745 | 0.775 | 0.755
| 12 | 0.672 | 0.623 | 0.605 | 0.610 | 0.593 | 0.577 | 0.620 | 0.622 | 0.656 | 0.651
| 18 | 0.604 | 0.558 | 0.530 | 0.536 | 0.513 | 0.501 | 0.547 | 0.544 | 0.589 | 0.577
Dongsi | 24 | 0.550 | 0.518 | 0.481 | 0.488 | 0.472 | 0.449 | 0.499 | 0.493 | 0.548 | 0.532
| 6 | 0.766 | 0.728 | 0.721 | 0.724 | 0.719 | 0.664 | 0.739 | 0.744 | 0.756 | 0.751
| 12 | 0.671 | 0.619 | 0.600 | 0.605 | 0.569 | 0.578 | 0.621 | 0.630 | 0.654 | 0.648
| 18 | 0.605 | 0.558 | 0.528 | 0.535 | 0.510 | 0.506 | 0.553 | 0.557 | 0.591 | 0.583
Guan. | 24 | 0.553 | 0.520 | 0.482 | 0.489 | 0.473 | 0.459 | 0.506 | 0.507 | 0.546 | 0.535
| 6 | 0.771 | 0.738 | 0.732 | 0.733 | 0.697 | 0.681 | 0.743 | 0.753 | 0.766 | 0.755
| 12 | 0.671 | 0.625 | 0.606 | 0.610 | 0.544 | 0.575 | 0.628 | 0.636 | 0.657 | 0.653
| 18 | 0.604 | 0.559 | 0.529 | 0.534 | 0.501 | 0.501 | 0.554 | 0.558 | 0.602 | 0.580
Guch. | 24 | 0.554 | 0.518 | 0.479 | 0.485 | 0.473 | 0.445 | 0.505 | 0.507 | 0.567 | 0.524
| 6 | 0.748 | 0.705 | 0.698 | 0.701 | 0.672 | 0.614 | 0.717 | 0.722 | 0.747 | 0.731
| 12 | 0.647 | 0.586 | 0.566 | 0.573 | 0.514 | 0.497 | 0.594 | 0.599 | 0.616 | 0.626
| 18 | 0.576 | 0.517 | 0.488 | 0.497 | 0.397 | 0.433 | 0.520 | 0.522 | 0.551 | 0.552
Huai. | 24 | 0.526 | 0.475 | 0.438 | 0.449 | 0.411 | 0.385 | 0.469 | 0.470 | 0.513 | 0.498
| 6 | 0.773 | 0.739 | 0.732 | 0.734 | 0.731 | 0.662 | 0.744 | 0.752 | 0.775 | 0.758
| 12 | 0.672 | 0.620 | 0.601 | 0.606 | 0.595 | 0.574 | 0.620 | 0.625 | 0.669 | 0.647
| 18 | 0.606 | 0.556 | 0.527 | 0.533 | 0.510 | 0.499 | 0.548 | 0.549 | 0.596 | 0.573
Nong. | 24 | 0.556 | 0.517 | 0.481 | 0.488 | 0.458 | 0.453 | 0.499 | 0.496 | 0.557 | 0.511
| Avg. rank | 2.30 | 5.32 | 6.40 | 5.87 | 9.05 | 8.98 | 4.40 | 4.98 | 4.07 | 3.63
Sum | # of wins | 37 | 0 | 4 | 0 | 0 | 2 | 5 | 2 | 10 | 0

Table 9. $R^{2}$ ($\uparrow$) comparison among NFCL and nine benchmarks.

##  Appendix G NFCL Intra-comparison

We only included the rankings of NFCL variations in the main content of this
section. To present our complete comparisons, we have added an additional
section with long tables. The title of each table indicates the evaluation
metric. These tables are summarized in Table 3 in Section 5.7.

Variation | NFCL-V | NFCL-C | NFCL-D
---|---|---|---
Datasets | $T$ | - | 16 | 32 | 64 | 128 | 16,16 | 32,32 | 64,64 | 128,128 | 128,128,64 | 64 | 128 | 64,64 | 128,128
| 6 | 0.071 | 0.070 | 0.070 | 0.070 | 0.070 | 0.070 | 0.070 | 0.071 | 0.070 | 0.070 | 0.071 | 0.071 | 0.071 | 0.072
| 12 | 0.091 | 0.090 | 0.090 | 0.090 | 0.090 | 0.090 | 0.090 | 0.090 | 0.090 | 0.090 | 0.092 | 0.091 | 0.091 | 0.092
| 18 | 0.109 | 0.108 | 0.108 | 0.108 | 0.108 | 0.108 | 0.108 | 0.109 | 0.109 | 0.109 | 0.111 | 0.110 | 0.111 | 0.111
Weather | 24 | 0.128 | 0.125 | 0.125 | 0.125 | 0.125 | 0.125 | 0.125 | 0.126 | 0.125 | 0.126 | 0.127 | 0.127 | 0.128 | 0.128
| 6 | 0.746 | 0.899 | 0.899 | 0.926 | 0.930 | 0.851 | 0.845 | 0.897 | 0.853 | 0.852 | 1.095 | 1.110 | 1.108 | 1.144
| 12 | 1.065 | 1.191 | 1.181 | 1.205 | 1.179 | 1.131 | 1.106 | 1.136 | 1.135 | 1.067 | 1.423 | 1.452 | 1.496 | 1.561
| 18 | 1.234 | 1.337 | 1.322 | 1.328 | 1.325 | 1.269 | 1.263 | 1.246 | 1.220 | 1.154 | 1.538 | 1.550 | 1.583 | 1.657
ILL | 24 | 1.328 | 1.435 | 1.444 | 1.435 | 1.413 | 1.347 | 1.348 | 1.342 | 1.332 | 1.257 | 1.607 | 1.597 | 1.483 | 1.577
| 6 | 0.065 | 0.064 | 0.064 | 0.064 | 0.064 | 0.064 | 0.064 | 0.063 | 0.064 | 0.063 | 0.066 | 0.066 | 0.066 | 0.066
| 12 | 0.088 | 0.088 | 0.088 | 0.088 | 0.087 | 0.087 | 0.087 | 0.087 | 0.087 | 0.087 | 0.089 | 0.089 | 0.090 | 0.090
| 18 | 0.107 | 0.107 | 0.106 | 0.106 | 0.106 | 0.107 | 0.106 | 0.106 | 0.106 | 0.106 | 0.108 | 0.107 | 0.108 | 0.109
Exchange | 24 | 0.123 | 0.122 | 0.122 | 0.122 | 0.122 | 0.122 | 0.122 | 0.122 | 0.121 | 0.122 | 0.122 | 0.122 | 0.123 | 0.123
| 6 | 0.368 | 0.366 | 0.368 | 0.363 | 0.366 | 0.367 | 0.365 | 0.365 | 0.368 | 0.365 | 0.360 | 0.360 | 0.365 | 0.364
| 12 | 0.408 | 0.419 | 0.418 | 0.419 | 0.420 | 0.419 | 0.418 | 0.416 | 0.420 | 0.418 | 0.406 | 0.410 | 0.410 | 0.410
| 18 | 0.423 | 0.440 | 0.438 | 0.438 | 0.435 | 0.437 | 0.436 | 0.437 | 0.434 | 0.435 | 0.427 | 0.426 | 0.427 | 0.427
ETTh1 | 24 | 0.425 | 0.440 | 0.438 | 0.439 | 0.439 | 0.440 | 0.440 | 0.438 | 0.440 | 0.440 | 0.431 | 0.431 | 0.431 | 0.433
| 6 | 0.211 | 0.212 | 0.212 | 0.212 | 0.212 | 0.212 | 0.212 | 0.213 | 0.213 | 0.213 | 0.215 | 0.215 | 0.216 | 0.215
| 12 | 0.239 | 0.240 | 0.239 | 0.238 | 0.240 | 0.239 | 0.240 | 0.241 | 0.242 | 0.240 | 0.239 | 0.240 | 0.241 | 0.241
| 18 | 0.253 | 0.253 | 0.253 | 0.253 | 0.254 | 0.254 | 0.255 | 0.255 | 0.254 | 0.255 | 0.253 | 0.253 | 0.254 | 0.254
ETTh2 | 24 | 0.261 | 0.261 | 0.261 | 0.261 | 0.262 | 0.262 | 0.263 | 0.262 | 0.263 | 0.263 | 0.261 | 0.262 | 0.261 | 0.263
| 6 | 0.246 | 0.240 | 0.239 | 0.240 | 0.240 | 0.240 | 0.240 | 0.239 | 0.240 | 0.241 | 0.243 | 0.244 | 0.244 | 0.245
| 12 | 0.338 | 0.326 | 0.325 | 0.325 | 0.326 | 0.325 | 0.326 | 0.327 | 0.327 | 0.326 | 0.330 | 0.331 | 0.334 | 0.333
| 18 | 0.414 | 0.399 | 0.397 | 0.397 | 0.398 | 0.398 | 0.398 | 0.397 | 0.397 | 0.400 | 0.405 | 0.405 | 0.404 | 0.402
ETTm1 | 24 | 0.477 | 0.459 | 0.459 | 0.458 | 0.457 | 0.459 | 0.458 | 0.459 | 0.459 | 0.459 | 0.467 | 0.467 | 0.468 | 0.465
| 6 | 0.148 | 0.148 | 0.148 | 0.148 | 0.147 | 0.147 | 0.147 | 0.147 | 0.148 | 0.148 | 0.148 | 0.149 | 0.149 | 0.149
| 12 | 0.177 | 0.176 | 0.176 | 0.175 | 0.175 | 0.176 | 0.176 | 0.176 | 0.177 | 0.177 | 0.178 | 0.178 | 0.178 | 0.178
| 18 | 0.200 | 0.197 | 0.198 | 0.197 | 0.197 | 0.197 | 0.197 | 0.198 | 0.198 | 0.198 | 0.200 | 0.200 | 0.201 | 0.200
ETTm2 | 24 | 0.219 | 0.216 | 0.216 | 0.216 | 0.215 | 0.216 | 0.216 | 0.216 | 0.215 | 0.216 | 0.219 | 0.219 | 0.219 | 0.219
| 6 | 0.261 | 0.260 | 0.260 | 0.260 | 0.261 | 0.260 | 0.261 | 0.261 | 0.262 | 0.262 | 0.271 | 0.271 | 0.272 | 0.272
| 12 | 0.330 | 0.328 | 0.327 | 0.327 | 0.329 | 0.328 | 0.328 | 0.328 | 0.328 | 0.329 | 0.336 | 0.337 | 0.337 | 0.337
| 18 | 0.372 | 0.369 | 0.369 | 0.370 | 0.371 | 0.369 | 0.370 | 0.371 | 0.371 | 0.371 | 0.377 | 0.378 | 0.378 | 0.378
Aotiz. | 24 | 0.403 | 0.399 | 0.399 | 0.400 | 0.399 | 0.399 | 0.400 | 0.399 | 0.400 | 0.400 | 0.406 | 0.407 | 0.408 | 0.408
| 6 | 0.263 | 0.262 | 0.261 | 0.261 | 0.261 | 0.261 | 0.261 | 0.262 | 0.262 | 0.263 | 0.268 | 0.269 | 0.269 | 0.269
| 12 | 0.321 | 0.320 | 0.320 | 0.320 | 0.320 | 0.321 | 0.320 | 0.321 | 0.321 | 0.322 | 0.324 | 0.324 | 0.323 | 0.324
| 18 | 0.358 | 0.357 | 0.357 | 0.356 | 0.358 | 0.357 | 0.357 | 0.357 | 0.359 | 0.358 | 0.361 | 0.361 | 0.362 | 0.362
Chang. | 24 | 0.386 | 0.384 | 0.385 | 0.384 | 0.385 | 0.384 | 0.385 | 0.384 | 0.387 | 0.385 | 0.389 | 0.389 | 0.388 | 0.388
| 6 | 0.241 | 0.238 | 0.238 | 0.238 | 0.238 | 0.239 | 0.238 | 0.239 | 0.240 | 0.240 | 0.245 | 0.245 | 0.246 | 0.247
| 12 | 0.309 | 0.305 | 0.305 | 0.306 | 0.306 | 0.306 | 0.306 | 0.306 | 0.307 | 0.308 | 0.310 | 0.311 | 0.312 | 0.312
| 18 | 0.354 | 0.351 | 0.351 | 0.351 | 0.352 | 0.352 | 0.352 | 0.353 | 0.353 | 0.354 | 0.356 | 0.356 | 0.357 | 0.358
Ding. | 24 | 0.387 | 0.384 | 0.384 | 0.384 | 0.384 | 0.385 | 0.385 | 0.385 | 0.385 | 0.386 | 0.388 | 0.388 | 0.389 | 0.389
| 6 | 0.260 | 0.259 | 0.258 | 0.259 | 0.259 | 0.259 | 0.259 | 0.259 | 0.260 | 0.261 | 0.267 | 0.268 | 0.270 | 0.270
| 12 | 0.330 | 0.327 | 0.327 | 0.327 | 0.328 | 0.328 | 0.328 | 0.329 | 0.329 | 0.330 | 0.334 | 0.334 | 0.335 | 0.336
| 18 | 0.373 | 0.371 | 0.370 | 0.371 | 0.372 | 0.371 | 0.372 | 0.373 | 0.373 | 0.373 | 0.377 | 0.377 | 0.378 | 0.378
Dongsi | 24 | 0.405 | 0.403 | 0.403 | 0.403 | 0.404 | 0.404 | 0.404 | 0.405 | 0.405 | 0.407 | 0.408 | 0.408 | 0.408 | 0.409
| 6 | 0.266 | 0.264 | 0.264 | 0.265 | 0.264 | 0.264 | 0.265 | 0.265 | 0.266 | 0.266 | 0.274 | 0.274 | 0.275 | 0.276
| 12 | 0.334 | 0.332 | 0.332 | 0.332 | 0.332 | 0.333 | 0.334 | 0.333 | 0.333 | 0.334 | 0.341 | 0.341 | 0.341 | 0.341
| 18 | 0.377 | 0.375 | 0.375 | 0.375 | 0.376 | 0.376 | 0.375 | 0.376 | 0.377 | 0.378 | 0.384 | 0.384 | 0.385 | 0.385
Guan. | 24 | 0.408 | 0.406 | 0.406 | 0.406 | 0.407 | 0.407 | 0.407 | 0.408 | 0.408 | 0.408 | 0.414 | 0.415 | 0.416 | 0.415
| 6 | 0.292 | 0.291 | 0.291 | 0.290 | 0.291 | 0.292 | 0.291 | 0.292 | 0.293 | 0.292 | 0.298 | 0.298 | 0.299 | 0.300
| 12 | 0.361 | 0.358 | 0.358 | 0.358 | 0.359 | 0.359 | 0.359 | 0.359 | 0.358 | 0.359 | 0.364 | 0.365 | 0.365 | 0.366
| 18 | 0.405 | 0.400 | 0.400 | 0.400 | 0.401 | 0.400 | 0.400 | 0.401 | 0.400 | 0.401 | 0.407 | 0.407 | 0.408 | 0.409
Guch. | 24 | 0.436 | 0.431 | 0.430 | 0.430 | 0.430 | 0.430 | 0.430 | 0.431 | 0.430 | 0.431 | 0.437 | 0.438 | 0.436 | 0.438
| 6 | 0.256 | 0.253 | 0.254 | 0.253 | 0.254 | 0.254 | 0.254 | 0.255 | 0.254 | 0.255 | 0.263 | 0.264 | 0.265 | 0.265
| 12 | 0.317 | 0.315 | 0.314 | 0.316 | 0.315 | 0.315 | 0.315 | 0.316 | 0.316 | 0.316 | 0.322 | 0.322 | 0.323 | 0.323
| 18 | 0.357 | 0.354 | 0.355 | 0.355 | 0.355 | 0.356 | 0.355 | 0.356 | 0.357 | 0.357 | 0.361 | 0.361 | 0.362 | 0.361
Huai. | 24 | 0.387 | 0.384 | 0.383 | 0.384 | 0.384 | 0.384 | 0.384 | 0.385 | 0.385 | 0.387 | 0.388 | 0.389 | 0.390 | 0.390
| 6 | 0.248 | 0.247 | 0.247 | 0.247 | 0.248 | 0.248 | 0.248 | 0.248 | 0.249 | 0.249 | 0.257 | 0.257 | 0.259 | 0.260
| 12 | 0.314 | 0.313 | 0.312 | 0.312 | 0.313 | 0.313 | 0.314 | 0.314 | 0.315 | 0.315 | 0.321 | 0.322 | 0.322 | 0.322
| 18 | 0.355 | 0.354 | 0.353 | 0.353 | 0.354 | 0.354 | 0.354 | 0.354 | 0.355 | 0.355 | 0.361 | 0.361 | 0.361 | 0.362
Nong. | 24 | 0.384 | 0.382 | 0.382 | 0.382 | 0.383 | 0.383 | 0.383 | 0.385 | 0.385 | 0.385 | 0.389 | 0.389 | 0.390 | 0.390
| Avg. rank | 8.42 | 4.57 | 3.93 | 4.13 | 5.15 | 5.13 | 5.72 | 6.47 | 7.38 | 7.88 | 10.45 | 11.08 | 12.12 | 12.57
Sum | # of wins | 5 | 9 | 15 | 13 | 7 | 1 | 0 | 2 | 3 | 3 | 1 | 1 | 0 | 0

Table 10. MAE ($\downarrow$) comparison among hyperparameter tuning.

Variation | NFCL-V | NFCL-C | NFCL-D
---|---|---|---
Datasets | $T$ | - | 16 | 32 | 64 | 128 | 16,16 | 32,32 | 64,64 | 128,128 | 128,128,64 | 64 | 128 | 64,64 | 128,128
| 6 | 0.057 | 0.058 | 0.058 | 0.058 | 0.057 | 0.058 | 0.058 | 0.058 | 0.057 | 0.057 | 0.058 | 0.058 | 0.059 | 0.058
| 12 | 0.075 | 0.075 | 0.075 | 0.074 | 0.074 | 0.075 | 0.075 | 0.074 | 0.074 | 0.074 | 0.076 | 0.076 | 0.075 | 0.075
| 18 | 0.090 | 0.089 | 0.089 | 0.089 | 0.087 | 0.089 | 0.089 | 0.089 | 0.090 | 0.089 | 0.093 | 0.090 | 0.094 | 0.093
Weather | 24 | 0.105 | 0.101 | 0.102 | 0.102 | 0.101 | 0.103 | 0.103 | 0.103 | 0.102 | 0.103 | 0.106 | 0.104 | 0.107 | 0.106
| 6 | 1.714 | 3.293 | 3.229 | 3.517 | 3.674 | 3.047 | 2.983 | 3.734 | 3.184 | 3.269 | 7.031 | 7.180 | 7.615 | 8.671
| 12 | 3.712 | 5.721 | 5.764 | 5.991 | 5.412 | 5.095 | 4.666 | 5.295 | 4.953 | 3.874 | 10.610 | 11.279 | 13.380 | 15.650
| 18 | 4.936 | 7.400 | 7.569 | 7.427 | 7.097 | 6.331 | 6.416 | 6.016 | 5.310 | 4.083 | 12.369 | 12.442 | 13.932 | 16.507
ILL | 24 | 5.426 | 8.143 | 8.835 | 8.315 | 8.348 | 6.964 | 7.191 | 6.810 | 6.077 | 5.076 | 12.306 | 11.902 | 9.575 | 12.028
| 6 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.011 | 0.010 | 0.011 | 0.010 | 0.011 | 0.011 | 0.011 | 0.011
| 12 | 0.020 | 0.019 | 0.019 | 0.019 | 0.019 | 0.019 | 0.019 | 0.019 | 0.019 | 0.019 | 0.020 | 0.020 | 0.020 | 0.020
| 18 | 0.029 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027 | 0.027
Exchange | 24 | 0.036 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034 | 0.034
| 6 | 0.320 | 0.318 | 0.319 | 0.312 | 0.319 | 0.319 | 0.317 | 0.317 | 0.323 | 0.317 | 0.313 | 0.314 | 0.320 | 0.320
| 12 | 0.388 | 0.407 | 0.406 | 0.408 | 0.408 | 0.408 | 0.406 | 0.406 | 0.413 | 0.407 | 0.391 | 0.399 | 0.400 | 0.399
| 18 | 0.411 | 0.439 | 0.437 | 0.437 | 0.434 | 0.439 | 0.435 | 0.437 | 0.431 | 0.434 | 0.428 | 0.425 | 0.429 | 0.424
ETTh1 | 24 | 0.414 | 0.438 | 0.437 | 0.438 | 0.438 | 0.440 | 0.440 | 0.437 | 0.441 | 0.442 | 0.433 | 0.433 | 0.433 | 0.436
| 6 | 0.106 | 0.105 | 0.105 | 0.105 | 0.105 | 0.105 | 0.105 | 0.105 | 0.106 | 0.105 | 0.107 | 0.107 | 0.108 | 0.107
| 12 | 0.132 | 0.131 | 0.131 | 0.130 | 0.131 | 0.131 | 0.131 | 0.132 | 0.133 | 0.132 | 0.132 | 0.133 | 0.134 | 0.134
| 18 | 0.147 | 0.146 | 0.146 | 0.145 | 0.147 | 0.146 | 0.146 | 0.148 | 0.147 | 0.147 | 0.147 | 0.147 | 0.148 | 0.148
ETTh2 | 24 | 0.157 | 0.156 | 0.155 | 0.155 | 0.156 | 0.156 | 0.156 | 0.156 | 0.157 | 0.157 | 0.156 | 0.157 | 0.156 | 0.158
| 6 | 0.167 | 0.159 | 0.159 | 0.159 | 0.159 | 0.159 | 0.160 | 0.159 | 0.160 | 0.160 | 0.162 | 0.163 | 0.163 | 0.164
| 12 | 0.309 | 0.287 | 0.285 | 0.285 | 0.286 | 0.286 | 0.287 | 0.288 | 0.287 | 0.287 | 0.294 | 0.295 | 0.298 | 0.297
| 18 | 0.463 | 0.429 | 0.426 | 0.426 | 0.429 | 0.427 | 0.428 | 0.427 | 0.427 | 0.431 | 0.441 | 0.441 | 0.439 | 0.435
ETTm1 | 24 | 0.614 | 0.570 | 0.571 | 0.569 | 0.568 | 0.571 | 0.568 | 0.569 | 0.570 | 0.571 | 0.585 | 0.586 | 0.586 | 0.582
| 6 | 0.060 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059 | 0.059
| 12 | 0.081 | 0.080 | 0.080 | 0.079 | 0.079 | 0.080 | 0.080 | 0.080 | 0.080 | 0.080 | 0.081 | 0.081 | 0.081 | 0.081
| 18 | 0.100 | 0.097 | 0.097 | 0.097 | 0.097 | 0.097 | 0.097 | 0.097 | 0.097 | 0.097 | 0.099 | 0.099 | 0.099 | 0.099
ETTm2 | 24 | 0.116 | 0.113 | 0.113 | 0.113 | 0.113 | 0.113 | 0.113 | 0.113 | 0.112 | 0.113 | 0.116 | 0.116 | 0.115 | 0.115
| 6 | 0.216 | 0.213 | 0.213 | 0.214 | 0.214 | 0.213 | 0.213 | 0.214 | 0.215 | 0.215 | 0.223 | 0.223 | 0.224 | 0.223
| 12 | 0.321 | 0.315 | 0.314 | 0.314 | 0.315 | 0.314 | 0.315 | 0.315 | 0.315 | 0.316 | 0.327 | 0.329 | 0.328 | 0.328
| 18 | 0.393 | 0.383 | 0.382 | 0.383 | 0.385 | 0.382 | 0.382 | 0.385 | 0.384 | 0.382 | 0.396 | 0.397 | 0.398 | 0.400
Aotiz. | 24 | 0.448 | 0.434 | 0.434 | 0.433 | 0.431 | 0.433 | 0.432 | 0.430 | 0.433 | 0.433 | 0.450 | 0.452 | 0.453 | 0.453
| 6 | 0.217 | 0.215 | 0.214 | 0.214 | 0.214 | 0.214 | 0.214 | 0.215 | 0.215 | 0.215 | 0.222 | 0.223 | 0.222 | 0.222
| 12 | 0.304 | 0.302 | 0.303 | 0.303 | 0.303 | 0.305 | 0.302 | 0.304 | 0.304 | 0.304 | 0.308 | 0.308 | 0.307 | 0.308
| 18 | 0.366 | 0.361 | 0.362 | 0.361 | 0.362 | 0.361 | 0.360 | 0.361 | 0.364 | 0.362 | 0.369 | 0.369 | 0.371 | 0.369
Chang. | 24 | 0.414 | 0.407 | 0.407 | 0.407 | 0.407 | 0.407 | 0.408 | 0.406 | 0.411 | 0.408 | 0.417 | 0.417 | 0.416 | 0.417
| 6 | 0.199 | 0.195 | 0.196 | 0.195 | 0.196 | 0.196 | 0.195 | 0.197 | 0.198 | 0.197 | 0.203 | 0.203 | 0.203 | 0.203
| 12 | 0.305 | 0.299 | 0.299 | 0.300 | 0.300 | 0.300 | 0.300 | 0.300 | 0.302 | 0.301 | 0.305 | 0.306 | 0.307 | 0.306
| 18 | 0.385 | 0.376 | 0.377 | 0.377 | 0.377 | 0.378 | 0.378 | 0.379 | 0.380 | 0.380 | 0.385 | 0.386 | 0.385 | 0.388
Ding. | 24 | 0.444 | 0.434 | 0.433 | 0.434 | 0.434 | 0.435 | 0.435 | 0.435 | 0.436 | 0.438 | 0.442 | 0.443 | 0.444 | 0.444
| 6 | 0.217 | 0.212 | 0.213 | 0.213 | 0.213 | 0.213 | 0.213 | 0.213 | 0.214 | 0.215 | 0.221 | 0.221 | 0.223 | 0.222
| 12 | 0.323 | 0.316 | 0.315 | 0.316 | 0.316 | 0.316 | 0.316 | 0.317 | 0.318 | 0.319 | 0.325 | 0.325 | 0.326 | 0.326
| 18 | 0.398 | 0.389 | 0.388 | 0.389 | 0.391 | 0.390 | 0.390 | 0.391 | 0.392 | 0.393 | 0.398 | 0.398 | 0.399 | 0.400
Dongsi | 24 | 0.455 | 0.447 | 0.446 | 0.447 | 0.448 | 0.448 | 0.448 | 0.450 | 0.450 | 0.453 | 0.456 | 0.456 | 0.456 | 0.457
| 6 | 0.223 | 0.219 | 0.219 | 0.221 | 0.219 | 0.219 | 0.220 | 0.221 | 0.221 | 0.221 | 0.229 | 0.229 | 0.229 | 0.230
| 12 | 0.325 | 0.321 | 0.320 | 0.321 | 0.320 | 0.321 | 0.323 | 0.322 | 0.322 | 0.324 | 0.336 | 0.335 | 0.334 | 0.335
| 18 | 0.397 | 0.391 | 0.392 | 0.390 | 0.393 | 0.392 | 0.393 | 0.393 | 0.395 | 0.398 | 0.413 | 0.411 | 0.412 | 0.412
Guan. | 24 | 0.454 | 0.449 | 0.448 | 0.447 | 0.451 | 0.450 | 0.452 | 0.450 | 0.453 | 0.452 | 0.469 | 0.475 | 0.474 | 0.469
| 6 | 0.298 | 0.295 | 0.296 | 0.295 | 0.295 | 0.296 | 0.295 | 0.297 | 0.297 | 0.295 | 0.301 | 0.301 | 0.302 | 0.304
| 12 | 0.421 | 0.415 | 0.415 | 0.414 | 0.416 | 0.414 | 0.415 | 0.416 | 0.413 | 0.417 | 0.423 | 0.425 | 0.425 | 0.426
| 18 | 0.506 | 0.494 | 0.494 | 0.495 | 0.496 | 0.494 | 0.494 | 0.495 | 0.493 | 0.496 | 0.508 | 0.508 | 0.509 | 0.510
Guch. | 24 | 0.567 | 0.554 | 0.552 | 0.553 | 0.554 | 0.551 | 0.553 | 0.555 | 0.553 | 0.553 | 0.567 | 0.569 | 0.566 | 0.570
| 6 | 0.220 | 0.216 | 0.217 | 0.216 | 0.217 | 0.216 | 0.216 | 0.218 | 0.217 | 0.217 | 0.227 | 0.228 | 0.229 | 0.229
| 12 | 0.318 | 0.312 | 0.310 | 0.312 | 0.313 | 0.312 | 0.311 | 0.313 | 0.312 | 0.313 | 0.324 | 0.324 | 0.325 | 0.324
| 18 | 0.387 | 0.377 | 0.379 | 0.378 | 0.377 | 0.379 | 0.378 | 0.380 | 0.380 | 0.380 | 0.391 | 0.391 | 0.392 | 0.391
Huai. | 24 | 0.442 | 0.430 | 0.428 | 0.430 | 0.430 | 0.429 | 0.429 | 0.430 | 0.431 | 0.433 | 0.442 | 0.444 | 0.444 | 0.445
| 6 | 0.192 | 0.192 | 0.192 | 0.192 | 0.192 | 0.192 | 0.191 | 0.193 | 0.193 | 0.192 | 0.200 | 0.200 | 0.201 | 0.201
| 12 | 0.288 | 0.285 | 0.285 | 0.285 | 0.286 | 0.285 | 0.286 | 0.287 | 0.287 | 0.286 | 0.295 | 0.296 | 0.296 | 0.296
| 18 | 0.351 | 0.349 | 0.348 | 0.349 | 0.350 | 0.349 | 0.350 | 0.349 | 0.350 | 0.350 | 0.361 | 0.360 | 0.360 | 0.360
Nong. | 24 | 0.400 | 0.397 | 0.398 | 0.397 | 0.399 | 0.398 | 0.399 | 0.399 | 0.401 | 0.400 | 0.410 | 0.410 | 0.412 | 0.411
| Avg. rank | 9.60 | 5.08 | 4.42 | 4.48 | 5.25 | 4.68 | 4.90 | 6.23 | 6.78 | 7.05 | 10.85 | 11.35 | 12.08 | 12.23
Sum | # of wins | 5 | 9 | 14 | 7 | 5 | 4 | 3 | 3 | 4 | 6 | 0 | 0 | 0 | 0

Table 11. MSE ($\downarrow$) comparison among hyperparameter tuning.

Variation | NFCL-V | NFCL-C | NFCL-D
---|---|---|---
Datasets | $T$ | - | 16 | 32 | 64 | 128 | 16,16 | 32,32 | 64,64 | 128,128 | 128,128,64 | 64 | 128 | 64,64 | 128,128
| 6 | 12.382 | 12.304 | 12.280 | 12.299 | 12.294 | 12.316 | 12.298 | 12.429 | 12.336 | 12.475 | 12.485 | 12.518 | 12.562 | 12.629
| 12 | 14.744 | 14.708 | 14.712 | 14.716 | 14.692 | 14.810 | 14.787 | 14.778 | 14.797 | 14.853 | 14.906 | 14.911 | 14.941 | 14.994
| 18 | 16.745 | 16.768 | 16.706 | 16.700 | 16.733 | 16.761 | 16.769 | 16.840 | 16.847 | 16.907 | 16.968 | 17.006 | 16.968 | 17.008
Weather | 24 | 18.776 | 18.483 | 18.467 | 18.525 | 18.467 | 18.498 | 18.489 | 18.561 | 18.574 | 18.587 | 18.621 | 18.731 | 18.745 | 18.738
| 6 | 32.565 | 33.468 | 33.436 | 33.840 | 34.015 | 33.065 | 32.836 | 33.220 | 33.175 | 32.929 | 34.388 | 34.501 | 34.242 | 33.952
| 12 | 39.430 | 38.622 | 38.362 | 38.623 | 38.601 | 37.713 | 37.287 | 37.110 | 37.800 | 37.456 | 38.915 | 38.992 | 38.282 | 38.098
| 18 | 42.895 | 40.170 | 39.817 | 39.669 | 40.111 | 39.034 | 38.807 | 38.749 | 38.705 | 38.935 | 40.334 | 40.479 | 39.647 | 39.490
ILL | 24 | 44.372 | 41.428 | 40.979 | 41.182 | 40.888 | 40.040 | 39.811 | 40.020 | 40.600 | 40.227 | 40.745 | 41.099 | 40.673 | 40.715
| 6 | 5.413 | 5.340 | 5.312 | 5.311 | 5.301 | 5.294 | 5.291 | 5.312 | 5.322 | 5.298 | 5.480 | 5.468 | 5.484 | 5.486
| 12 | 6.755 | 6.686 | 6.672 | 6.669 | 6.630 | 6.637 | 6.637 | 6.645 | 6.677 | 6.684 | 6.772 | 6.778 | 6.792 | 6.810
| 18 | 7.808 | 7.741 | 7.719 | 7.718 | 7.688 | 7.739 | 7.726 | 7.700 | 7.742 | 7.727 | 7.786 | 7.802 | 7.837 | 7.859
Exchange | 24 | 8.588 | 8.544 | 8.563 | 8.534 | 8.547 | 8.548 | 8.541 | 8.550 | 8.540 | 8.545 | 8.598 | 8.611 | 8.630 | 8.651
| 6 | 33.718 | 33.488 | 33.694 | 33.394 | 33.540 | 33.591 | 33.521 | 33.582 | 33.687 | 33.492 | 33.374 | 33.302 | 33.677 | 33.579
| 12 | 36.193 | 36.848 | 36.836 | 36.866 | 36.928 | 36.807 | 36.854 | 36.539 | 36.866 | 36.853 | 36.232 | 36.417 | 36.436 | 36.382
| 18 | 37.151 | 38.178 | 38.101 | 38.082 | 37.814 | 37.928 | 37.984 | 38.050 | 37.908 | 38.019 | 37.429 | 37.442 | 37.525 | 37.573
ETTh1 | 24 | 37.245 | 38.148 | 38.065 | 38.125 | 38.161 | 38.148 | 38.267 | 38.084 | 38.261 | 38.290 | 37.694 | 37.671 | 37.720 | 37.785
| 6 | 21.461 | 21.804 | 21.834 | 21.905 | 21.723 | 21.875 | 21.819 | 22.035 | 21.905 | 21.988 | 22.292 | 22.167 | 22.353 | 22.198
| 12 | 23.648 | 23.796 | 23.742 | 23.847 | 23.715 | 23.822 | 23.864 | 24.001 | 23.951 | 24.056 | 23.834 | 23.938 | 24.008 | 23.964
| 18 | 24.467 | 24.556 | 24.566 | 24.524 | 24.584 | 24.702 | 24.723 | 24.655 | 24.795 | 24.841 | 24.762 | 24.692 | 24.684 | 24.796
ETTh2 | 24 | 24.926 | 25.125 | 25.052 | 25.068 | 25.207 | 25.141 | 25.246 | 25.242 | 25.221 | 25.339 | 25.170 | 25.214 | 25.251 | 25.324
| 6 | 25.361 | 24.973 | 24.919 | 25.021 | 25.011 | 24.989 | 25.002 | 24.977 | 24.994 | 25.015 | 25.177 | 25.316 | 25.314 | 25.342
| 12 | 31.003 | 30.353 | 30.357 | 30.345 | 30.445 | 30.336 | 30.371 | 30.443 | 30.457 | 30.404 | 30.585 | 30.646 | 30.798 | 30.724
| 18 | 35.198 | 34.446 | 34.331 | 34.339 | 34.316 | 34.360 | 34.412 | 34.309 | 34.411 | 34.454 | 34.597 | 34.659 | 34.598 | 34.609
ETTm1 | 24 | 38.422 | 37.583 | 37.598 | 37.518 | 37.517 | 37.615 | 37.560 | 37.636 | 37.646 | 37.595 | 37.934 | 37.972 | 38.149 | 37.839
| 6 | 14.262 | 14.282 | 14.332 | 14.367 | 14.419 | 14.422 | 14.397 | 14.536 | 14.731 | 14.611 | 14.644 | 14.746 | 14.711 | 14.714
| 12 | 17.124 | 17.167 | 17.136 | 17.091 | 17.202 | 17.139 | 17.227 | 17.362 | 17.458 | 17.518 | 17.460 | 17.500 | 17.506 | 17.610
| 18 | 19.398 | 19.245 | 19.247 | 19.247 | 19.337 | 19.274 | 19.330 | 19.323 | 19.465 | 19.504 | 19.614 | 19.606 | 19.592 | 19.620
ETTm2 | 24 | 21.098 | 20.886 | 20.930 | 20.878 | 20.895 | 20.960 | 20.988 | 20.929 | 20.956 | 21.076 | 21.245 | 21.257 | 21.310 | 21.323
| 6 | 25.203 | 25.228 | 25.255 | 25.218 | 25.277 | 25.265 | 25.275 | 25.331 | 25.359 | 25.376 | 26.032 | 26.038 | 26.076 | 26.165
| 12 | 29.678 | 29.701 | 29.692 | 29.727 | 29.755 | 29.760 | 29.782 | 29.739 | 29.707 | 29.778 | 30.183 | 30.252 | 30.257 | 30.333
| 18 | 32.289 | 32.363 | 32.372 | 32.382 | 32.410 | 32.396 | 32.415 | 32.394 | 32.378 | 32.439 | 32.743 | 32.748 | 32.824 | 32.787
Aotiz. | 24 | 34.079 | 34.169 | 34.156 | 34.282 | 34.258 | 34.205 | 34.287 | 34.238 | 34.235 | 34.289 | 34.505 | 34.525 | 34.592 | 34.579
| 6 | 26.488 | 26.457 | 26.410 | 26.348 | 26.406 | 26.355 | 26.431 | 26.488 | 26.524 | 26.536 | 26.875 | 26.959 | 27.047 | 27.045
| 12 | 30.469 | 30.595 | 30.522 | 30.621 | 30.597 | 30.709 | 30.559 | 30.655 | 30.724 | 30.722 | 30.747 | 30.778 | 30.729 | 30.749
| 18 | 32.879 | 33.031 | 33.102 | 32.972 | 33.082 | 33.042 | 33.007 | 33.030 | 33.270 | 33.105 | 33.080 | 33.096 | 33.149 | 33.202
Chang. | 24 | 34.585 | 34.756 | 34.877 | 34.744 | 34.795 | 34.767 | 34.891 | 34.796 | 35.013 | 34.908 | 34.819 | 34.817 | 34.766 | 34.800
| 6 | 24.553 | 24.378 | 24.380 | 24.392 | 24.413 | 24.470 | 24.451 | 24.478 | 24.587 | 24.591 | 24.913 | 24.919 | 25.089 | 25.117
| 12 | 29.134 | 29.085 | 29.148 | 29.147 | 29.084 | 29.142 | 29.158 | 29.134 | 29.220 | 29.322 | 29.374 | 29.437 | 29.493 | 29.603
| 18 | 31.966 | 32.171 | 32.151 | 32.056 | 32.245 | 32.155 | 32.142 | 32.335 | 32.210 | 32.343 | 32.304 | 32.315 | 32.460 | 32.374
Ding. | 24 | 34.044 | 34.179 | 34.212 | 34.170 | 34.152 | 34.308 | 34.280 | 34.272 | 34.241 | 34.232 | 34.315 | 34.295 | 34.374 | 34.340
| 6 | 25.677 | 25.641 | 25.611 | 25.628 | 25.778 | 25.712 | 25.707 | 25.746 | 25.761 | 25.951 | 26.321 | 26.348 | 26.454 | 26.521
| 12 | 30.375 | 30.416 | 30.417 | 30.391 | 30.476 | 30.481 | 30.521 | 30.578 | 30.557 | 30.656 | 30.811 | 30.782 | 30.826 | 30.941
| 18 | 33.115 | 33.228 | 33.216 | 33.212 | 33.240 | 33.242 | 33.304 | 33.327 | 33.326 | 33.367 | 33.451 | 33.517 | 33.549 | 33.592
Dongsi | 24 | 35.041 | 35.138 | 35.129 | 35.141 | 35.247 | 35.234 | 35.246 | 35.338 | 35.340 | 35.345 | 35.408 | 35.334 | 35.422 | 35.429
| 6 | 26.203 | 26.143 | 26.164 | 26.204 | 26.198 | 26.165 | 26.228 | 26.244 | 26.292 | 26.292 | 26.968 | 27.026 | 27.035 | 27.051
| 12 | 30.813 | 30.807 | 30.847 | 30.861 | 30.912 | 30.894 | 30.907 | 30.970 | 30.941 | 30.978 | 31.347 | 31.426 | 31.455 | 31.460
| 18 | 33.539 | 33.641 | 33.670 | 33.699 | 33.695 | 33.746 | 33.732 | 33.738 | 33.774 | 33.829 | 34.150 | 34.115 | 34.171 | 34.174
Guan. | 24 | 35.413 | 35.621 | 35.639 | 35.632 | 35.649 | 35.648 | 35.676 | 35.742 | 35.729 | 35.690 | 35.944 | 35.986 | 36.034 | 36.025
| 6 | 26.256 | 26.174 | 26.206 | 26.185 | 26.224 | 26.285 | 26.246 | 26.253 | 26.282 | 26.321 | 26.838 | 26.833 | 26.998 | 27.011
| 12 | 30.630 | 30.562 | 30.526 | 30.576 | 30.605 | 30.615 | 30.597 | 30.570 | 30.638 | 30.682 | 30.962 | 31.025 | 31.048 | 31.092
| 18 | 33.222 | 33.140 | 33.139 | 33.136 | 33.161 | 33.216 | 33.091 | 33.155 | 33.085 | 33.202 | 33.531 | 33.517 | 33.592 | 33.608
Guch. | 24 | 35.016 | 34.924 | 34.871 | 34.878 | 34.913 | 34.981 | 34.897 | 34.876 | 34.968 | 35.027 | 35.263 | 35.256 | 35.285 | 35.323
| 6 | 25.204 | 25.150 | 25.187 | 25.110 | 25.246 | 25.200 | 25.230 | 25.291 | 25.289 | 25.278 | 25.913 | 25.934 | 26.069 | 26.060
| 12 | 29.364 | 29.369 | 29.335 | 29.387 | 29.397 | 29.387 | 29.300 | 29.460 | 29.461 | 29.545 | 29.883 | 29.930 | 29.883 | 29.955
| 18 | 31.929 | 31.983 | 31.965 | 31.951 | 32.015 | 32.055 | 31.981 | 32.004 | 32.179 | 32.246 | 32.308 | 32.361 | 32.411 | 32.392
Huai. | 24 | 33.729 | 33.776 | 33.809 | 33.716 | 33.771 | 33.820 | 33.846 | 33.889 | 33.915 | 33.928 | 34.003 | 34.006 | 34.066 | 34.094
| 6 | 25.166 | 25.133 | 25.097 | 25.101 | 25.192 | 25.188 | 25.134 | 25.215 | 25.216 | 25.249 | 25.918 | 25.945 | 26.083 | 26.140
| 12 | 29.728 | 29.682 | 29.701 | 29.711 | 29.724 | 29.705 | 29.802 | 29.805 | 29.850 | 29.854 | 30.225 | 30.275 | 30.313 | 30.326
| 18 | 32.347 | 32.377 | 32.376 | 32.366 | 32.426 | 32.430 | 32.475 | 32.438 | 32.449 | 32.517 | 32.788 | 32.836 | 32.831 | 32.855
Nong. | 24 | 34.145 | 34.209 | 34.201 | 34.250 | 34.336 | 34.289 | 34.289 | 34.405 | 34.353 | 34.323 | 34.520 | 34.542 | 34.582 | 34.568
| Avg. rank | 5.27 | 4.45 | 4.27 | 4.28 | 5.62 | 5.83 | 5.87 | 6.80 | 8.12 | 9.10 | 10.28 | 10.95 | 11.82 | 12.35
Sum | # of wins | 26 | 6 | 7 | 7 | 5 | 1 | 3 | 2 | 2 | 0 | 0 | 1 | 0 | 0

Table 12. SMAPE ($\downarrow$) comparison among hyperparameter tuning.

Variation | NFCL-V | NFCL-C | NFCL-D
---|---|---|---
Datasets | $T$ | - | 16 | 32 | 64 | 128 | 16,16 | 32,32 | 64,64 | 128,128 | 128,128,64 | 64 | 128 | 64,64 | 128,128
| 6 | 0.821 | 0.821 | 0.822 | 0.824 | 0.827 | 0.822 | 0.824 | 0.823 | 0.826 | 0.825 | 0.824 | 0.824 | 0.822 | 0.822
| 12 | 0.762 | 0.769 | 0.772 | 0.773 | 0.775 | 0.768 | 0.771 | 0.774 | 0.775 | 0.774 | 0.768 | 0.769 | 0.769 | 0.768
| 18 | 0.704 | 0.714 | 0.717 | 0.718 | 0.719 | 0.714 | 0.715 | 0.714 | 0.713 | 0.712 | 0.707 | 0.711 | 0.706 | 0.709
Weather | 24 | 0.639 | 0.658 | 0.659 | 0.659 | 0.661 | 0.655 | 0.658 | 0.656 | 0.657 | 0.654 | 0.651 | 0.651 | 0.648 | 0.649
| 6 | 0.560 | 0.198 | 0.208 | 0.137 | 0.108 | 0.270 | 0.281 | 0.117 | 0.239 | 0.232 | -0.646 | -0.702 | -0.836 | -1.172
| 12 | 0.110 | -0.374 | -0.391 | -0.445 | -0.325 | -0.237 | -0.146 | -0.284 | -0.203 | 0.044 | -1.506 | -1.661 | -2.288 | -2.921
| 18 | -0.151 | -0.720 | -0.760 | -0.737 | -0.669 | -0.487 | -0.509 | -0.417 | -0.272 | -0.014 | -1.789 | -1.823 | -2.260 | -2.912
ILL | 24 | -0.257 | -0.843 | -0.991 | -0.887 | -0.895 | -0.595 | -0.634 | -0.567 | -0.423 | -0.201 | -1.691 | -1.631 | -1.161 | -1.686
| 6 | 0.950 | 0.958 | 0.958 | 0.958 | 0.959 | 0.959 | 0.958 | 0.959 | 0.957 | 0.959 | 0.960 | 0.960 | 0.960 | 0.960
| 12 | 0.925 | 0.937 | 0.938 | 0.938 | 0.940 | 0.940 | 0.940 | 0.940 | 0.939 | 0.941 | 0.940 | 0.941 | 0.941 | 0.941
| 18 | 0.885 | 0.920 | 0.920 | 0.921 | 0.922 | 0.923 | 0.922 | 0.922 | 0.921 | 0.922 | 0.924 | 0.924 | 0.925 | 0.925
Exchange | 24 | 0.858 | 0.906 | 0.909 | 0.907 | 0.907 | 0.907 | 0.906 | 0.909 | 0.907 | 0.907 | 0.913 | 0.912 | 0.911 | 0.912
| 6 | 0.663 | 0.661 | 0.658 | 0.665 | 0.657 | 0.659 | 0.661 | 0.659 | 0.652 | 0.660 | 0.664 | 0.664 | 0.656 | 0.657
| 12 | 0.591 | 0.562 | 0.565 | 0.560 | 0.562 | 0.561 | 0.564 | 0.562 | 0.552 | 0.561 | 0.583 | 0.576 | 0.575 | 0.574
| 18 | 0.565 | 0.526 | 0.527 | 0.526 | 0.529 | 0.523 | 0.529 | 0.526 | 0.535 | 0.531 | 0.535 | 0.541 | 0.538 | 0.546
ETTh1 | 24 | 0.561 | 0.529 | 0.531 | 0.525 | 0.525 | 0.524 | 0.525 | 0.528 | 0.523 | 0.520 | 0.531 | 0.529 | 0.532 | 0.527
| 6 | 0.699 | 0.696 | 0.694 | 0.694 | 0.696 | 0.697 | 0.697 | 0.692 | 0.693 | 0.692 | 0.689 | 0.689 | 0.685 | 0.688
| 12 | 0.628 | 0.626 | 0.628 | 0.629 | 0.628 | 0.628 | 0.628 | 0.622 | 0.622 | 0.623 | 0.626 | 0.624 | 0.622 | 0.622
| 18 | 0.590 | 0.592 | 0.593 | 0.594 | 0.590 | 0.588 | 0.591 | 0.589 | 0.589 | 0.587 | 0.588 | 0.588 | 0.586 | 0.586
ETTh2 | 24 | 0.566 | 0.566 | 0.569 | 0.570 | 0.564 | 0.568 | 0.565 | 0.566 | 0.565 | 0.565 | 0.567 | 0.564 | 0.566 | 0.562
| 6 | 0.823 | 0.828 | 0.828 | 0.827 | 0.828 | 0.827 | 0.827 | 0.827 | 0.827 | 0.827 | 0.825 | 0.824 | 0.824 | 0.824
| 12 | 0.704 | 0.717 | 0.718 | 0.717 | 0.716 | 0.717 | 0.716 | 0.715 | 0.715 | 0.716 | 0.712 | 0.711 | 0.709 | 0.709
| 18 | 0.589 | 0.606 | 0.609 | 0.609 | 0.606 | 0.608 | 0.607 | 0.609 | 0.608 | 0.606 | 0.598 | 0.600 | 0.597 | 0.601
ETTm1 | 24 | 0.481 | 0.504 | 0.503 | 0.506 | 0.508 | 0.505 | 0.506 | 0.506 | 0.506 | 0.505 | 0.492 | 0.493 | 0.492 | 0.493
| 6 | 0.830 | 0.832 | 0.832 | 0.831 | 0.833 | 0.833 | 0.832 | 0.832 | 0.832 | 0.831 | 0.831 | 0.830 | 0.830 | 0.830
| 12 | 0.762 | 0.767 | 0.767 | 0.768 | 0.768 | 0.768 | 0.766 | 0.767 | 0.766 | 0.766 | 0.763 | 0.763 | 0.762 | 0.762
| 18 | 0.705 | 0.713 | 0.712 | 0.713 | 0.713 | 0.714 | 0.712 | 0.713 | 0.712 | 0.712 | 0.705 | 0.705 | 0.704 | 0.705
ETTm2 | 24 | 0.657 | 0.664 | 0.664 | 0.664 | 0.665 | 0.665 | 0.664 | 0.664 | 0.666 | 0.664 | 0.653 | 0.654 | 0.655 | 0.655
| 6 | 0.779 | 0.781 | 0.781 | 0.781 | 0.781 | 0.781 | 0.782 | 0.780 | 0.779 | 0.780 | 0.770 | 0.770 | 0.770 | 0.771
| 12 | 0.676 | 0.681 | 0.682 | 0.682 | 0.681 | 0.682 | 0.682 | 0.681 | 0.681 | 0.680 | 0.667 | 0.666 | 0.666 | 0.667
| 18 | 0.607 | 0.615 | 0.616 | 0.615 | 0.613 | 0.616 | 0.617 | 0.614 | 0.615 | 0.616 | 0.601 | 0.600 | 0.600 | 0.598
Aotiz. | 24 | 0.554 | 0.566 | 0.566 | 0.567 | 0.568 | 0.567 | 0.567 | 0.570 | 0.567 | 0.567 | 0.549 | 0.548 | 0.547 | 0.547
| 6 | 0.748 | 0.751 | 0.751 | 0.752 | 0.752 | 0.752 | 0.751 | 0.751 | 0.751 | 0.751 | 0.743 | 0.742 | 0.743 | 0.743
| 12 | 0.655 | 0.659 | 0.658 | 0.657 | 0.658 | 0.656 | 0.658 | 0.656 | 0.656 | 0.656 | 0.652 | 0.651 | 0.652 | 0.652
| 18 | 0.591 | 0.597 | 0.595 | 0.598 | 0.596 | 0.597 | 0.598 | 0.597 | 0.594 | 0.596 | 0.588 | 0.588 | 0.587 | 0.588
Chang. | 24 | 0.541 | 0.550 | 0.549 | 0.550 | 0.550 | 0.550 | 0.549 | 0.551 | 0.545 | 0.549 | 0.539 | 0.539 | 0.540 | 0.539
| 6 | 0.789 | 0.793 | 0.793 | 0.793 | 0.792 | 0.792 | 0.793 | 0.792 | 0.791 | 0.791 | 0.786 | 0.785 | 0.785 | 0.784
| 12 | 0.686 | 0.693 | 0.692 | 0.692 | 0.691 | 0.692 | 0.691 | 0.691 | 0.690 | 0.690 | 0.685 | 0.684 | 0.683 | 0.684
| 18 | 0.608 | 0.618 | 0.617 | 0.617 | 0.617 | 0.616 | 0.615 | 0.614 | 0.615 | 0.614 | 0.608 | 0.607 | 0.608 | 0.606
Ding. | 24 | 0.552 | 0.562 | 0.562 | 0.561 | 0.561 | 0.561 | 0.561 | 0.560 | 0.560 | 0.559 | 0.553 | 0.552 | 0.551 | 0.551
| 6 | 0.767 | 0.772 | 0.771 | 0.772 | 0.771 | 0.772 | 0.771 | 0.771 | 0.770 | 0.769 | 0.763 | 0.763 | 0.761 | 0.762
| 12 | 0.663 | 0.671 | 0.672 | 0.670 | 0.671 | 0.671 | 0.670 | 0.670 | 0.669 | 0.667 | 0.662 | 0.662 | 0.661 | 0.661
| 18 | 0.594 | 0.602 | 0.604 | 0.603 | 0.600 | 0.602 | 0.601 | 0.600 | 0.599 | 0.598 | 0.594 | 0.593 | 0.593 | 0.592
Dongsi | 24 | 0.541 | 0.549 | 0.550 | 0.549 | 0.548 | 0.548 | 0.549 | 0.547 | 0.546 | 0.544 | 0.542 | 0.541 | 0.542 | 0.540
| 6 | 0.761 | 0.766 | 0.766 | 0.765 | 0.766 | 0.766 | 0.766 | 0.765 | 0.765 | 0.766 | 0.756 | 0.756 | 0.757 | 0.756
| 12 | 0.665 | 0.670 | 0.671 | 0.670 | 0.671 | 0.670 | 0.669 | 0.670 | 0.669 | 0.668 | 0.655 | 0.656 | 0.657 | 0.657
| 18 | 0.598 | 0.605 | 0.605 | 0.607 | 0.603 | 0.605 | 0.604 | 0.604 | 0.603 | 0.600 | 0.585 | 0.586 | 0.585 | 0.585
Guan. | 24 | 0.546 | 0.554 | 0.553 | 0.554 | 0.552 | 0.553 | 0.552 | 0.552 | 0.550 | 0.552 | 0.534 | 0.529 | 0.530 | 0.534
| 6 | 0.769 | 0.772 | 0.771 | 0.771 | 0.771 | 0.771 | 0.771 | 0.770 | 0.769 | 0.771 | 0.768 | 0.767 | 0.767 | 0.766
| 12 | 0.666 | 0.670 | 0.671 | 0.671 | 0.670 | 0.671 | 0.670 | 0.669 | 0.672 | 0.669 | 0.665 | 0.663 | 0.664 | 0.663
| 18 | 0.594 | 0.603 | 0.604 | 0.603 | 0.602 | 0.603 | 0.603 | 0.602 | 0.604 | 0.601 | 0.593 | 0.593 | 0.592 | 0.591
Guch. | 24 | 0.541 | 0.553 | 0.554 | 0.553 | 0.553 | 0.555 | 0.554 | 0.552 | 0.553 | 0.553 | 0.542 | 0.541 | 0.543 | 0.540
| 6 | 0.744 | 0.749 | 0.748 | 0.749 | 0.748 | 0.748 | 0.748 | 0.746 | 0.747 | 0.747 | 0.737 | 0.736 | 0.736 | 0.735
| 12 | 0.639 | 0.645 | 0.647 | 0.645 | 0.644 | 0.645 | 0.645 | 0.644 | 0.645 | 0.643 | 0.634 | 0.633 | 0.632 | 0.633
| 18 | 0.568 | 0.578 | 0.576 | 0.577 | 0.577 | 0.576 | 0.577 | 0.575 | 0.574 | 0.574 | 0.565 | 0.565 | 0.563 | 0.565
Huai. | 24 | 0.513 | 0.524 | 0.526 | 0.524 | 0.524 | 0.525 | 0.525 | 0.523 | 0.523 | 0.522 | 0.514 | 0.512 | 0.511 | 0.512
| 6 | 0.773 | 0.773 | 0.773 | 0.773 | 0.772 | 0.773 | 0.774 | 0.772 | 0.772 | 0.772 | 0.765 | 0.765 | 0.764 | 0.763
| 12 | 0.669 | 0.671 | 0.672 | 0.671 | 0.670 | 0.672 | 0.670 | 0.669 | 0.669 | 0.669 | 0.661 | 0.660 | 0.660 | 0.661
| 18 | 0.603 | 0.605 | 0.606 | 0.605 | 0.603 | 0.605 | 0.604 | 0.604 | 0.604 | 0.603 | 0.594 | 0.594 | 0.595 | 0.593
Nong. | 24 | 0.554 | 0.556 | 0.556 | 0.556 | 0.554 | 0.556 | 0.555 | 0.554 | 0.552 | 0.553 | 0.544 | 0.544 | 0.542 | 0.543
| Avg. rank | 9.37 | 5.37 | 4.65 | 4.68 | 5.52 | 4.78 | 5.18 | 6.80 | 7.28 | 7.50 | 10.12 | 10.73 | 11.32 | 11.70
Sum | # of wins | 6 | 8 | 11 | 9 | 6 | 5 | 4 | 2 | 3 | 2 | 1 | 0 | 3 | 0

Table 13. $R^{2}$ ($\uparrow$) comparison among hyperparameter tuning.

##  Appendix H Full output layer weight map

Every NFCL performs the operation $X^{\prime}\cdot w$ (as defined in Section
4, where $w\in\mathbb{R}^{k,l,k,t}$, and $X^{\prime}\subset{X,h(X)}$). The
full map is obtained by reshaping $w$ into $\mathbb{R}^{k,l,kt}$, and then
conducting an element-wise product with $X^{\prime}$. Finally, we extract the
result of $k\cdot L+t$ rows from the full map to predict a specific time-
series $k$ at a specific time $t$. In the main context, the sub-sections 5.8
and 5.9 used the figure 8. The sub-sections 5.8 used the part of the blue-
colored wide-shallow box in Figure 8. Moreover, blued boxes reshaped into
$\mathbb{R}^{k,l}$ only to provide forecasting reasons in a specific time-
series $k$ at a specific time $t$. Also, the sub-sections 5.9 used the part of
the red and green colored boxes in Figure 8.

Figure 8. Fully illustrated the output layer’s weight $h(X)\odot w$.

##  Appendix I Analysis of the number of model’s parameters

Table 14 shows the number of learnable parameters. From the first to the third
columns denote the dataset, the number of time-series $K$, and the forecasting
length $T$. The fourth column onward shows the number of training parameters
per data shape in the model specified in the first row.

Dataset | $K$ | $T$ | NFCL-V | NFCL-C | DLinear | NLinear | TiDE | Informer | Autoformer | PatchTST | iTransformer | SCINet | TimesNet
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Weather | 21 | 6 | 63672 | 112560 | 300 | 150 | 1585331 | 11375125 | 10604565 | 6324230 | 6321670 | 100104 | 1174579
12 | 127302 | 176190 | 600 | 300 | 1727297 | 11375125 | 10604565 | 6333452 | 6324748 | 100248 | 1174729
18 | 190932 | 239820 | 900 | 450 | 1869263 | 11375125 | 10604565 | 6342674 | 6327826 | 100392 | 1174879
24 | 254562 | 303450 | 1200 | 600 | 2011229 | 11375125 | 10604565 | 6351896 | 6330904 | 100536 | 1175029
Nong. | 10 | 6 | 14480 | 37760 | 300 | 150 | 1496486 | 11338762 | 10551306 | 6324230 | 6321670 | 23104 | 1173912
12 | 28940 | 52220 | 600 | 300 | 1570604 | 11338762 | 10551306 | 6333452 | 6324748 | 23248 | 1174062
18 | 43400 | 66680 | 900 | 450 | 1644722 | 11338762 | 10551306 | 6342674 | 6327826 | 23392 | 1174212
24 | 57860 | 81140 | 1200 | 600 | 1718840 | 11338762 | 10551306 | 6351896 | 6330904 | 23536 | 1174362
Exchange | 8 | 6 | 9280 | 27904 | 300 | 150 | 1479538 | 11330568 | 10540040 | 6324230 | 6321670 | 14928 | 1173766
12 | 18544 | 37168 | 600 | 300 | 1541320 | 11330568 | 10540040 | 6333452 | 6324748 | 15072 | 1173916
18 | 27808 | 46432 | 900 | 450 | 1603102 | 11330568 | 10540040 | 6342674 | 6327826 | 15216 | 1174066
24 | 37072 | 55696 | 1200 | 600 | 1664884 | 11330568 | 10540040 | 6351896 | 6330904 | 15360 | 1174216
ILL | 7 | 6 | 7112 | 23408 | 300 | 150 | 1471321 | 11326983 | 10534919 | 6324230 | 6321670 | 11512 | 1173701
12 | 14210 | 30506 | 600 | 300 | 1526935 | 11326983 | 10534919 | 6333452 | 6324748 | 11656 | 1173851
18 | 21308 | 37604 | 900 | 450 | 1582549 | 11326983 | 10534919 | 6342674 | 6327826 | 11800 | 1174001
24 | 28406 | 44702 | 1200 | 600 | 1638163 | 11326983 | 10534919 | 6351896 | 6330904 | 11944 | 1174151

Table 14. The number of learnable parameters according to the nine baselines
and two NFCLs. Note that NFCL-D has the number of learnable parameter as three
times of NFCL-C’s since we hired the three decomposition blocks.

##  Appendix J Comparison of NFCLs generalization.

Figure 9. Illustration of NFCLs forecasting when $T=0$.

We meticulously prepared figures and tables to systematically analyze the
performance degradation of NFCL-D. Initially, Figure 9 was constructed by
training NFCL-V, C, and D adequately on the ILL dataset, and NFCLs set up the
$T$ to 18. Subsequently, upon completion of the training process, only the
values corresponding to $T=0$ among the predicted results for the test set
were imported and visualized. In the figure 9, the blue, yellow, green, and
red lines represent the real $Y$ and the predictions of NFCL-V, C, and D,
respectively.

Figure 9 illustrates three instances of prediction failure. Firstly, in area
1, NFCL-D exhibits a failure in prediction compared to the successful