# FITS: Modeling Time Series with $10k$ Parameters

Xu Zhijian, Zeng Ailing, Xu Qiang
Department of Computer Science and Engineering
The Chinese University of Hong Kong
Shatin, NT, Hong Kong
{zjxu21, qxu<EMAIL_ADDRESS>

###### Abstract

In this paper, we introduce FITS, a lightweight yet powerful model for time
series analysis. Unlike existing models that directly process raw time-domain
data, FITS operates on the principle that time series can be manipulated
through interpolation in the complex frequency domain, achieving performance
comparable to state-of-the-art models for time series forecasting and anomaly
detection tasks. Notably, FITS accomplishes this with a svelte profile of just
about $10k$ parameters, making it ideally suited for edge devices and paving
the way for a wide range of applications. The code is available:
https://github.com/VEWOXIC/FITS.

##  1 Introduction

Time series analysis plays a pivotal role in a myriad of sectors, from
healthcare appliances to smart factories. Within these domains, the reliance
is often on edge devices like smart sensors, driven by MCUs with limited
computational and memory resources. Time series data, marked by its inherent
complexity and dynamism, typically presents information that is both sparse
and scattered within the time domain. To effectively harness this data, recent
research has given rise to sophisticated models and methodologies (Zhou et
al., 2021; Liu et al., 2022a; Zeng et al., 2023; Nie et al., 2023; Zhang et
al., 2022). Yet, the computational and memory costs of these models makes them
unsuitable for resource-constrained edge devices.

On the other hand, the frequency domain representation of time series data
promises a more compact and efficient portrayal of inherent patterns. While
existing research has indeed tapped into the frequency domain for time series
analysis — FEDformer (Zhou et al., 2022a) enriches its features using spectral
data, and TimesNet (Wu et al., 2023) harnesses high-amplitude frequencies for
feature extraction via CNNs — a comprehensive utilization of the frequency
domain’s compactness remains largely unexplored. Specifically, the ability of
the frequency domain to employ complex numbers in capturing both amplitude and
phase information is not utilized, resulting in the continued reliance on
compute-intensive models for temporal feature extraction.

In this study, _we reinterpret time series analysis tasks, such as forecasting
and reconstruction, as interpolation exercises within the complex frequency
domain_. Essentially, we produce an extended time series segment by
interpolating the frequency representation of a provided segment.
Specifically, for forecasting, we can obtain the forecasting results by simply
extending the given look-back window with frequency interpolation; for
reconstruction, we recover the original segment by interpolating the frequency
representation of its downsampled counterpart. Building on this insight, we
introduce FITS (Frequency Interpolation Time Series Analysis Baseline). The
core of FITS is a complex-valued linear layer, meticulously designed to learn
amplitude scaling and phase shift, thereby facilitating interpolation within
the complex frequency domain.

Notably, while FITS operates interpolations in the frequency domain, it
fundamentally remains a time domain model, integrating the rFFT (Brigham &
Morrow, 1967) operation. That is, we transform the input segment into the
complex frequency domain using rFFT for frequency interpolation. This
interpolated frequency data is then mapped back to the time domain, resulting
in an elongated segment ready for supervision. This innovative design allows
FITS to be highly adaptable, fitting seamlessly into a plethora of downstream
time domain tasks such as forecasting and anomaly detection.

Apart from its streamlined linear architecture, FITS incorporates a low-pass
filter. This ensures a compact representation while preserving essential
information. Despite its simplicity, FITS consistently achieves state-of-the-
art (SOTA) performance. Remarkably, in most scenarios, FITS achieves this feat
with fewer than 10k parameters. This makes it 50 times more compact than the
lightweight temporal linear model DLinear (Zeng et al., 2023) and
approximately 10,000 times smaller than other mainstream models. Given its
efficiency in memory and computation, FITS stands out as an ideal candidate
for deployment, or even for training directly on edge devices, be it for
forecasting or anomaly detection.

In summary, our contributions can be delineated as follows:

  * •

We present FITS, an exceptionally lightweight model for time series analysis,
boasting a modest parameter count in the range of 5k$\sim$10k.

  * •

FITS offers a pioneering approach to time series analysis by employing a
complex-valued neural network. This simultaneously captures both amplitude and
phase information, paving the way for a more comprehensive and efficient
representation of time series data.

  * •

Despite being orders of magnitude smaller than most mainstream models, FITS
consistently delivers top-tier performance across a range of time series
analysis tasks.

##  2 Related Work and Motivation

###  2.1 Frequency-aware Time Series Analysis Models

Recent advancements in time series analysis have witnessed the utilization of
frequency domain information to capture and interpret underlying patterns.
FNet (Lee-Thorp et al., 2022) leverages a pure attention-based architecture to
efficiently capture temporal dependencies and patterns solely in the frequency
domain, eliminating the need for convolutional or recurrent layers. On the
other hand, FEDFormer (Zhou et al., 2022a) and FiLM (Zhou et al., 2022b)
incorporate frequency information as supplementary features to enhance the
model’s capability in capturing long-term periodic patterns and speed up
computation.

The other line of work aims to capture the periodicity inherent in the data.
For instance, DLinear (Zeng et al., 2023) adopts a single linear layer to
extract the dominant periodicity from the temporal domain and surpasses a
range of deep feature extraction-based methods. More recently, TimesNet (Wu et
al., 2023) achieves state-of-the-art results by identifying several dominant
frequencies instead of relying on a single dominant periodicity. Specifically,
they use the Fast Fourier Transform (FFT) to find the frequencies with the
largest energy and reshape the original 1D time series into 2D images
according to their periods.

However, these approaches still rely on feature engineering to identify the
dominant period set. Selecting this set based on energy may only consider the
dominant period and its harmonics, limiting the information captured.
Moreover, these methodologies are still considered inefficient and prone to
overfitting.

###  2.2 Divide and Conquer the Frequency Components

Treating a time series as a signal allows us to break it down into a linear
combination of sinusoidal components without any information loss. Each
component possesses a unique frequency, initial phase, and amplitude.
Forecasting directly on the original time series can be challenging, but
forecasting each frequency component is comparatively straightforward, as we
only need to apply a phase bias to the sinusoidal wave based on the time
shift. Subsequently, we linearly combine these shifted sinusoidal waves to
obtain the forecasting result.

This approach effectively preserves the frequency characteristics of the given
look-back window while maintaining semantic consistency between the look-back
window and the forecasting horizon. Specifically, the resulting forecasted
values maintain the frequency features of the original time series with a
reasonable time shift, ensuring that semantic consistency is maintained.

However, forecasting each sinusoidal component in the time domain can be
cumbersome, as the sinusoidal components are treated as a sequence of data
points. To address this, we propose conducting this manipulation in the
complex frequency domain, which offers a more compact and information-rich
representation, as described below.

##  3 Method

###  3.1 Preliminary: FFT and Complex Frequency Domain

The Fast Fourier Transform (FFT, (Brigham & Morrow, 1967)) efficiently
computes the Discrete Fourier Transform (DFT) of complex number sequences. The
DFT transforms discrete-time signals from the time domain to the complex
frequency domain. In time series analysis, the Real FFT (rFFT) is often
employed when working with real input signals. It condenses an input of N real
numbers into a sequence of N/2+1 complex numbers, representing the signal in
the complex frequency domain.

Complex Frequency Domain

In Fourier analysis, the complex frequency domain is a representation of a
signal in which each frequency component is characterized by a complex number.
This complex number captures both the amplitude and phase of the component,
providing a comprehensive description. The amplitude of a frequency component
represents the magnitude or strength of that component in the original time-
domain signal. In contrast, the phase represents the temporal shift or delay
introduced by that component. Mathematically, the complex number associated
with a frequency component can be represented as a complex exponential element
with a given amplitude and phase:

$X(f)=|X(f)|e^{j\theta(f)},$

where $X(f)$ is the complex number associated with the frequency component at
frequency $f$, $|X(f)|$ is the amplitude of the component, and $\theta(f)$ is
the phase of the component. As shown in Fig. 1(a), in the complex plane, the
complex exponential element can be visualized as a vector with a length equal
to the amplitude and angle equal to the phase:

$X(f)=|X(f)|(\cos{\theta(f)}+j\sin{\theta(f)})$

Therefore, the complex number in the complex frequency domain provides a
concise and elegant means of representing the amplitude and phase of each
frequency component in the Fourier transform.

(a) Complex number on the complex plane

(b) Complex number multiplication

Figure 1: Illustration of Complex Number Visualization and Multiplication

Time Shift and Phase Shift. The time shift of a signal corresponds to the
phase shift in the frequency domain. Especially in the complex frequency
domain, we can express such phase shift by multiplying a unit complex
exponential element with the corresponding phase. Mathematically, if we shift
a signal $x(t)$ forward in time by a constant amount $\tau$, resulting in the
signal $x(t-\tau)$, the Fourier transform is given by:

$X_{\tau}(f)=e^{-j2\pi f\tau}X(f)=|X(f)|e^{j(\theta(f)-2\pi f\tau)}\\\
=[cos(-2\pi f\tau)+jsin(-2\pi f\tau)]X(f)$

The shifted signal still has an amplitude of $|X(f)|$, while the phase
$\theta_{\tau}(f)=\theta(f)-2\pi f\tau$ shows a shift which is linear to the
time shift.

In summary, the amplitude scaling and phase shifting can be simultaneously
expressed as the multiplication of complex numbers, as shown in Fig. 1(b).

###  3.2 FITS Pipeline

Motivated by the fact that a longer time series provides a higher frequency
resolution in its frequency representation, we train FITS to extend time
series segment by interpolating the frequency representation of the input time
series segment. We use a single layer of complex-valued linear layer to learn
such interpolation, so that it can learn amplitude scaling and phase shifting
as the multiplication of complex numbers during the interpolation process. As
shown in Fig. 2, we use rFFT to project time series segments to the complex
frequency domain. After the interpolation, the frequency representation is
projected back with inverse rFFT (irFFT).

Figure 2: Pipeline of FITS, with a focus on the forecasting task. Initially,
the time series is normalized to zero-mean, followed by rFFT for frequency
domain projection. After LPF, a single complex-valued linear layer
interpolates the frequency. Zero padding and irFFT then revert this back to
the time domain, with iRIN finally reversing the normalization. The
reconstruction task follows the same pipeline, except for the reconstruction
supervision loss. Please check appendix for detail.

However, the mean of such segments will result in a very large 0-frequency
component in its complex frequency representation. To address this, we pass it
through reversible instance-wise normalization (RIN) (Kim et al., 2022) to
obtain a zero-mean instance. As a result, the normalized complex frequency
representation now has a length of $N/2$, where $N$ represents the original
length of the time series.

Additionally, FITS integrates a low-pass filter (LPF) to further reduce its
model size. The LPF effectively eliminates high-frequency components above a
specified cutoff frequency, compacting the model representation while
preserving essential time series information. Despite operating in the
frequency domain, FITS is supervised in the time domain using standard loss
functions like Mean Squared Error (MSE) after the inverse real-to-complex Fast
Fourier Transform (irFFT). This allows for versatile supervision tailored to
various downstream time series tasks.

In the case of forecasting tasks, we generate the look-back window along with
the horizon as shown in Fig. 2. This allows us to provide supervision for
forecasting and backcasting, where the model is encouraged to accurately
reconstruct the look-back window. Our ablation study reveals that combining
backcast and forecast supervision can yield improved performance in certain
scenarios.

For reconstruction tasks, we downsample the original time series segment based
on a specific downsampling rate. Subsequently, FITS is employed to perform
frequency interpolation, enabling the reconstruction of the downsampled
segment back to its original form. Thus, direct supervision is applied using
reconstruction loss to ensure faithful reconstruction. The reconstruction
tasks also follow the pipeline in Fig. 2 with the supervision replaced with
reconstruction loss.

###  3.3 Key Mechanisms of FITS

Complex Frequency Linear Interpolation. To control the output length of the
model, we introduce an interpolation rate denoted as $\eta$, which represents
the ratio of the model’s output length $L_{o}$ to its corresponding input
length $L_{i}$. Frequency interpolation operates on the normalized complex
frequency representation, which has half the length of the original time
series. Importantly, this interpolation rate can also be applied to the
frequency domain, as indicated by the equation:

$\eta_{freq}=\frac{L_{o}/2}{L_{i}/2}=\frac{L_{o}}{L_{i}}=\eta$

Based on this formula, with an arbitrary frequency $f$, the frequency band
$1\sim f$ in the original signal is linearly projected to the frequency band
$1\sim\eta f$ in the output signal. As a result, we define the input length of
our complex-valued linear layer as $L$ and the interpolated output length as
$\eta L$. Notably, when applying the Low Pass Filter (LPF), the value of $L$
corresponds to the cutoff frequency (COF) of the LPF. After performing
frequency interpolation, the complex frequency representation is zero-padded
to a length of $L_{o}/2$, where $L_{o}$ represents the desired output length.
Prior to applying the irFFT, an additional zero is introduced as the
representation’s zero-frequency component.

Low Pass Filter (LPF). The primary objective of incorporating the LPF within
FITS is to compress the model’s volume while preserving essential information.
The LPF achieves this by discarding frequency components above a specified
cutoff frequency (COF), resulting in a more concise frequency domain
representation. The LPF retains the relevant information in the time series
while discarding components beyond the model’s learning capability. This
ensures that a significant portion of the original time series’ meaningful
content is preserved. As demonstrated in Fig. 3, the filtered waveform
exhibits minimal distortion even when only preserving a quarter of the
original frequency domain representation. Furthermore, the high-frequency
components filtered out by the LPF typically comprise noise, which are
inherently irrelevant for effective time series modeling.

(a) Original

(b) COF at 6th harmonic (c) COF at 3rd harmonic (d) COF at 2nd harmonic

Figure 3: Waveform (1st row) and amplitude spectrum (2nd row) of a time series
segment selected from the ’OT’ channel of the ETTh1 dataset, spanning from the
1500th to the 1980th data point. The segment has a length of 480, and its
dominant periodicity is 24, corresponding to a base frequency of 20. The blue
lines represent the waveform/spectrum with no applied filter, while the orange
lines represent the waveform/spectrum with the filter applied. The filter
cutoff frequency is chosen based on a harmonic of the original time series.

Selecting an appropriate cutoff frequency (COF) remains a nontrivial
challenge. To address this, we propose a method based on the harmonic content
of the dominant frequency. Harmonics, which are integer multiples of the
dominant frequency, play a significant role in shaping the waveform of a time
series. By aligning the cutoff frequency with these harmonics, we keep
relevant frequency components associated with the signal’s structure and
periodicity. This approach leverages the inherent relationship between
frequencies to extract meaningful information while suppressing noise and
irrelevant high-frequency components. The impact of COF on different
harmonics’ waveforms is shown in Fig. 3. We further elaborate on the impact of
COF in our experimental results.

Weight Sharing. FITS handles multivariate tasks by sharing weights as in (Zeng
et al., 2023), balancing performance and efficiency. In practice, channels
often share a common base frequency when originating from the same physical
system, such as 50/60Hz for electrical appliances or daily base frequencies
for city traffic. Most of the datasets used in our experiments belong to this
category. For datasets that indeed contain channels with different base
frequencies, we can cluster those channels according to the base frequency and
train an individual FITS model for each cluster.

##  4 Experiments for Forecasting

###  4.1 Forecasting as Frequency Interpolation

Typically, the forecasting horizon is shorter than the given look-back window,
rendering direct interpolation unsuitable. Instead, we formulate the
forecasting task as the interpolation of a look-back window, with length $L$,
to a combination of the look-back window and forecasting horizon, with length
$L+H$. This design enables us to provide more supervision during training.
With this approach, we can supervise not only the forecasting horizon but also
the backcast task on the look-back window. Our experimental results
demonstrate that this unique training strategy contributes to the improved
performance of FITS. The interpolation rate of the forecasting task is
calculated by:

$\eta_{Fore}=1+\frac{H}{L},$

where $L$ represents the length of the look-back window and $H$ represents the
length of the forecasting horizon.

###  4.2 Experiment Settings

Datasets. All datasets used in our experiments are widely-used and publicly
available real-world datasets, including, Traffic, Electricity, Weather, ETT
(Zhou et al., 2021). We summarize the characteristics of these datasets in
appendix. Apart from these datasets for long-term time series forecasting, we
also use the M4 dataset to test the short-term forecasting performance.

Baselines. To evaluate the performance of FITS in comparison to state-of-the-
art time series forecasting models, including PatchTST (Nie et al., 2023),
TimesNet (Wu et al., 2023), FEDFormer (Zhou et al., 2022a) and LTSF-Linear
(Zeng et al., 2023), we rerun all the experiment with code and scripts
provided by their official implementation 111With a long-standing bug in the
coding architecture fixed, see README file in our codebase.. We report the
comparison with NBeats (Oreshkin et al., 2019), NHits (Challu et al., 2023)
and other transformer-based methods in the appendix.

Evaluation metrics. We follow the previous works (Zhou et al., 2022a; Zeng et
al., 2023; Zhang et al., 2022) to compare forecasting performance using Mean
Squared Error (MSE) as the core metrics. Moreover, to evaluate the short-term
forecasting, we symmetric Mean Absolute Percentage Error (SMAPE) following
TimesNet (Wu et al., 2023).

Implementation details. We conduct grid search on the look-back window of 90,
180, 360, 720 and cutoff frequency, the only hyper-parameter. Further
experiments also show that a longer look-back window can result in better
performance in most cases. To avoid information leakage, We choose the hyper-
parameter based on the performance of the validation set. We report the result
of FITS as the mean and standard deviation of 5 runs with random chosen random
seeds.

###  4.3 Comparisons with SOTAs

Competitive Performance with High Efficiency

We present the results of our experiments on long-term forecasting in Tab. 1
and Tab. 2. The results for short-term forecasting on the M4 dataset are
provided in the Appendix. Remarkably, our FITS consistently achieves
comparable or even superior performance across all experiments.

Tab. 3 presents the number of trainable parameters and MACs 222MACs (Multiply-
Accumulate Operations) is a commonly used metric that counts the total number
of multiplication and addition operations in a neural network.  for various
TSF models using a look-back window of 96 and a forecasting horizon of 720 on
the Electricity dataset. The table clearly demonstrates the exceptional
efficiency of FITS compared to other models.

Table 1: Long-term forecasting results on ETT dataset in MSE. The best result
is highlighted in bold, and the second best is highlighted with underline. IMP
is the improvement between FITS and the second best/ best result, where a
larger value indicates a better improvement. Most of the STD are under 5e-4
and shown as 0.000 in this table.

Dataset ETTh1 ETTh2 ETTm1 ETTm2 Horizon 96 192 336 720 96 192 336 720 96 192
336 720 96 192 336 720 PatchTST 0.385 0.413 0.440 0.456 0.274 0.338 0.367
0.391 0.292 0.330 0.365 0.419 0.163 0.219 0.276 0.368 Dlinear 0.384 0.443
0.446 0.504 0.282 0.350 0.414 0.588 0.301 0.335 0.371 0.426 0.171 0.237 0.294
0.426 FedFormer 0.375 0.427 0.459 0.484 0.340 0.433 0.508 0.480 0.362 0.393
0.442 0.483 0.189 0.256 0.326 0.437 TimesNet 0.384 0.436 0.491 0.521 0.340
0.402 0.452 0.462 0.338 0.374 0.410 0.478 0.187 0.249 0.321 0.408 FITS 0.372
0.404 0.427 0.424 0.271 0.331 0.354 0.377 0.303 0.337 0.366 0.415 0.162 0.216
0.268 0.348 STD 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 IMP 0.003 0.009 0.013 0.032 0.003 0.007
0.013 0.014 -0.011 -0.007 -0.001 0.004 0.001 0.003 0.008 0.020

Table 2: Long-term forecasting results on three popular datasets in MSE. The
best result is highlighted in bold and the second best is highlighted with
underline. IMP is the improvement between FITS and the second best/ best
result, where a larger value indicates a better improvement. Most of the STD
are under 5e-4 and shown as 0.000 in this table.

Dataset | Weather | Electricity | Traffic
---|---|---|---
Horizon | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720
PatchTST | 0.151 | 0.195 | 0.249 | 0.321 | 0.129 | 0.149 | 0.166 | 0.210 | 0.366 | 0.388 | 0.398 | 0.457
Dlinear | 0.174 | 0.217 | 0.262 | 0.332 | 0.140 | 0.153 | 0.169 | 0.204 | 0.413 | 0.423 | 0.437 | 0.466
Fedformer | 0.246 | 0.292 | 0.378 | 0.447 | 0.188 | 0.197 | 0.212 | 0.244 | 0.573 | 0.611 | 0.621 | 0.630
TimesNet | 0.172 | 0.219 | 0.280 | 0.365 | 0.168 | 0.184 | 0.198 | 0.220 | 0.593 | 0.617 | 0.629 | 0.640
FITS | 0.143 | 0.186 | 0.236 | 0.307 | 0.134 | 0.149 | 0.165 | 0.203 | 0.385 | 0.397 | 0.410 | 0.448
STD | 0.001 | 0.001 | 0.001 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | 0.001
IMP | 0.008 | 0.009 | 0.013 | 0.014 | -0.005 | 0.000 | 0.001 | 0.001 | -0.019 | -0.009 | -0.012 | 0.009

Among the listed models, the parameter counts range from millions down to
thousands. Notably, large models such as TimesNet and Pyraformer require a
staggering number of parameters, with 300.6M and 241.4M, respectively.
Similarly, popular models like Informer, Autoformer, and FEDformer have
parameter counts in the range of 13.61M to 20.68M. Even the lightweight yet
state-of-the-art model PatchTST has a parameter count of over 1 million.

In contrast, FITS stands out as a highly efficient model with an impressively
low parameter count. With only 4.5K to 16K parameters, FITS achieves
comparable or even superior performance compared to these larger models. It is
worth highlighting that FITS requires significantly fewer parameters compared
to the next smallest model, Dlinear, which has 139.7K parameters. For
instance, when considering a 720 look-back window and a 720 forecasting
horizon, the Dlinear model requires over 1 million parameters, whereas FITS
achieves similar performance with only 10k-50k parameters.

Table 3: Number of trainable parameters, MACs, and inference time of TSF
models under look-back window=96 and forecasting horizon=720 on the
Electricity dataset.

Model | Parameters | MACs | Infer. Time
---|---|---|---
TimesNet | 301.7M | 1226.49G | N/A
Pyraformer | 241.4M | 0.80G | 3.4ms
Informer | 14.38M | 3.93G | 49.3ms
Autoformer | 14.91M | 4.41G | 164.1ms
FiLM | 14.91M | 5.97G | 123.0ms
FEDformer | 20.68M | 4.41G | 40.5ms
PatchTST | 1.5M | 5.07G | 3.3ms
DLinear | 139.7K | 40M |  | _0.4ms_
---
(3.05ms CPU)
FITS (Ours) | 4.5K$\sim$10K | 1.6M$\sim$8.9M |  | 0.6ms
---
(_2.55ms_ CPU)

This analysis showcases the remarkable efficiency of FITS. Despite its small
size, FITS consistently achieves competitive results, making it an attractive
option for time series analysis tasks. FITS demonstrates that achieving state-
of-the-art or close to state-of-the-art performance with a considerably
reduced parameter footprint is possible, making it an ideal choice for
resource-constrained environments.

Case Study on ETTh2 Dataset

We conduct a comprehensive case study on the performance of FITS using the
ETTh2 dataset, which further highlights the impact of the look-back window and
cutoff frequency on model performance. We provide a case study on other
datasets in the Appendix. In our experiments, we observe that increasing the
look-back window generally leads to improved performance, while the effect of
increasing the cutoff frequency is minor.

Table 4: The results on the ETTh2 dataset. Values are visualized with a green
background, where darker background indicates worse performance. The top-5
best results are highlighted with a red background, and the absolute best
result is highlighted with red bold font. F represents supervision on the
forecasting task, while B+F represents supervision on backcasting and
forecasting tasks.

| Look-back Window | 90 | 180 | 360 | 720
---|---|---|---|---|---
Horizon | COF/nth Harmonic | F | B+F | F | B+F | F | B+F | F | B+F
| 2 | 0.293889 | 0.291371 | 0.290314 | 0.288107 | 0.279141 | 0.276635 | 0.275600 | 0.274817
| 3 | 0.293242 | 0.291333 | 0.289803 | 0.287171 | 0.278128 | 0.275723 | 0.273972 | 0.273567
| 4 | 0.292438 | 0.290559 | 0.288541 | 0.286174 | 0.277293 | 0.274494 | 0.272384 | 0.272031
| 5 | 0.292387 | 0.290369 | 0.288530 | 0.285527 | 0.276594 | 0.274042 | 0.272085 | 0.271719
96 | 6 | 0.292517 | 0.290466 | 0.287814 | 0.285384 | 0.275930 | 0.273883 | 0.271312 | 0.271028
| 2 | 0.379401 | 0.377047 | 0.361995 | 0.359322 | 0.337767 | 0.336419 | 0.334493 | 0.334621
| 3 | 0.379080 | 0.376874 | 0.360790 | 0.358059 | 0.337391 | 0.335736 | 0.333573 | 0.333758
| 4 | 0.378816 | 0.376472 | 0.360524 | 0.357973 | 0.336085 | 0.334531 | 0.332310 | 0.332475
| 5 | 0.378529 | 0.376429 | 0.360234 | 0.357533 | 0.336286 | 0.334475 | 0.332122 | 0.332281
192 | 6 | 0.378581 | 0.376481 | 0.360049 | 0.357478 | 0.335526 | 0.333846 | 0.331421 | 0.331667
| 2 | 0.419131 | 0.417096 | 0.391167 | 0.388905 | 0.360300 | 0.359665 | 0.356390 | 0.356319
| 3 | 0.419264 | 0.416645 | 0.389740 | 0.387614 | 0.359802 | 0.359291 | 0.355825 | 0.355972
| 4 | 0.419237 | 0.416085 | 0.389790 | 0.387815 | 0.358774 | 0.358096 | 0.354695 | 0.354880
| 5 | 0.418985 | 0.416009 | 0.388972 | 0.387115 | 0.358652 | 0.358093 | 0.354805 | 0.354794
336 | 6 | 0.418359 | 0.416369 | 0.388943 | 0.387183 | 0.358011 | 0.357432 | 0.354055 | 0.354205
| 2 | 0.420888 | 0.418226 | 0.405711 | 0.404412 | 0.387592 | 0.386235 | 0.379710 | 0.380367
| 3 | 0.420441 | 0.418290 | 0.404405 | 0.403520 | 0.386570 | 0.385907 | 0.379501 | 0.380132
| 4 | 0.420404 | 0.417756 | 0.404631 | 0.403425 | 0.386556 | 0.384828 | 0.378209 | 0.378890
| 5 | 0.419888 | 0.417725 | 0.403562 | 0.402755 | 0.385489 | 0.384758 | 0.378227 | 0.378810
720 | 6 | 0.419376 | 0.417854 | 0.403643 | 0.402616 | 0.384709 | 0.383960 | 0.377463 | 0.378101

Tab. 4 showcases the performance results obtained with different look-back
window sizes and cutoff frequencies. Larger look-back windows tend to yield
better performance across the board. On the other hand, increasing the cutoff
frequency only results in marginal performance improvements. However, it is
important to note that higher cutoff frequencies come at the expense of
increased computational resources, as illustrated in Tab. 5.

Considering these observations, we find utilizing a longer look-back window in
combination with a low cutoff frequency to achieve near state-of-the-art
performance with minimal computational cost. For instance, FITS surpasses
other methods when employing a 720 look-back window and setting the cutoff
frequency to the second harmonic. Remarkably, FITS achieves state-of-the-art
performance with a parameter count of only around 10k. Moreover, by reducing
the look-back window to 360, FITS already achieves close-to-state-of-the-art
performance by setting the cutoff frequency to the second harmonic, resulting
in a further reduction of the model’s parameter count to under 5k (as shown in
Tab. 5).

Table 5: The number of parameters under different settings on ETTh1 & ETTh2
dataset.

|  | Look-back Window
---|---|---
Horizon |  | COF/nth
---
Harmonic
90 | 180 | 360 | 720
96 | 2 | 703 | 1053 | 2279 | 5913
3 | 1035 | 1820 | 4307 | 12064
4 | 1431 | 2752 | 6975 | 20385
5 | 1922 | 3876 | 10374 | 31042
6 | 2450 | 5192 | 14338 | 43734
192 | 2 | 1064 | 1431 | 2752 | 6643
3 | 1564 | 2450 | 5192 | 13520
4 | 2187 | 3698 | 8475 | 22815
5 | 2914 | 5253 | 12558 | 34694
6 | 3710 | 7021 | 17334 | 48856
336 | 2 | 1615 | 1998 | 3483 | 7665
3 | 2392 | 3395 | 6608 | 15704
4 | 3321 | 5160 | 10725 | 26460
5 | 4402 | 7293 | 15834 | 40006
6 | 5600 | 9794 | 21828 | 56539
720 | 2 | 3078 | 3510 | 5418 | 10512
3 | 4554 | 5950 | 10266 | 21424
4 | 6318 | 9030 | 16650 | 36180
5 | 8370 | 12750 | 24570 | 54780
6 | 10710 | 17110 | 34026 | 77224

These results emphasize the lightweight nature of FITS, making it highly
suitable for deployment and training on edge devices with limited
computational resources. By carefully selecting the look-back window and
cutoff frequency, FITS can achieve excellent performance while maintaining
computational efficiency, making it an appealing choice for real-world
applications.

##  5 Experiment for Anomaly Detection

###  5.1 Reconstruction as Frequency Interpolation

As discussed before, we tackle the anomaly detection tasks in the self-
supervised reconstructing approach. Specifically, we make a $N$ time
equidistant sampling on the input and train a FITS network with an
interpolation rate of $\eta_{Rec}=N$ to up-sample it. Please check appendix A
for detail.

###  5.2 Experiment Settings

Datasets. We use five commonly used benchmark datasets: SMD (Server Machine
Dataset (Su et al., 2019)), PSM (Polled Server Metrics (Abdulaal et al.,
2021)), SWaT (Secure Water Treatment (Mathur & Tippenhauer, 2016)), MSL (Mars
Science Laboratory rover), and SMAP (Soil Moisture Active Passive satellite)
(Hundman et al., 2018). We report the performance on the synthetic dataset
(Lai et al., 2021) in the appendix F.

Baselines. We compare FITS with models such as TimesNet (Wu et al., 2023),
Anomaly Transformer (Xu et al., 2022), THOC (Shen et al., 2020), Omnianomaly
(Su et al., 2019), DGHL (Challu et al., 2022b). Following TimesNet (Wu et al.,
2023), we also compare the anomaly detection performance with other models
(Zeng et al., 2023; Zhang et al., 2022; Woo et al., 2022; Zhou et al., 2022a).

Evaluation metrics. Following the previous works (Xu et al., 2022; Shen et
al., 2020; Wu et al., 2023), we use Precision, Recall, and F1-score as
metrics.

Implementation details. We use a window size of 200 and downsample the time
series segment by a factor of 4 as the input to train FITS to reconstruct the
original segment. We follow the methodology of the Anomaly Transformer (Xu et
al., 2022), where time points exceeding a certain reconstruction loss
threshold are classified as anomalies. The threshold is selected based on the
highest F1 score achieved on the validation set. To handle consecutive
abnormal segments, we adopt a widely-used adjustment strategy (Su et al.,
2019; Xu et al., 2018; Shen et al., 2020), considering all anomalies within a
specific successive abnormal segment as correctly detected when one anomalous
time point is identified. This approach aligns with real-world applications,
where an abnormal time point often triggers the attention to the entire
segment.

Table 6: Anomaly detection result of F1-scores on 5 datasets. The best result
is highlighted in bold, and the second best is highlighted with underline.
Full results are reported in the Appendix.

Models | FITS | TimesNet |  | Anomaly
---
Transformer
THOC |  | Omni
---
Anomaly
| Stationary
---
Transformer
DGHL | OCSVM | IForest | LightTS | Dlinear | IMP
SMD | 99.95 | 85.81 | 92.33 | 84.99 | 85.22 | 84.72 | N/A | 56.19 | 53.64 | 82.53 | 77.1 | 7.62
PSM | 93.96 | 97.47 | 97.89 | 98.54 | 80.83 | 97.29 | N/A | 70.67 | 83.48 | 97.15 | 93.55 | -3.93
SWaT | 98.9 | 91.74 | 94.07 | 85.13 | 82.83 | 79.88 | 87.47 | 47.23 | 47.02 | 93.33 | 87.52 | 4.83
SMAP | 70.74 | 71.52 | 96.69 | 90.68 | 86.92 | 71.09 | 96.38 | 56.34 | 55.53 | 69.21 | 69.26 | -25.95
MSL | 78.12 | 85.15 | 93.59 | 89.69 | 87.67 | 77.5 | 94.08 | 70.82 | 66.45 | 78.95 | 84.88 | -15.96

###  5.3 Comparisons with SOTAs

In Table 6, FITS stands out with outstanding results on various datasets.
Particularly, on SMD and SWaT datasets, FITS achieves nearly perfect
F1-scores, around 99.95% and 98.9%, respectively, showcasing its precision in
anomaly detection and classification. In contrast, models like TimesNet,
Anomaly Transformer, and Stationary Transformer struggle to match FITS’
performance on these datasets.

However, FITS shows comparatively lower performance on the SMAP and MSL
datasets. These datasets present a challenge due to their binary event data
nature, which may not be effectively captured by FITS’ frequency domain
representation. In such cases, time-domain modeling is preferable as the raw
data format is sufficiently compact. Thus, models specifically designed for
anomaly detection, such as THOC and Omni Anomaly, achieve higher F1-scores on
these datasets.

For a more comprehensive evaluation, waveform visualizations and detailed
analysis can be found in the appendix, providing deeper insights into FITS’
strengths and limitations in different anomaly detection scenarios. It is
important to note that the reported results are achieved with a parameter
range of 1-4K and MACs (Multiply-Accumulate Operations) of 10-137K, which will
be further detailed in the appendix.

While the datasets in use are instrumental, it is imperative to acknowledge
their limitations as delineated in (Lai et al., 2021). Particularly on the
synthetic dataset from (Lai et al., 2021), FITS demonstrates impeccable
detection capabilities, registering a flawless 100% F1 score. For a detailed
breakdown, readers can refer to the table in appendix F. This dataset marries
a sinusoidal wave of a single frequency with intricately introduced anomaly
patterns, which pose challenges for identification in the time domain. Yet,
FITS, leveraging the frequency domain, adeptly discerns these anomalies,
particularly those introducing unexpected frequency components.

Moreover, FITS boasts an impressive sub-millisecond inference speed — a marked
distinction when compared to the latency typical of larger models or
communication overheads. This speed underscores FITS’s suitability as a first-
responder tool for promptly spotting critical errors. When paired as a
preliminary filter with a specialized AD algorithm geared for detailed
detection, the combined system stands as a paragon of both robustness and
swift responsiveness facing diverse anomalies.

##  6 Conclusions and Future Work

In this paper, we propose FITS for time series analysis, a low-cost model with
$10k$ parameters that can achieve performance comparable to state-of-the-art
models that are often several orders of magnitude larger. As the future work,
we plan to evaluate FITS on more real-world scenario and improve the
interpretability of it. Further, we also aim to explore the frequency domain
large-scale complex-valued neural network such as complex-valued Transformers.

## References

  * Abdulaal et al. (2021) Ahmed Abdulaal, Zhuanghua Liu, and Tomer Lancewicki.  Practical approach to asynchronous multivariate time series anomaly detection and localization.  In _Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery; Data Mining_ , KDD ’21, pp. 2485–2494, New York, NY, USA, 2021. Association for Computing Machinery.  ISBN 9781450383325.  doi: 10.1145/3447548.3467174.  URL https://doi.org/10.1145/3447548.3467174.
  * Brigham & Morrow (1967) E. O. Brigham and R. E. Morrow.  The fast fourier transform.  _IEEE Spectrum_ , 4(12):63–70, 1967.  doi: 10.1109/MSPEC.1967.5217220.
  * Challu et al. (2022a) Cristian Challu, Kin G Olivares, Boris N Oreshkin, Federico Garza, Max Mergenthaler, and Artur Dubrawski.  N-hits: Neural hierarchical interpolation for time series forecasting.  _arXiv preprint arXiv:2201.12886_ , 2022a.
  * Challu et al. (2023) Cristian Challu, Kin G Olivares, Boris N Oreshkin, Federico Garza Ramirez, Max Mergenthaler Canseco, and Artur Dubrawski.  Nhits: neural hierarchical interpolation for time series forecasting.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 37, pp. 6989–6997, 2023.
  * Challu et al. (2022b) Cristian I Challu, Peihong Jiang, Ying Nian Wu, and Laurent Callot.  Deep generative model with hierarchical latent factors for time series anomaly detection.  In _International Conference on Artificial Intelligence and Statistics_ , pp. 1643–1654. PMLR, 2022b.
  * Hundman et al. (2018) Kyle Hundman, Valentino Constantinou, Christopher Laporte, Ian Colwell, and Tom Soderstrom.  Detecting spacecraft anomalies using LSTMs and nonparametric dynamic thresholding.  In _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery &amp Data Mining_. ACM, jul 2018.  doi: 10.1145/3219819.3219845.  URL https://doi.org/10.11452F3219819.3219845.
  * Kim et al. (2022) Taesung Kim, Jinhee Kim, Yunwon Tae, Cheonbok Park, Jang-Ho Choi, and Jaegul Choo.  Reversible instance normalization for accurate time-series forecasting against distribution shift.  In _International Conference on Learning Representations_ , 2022.  URL https://openreview.net/forum?id=cGDAkQo1C0p.
  * Lai et al. (2021) Kwei-Herng Lai, Daochen Zha, Junjie Xu, Yue Zhao, Guanchu Wang, and Xia Hu.  Revisiting time series outlier detection: Definitions and benchmarks.  In _Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 1)_ , 2021.  URL https://openreview.net/forum?id=r8IvOsnHchr.
  * Lee-Thorp et al. (2022) James Lee-Thorp, Joshua Ainslie, Ilya Eckstein, and Santiago Ontanon.  Fnet: Mixing tokens with fourier transforms, 2022.
  * Liu et al. (2022a) Minhao Liu, Ailing Zeng, Muxi Chen, Zhijian Xu, Qiuxia Lai, Lingna Ma, and Qiang Xu.  Scinet: Time series modeling and forecasting with sample convolution and interaction.  In _Advances in Neural Information Processing Systems_ , 2022a.
  * Liu et al. (2022b) Shizhan Liu, Hang Yu, Cong Liao, Jianguo Li, Weiyao Lin, Alex X. Liu, and Schahram Dustdar.  Pyraformer: Low-complexity pyramidal attention for long-range time series modeling and forecasting.  In _International Conference on Learning Representations_ , 2022b.  URL https://openreview.net/forum?id=0EXmFzUn5I.
  * Mathur & Tippenhauer (2016) Aditya P. Mathur and Nils Ole Tippenhauer.  Swat: a water treatment testbed for research and training on ics security.  In _2016 International Workshop on Cyber-physical Systems for Smart Water Networks (CySWater)_ , pp. 31–36, 2016.  doi: 10.1109/CySWater.2016.7469060.
  * Nie et al. (2023) Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam.  A time series is worth 64 words: Long-term forecasting with transformers.  In _International Conference on Learning Representations_ , 2023.
  * Oreshkin et al. (2019) Boris N. Oreshkin, Dmitri Carpov, Nicolas Chapados, and Yoshua Bengio.  N-BEATS: neural basis expansion analysis for interpretable time series forecasting.  _CoRR_ , abs/1905.10437, 2019.  URL http://arxiv.org/abs/1905.10437.
  * Shen et al. (2020) Lifeng Shen, Zhuocong Li, and James Kwok.  Timeseries anomaly detection using temporal hierarchical one-class network.  In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), _Advances in Neural Information Processing Systems_ , volume 33, pp. 13016–13026. Curran Associates, Inc., 2020.  URL https://proceedings.neurips.cc/paper_files/paper/2020/file/97e401a02082021fd24957f852e0e475-Paper.pdf.
  * Su et al. (2019) Ya Su, Youjian Zhao, Chenhao Niu, Rong Liu, Wei Sun, and Dan Pei.  Robust anomaly detection for multivariate time series through stochastic recurrent neural network.  In _Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery; Data Mining_ , KDD ’19, pp. 2828–2837, New York, NY, USA, 2019. Association for Computing Machinery.  ISBN 9781450362016.  doi: 10.1145/3292500.3330672.  URL https://doi.org/10.1145/3292500.3330672.
  * Woo et al. (2022) Gerald Woo, Chenghao Liu, Doyen Sahoo, Akshat Kumar, and Steven Hoi.  Etsformer: Exponential smoothing transformers for time-series forecasting, 2022.
  * Wu et al. (2021) Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long.  Autoformer: Decomposition transformers with auto-correlation for long-term series forecasting.  _Advances in Neural Information Processing Systems_ , 34:22419–22430, 2021.
  * Wu et al. (2023) Haixu Wu, Tengge Hu, Yong Liu, Hang Zhou, Jianmin Wang, and Mingsheng Long.  Timesnet: Temporal 2d-variation modeling for general time series analysis.  In _International Conference on Learning Representations_ , 2023.
  * Xu et al. (2018) Haowen Xu, Yang Feng, Jie Chen, Zhaogang Wang, Honglin Qiao, Wenxiao Chen, Nengwen Zhao, Zeyan Li, Jiahao Bu, Zhihan Li, Ying Liu, Youjian Zhao, and Dan Pei.  Unsupervised anomaly detection via variational auto-encoder for seasonal KPIs in web applications.  In _Proceedings of the 2018 World Wide Web Conference on World Wide Web - WWW '18_. ACM Press, 2018.  doi: 10.1145/3178876.3185996.  URL https://doi.org/10.1145/2F3178876.3185996.
  * Xu et al. (2022) Jiehui Xu, Haixu Wu, Jianmin Wang, and Mingsheng Long.  Anomaly transformer: Time series anomaly detection with association discrepancy, 2022.
  * Zeng et al. (2023) Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu.  Are transformers effective for time series forecasting?  2023\.
  * Zhang et al. (2022) Tianping Zhang, Yizhuo Zhang, Wei Cao, Jiang Bian, Xiaohan Yi, Shun Zheng, and Jian Li.  Less is more: Fast multivariate time series forecasting with light sampling-oriented mlp structures.  _arXiv preprint arXiv:2207.01186_ , 2022.
  * Zhou et al. (2021) Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang.  Informer: Beyond efficient transformer for long sequence time-series forecasting.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 35, pp. 11106–11115, 2021.
  * Zhou et al. (2022a) Tian Zhou, Ziqing Ma, Qingsong Wen, Xue Wang, Liang Sun, and Rong Jin.  Fedformer: Frequency enhanced decomposed transformer for long-term series forecasting.  In _International Conference on Machine Learning_ , 2022a.
  * Zhou et al. (2022b) Tian Zhou, Ziqing Ma, xue wang, Qingsong Wen, Liang Sun, Tao Yao, Wotao Yin, and Rong Jin.  FiLM: Frequency improved legendre memory model for long-term time series forecasting.  In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), _Advances in Neural Information Processing Systems_ , 2022b.  URL https://openreview.net/forum?id=zTQdHSQUQWc.

##  Appendix A Pipeline for Reconstruction

The pipeline for the reconstruction task is shown in Fig. 4. In this process,
the model input $x$ is derived from a segment of the time series $y$ using an
equidistant sampling technique with a specified downsample rate $\eta$.
Subsequently, FITS performs frequency interpolation, generating an upsampled
output $\hat{x}_{up-sampled}$ with the same length as $y$. The reconstruction
loss is computed by comparing the original $y$ and the upsampled $\hat{x}_{up-
sampled}$. Please note that, due to space constraints, the depicted
downsample/upsample rate $\eta$ in the figure is shown as 1.5, which is not a
practical value. In our actual experiments, we employ a $\eta$ value of 4.

Figure 4: Pipeline of FITS, with a focus on the Reconstruction task.

##  Appendix B Details of forecasting datasets

We report the characteristics in the tab.7.

Table 7: The statistics of the seven used forecasting datasets.

Dataset | Traffic | Electricity | Weather | ETTh1$\&$ETTh2 | ETTm1 $\&$ETTm2
---|---|---|---|---|---
Channels | 862 | 321 | 21 | 7 | 7
Sampling Rate | 1hour | 1hour | 10min | 1hour | 15min
Total Timesteps | 17,544 | 26,304 | 52,696 | 17,420 | 69,680

##  Appendix C More Results on Forecasting Task

We show the comparison with transformer-based models, short-term forecasting
on M4, and the impact of random seeds below.

###  C.1 Comparison with Transformer-based Methods

We further compare FITS with Autoformer (Wu et al., 2021), Informer (Zhou et
al., 2021), FiLM (Zhou et al., 2022b) and Pyraformer (Liu et al., 2022b). The
results are shown in Tab. 8 and Tab. 9. Note that the results in these tables
are directly reterived from the original paper and may still suffer from the
bug mentioned above. We cannot rerun these models because of the incomplete
codebase or the extereme large time consumption.

Table 8: Long-term forecasting results on ETT datasets in MSE. The best result
is highlighted in bold.

Dataset | ETTh1 | ETTh2 | ETTm1 | ETTm2
---|---|---|---|---
Horizon | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720
Autoformer | 0.449 | 0.500 | 0.521 | 0.514 | 0.358 | 0.456 | 0.482 | 0.515 | 0.505 | 0.553 | 0.621 | 0.671 | 0.255 | 0.281 | 0.339 | 0.433
Informer | 0.865 | 1.008 | 1.107 | 1.181 | 3.755 | 5.602 | 4.721 | 3.647 | 0.672 | 0.795 | 1.212 | 1.166 | 0.365 | 0.533 | 1.363 | 3.379
FEDFormer | 0.376 | 0.420 | 0.459 | 0.506 | 0.346 | 0.429 | 0.496 | 0.463 | 0.379 | 0.426 | 0.445 | 0.543 | 0.203 | 0.269 | 0.325 | 0.421
Pyraformer | 0.664 | 0.790 | 0.891 | 0.963 | 0.645 | 0.788 | 0.907 | 0.963 | 0.543 | 0.557 | 0.754 | 0.908 | 0.435 | 0.730 | 1.201 | 3.625
FiLM | 0.371 | 0.414 | 0.442 | 0.465 | 0.284 | 0.357 | 0.377 | 0.439 | 0.302 | 0.338 | 0.373 | 0.420 | 0.165 | 0.222 | 0.277 | 0.371
FITS | 0.368 | 0.404 | 0.405 | 0.425 | 0.255 | 0.307 | 0.306 | 0.368 | 0.305 | 0.339 | 0.366 | 0.414 | 0.164 | 0.217 | 0.269 | 0.347

Table 9: Long-term forecasting results on three popular datasets in MSE. The
best result is highlighted in bold.

Dataset | Electricity | Traffic | Weather
---|---|---|---
Horizon | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720 | 96 | 192 | 336 | 720
Autoformer | 0.201 | 0.222 | 0.231 | 0.254 | 0.613 | 0.616 | 0.622 | 0.660 | 0.266 | 0.307 | 0.359 | 0.419
Informer | 0.274 | 0.296 | 0.300 | 0.373 | 0.719 | 0.696 | 0.777 | 0.864 | 0.300 | 0.598 | 0.578 | 1.059
FEDFormer | 0.193 | 0.201 | 0.214 | 0.246 | 0.587 | 0.604 | 0.621 | 0.626 | 0.217 | 0.276 | 0.339 | 0.403
Pyraformer | 0.386 | 0.386 | 0.378 | 0.376 | 2.085 | 0.867 | 0.869 | 0.881 | 0.896 | 0.622 | 0.739 | 1.004
FiLM | 0.154 | 0.164 | 0.188 | 0.236 | 0.416 | 0.408 | 0.425 | 0.520 | 0.199 | 0.228 | 0.267 | 0.319
FITS | 0.137 | 0.142 | 0.165 | 0.202 | 0.381 | 0.381 | 0.410 | 0.446 | 0.145 | 0.188 | 0.236 | 0.308

###  C.2 Comparison with NBeats & NHITS

We show the comparison with mentioned N-HiTS and N-BEATS on MSE in the
following table. FITS outperforms these two models in most cases while
maintaining a compact model size. We will consider adding the following
results to our main result. The results for N-HiTS and N-BEATS are retrieved
from the paper of N-HiTS (Challu et al., 2023).

Table 10: Comparison with N-HiTS and N-BEATS on MSE

Dataset | Horizon | FITS | N-BEATS | N-HiTS
---|---|---|---|---
Electricity | 96 | 0.138 | 0.145 | 0.147
192 | 0.152 | 0.180 | 0.167
336 | 0.166 | 0.200 | 0.186
720 | 0.205 | 0.266 | 0.243
Traffic | 96 | 0.401 | 0.398 | 0.402
192 | 0.407 | 0.409 | 0.420
336 | 0.420 | 0.449 | 0.448
720 | 0.456 | 0.589 | 0.539
Weather | 96 | 0.145 | 0.167 | 0.158
192 | 0.188 | 0.229 | 0.211
336 | 0.236 | 0.287 | 0.274
720 | 0.308 | 0.368 | 0.351
ETTm2 | 96 | 0.164 | 0.184 | 0.176
192 | 0.217 | 0.273 | 0.245
336 | 0.269 | 0.309 | 0.295
720 | 0.347 | 0.411 | 0.401

###  C.3 Short-term Forecasting on M4

We evaluate FITS’ performance on the M4 dataset following the TimesNet (Wu et
al., 2023). We retrieve the following results from the TimesNet paper. As
shown in Tab.11, FITS shows the suboptimal results on the M4 dataset. The
reason for this outcome is threefold. First, the M4 dataset is a collection of
many time series from different domains. These time series have different
temporal information and periodicity, and no correlations exist among them. We
can not regard them as simple multivariate forecasting tasks. Second, other
models have a very large amount of parameters, especially TimesNet, which
makes them have enough capability to model such diverse datasets with one
model. However, considering the lightweight of FITS, it is hard for it to
achieve ideal results. Finally, the setting for the M4 dataset is not suitable
for FITS. The look-back window is set to 12, 16, and 36 for yearly, quarterly,
and monthly prediction accordingly, which is twice the length of the
forecasting horizon. Such a short look-back window is very difficult to
extract meaningful frequency representation, which further worsens the FITS’
performance. We compare FITS with lightweight model DLinear (Zeng et al.,
2023), state-of-the-art model TimesNet (Wu et al., 2023) and two hierarchical
time series modeling model N-Hits (Challu et al., 2022a) and N-Beats (Oreshkin
et al., 2019).

Table 11: Results on M4 dataset in SMAPE.  | FITS | DLinear | TimesNet | N-Hits | N-Beats
---|---|---|---|---|---
Yearly | 14.00 | 16.96 | 13.38 | 13.41 | 13.43
Quarterly | 10.72 | 12.14 | 10.1 | 10.2 | 10.12
Monthly | 13.49 | 13.51 | 12.67 | 12.7 | 12.67

##  Appendix D Case Study on Other Datasets

We show the parameter table and performance on other datasets below.

###  D.1 ETTh1, ETTm1 & m2

Tab.12 shows the corresponding results on ETTh1 dataset with different
settings. ETTh1 shows a abnormal behavior since FITS does not benefits form
the longer look-back window, i.e. 720. Instead, it achieves the sota
performance at look-back window of 360. We also find this phenomenon in the
ETTm1 dataset. We attribute this phenomenon to the distribution shift that
exist in the datasets. The longer look-back window will introduce more
information from a shifted distribution and sabotage the forecasting result.

Table 12: The results on the ETTh1 dataset. Values are visualized with a green
background, where darker background indicates worse performance. The top-5
best results are highlighted with a red background, and the absolute best
result is highlighted with red bold font. F represents supervision on the
forecasting task, while B+F represents supervision on backcasting and
forecasting tasks.

| Look-back Window | 90 | 180 | 360 | 720
---|---|---|---|---|---
Horizon | COF/nth Harmonic | F | B+F | F | B+F | F | B+F | F | B+F
| 2 | 0.391210 | 0.391987 | 0.388999 | 0.389400 | 0.396473 | 0.396321 | 0.402964 | 0.403307
| 3 | 0.390959 | 0.389857 | 0.385510 | 0.386088 | 0.380594 | 0.380825 | 0.386660 | 0.387143
| 4 | 0.387971 | 0.388607 | 0.383368 | 0.384021 | 0.377323 | 0.377619 | 0.383072 | 0.383700
| 5 | 0.386867 | 0.387619 | 0.382036 | 0.382446 | 0.374568 | 0.375034 | 0.380786 | 0.381331
96 | 6 | 0.386679 | 0.386013 | 0.380810 | 0.381219 | 0.372101 | 0.372712 | 0.379216 | 0.379815
| 2 | 0.441343 | 0.442006 | 0.432260 | 0.432547 | 0.426438 | 0.426715 | 0.435860 | 0.440504
| 3 | 0.440369 | 0.440186 | 0.429132 | 0.429812 | 0.411548 | 0.411818 | 0.420554 | 0.422348
| 4 | 0.438746 | 0.439231 | 0.427605 | 0.427898 | 0.409010 | 0.409368 | 0.417933 | 0.418202
| 5 | 0.437784 | 0.438436 | 0.426070 | 0.426605 | 0.406772 | 0.407058 | 0.415545 | 0.416382
192 | 6 | 0.436423 | 0.437092 | 0.425226 | 0.425557 | 0.404390 | 0.404649 | 0.414150 | 0.415508
| 2 | 0.482840 | 0.483552 | 0.458024 | 0.460152 | 0.448917 | 0.448952 | 0.455474 | 0.462104
| 3 | 0.482419 | 0.482874 | 0.455709 | 0.458133 | 0.433826 | 0.434296 | 0.440474 | 0.447663
| 4 | 0.481796 | 0.482031 | 0.454570 | 0.456531 | 0.431753 | 0.432849 | 0.437971 | 0.447336
| 5 | 0.480466 | 0.480184 | 0.453879 | 0.455248 | 0.430369 | 0.430036 | 0.434967 | 0.440387
336 | 6 | 0.477769 | 0.478893 | 0.452324 | 0.454378 | 0.427425 | 0.427670 | 0.434515 | 0.439989
| 2 | 0.471969 | 0.474663 | 0.442253 | 0.444961 | 0.445856 | 0.447131 | 0.450286 | 0.451247
| 3 | 0.470309 | 0.472660 | 0.440675 | 0.443468 | 0.430920 | 0.432436 | 0.435924 | 0.436795
| 4 | 0.469793 | 0.471439 | 0.438692 | 0.442282 | 0.428483 | 0.430407 | 0.433446 | 0.438730
| 5 | 0.470391 | 0.471038 | 0.437598 | 0.441572 | 0.426720 | 0.428304 | 0.431934 | 0.433103
720 | 6 | 0.468232 | 0.469044 | 0.438001 | 0.440176 | 0.424749 | 0.425963 | 0.430258 | 0.431978

Table 13: The number of parameters under different settings on ETTm1 & ETTm2
dataset.

|  | Look-back Window
---|---|---
Horizon | COF/nth Harmonic | 90 | 180 | 360 | 720
96 | 4 | 420 | 513 | 621 | 1330
6 | 561 | 759 | 1015 | 2444
8 | 703 | 1053 | 1505 | 3835
10 | 861 | 1426 | 2050 | 5609
12 | 1035 | 1820 | 2726 | 7636
14 | 1225 | 2262 | 5561 | 16974
192 | 4 | 645 | 703 | 759 | 1505
6 | 850 | 1035 | 1218 | 2726
8 | 1064 | 1431 | 1820 | 4307
10 | 1302 | 1922 | 2501 | 6248
12 | 1564 | 2450 | 3290 | 8549
14 | 1875 | 3042 | 6767 | 18942
336 | 4 | 990 | 969 | 966 | 1715
6 | 1275 | 1449 | 1566 | 3149
8 | 1615 | 1998 | 2275 | 5015
10 | 1974 | 2666 | 3157 | 7242
12 | 2392 | 3395 | 4136 | 9960
14 | 2825 | 4212 | 8509 | 21894
720 | 4 | 1890 | 1710 | 1518 | 2380
6 | 2448 | 2530 | 2436 | 4324
8 | 3078 | 3510 | 3570 | 6844
10 | 3780 | 4650 | 4920 | 9940
12 | 4554 | 5950 | 6486 | 13612
14 | 5400 | 7410 | 13266 | 30012

Table 14: The results on the ETTm1 dataset. Values are visualized with a green
background, where darker background indicates worse performance. The top-5
best results are highlighted with a red background, and the absolute best
result is highlighted with red bold font. F represents supervision on the
forecasting task, while B+F represents supervision on backcasting and
forecasting tasks.

| Look-back Window | 90 | 180 | 360 | 720
---|---|---|---|---|---
Horizon | COF/nth Harmonic | F | B+F | F | B+F | F | B+F | F | B+F
| 6 | 0.365445 | 0.364920 | 0.312641 | 0.312953 | 0.305685 | 0.306146 | 0.314674 | 0.314514
| 8 | 0.364851 | 0.364060 | 0.312192 | 0.312162 | 0.304728 | 0.304859 | 0.311096 | 0.311866
| 10 | 0.364031 | 0.363901 | 0.311927 | 0.312576 | 0.303561 | 0.304319 | 0.310334 | 0.310245
| 12 | 0.363204 | 0.363368 | 0.311663 | 0.311027 | 0.303412 | 0.303837 | 0.309762 | 0.309719
96 | 14 | 0.362444 | 0.362795 | 0.311292 | 0.311589 | 0.303153 | 0.303350 | 0.309248 | 0.309280
| 6 | 0.400734 | 0.400507 | 0.348337 | 0.348248 | 0.339300 | 0.339427 | 0.342016 | 0.342371
| 8 | 0.400492 | 0.400347 | 0.347860 | 0.347737 | 0.338135 | 0.338368 | 0.339153 | 0.339648
| 10 | 0.399691 | 0.399442 | 0.347713 | 0.348003 | 0.337680 | 0.337875 | 0.338802 | 0.338988
| 12 | 0.399838 | 0.398958 | 0.347586 | 0.347586 | 0.337414 | 0.337599 | 0.338334 | 0.338354
192 | 14 | 0.399357 | 0.399249 | 0.347410 | 0.347304 | 0.337172 | 0.337290 | 0.337920 | 0.338687
| 6 | 0.431620 | 0.431774 | 0.384310 | 0.384216 | 0.372982 | 0.373155 | 0.369938 | 0.369875
| 8 | 0.431072 | 0.431733 | 0.384172 | 0.384051 | 0.371957 | 0.372328 | 0.367226 | 0.367214
| 10 | 0.431126 | 0.431569 | 0.383794 | 0.383776 | 0.371696 | 0.371917 | 0.366873 | 0.367586
| 12 | 0.430388 | 0.431283 | 0.383467 | 0.383656 | 0.371289 | 0.371662 | 0.366383 | 0.366726
336 | 14 | 0.430749 | 0.431064 | 0.383719 | 0.383675 | 0.371363 | 0.371460 | 0.366238 | 0.366492
| 6 | 0.492552 | 0.492473 | 0.443358 | 0.443911 | 0.427787 | 0.428065 | 0.418358 | 0.418436
| 8 | 0.491822 | 0.492750 | 0.443295 | 0.443706 | 0.427054 | 0.427269 | 0.415964 | 0.416064
| 10 | 0.492088 | 0.492691 | 0.443319 | 0.443260 | 0.426868 | 0.427377 | 0.415702 | 0.416024
| 12 | 0.491904 | 0.492589 | 0.443103 | 0.443476 | 0.426748 | 0.426819 | 0.415402 | 0.415881
720 | 14 | 0.491202 | 0.491228 | 0.443076 | 0.443384 | 0.426571 | 0.427071 | 0.415266 | 0.415599

Table 15: The results on the ETTm2 dataset. Values are visualized with a green
background, where darker background indicates worse performance. The top-5
best results are highlighted with a red background, and the absolute best
result is highlighted with red bold font. F represents supervision on the
forecasting task, while B+F represents supervision on backcasting and
forecasting tasks.

| Look-back Window | 90 | 180 | 360 | 720
---|---|---|---|---|---
Horizon | COF/nth Harmonic | F | B+F | F | B+F | F | B+F | F | B+F
| 6 | 0.185981 | 0.185774 | 0.174363 | 0.174277 | 0.166817 | 0.166589 | 0.164719 | 0.164500
| 8 | 0.185601 | 0.185615 | 0.174002 | 0.173797 | 0.166025 | 0.165700 | 0.164127 | 0.163744
| 10 | 0.185515 | 0.185338 | 0.173717 | 0.173536 | 0.165634 | 0.165413 | 0.163178 | 0.162928
| 12 | 0.185489 | 0.185038 | 0.173951 | 0.173368 | 0.165369 | 0.165113 | 0.162719 | 0.162550
96 | 14 | 0.185646 | 0.185048 | 0.173502 | 0.173315 | 0.165362 | 0.165198 | 0.162575 | 0.162346
| 6 | 0.249365 | 0.249119 | 0.233717 | 0.233428 | 0.221364 | 0.220948 | 0.218563 | 0.218267
| 8 | 0.249366 | 0.248945 | 0.233405 | 0.233171 | 0.220596 | 0.220165 | 0.218115 | 0.217808
| 10 | 0.248919 | 0.248686 | 0.233276 | 0.232929 | 0.220104 | 0.220518 | 0.217338 | 0.216963
| 12 | 0.248677 | 0.248890 | 0.233005 | 0.233042 | 0.220007 | 0.219895 | 0.216927 | 0.216714
192 | 14 | 0.248678 | 0.248454 | 0.233162 | 0.232763 | 0.219897 | 0.219597 | 0.216879 | 0.216650
| 6 | 0.309083 | 0.308863 | 0.286387 | 0.286170 | 0.273920 | 0.273816 | 0.269833 | 0.269620
| 8 | 0.309234 | 0.308577 | 0.286186 | 0.286041 | 0.273418 | 0.273171 | 0.269393 | 0.269252
| 10 | 0.308768 | 0.308713 | 0.286102 | 0.285768 | 0.273038 | 0.272893 | 0.268921 | 0.268596
| 12 | 0.308741 | 0.308568 | 0.286529 | 0.285881 | 0.272931 | 0.272763 | 0.268468 | 0.268273
336 | 14 | 0.308759 | 0.308393 | 0.286232 | 0.285711 | 0.272905 | 0.272804 | 0.268366 | 0.268219
| 6 | 0.408977 | 0.408844 | 0.384164 | 0.383995 | 0.366645 | 0.366654 | 0.350173 | 0.349770
| 8 | 0.409260 | 0.408714 | 0.383980 | 0.383899 | 0.366070 | 0.366085 | 0.349659 | 0.349619
| 10 | 0.408793 | 0.408703 | 0.383886 | 0.383827 | 0.365909 | 0.365935 | 0.349019 | 0.348881
| 12 | 0.408698 | 0.408639 | 0.383921 | 0.383664 | 0.365810 | 0.365805 | 0.348831 | 0.348863
720 | 14 | 0.408765 | 0.408479 | 0.383926 | 0.383620 | 0.365790 | 0.365801 | 0.348938 | 0.348766

Tab. 13 shows the parameter count of parameters of FITS with different
settings on the ETTm1 & 2 datasets. Tab. 14 and Tab.15 show the corresponding
results on ETTm1 and ETTm2 datasets with different settings. Note that FITS
constantly achieves SOTA performance on the ETTm2 dataset with under 10k
parameters.

###  D.2 Traffic

Tab. 16 shows the parameter count of parameters of FITS with different
settings on the Traffic dataset. Tab. 17shows the result on the Traffic
dataset with different settings correspondingly. The traffic dataset has a
very large amount of channels, making many models need many parameters to
model the temporal information. FITS only needs 50k parameters to achieve
comparable performance.

Table 16: The number of parameters under different settings on Traffic
dataset.

|  | Look-back Window
---|---|---
Horizon | COF/nth Harmonic | 90 | 180 | 360 | 720
96 | 3 | 1035 | 1820 | 4307 | 12064
| 5 | 1922 | 3876 | 10374 | 31042
| 8 | 3698 | 8475 | 24186 | 75628
| 10 | N/A | 12558 | 36765 | 116202
192 | 3 | 1564 | 2450 | 5192 | 13520
| 5 | 2914 | 5253 | 12558 | 34694
| 8 | 5633 | 11400 | 29329 | 84434
| 10 | N/A | 16926 | 44460 | 130005
336 | 3 | 2392 | 3395 | 6608 | 15704
| 5 | 4402 | 7293 | 15834 | 40006
| 8 | 8514 | 15900 | 36974 | 97902
| 10 | N/A | 23478 | 56088 | 150549
720 | 3 | 4554 | 5950 | 10266 | 21424
| 5 | 8370 | 12750 | 24570 | 54780
| 8 | 16254 | 27750 | 57546 | 133644
| 10 | N/A | 40950 | 87210 | 205440

Table 17: The results on the Traffic dataset. Values are visualized with a
green background, where darker background indicates worse performance. The
top-5 best results are highlighted with a red background, and the absolute
best result is highlighted with red bold font. F represents supervision on the
forecasting task, while B+F represents supervision on backcasting and
forecasting tasks.

| Look-back Window | 90 | 180 | 360 | 720
---|---|---|---|---|---
Horizon | COF/nth Harmonic | F | B+F | F | B+F | F | B+F | F | B+F
| 3 | 0.694065 | 0.694425 | 0.474606 | 0.475881 | 0.455815 | 0.457292 | 0.436317 | 0.436616
| 5 | 0.686110 | 0.684290 | 0.457057 | 0.456547 | 0.419903 | 0.419748 | 0.397558 | 0.397210
| 8 | 0.682876 | 0.681464 | 0.452024 | 0.451470 | 0.410857 | 0.410458 | 0.387791 | 0.388830
96 | 10 | N/A | N/A | 0.451340 | 0.450850 | 0.409948 | 0.409614 | 0.385763 | 0.386596
| 3 | 0.627212 | 0.636434 | 0.481686 | 0.485085 | 0.463516 | 0.464170 | 0.442661 | 0.443547
| 5 | 0.622882 | 0.622451 | 0.467921 | 0.467470 | 0.431367 | 0.431097 | 0.407908 | 0.407850
| 8 | 0.620314 | 0.620003 | 0.463119 | 0.462929 | 0.422517 | 0.422336 | 0.399032 | 0.399101
192 | 10 | N/A | N/A | 0.462595 | 0.461907 | 0.421705 | 0.421677 | 0.397286 | 0.398034
| 3 | 0.635301 | 0.662283 | 0.496200 | 0.510793 | 0.473090 | 0.476491 | 0.454243 | 0.456989
| 5 | 0.632244 | 0.631760 | 0.481267 | 0.480599 | 0.442476 | 0.442128 | 0.420268 | 0.420239
| 8 | 0.629962 | 0.629700 | 0.477111 | 0.476673 | 0.434504 | 0.434124 | 0.411608 | 0.412114
336 | 10 | N/A | N/A | 0.476248 | 0.476044 | 0.433656 | 0.433456 | 0.410500 | 0.410417
| 3 | 0.685472 | 0.732168 | 0.529004 | 0.606921 | 0.500635 | 0.587891 | 0.488116 | 0.489934
| 5 | 0.670385 | 0.669979 | 0.507742 | 0.507104 | 0.469442 | 0.469337 | 0.456470 | 0.456207
| 8 | 0.668054 | 0.668322 | 0.504645 | 0.536565 | 0.463208 | 0.463744 | 0.449778 | 0.449220
720 | 10 | N/A | N/A | 0.503795 | 0.503702 | 0.462643 | 0.463091 | 0.448882 | 0.448182

###  D.3 Weather

Tab. 18 shows the parameter count of parameters of FITS with different
settings on the Weather dataset. Tab. 17shows the result on the Traffic
dataset with different settings correspondingly. Note that we achieve the
result in the main table by setting the COF as 75 and the look-back window as
700.

Table 18: The number of parameters per channel under different settings on
Weather dataset.

|  | Look-back Window
---|---|---
Horizon | COF/nth Harmonic | 90 | 180 | 360 | 720
96 | 5 | 496 | 630 | 806 | 1845
| 8 | 703 | 1053 | 1505 | 3835
| 10 | 861 | 1426 | 2050 | 5609
| 12 | 1035 | 1820 | 2726 | 7636
192 | 5 | 752 | 861 | 988 | 2050
| 8 | 1064 | 1431 | 1820 | 4307
| 10 | 1302 | 1922 | 2501 | 6248
| 12 | 1564 | 2450 | 3290 | 8549
336 | 5 | 1136 | 1197 | 1248 | 2378
| 8 | 1615 | 1998 | 2275 | 5015
| 10 | 1974 | 2666 | 3157 | 7242
| 12 | 2392 | 3395 | 4136 | 9960
720 | 5 | 2160 | 2100 | 1950 | 3280
| 8 | 3078 | 3510 | 3570 | 6844
| 10 | 3780 | 4650 | 4920 | 9940
| 12 | 4554 | 5950 | 6486 | 13612

Table 19: The results on the Weather dataset. Values are visualized with a
green background, where darker background indicates worse performance. The
top-5 best results are highlighted with a red background, and the absolute
best result is highlighted with red bold font. F represents supervision on the
forecasting task, while B+F represents supervision on backcasting and
forecasting tasks.

| Look-back Window | 90 | 180 | 360 | 720
---|---|---|---|---|---
Horizon | COF/nth Harmonic | F | B+F | F | B+F | F | B+F | F | B+F
| 5 | 0.168999 | 0.167446 | 0.154289 | 0.153895 | 0.146525 | 0.147573 | 0.145413 | 0.145591
| 8 | 0.168489 | 0.167466 | 0.154478 | 0.152689 | 0.150962 | 0.145706 | 0.150321 | 0.144873
| 10 | 0.167841 | 0.16671 | 0.157372 | 0.154233 | 0.144939 | 0.145926 | 0.14378 | 0.143663
96 | 12 | 0.167211 | 0.167596 | 0.153383 | 0.152575 | 0.145388 | 0.144406 | 0.144109 | 0.143549
| 5 | 0.216142 | 0.214978 | 0.199501 | 0.20005 | 0.190332 | 0.188983 | 0.188193 | 0.187653
| 8 | 0.215683 | 0.214616 | 0.2003 | 0.197924 | 0.190594 | 0.188961 | 0.187488 | 0.187293
| 10 | 0.216368 | 0.215074 | 0.198559 | 0.196907 | 0.188374 | 0.188846 | 0.189149 | 0.187211
192 | 12 | 0.215767 | 0.214471 | 0.19829 | 0.196227 | 0.188737 | 0.188946 | 0.186627 | 0.187315
| 5 | 0.271176 | 0.268613 | 0.252509 | 0.250649 | 0.2425 | 0.241132 | 0.238817 | 0.237748
| 8 | 0.271297 | 0.269261 | 0.254065 | 0.251179 | 0.242187 | 0.241113 | 0.237223 | 0.236302
| 10 | 0.270489 | 0.26817 | 0.25271 | 0.25147 | 0.241169 | 0.241497 | 0.237585 | 0.23675
336 | 12 | 0.270616 | 0.268199 | 0.252692 | 0.250244 | 0.241046 | 0.240129 | 0.236639 | 0.236732
| 5 | 0.350177 | 0.347762 | 0.331098 | 0.329988 | 0.317609 | 0.317085 | 0.307792 | 0.308279
| 8 | 0.350805 | 0.348277 | 0.332104 | 0.329343 | 0.317169 | 0.316756 | 0.307681 | 0.307556
| 10 | 0.350146 | 0.347992 | 0.331052 | 0.329464 | 0.317204 | 0.316535 | 0.307876 | 0.307997
720 | 12 | 0.349919 | 0.347507 | 0.330589 | 0.328929 | 0.31728 | 0.317051 | 0.307549 | 0.307695

###  D.4 Electricity

Tab. 20 shows the parameter count of parameters of FITS with different
settings on the Electricity dataset. Tab. 21 shows the result on the
Electricity dataset with different settings correspondingly. We find that the
Electricity dataset is sensitive to the COF. This is because this dataset
shows significant multi-periodicity, which requires capturing high-frequency
components. Otherwise, FITS will not learn such information.

Table 20: The number of parameters under different settings on Electricity
dataset.

|  | Look-back Window
---|---|---
Horizon | COF/nth Harmonic | 90 | 180 | 360 | 720
96 | 4 | 1431 | 2752 | 6975 | 20385
6 | 2450 | 5192 | 14338 | 43734
8 | 3698 | 8475 | 24186 | 75628
10 | N/A | 12558 | 36765 | 116202
192 | 4 | 2187 | 3698 | 8475 | 22815
6 | 3710 | 7021 | 17334 | 48856
8 | 5633 | 11400 | 29329 | 84434
10 | N/A | 16926 | 44460 | 130005
336 | 4 | 3321 | 5160 | 10725 | 26460
6 | 5600 | 9794 | 21828 | 56539
8 | 8514 | 15900 | 36974 | 97902
10 | N/A | 23478 | 56088 | 150549
720 | 4 | 6318 | 9030 | 16650 | 36180
6 | 10710 | 17110 | 34026 | 77224
8 | 16254 | 27750 | 57546 | 133644
10 | N/A | 40950 | 87210 | 205440

Table 21: The results on the Electricity dataset. Values are visualized with a
green background, where darker background indicates worse performance. The
top-5 best results are highlighted with a red background, and the absolute
best result is highlighted with red bold font. F represents supervision on the
forecasting task, while B+F represents supervision on backcasting and
forecasting tasks.

| Look-back Window | 90 | 180 | 360 | 720
---|---|---|---|---|---
Horizon | COF/nth Harmonic | F | B+F | F | B+F | F | B+F | F | B+F
| 4 | 0.211863 | 0.211541 | 0.164613 | 0.164616 | 0.156359 | 0.156297 | 0.150566 | 0.150438
| 6 | 0.207940 | 0.207753 | 0.158720 | 0.158540 | 0.145222 | 0.145141 | 0.141353 | 0.141240
| 8 | 0.205725 | 0.205731 | 0.155645 | 0.155360 | 0.141808 | 0.141797 | 0.136764 | 0.136518
96 | 10 | N/A | N/A | 0.153865 | 0.153727 | 0.140404 | 0.140146 | 0.134548 | 0.134512
| 4 | 0.208883 | 0.208732 | 0.177320 | 0.177169 | 0.170017 | 0.169900 | 0.164768 | 0.164958
| 6 | 0.204794 | 0.204745 | 0.171682 | 0.171478 | 0.159142 | 0.158998 | 0.155794 | 0.155677
| 8 | 0.202812 | 0.202812 | 0.168498 | 0.168369 | 0.155829 | 0.155772 | 0.151284 | 0.151151
192 | 10 | N/A | N/A | 0.166973 | 0.166850 | 0.154208 | 0.154133 | 0.149191 | 0.149113
| 4 | 0.223752 | 0.223654 | 0.193835 | 0.193764 | 0.185941 | 0.185745 | 0.180178 | 0.180076
| 6 | 0.225369 | 0.228277 | 0.188469 | 0.188316 | 0.175312 | 0.175207 | 0.171449 | 0.171408
| 8 | 0.217910 | 0.217896 | 0.185530 | 0.185429 | 0.172143 | 0.172221 | 0.167087 | 0.167156
336 | 10 | N/A | N/A | 0.184023 | 0.183940 | 0.170639 | 0.170568 | 0.165106 | 0.165353
| 4 | 0.264973 | 0.264844 | 0.232324 | 0.231985 | 0.223079 | 0.222988 | 0.217765 | 0.217644
| 6 | 0.262983 | 0.262587 | 0.227060 | 0.227014 | 0.213388 | 0.213266 | 0.209735 | 0.209591
| 8 | 0.261890 | 0.262908 | 0.224484 | 0.224334 | 0.210542 | 0.210451 | 0.205780 | 0.205604
720 | 10 | N/A | N/A | 0.223122 | 0.223114 | 0.209192 | 0.209104 | 0.204054 | 0.203816

##  Appendix E Full Anomaly Detection Results

The full results with Accuracy, Precision, Recall, and F1-score are shown in
Tab. 22. For better performance, we also conduct experiments only on the first
channel of the SML dataset, denoted as (C0). We also trained FITS using only
the analog channels of SWaT, denoted as (analog).

Table 22: Full results on five datasets.

Datasets | Accuracy | Precision | Recall | F1-score
---|---|---|---|---
SMD | 99.92 | 99.9 | 100 | 99.95
PSM | 94.43 | 97.2 | 90.43 | 93.69
SWaT | 99.42 | 97.84 | 100 | 98.9
SWaT(analog) | 97.81 | 91.74 | 100 | 95.69
SMAP | 89.39 | 77.52 | 65.05 | 70.74
MSL | 81.52 | 61.38 | 80.16 | 69.52
MSL(C0) | 83.77 | 81.34 | 75.15 | 78.12

##  Appendix F Anomaly Detection Results on Synthetic Dataset

We generate the synthetic dataset using the script provided in the benchmark
with the default setting, i.e., 5% outlier on each channel with different
outlier types. We generate 4000 time-steps as our dataset, in which we take
2500 for training and the rest 1500 for testing. For our FITS model, we use
four different reconstruction windows, labeled as FITS-winxxx. We compare with
the results retrieved from Table 17 of the original paper (Lai et al., 2021).
The result is shown in 23.

Model | Precision | Recall | F1-score
---|---|---|---
FITS-win24 | 1 | 1 | 1
FITS-win50 | 1 | 1 | 1
FITS-win100 | 1 | 0.9993 | 0.9996
FITS-win400 | 1 | 0.9991 | 0.9995
AR | 0.59 | 0.77 | 0.64
GBRT | 0.47 | 0.56 | 0.51
LSTM-RNN | 0.22 | 0.26 | 0.24
IForest | 0.48 | 0.57 | 0.52
OCSVM | 0.62 | 0.74 | 0.67
AutoEncoder | 0.20 | 0.24 | 0.22
GAN | 0.15 | 0.15 | 0.15
Table 23: Results on the synthetic dataset.

##  Appendix G Datasets Visualization on Anomaly Detection

As shown in Fig. 5 and Fig. 6, most PSM and SMD datasets channels are analog
values. Especially the PSM dataset shows great periodicity.

Figure 5: Waveform of PSM dataset.

Figure 6: Waveform of SMD dataset.

While some channels in the SWaT dataset are binary event values, as shown in
Fig. 7.

Figure 7: Waveform of SWAT dataset.

However, as shown in Fig. 10 and Fig. 9, for SMAP and MSL datasets, most
channels are binary event values that are hard for FITS to learn frequency
representation.

Figure 8: Waveform of SMAP dataset.

Figure 9: Waveform of MSL dataset.

##  Appendix H Parameter Counts for Anomaly Detection

We use a fixed sliding window of 200 and 400 for all the datasets and do not
apply any frequency filter. The downsample rate is set as 4 for any dataset.
Thus, the number of parameters is as Tab. 24.

Table 24: MACs and parameter count of FITS on Anomaly Detection task. We report the MACs on the SWaT dataset which has 55 channels.  Window | Params | MACs
---|---|---
200 | 2600 | 137.5k
400 | 10200 | 550k

##  Appendix I Critical Difference Plot

We generate the critical difference plot on our result with the default alpha
as 0.05. FITS’s placement at the top of the critical difference plot, without
intersecting with other lines, demonstrates its consistent and superior
performance in terms of MSE compared to the other models. This signifies the
effectiveness of FITS in forecasting tasks. Moreover, the absence of
intersection indicates the statistical significance of the performance
difference, indicating that the disparity in MSE between FITS and others is
unlikely due to chance alone. The critical difference plot also showcases the
robustness of FITS’s performance across various evaluation metrics,
reinforcing its reliability. As the top performer in terms of MSE, FITS
emerges as a strong contender for model selection when tackling regression
problems. The statistical significance illustrated by the critical difference
plot further bolsters the confidence in the performance comparison, providing
substantial evidence that FITS outperforms the alternatives significantly.

Figure 10: The Critical Difference Plot on the FITS and other baselines with
alpha=0.05.