FITS: Modeling Time Series with $10k$ Parameters
Authors: Zhijian Xu
Ailing Zeng
Qiang Xu
Venue: ICLR 2024 spotlight (2024)
Keywords: Time series analysis
Time series forecasting
Complex-valued neural network
Primary area: general machine learning (i.e., none of the above)

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 for review at: \url{https://anonymous.4open.science/r/FITS}.

Published as a conference paper at ICLR 2024
FITS: MODELING TIME SERIES WITH 10k PARAME- TERS
Zhijian Xul, Ailing Zeng', Qiang Xuæl
Department of Computer Science and Engineering, CUHK Interraatioaal Digital Economy Academy (IDEA)
zjxu21 Ccse. cuhk. edu. hk,
quuêcse. culicse.cuhk..o cuhk. edu.
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 state-of- the-art models for time series forecasting and anomaly detection tasks. Notably, FITS accomplishes this with svelte profile of just about 10k parameters, making it ideally suited for edge devices and paving the way for wide range of applications. The code is available at: https: / tht.b.gggguuuu...ooo...o com /w///VVXOXC//FTS
INTRODUCTION
Time series analysis plays pivotal role in myriad of sectors, from healthcare appliances smart factories. Within these domains, the reliance 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 sophisticated models and methodologies (Zhou et al.) 2021] Liu et al.) [2022a] Zeng et al.) 2023 Nie et al.) 2023) Zhang 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 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 complex- vaauud linear layer, meticulously designed learn amplitude scaling and phase shift, thereby facilitating interpolation within the complex frequency domain.
Notably, while FITS operates interpolations 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 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.

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Published as a conference paper at ICLR 2024
Apart from its streamlined linear architecture, FITS incorporates a low-pass filter. This ensures pact representation while preserving essential information. Despite its simplicity, FITS consistently achieves ofte--o-thhearr (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 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 for forecasting or anomaly detection.
In summary, our contributions can be delineated follows:
We present FITS, an exceptionally lightweight model for time series analysis, boasting a modest parameter count in the range of 5k~10k.
FITS offers pioneering approach to time series analysis by employing a complex- -valud neural network. This simultaneously captures both amplitude and phase information, paving the way for 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 range of time series analysis tasks.
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 pure attention-bar 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 capturing long-term periodic patterns and speed up computation.
The other line of work aims to capture the periodicity inherent the data. For instance, DLinear ZZeng eet al.] (202)) adopts a single linear layer to extract the dominant periodicity from the temporal domain and surpasses a range of deep feature extraction-bars based methods. More recently. TimesNet (Wu et al. 2023) achieves state-of-the 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 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 signal allows us to break it down into a linear combination of sinusoidal components without any information loss. Each component possesses unique frequency, initial phase, and amplitude. Forecasting directly on the original time series can be challenging, but forecasting each frequency component is comparatively straightforward, we only need to apply 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 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 the original time series with reasonable time shift, ensuring that semantic consistency is maintained.
However, forecasting each sinusoidal component the time domain can be cumbersome, as the sinusoidal components treated sequence of data points. To address this, propose conducting
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this manipulation in the complex frequency domain, which offers compact and information- rich representation, as described below.
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) often employed when working with real input signals. condenses an input of real numbers into 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 representation of 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 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 delay introduced by that component. Mathematically, the complex number associated with a frequency component can be represented complex exponential element with given amplitude and phase:
x(f) (X()))e**())
where X(f) is the complex number associated with the frequency component frequency f. (f)) is the amplitude of the component, and f(f) is the phase of the component. As shown in Fig. I(a) in the complex plane, the complex exponential element can be visualized as a vector with length equal to the amplitude and angle equal to the phase:
X()) ()) + sin O(J))
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.
Imaginary
Imaginary
(IX,) I& (e
XX Jonsd + [[X]sm
Rx
Real
Real
(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 signal corresponds to the phase shift 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. we shift signal xff) forward in time by constant amount т, resulting in the signal x(t T), the Fourier transform is given by:
= = -2 = (X0)
[cos(-2nff) + j
The shifted signal still has an amplitude of [X()),, while the phase (f( = f(f) - f shows a shift which is linear to the time shift.
In summary, the amplitude scaling and phase shifting can be simultancously expressed as the multiplication of complex numbers, shown in Fig. 1(b)
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3.2 FITS PIPELINE
Motivated by the fact that a longer time series provides a higher frequency resolution in its fre- quency 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).
LET
LPF
Complexx-aluud Linear Layer
RIN
X
Valla
Arppa
Arpp
irFFT
0 Padding
IRIN
1000
0
1/ow
8n
8
X
Figure 2: Pipeline of FITS, with focus the forecasting task. Initially, the time series is normalized to zero-mean, followed by rFFT for frequency domain projection. After LPF, a single complex-valune 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 very large O-frequency component in its complex frequency representation. To address this, we pass through reversible instance-wise normalization (RIN) (Kim et al. 2022) to obtain zero-mean instance. As 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 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 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 provide supervision for forecasting and backcasting, where the model is encouraged to accurately reconstruct the look-back window. Our ablation study reveals that combining backeast and forecast supervision can yield improved performance in certain scenarios.
For reconstruction tasks, we downsample the original time series segment based on specific downsampling rate. Subsequently, FITS is employed perform frequency interpolation, enabling the reconstruction of the downsampled segment back to its original form. Thus, direct supervision applied using reconstruction loss to ensure faithful reconstruction. The reconstruction tasks also follow the pipeline in Fig. 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 y7, which represents the ratio of the model's output length L, to its corresponding input length L, 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:
Lo/2
Lo
Mfrrg L./2
L
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Based on this formula, with arbitrary frequency f. the frequency band ~ ~f in the original signal is limearly projected to the frequency band ~ 'ff 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 nL. Notably, when applying the Low Pass Filter (L.P),, the value of I. corresponds to the cutoff frequency (COF) of the IPF. After performing frequency interpolation, the complex frequency representation is zero-padded to a length of 4,/2, where L, represents the desired output length. Prior to applying the irFFT, an additional zero is introduced as the representation's zero-frequenge 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 specified cutoff' frequency (COF), resulting in a more concise fre- quency domain representation. The LPF retains the relevant information the time series while discarding components beyond the model's learning capability. This ensures that 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 quarter of the original frequency domain representation. Furthermore, the high- frgu-fryeeency components filtered out by the LPF typically comprise noise, which are inherently irrelevant for effective time series modeling.
No Filter MSE: 0.0000
car: 120 MSE: 0.0727
MSE: 0.1660
40 MSE: 0.4298
30 Myy
MW
30-
Joh M
MW
25-
25.
25-
25-
20-
15
15-
200
200
25c
100
160
200
230
(a) Original
(b) COF 6th
harmonic
(c) COF at 3dd harmonic (d) COF at 2dd harmonic
Figure 3: Waveform (1xt 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 length of 480, and its dominant periodicity is 24, corresponding to 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 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 harmonies' waveforms shown in Fig. 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 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 look-back
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window, with length L, combination of the look-back window and forecasting horizon, with length L+H H. This design enables us to provide more supervision during training. With this approach, 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:
H
More = 1 +
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 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-ar time series forecast- ing models, including PatchTST (Nie et al. 2023), TimesNet (Wu et al. 2023), FEDFormer (Zhou al. 2022a) and LTSF-Linear (Zeng et al. 2023),
rerun all the experiment with code and scripts
provided by their official implementation We report the comparison with NBeats (Oreshkin et al. 2019) NHits (Challu et al. 2023) and other transformer-basee methods in the appendix.
Evaluation metrics. We follow the previous works (Zhou et al. 2022a Zeng et al. [203 Zhang 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 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. [ Tab. 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.
The results
Tab. presents the number of trainable parameters and MACs for various TSF models using look-back window of 96 and forecasting horizon of 720 on the Electricity dataset. The table clearly demonstrates the exceptional efficiency of FITS compared to other models.
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-ard model PatchTST has a parameter count over 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 720 back window and 720 forecasting horizon, the Dlinear model requires over 1 million parameters, whereas FITS achieves similar performance with only 10k-50k parameters.
With long-standing bug in the coding architecture fixed, see README file in our codebase. *MACs Multiply-Accumulat Operations) is commonly used metric that counts the total number of multiplication and addition operations neural network.

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Published as a conference paper at ICLR 2024
Table 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
Fh
FTTh2
FTTml
FTTm2
Horizon
336
720
336
720
720
336
720
P
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
0.468
Dlincar
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
YedFormer
0.375
0.459
0.484
0.508
0.480
0.483
0.326
0.437
TimesNet
0.384
0.136
0.491
0.521
0.340
0.402
0.152
0.462
0.338
0.374
0.110
0.478
0.187
0.249
0.321
0.108
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
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
Horizoo
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
0.188
0.197
0.212
0.573
0.611
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
0.134
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
Table 3: Number of trainable parameters, MACs, and inference time of TSF models under look-back
This analysis showcases the remarkable effi- ciency of FITS. Despite its small size, FITS consistently achieves competitive results, mak- ing an attractive option for time series anal- ysis tasks. FITS demonstrates that achieving state-of-thh- thet-tt--rtttt or close to state-of-the-arr perfor- mance with considerably reduced parameter footprint is possible, making it an ideal choice for resource-constrained environments.
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, Ims
FiLM
14.91M
5,97G
123.0ms
FEDformer
20.68M
4.41G
40.5ms
Case Study on ETTh2 Dataset
PatchTST
1.5M
5.07G
3.3ms
0.4ms
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.
DLinear
139.7K
40M
(3.05ms CPU)
0.6ms
FITS (Ours)
4.5K-10K
1.6--8.MM
(2.55ms CPU)
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-as 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 sfate-of-the-art performance with parameter count of only around 10k. Moreover, by reducing the look-back window to 360, FITS already achieves totssett--tttt----the performance by setting the cutoff frequency to the second harmonic, resulting in further reduction of the model's parameter count to under 5k (as shown in Tab. 55.
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Table 4: The results on the ETTh2 dataset. Values are visualized with green background, where darker background indicates worse performance. The top-5 best results are highlighted with 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
COPhthh Harmonic
F
B+F
F
B+F
B+F
B+F
0.399889
0.291371
0:290314
0.288107
0.279141
0.276635
0.275600
0.274817
0.293242
0.291333
0.289803
0.287171
0.278128
0.275723
0.273972
0.273567
96
0.292438
0,2901859
0.288541
0.286174
0.277293
0.274494
0.272384
0.272031
0.292387
0.290369
0,288530
0.285527
0.274042
0.272085
0:271719
0.292517
0,291466
0.287814
08285984
0,275930
0.273883
0.271512
0.271028
0:379401
0. 0.377047
0.361995
0. 399322
0.337767
0.336419
0.334493
0.334621
0.579080
0.376874
0.360790
0. 0.358059
0.337391
0,335736
0.333573
0.333758
192
5.878816
0.376472
0.3605344
0 0577773
0.336085
0,334531
0.492110
0.352475
0.378529
0.376129
0.360234
0 0.357533
0.336286
0.334475
0.332122
0.332281
0.378581
0.376481
0,360049
0.357478
0.335526
0,333846
0.331421
0.331667
0141133
0/177096
0.391167
08388505
0,360300
0,359665
0. 0,166390
1.356319
3
0.419264
0.416645
0.389740
0.387614
0.359802
0.359291
0.355825
0.355972
336
0.192337
0,416085
0,389790
0.387815
0.358774
0.358096
0.354695
0.354800
0.418985
0.416009
0,388972
0.387115
0.358652
0,358093
0.354805
0.354794
0.418359
0,416499
D.388945
0.887183
0.358011 1
0.357452
0.354055
0.35420S
0.120888
0.4182266
0:405711
0.404112
0.387592
0,386235
0.379710
0.380367
0.420441
0,418290
0.404405
0,403520
0.386570
0,385907
0.379501
0.380132
720
0.4204134
0.417756
D.4044631
0.405425
0. 0.386556
0.384828
0. 0.378209
D. 0.578890
0.419888
0.417725
0.403562
0. 0.402755
0.385489
0,38-1558
0.378227
0.578810
0.419376
0,417854
D.403643
0,402616
0,384709
0,383960
0.377463
0.378101
Table 5: The number of parameters under different settings ETThl & ETTh2 dataset.
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 win- dow and cutoff frequency, FITS can achieve excellent performance while maintaining computational effi- ciency, making it an appealing choice for real-world applications.
Look-back Windows
COI-ath
Horizon
Harmonic
90
180
2
703
1053
2279
5913
1035
1820
-3077
12064
1431
2752
20385
1922
3876
105744
31042
2450
5192
14338
43734
1064
1564
2450
5192
13520
192
2187
3698
8475
22815
5 EXPERIMENT
2914
5253
3710
7021
17334
48856
FOR ANOMALY DETECTION
1615
1998
3483
7665
2392
336
3521
5160
10725
26460
5.1 RECONSTRUCTION
4402
7293
15834
40006
5600
9794
21828
56539
AS FREQUENCY INTERPOLATION
3078
3510
5418
10512
4554
5950
10266
21424
As discussed before, we tackle the anomaly detection tasks in the self-supervised reconstructing approach. Specifically, make N time equidistant sampling on the input and train a FITS network with an interpolation rate of 1|ec = N to up-sample it. Please
720
6318
9030
16650
36180
8370
12750
24570
54780
10710
17110
34026
77224
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. (202I)) 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 the synthetic dataset (Lai 202 in the appendix G ct al.
Baselines. We compare FITS with models such TimesNet (Wu et al. 2023), Anomaly Trans-
former [u et al. 2022). THOC (Shen et al., 2020), Omnianomaly (Su et al. 2019). DGHL (Challu
et al. 2022). Following TimesNet ( (Wu et al. (023), ) also compare the anomaly detection per-
formance 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. Wu ct al.
2020
2023), use Precision, Recall, and F1-score metrics.
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Implementation details. We window size of 200 and downsample the time series segment by factor of 4 as the imput to train FITS to reconstruct the original segment. We follow the methodology of the Anomaly Transformer (Xu et al. 2022), where time points exceeding certain reconstruction loss threshold are classified as anomalies. The threshold is selected based on the highest F1 score achieved the validation set. To handle consecutive abnormal segments, adopt widely-used adjustment strategy (Su et al. 2019| 9 Xu et al. [0018] Shen et al. 2022). 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 Fl-scores on 5 datasets. The best result is highlighted in bold, and the second best highlighted with underline. Full results are reported in the Appendix.
Models
PITS
TimesNet
Anomaly Transfomer
THOC
Onnt Anomaly
Stationary Transformer
DIGHT.
OCSVM
[Forest
LighTTS
Dliiear
SMD
99.95
35.81
92.33
84.99
85.22
84.72
NIA
56.19
53.64
82.53
77.1
7.62
PSM
93.96
97.47
07 39
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
90:68
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
81.58
-15.96
5.3
COMPARISONS WITH SOTAS
In T FITS stands out with outstanding results on various datasets. Particularly, on SMD and SWaT datasets, FITS achieves nearly perfect Fl-scores, around 99. 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 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 Fl-scores on these datasets.
For more comprehensive evaluation, waveform visualizations and detailed analysis can be found in the appendix, providing deeper insights into FITS' strengths and limitations different anomaly detection scenarios. It is important to note that the reported results are achieved with parameter range of 1-4K and MACs (Multipyy-Acummull Operations) of 10-137K, which will be further detailed in the appendix.
While the datasets in use are instrumental, is imperative to acknowledge their limitations as delineated in (Lai et al. [2021). Particularly on the synthetic dataset from (Lai al., 2022I). FITS demonstrates impeccable detection capabilities, registering a flawless 100% Fl score. For a detailed breakdown, readers can refer the table in appendix G This dataset marries a sinusoidal wave of
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 - 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 specialized AD algorithm geared for detailed detection, the combined system stands as a paragon of both robustness and swift responsiveness facing diverse anomalies.
CONCLUSIONS AND FUTURE WORK
In this paper, we propose FITS for time series analysis, low-cost model with 10k parameters that can achieve performance comparable to ofate-of-the-art models that are often several orders of magnitude larger. As the future work, 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.
plan to evaluate FITS on more real-world scenario and

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Published as a conference paper at ICLR 2024
ACKNOWLEDGEMENTS
This work is supported part by the CUHK SSFCRS funding under Grant No. 3136023, and in part by the Research Matching Grant Scheme under Grant No. 7106937, 8601 130, and 8601440.
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A PIPELINE FOR RECONSTRUCTION
The pipeline for the reconstruction task is shown in Fig. 4 In this process, the model input æ is derived from segment the time series y using an equidistant sampling technique with specified downsample rate 7) Subsequently, FITS performs frequency interpolation, generating an upsampled output hup-sampled with the same length as y. The reconstruction loss computed by comparing the original y and the upsampled Tup Tup-sumplee sampled. Please note that, due space constraints, the depicted downsample/upsampp rate 7) in the figure is shown as 1.5, which is not a practical value. In our actual experiments, employ a r value of 4.
Volue
Amp
Read Froq
Amp
Hanno
RIN #FFT
LPF
Complex-valued Linear Layer
0
X
X'
Value
Amp
Amp
irFFT
O Padding
iRIN
Fro
Frop
0
Rup Pup-sampled
8
X"
Figure 4: Pipeline of FITS, with a focus on the Reconstruction task.
B DETAILS OF FORECASTING DATASETS
We report the characteristics in the tab
Table The statistics of the seven used forecasting datasets.
Dataset
Traffic
Electricity
Weather
ETTh1&ETTh2
ETTml &ETTm2
Channels
862
321
21
7
Sampling Rate
lhour
lhour
10min
lhour
15min
Total Timesteps
17.544
26,304
52,696
17,420
69,680
C CRITICAL DIFFERENCE PLOT
We generate the critical difference plot on our result with the default alpha as 0.05 as shown in Fig. 5 FITS's placement at the top 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 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 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.
D 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.
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MSE
5
4
3
2
1
FITS
FedFormer TimesNet
PatchTST Dlinear
Figure The Critical Difference Plot on the FITS and other baselines with alpha=0.05.
D.1 COMPARISON WITH TRANSFORMER-BANE BASEDD METHODS
We further compare FITS with Autoformer Wu et al. 2021). Informer (Zhou et al. 2021 FiLM (Zhou al. 2022b) and Pyraformer (Liu et al. 2022b). The results are shown in Tab. and Tab. 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.
Datnset
Horizon
96
192
330
720
96
192
330
720
96
192
335
720
96
192
3.56
720
Autoformer
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
0.120
0.159
0.506
0.346
0.129
0.196
0:163
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
0.465
0.377
0.439
0.302
0.165
BITS
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
Traflic
Weather
Horizon
192
336
720
96
192
720
96
192
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
D.2 COMPARISON WITH NBEATS & NHITS
We show the comparison with mentioned N-HiTS and N-BEATS 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 our main result. The results for N-HiTS and N-BEATS are retrieved from the paper of N-HiTS (Challu et al. 2023).
D.3 SHORT-TERY FORECASTING 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. shown Tab FITS shows the suboptimal results on the M4 dataset. The reason for this outcome is threefold. First, the M4 dataset a collection of many time series from different domains. These time series have different temporal information and periodicity, and 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 model such diverse datasets with one model. However, considering the lightweight of FITS, is hard for to achieve ideal results. Finally, the setting for the M4 dataset not suitable for FITS. The look-back window is set to
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Table 10: Comparison with N-HiTSS and N-BEATS MSE
Dataset
Horizon
FITS
N-BEATS
N-HITS
96
0.138
0.145
0.147
Electricity
192
0.152
0.180
0.167
336
0.166
0.200
0.186
720
0.205
0.266
0.243
96
0.401
0.398
0.402
Traffic
192
0.407
0,420
336
0.420
0.449
0.448
720
0.456
0.539
96
0.145
0.167
0.158
Weather
336
0.236
0.288
0.274
96
0.164
0.184
0.176
ETTm2
192
0.211
0.273
336
0.269
0.309
0.295
720
0.347
Table 11: Results on M4 dataset in SMAPE
FITS
TimesNet DLinear
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
12. 16, and 36 for yearly, quarterly, and monthly prediction accordingly. which twice the length of the forecasting horizon. Such short look-back window is very difficult to extract meaningful frequency representation, which further worsens the FITS' performance. We compare FTTS with lightweight model DLinear (Zeng et al., (023). state-of-the-art model TimesNet (Wu et al.) 2023) and two hierarchical time series modeling model N-Hits (Challu et al. 2023) and N-Beats (Oreshkin
al. 2019).
E CASE STUDY ON OTHER DATASETS
We show the parameter table and performance other datasets below.
E.1 ETTH1, ETTMI & M2
Tab[1] shows the corresponding results on ETThl dataset with different settings. ETTh] shows 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 ETTml dataset. We attribute this phenomenon the distribution shift that exist in the datasets. The longer look-back window will introduce more information from shifted distribution and sabotage the forecasting result.
Tab. shows the parameter count of parameters of FITS with different settings the ETTml & 2 datasets. Tab. [4] and Tab Tab[55 show the corresponding results on ETTml and ETTm2 datasets with different settings. Note that FITS constantly achieves SOTA performance the ETTm2 dataset with under 10k parameters.
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Table 12: The results the ETThl dataset. Values visualized with green background, where darker background indicates worse performance. The top-5 best results are highlighted with 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.
back Window
50
ISU
360
720
Horizon
COFmih Harmonic
B+F
B+F
B+F
B+F
0.391210
0.391987
0.358999
0.389400
0.396873
0.396321
0.402544
0.403307
0.390959
0.389857
0.385510
0.386088
0.380894
0.380825
0.386660
0.387143
96
0.377619
0.385072
0.366867
0.387619
D.382036
0.374568
3.387786
0:381331
0.3 0.386679
0.386013
0.390810
0.381219
0,372101
0.572712
0,379216
0.579815
04441343
0.442006
03325477
0.467155
0.455800
0.441186
422812
04118I
0,421554
192
04437466
0.439231
D442605
0427898
0. 0.409010
0.409368
0.417933
0.418202
0.437844
0.438436
0.426070
0 0426605
0.406772
0,407058
0.415545
0.416382
0,425226
425557
0,406699
0,414150
DX22840
0.283552
D:458024
0 06112
0:448917
0,448952
0:455474
0:462104
0.182419
0.182874
0.455709
O 0458153
0.133826
0,434296
0,410474
0.447663
336
0.481796
0.482031
0,454570
0456531
0.431753
0,432849
0,437971
4.400666
0,480184
D.453879
0555248
0.480369
04400036
0-444967
44449887
0.477669
0,478893
0.452134
0 0544378
0.427425
0427670
0.445115
0.499899
0,444%6
0,450286
0.4703009
0.472660
D.440675
0 0444466
0.438920
0.4244
0.439924
0.462995
720
0-469793
0.4711439
D.438692
0.442222
0.4284833
0,404407
0.435446
0.436730
0.470391
0,4771038
D.457598
0415572
B.436720
0.428304
0,431934
0:433105
0.468232
0.169044
0.438001
0.410176
0.424749
0.425963
0.430258
0,431988
Table 13: The number of parameters under different settings on ETTml & ETTm2 dataset.
Look- back Window
Horizon
Harmonic
90
720
420
513
621
1330
561
759
1015
2444
3835
10
861
1426
2050
5609
12
1035
1820
2726
7636
645
703
759
1508
850
1035
1218
2726
1064
10
1302
1922
2501
6248
12
1564
2450
3290
8544
14
1875
3042
6767
18942
990
969
966
1715
1275
1449
1566
31499
5015
10
1974
2666
3157
7242
12
2392
3395
4136
9960
14
2825
4212
8509
21894
1890
1710
2380
2448
2530
2136
4524
3078
3510
3570
6844
3780
9940
12
4554
5950
6486
13612
14
5400
7410
13266
30012
E.2 TRAFFIC
Tab. 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.
15

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Published as a conference paper at ICLR 2024
Table 14: The results on the ETTml dataset. Values are visualized with a green background, where darker background indicates worse performance. The top-5 best results are highlighted with 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
Hurizon
COF/ath Harmonic
B+F
0.365445
0.3649220
0.312641
0.312953
0.305685
0.306146
0.314674
0.314514
0.364851
0.364060
0.312192
0.312162
0,304728
0,304859
0.311096
0.311866
96
10
0.364031
0. 0.363901
0.31
0.312576
0.303561
0.304319
0.310334
0.310245
0.363204
0.311027
0.303412
0.303837
0.309762
0.309719
14
0, 0,362444
0. 0.362795
0,31
0.311589
0.303153
0,303350
0.309248
0,309280
0.400734
0.400507
0.348337
0.348248
0.339300
0.339427
0,342016
0.342371
0,400492
0,400347
0.347860
0,347737
0.338135
0,338368
0,389153
0.339648
192
0,399691
0.399441
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
0.399249
0.347410
0.347304
0.337172
0.337290
0.337920
0.318687
0,431620
0.431774
0.384310
0,384216
0.372982
0.373155
0. 0.369938
0.369875
0.431072
0.431733
0.384172
0.384051
0.371957
0.372328
0.367226
0.367214
336
0,431126
0,431569
0.383794
0.383776
0.371696
0.371917
0,366873
0.367586
0.430388
0,431283
0.383467
0.383656
0.371289
0.371662
0.366383
0.366726
14
0,430749
0,431064
0,383719
0.383675
0.371363
0,371460
0,366238
0,366492
0.492552
0.492473
0,443358
0.44391
0.427787
0.428065
0.488358
0.418436
0,491822
0.492750
0. 0,443295
0,443706
0.427054
0,427269
0.15964
0,416064
720
0.492088
0.492691
0,443319
0.443260
0.426868
0.427377
0.417002
0.416024
0,491904
0.492589
0,443103
0.443476
0.426748
0.426819
0,415402
0.415881
0.491
0.491228
0. 0,443076
0.443384
0.426571
0.427071
0.415266
0.415599
Table 15: The results on the ETTm2 dataset. Values are visualized with green background, where darker background indicates worse performance. The top-5 best results are highlighted with 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-hod Window
90
Horizon
COH/ath Harmonic
B+F
F
B+F
F
B-F
F
B+F
0.185981
0.185774
0.174363
0.174277
0.166817
0.166589
0.164719
0. 0.164500
0.185601
0.165615
074002
0.173797
166025
0,165700
0.164127
0. 1616374
96
0.185515
0.185338
0.173717
0.173536
0.165634
0.165413
0.163178
0.162928
0. 185489
0. 08150338
173951
0.173368
0,166369
0,165113
0.162719
0. 1625500
0. 085666
08185448
0. 0.133502
0,173315
0.165362
0,165198
0.162575
0.162346
0.249365
1.249119
0.233717
0.233428
0.221364
0.220948
0.218563
0.218267
0.249366
0.2489-15
0.235-405
0.233171
0.220596
0.220165
0,218115
0,217808
192
0.248919
0.248686
0.233276
0.232929
0.220104
0.220518
0,217338
0.216963
12
0.248677
0. 0,248890
0.233005
0.233042
0.220007
0.219895
0.216927
0.216714
0,248678
0.248454
0,233162
0,232763
0.219897
0.219597
0.216879
0.216650
0,309083
D. 0.308863
0,286187
0.286170
0,273920
0,273816
0.269833
0,269620
0.308577
0.286186
0.286041
0.273418
0.273171
0.269393
0.269252
336
10
0,308768
0. 0.308713
0,286102
0.285768
0.273038
0,272893
0.268921
0,268396
0.3088741
0.308568
0.286529
0.285881
0.272931
0.272763
0.2684488
0.268273
14
0. 0,308759
0. 303393
0. 0.286232
0.285711
0.272905
0.272804
0. 0,268366
0,268219
0.408977
0 4048844
0.384164
0.383995
0,366645
0.366654
0.350173
0.349770
0.409260
0 0.408714
0.383980
0.383899
0.366070
0.366085
0.349659
0.349619
720
0.4088793
0.383886
0.383827
0.365909
0.365935
0.349019
0.348881
12
0.408698
0. 0.408679
0.383921
0.383664
0.365810
0.365805
0.348831
0.348863
0.408479
0.383620
0.365790
0.365801
0.348938
0.348766
E.3 WEATHER
Tab. 18 shows the parameter count of parameters of FITS with different settings on the Weather dataset. Tab. [7shows 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- window as 700.
16

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Published as a conference paper at ICLR 2024
Table 16: The number of parameters under different settings on Traffic dataset.
Look-back Window
Horizon
COF/nth Harmonic
90
180
360
720
1820
12064
1922
3876
10574
31042
75628
10
NIA
12558
36765
116202
192
13520
2914
5253
12558
34694
10
N/A
16926
44460
130005
336
2392
3395
6608
15704
4402
7293
15834
40006
36974
97902
10
NIA
23478
5608S
150549
720
4554
5950
10266
21424
8370
12750
24570
54780
16254
27750
57546
133644
10
N/A
40950
87210
205440
Table 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 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
COHlath Harmonic
B+F
B+F
B+F
B+F
0.694065
0.694425
0.474606
0,475881
0,455815
0,457292
0.436317
0.436616
96
0.686110
0.684290
0.457057
0.456547
0.419903
0.419748
0,397558
0.397210
0.682876
0.681464
0.452024
0,451470
0.410857
0.410458
0.387791
0.388830
N/A
N/A
0,451340
0.450850
0.409948
0.409614
0.385763
0.386596
0.636434
0.481686
0.485085
0.463516
0.464170
0.442661
0.443547
0.622882
0.622451
0.467921
0.467470
0.431367
0.431097
0.407908
0.407850
192
0.620814
0.620003
0.463119
0.462929
0.422511
0.422336
399032
0,399101
10
N/A
NA
0.462595
0.461907
0.421705
0.421677
0,397286
0,398034
0.473090
0,476491
0.454243
0.632244
0.631760
0.481267
0.480599
0.442476
0.442128
0.420268
0.420239
336
0.629962
0,434124
10
N/A
NA
0.476248
0.476044
0.433656
0.4334566
0.410500
0.410417
0.500635
0.587891
0.670385
0. 0.669979
0.507742
0.507104
0.469442
0.469337
0,456470
0.456207
720
0.668322
0.536565
0.463208
0.463744
0.449220
10
N/A
N/A
0.503795
0.503702
0.462643
0.463091
0.448882
0.448182
E.4 ELECTRICITY
Tab. 20 shows the parameter count of parameters of FITS with different settings on the Electricity dataset. Tab. shows the result on the Electricity dataset with different settings correspondingly. We find that the Electricity dataset sensitive the COF This because this dataset shows significant multi-pen multii-priiodicity which requires capturing high-frequency components. Otherwise, FITS will not learn such information.
17

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Published as a conference paper at ICLR 2024
Table 18: The number of parameters per channel under different settings on Weather dataset.
Look-back Window
Horizon
COFlnth Harmonic
180
360
96
5
496
630
806
1845
1053
10
861
1426
2050
5609
1035
192
5
752
861
988
2050
8
1064
1431
1820
10
1302
1922
2501
6248
12
1564
2450
336
5
1136
1197
1248
2378
8
1615
1998
2275
5015
10
1974
2666
31577
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 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.
Window
Horizon
COF/ath Harmonic
B+F
B+F
B+F
B4F
0.168999
0.167446
0.154289
0.153895
0.146525
0.147573
0.145413
0.145591
0.168489
0.167466
0.154478
0,152689
0.150962
0.145706
0.150321
0.144873
0.167841
0.16671
0.157372
0.154233
0.144939
0.145926
0.14378
0.143663
12
0.167211
0.167596
0.153383
0,152575
0.145388
0.144406
0.144109
0.143549
0,216142
0.214978
0.199501
0.20005
0. 190332
0. 0.188983
0.188193
0.166653
0.215683
0.214661
0.2003
0,197924
0.190994
0.188961
0.187488
0.199993
0,216368
0.215074
0.198559
0.196907
0.188374
0.188846
0.189149
0.187211
12
0.215767
0.214471
0.19829
0,196227
0.188737
0.188946
0.186627
0.187315
0.27
0.268613
0.252509
0.250649
0.2425
0.241132
0.238817
0.237748
0.271297
0.269261
0.254065
0,251179
0.242187
0,241113
0.237223
0.236302
0,270489
0.26817
0.25271
0.25147
0.241169
0.241497
0.237585
0.23675
12
0.270616
0.268199
0.252692
0,250244
0.241046
0,240129
0.236639
0.236732
0.350177
0.347762
0.331098
0.329988
0.317609
0.317085
0.307792
0.308279
0.350805
0.348277
0.332104
0,329343
0.317169
0.316756
0.30768]
0.307556
0,350146
0.347992
0.331052
0.329464
0.317204
0.316535
0.307876
0.307997
12
0,349919
0.347507
0.330589
0.328929
0.31728
0.317051
0.307549
0.307695
18

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Published as a conference paper at ICLR 2024
Table 20: The number of parameters under different settings Electricity dataset.
Look-back Window
Horizon
COF/nth Harmonic
90
180
360
720
20385
6
2450
5192
14338
43734
96
75628
10
N/A
12558
36763
116202
22815
192
6
3710
7021
17334
48856
11400
84434
10
N/A
16926
44460
130005
4
3321
5160
10725
26460
336
5600
9794
21828
56539
15900
36974
97902
10
N/A
23478
50088
150549
6318
9030
16650
36180
720
10710
17110
34026
77224
8
16254
27750
57546
133644
10
N/A 40350
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 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/nhh Harmonic
0.211863
0.211541
0,164613
0,164616
0.156359
0.156297
0.150566
0.150438
96
0,207940
0.207753
0.158720
0.158540
0.145222
0.145141
0.141353
0.141240
0,205725
0.205731
0,155645
0,155360
0.141808
0.141797
0.136764
0.136518
N/A
NA
0.153865
0,153727
0.140404
0.140146
0.134548
0.134512
0.208883
0.208732
0,177320
0,177169
0.1 0.170017
0. 169900
0.164768
164958
0.204794
0.204745
0.171682
0.171478
0.199142
0. 158998
0.155794
155677
192
0,202812
0,202812
0,168498
0,168369
0.155829
0.155772
0.151284
0.151151
10
N/A
N/A
0.1669
0.166850
D. 154208
0,154133
0.149191
0.149113
0.225752
0.223654
0.193835
0.193764
0.185941
0.185745
0.180178
0.180076
0.225369
0,228277
188469
0.1883
0. 0.135312
0. 175207
0.171449
171408
336
0,217910
0,217896
0,185530
0,185429
0.172143
0.172221
0.167156
10
N/A
N/A
184023
0.1939
0. 0.199639
0. 170568
0.165106
165353
0.264973
0.232324
0.2229888
0.217644
720
0,2629883
0.262587
0,227060
0,227014
0.213388
0,213266
0.209735
0.209591
0.262908
0.224484
0.224334
0.210542
10
N/A
N/
0.223122
0,223114
0.209192
0.209104
0.204054
0,203816
F 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 (CO). We also trained FITS using only the analog channels of SWaT, denoted as (analog).
Table 22: Full results on five datasets.
Datasets
Accuracy
Precision
Rocall
Fi-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
(WaT(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/CO)
83.77
81.34
75.15
78.12
G ANOMALY DETECTION RESULTS ON SYNTHETIC DATASET
We generate the synthetic dataset using the script provided in the benchmark with the default setting, ie., 5% outlier each channel with different outlier types. We generate 4000 time-steps
19

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Published as a conference paper at ICLR 2024
Model
Precision
Recall
Fl-score
FITS-win24
FITS-win50
I
FITS-win100
1
0.9993
0,9996
FITS-win400
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
Forest
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.
dataset, in which take 2500 for training and the rest 1500 for testing. For our FITS model, 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
H DATASETS VISUALIZATION ON ANOMALY DETECTION
As shown in Fig. and Fig. 7 most PSM and SMD datasets channels analog values. Especially the PSM dataset shows great periodicity.
300000
Renur
030
Rosure
84
Sature
0.2
Nature
220
Satur
Natum 31
2000
00000
Figure Waveform PSM dataset.
While some channels in the SWaT dataset are binary event values, as shown in Fig. 8
20

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Published as a conference paper at ICLR 2024
100
05
08
1.0
00
00
1.0
035
05
0.0
00
-0.9
100
05-
64-
1.0
64-
05-
65-
1.0
50-
6--
10-
08
100
05-
68
100
62-
1.0
03
08
35
08
120000
Figure 7: Waveform of SMD dataset.
21

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Published as a conference paper at ICLR 2024
300
300
140
001
135
221
283
X000)
00320
80330
Figure 8: Waveform of SWAT dataset.
However, shown in Fig. 9 and Fig. 10 for SMAP and MSL datasets, most channels are binary event values that are hard for FITS to learn frequency representation.
Figure Waveform of SMAP dataset.
22

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Published as a conference paper at ICLR 2024
8.5
8.0
880
8.0
8.5
8.0
8.0
840
880
889
839
8.9
8.0
8.20
83%
880
8.00
8.9
0.59
6.5
8.00
J0622
$2500
Figure 10: Waveform of MSL dataset.
23

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Published as a conference paper at ICLR 2024
I 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 as 4 for any dataset. Thus, the number of parameters is Tab, 24
Table 24: MACS and parameter count of FITS 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
24


Reviews:
[{"review_id": "PeE9zjvuA1", "reviewer": "Reviewer_YAqa", "summary": "This paper proposes an impressive compact model, named FITS, for time series tasks, including forecasting and anomaly detection. FITS achieves manipulation to time series through interpolation in the frequency domain. The whole framework is quite simple and has remarkably few parameters. FITS achieves competitive performance to SOTA baselines on both forecasting and anomaly detection with about 50 times fewer parameters. With such impressive performance, the proposed model would have a certain impact on the community.", "questions": "1. How do author deploy the model on devices that do not support the complex computation?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The experiment result is surprisingly good considering the tiny footprint of the model. The standard deviation of the error is very small which may indicate that the model is very stable because of its simplicity. \n2. Authors provide comprehensive ablation analysis to show light-weightness of the model and its superior performance across the hyper-parameters.\n3. Authors use a synthetic dataset to show the key idea of FITS which is devide and conquer the different frequency. It also explained the effectiveness of the model on the AD task.", "weaknesses": "1. The result on anomaly detection is not remarkable. Especially on the SMAP and MSL dataset. Some more in depth analysis is needed to find out the reason.\n2. Even though the FITS have far fewer parameters comparing to the DLinear, it still need more time to inference on the GPU (0.6 and 0.4 ms respectively). Does this mean the DLinear is still the best choice for the real-time application?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZUufs1ESxO", "reviewer": "Reviewer_HyfR", "summary": "The paper presents FITS, a lightweight model designed for time series analysis. In contrast to conventional models that work directly with raw time-domain data, FITS operates within the complex frequency domain. It utilizes a streamlined linear layer and an efficient low-pass filter, achieving state-of-the-art performance in forecasting and anomaly detection tasks with a mere 10k parameters. FITS offers a novel perspective on these tasks, viewing them as interpolation exercises within the frequency domain. This approach extends time series segments for forecasting and reconstructing downsampled data. FITS employs a complex-valued linear layer to master amplitude scaling and phase shift, facilitating efficient complex frequency domain interpolation. Its compact size and competitive performance render FITS an excellent choice for edge devices, unlocking a wide array of applications in time series analysis.", "questions": "Did you try fitting a larger neural network for frequency interpolation task? It would be interesting to see if the performance of this architecture scales with the size.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper stands out for its clear and well-structured writing, making it easy to grasp the presented ideas. Addressing a critical issue, the paper tackles the challenge of developing time series analysis models suitable for deployment on resource-constrained edge devices, ensuring optimal performance. Furthermore, the paper impressively conducts a comprehensive evaluation. It includes a comparison with various state-of-the-art methods, even though these methods vastly differ in size from the proposed approach. The paper also introduces an efficient method for selecting the cutoff frequency, explores the impact of different lookback window sizes through ablation studies, and offers detailed insights into the training process. Notably, the approach ingeniously combines simple components such as RevIN, LPF, and the Complex-valued Linear layer to create a robust architecture that can be applied to industrial-grade time series analysis tasks.", "weaknesses": "The method's limitation lies in its inability to generate probabilistic forecasts, a crucial requirement for numerous industrial applications. Moreover, the evaluations conducted on benchmark datasets may not accurately reflect the real-world scenarios of edge devices. These benchmark datasets, such as those related to traffic and weather, typically do not require processing on edge devices and can be handled in offline settings. It remains uncertain how well the model will perform when applied to data from edge devices, like healthcare devices or industrial sensors, which present different challenges and requirements.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nPIeK8paZD", "reviewer": "Reviewer_r1EX", "summary": "The authors propose a parameter-efficient model architecture for time series anomaly detection and forecasting. The model's pipeline consists of the Fourier transform of the original time series, followed by a low-pass filter in the freq. domain, a linear layer (complex-valued), padding, and the inverse transform to map back into the time domain. For forecasting, the result of the inverse transform can be used directly, while for anomaly detection, a reconstruction error threshold, which increases the F1-score, is determined on the validation set. The authors show competitive performance on various benchmark data sets.", "questions": "* Can you explain why $x(t-\\tau)$ is a shift forward in time, rather than backward?\n* Can you add a more thorough caption to Fig. 2? As it stands, it is not comprehensible enough to fully understand your method. Is it just a single linear layer/what is the output dimension? What are the typical dimensions of $\\tilde{X}$? Are (i)RIN and (i)rFFT performed in parallel or sequentially? Please elaborate more on how RIN works, either in the Figure or its caption. \n* Have you experimented with more sophisticated layers than just linear? \n* What is the back-casting task, and how is it used in training?\n* Can you motivate in 1-2 sentences why the output length needs to be controlled (Section 3.3)?\n* You say the LPF discards components beyond the model's learning capabilities. What do you mean by this? The model's capabilities are independent of the signal, and high frequencies are not always noise or irrelevant. It seems domain-specific. Furthermore, you say high-frequency components such as trends are filtered out. First, trend signals can be low-frequency, and second, they are particularly important in the forecasting realm.\n* The method is particularly interesting because of its parameter-effectiveness. Can you perform experiments on real-time inference time on simple hardware (not necessarily edge devices) to substantiate this (e.g., real-time anomaly detection)? What are the challenges regarding domain shifts and refitting, and how could they be addressed? This could substantially improve the focus and contribution of the paper.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The manuscript is well-organized and clearly written. I appreciate the preliminary section, which is very instructive. The experimental evaluation is thorough, and many competing methods are compared in a rigorous manner. Lastly, I appreciate the different ablation studies the authors performed. Overall, the results are convincing, and I believe the proposed method sets a great benchmark for such a parameter-efficient method.", "weaknesses": "The presentation of FITS could be more comprehensive (more details in the questions). Furthermore, I would like to see more \"basic\" baselines (e.g., ARIMA, DeepAR for forecasting, and Random Cut Forest for Anomaly detection). To further substantiate the claims about performance (and to put them on a statistically sound foundation), a critical difference plot[1] may be a useful analysis tool. Lastly, the method employs several heuristics that need to be tuned in the real world, a drawback it shares with other methods.\n\n1. Demšar, J. (2006). Statistical comparisons of classifiers over multiple data sets. The Journal of Machine learning research, 7, 1-30.\n\n\n**Rebuttal:** I increased my score by one after the rebuttal, as all my concerns have been addressed appropriately.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "d7WmULBDAR", "reviewer": "Reviewer_u3QE", "summary": "Modern time series forecasting methods are heavily-parametrised, and in spite of their compelling performance on benchmark datasets, may not be appropriate in resource-constrained settings. In the spirit of recent work leveraging representations of time series in the frequency domain, the authors propose a new methodology that captures amplitude and phase information using a real-valued neural network. A low-pass filter is also included to further reduce the size of the model. An extensive experimental evaluation demonstrates how model performance is comparable (and sometimes better) to other SOTA techniques while also having orders of magnitude less parameters. The suitability of the model to anomaly detection via time series reconstruction is also validated on a variety of datasets.", "questions": "Please refer to comments in the section on *Weaknesses* as a guide for the rebuttal.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The reduction in model size enabled by the proposed architecture is commendable, and the minor trade-off with predictive performance (if any at all in some cases) makes this a compelling model for practitioners working with time series forecasting models.\n- The paper is well-written and nicely structured. Although there has been a recent resurgence in interest on how analysis in the frequency domain can be applied to forecasting, the authors adequately position this paper in the context of related work.\n- The experimental evaluation is extensive, and I especially appreciated the extension to anomaly detection, which is one of the more common downstream tasks stemming from time series forecasting. The insights on why the model works better on some datasets than others (e.g. citing the binary nature of some time series, as well as overall limitations with benchmarks in the field) was also appreciated.", "weaknesses": "- There are quite a few bits and pieces to the model pipeline, and at times the extent to which each component contributes towards overall performance is unclear. Perhaps a few ablation studies in the supplementary material can address this?\n- The concluding remarks of the paper focus on a few settings where the method may not be most appropriate, but I would have liked to see broader thoughts on future work, such as further reductions to model size, and possibly interpretability. \n- While I appreciated the experiment on anomaly detection, this is a field of study where (as referenced by the authors themselves), a few metrics may not be sufficient for fully capturing the success of different techniques. There is consequently a risk that the experiment featured here might not be sufficiently conclusive (although the results obtained on synthetic data are a step in the right direction).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (spotlight)