Enhancing SSVEP Identification With Less Individual Calibration Data Using Periodically Repeated Component Analysis

Objective: Spatial filtering and template matching-based steady-state visually evoked potentials (SSVEP) identification methods usually underperform in SSVEP identification with small-sample-size calibration data, especially when a single trial of data is available for each stimulation frequency. Methods: In contrast to the state-of-the-art task-related component analysis (TRCA)-based methods, which construct spatial filters and SSVEP templates based on the inter-trial task-related components in SSVEP, this study proposes a method called periodically repeated component analysis (PRCA), which constructs spatial filters to maximize the reproducibility across periods and constructs synthetic SSVEP templates by replicating the periodically repeated components (PRCs). We also introduced PRCs into two improved variants of TRCA. Performance evaluation was conducted in a self-collected 16-target dataset, a public 40-target dataset, and an online experiment. Results: The proposed methods show significant performance improvements with less training data and can achieve comparable performance to the baseline methods with 5 trials by using 2 or 3 training trials. Using a single trial of calibration data for each frequency, the PRCA-based methods achieved the highest average accuracies of over 95% and 90% with a data length of 1 s and maximum average information transfer rates (ITR) of 198.8±57.3 bits/min and 191.2±48.1 bits/min for the two datasets, respectively. Averaged online accuracy of 94.00 ± 7.35% and ITR of 139.73±21.04 bits/min were achieved with 0.5-s calibration data per frequency. Significance: Our results demonstrate the effectiveness and robustness of PRCA-based methods for SSVEP identification with reduced calibration effort and suggest its potential for practical applications in SSVEP-BCIs.


Enhancing SSVEP Identification With Less Individual Calibration Data Using Periodically Repeated Component Analysis
Yufeng Ke , Member, IEEE, Shuang Liu , and Dong Ming , Member, IEEE Abstract-Objective: Spatial filtering and template matching-based steady-state visually evoked potentials (SSVEP) identification methods usually underperform in SSVEP identification with small-sample-size calibration data, especially when a single trial of data is available for each stimulation frequency.Methods: In contrast to the state-of-the-art task-related component analysis (TRCA)-based methods, which construct spatial filters and SSVEP templates based on the inter-trial task-related components in SSVEP, this study proposes a method called periodically repeated component analysis (PRCA), which constructs spatial filters to maximize the reproducibility across periods and constructs synthetic SSVEP templates by replicating the periodically repeated components (PRCs).We also introduced PRCs into two improved variants of TRCA.Performance evaluation was conducted in a self-collected 16-target dataset, a public 40-target dataset, and an online experiment.Results: The proposed methods show significant performance improvements with less training data and can achieve comparable performance to the baseline methods with 5 trials by using 2 or 3 training trials.Using a single trial of calibration data for each frequency, the PRCA-based methods achieved the highest average accuracies of over 95% and 90% with a data length of 1 s and maximum average information transfer rates (ITR) of 198.8±57.3bits/min and 191.2±48.1 bits/min for the two datasets, respectively.Averaged online accuracy of 94.00 ± 7.35% and ITR of 139.73±21.04bits/min were achieved with 0.5-s calibration data per frequency.Significance: Our results demonstrate the effectiveness and robustness of PRCA-based methods for SSVEP identification with reduced calibration effort and suggest its potential for practical applications in SSVEP-BCIs.

I. INTRODUCTION
S TEADY-STATE visually evoked potential (SSVEP) is a brain response elicited by frequency-specific flickering visual stimuli [1].SSVEP-based brain-computer interfaces (BCIs) have the advantages of a high information transfer rate (ITR) and low user training [2], [3], which makes them potentially applicable in a variety of fields, such as rehabilitation, entertainment, and machine control.Several factors, including stimulation presentation, multiple target coding, and target identification algorithms, affect the performance of an SSVEP-based BCI.Among them, target identification algorithm plays an important role.Spatial filtering methods like Canonical Correlation Analysis (CCA) [4], Task-Related Component Analysis (TRCA) [3], and their variations are the most common SSVEP frequency identification methods in existing studies.The core of a spatial filtering method is to find reliable spatial filters, namely linear combinations of multichannel EEG, using knowledge of the visual stimuli or SSVEPs through the construction of artificial sine-cosine reference signals, the collection of individual training data, or both [5].By enhancing the desired SSVEP components and suppressing the non-SSVEP components, spatial filtering-based methods effectively improve the signal-to-noise ratio (SNR) of multichannel SSVEPs and the performance of SSVEP decoding.In the past decade, CCA with SSVEP templates [6], task-related component analysis (TRCA) [3], correlated component analysis [7], sum of squared correlations [8], task-discriminant component analysis (TDCA) [9], and their variants, have achieved state-of-the-art (SOTA) performance by using individual training data and prior knowledge of SSVEP in various ways.
These spatial filtering methods based on individual training data all adopt a common method to achieve SSVEP frequency identification, i.e., frequency identification is achieved by template matching after spatially filtering test data and templates from individual training data and/or artificial sine-cosine reference signals.Although these supervised spatial filtering-based methods have shown excellent performance in SSVEP identification, collecting training data is still an urgent challenge in real-world applications since they need sufficient individual training data to obtain valid spatial filters and individual SSVEP templates.Without sufficient training data for each stimulation frequency, the covariance matrix used to estimate the spatial filters would be unreliable, and the individual SSVEP templates obtained by averaging over multiple trials would have low SNRs [3], [9].As a result, SSVEP identification performance would be significantly degraded.In the extreme case of only one training trial per stimulation frequency, some algorithms cannot estimate valid spatial filters, and individual templates can only use the single-trial data with low SNRs.However, it takes time and effort for users to collect multiple trials of training data in practical applications, especially for SSVEP-BCIs with large numbers of targets.Therefore, it is of great value for the practical application of supervised spatial filtering algorithms to reduce the calibration time while maintaining good SSVEP identification performance.
In recent years, transfer learning-based methods have demonstrated the ability to reduce the calibration effort for SSVEP-BCIs by transferring information across subjects [10], [11], [12], [13], [14], across sessions [15], across stimulation frequencies [16], [17], [18], and across devices [19], [20], [21].However, to achieve good enough recognition performance, these transfer learning-based methods need not only source domain training data but also enough target domain data, i.e., calibration data acquisition is still required for a new user or device.In addition to the amount of data in the source and target domains, the similarity of the target domain to the source domain data also affects the transfer learning performance.Furthermore, to effectively transfer information across domains, the amount of source domain data required is usually relatively large, and the algorithm is often more complicated than the pure spatial filtering algorithms, thus significantly increasing the cost of data acquisition, storage, and processing.
Recent studies have developed extended versions of eTRCA, i.e., the multi-stimulus eTRCA (ms-eTRCA), the similarityconstrained TRCA (scTRCA), and the TRCA with sine-cosine reference signal (TRCA-R), to improve its performance of spatial filtering with insufficient calibration data by learning across multiple neighboring stimulation frequencies [22], by introducing the similarity-constraint covariance matrices between training trials and the artificial sine-cosine templates [23], and by introducing an orthogonal projection matrix to make use of the knowledge of the sine-cosine templates [5], respectively.However, if the calibration data is insufficient, especially if there is only a single-trial calibration data per frequency, their performance is still deteriorated because the reliable individual SSVEP templates required for template-matching cannot be derived from these methods.
Another promising approach to overcome the problem of insufficient training data is data augmentation by generating artificial EEG data using deep learning-based methods [24], [25], [26], [27].However, deep learning-based models are usually not applicable to insufficient data situations because they require a large amount of calibration data for the model to converge.Recent studies have proposed to reconstruct SSVEP templates for stimulation frequencies without training data by extracting and transferring the common periodic response across stimulation frequencies, thus generating templates for untrained frequencies and achieving remarkable SSVEP identification performance [17], [18].These methods can effectively reduce the calibration time when there is sufficient training data to obtain reliable templates for source stimulation frequencies because they can generate templates for target stimulation frequencies and share spatial filters with them.Another recent study by Luo et al. proposed to produce reliable individual SSVEP templates and spatial filters with insufficient calibration data by generating artificial EEG data using the source aliasing matrix estimationbased method [28].The augmented calibration data with the artificial EEG data enabled eTRCA and TDCA to work well with a single calibration trial, achieving an average accuracy above 90% for the Benchmark dataset with a data length of 1s.These studies provide potential methods to reduce calibration time significantly, but these methods rely on other relatively complex algorithms to generate data.How to use insufficient data to achieve remarkable performance without increasing the algorithm's complexity has yet to be explored and is an area of ongoing research.
Inspired by the fact that an SSVEP is a sinusoid-like signal formed by a periodically repeated pattern and that it can be explained by the temporal superposition of transient visually evoked potentials (VEPs) [29], this study is dedicated to solving the problem of insufficient training data for the SOTA spatial filter-based SSVEP identification methods using the periodically repeated pattern of SSVEPs.We propose directly using the periodically repeated components (PRCs) of the SSVEPs to estimate spatial filters and to reconstruct the synthetic individual SSVEP templates.The synthetic SSVEP templates reconstructed using PRCs in this study hold promise for high-SNR SSVEP templates with only a single calibration trial available for each stimulation frequency.Thus, SSVEP identification performance is expected to be significantly improved with a single calibration trial.Therefore, we introduced the synthetic SSVEP templates into the SOTA algorithms, TRCA, and its variants, to verify the performance of the synthetic template in different methods.The characteristics of the synthetic SSVEP templates and possible factors that may affect the performance of the proposed methods will also be discussed.

A. Problem Description
Although the SOTA supervised spatial filtering-based methods, like TRCA and its variants, are efficient in SSVEP identification in many studies, their performance drops sharply when the training data is insufficient.Especially when there is only one calibration trial for each stimulation frequency, these methods usually fail to identify the target frequency of test data because they fail to obtain i) efficient spatial filters since the inter-trial covariance matrix is degraded to a zero matrix and ii) the individual SSVEP template signals with high signal-to-noise ratios (SNR) since the method of multi-trial averaging to improve SNR is not available.Inspired by the fact that an SSVEP is a sinusoid-like signal formed by a periodically repeated pattern and that SSVEP can be explained by temporal superposition Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
of transient VEPs [29], we believe that these methods can be adapted to cope with the single-trial training problem and to improve the performance of these methods with small individual calibration data by introducing the PRCs of SSVEP and the synthetic individual SSVEP templates reconstructed from the PRCs into these methods.
In the following section, we will show how to construct the PRCs and PRC-based synthetic individual SSVEP templates and use them in these methods to construct effective spatial filters and realize SSVEP identification when only single-trial calibration data is available for each stimulation frequency.In summary, the work of this study includes: i) The method to construct the PRCs and to reconstruct the PRC-based synthetic individual SSVEP templates from raw SSVEP data was proposed; ii) TRCA was adapted to Periodically Repeated Component Analysis (PRCA) to address the single-trial training problem by introducing the PRCs and the synthetic SSVEP templates into TRCA; iii) The PRCs and the synthetic SSVEP template were introduced into the variants of TRCA, the similarity-constrained TRCA (scTRCA) and the TRCA with sine-cosine reference signal (TRCA-R), to form improved algorithms, similarity-constrained PRCA (scPRCA) and TRCA-R using the PRC-based synthetic individual SSVEP templates (PRCA-R); iv) SSVEP identification performance was evaluated and compared between different methods.

B. Target Identification Algorithm
This study employed spatial filter-based methods with individual calibration data to identify the target stimuli of given SSVEP data.Individual calibration data and single-trial test data are denoted by a four-way tensor X = (X ) njkh ∈ R N f ×N c ×N s ×N t and a two-way tensor X ∈ R N c ×N s , respectively.Here, n, j, k, and h indicate the index of the stimulation frequency, the channel index, the index of sample points, and the index of training trials, respectively; N f , N c , N s , and N t indicate the number of stimuli, the number of channels, the number of sampling points in each trial, and the number of training trials, respectively.The target identification of SSVEP is to assign a testing trial X to one of the N f classes, According to the study [30], the filter bank analysis with 5 sub-bands was applied to extract the harmonic components of SSVEP response.The lower and upper cut-off frequencies of the m-th (m = 1, 2, . . . . . ., 5) sub-band were set to [m × 8 − 2] Hz and 90 Hz, respectively.The m-th sub-band of the calibration data and the test data are denoted as For the spatial filter-based methods, an ensemble spatial filter is the spatial filter corresponding to the m-th sub-band of the n-th frequency obtained through an algorithm from the calibration data X (m)   n ∈ R N c ×N s ×N t .The correlation-based feature for the m-th sub-band is calculated between single-trial test data X (m) ∈ R N c ×N s and averaged calibration data across trials for n-th frequency X (m) n ∈ R N c ×N s as follows: where, ρ(a, b) indicates the two-dimensional correlation analysis between a and b.A weighted sum of the correlation values corresponding to all sub-bands was calculated as the feature for target identification: where, N m = 5 is the total number of sub-bands, and a(m) was defined as a(m) = m −1.25 + 0.25 according to [2].Then, the target frequency f τ of the test data can be obtained by the following equation: 1) TRCA and PRCA: By maximizing the reproducibility of task-related components (TRCs) across multiple trials, TRCA can efficiently extract SSVEP from noisy multichannel EEG and has been widely used in SSVEP BCI studies due to its performance superiority.Given the individual calibration data Here, S (m) n is the symmetric matric that is obtained as the sum of all possible combinations of the inter-trial covariances and Q (m) n is defined as the summations of the autocovariance across each calibration trial.They can be calculated as follows [3], [31]: where Cov(•, •) indicates covariance operation.To compel a finite solution for (4), the denominator of ( 4) is constrained to 1.In the implementation of the algorithm, we used the method in the reformulated fast TRCA algorithm [32] Here, X (m) n is the summation of zero-centered X (m) n across trials.Then the optimal spatial filter that can extract SSVEP component is the eigenvector with the largest eigenvalue of the matrix, (Q n .Once spatial filters for all the stimulation frequencies and sub-bands are obtained, features for target identification of a given test data X ∈ R N c ×N s can be calculated according to (1) and (2).Finally, the target frequency f τ of the test data is identified according to (3).x(m) Although the TRCA-based method has achieved SOTA performance in SSVEP identification in many studies, its performance drops sharply when the training data is insufficient.In particular, when there is only one training trial for each stimulation frequency, the TRCA-based method fails to identify the target frequency of test data because it fails to obtain valid spatial filters since the inter-trial covariance matrix in ( 5) is degraded to a zero matrix and the individual SSVEP templates with high SNR since the method of multi-trial averaging to improve SNR is not available.We proposed to address this problem by maximizing the PRCs of SSVEPs and applying the synthetic individual SSVEP templates in template matching.
As shown in Fig. 1, given calibration data for the m-th sub-band of n-th stimulation frequency f n , there will be P n = floor(N s f n /F s ) periods in each trial.Here, F s is the sampling frequency, floor(a) rounds a to the nearest integer less than or equal to a.The number of sampling points in each period is L n = round(F s /f n ), here round(a) rounds a to the nearest integer.The period lengths L n were calculated using f n for all sub-bands of the n-th stimulation frequency since the period lengths of the harmonics would be too short to be easily affected by noise when calculating covariance matrixes.Through segmenting X (m) n according to the length of a period L n by window [1 + (p − 1) * L n , p * L n ], p = 1, . . ., P n is the index of the window, we can obtain a new three-dimension tensor Then, the spatial filter ω(m) n ∈ R N c that can maximize the reproducibility of PRC across multiple periods is calculated as: Here,

S(m)
n is the summation of all possible combinations of the inter-period covariances, Q(m) n is the summation of the autocovariance across each period.They can be obtained as follows: Here, p 1 and p 2 are the indexes of periods in X (m) n .

S(m)
n can also be calculated using a fast implementation like S (m) n in fast TRCA [32].Once the spatial filters for all sub-bands and all stimulation frequencies were obtained using the above method, an ensemble spatial filter for the m-th sub-band is obtained as follows: The template of a period, namely the PRC, n,p .To apply the template matching method based on individual SSVEP template like the TRCA-based method, a synthetic SSVEP template of desired length X (m) n can be reconstructed by duplicating and concatenating the PRC since an SSVEP essentially consists of PRCs when disregarding possible transient responses. Here, n from the first sample point to the l-th sample point.For a coming test data 5) of N s length are reconstructed.The correlation-based feature for the m-th sub-band is calculated between single-trial test data X (m) ∈ R N c ×N s and the synthetic SSVEP template for n-th frequency The target frequency f τ of the test data can be identified according to (2) and (3).Unlike TRCA, PRCA holds promise for efficient spatial filters and high-SNR SSVEP templates with only a single calibration trial available for each stimulation frequency.
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2) scTRCA and scPRCA:
The scTRCA introduces similarity constraints based on artificial sine-cosine templates to find better spatial filters to maximize the inter-trial covariance and the covariance between training trials and the artificial sine-cosine signal [23].In scTRCA, a block covariance matrix for the m-th sub-band of n-th stimulation frequency is defined as: where S 11 is the inter-trial covariance, namely, A block covariance matrix Q(m) n is defined as: where, Q 1 and Q 2 denote the autocovariance matrixes of X (m)   n and Y n , respectively.Like (4), a spatial filter ω maximize the similarity between training data and the artificial sine-cosine template.After spatial filters corresponding to all stimulation frequencies are obtained, the ensemble spatial filters can be constructed as Then, the correlation-based feature between single-trial test data X (m) ∈ R N c ×N s and averaged calibration data across trials for n-th frequency X (m) n ∈ R N c ×N s for the m-th sub-band can be calculated as: Same as TRCA, the inter-trial covariance matrix S 11 in (14) will degrade to zero matrix when there is only one calibration trial available.So, we proposed scPRCA by replacing the training data (14) and (15).The sine-cosine template in ( 15) and ( 16) remains only one period, In (20), we replace the multi-trial average template X (m) n with the PRC-based synthetic template X (m) n in correlation-based feature for scPRCA.Thus, scPRCA can work with single-trial calibration data by maximizing the inter-period reproducibility and the similarity between PRC and the sine-cosine template at single-period level.
3) TRCA-R and PRCA-R: Wong et al. proposed TRCA-R by introducing an orthogonal projection matrix P n into the TRCA algorithm to make use of the knowledge of sine-cosine reference signal Y n in solving spatial filters for the n-th stimulation frequency [5].Here, P n is formed by: Q n derives from the QR decomposition of the sine-cosine reference signal of the n-th stimulation frequency Y n .More details about TRCA-R can be found in [5].
It should be noted that in the implementation of TRCA-R, the relationship between trials is not quantified by directly calculating the covariance between trials like that in (5), so if there are only single-trial training data, the spatial filter solution will not be invalid.So, we proposed to replace the multi-trial average template X (m) n with the PRC-based synthetic template X (m) n in correlation-based feature calculation without changing the calculation method of the spatial filters in the method using TRCA-R with PRC-based synthetic templates (PRCA-R).

1) Datasets:
The performance of the proposed methods was evaluated on two datasets, i.e., a self-collected 16-frequency SSVEP dataset (Dataset I) and the public 40-frequency Benchmark dataset [33] (Dataset II).Sixty-four channels of EEG data were recorded from healthy subjects with normal or correctedto-normal vision by SynAmps2 (Neuroscan Inc.) at a sampling rate of 1000 Hz in laboratory settings in both datasets.Each target was encoded with a single-frequency sine wave using a joint frequency and phase modulation (JFPM) method [2].
For Dataset I, 29 subjects (15 males, aged 18-26 years) participated in six blocks of a cued-spelling task on a 4 × 4 matrix of a virtual keyboard on a LED monitor with a refresh rate of 240 Hz in a room without electromagnetic shielding.Each target was encoded by one of the 16 stimulation frequencies ranging from 8 Hz to 15.5 Hz with an interval of 0.5 Hz, and phase ranges were 0 π to 1.5 π with an interval of 0.5 π.For each subject, the experiment included six blocks, each containing 16 trials corresponding to all 16 target frequencies.Each trial started with a 0.5-s target cue (a red cross at the center of the target location).Subjects were asked to shift their gaze to the target as soon as possible.Then, all targets started flickering on the screen concurrently for 2 s.After the stimulation ended, the screen was blank for 1.5 s until the next cue began.Between two consecutive blocks, subjects were allowed to rest for several minutes.All participants read and signed an informed consent form before the experiment.The experiment protocol was approved by the Ethics Committee of Tianjin University (under Application No. TJUE-2022-189, 2022-03-08).
Dataset II consists of SSVEP data with 40 targets from 35 participants.Each subject completed six blocks of a cuedspelling task on a 5 × 8 matrix of a virtual keyboard in a room with electromagnetic shielding.Each block consists of 40 trials corresponding to all 40 characters.Each target was encoded by one of the 40 stimulation frequencies ranging from 8 Hz to 15.8 Hz with an interval of 0.2 Hz, and phase ranges were 0 π to 1.5 π with an interval of 0.5 π.During each trial, subjects were asked to gaze at one of the 40 flickering targets for 5 s following a 0.5-s target cue.All data in Dataset II was sampled at 1000 Hz during recording but downsampled to 250 Hz before release.More detailed information on Dataset II can be found in [33].
2) EEG Preprocessing: Eleven EEG channels (Pz, POz, PO3/4, PO5/6, PO7/8, Oz, O1/2) were used in the performance evaluation for both datasets.A primary study showed that the sampling frequency had a significant effect on the performance of the ePRCA-based method, with better performance at higher sampling frequencies (see Fig. S1).Considering the impact of sampling frequency on computational cost and performance, we decided to use a sampling rate of 1000 Hz for both datasets.For this reason, dataset II was upsampled to 1000 Hz before further processing.To evaluate performance at different data lengths, the data were segmented into epochs [0.14 s (0.14+T w ) s], where the time 0 s indicated stimulation onset, and T w was the data length used in performance evaluation (T w = 0.2 s, 0.3 s, . . . . .., 1.5 s).Then, filter bank analysis was performed to decompose EEG segments into sub-band components by performing zero-phase forward and reverse filtering using 6-order Chebyshev type I filters.The lower and upper cut-off frequencies of the m − th sub-band were set to [m × 8 − 2] Hz and 90 Hz, respectively.The data after preprocessing were used in target identification and performance evaluation.
3) Performance evaluation: Spatial filters corresponding to multi-frequency were integrated for target identification in the methods based on TRCA and its variants according to the ensemble method in Section II-B.Namely, the performance of methods based on ensemble TRCA (eTRCA), ensemble PRCA (ePRCA), ensemble scTRCA (escTRCA)), ensemble scPRCA (escPRCA), ensemble TRCA-R (eTRCA-R), and ensemble PRCA-R (ePRCA-R) were evaluated in this study.SSVEP identification accuracy and ITR were used for performance evaluation.ITR (in bits/min) is defined as follows [34]: where, N f is the number of target frequencies, P is the accuracy of target identification, and T is the time required for a target selection in a BCI system, including the gaze shifting time (0.5 s in this study) and the data length T w used for target identification.
To examine the performance of the proposed method, target identification accuracy was first compared between eTRCA and ePRCA at different numbers of training trials (N train = 1, 2, . . . . .., 5) and different EEG channel montages.Here, four subsets of montage configuration using data lengths of 0.5 s and 1 s were evaluated, including 3-channel montage (N ch = 3: Oz, O1/2), 6-channel montage (N ch = 6: POz, PO3/4, Oz, O1/2), 8-channel montage (N ch = 8: POz, PO3/4, PO5/6, Oz, O1/2), and 11-channel montage (N ch = 11: Pz, POz, PO3/4, PO5/6, PO7/8, Oz, O1/2).Then, we mainly focus on the performance of single-trial calibration.Target identification accuracies using ePRCA were examined for the effect of training data length (T train = 0.5 s, 1 s, 1.5 s, and 2 s) under single-trial calibration.We compared target identification accuracies and ITRs with single-trial training data between the SOTA methods and their improved versions using test data lengths (T test ) from 0.2 s to 1.5 s with an interval of 0.1 s.Repeated-measure analysis of variance (RMANOVA) and paired t-tests were performed for comparisons.The Geisser-Greenhouse correction was employed if sphericity was not satisfied.The p-values of multiple comparisons were corrected by controlling the False Discovery Rate (FDR).

D. Online Performance Evaluation
An online experiment was conducted to examine the performance of a 16-target SSVEP-BCI using ePRCA with single-trial calibration data per target.The EEG recording system, the layout of the visual stimuli and the stimulation frequencies and phases were identical to those in Dataset I collection.11 posterior channels (Pz, POz, PO3/4, PO5/6, PO7/8, Oz, O1/2) were employed in the online evaluation experiment.A training stage of 16 trials was conducted to collect one trial of calibration data for each target.To examine the performance of BCI performance with short training time, each training trial consisted of a 0.5-s blank, a 0.5-s visual cue, and a 0.5-s flickering.A single-trial of 0.5-s data per target collected in the training stage were used as individual training data to produce the spatial filters and the synthetic SSVEP templates using ePRCA-based method for the subsequent online testing stage.The testing stage consisted of 5 blocks of cue-guided BCI experiments, each including 16 trials corresponding to the 16 targets in random order.Each trial lasted 2 s, including 0.5-s blank, a 0.5-s gaze shifting, and a 1-s visual flickering.The accuracy of target identification and ITR were used for performance evaluation.15 subjects (9 males, aged 20-25 years) participated in this experiment with informed consents and payments according to the experiment protocol Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

A. Characteristics of PRCs and Synthetic SSVEP Templates
Fig. 2(a) shows the grand-average PRCs obtained from 1-s single-trial calibration data for channel Oz's first sub-band of all stimulation frequencies.The curves of PRCs demonstrate that averaging across periods can yield the signal pattern of PRCs of SSVEP.The PRCs at different stimulation frequencies showed different temporal patterns.Consistent with previous studies [29], the higher the stimulation frequency, the closer the PRC waveform is to the single-period waveform of the sine wave.Fig. 2(b) compares the grand-average real SSVEP templates generated by 5-trial averaging and the synthetic SSVEP templates generated by duplicating and concatenating the PRCs from single-trial calibration data.Overall, the synthetic templates show a high similarity to the real templates.Thus, the feasibility of constructing synthetic templates based on single-trial data is preliminarily demonstrated.However, some phenomena deserve further discussion.For example, the similarity between real and synthetic templates has visible differences between different frequencies, and the higher the frequency, the higher the similarity seems to be.Also, the waveform similarity is stronger for later time windows (after 0.3 s) than for earlier ones (before 0.3 s).Fig. 2(c) and (d) shows the average SNRs for 15.5 Hz synthetic and real SSVEP templates and testing data after spatially filtered using ePRCA and eTRCA-based filters calibrated with 2 trials 1-s SSVEP data, respectively.This study defines the SNR as the ratio of the amplitude at a given frequency f to the mean amplitude of signals within 14 Hz neighboring frequency band [f − 7, f + 7] Hz.The SNR spectrums show that the ePRCA-based method achieved hinger average SNRs at 15.5 Hz and its harmonics for both SSVEP templates and test data.Fig. 2(e) and (f) show the SNRs averaged across all stimulation frequencies and their harmonic frequencies for SSVEP templates and single-trial testing data respectively.Compared with eTRCA-based method, paired t-tests revealed that ePRCA achieved significantly higher SNRs for both SSVEP templates [t(28) = 20.90, p<10e-16] and testing data [t(28) = 5.96, p<0.00001] after spatially filtered.These results suggest that the proposed ePRCA-based method can not only improve the SNRs of SSVEP templates, but also can improve the SNRs of test data by providing more efficient spatial filters.

B. Effects of Number of Training Trials and EEG Channel
Montages: eTRCA vs. ePRCA Fig. 3(a) and (b) compare SSVEP identification accuracy with data lengths (i.e., T train and T test ) of 0.5 s and 1 s between eTRCA and ePRCA using different numbers of training trials (i.e., N train ) and numbers of EEG channels (i.e., N ch ) for Datasets I and II, respectively.Average accuracies with ePRCA outperformed eTRCA regardless of data length, the number of training trials, and the number of channels.Paired t-tests indicate that ePRCA's accuracy was significantly better than eTRCA's in most cases, as shown in Fig. 3(c) and (d).Especially when these parameters are small, i.e., the amount of training data is small, the advantages of ePRCA are particularly obvious.When using single-trial training data, the accuracy of Dataset I with ePRCA is 38.36% to 64.16% higher than that of eTRCA, and the highest average accuracies using ePRCA with data lengths of 0.5 s and 1 s reach 84.81% and 95.26%, respectively.For dataset II, the accuracy of ePRCA under single-trial training is higher than that of eTRCA by 19.57% to 75.37%, and the highest accuracies with data lengths of 0.5 s and 1 s when using ePRCA reach 57.21% and 88.60%, respectively.The above results suggest that i) ePRCA outperformed eTRCA in SSVEP identification, ii) ePRCA can achieve comparable performance to eTRCA with much less training data, and more importantly, iii) PRCA-based method can identify SSVEPs in single-trial training situations efficiently.

C. Effects of Training Data Length
The results in the last section show that the ePRCA-based method achieved impressive performance with single-trial calibration data per frequency.In this section, we focus on the effect of training data length (T train ) on single-trial training   For Dataset I, all the proposed methods achieved average accuracies of above 90% with a data length of 0.6 s or 0.7 s and above 95% with a data length of 1 s.For Dataset II, average accuracies of 88.6%, 91.3%, and 88.7% were achieved by ePRCA, escPRCA, and ePRCA-R, respectively, with a data length of 1 s.Using escPRCA with a data length of 1 s, 27 out of 35 subjects in Dataset II achieved accuracies above 90%.The highest average ITRs and the corresponding test data lengths with single-trial training data obtained by all the proposed methods, as well as the eCCA-based method [2] for comparison since eCCA can handle the singletrial calibration problem, are shown in Table I

E. Online BCI Performance With Single-trial Calibration
Table II shows the results of the online experiments.The averaged accuracy across all subjects was 94.00 ± 7.35%, leading to an averaged ITR of 139.73 ± 21.04 bits/min.With a calibration data of 0.5 s for each target frequency, 11 out of the 15 subjects achieved accuracy above 90% with testing data of 1 s.The minimal and maximal ITR was 83.08 bits/min (S3) and 160 bits/min (S6 and S12) respectively.

IV. DISCUSSION
Spatial filtering and template matching-based algorithms have performed excellently in extracting SSVEP features and improving SSVEP-BCI decoding accuracies.However, improving the performance of SSVEP identification methods with a small sample size of training data is a critical issue for the practical application of SSVEP-BCI since it is still a challenge to train an efficient supervised individual model using the SOTA spatial filtering algorithms with insufficient training data.To overcome this challenge, we proposed to improve the performance of spatial filtering algorithms and individual SSVEP templates under small training data size by exploiting periodically repeated pattern of SSVEP data.

A. PRC-Based Methods Enhance SSVEP Identification With Small-Sample-Size Training Data
To the best of our knowledge, this is the first study to employ the periodically repeated pattern of SSVEP to derive the spatial filter and reconstruct the synthetic individual SSVEP template for SSVEP identification.Fig. S2 illustrates the average accuracy against the number of training trials per stimulation frequency for both datasets using different methods with 1-s training and test data.Statistical analyses show that the performance of PRCA-based methods using 2 or 3 trials per stimulation frequency is comparable to that of TRCA-based methods using 5 trials per frequency.Thus, the proposed method can greatly Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
shorten the calibration time by 40-60% and facilitate real-world applications of SSVEP-based BCIs.More importantly, with only one-trial calibration data per target, our proposed methods greatly outperformed the SOTA algorithms, achieving average accuracies of over 95% and 90% on 1-s test data for a 16-target dataset and the Benchmark dataset, respectively.In the online cue-guided experiment, the ePRCA-based method achieved an averaged accuracy of 94.00 ± 7.35% using 0.5-s single-trial calibration data for each target.These findings demonstrate the potential of the proposed method to shorten calibration for high-performance SSVEP-BCIs in practical applications.
Compared to studies that have attempted to shorten calibration time by improving spatial filters with insufficient data [5], [22], [23], the method proposed in the current study can generate more reliable individual SSVEP templates required for templatematching, thus further improving performance, especially when there is only one calibration trial per frequency.In comparison to the stimulus-stimulus template transfer methods, which generate SSVEP templates for target frequencies by transferring common components across stimuli [17], [18], the proposed PRC-based method not only generates reliable individual SSVEP templates for all frequencies but can also improve the reliability of spatial filters, thus achieving better performance with 40 trials of entire calibration data.[17] achieved a maximum average ITR of 123.7 bits/min with five trials per target for eight source frequencies, while the current study achieved a maximum average ITR of over 176.9 bits/min with one trial per target for all 40 frequencies on the Benchmark dataset.The performance of the method proposed in the present study is comparable to the data augmentation method in [28] with single-trial calibration data per frequency, but in contrast to the method in the study [28], which requires the use of the source aliasing matrix estimation algorithm to generate artificial data, the present study essentially proposes a novel method to make use of the limited SSVEP data without using other algorithms and thus without increasing the computational complexity of the algorithm (see section S-II in Supplementary Materials).It is also worth noting that the periodically repeated pattern and individual template reconstruction method in this study have potential applications in other spatial filtering-and template-matching-based SSVEP identification methods, like TDCA and eCCA, and signal denoising methods for multivariate data with periodic characteristics.

B. Potential Issues Affecting the Performance of PRC-Based Methods
Although the method proposed in this study has improved SSVEP detection performance using small sample size training data without significantly increasing computational complexity, several issues are worth discussing to provide a possible basis for further improving the method in future studies.Here, we mainly focus on the characteristics of synthetic SSVEP templates and their association with SSVEP identification performance.
First, because the number of periods obtained from the same length of training data is different for different stimulation frequencies, the reliability of the spatial filter and the SSVEP template may be different, resulting in different identification accuracies for different frequencies.To verify this issue, we presented the average SSVEP identification accuracies against stimulation frequency for both datasets using ePRCA with singletrial calibration data in Fig. S3.Repeated measure ANOVAs revealed that the effects of stimulation frequency were significant (ps<0.05) in all cases except the Dataset I with T train = T test = 1 s.The phenomenon of lower recognition accuracy for lower frequencies was revealed when the test data length was 0.5 s for Dataset I.A similar phenomenon can be found for Dataset II when T train = T test = 0.5 s, but not as explicitly as Dataset I.These findings suggest that stimulation frequency may affect performance, especially when the length of training data is short, possibly because enough periods cannot be extracted from short training data for lower frequencies to generate reliable spatial filters and synthetic SSVEP templates.This needs to be further investigated in future research to improve the performance of PRC-based methods.
Second, the inverted U-shaped relationship between the accuracy and the training data length suggests that it is not the longer the better for the length of a single calibration trial, even though a sufficient length of training data is required to obtain enough periods.This may be because the sampling rate of the EEG signal, 1000 Hz, cannot be an integer multiple of all the stimulation frequencies, resulting in temporal errors in extracting the data segment according to the period lengths.With long single trial data lengths, errors accumulate in the late periods, resulting in less reliable single-period signals and, thus, less reliable spatial filters and templates.A higher sampling rate can alleviate this problem, but the performance benefits of the higher sampling rate must be weighed against the cost of data acquisition and storage.
Further discussion is warranted regarding the characteristics of synthetic SSVEP templates and their relation to SSVEP identification performance.The real template, which is the result of the average of multiple trials, can be used as a reference for understanding the characteristics of the synthetic template.The relationship between the synthetic and real templates can help us to understand the temporal dynamics of the synthetic template.We have seen higher similarity between real and synthetic templates for higher frequencies and later time windows in Dataset I in Section III-A.Here, we further analyze the similarity by calculating the correlation coefficient between 1-s real and synthetic SSVEP templates for each stimulation frequency and time interval, as shown in Fig. S4.The correlation coefficients were calculated between real SSVEP templates, which were derived by averaging over five-trial 1-s calibration data, and synthetic SSVEP templates, which were derived by reconstructing single-period templates from single-trial 1-s calibration data.The tendency of a higher correlation coefficient corresponding to a higher stimulation frequency is obvious in dataset I but not in Dataset II, while the relationship between correlation and time interval shows an inverted U-shaped relationship in both datasets.The inverted U-shape of the correlation here can be explained from the following two points of view: i) in the early time windows, the weak correlation may be due to the fact that the real SSVEP templates contain the transient event-related responses that appeared only in the early phase, whereas the synthetic template contains only the periodically repeated patterns, i.e., steady-state responses; and ii) in the late time windows, the correlation may be affected by the temporal precision of the synthetic templates, that is, the temporal errors of the period lengths will accumulate as the number of periods increases during reconstruction of the synthetic templates by duplicating and concatenating the PRCs.
Through the above analysis, we have a better understanding of the characteristics of synthetic templates, but more importantly, how these characteristics affect the SSVEP identification performance of PRC-based methods.To investigate this question, we examined the association between the accuracy of SSVEP identification with single-trial calibration data and the correlation coefficient between the real templates and the synthetic templates.The results of correlation analyses under different test data lengths for Dataset I and II, indicate strong positive associations between SSVEP identification accuracies and the similarity between the synthetic templates and the real templates, especially when the test data length is less than 0.5 s [Dataset I: Pearson's rs = 0.632, 0.644, 0.547, and 0.427 for T test = 0.2 s, 0.3 s, 0.5 s, and 1 s, respectively; Dataset II: Pearson's rs = 0.702, 0.732, 0.703, and 0.575 for T test = 0.2 s, 0.3 s, 0.5 s, and 1 s, respectively].It should be noted that the real template obtained by averaging over multiple trials can preserve the transient response, which occurs mainly in the early phase of a single-trial SSVEP.In contrast, the spatial filtering and synthetic template generation method proposed in this study preserves only the periodically repeated components, i.e., the steady-state response.Taking together with the results of low correlation between real and synthetic templates at lower stimulation frequencies in Fig. S4, the strong correlation between SSVEP recognition accuracy and the similarity of real and synthetic templates may imply that the performance of PRC-based methods using short test data may have been affected by transient VEPs that are present in single-trial test data but not in synthetic templates.

C. Future Work
Although the proposed method significantly improved SSVEP recognition performance with less training data, further improvements may be made in the future.First, the method proposed in this study requires at least one trial of training data for each stimulation frequency.In the future, it is possible to consider constructing a common periodic pattern for different frequencies based on single-trial data, and constructing templates for stimulation frequencies without training data based on the common periodic pattern, thereby further reducing the required training data.Second, extracting transient responses that are separable across stimulation frequencies from single trial data may be a potential way to improve recognition performance with short data lengths.Furthermore, addressing the impact of temporal precision in extracting periodically repeated components may also further improve the reliability of spatial filters and templates.

V. CONCLUSION
This study proposes an efficient spatial filtering-and template matching-based SSVEP identification method named PRCA to obtain more reliable classification models with less calibration data by maximizing the PRCs in SSVEP and constructing synthetic SSVEP templates.In comparison with the existing TRCAbased methods that maximize the reproducibility of SSVEP across trials and underperform with insufficient calibration data, PRCA-based methods maximize the reproducibility of PRCs across periods in SSVEP and can improve the SNRs of SSVEP templates, provide more efficient spatial filters, and thus enhance SSVEP identification performance with insufficient calibration data.The results on two datasets show that PRCA-based methods perform significantly better with less calibration data, especially when single-trial calibration data is available for each stimulation frequency.In general, these findings suggest that the proposed PRCA-based methods have the advantages of high performance, less calibration, and ease of implementation, making PRCA a suitable algorithm for SSVEP-based BCI applications.

Fig. 1 .
Fig. 1.Diagram illustrates PRCA and the method to reconstruct synthetic SSVEP template from a single-trial calibration data in SSVEP analysis.x(m)

( 14 )
S 12 and S 21 represent the covariance between training trials X (m) n ∈ R N c ×N s ×N t and the sine-cosine template Y n for the n-th stimulation frequency.N h is the number of harmonic frequencies used in Y n .S 22 is the autocovariance of Y n .
be obtained by calculating the eigenvector of the matrix ( Q(m) n ) −1 Ŝ(m) n corresponding to the largest eigenvalue.Here, u (m) n maximize the inter-trial reproducibility and v (m) n

Fig. 2 .
Fig. 2. Characteristics of PRCs and synthetic SSVEP templates of Dataset I. (a) The grand-average PRCs obtained from 1-s single-trial calibration data for the first sub-band of all stimulation frequencies at channel Oz.(b) Average real SSVEP templates were obtained by 5-trial averaging, and the synthetic SSVEP templates were obtained by duplicating and concatenating the PRCs for the four typical frequencies at channel Oz.(c) Average SNRs of 15.5 Hz synthetic SSVEP template and real SSVEP template after spatially filtered using ePRCA and eTRCA-based filters.(d) Average SNRs of 15.5 Hz single-trial test data after spatially filtered using ePRCA and eTRCA-based filters.(e) and (f) show the SNRs averaged across stimulation frequencies and their harmonic frequencies for SSVEP templates and single-trial testing data respectively.

Fig. 3 .
Fig. 3. Comparison of SSVEP identification accuracy between eTRCA and ePRCA using different numbers of training trials (i.e., N train ) and numbers of EEG channels (i.e., N ch ).(a) and (b) show the group-averaged SSVEP identification accuracies against the number of training trials and channels using different data lengths for Datasets I and II, respectively.(c) and (d) illustrate the t-values derived from paired t-tests between ePRCA and eTRCA.The cells in white demonstrate insignificant differences (p>0.05), and the cells in pink demonstrate significant differences (p<0.05) in (c) and (d).

Fig. 4 (
a) and (b) compare the accuracies between eTRCA with 5-trial training data (N train = 5) and ePRCA with different numbers of training trials (N train = 1, 2, 3, 4, 5) for Datasets I (a) and II (b) with 11 channels and data lengths of 0.5 s and 1 s, respectively.The results show that the accuracies of ePRCA for Dataset I with N train ≥2 and Dataset II with N train ≥3 are comparable to those of eTRCA with N train = 5, regardless of data length.These results demonstrate that ePRCA can achieve comparable performance to eTRCA with much less training data than eTRCA.

Fig. 5 .
Fig. 5. Accuracies against training data length (T train ) using ePRCA with single-trial calibration data for Dataset I (a) and II (b).The asterisks in the subfigures indicate significant differences between different training data lengths obtained by paired t-tests after correction for multiple comparisons using the FDR approach ( * * p<0.01, * * * p<0.001, * * * * p<0.0001).

Fig. 6 .
Fig. 6.Accuracies and ITRs against testing data length (T test ) using different methods with single-trial training data.(a), (c), and (e) are results for Dataset I. (b), (d), and (f) are results for Dataset II.The asterisks in the subfigures indicate significant differences between different training data lengths (T train = T test vs.T train = 1 s) for the proposed methods by paired t-tests after correction for multiple comparisons using the FDR approach ( * p<0.01, * * p<0.01, * * * p<0.001, * * * * p<0.0001).