Analysis of Postural Control Using Principal Component Analysis: The Relevance of Postural Accelerations and of Their Frequency Dependency for Selecting the Number of Movement Components

One criterion when selecting the number of principal components (PCs) to be considered in a principal component analysis (PCA) is the fraction of overall variance that each PC represents. When applying a PCA to kinematic marker data in postural control research, this criterion relates to the amplitude of postural changes, recently often called “principal (postural) positions” (PPs). However, in the assessment of postural control, important aspects are also how fast posture changes and the acceleration of postural changes, i.e., “principal accelerations” (PAs). The current study compared how much of the total position variance each PP explained (PP_rVAR) and how much of the total acceleration variance each PA explained (PA_rVAR). Furthermore, the frequency content of PP and PA signals were evaluated. Postural movements of 26 participants standing on stable ground or balancing on a multiaxial balance board were analyzed by applying a PCA on 90 marker coordinates. For each PC, PP_rVAR, PA_rVAR, and the Fourier transformations of the PP and PA time series were calculated. The PP_rVAR and the PA_rVAR-distributions differed substantially. The PP-frequency domain was observed well below 5 Hz, the PA-frequency domain up to 5 Hz for stable standing and up to 10 Hz on the balance board. These results confirm that small-amplitude but fast movement components can have a higher impact on postural accelerations—and thus on the forces active in the system—than large-amplitude but slow lower-order movement components. Thus, PA variance and its dependence on filter frequencies should be considered in dimensionality reduction decisions.

One criterion when selecting the number of principal components (PCs) to be considered in a principal component analysis (PCA) is the fraction of overall variance that each PC represents. When applying a PCA to kinematic marker data in postural control research, this criterion relates to the amplitude of postural changes, recently often called "principal (postural) positions" (PPs). However, in the assessment of postural control, important aspects are also how fast posture changes and the acceleration of postural changes, i.e., "principal accelerations" (PAs). The current study compared how much of the total position variance each PP explained (PP_rVAR) and how much of the total acceleration variance each PA explained (PA_rVAR). Furthermore, the frequency content of PP and PA signals were evaluated. Postural movements of 26 participants standing on stable ground or balancing on a multiaxial balance board were analyzed by applying a PCA on 90 marker coordinates. For each PC, PP_rVAR, PA_rVAR, and the Fourier transformations of the PP and PA time series were calculated. The PP_rVAR and the PA_rVAR-distributions differed substantially. The PP-frequency domain was observed well below 5 Hz, the PA-frequency domain up to 5 Hz for stable standing and up to 10 Hz on the balance board. These results confirm that small-amplitude but fast movement components can have a higher impact on postural accelerations-and thus on the forces active in the system-than large-amplitude but slow lower-order movement components. Thus, PA variance and its dependence on filter frequencies should be considered in dimensionality reduction decisions.

INTRODUCTION
Principal component analysis (PCA) is an unsupervised data analysis procedure often used as a preprocessing step, e.g., to improve performance or for dimensionality reduction, before more complex machine learning procedures are applied. If applied in the analysis of human motion, a PCA can by itself reveal interesting information about the coordinative structure of complex whole-body movements. Accordingly, applying a PCA on kinematic data has received increasing attention in research on several kinds of human movements, such as reaching , karate kicking (Zago et al., 2017a), juggling (Zago et al., 2017b), skiing (Federolf et al., 2014;Gløersen et al., 2018;Pellegrini et al., 2018), or walking (Troje, 2002;Daffertshofer et al., 2004;Verrel et al., 2009;Zago et al., 2017c).
One of the main purposes for performing a PCA on kinematic data-or in fact on any dataset-is the idea that the entire variance in the data can often be approximated to high accuracy with only a limited number of principal components (PCs). One of the most common criteria for choosing the number of PCs is the eigenvalue spectrum, which represents the variance explained by each PC and which can be expressed in relative values, i.e., as a percentage of the entire variance in the data.
A research area where PCA has been particularly frequently applied on kinematic human movement data is research on postural control Federolf, 2016;Promsri et al., 2018Promsri et al., , 2019Promsri et al., , 2020aWachholz et al., 2019a,b). In postural control studies, when PCA is applied to kinematic data, it decomposes the complex multi-segment whole-body movements into a set of one-dimensional movement components, called "principal movements" PM k , where k is the order of the movement component Federolf, 2016). Previous research has shown that the lower-order PM k represent in close approximation the classical motor strategies (Horak and Nashner, 1986;Winter, 1995), i.e., the ankle or hip strategies (Federolf, 2013). If PCA is calculated on normalized data from different volunteers, then a subject-specific relative explained variance can be calculated in analogy to the eigenvalues, which quantify the explained variance for the whole dataset and are thus not subject-specific. The relative explained variance-spectra provide one criterion for how many movement components PM k one wants to consider in the analysis (Federolf, 2013;. However, analyzing the different postures observed during a measurement sequence may not be the only variable of interest. How fast the posture changes and how much a postural change is accelerated, also provide valuable information. We have shown in previous papers, that Newton's mechanics can be applied to the PCA-based posture space by defining a "principal (postural) position" (PP k ) for each PM and their time derivatives, principal velocity (PV k ) and principal acceleration (PA k ) (Federolf, 2016;Longo et al., 2019). The PA k are of interest, since they relate to forces acting in the system and thus to the neuromuscular control of the postural movements (Federolf, 2016;Promsri et al., 2018Promsri et al., , 2019Promsri et al., , 2020aWachholz et al., 2019aWachholz et al., ,b, 2020. We want to emphasize here that the PA k obtained by double-differentiation of the PP k time series (Federolf, 2016;Longo et al., 2019) are different variables than when a PCA is performed directly on acceleration data (Verheul et al., 2019): The former can be seen as an expansion of the movement strategy concept (Horak and Nashner, 1986;Winter, 1995), since the PA k quantify the acceleration of the considered movement components/movement strategies; the latter PCA identifies correlated patterns directly in acceleration data, which yields a different solution.
Differentiation is a non-linear operation and, consequently, the relative variance spectra of the PP k differ from the PA k relative variance spectra . Particularly in postural control it is likely that large-amplitude, yet slow movement components influence the PA-spectrum less than small-amplitude, but fast movement components. The PAexplained variance spectrum could be a second important criterion for the decision on how many PM k should be considered in an analysis . Unfortunately, noise amplification in differentiation makes a filtering of the PP k signals necessary before PV k and PA k are calculated (Winter et al., 1974), and since the PA k variance spectra are speeddependent, they will change with the filter cut-off frequency used before the differentiation.
In summary, when applying a PCA to investigate the coordinative structure of postural control movements, both the principal positions (PP k ) and the principal accelerations (PA k ) are of interest since they provide relevant information on the composition of the postural movements and on the control of the movement components, respectively (Promsri et al., 2020a). Both the PP k -and PA k -spectra should be considered when selecting the number of movement components to be analyzed, however, the PA k -spectra are speed-and thus filter frequency-dependent. Thus, the purposes of the current Brief Research Report were (i) to compare the PP k and PA k relative variance spectra for postural control data; (ii) to evaluate the frequency content of the PP k and PA k time series; and (iii) to assess how the PA relative variance spectrum depends on the filtering cut-off frequency.

Participants
Twenty-six physically active young adults (14/12 males/females, age 25.3 ± 4.2 years, weight 70.7 ± 11.4 kg, height 175.0 ± 8.1 cm, physical activity participation 8.4 ± 5.4 h/weeks [mean ± SD]) with no neuromuscular injuries/disorders and no specific balance training participated in the current study. All volunteers provided informed consent and the study protocol had been approved by the Board of Ethical Questions in Science of the University of Innsbruck, Austria (Certificate 16/2016).

Measurement Procedures
Participants were equipped with 39 reflective-markers according to the "Plug-In Gait" marker setup (Vicon Motion Systems Ltd., Oxford, UK). Two 80-s barefooted-bipedal balancing trials, one for each support surface, were completed in randomized order on a firm surface (FS) and on a wobble board (WB; Powrx Balance Board; POWRX GmbH., Germany). After completing the first trial, participants could rest for up to 3 min. For the WB condition, volunteers had a 15-s familiarization trial with no instruction or feedback. Postural movement trajectories were captured by a standard 8-camera motion tracking system (Vicon Bonita B10 cameras with Nexus 2.2.3 software; Vicon Motion Systems Ltd., Oxford, UK) using a sampling rate of 250 Hz.
To standardize the standing position (Supplementary Figure 1), participants were asked to place two marked points (base of each 2nd metatarsal bone) over a horizontal line taped on the floor for FS or over a horizontal diameter of the WB; to align the inside of the feet (the medial borders of each distal end of the first metatarsal bone) with tapes defining an individual inter-feet distance (15% of biacromial diameter); to rest their hands on the hips; and to look straight ahead at a 10-cm-diameter red-circle target on a wall at the individual eye level ∼5 m away. To standardize the position of the wobble board, we placed the center of the wobble board over the center of a reticle cross-line marked on the floor. During testing, volunteers were asked to stand still for the FS or to keep the board horizontal for the WB; to avoid any voluntary movements; and to keep their eyes on the target.

Data Analysis
Kinematic Data Pre-processing All data processing was conducted in Matlab (MathWorks Inc., Natrick, MA, USA). The pre-processing steps and the PCA analysis were conducted based on earlier studies (Federolf, 2016;Promsri et al., 2018Promsri et al., , 2019Promsri et al., , 2020a. Briefly, any gaps in marker trajectories were filled by a PCA-based reconstruction technique (Federolf, 2013;Gløersen and Federolf, 2016). Two PCAs were performed, one for each balancing condition (Promsri et al., 2020a). The middle 60 seconds of each balancing trial were extracted and nine asymmetrical markers placed on the upper arms, lower arms, right scapular, upper thighs, and the lower thighs were omitted. In analogy to previous studies (Troje, 2002;Daffertshofer et al., 2004;Verrel et al., 2009;Federolf, 2016), the 3D coordinates (x, y, z) of the remaining 30 markers of each dataset at a given time t were interpreted as 90-dimensional posture vectors: Three pre-processing steps were then conducted. First, the posture vectors were centered by subtracting the subject's mean posture vector. For each subject, subj, a mean posture vector: where the bar over the variable indicates the mean over time, x = mean t (x (t) ), was subtracted from each posture vector: This procedure is the first step toward removing anthropometric differences (Federolf, 2016). The PCA was, therefore, conducted on deviations from a subject's mean posture, i.e., on postural movements. Second, the centered posture vectors were normalized to the mean Euclidean distance d subj (Federolf, 2013(Federolf, , 2016. Thus, for each posture vector − − → p ′ (t) the Euclidean norm: was calculated and the − − → p ′ (t) were then divided by the mean of these Euclidian distances: Third, the normalized posture vectors were weighted using sexspecific mass distributions (Gløersen et al., 2018). Specifically, for each marker i a weight factor w i was calculated by dividing the relative weight of the segment to which the marker was attached, m s , by the number n s of markers on this segment. For markers placed on joints, the masses of both segments were added. For example, w i for the knee markers was calculated as w i = m thigh n thigh + m shin n shin with n thigh = n shin = 3, m thigh = 14.16%, and m shin = 4.33% for men (de Leva, 1996). Thus, the normalized postural movement vectors had the form:

Principal Component Analysis
The PCA was calculated by a singular-value decomposition of the input matrix's covariance matrix and produced a set of PC-eigenvectors, − − → PC k , which form a new basis for the vector space of marker positions . All PC-eigenvectors are linear combinations of the original marker coordinates. Animated stick figures can be created from the mean postures and from each eigenvector to characterize the principal movements PM k Federolf, 2016). The time evolution of each PM k , i.e., the PP k (t), were obtained by a coordinate transformation of the normalized posture vectors onto FIGURE 1 | Illustration of the first ten principal movements (PM 1−10 ) of bipedal standing on (A) the firm surface and bipedal balancing on (B) the wobble board. Gray and black lines/dots show the extreme posture in opposite directions. Movement amplitudes are amplified using the indicated factor for a better visualization (Firm surface: amplification 10× for PM 1−5 , and 20× for PM 6−10 ; Wobble board: amplification 1× for PM 1−5 , and 2× for PM 6−10 ). Movements are clearer and can be more easily characterized when viewed in animated stick figure videos: Supplementary Videos S1, S2 for balancing on the firm and soft surfaces, respectively.
Frontiers in Bioengineering and Biotechnology | www.frontiersin.org the PCA-eigenvectors.
The PP k (t) represent positions in posture space, i.e., how much the posture at time t deviates in the direction of the PC keigenvector from the mean posture (Federolf, 2016). In other words, the PP k (t) represent the amplitude of each movement component PM k . The variance of each PP k (t), divided by the sum of the variances of all PP k (t), results in a variable relative explained variance of principal position PP_rVAR k that quantifies for each volunteer and each order k, how much the specific PM k contributed to the whole postural movements of the subject.
In analogy to Newton's mechanics and differentiation rules, the rate of postural change can be quantified by principal velocities PV k (t), i.e., by the first time derivative of the PP k (t), PV k = d dt PP k ; and the acceleration of postural movements can be quantified by principal accelerations PA k (t), i.e., by the second time derivative of the PP k (t), PA k = d 2 dt 2 PP k (Federolf, 2016). In case of unperturbed human postural control, PA k (t) are either a direct result of muscle activation, a result of the neuromuscular system utilizing gravity to produce desired accelerations, or an undesired result of gravity which the neuromuscular system was not able to prevent e.g., loss of stability (Promsri et al., 2020a). In this sense, the PA k (t) are the essential mechanical variables that the sensorimotor system must control in order to govern the body's motion and maintain its stability. Thus, each PA k (t) represents a variable that quantifies how the mechanical system is controlled (Federolf, 2016;Promsri et al., 2020a). In analogy to PP_rVAR k , we calculated the variable relative explained variance of principal acceleration PA_rVAR k to assess how much each movement component contributed to the overall postural accelerations in the individual subjects.
Due to noise amplification in the differentiation processes (Winter et al., 1974), filtering of the PP k (t) is needed before computing PV k (t) and PA k (t). The current study examined the effect of low-pass filtering using a 3rd-order, zero-phase, low-pass Butterworth filter. The Butterworth filter was selected, since it is free of ripples in the pass and stop band. The filter order (3rd) was selected arbitrarily, however, preliminary tests suggested that the filter order has a very small effect on the PA time series. Prior FIGURE 2 | Box plots representing the data from all 26 participants of (A) the relative explained variance of principal postural positions (PP_rVAR k ) and (B) the relative explained variance of principal postural accelerations (PA_rVAR k ) of standing on the firm surface (FS) and balancing on the wobble board (WB). The PA_rVAR k were determined after filtering the data with a 3rd-order 10 Hz low-pass Butterworth filter.
FIGURE 3 | Box plots of the relative explained variance of principal postural acceleration (PA_rVAR k ) of standing on the firm surface (FS) and balancing on the wobble board (WB) with different cut-off frequencies, including (A) 1 Hz, (B) 2 Hz, (C) 5 Hz, (D) 10 Hz, (E) 20 Hz, and (F) no filtering, which were observed from 26 participants (k displays order of principal components, PMs; k = 1 to 25). The letter, "A," and its arrows point to lower-order PAs, PA_rVAR 1 , and PA_rVAR 2 , whose contribution to the overall acceleration variance decrease with increasing cutoff frequencies. The letter "B" and its arrows highlight two medium-order PAs, PA_rVAR 8 , and PA_rVAR 10 , whose contribution to the overall acceleration variance increase as cutoff frequencies are increased.
to filtering, the frequency contents of the raw PP k (t) and PA k (t) were evaluated using a Fourier transformation. Then, the effect of cut-off frequency on PA_rVAR k was evaluated for both balancing situations, FS and WB, with cut-off frequencies of 1, 2, 5, 10, and 20 Hz and with no filtering. Finally, explained variance spectra of PP_rVAR k and PA_rVAR k (10 Hz) were compared.

RESULTS
The first 10 principal movements (PM 1−10 ) of standing on a firm surface (FS) and balancing on a wobble board (WB) are described and shown in (Table 1, Figure 1), and in (Supplementary Videos 1, 2). Higher-order movement components were not included for the visualization and description, since their small movement amplitudes make them difficult to characterize, however, higher-order components were considered in the evaluation of the variance spectra. The spectra of explained variance, PP_rVAR k and PA_rVAR k (for a cut-off frequency of 10 Hz) are shown in (Figure 2). As expected, several movement components that contributed little to the postural variance did have an over-proportional contribution to the acceleration variance. Specifically, for standing on the FS, PM 3 , PM 8 , and PM 10 which predominantly represented hip strategy and upper body movements, and for balancing on the WB PM 8 which predominantly quantified ankle plantar/dorsiflexion, were of particular interest.
Fourier transformations of the raw PP and PA time series of one arbitrarily selected, representative volunteer are shown in (Supplementary Figure 2) for FS and in (Supplementary Figure 3) for WB. The PP-frequency domain of both FS and WB conditions was observed well below 5 Hz. In contrast, in the PA-spectra, despite the strong and blue-shifted noise, signals are visible in the ranges 0-5 Hz for FS and in the range up to ∼10 Hz for WB. In addition, (Figure 3) illustrates how the spectrum of explained variance PA_rVAR k changes with increasing filter cut-off frequency.

DISCUSSION
Our analysis demonstrates that PA k and PA k -based variables, here PA_rVAR k , depend on the filter cut-off applied in the PA calculation. Low cut-off frequencies (<5 Hz) lead to over-pronunciation of slow movement components. As filter frequency is increased (5-20 Hz) a new pattern emerges, in line with the expectation that some of the higher-order movement components contribute more than other movement components to the accelerations. The Fourier analysis of the underlying signals suggests that the pattern emerging with increasing filter cut-offs is not (not only) a consequence of noise increasingly affecting the signal: while the PP k (t) live in a very low frequency range (<3 Hz), several of the PA k (t) show a relevant frequency content up to ∼5 Hz in FS and up to 10 Hz in the WB conditions. These observations suggest that filter cut-off frequencies of 5-7 Hz for FS and around 10 Hz for WB would be appropriate.
The current findings underpin that (i) when focusing only on the classical movement strategies (lower-order PMs), one might overlook movement components that are small in postureamplitude, but that can be accelerated fast and thus provide an important contribution to postural control. Spectra of PA-explained variance should be considered when deciding on how many PC-components are included in an analysis. (ii) When interested in neuromuscular control and thus in the accelerations and forces controlling postural movements, then filter frequencies should not be selected below 5 Hz for stable situations and not below 10 Hz for more dynamic balancing trials. The current findings corroborate the findings of Longo et al. (2019), who assessed PA-relative variance in a cyclic upper-body motion. Moreover, Longo et al. (2019) also mathematically validated that all PA k together (i.e., the sum of all PA k ) represent the entire marker accelerations present in the dataset. The current results also agree with previous studies in which the dependence of PA k variables on filter cutoff frequencies was assessed, and which reported consistent results for cut-off frequencies in the range 5 to 12 Hz Promsri et al., 2018Promsri et al., , 2019Promsri et al., , 2020a. Furthermore, recent studies on muscle synergies and on coherence between electromyographic signals from different muscles also reported spectra peaking around 9 Hz and posture-related coherence in frequency bands 5-20 Hz (Boonstra et al., 2008(Boonstra et al., , 2015, which supports the assumption that the PA k signals in this frequency range are of physiological origin and probably not an artifact or noise phenomenon. The role of movement analysis in monitoring and diagnosing neurodegenerative conditions is increasingly recognized, particularly when combined with machine/deep learning approaches (Buckley et al., 2019). However, how successful such approaches can become depends largely on the information contained in the input data to these algorithms. Disregarding information at an early stage, e.g., due to dimensionality reduction or through filtering, is a form of investigator bias that likely affects even the performance of so-called unsupervised methods. Driven by biomechanical considerations, the current study evaluated what information might be contained in the often disregarded higher-order PC-components. The question, which specific PC k components are relevant, depends on the specific movement, the specific boundary conditions that are present, and the research question that is studied. However, as general advise the current study suggest that PA_rVAR kspectra should be analyzed when deciding on how many PC components are to be considered; and the frequency content and suitable filters should be carefully assessed in the calculation of PAs.

DATA AVAILABILITY STATEMENT
The datasets generated for this study are available on request to the corresponding author.

ETHICS STATEMENT
The study reported here was approved by the Board of Ethical Questions in Science of the University of Innsbruck, Austria (Certificate 16/2016). All participants provided informed written consent prior to their participation.

AUTHOR CONTRIBUTIONS
AP and PF have contributed equally to the design and implementation of the research and to the writing of the manuscript.

FUNDING
This study was supported by the University of Phayao, Phayao, Thailand [grant number 28082015] as the educational grant to the first author. Parts of the open access publishing costs after publication may be covered by internal university grants from the University of Innsbruck, Innsbruck, Austria.

ACKNOWLEDGMENTS
We are gratefully acknowledge all volunteers for their participation, Carina Zöhrer and Elena Pocecco for their help with recruiting participants, and Armin Niederkofler for technical advice on data acquisition.