Agenda

Rambling, trembling
1. questions: mean, is \(F_hor = 0\) the right assumption? )
2. observation: mean, seems like tonic activation allow for keep the COP a bit off the COM.
Fourier analysis
1. questions (rounding, expected frequencies retrieved from papers)
Trajectory, speed, acceleration
1. questions (parametrization of rambling, trembling ?)
Next steps / questions
1. What is the research question?
2. Which data set are we using, how do I access?
3. Which features should be extracted?

Data Set

Physionet data set, 6000 rows, 9 columns (Time, Fx, Fy, Fz, Mx, My, Mz, COPx, COPy). 163 subjects, 116 female, 47 males. 16 with severe disability, eight hearing and vestibular deficits, two with visual deficits. Sampling frequency 1/0.01 = 100Hz. Link to article https://peerj.com/articles/2648.pdf

\[\begin{align} COP_x &= - \frac{M_y}{F_z} \\ COP_y &= \frac{M_x}{F_z} \\ \end{align}\]

Time.s.	Fx.N.	Fy.N.	Fz.N.	Mx.Nm.	My.Nm.	Mz.Nm.	COPx.cm.	COPy.cm.
0.01	1.504028	0.123810	446.1493	3.595200	17.77963	-0.751617	-3.985130	0.805829
0.02	1.386111	-0.003157	446.2276	3.457270	17.99393	-0.756481	-4.032455	0.774777
0.03	1.266321	-0.122046	446.3541	3.318239	18.20646	-0.761518	-4.078928	0.743409
0.04	1.142307	-0.225587	446.5613	3.177267	18.41523	-0.766670	-4.123785	0.711496
0.05	1.011086	-0.309218	446.8543	3.034027	18.61742	-0.771594	-4.166329	0.678975

Pipeline

Filter raw signal with butterworth filter
Decompose signal into rambling, trembling components
Discrete fourier analysis determining frequency spectrum of forces, center of pressure
Integrate distance travel to reveal velocity, acceleration profile

Butterworth filter

The data provided are filtered using low pass Butterworth filter. The specifications are:

4 order, low pass filter
zero-lag phase
cutoff frequency of 8 Hz, most of the signal power is below 1 Hz ( Winter 1995)
sampling frequency = 100Hz
Nyquist Frequency = sampling frequency / 2

Decomposing COP trajectory into rambling and trembling

The COP is a bivariate distribution, jointly defined by anterior-posterior “AP” and medio-lateral “ML” coordinates. The COPx and COPy time series define the COP path relative to the origin of the force platform coordinate system. These time series were computed using the digitized output signals of the force platform amplifiers

The CoP displacement was decomposed into two components, rambling and trembling in AP and ML direction, using the method described by Zatsiorsky and Duarte [xx]. The method assumes two contributing components of CoP; the sum of which equals the total CoP. Rambling quantifies the migration of a moving reference point in relation to the body’s equilibrium, and has been suggested to represent the supra-spinal component of the CoP trajectory. Trembling quantifies the oscillations around the reference point, and has been suggested to represent the spinal and peripheral component of the CoP trajectory.

To estimate the rambling trajectory the instant equilibrium positions were identified as the CoP positions when the horizontal force Fhor = 0, and interpolated using a cubic spline function. To obtain the trembling trajectory, the deviations of the CoP trajectory from the interpolated instant equilibrium positions trajectory were determined. R

Approach

identify IEP (instant equilibrium points) through \(t_{1:n} for F_{hor} = 0\)
identify COP positions at IEP \(F_{hor} = 0\) by \(COP_{t_{1:n}}\)
Rambling trajectory: interpolate COP at IEP \(F_{hor} = 0\) using cubic splines between data points
Trembling trajectory: Subtract from COP trajectory the rambling trajectory

IEP identification from horizontal force

COPx trajectory with IEPs projected

For each IEP point in time the COPx is identified. At those positions postural equilibrium is assumed, since horizontal forces are zero. The rambling trajectory provides likely the equilibrium path.

Extracting rambling trajectory

R basic offers fmm, periodic, natural cubic splining methods. No visual difference between those three methods. See below:

Rambling and trembling decomposed from COPx

Rambling (supraspinal contribution)

Rambling is the motion of a moving reference point with respect to which the body’s equilibrium is instantly maintained \(F_{hor} = 0, \ (F_x, F_y)\). Anterior - posterior

Trembling (muscle stiffness, reflexive properties contribution)

The oscillation around that moving equilibrium trajectory (rambling) is termed trembling.

Features

n = amount of IEPs
mean.max = mean max distance ? (AP, ML)
mean.min = mean min distance ? (AP, ML)
mean CPO = arithmetic means of AP, and ML time series
\[\eq{ \bar{AP} &= \frac{1}{N}\sum{AP_O[n]}, \ \ \ \bar{ML} = \frac{1}{N}\sum{ML_O[n]} &(1) \\ AP[n] &= AP_O[n] - \bar{AP} , \ \ ML[n] = ML_O[n[- \bar{ML}]] &(2) \\ RD[n] &= \sqrt{AP[n]^2 + ML[n]^2 } &(3) \\ MDIST &= \frac{1}{N} \sum {RD[n]} &(4) \\ MDIST_{AP} &= \frac{1}{N} \sum{|AP[n]|} &(5) \\ RDIST &= \sqrt{\frac{1}{N}\sum{RD[n]^2}} &(6) \\ RDIST_{AP} &= s_{AP} = \sqrt{\frac{1}{N}\sum{AP[n]^2}} &(7) \\ TOTEX &= \sum^{N-1}_{n=1} = \Bigg(AP[n+1] - AP[n]\Bigg)^2 + \Bigg(ML[n+1] - ML[n]\Bigg)^2 &(8) \\ TOTEX_{AP} &= \sum^{N-1}_{n=1} \Bigg|AP[n+1] - AP[n] \Bigg| &(9) \\ MVELO &= \frac{TOTEX}{T} &(10) \\ MVELO_{AP} &= \frac{TOTEX_{AP}}{T} &(11) \\ AREA - CC &= \pi(MDIST + Z_{0.05} \ S_{RD}) &(12) \\ s_{RD} &= \sqrt{\frac{1}{N}\sum{RD^2[n] - MDIST^2}} &(13) \\ a &= \sqrt{F_{.05[2, n-2]}(s^2_{AP} + s^2_{ML} + D)} &(14) \\ b &= \sqrt{F_{.05[2, n-2]}(s^2_{AP} + s^2_{ML} - D)} &(15) \\ D &= \sqrt{(s^2_{AP} + s^2_{ML}) - 4(s^2_{AP} \ s^2_{ML} - s^2_{AP,ML})} &(16) \\ S_{APML} &= \frac{1}{N}\sum{AP[n] \ ML[n]} &(17) \\ AREA - CE &= \ \pi ab = 2\pi \ F_{.05[2,n-2]}\sqrt{s^2_{AP} \ s^2 {ML} - s^2_{APML}} &(18) \\ AREA - SW &= \frac{1}{2T} \sum^{N-1}_{n=1}|AP[n+1] \ ML[n] - AP[n] \ ML[n+1]| &(19) \\ MFREQ &= \frac{TOTEX}{2\pi \ MDIST \ T} = \frac{MVELO}{2\pi \ MDIST} &(20) \\ MFREQ_{AP} &= \frac{TOTEX_{AP}}{4 \sqrt2 \ MDIST_{AP} \ T} = \frac{MVELO_{AP}}{4 \sqrt2\ MDIST_{AP}} &(21) \\ FD &= \frac{log(N)}{log(\frac{Nd}{TOTEX})} &(22) \\ d_{FD-CC} &= 2(MDIST + Z_{.05} \ S_{RD}) &(23) \\ d_{FD - CE} &= \sqrt{(2a \ 2b} = \sqrt{8F_{0.05[2 ,ne-2]}\sqrt{(s^2_{AP} \ s^2_{ML} - s^2_{AP, ML})}} &(24) \\ \mu_k &= \sum^j_{m=i}{(m\Delta \ f)^k \ G[m]} &(25) \\ POWER &= \mu_0 &(26) \\ \sum^u_{m=i} G[m] &\geq 0.05 \ \mu_0 &(27) \\ \sum^v_{m=i} G[m] &\geq 0.95 \ \mu_0 &(28) \\ CFREQ &= \sqrt\frac{\mu_2}{\mu_0} &(29) \\ FREQD &= \sqrt{1 - \frac{\mu_1^2}{\mu_0 \ mu_2}} &(30) \\ }\] (2) AP and ML time services are referenced tot he mean COP

Trajectory RMS

Reference:
1. Perperoglou, A., Sauerbrei, W., Abrahamowicz, M. et al. A review of spline function procedures in R. BMC Med Res Methodol 19, 46 (2019). https://doi.org/10.1186/s12874-019-0666-3
2. Zatsiorsky, V. M., & Duarte, M. (1999). Instant equilibrium point and its migration in standing tasks: rambling and trembling components of the stabilogram. In Motor control (Vol. 3, Issue 1, pp. 28–38). https://doi.org/10.1123/mcj.3.1.28
3. Zatsiorsky, V. M., & Duarte, M. (2000). Rambling and trembling in quiet standing. Motor Control, 4(2), 185–200. https://doi.org/10.1123/mcj.4.2.185
4. Santos, D. A., & Duarte, M. (2016). A public data set of human balance evaluations. PeerJ, 2016(11). https://doi.org/10.7717/peerj.2648
5. Prieto, T. E., Myklebust, J. B., Hoffmann, R. G., Lovett, E. G., & Myklebust, B. M. (1996). Measures of postural steadiness: Differences between healthy young and elderly adults. IEEE Transactions on Biomedical Engineering, 43(9), 956–966. https://doi.org/10.1109/10.532130

Fourier Transformation

\[\begin{align} d(\omega_j = \sqrt{\frac{1}{n}}\sum_{t=1}^{n}x_td^{-2\pi\omega_jt}\\ for \ j &= 0, 1, ..., n-1\\ \omega_j &= \frac{j}{n} \end{align}\]

Scaled Periodigram

\[\begin{aligned} |d(j/n)^2 &= \frac{1}{n}\bigg({\sum_{t=1}^n x_t cos(2\pi t j/n)\bigg)^2} + \frac{1}{n}\bigg({\sum_{t=1}^n x_t sin(2\pi t j/n)\bigg)^2} \\ P(j/n) &= \frac{4}{n}|d(j/n)|^2 \\ \end{aligned}\]

COPx, COPy speed

COPx speed

\[\begin{align} Central\:Difference\:Method:&\\ \hat{v} = \frac{ds}{dt} &= \frac{s_{i+1}-s_{i-1}}{2\varDelta t} \\ \hat{a} = \frac{dv}{dt} &= \frac{v_{i+1}-v_{i-1}}{2\varDelta t} = \frac{s_{i+2}-2s_i + s_{i-2}}{4(\varDelta t)^2} \end{align}\]

COPx velocity values

time	dx	vx	std.vx
0.01	0.051654	2.58270	1,2
0.02	0.059505	2.97525	1,2
0.03	0.069383	3.46915	1,2
0.04	0.078617	3.93085	1,2
0.05	0.085044	4.25220	2,3
0.06	0.087564	4.37820	2,3

COPx velocity distribution

Var1	<-4	-3,-4	-2,-3	-1,-2	-1, mu	mu,1	1,2	2,3	3,4	>4
Freq	7	46	126	552	2134	2402	590	94	35	12

COPy speed