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Smooth GPS Tuple

Description

κ°€κ³΅λ˜μ§€ μ•Šμ€ GPS tuple μ€‘μ—λŠ” noiseκ°€ μ‘΄μž¬ν•˜λ©°, 뢄석에 κ·ΈλŒ€λ‘œ μ‚¬μš©ν•˜λ©΄ 였차λ₯Ό λ°œμƒμ‹œν‚¬ 수 μžˆλ‹€. λ”°λΌμ„œ ν•΄λ‹Ή 데이터λ₯Ό λ¨Όμ € 2Hz둜 down-samplingν•œ λ’€(5κ°œμ”© 평균 계산) 10Hz둜 λ‹€μ‹œ up-samplingν•˜μ—¬(cubic spline interpolation) noiseκ°€ 적은 λΆ€λ“œλŸ¬μš΄ λ°μ΄ν„°λ‘œ λ§Œλ“ λ‹€. κ·Έ κ²°κ³Όλ₯Ό smooth GPS tuple이라 ν•˜λ©°, ν•΄λ‹Ή tuple의 μœ„μΉ˜ 및 속λ ₯을 뢄석에 μ‚¬μš©ν•œλ‹€.

Formal Definition

Before the analysis, we smooth the raw GPS data through the following two steps for noise reduction.
1. Average Down-Sampling
We down-sample the time-series xloc=(sloc,vGPS):PΓ—T[Ξ»GPS]β†’R2Γ—V\mathbf{x}^{\mathsf{loc}} = (\mathbf{s}^{\mathsf{loc}}, v^{\mathsf{GPS}}): \mathscr{P} \times \mathbb{T}[\lambda^{\mathsf{GPS}}] \rightarrow \mathbb{R}^2 \times \mathcal{V} of GPS tuples by averaging every mGPS=(2kGPS+1)Β (kGPS∈N)m^{\mathsf{GPS}} = (2k^{\mathsf{GPS}}+1) \ (k^{\mathsf{GPS}} \in \mathbb{N}) consecutive tuples. The down-sampling results in the time-series xΛ‰:PΓ—T[mGPSΞ»GPS]β†’R2Γ—V\mathbf{\bar{x}}: \mathscr{P} \times \mathbb{T}[m^{\mathsf{GPS}}\lambda^{\mathsf{GPS}}] \rightarrow \mathbb{R}^2 \times \mathcal{V} such that
xΛ‰(P,t):=(sΛ‰(P,t),vΛ‰(P,t))=1mGPSβˆ‘i=βˆ’kGPSkGPS(sloc(P,t+iΞ»GPS),vGPS(P,t+iΞ»GPS))\mathbf{\bar{x}}(P,t) := (\mathbf{\bar{s}}(P,t), \bar{v}(P,t)) =\\ \frac{1}{m^{\mathsf{GPS}}} \sum_{i=-k^{\mathsf{GPS}}}^{k^{\mathsf{GPS}}} (\mathbf{s}^{\mathsf{loc}} (P,t +i\lambda^{\mathsf{GPS}}), v^{\mathsf{GPS}}(P,t+i\lambda^{\mathsf{GPS}}))
2. Cubic Spline Interpolation
We up-sample xΛ‰(P,t)\mathbf{\bar{x}}(P,t) to a time-series per Ξ”t\Delta t such that mGPSΞ»GPS/Ξ”t∈Nm^{\mathsf{GPS}}\lambda^{\mathsf{GPS}} / \Delta t \in \mathbb{N} by applying the cubic spline interpolation.
In other words, we define the spline SP:Tβ†’R2Γ—VS_P: \mathbb{T} \rightarrow \mathbb{R}^2 \times \mathcal{V} for each P∈PP \in \mathscr{P} satisfying
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SP(t)∈C2[T]S_P(t) \in C^2[\mathbb{T}] (i.e., SPS_P is twice continuously differentiable on T\mathbb{T}.)
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On each subinterval [t0+nmGPSΞ»GPS,t0+(n+1)mGPSΞ»GPS]∈TΒ (n∈Z)[t_0 + nm^{\mathsf{GPS}}\lambda^{\mathsf{GPS}}, t_0 + (n+1)m^{\mathsf{GPS}}\lambda^{\mathsf{GPS}}] \in \mathbb{T}\ (n \in \mathbb{Z}), SPS_P is an (R2Γ—V)(\mathbb{R}^2 \times \mathcal{V})-valued polynomial of degree 3.
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SP(t)=xΛ‰(P,t)Β Β βˆ€t∈T[mGPSΞ»GPS]S_P(t) = \bar{\mathbf{x}}(P,t)\ \ \forall t \in \mathbb{T}[m^{\mathsf{GPS}}\lambda^{\mathsf{GPS}}].
and define the smooth GPS tuples x:PΓ—T[Ξ”t]β†’R2Γ—V\mathbf{x}: \mathscr{P} \times \mathbb{T}[\Delta t] \rightarrow \mathbb{R}^2 \times \mathcal{V} as
x(P,t)=(sx(P,t),sy(P,t),v(P,t)):=SP(t)\mathbf{x}(P,t) = (s_x(P,t), s_y(P,t), v(P,t)) := S_P(t)
on t∈T[Ξ”t]t \in \mathbb{T}[\Delta t].

Parameter Setting

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[1.8.3] Ξ»GPS=0.1 s\lambda^{\mathsf{GPS}} = 0.1\,\text{s}, mGPS=1m^{\mathsf{GPS}} = 1, and Ξ”t=0.1 s\Delta t = 0.1\,\text{s} (No down-sampling or up-sampling.)
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[1.9.0] mGPS=5m^{\mathsf{GPS}} = 5 and Ξ”t=0.5 s\Delta t = 0.5\,\text{s} (Raw GPS data are down-sampled by 2 Hz without additional up-sampling.)
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[1.9.2] mGPS=5m^{\mathsf{GPS}} = 5 and Ξ”t=0.1 s\Delta t = 0.1\,\text{s} (Raw GPS data are smoothed by consecutive 2 Hz down-sampling and 10 Hz up-sampling.)