Fitting a ballistic trajectory to noisy data where both spacial and temporal domains observations are noisyFitting a curve to noisy data is somewhat trivial. However it generally assumes that data abscissa is fixed, and the error is computed on the ordinate.In my setup, I have 3D spacial observations of ballistic trajectories (that I model with a simple parabola), but the observations time are also noisy.Therefore, I have to estimate the initial position&nbsp;y0,y0,z0&nbsp;and initial speed&nbsp;vx0,vy0,vz0, based on 4D (noisy) observations&nbsp;(Xi,Yi,Zi,Ti),i∈[0,N], such that they fit the following model:{x(t)=x0+vx0ty(t)=y0+vy0tz(t)=z0+vz0t−g2t2with t monotonically increasing with i.I'm not sure how to formulate such optimization problem because I have 6 parameters to estimate, but also 4N variables with only 3N equations… My intuition tells me there's only one single parabola that minimizes the error (MSE for example), but I can't formulate the problem.

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Fitting a ballistic trajectory to noisy data where both spacial and temporal domains observations are noisyFitting a curve to noisy data is somewhat trivial. However it generally assumes that data abscissa is fixed, and the error is computed on the ordinate.In my setup, I have 3D spacial observations of ballistic trajectories (that I model with a simple parabola), but the observations time are also noisy.Therefore, I have to estimate the initial position&amp;nbsp;y0,y0,z0&amp;nbsp;and initial speed&amp;nbsp;vx0,vy0,vz0, based on 4D (noisy) observations&amp;nbsp;(Xi,Yi,Zi,Ti),i∈[0,N], such that they fit the following model:{x(t)=x0+vx0ty(t)=y0+vy0tz(t)=z0+vz0t−g2t2with t monotonically increasing with i.I&#039;m not sure how to formulate such optimization problem because I have 6 parameters to estimate, but also 4N variables with only 3N equations… My intuition tells me there&#039;s only one single parabola that minimizes the error (MSE for example), but I can&#039;t formulate the problem.

cab65699m · Accepted Answer

Hint.Define an error function likeE(x0,y0,z0,vx0,vy0,vz0)=∑k(xk−x0−vx0tk)2+(yk−y0−vy0tk)2+(zk−z0−vz0tk−g2tk2)2so the minimum can easily be obtained by solving ∇E=0

Fitting a ballistic trajectory to noisy data where

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