From the references listed below, a formula is given for the best least square line fit to a traveling charged particle with sensor detections. This is fairly simple and does not take into account Cherenkov radiation and its cone of radiation, but it is a good starting point. I will show the derivation of the formula.
\nEd: The notebook implementing this scores 1.214 least squares notebook
\nWe assume that the particle is traveling such that its position at time t is
\n$$pos(t) = r - v*t$$
\nwhere \\(r\\) and \\(v\\) are 3-dimensional vectors representing the position at time 0 and the velocity, respectively.
\nThen we have a series of \\(N\\) measurements from sensors at positions \\(r_i\\) at times \\(t_i\\). We assume that the particle reaches nearest to the sensor at that time (this is the simplification). Therefore, the total error in 3-dimensions across all the sensors is
\n$$E = \\sum_{i=0}^{N}(r_i-r-vt_i)^T(r_i-r-vt_i)$$
\nTo find the values of v and r that make this minimal, we take the derivative with respect to \\(r\\) and \\(v\\):
\n$$\\frac {\\partial E}{\\partial r} = \\sum_{i=0}^{N}2(r_i-r-vt_i)(-1) = 0$$
\nand so
\n$$r = \\frac {1}{N}\\sum_{i=0}^{N}(r_i - vt_i) = \\langle r_i \\rangle - v \\langle t_i \\rangle$$
\nwhere we use the \\(\\langle \\rangle \\) notation to denote the average over the data.
\nTaking the derivative with respect to \\(v\\):
\n$$\\frac {\\partial E}{\\partial v} = \\sum_{i=0}^{N}2(r_i-r-vt_i)(-t_i) = 0$$
\nor
\n$$\\sum t_i r = \\sum (r_i t_i - v t_i^2) $$
\nor
\n$$\\langle t_i \\rangle r = \\langle r_i t_i \\rangle - v \\langle t_i^2 \\rangle$$
\nSubstituting for r in this equation with the value obtained above and some simple manipulations produces
\nthe answer
$$v_{est} = \\frac{ \\langle r_i * t_i \\rangle - \\langle r_i \\rangle * \\langle t_i \\rangle}{\\langle t_i^2 \\rangle - \\langle t_i \\rangle ^2}$$
\nReferences:
\nMirco Hunnefield Masters Thesis, Online Reconstruction of Muon-Neutrino Events in IceCube using Deep Learning Techniques
\nKai Schatto, PhD Thesis, Stacked searches for high-energy neutrinos from blazars with IceCube
From the references listed below, a formula is given for the best least square line fit to a traveling charged particle with sensor detections. This is fairly simple and does not take into account Cherenkov radiation and its cone of radiation, but it is a good starting point. I will show the derivation of the formula.
\nEd: The notebook implementing this scores 1.214 least squares notebook
\nWe assume that the particle is traveling such that its position at time t is
\n$$pos(t) = r - v*t$$
\nwhere \\(r\\) and \\(v\\) are 3-dimensional vectors representing the position at time 0 and the velocity, respectively.
\nThen we have a series of \\(N\\) measurements from sensors at positions \\(r_i\\) at times \\(t_i\\). We assume that the particle reaches nearest to the sensor at that time (this is the simplification). Therefore, the total error in 3-dimensions across all the sensors is
\n$$E = \\sum_{i=0}^{N}(r_i-r-vt_i)^T(r_i-r-vt_i)$$
\nTo find the values of v and r that make this minimal, we take the derivative with respect to \\(r\\) and \\(v\\):
\n$$\\frac {\\partial E}{\\partial r} = \\sum_{i=0}^{N}2(r_i-r-vt_i)(-1) = 0$$
\nand so
\n$$r = \\frac {1}{N}\\sum_{i=0}^{N}(r_i - vt_i) = \\langle r_i \\rangle - v \\langle t_i \\rangle$$
\nwhere we use the \\(\\langle \\rangle \\) notation to denote the average over the data.
\nTaking the derivative with respect to \\(v\\):
\n$$\\frac {\\partial E}{\\partial v} = \\sum_{i=0}^{N}2(r_i-r-vt_i)(-t_i) = 0$$
\nor
\n$$\\sum t_i r = \\sum (r_i t_i - v t_i^2) $$
\nor
\n$$\\langle t_i \\rangle r = \\langle r_i t_i \\rangle - v \\langle t_i^2 \\rangle$$
\nSubstituting for r in this equation with the value obtained above and some simple manipulations produces
\nthe answer
$$v_{est} = \\frac{ \\langle r_i * t_i \\rangle - \\langle r_i \\rangle * \\langle t_i \\rangle}{\\langle t_i^2 \\rangle - \\langle t_i \\rangle ^2}$$
\nReferences:
\nMirco Hunnefield Masters Thesis, Online Reconstruction of Muon-Neutrino Events in IceCube using Deep Learning Techniques
\nKai Schatto, PhD Thesis, Stacked searches for high-energy neutrinos from blazars with IceCube
A couple interesting things I found when comparing this formula to the one I was using:
\nSee this and much more in my new notebook.. I can run the above formula in just 7 seconds per batch file! And can get a better score (though takes about 70-90 seconds per batch file) by slightly optimizing which points we treat as aux=False.
", "rawMarkdown": "A couple interesting things I found when comparing this formula to the one I was using:\n\n1. You can weight it - for instance weight by 'charge' column - if you convert theThere is an easier derivation of the formula.
\nLet the charge move along the trajectory r=q+ u t and N sensors fire only along it (t,r). Then (the second equation is the multiplication of the first by t):
\n$$
\n\\mathbf{r}_i =\\mathbf{q} + \\mathbf{u}\\,t_i,\\,\\,\\,\\,\\,\\,\\mathbf{r}_i\\,t_i= \\mathbf{q}\\,t_i + \\mathbf{u}\\,t^2_i
\n$$
\nAveraging over i and solving the system of equations for u, q, we obtain what we are looking for.
Interesting. While your derivation ends up with the same formula, you are solving a different problem. I was solving for the q, u that gives the smallest error. You are solving the problem where the error is always 0. You wrote \\(r_i = q + u t_i\\) as an equality.
", "rawMarkdown": "Interesting. While your derivation ends up with the same formula, you are solving a different problem. I was solving for the q, u that gives the smallest error. You are solving the problem where the error is always 0. You wrote \\\\(r_i = q + u t_i\\\\) as an equality.", "votes": null }, { "id": "2152185", "postDate": "02/20/2023 16:26:46", "content": "Of course, the method of least squares (which you used) is a mathematically more correct procedure. But sometimes you can make your life easier :)
", "rawMarkdown": "Of course, the method of least squares (which you used) is a mathematically more correct procedure. But sometimes you can make your life easier :)", "votes": null }, { "id": "2222242", "postDate": "04/15/2023 04:41:06", "content": "interesting work
", "rawMarkdown": "interesting work", "votes": null } ], "comments": [ { "id": 2119872, "author_name": "roberthatch", "author_url": "", "post_date": "01/29/2023 07:03:17", "content": "A couple interesting things I found when comparing this formula to the one I was using:
\nSee this and much more in my new notebook.. I can run the above formula in just 7 seconds per batch file! And can get a better score (though takes about 70-90 seconds per batch file) by slightly optimizing which points we treat as aux=False.
", "votes": null, "replies": [] }, { "id": 2151619, "author_name": "synset", "author_url": "", "post_date": "02/20/2023 07:57:14", "content": "There is an easier derivation of the formula.
\nLet the charge move along the trajectory r=q+ u t and N sensors fire only along it (t,r). Then (the second equation is the multiplication of the first by t):
\n$$
\n\\mathbf{r}_i =\\mathbf{q} + \\mathbf{u}\\,t_i,\\,\\,\\,\\,\\,\\,\\mathbf{r}_i\\,t_i= \\mathbf{q}\\,t_i + \\mathbf{u}\\,t^2_i
\n$$
\nAveraging over i and solving the system of equations for u, q, we obtain what we are looking for.
Interesting. While your derivation ends up with the same formula, you are solving a different problem. I was solving for the q, u that gives the smallest error. You are solving the problem where the error is always 0. You wrote \\(r_i = q + u t_i\\) as an equality.
", "votes": null, "replies": [ { "id": 2152185, "author_name": "synset", "author_url": "", "post_date": "02/20/2023 16:26:46", "content": "Of course, the method of least squares (which you used) is a mathematically more correct procedure. But sometimes you can make your life easier :)
", "votes": null, "replies": [] } ] } ] }, { "id": 2222242, "author_name": "hassaandaoud", "author_url": "", "post_date": "04/15/2023 04:41:06", "content": "interesting work
", "votes": null, "replies": [] } ], "raw_markdown_by_id": { "2118414": "From the references listed below, a formula is given for the best least square line fit to a traveling charged particle with sensor detections. This is fairly simple and does not take into account Cherenkov radiation and its cone of radiation, but it is a good starting point. I will show the derivation of the formula.\n\nEd: The notebook implementing this scores 1.214 [least squares notebook](https://www.kaggle.com/code/solverworld/icecube-neutrino-path-least-squares-1-214)\n\nWe assume that the particle is traveling such that its position at time t is\n$$pos(t) = r - v*t$$\nwhere \\\\(r\\\\) and \\\\(v\\\\) are 3-dimensional vectors representing the position at time 0 and the velocity, respectively.\nThen we have a series of \\\\(N\\\\) measurements from sensors at positions \\\\(r_i\\\\) at times \\\\(t_i\\\\). We assume that the particle reaches nearest to the sensor at that time (this is the simplification). Therefore, the total error in 3-dimensions across all the sensors is\n$$E = \\sum_{i=0}^{N}(r_i-r-vt_i)^T(r_i-r-vt_i)$$\nTo find the values of v and r that make this minimal, we take the derivative with respect to \\\\(r\\\\) and \\\\(v\\\\):\n$$\\frac {\\partial E}{\\partial r} = \\sum_{i=0}^{N}2(r_i-r-vt_i)(-1) = 0$$\nand so\n$$r = \\frac {1}{N}\\sum_{i=0}^{N}(r_i - vt_i) = \\langle r_i \\rangle - v \\langle t_i \\rangle$$\nwhere we use the \\\\(\\langle \\rangle \\\\) notation to denote the average over the data.\nTaking the derivative with respect to \\\\(v\\\\):\n\n$$\\frac {\\partial E}{\\partial v} = \\sum_{i=0}^{N}2(r_i-r-vt_i)(-t_i) = 0$$\nor\n$$\\sum t_i r = \\sum (r_i t_i - v t_i^2) $$\nor\n$$\\langle t_i \\rangle r = \\langle r_i t_i \\rangle - v \\langle t_i^2 \\rangle$$\nSubstituting for r in this equation with the value obtained above and some simple manipulations produces\nthe answer\n \n \n$$v_{est} = \\frac{ \\langle r_i * t_i \\rangle - \\langle r_i \\rangle * \\langle t_i \\rangle}{\\langle t_i^2 \\rangle - \\langle t_i \\rangle ^2}$$\n \nReferences:\n\nMirco Hunnefield Masters Thesis, Online Reconstruction of Muon-Neutrino Events in IceCube using Deep Learning Techniques\nKai Schatto, PhD Thesis, Stacked searches for high-energy neutrinos from blazars with IceCube", "2119872": "A couple interesting things I found when comparing this formula to the one I was using:\n\n1. You can weight it - for instance weight by 'charge' column - if you convert the