![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F1394009%2F7a1bae323673469a6ffe29d6d2726ceb%2Ferr_angles.png?generation=1678811788376942&alt=media\")
As we known, the competition dataset has a fairly isotropic distribution of track directions. On top of the isotropically distributed neutrino tracks, the noise is added, in the form of cosmic ray muon tracks. This noise should not correlate with the direction of the target track.
\nTherefore, it is natural to expect that the model errors would weakly depend on the target angles. However, it turns out that this is not the case. Below are the average error values as a function of the true zenith and azimuth angles (the cosine is taken from the zenith angle to obtain a uniform distribution). As you can see, there is a significant monotonic drop in the error when going from zenith=π (particle comes from below) to zenith=0 (particle comes from above). The angle zenith=π/2 roughly corresponds to the total error over all angles (the horizontal line):
\n
This observation forced us to conduct a more detailed analysis of the data in terms of the value of the target zenith angle (its cosine). This analysis shows significant data anisotropy (see notebook). For example, below are the average values of the number of active strings, triggered sensors and pulses at various values of the zenith angle. Each point has error bars; they are obtained by dividing the standard deviation from the mean by the square root of the number of events with a given angle. The horizontal line is the corresponding average value for all angles. The dotted line is the probability density of the number of events as a function of the zenith:
\n![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F1394009%2F84b92d35b673a9533ade44007ae48487%2Fim01.png?generation=1678811833894937&alt=media\")
The number of active strings, sensors and pulses is much larger for tracks coming from above. It is clear that the more pulses in the event, the easier it is usually for the model to reconstruct the track. This explains the behavior of the model error, but not the reasons leading to such anisotropy.
\nNote that \"noise\" (background) muons enter the IceCube only from above (the Earth screens them completely). It is clear that if background tracks of muons (also coming from above) are superimposed on the track from a neutrino coming from above, then it is easier for the model to reconstruct the track than with counter flows. But it doesn't explain either why there should be more pulses on top tracks.
\nSo, what's the reason for it? Most of the events present in the dataset are neutrinos produced by the collision
\ncosmic rays with the Earth's atmosphere. These cosmic rays and, thus, atmospheric neutrinos mostly have an isotropic distribution. However, in reality the Earth is not completely transparent to neutrinos (and fortunately for IceCube). Below is the number of events from atmospheric neutrinos depending on the cosine of the zenith angle [source]:
![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F1394009%2Fb1b1eef85dda002a1d4f45fcb93d39ff%2Feath.png?generation=1678811913097185&alt=media\")
As can be seen, there are significantly fewer atmospheric neutrinos arriving along the Earth's axis than those arriving at an angle. Apparently, it is this effect that contributes to the non-uniformity of the zenith angle, noted in this message (see also comment Philipp Eller - @pellerphys). In the diagrams of the distribution density of the cosine of the zenith angle, given above (dashed line), the difference between the minimum and the maximum is almost 30%. However, the number of events with a given angle does not affect the average values obtained at a given angle.
\nThe only reasonable explanation that we see so far is as follows. The cross section of the neutrino interaction reaction increases quite strongly with its energy. This means that the Earth absorbs high-energy neutrinos more strongly. Therefore, on average, neutrinos with lower energy enter the IceCube from below, and atmospheric neutrinos arriving from above have, on average, higher energy. Perhaps this is what increases the number of pulses at a small zenith, since more leptons with high energy are born, which generate more powerful Cherenkov radiation.
\nLet's note other factors that make space anisotropic in the vertical direction. They are unlikely to explain the observed effect, but may be useful for understanding the general background of the problem.
\n![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F1394009%2F198218062ade8a5e795c3b0c1c6daf77%2Fdust_kaggle.png?generation=1678812007702392&alt=media\")
Below are a few more graphs for various event characteristics. The logarithm of the total charge, the average time (after the first pulse shifted to t=0) and the proportion of auxiliary=True pulses in the event:
\n![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F1394009%2F3ae61425db7c75b5c9e6ee8599ca0d10%2Fim02.png?generation=1678812063990501&alt=media\")
Average values of event pulse coordinates (in kilometers):
\n![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F1394009%2F0fd0053e95254f148d5aacc06b4be247%2Fim03.png?generation=1678812089871467&alt=media\")
Standard deviations in kilometers (track compactness):
\n![\"\"](\"https://www.googleapis.com/download/storage/v1/b/kaggle-forum-message-attachments/o/inbox%2F1394009%2Fd3907cad6ca356cdfa5270048779b27d%2Fim04.png?generation=1678812111389928&alt=media\")
Happy kaggling!
", "messages": [ { "id": 2181643, "postDate": "2023-03-14T16:56:22.463Z", "content": "As we known, the competition dataset has a fairly isotropic distribution of track directions. On top of the isotropically distributed neutrino tracks, the noise is added, in the form of cosmic ray muon tracks. This noise should not correlate with the direction of the target track.
\nTherefore, it is natural to expect that the model errors would weakly depend on the target angles. However, it turns out that this is not the case. Below are the average error values as a function of the true zenith and azimuth angles (the cosine is taken from the zenith angle to obtain a uniform distribution). As you can see, there is a significant monotonic drop in the error when going from zenith=π (particle comes from below) to zenith=0 (particle comes from above). The angle zenith=π/2 roughly corresponds to the total error over all angles (the horizontal line):
\n
This observation forced us to conduct a more detailed analysis of the data in terms of the value of the target zenith angle (its cosine). This analysis shows significant data anisotropy (see notebook). For example, below are the average values of the number of active strings, triggered sensors and pulses at various values of the zenith angle. Each point has error bars; they are obtained by dividing the standard deviation from the mean by the square root of the number of events with a given angle. The horizontal line is the corresponding average value for all angles. The dotted line is the probability density of the number of events as a function of the zenith:
\n
The number of active strings, sensors and pulses is much larger for tracks coming from above. It is clear that the more pulses in the event, the easier it is usually for the model to reconstruct the track. This explains the behavior of the model error, but not the reasons leading to such anisotropy.
\nNote that \"noise\" (background) muons enter the IceCube only from above (the Earth screens them completely). It is clear that if background tracks of muons (also coming from above) are superimposed on the track from a neutrino coming from above, then it is easier for the model to reconstruct the track than with counter flows. But it doesn't explain either why there should be more pulses on top tracks.
\nSo, what's the reason for it? Most of the events present in the dataset are neutrinos produced by the collision
\ncosmic rays with the Earth's atmosphere. These cosmic rays and, thus, atmospheric neutrinos mostly have an isotropic distribution. However, in reality the Earth is not completely transparent to neutrinos (and fortunately for IceCube). Below is the number of events from atmospheric neutrinos depending on the cosine of the zenith angle [source]:
As can be seen, there are significantly fewer atmospheric neutrinos arriving along the Earth's axis than those arriving at an angle. Apparently, it is this effect that contributes to the non-uniformity of the zenith angle, noted in this message (see also comment Philipp Eller - @pellerphys). In the diagrams of the distribution density of the cosine of the zenith angle, given above (dashed line), the difference between the minimum and the maximum is almost 30%. However, the number of events with a given angle does not affect the average values obtained at a given angle.
\nThe only reasonable explanation that we see so far is as follows. The cross section of the neutrino interaction reaction increases quite strongly with its energy. This means that the Earth absorbs high-energy neutrinos more strongly. Therefore, on average, neutrinos with lower energy enter the IceCube from below, and atmospheric neutrinos arriving from above have, on average, higher energy. Perhaps this is what increases the number of pulses at a small zenith, since more leptons with high energy are born, which generate more powerful Cherenkov radiation.
\nLet's note other factors that make space anisotropic in the vertical direction. They are unlikely to explain the observed effect, but may be useful for understanding the general background of the problem.
\n
Below are a few more graphs for various event characteristics. The logarithm of the total charge, the average time (after the first pulse shifted to t=0) and the proportion of auxiliary=True pulses in the event:
\n
Average values of event pulse coordinates (in kilometers):
\n
Standard deviations in kilometers (track compactness):
\n
Happy kaggling!
", "rawMarkdown": "As we known, the competition dataset has a fairly isotropic distribution of track directions. On top of the isotropically distributed neutrino tracks, the noise is added, in the form of cosmic ray muon tracks. This noise should not correlate with the direction of the target track.\n\nTherefore, it is natural to expect that the model errors would weakly depend on the target angles. However, it turns out that this is not the case. Below are the average error values as a function of the true zenith and azimuth angles (the cosine is taken from the zenith angle to obtain a uniform distribution). As you can see, there is a significant monotonic drop in the error when going from zenith=π (particle comes from below) to zenith=0 (particle comes from above). The angle zenith=π/2 roughly corresponds to the total error over all angles (the horizontal line):\n\n\n\nThis observation forced us to conduct a more detailed analysis of the data in terms of the value of the target zenith angle (its cosine). This analysis shows significant data anisotropy (see notebook). For example, below are the average values of the number of active strings, triggered sensors and pulses at various values of the zenith angle. Each point has error bars; they are obtained by dividing the standard deviation from the mean by the square root of the number of events with a given angle. The horizontal line is the corresponding average value for all angles. The dotted line is the probability density of the number of events as a function of the zenith:\n\n\n\nThe number of active strings, sensors and pulses is much larger for tracks coming from above. It is clear that the more pulses in the event, the easier it is usually for the model to reconstruct the track. This explains the behavior of the model error, but not the reasons leading to such anisotropy.\n\nNote that \"noise\" (background) muons enter the IceCube only from above (the Earth screens them completely). It is clear that if background tracks of muons (also coming from above) are superimposed on the track from a neutrino coming from above, then it is easier for the model to reconstruct the track than with counter flows. But it doesn't explain either why there should be more pulses on top tracks.\n\nSo, what's the reason for it? Most of the events present in the dataset are neutrinos produced by the collision\ncosmic rays with the Earth's atmosphere. These cosmic rays and, thus, atmospheric neutrinos mostly have an isotropic distribution. However, in reality the Earth is not completely transparent to neutrinos (and fortunately for IceCube). Below is the number of events from atmospheric neutrinos depending on the cosine of the zenith angle [source]:\n\n\n\nAs can be seen, there are significantly fewer atmospheric neutrinos arriving along the Earth's axis than those arriving at an angle. Apparently, it is this effect that contributes to the non-uniformity of the zenith angle, noted in this message (see also comment Philipp Eller - @pellerphys). In the diagrams of the distribution density of the cosine of the zenith angle, given above (dashed line), the difference between the minimum and the maximum is almost 30%. However, the number of events with a given angle does not affect the average values obtained at a given angle.\n\nThe only reasonable explanation that we see so far is as follows. The cross section of the neutrino interaction reaction increases quite strongly with its energy. This means that the Earth absorbs high-energy neutrinos more strongly. Therefore, on average, neutrinos with lower energy enter the IceCube from below, and atmospheric neutrinos arriving from above have, on average, higher energy. Perhaps this is what increases the number of pulses at a small zenith, since more leptons with high energy are born, which generate more powerful Cherenkov radiation.\n\nLet's note other factors that make space anisotropic in the vertical direction. They are unlikely to explain the observed effect, but may be useful for understanding the general background of the problem.\n\n