I would like to explain mathematic background for Akio Saito's excellent notebook here.
\nI hope that understanding math for it may lead you to improve more sophisticated postprocessing!
This notebook aims to minimize the cost function below.
\n$$
\nL(X_{1:N}) = \\sum_{i=1}^{N} \\alpha_i | X_i - \\hat{X}i |^2 + \\sum{i=1}^{N-1} \\beta_i | (X_{i+1} - X_{i}) - \\Delta \\hat{X}_i |^2
\n$$
where \\(\\hat{X}_i\\) is absolute position predicted by your model and \\(\\Delta \\hat{X}_i\\) is relative position predicted by sensor data.
\nThe above function can be written as below by matrix.
\n$$
\n(X-\\hat{X})^T A (X-\\hat{X}) + (DX-\\Delta \\hat{X})^T B (DX-\\Delta \\hat{X})
\n$$
\nwhere
\n$$
\n A = \\left(
\n \\begin{array}{ccc}
\n \\alpha_1 & 0 & 0 \\\\
\n 0 & \\ddots & 0 \\\\
\n 0 & 0 & \\alpha_n
\n \\end{array}
\n \\right),
\n$$
$$
\n B = \\left(
\n \\begin{array}{ccc}
\n \\beta_1 & 0 & 0 \\\\
\n 0 & \\ddots & 0 \\\\
\n 0 & 0 & \\beta_{n-1}
\n \\end{array}
\n \\right),
\n$$
$$
\n D = \\left(
\n \\begin{array}{cccc}
\n -1 & 1 & 0 & 0\\\\
\n 0 & \\ddots & \\ddots & 0 \\\\
\n 0 & 0 & -1 & 1
\n \\end{array}
\n \\right),
\n$$
You can get optimal X for cost function by differentiate the above function and find an X which makes the function equals to 0.
\nThat is, you need to solve the below equation
\n$$
\n(A+A^T)(X-\\hat{X}) + D^T(B+B^T)(DX-\\Delta \\hat{X}) = 0.
\n$$
\nSolve it and you get
\n$$
\n(A + D^TBD)X = A\\hat{X}+D^TB\\Delta \\hat{X},
\n$$
\nwhich matches to Saito's notebook.
\nNote that \\(A = A^T, B = B^T \\).
I hope this explanation help you to improve your score!
", "messages": [ { "id": "1287689", "postDate": "04/29/2021 09:58:55", "content": "I would like to explain mathematic background for Akio Saito's excellent notebook here.
\nI hope that understanding math for it may lead you to improve more sophisticated postprocessing!
This notebook aims to minimize the cost function below.
\n$$
\nL(X_{1:N}) = \\sum_{i=1}^{N} \\alpha_i | X_i - \\hat{X}i |^2 + \\sum{i=1}^{N-1} \\beta_i | (X_{i+1} - X_{i}) - \\Delta \\hat{X}_i |^2
\n$$
where \\(\\hat{X}_i\\) is absolute position predicted by your model and \\(\\Delta \\hat{X}_i\\) is relative position predicted by sensor data.
\nThe above function can be written as below by matrix.
\n$$
\n(X-\\hat{X})^T A (X-\\hat{X}) + (DX-\\Delta \\hat{X})^T B (DX-\\Delta \\hat{X})
\n$$
\nwhere
\n$$
\n A = \\left(
\n \\begin{array}{ccc}
\n \\alpha_1 & 0 & 0 \\\\
\n 0 & \\ddots & 0 \\\\
\n 0 & 0 & \\alpha_n
\n \\end{array}
\n \\right),
\n$$
$$
\n B = \\left(
\n \\begin{array}{ccc}
\n \\beta_1 & 0 & 0 \\\\
\n 0 & \\ddots & 0 \\\\
\n 0 & 0 & \\beta_{n-1}
\n \\end{array}
\n \\right),
\n$$
$$
\n D = \\left(
\n \\begin{array}{cccc}
\n -1 & 1 & 0 & 0\\\\
\n 0 & \\ddots & \\ddots & 0 \\\\
\n 0 & 0 & -1 & 1
\n \\end{array}
\n \\right),
\n$$
You can get optimal X for cost function by differentiate the above function and find an X which makes the function equals to 0.
\nThat is, you need to solve the below equation
\n$$
\n(A+A^T)(X-\\hat{X}) + D^T(B+B^T)(DX-\\Delta \\hat{X}) = 0.
\n$$
\nSolve it and you get
\n$$
\n(A + D^TBD)X = A\\hat{X}+D^TB\\Delta \\hat{X},
\n$$
\nwhich matches to Saito's notebook.
\nNote that \\(A = A^T, B = B^T \\).
I hope this explanation help you to improve your score!
", "rawMarkdown": "I would like to explain mathematic background for [Akio Saito's excellent notebook](https://www.kaggle.com/saitodevel01/indoor-post-processing-by-cost-minimization) here.\nI hope that understanding math for it may lead you to improve more sophisticated postprocessing!\n\nThis notebook aims to minimize the cost function below.\n\n$$\nL(X_{1:N}) = \\sum_{i=1}^{N} \\alpha_i \\| X_i - \\hat{X}_i \\|^2 + \\sum_{i=1}^{N-1} \\beta_i \\| (X_{i+1} - X_{i}) - \\Delta \\hat{X}_i \\|^2\n$$\n\nwhere \\\\(\\hat{X}_i\\\\) is absolute position predicted by your model and \\\\(\\Delta \\hat{X}_i\\\\) is relative position predicted by sensor data.\n\nThe above function can be written as below by matrix.\n$$\n(X-\\hat{X})^T A (X-\\hat{X}) + (DX-\\Delta \\hat{X})^T B (DX-\\Delta \\hat{X})\n$$\nwhere\n$$\n A = \\left(\n \\begin{array}{ccc}\n \\alpha_1 & 0 & 0 \\\\\\\\\n 0 & \\ddots & 0 \\\\\\\\\n 0 & 0 & \\alpha_n\n \\end{array}\n \\right),\n$$\n\n$$\n B = \\left(\n \\begin{array}{ccc}\n \\beta_1 & 0 & 0 \\\\\\\\\n 0 & \\ddots & 0 \\\\\\\\\n 0 & 0 & \\beta_{n-1}\n \\end{array}\n \\right),\n$$\n\n$$\n D = \\left(\n \\begin{array}{cccc}\n -1 & 1 & 0 & 0\\\\\\\\\n 0 & \\ddots & \\ddots & 0 \\\\\\\\\n 0 & 0 & -1 & 1\n \\end{array}\n \\right),\n$$\n\nYou can get optimal X for cost function by differentiate the above function and find an X which makes the function equals to 0.\nThat is, you need to solve the below equation\n$$\n(A+A^T)(X-\\hat{X}) + D^T(B+B^T)(DX-\\Delta \\hat{X}) = 0.\n$$\nSolve it and you get\n$$\n(A + D^TBD)X = A\\hat{X}+D^TB\\Delta \\hat{X},\n$$\nwhich matches to Saito's notebook.\nNote that \\\\(A = A^T, B = B^T \\\\).\n\nI hope this explanation help you to improve your score!", "votes": null }, { "id": "1287702", "postDate": "04/29/2021 10:06:43", "content": "my latex display is sadly broken…
", "rawMarkdown": "my latex display is sadly broken...", "votes": null }, { "id": "1287824", "postDate": "04/29/2021 12:50:17", "content": "Thank you for the post. Is the latex broken when you preview the post? I just copied your latex and it looks good in preview mode.
", "rawMarkdown": "Thank you for the post. Is the latex broken when you preview the post? I just copied your latex and it looks good in preview mode.", "votes": null }, { "id": "1287940", "postDate": "04/29/2021 14:44:28", "content": "Thank you for investigation.
\nActually it looks broken in my preview mode and I have no idea what to do…
it is fixed though I don't know why.
", "rawMarkdown": "it is fixed though I don't know why.", "votes": null }, { "id": "1289746", "postDate": "05/01/2021 10:28:07", "content": "Thanks for the explanations. It seems that the last formula has a small error. It should rather be
\n$$
\n(A + D^TBD)X = A\\hat{X}+D^TB\\Delta \\hat{X}
\n$$
Thank you for your comment!
\nYou are right and it is important point.
\nI fixed it.
I was a bit confused by the following fact: ML-predicted coordinates are in the coordinate system of the building, however the deltaX predicted by the sensor are related to the compass direction. Can we compare them? Or did I misunderstand anything? Thanks!
", "rawMarkdown": "I was a bit confused by the following fact: ML-predicted coordinates are in the coordinate system of the building, however the deltaX predicted by the sensor are related to the compass direction. Can we compare them? Or did I misunderstand anything? Thanks!", "votes": null }, { "id": "1302889", "postDate": "05/11/2021 19:07:27", "content": "It looks like the floor plans are scaled such that they match the compass directions. So yes, I think they are directly comparable.
", "rawMarkdown": "It looks like the floor plans are scaled such that they match the compass directions. So yes, I think they are directly comparable.", "votes": null } ], "comments": [ { "id": 1287702, "author_name": "iwatatakuya", "author_url": "", "post_date": "04/29/2021 10:06:43", "content": "my latex display is sadly broken…
", "votes": null, "replies": [ { "id": 1287824, "author_name": "jiweiliu", "author_url": "", "post_date": "04/29/2021 12:50:17", "content": "Thank you for the post. Is the latex broken when you preview the post? I just copied your latex and it looks good in preview mode.
", "votes": null, "replies": [] }, { "id": 1287940, "author_name": "iwatatakuya", "author_url": "", "post_date": "04/29/2021 14:44:28", "content": "Thank you for investigation.
\nActually it looks broken in my preview mode and I have no idea what to do…
it is fixed though I don't know why.
", "votes": null, "replies": [] } ] }, { "id": 1289746, "author_name": "helgith", "author_url": "", "post_date": "05/01/2021 10:28:07", "content": "Thanks for the explanations. It seems that the last formula has a small error. It should rather be
\n$$
\n(A + D^TBD)X = A\\hat{X}+D^TB\\Delta \\hat{X}
\n$$
Thank you for your comment!
\nYou are right and it is important point.
\nI fixed it.
I was a bit confused by the following fact: ML-predicted coordinates are in the coordinate system of the building, however the deltaX predicted by the sensor are related to the compass direction. Can we compare them? Or did I misunderstand anything? Thanks!
", "votes": null, "replies": [ { "id": 1302889, "author_name": "tvdwiele", "author_url": "", "post_date": "05/11/2021 19:07:27", "content": "It looks like the floor plans are scaled such that they match the compass directions. So yes, I think they are directly comparable.
", "votes": null, "replies": [] } ] } ], "raw_markdown_by_id": { "1287689": "I would like to explain mathematic background for [Akio Saito's excellent notebook](https://www.kaggle.com/saitodevel01/indoor-post-processing-by-cost-minimization) here.\nI hope that understanding math for it may lead you to improve more sophisticated postprocessing!\n\nThis notebook aims to minimize the cost function below.\n\n$$\nL(X_{1:N}) = \\sum_{i=1}^{N} \\alpha_i \\| X_i - \\hat{X}_i \\|^2 + \\sum_{i=1}^{N-1} \\beta_i \\| (X_{i+1} - X_{i}) - \\Delta \\hat{X}_i \\|^2\n$$\n\nwhere \\\\(\\hat{X}_i\\\\) is absolute position predicted by your model and \\\\(\\Delta \\hat{X}_i\\\\) is relative position predicted by sensor data.\n\nThe above function can be written as below by matrix.\n$$\n(X-\\hat{X})^T A (X-\\hat{X}) + (DX-\\Delta \\hat{X})^T B (DX-\\Delta \\hat{X})\n$$\nwhere\n$$\n A = \\left(\n \\begin{array}{ccc}\n \\alpha_1 & 0 & 0 \\\\\\\\\n 0 & \\ddots & 0 \\\\\\\\\n 0 & 0 & \\alpha_n\n \\end{array}\n \\right),\n$$\n\n$$\n B = \\left(\n \\begin{array}{ccc}\n \\beta_1 & 0 & 0 \\\\\\\\\n 0 & \\ddots & 0 \\\\\\\\\n 0 & 0 & \\beta_{n-1}\n \\end{array}\n \\right),\n$$\n\n$$\n D = \\left(\n \\begin{array}{cccc}\n -1 & 1 & 0 & 0\\\\\\\\\n 0 & \\ddots & \\ddots & 0 \\\\\\\\\n 0 & 0 & -1 & 1\n \\end{array}\n \\right),\n$$\n\nYou can get optimal X for cost function by differentiate the above function and find an X which makes the function equals to 0.\nThat is, you need to solve the below equation\n$$\n(A+A^T)(X-\\hat{X}) + D^T(B+B^T)(DX-\\Delta \\hat{X}) = 0.\n$$\nSolve it and you get\n$$\n(A + D^TBD)X = A\\hat{X}+D^TB\\Delta \\hat{X},\n$$\nwhich matches to Saito's notebook.\nNote that \\\\(A = A^T, B = B^T \\\\).\n\nI hope this explanation help you to improve your score!", "1287702": "my latex display is sadly broken...", "1287824": "Thank you for the post. Is the latex broken when you preview the post? I just copied your latex and it looks good in preview mode.", "1287940": "Thank you for investigation.\nActually it looks broken in my preview mode and I have no idea what to do...", "1288039": "it is fixed though I don't know why.", "1289746": "Thanks for the explanations. It seems that the last formula has a small error. It should rather be\n\n$$\n(A + D^TBD)X = A\\hat{X}+D^TB\\Delta \\hat{X}\n$$", "1290037": "Thank you for your comment!\nYou are right and it is important point.\nI fixed it.", "1302883": "I was a bit confused by the following fact: ML-predicted coordinates are in the coordinate system of the building, however the deltaX predicted by the sensor are related to the compass direction. Can we compare them? Or did I misunderstand anything? Thanks!", "1302889": "It looks like the floor plans are scaled such that they match the compass directions. So yes, I think they are directly comparable." }, "source": "meta" }