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HYChou0515/bayes_flow.ipynb

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Use optimization and bayes rule to calculate probability for each step of flows.
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bayes_flow.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "5ccb721a-5978-4809-a8ae-916ccf838ed3",
"metadata": {},
"outputs": [],
"source": [
"import scipy as sp\n",
"from scipy import optimize\n",
"import numpy as np\n",
"import pandas as pd\n",
"import itertools as it\n",
"\n",
"rng = np.random.default_rng(0)"
]
},
{
"cell_type": "markdown",
"id": "31b8a5fa-3bb2-4b6b-b896-0831d881b2a6",
"metadata": {
"tags": []
},
"source": [
"## Naive Bayes\n",
"This example shows if we see each step independent to other steps, we get some trouble.."
]
},
{
"cell_type": "markdown",
"id": "01f8f71d-ad94-4440-9603-269c9303afc2",
"metadata": {
"tags": []
},
"source": [
"### Indenpent flow"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "9a7856fa-e090-49fe-bc08-ddb97e5fb09f",
"metadata": {
"tags": []
},
"outputs": [],
"source": [
"tool_prob = [[p/sum(ps) for p in ps] for ps in [[30,2,1],[3,1],[9,1,1]]]\n",
"# tool_prob = [[p/sum(ps) for p in ps] for ps in [[1,1,1],[1,1],[1,1,1]]]\n",
"defect_prob = [[.6,0,0],[0,.2],[0,0,0]]\n",
"n_trial = 10000"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "dfa4b87d-1fde-492d-abbc-d169a86c3d0b",
"metadata": {},
"outputs": [],
"source": [
"def trial():\n",
" assert len(tool_prob) == len(defect_prob)\n",
" assert [len(ps) for ps in tool_prob] == [len(ps) for ps in defect_prob]\n",
" good_prob = [[1-p for p in ps] for ps in defect_prob]\n",
" result = []\n",
" gp = 1.0\n",
" for step in range(len(tool_prob)):\n",
" tps = tool_prob[step]\n",
" gps = good_prob[step]\n",
" tid = rng.choice(np.arange(len(tps)), p=tps)\n",
" gp *= gps[tid]\n",
" result.append(tid)\n",
" result.append(1 if rng.random() >= gp else 0)\n",
" return result"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "e7ee07de-22b0-4f85-851d-f5c09bfa5e27",
"metadata": {
"tags": []
},
"outputs": [
{
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"<div>\n",
"<style scoped>\n",
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"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
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" <th></th>\n",
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" <th>9996</th>\n",
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" <tr>\n",
" <th>9997</th>\n",
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" <th>9999</th>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"<p>10000 rows × 3 columns</p>\n",
"</div>"
],
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"\n",
"[10000 rows x 3 columns]"
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"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df = pd.DataFrame([trial() for _ in range(n_trial)])\n",
"defect = df.iloc[:,-1]\n",
"df = df.drop(columns=df.columns[-1])\n",
"df"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "95b689a3-656d-4c4e-90d1-8ab77898ebe1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[[0.6216989436619718, 0.0556464811783961, 0.04234527687296419],\n",
" [0.543721674220208, 0.6462829736211031],\n",
" [0.5704550985641275, 0.5609220636663008, 0.5678449258836944]]"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"naive_bayes = []\n",
"for step in range(len(tool_prob)):\n",
" ps = []\n",
" for tid in range(len(tool_prob[step])):\n",
" dom = sum(df[step]==tid)+2\n",
" num = sum((df[step]==tid) & (defect==0))+1\n",
" if dom == 0:\n",
" ps.append(np.nan)\n",
" else:\n",
" ps.append(1-num/dom)\n",
" naive_bayes.append(ps)\n",
"naive_bayes"
]
},
{
"cell_type": "markdown",
"id": "0a9a7e23-2d87-48c5-8287-309a62fdaded",
"metadata": {},
"source": [
"We see that even tool_00 is worse than tool_11, but we estimated tool_11 worse than tool_00\n",
"\n",
"This is because some bad result from tool_11 should be blamed on tool_00.\n",
"\n",
"Using this method, probability across steps cannot be compared."
]
},
{
"cell_type": "markdown",
"id": "daa44038-0bd2-40d1-b45e-4f9d5b897670",
"metadata": {
"tags": []
},
"source": [
"### Predefined Flow"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "d220ee43-46fa-4a70-afb7-fd2b1bbd5b3e",
"metadata": {},
"outputs": [],
"source": [
"tool_choice = [3,2,3]\n",
"all_flows = list(it.product(*(range(t) for t in tool_choice)))\n",
"flows = [*all_flows]\n",
"rng.shuffle(flows)\n",
"flows = flows[:8]\n",
"flow_prob = [1,1,3,1,2,14,1,1]\n",
"flow_prob = [p/sum(flow_prob) for p in flow_prob]\n",
"\n",
"defect_prob = [[.6,0,0],[0,.2],[0,0,0]]\n",
"n_trial = 10000"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "51afe661-d3a2-4331-8757-3e3aad457cc5",
"metadata": {},
"outputs": [],
"source": [
"def trial():\n",
" assert len(flow_prob) == len(flows)\n",
" good_prob = [[1-p for p in ps] for ps in defect_prob]\n",
" result = []\n",
" gp = 1.0\n",
" fl = rng.choice(flows, p=flow_prob)\n",
" for step, tid in enumerate(fl):\n",
" gp *= good_prob[step][tid]\n",
" result.append(tid)\n",
" result.append(1 if rng.random() >= gp else 0)\n",
" return result"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "dd73ebb0-ff30-425c-b4a4-987e07c29000",
"metadata": {
"tags": []
},
"outputs": [
{
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"<p>10000 rows × 3 columns</p>\n",
"</div>"
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" 0 1 2\n",
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"\n",
"[10000 rows x 3 columns]"
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"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"df = pd.DataFrame([trial() for _ in range(n_trial)])\n",
"defect = df.iloc[:,-1]\n",
"df = df.drop(columns=df.columns[-1])\n",
"df"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "ca314f3d-bd00-4ab2-bf1c-6c7bd8b0adfe",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[[0.6122881355932204, 0.0005882352941176672, 0.10718492343934038],\n",
" [0.42633443822151107, 0.3968127490039841],\n",
" [0.5628327149540249, 0.36520737327188935, 0.14392650561415443]]"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"naive_bayes = []\n",
"for step in range(len(tool_choice)):\n",
" ps = []\n",
" for tid in range(tool_choice[step]):\n",
" dom = sum(df[step]==tid)+2\n",
" num = sum((df[step]==tid) & (defect==0))+1\n",
" if dom == 0:\n",
" ps.append(np.nan)\n",
" else:\n",
" ps.append(1-num/dom)\n",
" naive_bayes.append(ps)\n",
"naive_bayes"
]
},
{
"cell_type": "markdown",
"id": "842f4206-5bb5-4537-b204-b5df93a1ee3e",
"metadata": {},
"source": [
"## Optimization"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "c33a9667-4a8c-48b6-ab19-b8c6aeb18c97",
"metadata": {
"tags": []
},
"outputs": [
{
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" </tr>\n",
" </tbody>\n",
"</table>\n",
"<p>1000 rows × 3 columns</p>\n",
"</div>"
],
"text/plain": [
" 0 1 2\n",
"0 0 0 2\n",
"1 0 0 2\n",
"2 0 0 0\n",
"3 0 0 0\n",
"4 0 0 0\n",
".. .. .. ..\n",
"995 0 0 2\n",
"996 0 0 0\n",
"997 2 0 0\n",
"998 0 0 0\n",
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"\n",
"[1000 rows x 3 columns]"
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},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\n",
"n_trial = 1000\n",
"df = pd.DataFrame([trial() for _ in range(n_trial)])\n",
"defect = df.iloc[:,-1]\n",
"df = df.drop(columns=df.columns[-1])\n",
"df"
]
},
{
"cell_type": "markdown",
"id": "90aa8115-baa6-4dcb-a971-6ce4e52b1f34",
"metadata": {},
"source": [
"`good_prob_each_flow` is the good probability for each flow based on the observed samples.\n",
"\n",
"Later we will use this as the target to assign each tool the bad probability through optimization."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "42405712-0cd2-4c6d-9e69-09c2a6792875",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"bad trial/ total trial for each flow\n"
]
},
{
"data": {
"text/plain": [
"flows\n",
"(0, 0, 0) 0.439655\n",
"(0, 0, 2) 0.415094\n",
"(0, 1, 1) 0.279070\n",
"(1, 0, 1) 1.000000\n",
"(1, 0, 2) 1.000000\n",
"(2, 0, 0) 1.000000\n",
"(2, 0, 2) 1.000000\n",
"(2, 1, 2) 0.880597\n",
"Name: defect, dtype: float64"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"rearrange to a 1D array\n"
]
},
{
"data": {
"text/plain": [
"array([0.43965517, nan, 0.41509434, nan, 0.27906977,\n",
" nan, nan, 1. , 1. , nan,\n",
" nan, nan, 1. , nan, 1. ,\n",
" nan, nan, 0.88059701])"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"flow_col_name = 'flows'\n",
"defect.name = 'defect'\n",
"_concats = pd.concat([df.apply(tuple, axis=1).rename(flow_col_name), defect], axis=1).groupby(flow_col_name)[defect.name]\n",
"df_good_prob_each_flow = 1-_concats.sum()/_concats.count()\n",
"print(\"bad trial/ total trial for each flow\")\n",
"display(df_good_prob_each_flow)\n",
"\n",
"print(\"rearrange to a 1D array\")\n",
"df_good_prob_each_flow = df_good_prob_each_flow.reset_index()\n",
"df_good_prob_each_flow[flow_col_name] = df_good_prob_each_flow[flow_col_name].apply(all_flows.index)\n",
"good_prob_each_flow = np.zeros(shape=(len(all_flows),))+np.nan\n",
"good_prob_each_flow[df_good_prob_each_flow[flow_col_name]] = df_good_prob_each_flow[defect.name]\n",
"good_prob_each_flow"
]
},
{
"cell_type": "markdown",
"id": "c3e36aba-e28c-40ab-8977-d83523d48ffa",
"metadata": {},
"source": [
"$A \\in \\{0,1\\}^{N \\times M}$ is an indicater matrix, where $N$ is the number of total possible flows, and $M$ is the number of all tools.\n",
"\n",
"Each row represent a possible choice of tools of a flow.\n",
"\n",
"For example, $A[1]$ is $[1,0,0,1,0,0,1,0]$, means flow[0] is tool_00 -> tool_10 -> tool_21.\n",
"\n",
"Note that the 2nd 1 in the row is in the 4th element of the row, which means the 1st tool in the 2nd step as the 1st step has 3 choices of tool."
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "82338f07-1408-4244-b240-e9dad1866e12",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1., 0., 0., 1., 0., 1., 0., 0.],\n",
" [1., 0., 0., 1., 0., 0., 1., 0.],\n",
" [1., 0., 0., 1., 0., 0., 0., 1.],\n",
" [1., 0., 0., 0., 1., 1., 0., 0.],\n",
" [1., 0., 0., 0., 1., 0., 1., 0.],\n",
" [1., 0., 0., 0., 1., 0., 0., 1.],\n",
" [0., 1., 0., 1., 0., 1., 0., 0.],\n",
" [0., 1., 0., 1., 0., 0., 1., 0.],\n",
" [0., 1., 0., 1., 0., 0., 0., 1.],\n",
" [0., 1., 0., 0., 1., 1., 0., 0.],\n",
" [0., 1., 0., 0., 1., 0., 1., 0.],\n",
" [0., 1., 0., 0., 1., 0., 0., 1.],\n",
" [0., 0., 1., 1., 0., 1., 0., 0.],\n",
" [0., 0., 1., 1., 0., 0., 1., 0.],\n",
" [0., 0., 1., 1., 0., 0., 0., 1.],\n",
" [0., 0., 1., 0., 1., 1., 0., 0.],\n",
" [0., 0., 1., 0., 1., 0., 1., 0.],\n",
" [0., 0., 1., 0., 1., 0., 0., 1.]])"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = np.zeros(shape=(len(all_flows),sum(tool_choice)))\n",
"for i, fl in enumerate(all_flows):\n",
" s = 0\n",
" for j, f in enumerate(fl):\n",
" A[i,s+f] = 1\n",
" s += tool_choice[j]\n",
"A"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "ff74b8f7-e784-4fe5-9e63-7d81c223d565",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1., 0., 0., 1., 0., 1., 0., 0.],\n",
" [1., 0., 0., 1., 0., 0., 0., 1.],\n",
" [1., 0., 0., 0., 1., 0., 1., 0.],\n",
" [0., 1., 0., 1., 0., 0., 1., 0.],\n",
" [0., 1., 0., 1., 0., 0., 0., 1.],\n",
" [0., 0., 1., 1., 0., 1., 0., 0.],\n",
" [0., 0., 1., 1., 0., 0., 0., 1.],\n",
" [0., 0., 1., 0., 1., 0., 0., 1.]])"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"array([-8.21764556e-01, -8.79249458e-01, -1.27629346e+00, 1.00000008e-09,\n",
" 1.00000008e-09, 1.00000008e-09, 1.00000008e-09, -1.27155174e-01])"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# use the log-likelihood instead of product of probability\n",
"c = np.log(good_prob_each_flow+1e-9)\n",
"# drop flow doesn't exist in the samples for efficiency\n",
"A = A[~np.isnan(c)]\n",
"c = c[np.nonzero(~np.isnan(c))[0]]\n",
"display(A)\n",
"display(c)"
]
},
{
"cell_type": "markdown",
"id": "0c4cf8b7-be39-4ca4-a5c9-e81ba0bfb9b9",
"metadata": {},
"source": [
"The objective is based on $x \\in \\mathbb{R}^M$, the hypothesized bad probability for each tool.\n",
"\n",
"Given $x$, we can find the theoretical bad probability for each flow, and then compare with the sampled probability $c$."
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "47a3a5bb-9b31-44ef-ad33-b5104d9ef8d5",
"metadata": {},
"outputs": [],
"source": [
"def opt(x):\n",
" y = A@x-c\n",
" return np.sqrt(sum(y**2)/y.shape[0])"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "1ca06a9d-57f8-4286-bbb7-f992ed411c5f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.6218390100158231"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# test\n",
"opt(np.zeros(A.shape[1]))"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "63095461-3915-45c2-8098-472b474f778b",
"metadata": {},
"outputs": [],
"source": [
"res = optimize.minimize(opt, np.zeros(A.shape[1]), constraints=optimize.LinearConstraint(np.eye(A.shape[1]), lb=-np.inf, ub=0))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "046e274b-1b3f-40b0-9f4e-540eec078d27",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[[0.6, 0, 0], [0, 0.2], [0, 0, 0]]"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"defect_prob"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "b31656a7-cfe3-4d32-bbdd-c4ef8607a4a8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([0.40935805, 1. , 1. , 1. , 0.8084688 ,\n",
" 1. , 0.91830942, 1. ])"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.exp(res.x)\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "63a3f7a1-f6f7-4069-a2ba-eaad0a4851c6",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.12"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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