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markov-models/markov.ipynb

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beginning markov code
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{
"metadata": {
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
finished code, beginning writeup
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"version": "3.8.6-final"
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},
"orig_nbformat": 2,
"kernelspec": {
"name": "python3",
"display_name": "Python 3",
"language": "python"
}
},
"nbformat": 4,
"nbformat_minor": 2,
"cells": [
{
"cell_type": "code",
working on forward/backward procedure
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"execution_count": 1,
beginning markov code
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"metadata": {},
"outputs": [],
"source": [
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"#IMPORTS AND COMMON VARIABLES\n",
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"import matplotlib.pyplot as plt\n",
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"from matplotlib import cm\n",
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"import numpy as np\n",
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"from math import sqrt\n",
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"\n",
"from constants import *\n",
"from maths import gaussian\n",
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"from markov import MarkovModel\n",
"from markovlog import LogMarkovModel\n",
"\n",
"x = np.linspace(-4, 8, 120) # x values for figures\n",
"x_label = \"Observation Space\"\n",
"y_label = \"Probability Density\""
beginning markov code
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]
},
{
"source": [
"State Probability Functions (1)\n",
"==================="
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
working on forward/backward procedure
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"execution_count": 2,
beginning markov code
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"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "<Figure size 432x288 with 1 Axes>",
finished code, beginning writeup
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beginning markov code
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"state_1_y = [gaussian(i, state1.mean, state1.std_dev) for i in x]\n",
"state_2_y = [gaussian(i, state2.mean, state2.std_dev) for i in x]\n",
"\n",
"plt.plot(x, state_1_y, c='r', label=\"State 1\")\n",
"plt.plot(x, state_2_y, c='b', label=\"State 2\")\n",
"\n",
"plt.legend()\n",
"plt.title(\"State Probability Density Functions\")\n",
"\n",
finished code, beginning writeup
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"plt.xlabel(x_label)\n",
"plt.ylabel(y_label)\n",
beginning markov code
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"plt.grid(linestyle=\"--\")\n",
"\n",
"plt.show()"
]
},
{
"source": [
"Output Probability Densities (2)\n",
"=========="
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
working on forward/backward procedure
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"execution_count": 3,
beginning markov code
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"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"3.8 -> State 1: 0.02185157424475792, State 2: 0.5471239427774459\n4.2 -> State 1: 0.009496655019831194, State 2: 0.5471239427774459\n3.4 -> State 1: 0.04499247209432338, State 2: 0.3947074079064296\n-0.4 -> State 1: 0.16833223796171576, State 2: 1.500652790137751e-09\n1.9 -> State 1: 0.2509478601290037, State 2: 0.006331212017054291\n3.0 -> State 1: 0.08289761566062391, State 2: 0.2054255182126689\n1.6 -> State 1: 0.2933877723035829, State 2: 0.001596702666402633\n1.9 -> State 1: 0.2509478601290037, State 2: 0.006331212017054291\n5.0 -> State 1: 0.0012852324969092556, State 2: 0.2054255182126689\n"
]
}
],
"source": [
"for obs in observations:\n",
" print(f'{obs} -> State 1: {gaussian(obs, state1.mean, state1.std_dev)}, State 2: {gaussian(obs, state2.mean, state2.std_dev)}')"
]
},
{
"cell_type": "code",
working on forward/backward procedure
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"execution_count": 4,
beginning markov code
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"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": "<Figure size 432x288 with 1 Axes>",
finished code, beginning writeup
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beginning markov code
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"state_1_y = [gaussian(i, state1.mean, state1.std_dev) for i in x]\n",
"state_2_y = [gaussian(i, state2.mean, state2.std_dev) for i in x]\n",
"\n",
"plt.plot(x, state_1_y, c='r', label=\"State 1\")\n",
"plt.plot(x, state_2_y, c='b', label=\"State 2\")\n",
"\n",
"plt.legend()\n",
finished code, beginning writeup
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"plt.title(\"State Probability Density Functions With Observations\")\n",
beginning markov code
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"\n",
finished code, beginning writeup
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"plt.xlabel(x_label)\n",
"plt.ylabel(y_label)\n",
beginning markov code
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"plt.grid(linestyle=\"--\")\n",
"\n",
"state1_pd = [gaussian(i, state1.mean, state1.std_dev) for i in observations]\n",
"state2_pd = [gaussian(i, state2.mean, state2.std_dev) for i in observations]\n",
"\n",
"#############################################\n",
"# Observation Marks \n",
"#############################################\n",
"\n",
"config = {\n",
" \"s\": 65,\n",
" \"marker\": 'x'\n",
"}\n",
"\n",
"plt.scatter(observations, state1_pd, color=(0.5, 0, 0), **config)\n",
"plt.scatter(observations, state2_pd, color=(0, 0, 0.5), **config)\n",
"\n",
"plt.show()"
]
},
working on forward/backward procedure
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{
"source": [
"# Forward Procedure (3)"
],
"cell_type": "markdown",
"metadata": {}
},
beginning markov code
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{
"cell_type": "code",
working on forward/backward procedure
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"execution_count": 5,
beginning markov code
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"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
finished code, beginning writeup
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"[[9.61469267e-03 2.00389806e-04 2.89433448e-04 4.30961030e-04\n 9.94968491e-05 7.58875028e-06 2.06308972e-06 4.76335758e-07\n 5.63267096e-10]\n [3.06389408e-01 1.56214298e-01 5.73475605e-02 8.00607227e-11\n 1.63710810e-07 1.25762776e-06 2.59451223e-09 7.98988065e-10\n 6.02373446e-09]]\n"
beginning markov code
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]
},
{
"output_type": "execute_result",
"data": {
"text/plain": [
finished code, beginning writeup
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"1.9197737567283167e-10"
beginning markov code
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]
},
"metadata": {},
working on forward/backward procedure
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"execution_count": 5
beginning markov code
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}
],
"source": [
"model = MarkovModel(states=[state1, state2], observations=observations, state_transitions=state_transition)\n",
"model.populate_forward()\n",
"\n",
finished code, beginning writeup
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"print(model.forward)\n",
beginning markov code
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"\n",
finished code, beginning writeup
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"forward = model.forward\n",
"model.calculate_p_obs_forward()\n"
beginning markov code
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]
},
working on forward/backward procedure
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{
"source": [
"# Backward Procedure (4)"
],
"cell_type": "markdown",
"metadata": {}
},
beginning markov code
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{
"cell_type": "code",
working on forward/backward procedure
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"execution_count": 6,
beginning markov code
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"metadata": {},
working on forward/backward procedure
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"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
finished code, beginning writeup
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"[[6.57892066e-11 2.92661737e-09 6.89869860e-08 4.45463420e-07\n 1.92898428e-06 2.51074677e-05 9.30059204e-05 3.93414211e-04\n 2.00000000e-02]\n [6.24515168e-10 1.22518178e-09 2.99943420e-09 2.11439342e-08\n 3.02470605e-07 1.14745634e-06 3.77015860e-05 5.73240014e-03\n 3.00000000e-02]]\n"
working on forward/backward procedure
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]
},
{
"output_type": "execute_result",
"data": {
"text/plain": [
finished code, beginning writeup
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"1.9197737567283167e-10"
working on forward/backward procedure
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]
},
"metadata": {},
"execution_count": 6
}
],
"source": [
"model = MarkovModel(states=[state1, state2], observations=observations, state_transitions=state_transition)\n",
"model.populate_backward()\n",
"\n",
initial occupation and transition likelihoods
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"print(model.backward)\n",
finished code, beginning writeup
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"\n",
"backward = model.backward\n",
initial occupation and transition likelihoods
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"model.calculate_p_obs_backward()\n"
working on forward/backward procedure
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]
},
{
"source": [
"# Compare Forward/Backward Final"
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
finished code, beginning writeup
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"forward: 1.9197737567283167e-10\nbackward: 1.9197737567283167e-10\ndiff: 0.0\n"
working on forward/backward procedure
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]
}
],
"source": [
"model = MarkovModel(states=[state1, state2], observations=observations, state_transitions=state_transition)\n",
"model.populate_forward()\n",
"model.populate_backward()\n",
"\n",
initial occupation and transition likelihoods
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"print(\"forward:\", model.calculate_p_obs_forward())\n",
finished code, beginning writeup
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"print(\"backward:\", model.calculate_p_obs_backward())\n",
"\n",
"print(\"diff: \", model.p_obs_forward - model.p_obs_backward)\n"
working on forward/backward procedure
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]
initial occupation and transition likelihoods
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},
{
"source": [
"# Occupation Likelihoods (5)"
],
"cell_type": "markdown",
"metadata": {}
},
finished code, beginning writeup
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{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[[3.29488306e-03 3.05486147e-03 1.04007783e-01 9.99999991e-01\n 9.99742065e-01 9.92483109e-01 9.99490475e-01 9.76142401e-01\n 5.86805704e-02]\n [9.96705117e-01 9.96945139e-01 8.95992217e-01 8.81769869e-09\n 2.57935122e-04 7.51689067e-03 5.09524759e-04 2.38575993e-02\n 9.41319430e-01]]\n"
]
}
],
"source": [
"model = MarkovModel(states=[state1, state2], observations=observations, state_transitions=state_transition).populate()\n",
"\n",
"occupation = model.occupation\n",
"print(model.occupation)"
]
},
{
"source": [
"# Re-estimate Mean & Variance (6)"
],
"cell_type": "markdown",
"metadata": {}
},
initial occupation and transition likelihoods
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{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
finished code, beginning writeup
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"mean: [1, 4]\nvariance: [1.44, 0.49]\n\nmean: [1.6747846620246418, 4.089148442531048]\nvariance: [1.8465047320872243, 0.37804421554121403]\n"
initial occupation and transition likelihoods
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]
}
],
"source": [
finished code, beginning writeup
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"model = MarkovModel(states=[state1, state2], observations=observations, state_transitions=state_transition).populate()\n",
"\n",
"print(\"mean: \", [state1.mean, state2.mean])\n",
"print(\"variance: \", [state1.variance, state2.variance])\n",
"print()\n",
"\n",
"print(\"mean: \", model.reestimated_mean())\n",
"print(\"variance: \", model.reestimated_variance())\n"
]
},
{
"source": [
"New PDFs (7)\n",
"==================="
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"data": {
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"model = MarkovModel(states=[state1, state2], observations=observations, state_transitions=state_transition).populate()\n",
initial occupation and transition likelihoods
2020-12-24 14:58:45 +00:00
"\n",
finished code, beginning writeup
2020-12-31 19:30:39 +00:00
"new_mean = model.reestimated_mean()\n",
"new_var = model.reestimated_variance()\n",
"new_std_dev = [sqrt(x) for x in new_var]\n",
initial occupation and transition likelihoods
2020-12-24 14:58:45 +00:00
"\n",
finished code, beginning writeup
2020-12-31 19:30:39 +00:00
"state_1_y = [gaussian(i, new_mean[0], new_std_dev[0]) for i in x]\n",
"state_2_y = [gaussian(i, new_mean[1], new_std_dev[1]) for i in x]\n",
initial occupation and transition likelihoods
2020-12-24 14:58:45 +00:00
"\n",
finished code, beginning writeup
2020-12-31 19:30:39 +00:00
"plt.plot(x, state_1_y, c='r', label=\"State 1\")\n",
"plt.plot(x, state_2_y, c='b', label=\"State 2\")\n",
"\n",
"plt.legend()\n",
"plt.title(\"Re-estimated Probability Density Functions\")\n",
"\n",
"plt.xlabel(x_label)\n",
"plt.ylabel(y_label)\n",
"plt.grid(linestyle=\"--\")\n",
"\n",
"plt.show()"
]
},
{
"source": [
"# Compare PDFs (7)"
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"output_type": "display_data",
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"model = MarkovModel(states=[state1, state2], observations=observations, state_transitions=state_transition).populate()\n",
"\n",
"new_mean = model.reestimated_mean()\n",
"new_var = model.reestimated_variance()\n",
"new_std_dev = [sqrt(x) for x in new_var]\n",
"\n",
"#######################################\n",
"# Original\n",
"#######################################\n",
"state_1_y = [gaussian(i, state1.mean, state1.std_dev) for i in x]\n",
"state_2_y = [gaussian(i, state2.mean, state2.std_dev) for i in x]\n",
"plt.plot(x, state_1_y, '--', c='r', label=\"State 1\", linewidth=1.0)\n",
"plt.plot(x, state_2_y, '--', c='b', label=\"State 2\", linewidth=1.0)\n",
"\n",
"#######################################\n",
"# Re-Estimated\n",
"#######################################\n",
"state_1_new_y = [gaussian(i, new_mean[0], new_std_dev[0]) for i in x]\n",
"state_2_new_y = [gaussian(i, new_mean[1], new_std_dev[1]) for i in x]\n",
"plt.plot(x, state_1_new_y, c='r', label=\"New State 1\")\n",
"plt.plot(x, state_2_new_y, c='b', label=\"New State 2\")\n",
"\n",
"plt.legend()\n",
"plt.title(\"Re-estimated Probability Density Functions\")\n",
"\n",
"plt.xlabel(x_label)\n",
"plt.ylabel(y_label)\n",
"plt.grid(linestyle=\"--\")\n",
"\n",
"plt.show()"
]
},
{
"source": [
"# Multiple Iterations"
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"mean (0): [1.6747846620246418, 4.089148442531048]\nvar (0): [1.8465047320872243, 0.37804421554121403]\n\nmean (1): [1.9929120009112011, 4.056666511439969]\nvar (1): [2.1156649116379684, 0.2918412170796056]\n\nmean (2): [2.1821733376587726, 3.9904324818569052]\nvar (2): [2.3495798563848562, 0.23463095910597057]\n\nmean (3): [2.2693898497202887, 3.9240609125392]\nvar (3): [2.461617864622651, 0.17784017241318095]\n\nmean (4): [2.301476335144215, 3.8740199830675355]\nvar (4): [2.53136226603993, 0.13079227312275157]\n\n"
]
},
{
"output_type": "display_data",
"data": {
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"iterations = 5\n",
"\n",
"mean = [state1.mean, state2.mean]\n",
"var = [state1.variance, state2.variance]\n",
"\n",
"plt.plot(x, [gaussian(i, mean[0], sqrt(var[0])) for i in x], '--', c='r', linewidth=1.0)\n",
"plt.plot(x, [gaussian(i, mean[1], sqrt(var[1])) for i in x], '--', c='b', linewidth=1.0)\n",
"\n",
"for i in range(iterations):\n",
" model = MarkovModel(states=[State(mean[0], var[0], state1.entry, state1.exit), State(mean[1], var[1], state2.entry, state2.exit)], \n",
" observations=observations, \n",
" state_transitions=state_transition)\n",
" model.populate()\n",
"\n",
" mean = model.reestimated_mean()\n",
" var = model.reestimated_variance()\n",
"\n",
" print(f\"mean ({i}): \", mean)\n",
" print(f\"var ({i}): \", var)\n",
" print()\n",
"\n",
" state_1_y = [gaussian(i, mean[0], sqrt(var[0])) for i in x]\n",
" state_2_y = [gaussian(i, mean[1], sqrt(var[1])) for i in x]\n",
"\n",
" style = '--'\n",
" linewidth = 1.0\n",
" if i == iterations - 1:\n",
" style = '-'\n",
" linewidth = 2.0\n",
"\n",
" plt.plot(x, state_1_y, style, c='r', linewidth=linewidth)\n",
" plt.plot(x, state_2_y, style, c='b', linewidth=linewidth)\n",
"\n",
"plt.title(\"Probability Density Function Iterations\")\n",
"\n",
"plt.xlabel(x_label)\n",
"plt.ylabel(y_label)\n",
"plt.grid(linestyle=\"--\")\n",
"\n",
"plt.show()\n"
]
},
{
"source": [
"# Baum-Welch State Transition Re-estimations"
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[[0.92 0.06]\n [0.04 0.93]]\n"
]
},
{
"output_type": "execute_result",
"data": {
"text/plain": [
"array([[0.8035064 , 0.18507025],\n",
" [0.26043109, 0.49589971]])"
]
},
"metadata": {},
"execution_count": 13
}
],
"source": [
"model = MarkovModel(states=[state1, state2], observations=observations, state_transitions=state_transition).populate()\n",
"\n",
"print(a_matrix)\n",
"model.reestimated_state_transitions()"
initial occupation and transition likelihoods
2020-12-24 14:58:45 +00:00
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
beginning markov code
2020-12-11 21:33:20 +00:00
}
]
}
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