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Refute my hypothesis of equal diagonals in A_optimal covariance.
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‎A_optimal_diagonals.ipynb

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"import diffnet as dn\n",
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"from cvxopt import matrix"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"import numpy as np"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"K = 5\n",
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"s = np.random.rand( K, K)\n",
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"s = 0.5*(s.T + s)\n",
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"# for i in range(K): s[i,i] = np.inf"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[0.31606839, 0.89819236, 0.89703422, 0.30095392, 0.54763611],\n",
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" [0.89819236, 0.70860277, 0.77928054, 0.47375313, 0.64210208],\n",
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" [0.89703422, 0.77928054, 0.8733601 , 0.08316998, 0.12200628],\n",
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" [0.30095392, 0.47375313, 0.08316998, 0.41130463, 0.29626953],\n",
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" [0.54763611, 0.64210208, 0.12200628, 0.29626953, 0.63852698]])"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"s"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" pcost dcost gap pres dres k/t\n",
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" 0: 0.0000e+00 -0.0000e+00 1e+02 4e+00 7e+02 1e+00\n",
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" 1: 3.3375e+00 3.6649e+00 1e+01 6e-01 1e+02 5e-01\n",
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" 2: 3.4839e+00 3.7049e+00 8e+00 4e-01 6e+01 3e-01\n",
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" 3: 3.4750e+00 3.5654e+00 3e+00 1e-01 2e+01 1e-01\n",
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" 4: 3.3033e+00 3.3647e+00 2e+00 7e-02 1e+01 8e-02\n",
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" 5: 3.3956e+00 3.4196e+00 1e+00 4e-02 6e+00 3e-02\n",
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" 6: 3.2196e+00 3.2352e+00 8e-01 2e-02 3e+00 2e-02\n",
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" 7: 3.1761e+00 3.1793e+00 2e-01 3e-03 6e-01 4e-03\n",
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" 8: 3.1702e+00 3.1708e+00 3e-02 6e-04 1e-01 7e-04\n",
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" 9: 3.1686e+00 3.1686e+00 4e-03 8e-05 1e-02 1e-04\n",
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"10: 3.1683e+00 3.1683e+00 4e-04 7e-06 1e-03 8e-06\n",
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"11: 3.1683e+00 3.1683e+00 8e-06 2e-07 3e-05 2e-07\n",
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"12: 3.1683e+00 3.1683e+00 6e-07 1e-08 2e-06 1e-08\n",
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"13: 3.1683e+00 3.1683e+00 4e-08 8e-10 1e-07 1e-09\n",
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"Optimal solution found.\n"
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]
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}
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],
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"source": [
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"n = dn.A_optimize( matrix(s))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[ 1.41e-01 1.94e-10 -3.02e-11 8.84e-02 4.21e-10]\n",
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"[ 1.94e-10 1.05e-01 9.93e-11 2.05e-01 5.44e-10]\n",
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"[-3.02e-11 9.93e-11 3.05e-11 6.61e-02 6.85e-02]\n",
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"[ 8.84e-02 2.05e-01 6.61e-02 3.26e-01 7.75e-11]\n",
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"[ 4.21e-10 5.44e-10 6.85e-02 7.75e-11 8.75e-10]\n",
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"\n"
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]
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}
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],
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"source": [
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"print(n)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"3.168258731683343"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"np.trace(dn.covariance( n/s**2))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[0.48064717, 0.12440851, 0.15276036, 0.15276036, 0.15276036],\n",
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" [0.12440851, 1.13881256, 0.30465855, 0.30465855, 0.30465855],\n",
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" [0.15276036, 0.30465855, 0.47877161, 0.37408816, 0.47877161],\n",
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" [0.15276036, 0.30465855, 0.37408816, 0.37408816, 0.37408816],\n",
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" [0.15276036, 0.30465855, 0.47877161, 0.37408816, 0.69593923]])"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"dn.covariance( n/s**2)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Clearly, the diagonal elements are not the same."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"s = np.ones( (K, K)) + 0.1*(np.random.rand( K, K) - 0.5)\n",
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"s = 0.5*(s + s.T)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[0.97351052, 0.96050154, 1.01524271, 1.00311652, 1.00580902],\n",
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" [0.96050154, 0.95744738, 1.00080002, 1.03008442, 0.97986122],\n",
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" [1.01524271, 1.00080002, 1.01532568, 1.01995957, 1.00801957],\n",
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" [1.00311652, 1.03008442, 1.01995957, 1.02903125, 1.01919704],\n",
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" [1.00580902, 0.97986122, 1.00801957, 1.01919704, 0.99439068]])"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"s"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" pcost dcost gap pres dres k/t\n",
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" 0: 0.0000e+00 -0.0000e+00 1e+02 4e+00 2e+01 1e+00\n",
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" 1: 4.1070e+00 4.4992e+00 2e+01 1e+00 4e+00 6e-01\n",
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" 2: 8.1702e+00 1.0088e+01 2e+02 2e+00 7e+00 2e+00\n",
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" 3: 1.3893e+01 1.4048e+01 1e+01 3e-01 1e+00 2e-01\n",
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" 4: 1.6870e+01 1.6949e+01 6e+00 1e-01 5e-01 1e-01\n",
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" 5: 1.8994e+01 1.9039e+01 3e+00 5e-02 2e-01 6e-02\n",
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" 6: 2.0385e+01 2.0392e+01 4e-01 7e-03 3e-02 8e-03\n",
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" 7: 2.0644e+01 2.0645e+01 5e-02 1e-03 4e-03 1e-03\n",
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" 8: 2.0685e+01 2.0685e+01 2e-03 3e-05 1e-04 4e-05\n",
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" 9: 2.0686e+01 2.0686e+01 1e-04 2e-06 7e-06 2e-06\n",
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"10: 2.0686e+01 2.0686e+01 4e-06 7e-08 3e-07 9e-08\n",
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"Optimal solution found.\n"
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]
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}
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],
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"source": [
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"n = dn.A_optimize( matrix( s))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([[4.04384771, 0.9226813 , 0.87485767, 0.94890855, 0.85672638],\n",
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" [0.9226813 , 3.97632199, 0.89016928, 0.82843275, 0.90830927],\n",
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" [0.87485767, 0.89016928, 4.23165798, 0.98485259, 0.94333631],\n",
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" [0.94890855, 0.82843275, 0.98485259, 4.29673937, 0.9415661 ],\n",
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" [0.85672638, 0.90830927, 0.94333631, 0.9415661 , 4.13744 ]])"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"dn.covariance( n/s**2)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The observed approximately equal diagonal elements may be attributable to similar $s_{ij}$ values in all the relative binding free energy calculations."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 2",
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"language": "python",
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"name": "python2"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 2
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython2",
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"version": "2.7.15"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}

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