examples/standard_msm.ipynb

{
  "cells": [
    {
      "cell_type": "code",
      "source": [
        "import Markov_Models as mm\n",
        "\n",
        "import numpy as np\n",
        "%matplotlib inline\n",
        "import matplotlib.pyplot as plt"
      ],
      "outputs": [],
      "execution_count": 1,
      "metadata": {
        "collapsed": true
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Generate  random data\n",
        "\nLet's create 10 simulations with 2 random variables (features). We'll define minima and have simulations randomly select a minima to start each simulation from. I've set it up to be a fair selection (uniform distribution), but you can play with this to change the distribution shape. Once we've made our random data, we'll initialize our **`MSM`** object. There are a lot of ways to generate random data, but I'm using `numpy` because it's fast as well as easy."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "minima = [[-0.5, 0], [0, -0.5], [0.5, 0], [0, 0.5]]\n",
        "N = len(minima)\n",
        "states = np.random.choice(np.arange(N), p=[1/N for i in range(N)], size=10)\n",
        "\ndata = [np.random.normal(loc=minima[states[i]], scale=0.2, size=(1000000, 2)) for i in range(10)]"
      ],
      "outputs": [],
      "execution_count": 22,
      "metadata": {
        "collapsed": true
      }
    },
    {
      "cell_type": "code",
      "source": [
        "model = mm.MSM(data)"
      ],
      "outputs": [],
      "execution_count": 23,
      "metadata": {
        "collapsed": true
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Base Class **`(Markov_Model.MSM)`** Attributes\n",
        "There are several attributes, and callable functions stored within the default model **`MSM`** object. A particularly useful one is the **`MSM.histogram`** function, which we can use to calculate the distribution. For visualization of the FES with `matplotlib`, we can define the range of the histogram matrix by the **`MSM.extent`**"
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "his0 = model.histogram(0, bins=50)\n",
        "his1 = model.histogram(1, bins=50)\n",
        "his = model.histogram(bins=50)"
      ],
      "outputs": [],
      "execution_count": 24,
      "metadata": {
        "collapsed": true
      }
    },
    {
      "cell_type": "code",
      "source": [
        "plt.figure(figsize=(8,4))\n",
        "plt.subplot(1,2,1)\n",
        "plt.plot(his0/his0.sum(), linewidth=3, alpha=0.5)\n",
        "plt.xlabel(r'$X$', fontsize=25)\n",
        "plt.ylabel(r'$P$', fontsize=25)\n",
        "\n",
        "plt.subplot(1,2,2)\n",
        "plt.plot(his1/his1.sum(), linewidth=3, alpha=0.5)\n",
        "plt.xlabel(r'$Y$', fontsize=25)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 25,
          "data": {
            "text/plain": [
              "<matplotlib.text.Text at 0x118d6b470>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
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              "rkJggg==\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x118b2fe80>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 25,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "plt.figure(figsize=(10,8))\n",
        "plt.contourf(-np.ma.log(his.T/his.sum()), 25, cmap=plt.cm.Spectral, extent=model.extent)\n",
        "plt.xlabel(r'$X$')\n",
        "plt.ylabel(r'$Y$')\n",
        "plt.colorbar()"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 81,
          "data": {
            "text/plain": [
              "<matplotlib.colorbar.Colorbar at 0x14363ca90>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
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            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x143868ac8>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 81,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Fitting microstate model  **`(MSM.microstates.fit)`**\n",
        "\n",
        "The method uses *k-means++* by default to generate 100 centroids **`(MSM.microstates.centroids)`**, fitting to 10% (**`fraction=0.1`**) of the simulation data.\n",
        "For very large models, it may be necessary to clustering using the batch method *K-Means Mini Batch*."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "model.microstates.fit(100, fraction=1e-2, lag=100, method='MiniBatchKMeans')"
      ],
      "outputs": [],
      "execution_count": 29,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "plt.figure(figsize=(10,8))\n",
        "plt.title('Microstate Centroid Positions')\n",
        "plt.contourf(-np.ma.log(his.T/his.sum()), 25, cmap=plt.cm.Spectral, extent=model.extent)\n",
        "plt.xlabel(r'$X$')\n",
        "plt.ylabel(r'$Y$')\n",
        "plt.colorbar()\n",
        "\n",
        "plt.plot(model.microstates.centroids[:,0], \n",
        "             model.microstates.centroids[:,1], \n",
        "             'o', color='k')\n",
        "for i in range(model.n_microstates):\n",
        "    plt.text(model.microstates.centroids[i,0], \n",
        "             model.microstates.centroids[i,1], \n",
        "             i, color='w')"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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              "AACoQSwBAADUIJYAAABq/B83Y9nmAZQQMgAAAABJRU5ErkJggg==\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x1431f3a58>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 82,
      "metadata": {
        "scrolled": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Check microstate attributes\n",
        "\nWe now have several attributes to our microstate model (**`model.microstates`**), containing the **`lag`**, **`centroids`**, **`count_matrix`**, **`transition_matrix`**. Additionally, we have our microstate **`labels`** of the data, which stores the discrete microstate time-dependently."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "print('The system lag time is '+str(model.microstates.lag)+' time steps')"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The system lag time is 100 time steps\n"
          ]
        }
      ],
      "execution_count": 31,
      "metadata": {
        "scrolled": true
      }
    },
    {
      "cell_type": "code",
      "source": [
        "model.microstates.labels"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 32,
          "data": {
            "text/plain": [
              "[array([86, 50, 36, ...,  1, 19, 65], dtype=int32),\n",
              " array([45, 18, 88, ..., 29, 45,  8], dtype=int32),\n",
              " array([22,  6, 37, ..., 28, 13, 22], dtype=int32),\n",
              " array([41, 75, 22, ...,  2,  2, 53], dtype=int32),\n",
              " array([97,  0, 39, ..., 59, 70,  7], dtype=int32),\n",
              " array([31, 67, 11, ...,  4, 30, 32], dtype=int32),\n",
              " array([50, 19, 96, ..., 49, 76, 78], dtype=int32),\n",
              " array([74, 76,  1, ..., 44, 26, 27], dtype=int32),\n",
              " array([46, 56,  6, ..., 33, 15,  2], dtype=int32),\n",
              " array([79, 10, 21, ..., 97, 35, 10], dtype=int32)]"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 32,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Predict microstates **`(MSM.microstates.predict)`**\n",
        "\nWe can predict what states, new data would be assigned using the **`predict`** method. Below, I've shown the assignment for randomly generated data."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "model.microstates.predict(np.random.randn(25,2))"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 33,
          "data": {
            "text/plain": [
              "array([61, 72, 58, 24, 54, 34, 58, 73, 63, 88, 58, 23, 58, 42, 94, 34, 39,\n",
              "       71, 57, 88, 58, 54, 17, 86, 48])"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 33,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Calculate microstate kinetics\n",
        "\nIn addition to the attributes, we also have a few functions that allow us to calculate the spectral decomposition of **`eigenvalues`**, **`left_eigenvectorss`**, and **`right_eigenvectorss`**. The implied timescales is calculated through pyemma, while **`MSM.microstates.timescales`** can be used, it may be easier to directly use `MSMBuilder`, `PyEMMA`, all of the analysis are calculated using `msmtools`."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "model.microstates.eigenvalues(10)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 34,
          "data": {
            "text/plain": [
              "array([ 1.        ,  0.89314624,  0.84759997,  0.75615667, -0.0051462 ,\n",
              "        0.00438146, -0.00415582,  0.00410685,  0.00390234, -0.00387402])"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 34,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "model.microstates.left_eigenvector(2)[:10,:]"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 35,
          "data": {
            "text/plain": [
              "array([[ 0.0031632 , -0.00300947],\n",
              "       [ 0.01609128,  0.02051178],\n",
              "       [ 0.01889836, -0.02481237],\n",
              "       [ 0.00945093,  0.0025631 ],\n",
              "       [ 0.01597982, -0.0004324 ],\n",
              "       [ 0.00820296,  0.01123121],\n",
              "       [ 0.02225625, -0.03026619],\n",
              "       [ 0.01318375,  0.00019968],\n",
              "       [ 0.00831654, -0.00017184],\n",
              "       [ 0.01092375,  0.0148493 ]])"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 35,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "model.microstates.right_eigenvector(2)[:10,:]"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 36,
          "data": {
            "text/plain": [
              "array([[ 1.        , -0.95140141],\n",
              "       [ 1.        ,  1.27471349],\n",
              "       [ 1.        , -1.31293765],\n",
              "       [ 1.        ,  0.27120064],\n",
              "       [ 1.        , -0.0270594 ],\n",
              "       [ 1.        ,  1.36916589],\n",
              "       [ 1.        , -1.35989619],\n",
              "       [ 1.        ,  0.01514615],\n",
              "       [ 1.        , -0.02066247],\n",
              "       [ 1.        ,  1.35935983]])"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 36,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "model.microstates.timescales(lags=[10,20,30,40,50,60,70,80,90,100], k=1)"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "application/vnd.jupyter.widget-view+json": {
              "model_id": "d94b89beacdd41e29c9cd091e947d87e"
            }
          },
          "metadata": {}
        },
        {
          "output_type": "execute_result",
          "execution_count": 37,
          "data": {
            "text/plain": [
              "array([[  88.578124  ],\n",
              "       [ 176.98741572],\n",
              "       [ 265.70169752],\n",
              "       [ 354.33191439],\n",
              "       [ 442.62200866],\n",
              "       [ 531.15367331],\n",
              "       [ 619.6170685 ],\n",
              "       [ 708.75637632],\n",
              "       [ 796.64735014],\n",
              "       [ 884.91691051]])"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 37,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "its = model.microstates.timescales(k=50)\n",
        "sep = its[0,:-1]/its[0,1:]"
      ],
      "outputs": [],
      "execution_count": 51,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "plt.figure(figsize=(10,10))\n",
        "plt.title('Separation in Implied Timescales', fontsize=25)\n",
        "plt.text(5, 10, 'You may need to play with the scaling to see them all!', color='r', fontsize=10, horizontalalignment='center')\n",
        "plt.hlines(1, 0, 10, linestyles='--', color='r', alpha=0.5, label='No separation in timescale')\n",
        "plt.plot(np.arange(2,51), sep, 'o', color='k', markersize=15)\n",
        "plt.xlabel(r'$i$')\n",
        "plt.ylabel(r'${t_{i}}/{t_{i+1}}$')\n",
        "plt.xlim(0,10)\n",
        "plt.legend()"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 70,
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x119005278>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
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              "hDgAAIAIEeIAAAAiRIgDAACIECEOAAAgQv8F7QBRzNRd4fIAAAAASUVORK5CYII=\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x119298c50>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 70,
      "metadata": {}
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Fit Macrostates Model **`(MSM.macrostates.fit)`**\n",
        "\n",
        "In the previous calculation we saw there were 3 large separations in timescale, therefore we can expect a 4 macrostate model. The method we use to coarse-grain is *Robust Perron Cluster Analysis (PCCA++)*. This is set as the default method to cluster microstate centroids **`(MSM.microstates.centroids)`**. I've additionally implemented a clustering method using *Hidden Markov Models* and *Gaussian Mixture Models* which can be selected by setting `method=HMM` or `method=GMM` respectively, however these have not been thoroughly tested. \n",
        "\nYou'll notice many of the same attributes present in **`MSM.microstates`** is available for **`MSM.macrostates`**. These perform the same methods, but are performed for micro and macroscopic models."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "model.macrostates.fit(4, method='PCCA')"
      ],
      "outputs": [],
      "execution_count": 71,
      "metadata": {
        "collapsed": true
      }
    },
    {
      "cell_type": "code",
      "source": [
        "model.macrostates.transition_matrix"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 72,
          "data": {
            "text/plain": [
              "array([[ 0.81196017,  0.0056581 ,  0.09105921,  0.09132252],\n",
              "       [ 0.00584748,  0.81826139,  0.09120542,  0.08468571],\n",
              "       [ 0.0655859 ,  0.06356365,  0.86375638,  0.00709407],\n",
              "       [ 0.06332421,  0.05682031,  0.00682969,  0.8730258 ]])"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 72,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "plt.figure(figsize=(10,8))\n",
        "plt.title('Macrostate Labeled Centroid Positions')\n",
        "plt.contourf(-np.ma.log(his.T/his.sum()), 25, cmap=plt.cm.Spectral, extent=model.extent)\n",
        "plt.xlabel(r'$X$')\n",
        "plt.ylabel(r'$Y$')\n",
        "plt.colorbar()\n",
        "\n",
        "colors = plt.cm.rainbow(np.linspace(0, 1, model.n_macrostates))\n",
        "s = model.macrostates.metastable_sets\n",
        "for i in range(model.n_macrostates):\n",
        "    plt.plot(model.microstates.centroids[s[i],0], \n",
        "             model.microstates.centroids[s[i],1], \n",
        "             'o', color=colors[i])"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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              "SUVORK5CYII=\n"
            ],
            "text/plain": [
              "<matplotlib.figure.Figure at 0x11f32e9e8>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 78,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Attributes specific to **`(MSM.macrostates)`**\n",
        "\n",
        "Here, I'll list a few of the attributes that are specific to the **`macrostates`** object.\n",
        "\n",
        "* **`metastable_labels`**: This contains the **crisp** assignments of each microstate to its corresponding macrostates\n",
        "* **`metastable_sets`**: This is another form of storing **crisp** assignments. Neither is different, but some are easier to work with, depending on the situation.\n",
        "* **`membership`**: Fuzzy membership probability for all microstates to belong in all macrostates. This attribute is not available if you coarse grain with *HMM*, as it is a result of the *PCCA++* method.\n",
        "* **`score`**: This will calculate the Silhouette Score that estimates effectively how much overlap there is between the assignment of microstates to a specific macrostates. *PCCA++* tries to reduce this value by minimizing how *fuzzy* the memberships are, but expect some overlap to exist. -1 value means all macrostates elements are overlapping, and +1 means memberships are perfectly crisp."
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "model.macrostates.metastable_labels"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 79,
          "data": {
            "text/plain": [
              "array([3, 4, 3, 2, 1, 4, 3, 2, 1, 4, 2, 1, 4, 3, 3, 3, 4, 1, 1, 4, 4, 2, 3,\n",
              "       4, 2, 3, 4, 4, 3, 1, 1, 1, 1, 3, 1, 2, 4, 3, 3, 2, 2, 3, 1, 3, 4, 1,\n",
              "       3, 1, 4, 4, 1, 4, 4, 3, 2, 4, 3, 2, 3, 2, 4, 2, 2, 1, 2, 4, 4, 1, 1,\n",
              "       4, 2, 1, 3, 4, 4, 3, 4, 3, 4, 2, 4, 3, 3, 3, 1, 3, 4, 1, 1, 4, 2, 4,\n",
              "       4, 1, 2, 1, 4, 2, 2, 3])"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 79,
      "metadata": {
        "scrolled": true
      }
    },
    {
      "cell_type": "code",
      "source": [
        "model.macrostates.memberships[:10,:]"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 80,
          "data": {
            "text/plain": [
              "array([[  1.31823334e-03,   2.99342353e-01,   6.99339414e-01,\n",
              "          0.00000000e+00],\n",
              "       [  6.71221184e-02,   1.51203784e-03,   8.44416731e-04,\n",
              "          9.30521427e-01],\n",
              "       [  3.50585978e-02,   2.65207949e-03,   9.61618190e-01,\n",
              "          6.71132430e-04],\n",
              "       [  3.72371392e-04,   8.06330540e-01,   1.53469034e-03,\n",
              "          1.91762398e-01],\n",
              "       [  9.71788212e-01,   1.34137895e-03,   1.81635164e-02,\n",
              "          8.70689281e-03],\n",
              "       [  1.11211590e-03,   3.12751401e-04,   3.77059553e-04,\n",
              "          9.98198073e-01],\n",
              "       [  1.38619858e-03,   1.81604658e-03,   9.96239452e-01,\n",
              "          5.58302535e-04],\n",
              "       [  2.91667132e-04,   9.97787229e-01,   2.68040252e-05,\n",
              "          1.89429969e-03],\n",
              "       [  9.91975766e-01,   1.32077742e-03,   5.61226616e-03,\n",
              "          1.09119078e-03],\n",
              "       [  1.19433302e-03,   7.59971960e-03,   2.98547895e-04,\n",
              "          9.90907399e-01]])"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 80,
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true
      }
    }
  ],
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