transition-counting.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Transition counting\n",
    "\n",
    "To begin estimating markov state models one needs to collect statistics over available data, in particular transitions between states at a given lag time $\\tau$.\n",
    "\n",
    "These statistics can be collected using deeptime's transition count estimator ([apidocs](../api/generated/deeptime.markov.TransitionCountEstimator.rst#deeptime.markov.TransitionCountEstimator)).\n",
    "\n",
    "When creating an instance of said estimator, one has to make two choices: The lagtime $\\tau > 0$ and the count mode. The lag time is the step size of the markov chain, i.e, an underlying transition matrix of the process with\n",
    "\n",
    "$$  P_{ij} = \\mathbb{P}(X_{t+\\tau}=j \\mid X_t = i). $$\n",
    "\n",
    "The count mode can be one of:\n",
    "\n",
    "- `\"sample\"`: Strides the trajectory with lagtime $\\tau$ and uses the strided counts as transitions, i.e., a trajectory of length $T$ has $\\lfloor T/\\tau \\rfloor$ counts at time indices\n",
    "   $$ (0 \\rightarrow \\tau), (\\tau \\rightarrow 2 \\tau), ..., (((T/ \\tau )-1) \\tau \\rightarrow T). $$\n",
    "   This means that potentially transitions can be missed.\n",
    "- `\"sliding\"`: Uses a sliding window approach for counting. A trajectory of length $T$ has $T-\\tau$ transition counts at \n",
    "   $$ (0 \\rightarrow \\tau), (1 \\rightarrow \\tau+1), ..., (T-\\tau-1 \\rightarrow T-1). $$\n",
    "   This introduces an overestimation of the actual count values by a factor of \"lagtime\". For maximum-likelihood MSMs this plays no role but it leads to wrong error bars in uncertainty estimation.\n",
    "- `\"sliding-effective\"`: Same as sliding mode, just that the resulting count matrix is divided by the lagtime after counting. This which can be shown to provide a likelihood that is the geometrical average over shifted subsamples of the trajectory, $(s_1,\\:s_{tau+1},\\:...),\\:(s_2,\\:t_{tau+2},\\:...)$, etc. This geometrical average converges to the correct likelihood in the statistical limit <cite data-footcite=\"trendelkamp2015estimation\">(Trendelkamp-Schroer, 2015)</cite>.\n",
    "- `\"effective\"`: Uses an estimate of the transition counts that are statistically uncorrelated. Recommended when used with a [Bayesian MSM](./mlmsm.ipynb) (or the respective [API docs](../api/generated/deeptime.markov.msm.BayesianMSM.rst#deeptime.markov.msm.BayesianMSM)). A description of the estimation procedure can be found in <cite data-footcite=\"noe2015statistical\">(Noé, 2015)</cite>."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:42:59.311387Z",
     "iopub.status.busy": "2020-12-02T13:42:59.303978Z",
     "iopub.status.idle": "2020-12-02T13:43:00.165703Z",
     "shell.execute_reply": "2020-12-02T13:43:00.166058Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from deeptime.markov import TransitionCountEstimator\n",
    "\n",
    "estimator = TransitionCountEstimator(\n",
    "    lagtime=1,\n",
    "    count_mode=\"sliding\"\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The estimator can be used to fit a count model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.170542Z",
     "iopub.status.busy": "2020-12-02T13:43:00.170004Z",
     "iopub.status.idle": "2020-12-02T13:43:00.172259Z",
     "shell.execute_reply": "2020-12-02T13:43:00.171830Z"
    }
   },
   "outputs": [],
   "source": [
    "trajectory = np.array([0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 7])\n",
    "counts = estimator.fit(trajectory).fetch_model()  # fit and fetch the model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The count model has several properties, which can be used to later on estimate markov state models. One of them is the count matrix $C$, which countains transition counts from one state to the other as encountered in the trajectory:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.180445Z",
     "iopub.status.busy": "2020-12-02T13:43:00.180029Z",
     "iopub.status.idle": "2020-12-02T13:43:00.182662Z",
     "shell.execute_reply": "2020-12-02T13:43:00.182208Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0., 2., 0., 0., 0., 0., 0., 0.],\n",
       "       [1., 0., 2., 0., 0., 0., 0., 1.],\n",
       "       [0., 2., 0., 2., 0., 0., 0., 0.],\n",
       "       [0., 0., 2., 0., 2., 0., 0., 0.],\n",
       "       [0., 0., 0., 2., 0., 2., 0., 0.],\n",
       "       [0., 0., 0., 0., 2., 0., 1., 0.],\n",
       "       [0., 0., 0., 0., 0., 1., 0., 0.],\n",
       "       [0., 0., 0., 0., 0., 0., 0., 0.]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "counts.count_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this example, the $(1,2)$-th element is $C_{12} = 2$, meaning there are two counted transitions from state $1$ to state $2$. Furthermore, one has access to a state histogram, which contains the information how often each state was visited within the presented data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.185669Z",
     "iopub.status.busy": "2020-12-02T13:43:00.185258Z",
     "iopub.status.idle": "2020-12-02T13:43:00.188013Z",
     "shell.execute_reply": "2020-12-02T13:43:00.188344Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "State 0: 2 occurances.\n",
      "State 1: 4 occurances.\n",
      "State 2: 4 occurances.\n",
      "State 3: 4 occurances.\n",
      "State 4: 4 occurances.\n",
      "State 5: 3 occurances.\n",
      "State 6: 1 occurances.\n",
      "State 7: 1 occurances.\n"
     ]
    }
   ],
   "source": [
    "for state, histogram_value in enumerate(counts.state_histogram):\n",
    "    print(f\"State {state}: {histogram_value} occurances.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Selecting a submodel\n",
    "\n",
    "Depending on what kind of markov model is estimated, it is required to have either a strongly or weakly connected set of states. This means that the count matrix is interpreted as an adjacency matrix\n",
    "$$ A_{ij} = \\begin{cases}1 &\\text{, if } C_{ij} > 0 \\\\ 0 &\\text{, otherwise.}\\end{cases} $$\n",
    "A graph induced by the adjacency matrix is called strongly connected if the edges are directed and there is a path from each state $i$ to each state $j$. It is called weakly connected if the edges are undirected and there is a path from each state $i$ to each state $j$.\n",
    "\n",
    "Plotting the graph (using [networkx](https://networkx.github.io/)), one can see that our example data consists of $8$ states in total where all except state $7$ are strongly connected."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.192731Z",
     "iopub.status.busy": "2020-12-02T13:43:00.192310Z",
     "iopub.status.idle": "2020-12-02T13:43:00.685703Z",
     "shell.execute_reply": "2020-12-02T13:43:00.685271Z"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import networkx as nx\n",
    "\n",
    "G = nx.from_numpy_matrix(counts.count_matrix, create_using=nx.DiGraph())\n",
    "pos = nx.circular_layout(G)\n",
    "nx.draw_circular(G, node_color='#90EE90')\n",
    "nx.draw_networkx_labels(G, pos, labels={i: i for i in G.nodes()}, font_size=15)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The connected sets (weak and strong) can be obtained from the model directly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.690767Z",
     "iopub.status.busy": "2020-12-02T13:43:00.690008Z",
     "iopub.status.idle": "2020-12-02T13:43:00.692386Z",
     "shell.execute_reply": "2020-12-02T13:43:00.692766Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Weakly connected sets: [array([0, 1, 2, 3, 4, 5, 6, 7])]\n",
      "Strongly connected sets: [array([0, 1, 2, 3, 4, 5, 6]), array([7])]\n"
     ]
    }
   ],
   "source": [
    "print(\"Weakly connected sets:\", counts.connected_sets(directed=False))\n",
    "print(\"Strongly connected sets:\", counts.connected_sets(directed=True))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And a submodel can be selected based on either an explicitly selected subset of states:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.694815Z",
     "iopub.status.busy": "2020-12-02T13:43:00.694398Z",
     "iopub.status.idle": "2020-12-02T13:43:00.697531Z",
     "shell.execute_reply": "2020-12-02T13:43:00.697142Z"
    }
   },
   "outputs": [],
   "source": [
    "submodel_explicit = counts.submodel(states=[0, 1, 7])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which then contains a subselected count matrix. The \"states\" of a count model always refer to the dimensions of the count matrix, i.e., if $C\\in\\mathbb{R}^{n\\times n}$, then there are $n$ states. To be able to trace back, which state in a submodel belongs to which state in the data, there are also \"symbols\", which are the states of the root model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.701253Z",
     "iopub.status.busy": "2020-12-02T13:43:00.700750Z",
     "iopub.status.idle": "2020-12-02T13:43:00.703055Z",
     "shell.execute_reply": "2020-12-02T13:43:00.703491Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "# states: 3\n",
      "symbols: [0 1 7]\n"
     ]
    }
   ],
   "source": [
    "print(\"# states:\", submodel_explicit.n_states)\n",
    "print(\"symbols:\", submodel_explicit.state_symbols)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If presented with a set of symbols which should be mapped to states, there is a convenience function (no longer present symbols are discarded):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.707363Z",
     "iopub.status.busy": "2020-12-02T13:43:00.706891Z",
     "iopub.status.idle": "2020-12-02T13:43:00.709637Z",
     "shell.execute_reply": "2020-12-02T13:43:00.709180Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "submodel_explicit.symbols_to_states([0, 1, 7, 8])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Furthermore one can restrict data to the state selection of the model. No longer present symbols are set to `-1`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.713242Z",
     "iopub.status.busy": "2020-12-02T13:43:00.712806Z",
     "iopub.status.idle": "2020-12-02T13:43:00.715408Z",
     "shell.execute_reply": "2020-12-02T13:43:00.715041Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[array([ 0,  1, -1, -1, -1, -1, -1, -1, -1,  1,  0,  1, -1, -1, -1, -1, -1,\n",
       "        -1, -1, -1, -1,  1,  2], dtype=int32)]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "submodel_explicit.transform_discrete_trajectories_to_submodel(trajectory)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Most of the time the state selection should be based on the connected sets. For this, one can call"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.719109Z",
     "iopub.status.busy": "2020-12-02T13:43:00.718671Z",
     "iopub.status.idle": "2020-12-02T13:43:00.720242Z",
     "shell.execute_reply": "2020-12-02T13:43:00.719824Z"
    }
   },
   "outputs": [],
   "source": [
    "submodel = counts.submodel_largest(connectivity_threshold=1., directed=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This selects the largest (strongly) connected set where only states with at least `1` transition count are considered connected."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2020-12-02T13:43:00.736687Z",
     "iopub.status.busy": "2020-12-02T13:43:00.734931Z",
     "iopub.status.idle": "2020-12-02T13:43:00.880910Z",
     "shell.execute_reply": "2020-12-02T13:43:00.881529Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G = nx.from_numpy_matrix(submodel.count_matrix, create_using=nx.DiGraph())\n",
    "pos = nx.circular_layout(G)\n",
    "nx.draw_circular(G, node_color='#90EE90')\n",
    "nx.draw_networkx_labels(G, pos, labels={i: i for i in G.nodes()}, font_size=15)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As one can see, the weakly connected state `7` is removed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Optionally, one can give probability constraints of states as well as selected the connected set by population and not by size. A description of this can be found [here](../api/generated/deeptime.markov.TransitionCountModel.rst#deeptime.markov.TransitionCountModel.submodel_largest)."
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "raw_mimetype": "text/restructuredtext"
   },
   "source": [
    ".. rubric:: References\n",
    "\n",
    ".. footbibliography::"
   ]
  }
 ],
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   "display_name": "Python 3 (ipykernel)",
   "language": "python",
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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  "toc": {
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   "title_cell": "Table of Contents",
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