lab4_discrete_dynamical_syatem.ipynb

{
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  {
   "cell_type": "markdown",
   "id": "eaa0d4c8",
   "metadata": {},
   "source": [
    "## Discrete Dynamical Systems: First-Order Difference Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdac2f7f",
   "metadata": {},
   "source": [
    "This lecture note will illustrate one common kind of application of eigenvalues: to the study of the long-term behavior (or steady-state response) of a dynamical system described by a **difference equation**.\n",
    "We will introduce stochastic matrics, which encode this type of difference equation.  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4106f3b3",
   "metadata": {},
   "source": [
    "### Difference Equations\n",
    "\n",
    "**Definition 18.1.** A *difference equation* is an an equation of the form\n",
    "$$ \\vec{v}_{k+1} = A\\vec{v}_k \\hspace{2in} $$\n",
    "for an $n\\times n$ matrix $A$ (called a transition matrix) and vectors $\\vec{v}_0, \\vec{v}_1,, \\vec{v}_2, ...$ in $\\mathbb{R}^n$. Here\n",
    "  * $\\vec{v}_k$ is the \"state at time $k$,\" and\n",
    "  * $\\vec{v}_{k+1}$ is the \"state at time $k+1$.\" \n",
    "\n",
    "Observe that\n",
    "$$ \\vec{v}_k = A\\vec{v}_{k-1} = A^2\\vec{v}_{k-2} = \\cdots = A^k\\vec{v}_0.$$\n",
    "This should hint to you that the long-term behavior of a difference equation (when $k\\rightarrow\\infty$) is an eigenvalue problem."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ed13fed",
   "metadata": {},
   "source": [
    "**Example 18.2.**  In a population of rabbits,\n",
    "   1. half of the newborn rabbits survive their first year;\n",
    "   2. of those, half survive their second year;\n",
    "   3. the maximum life span is three years;\n",
    "   4. rabbits produce 0, 6, 8 rabbits in their first, second, and third years, respectively.\n",
    "\n",
    "Let $\\vec{p}_k=(x_k, y_k, z_k)$ be the population vector whose entries $x_k, y_k, z_k$ are the number of rabbits aged 0, 1, and 2, respectively.  \n",
    "A population vector $\\vec{p}$ contains the abundance of rabbits in each age group. We can use the fertility and survival of each age group to predict the abundance of each group after one year:\n",
    "\n",
    "$$\\begin{align*}\n",
    "  x_{k+1} &=& 0\\,x_k + 6\\,y_k + 8\\,z_k \\\\\n",
    "  y_{k+1} &=& 0.5\\,x_k + 0\\,y_k + 0\\,z_k \\\\\n",
    "  z_{k+1} &=& 0\\,x_k + 0.5\\,y_k + 0\\,z_k\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "The superscript $k$ refers to the year. The coefficient matrix of this system of equations is:\n",
    "\n",
    "$$ A = \\begin{bmatrix} 0 & 6 & 8 \\\\\n",
    "                       0.5 & 0 & 0 \\\\\n",
    "                       0 & 0.5 & 0 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "This matrix represents the _age transition_ year on year. Multiply this matrix by the population vector to get the age-structured population on the next year. Let's write a bit of code to see the population in each age group grow over 10 years. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "37c56787",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "09dbdcaf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0. , 6. , 8. ],\n",
       "       [0.5, 0. , 0. ],\n",
       "       [0. , 0.5, 0. ]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[0,6,8], [0.5,0,0], [0,0.5,0]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5c75ca5e",
   "metadata": {},
   "outputs": [],
   "source": [
    "p = np.array([25, 10, 5]) # initial age-structured population"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "68785f5e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[100.   12.5   5. ]\n",
      "[115.    50.     6.25]\n",
      "[350.   57.5  25. ]\n",
      "[545.   175.    28.75]\n",
      "[1280.   272.5   87.5]\n",
      "[2335.    640.    136.25]\n",
      "[4930.  1167.5  320. ]\n",
      "[9565.   2465.    583.75]\n",
      "[19460.   4782.5  1232.5]\n"
     ]
    }
   ],
   "source": [
    "N = 10\n",
    "pn = np.zeros((N, len(p)))\n",
    "pn[0,:] = p.copy()\n",
    "for i in range(1,N):\n",
    "    pn[i,:] = A.dot(pn[i-1,:])\n",
    "    print(pn[i,:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "fd848008",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(12,4))\n",
    "plt.plot(pn[:,0], label='x')\n",
    "plt.plot(pn[:,1], label='y')\n",
    "plt.plot(pn[:,2], label='z')\n",
    "plt.legend()\n",
    "#plt.grid()\n",
    "plt.xlabel('Years')\n",
    "plt.title('Population in each age group');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d74e5e7e",
   "metadata": {},
   "source": [
    "Nothing to see here: the population in each age group grows over time. But one question ecologists may need to answer is whether the species reaches a stable growth pattern, and what is the _stable age distribution_. That occurs if the percentage in each age group remains the same, i.e., \n",
    "$\\vec{p}_{k+1}$ is a multiple of $\\vec{p}_k$:\n",
    "\n",
    "$$ \\vec{p}_{k+1} = A\\, \\vec{p}_k = r\\, \\vec{p}_k $$\n",
    "\n",
    "We have on the right an eigenvalue problem, and the stable age distribution is an eigenvector of the age transition matrix.\n",
    "\n",
    "**Example 18.3.** A classic example in this field is the hypothetical beetle population of Bernadelli (Bernardelli, H. (1941), Population waves, Journal of the Burma Research Society 31, Part 1. Also in _Mathematical Demography, Selected Papers_ (1977), pp. 215-219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-81046-6_24), also separated in three age groups, and with age transition matrix:\n",
    "\n",
    "$$ T = \\begin{bmatrix} 0 & 0 & 6 \\\\\n",
    "                       1/2 & 0 & 0 \\\\\n",
    "                       0 & 1/3 & 0 \\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0b4a57c7",
   "metadata": {},
   "outputs": [],
   "source": [
    "T = np.array([[0,0,6], [0.5,0,0], [0,1/3,0]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c811defd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = np.array([25, 10, 5])\n",
    "N = 24\n",
    "pn = np.zeros((N, len(p)))\n",
    "pn[0,:] = p.copy()\n",
    "for i in range(1,N):\n",
    "    pn[i,:] = T.dot(pn[i-1,:])\n",
    "    \n",
    "plt.figure(figsize=(12,4))\n",
    "plt.plot(pn[:,0], label='x')\n",
    "plt.plot(pn[:,1], label='y')\n",
    "plt.plot(pn[:,2], label='z')\n",
    "plt.legend()\n",
    "#plt.grid()\n",
    "plt.xlabel('Years')\n",
    "plt.title('Population in each age group');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "074967f8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[25. 10.  5.]\n"
     ]
    }
   ],
   "source": [
    "print(pn[6,:])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a603c9d",
   "metadata": {},
   "source": [
    "In this case, the initial population distribution repeats every few years! If we look at the eigenvalues for the matrix, in this case, they are complex numbers.\n",
    "In Python, complex numbers are represented as the sum of a real part and an imaginary part, using the symbol `j` for the imaginary unit corresponding to $\\sqrt{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "067e65cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.5+0.8660254j, -0.5-0.8660254j,  1. +0.j       ])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eig(T)[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd530055",
   "metadata": {},
   "source": [
    "#### Compute eigenthings in Python  \n",
    "\n",
    "You can compute the eigenvalues and eigenvectors of a matrix using [`numpy.linalg.eig()`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eig.html). It returns a tuple: its first element is an array with the eigenvalues, and its second element is a 2D array  where each column is an eigenvector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "518f242a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.        , -0.99999998, -1.00000002])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eig(A)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "9f2b53ba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.96836405,  0.87287156, -0.87287157],\n",
       "       [-0.24209101, -0.43643579,  0.43643577],\n",
       "       [-0.06052275,  0.2182179 , -0.21821788]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eig(A)[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "185d743e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.0000000000000013 [-0.96836405 -0.24209101 -0.06052275]\n",
      "-0.9999999789753345 [ 0.87287156 -0.43643579  0.2182179 ]\n",
      "-1.0000000210246673 [-0.87287157  0.43643577 -0.21821788]\n"
     ]
    }
   ],
   "source": [
    "# display each eigenvalue with the corresponding eigenvector, side by side\n",
    "eigenvalues, eigenvectors = np.linalg.eig(A)\n",
    "\n",
    "for eigenvalue, eigenvector in zip(eigenvalues, eigenvectors.T):\n",
    "    print(eigenvalue, eigenvector)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a500b8c",
   "metadata": {},
   "source": [
    "In the `for` statement above, each iteration picks an eigenvalue, and the corresponding eigenvector. Note that when iterating over a 2D array, the natural order advances over the first dimension: the rows; to get the columns, we apply the transpose of the `eigenvectors` matrix. Here, we used the [`numpy.ndarray.T`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.ndarray.T.html), which has the same effect as `.transpose`.\n",
    "\n",
    "**Remark.** To create the diagonal matrix of eigenvalues, use [`numpy.diag()`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.diag.html): if you give it a 1D array, it returns a 2D array with the elements of the input array in the diagonal. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "eb536693",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 0 0]\n",
      " [0 2 0]\n",
      " [0 0 3]]\n"
     ]
    }
   ],
   "source": [
    "L =np.array([1, 2, 3])\n",
    "print(np.diag(L))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "896ff998",
   "metadata": {},
   "source": [
    "**Activity 18.4.** In a certain town, 30 percent of the married women get divorced each year and 20 percent of the single women get married each year. There are 8000 married women and 2000 single women, \n",
    "and the total population remains constant. Let us investigate the long-range prospects if\n",
    "these percentages of marriages and divorces continue indefinitely into the future.\n",
    "\n",
    "  * (a) Set up a difference equation for this problem.\n",
    "  * (b) Find the number of married and single women after 5, 10, 15, 20, and 30 years. What do you observe?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43a378c4",
   "metadata": {},
   "source": [
    "**Solution**\n",
    "\n",
    "Let $m$ represent the number of married women and $s$ the number of single women. The corresponding mathematical model is\n",
    "$$\\begin{align*}\n",
    "  m_{k+1} &=& 0.7\\,m_k + 0.2\\,s_k\\\\\n",
    "  s_{k+1} &=& 0.3\\,m_k + 0.8\\,s_k\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "Let $\\vec{p}_k = (m_k, s_k)$ be the population vector. Then the difference equation has the form\n",
    "$$\n",
    " \\vec{p}_{k+1} = M\\vec{p}_k, \n",
    "$$\n",
    "where \n",
    "$$ M = \\begin{bmatrix} 0.7 & 0.2 \\\\\n",
    "                       0.3 & 0.8 \n",
    "       \\end{bmatrix}\n",
    "$$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "f467ba48",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "M = np.array([[0.7,0.2], [0.3,0.8]])\n",
    "\n",
    "p = np.array([8000, 2000])\n",
    "N = 30\n",
    "pn = np.zeros((N, len(p)))\n",
    "pn[0,:] = p.copy()\n",
    "\n",
    "for i in range(1,N):\n",
    "    pn[i,:] = M.dot(pn[i-1,:])\n",
    "    \n",
    "plt.figure(figsize=(12,4))\n",
    "plt.plot(pn[:,0], label='m')\n",
    "plt.plot(pn[:,1], label='s')\n",
    "plt.legend()\n",
    "#plt.grid()\n",
    "plt.xlabel('Years')\n",
    "plt.title('Population in each age group');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0f528f9a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k =   0    8000    2000\n",
      "k =   1    6000    4000\n",
      "k =   2    5000    5000\n",
      "k =   3    4500    5500\n",
      "k =   4    4250    5750\n",
      "k =   5    4125    5875\n",
      "k =   6    4062    5937\n",
      "k =   7    4031    5968\n",
      "k =   8    4015    5984\n",
      "k =   9    4007    5992\n",
      "k =  10    4003    5996\n",
      "k =  11    4001    5998\n",
      "k =  12    4000    5999\n",
      "k =  13    4000    5999\n",
      "k =  14    4000    5999\n",
      "k =  15    4000    5999\n",
      "k =  16    4000    5999\n",
      "k =  17    4000    5999\n",
      "k =  18    4000    5999\n",
      "k =  19    4000    5999\n",
      "k =  20    4000    5999\n",
      "k =  21    4000    5999\n",
      "k =  22    4000    5999\n",
      "k =  23    4000    5999\n",
      "k =  24    4000    5999\n",
      "k =  25    4000    5999\n",
      "k =  26    4000    5999\n",
      "k =  27    4000    5999\n",
      "k =  28    4000    5999\n",
      "k =  29    4000    5999\n"
     ]
    }
   ],
   "source": [
    "for i in range(0,N):\n",
    "    print('k = %3d  %6d  %6d'  % (i, pn[i,0], pn[i,1]) )"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15e81359",
   "metadata": {},
   "source": [
    "### Markov Chains"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d0d725c",
   "metadata": {},
   "source": [
    "In Activity 18.4, we studied a simple matrix model for predicting the number of married\n",
    "and single women in a certain town each year. Given an initial vector $\\vec{p}_0$ whose coordinates represent \n",
    "the current number of married and single women, we were able to\n",
    "predict the number of married and single women in future years by computing\n",
    "$$ \n",
    " \\vec{p}_1 = M\\vec{p}_0,\\ \\ \\ \\vec{p}_2 = M\\vec{p}_1,\\ \\ \\ \\vec{p}_3 = M\\vec{p}_2,\\ \\  ...\n",
    "$$\n",
    "\n",
    "If we scale the initial vector so that its entries indicate the **proportions of the population**\n",
    "that are married and single, then the coordinates of $\\vec{p}_k$ will indicate the proportions of\n",
    "married and single women after n years. The sequence of vectors that we generate in\n",
    "this manner is an example of a **Markov chain**. \n",
    "\n",
    "**Definition 18.5** A *stochastic process* is any sequence of experiments for which the outcome at\n",
    "any stage depends on chance. A Markov process is a stochastic process with the\n",
    "following properties:\n",
    "  1. The set of possible outcomes or states is finite.\n",
    "  2. The probability of the next outcome depends only on the previous outcome.\n",
    "  3. The probabilities are constant over time.\n",
    "  \n",
    "**Remark** The sum of entries of $\\vec{p}_k$ should be equal to $1$,\n",
    "since it’s a probability distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64b31ee1",
   "metadata": {},
   "source": [
    "**Example 18.6.** A car dealer leases four types of vehicles: *four-door sedans*, *sports cars*, \n",
    "*minivans*, and *SUV*. The term of the lease is 2\n",
    "years. At the end of the term, customers must renegotiate the lease and choose a new\n",
    "vehicle.\n",
    "\n",
    "The car leasing can be viewed as a process with four possible outcomes.\n",
    "The probability of each outcome can be estimated by reviewing records of previous\n",
    "leases. The records indicate that 80 percent of the customers currently leasing sedans\n",
    "will continue doing so in the next lease. Furthermore, 10 percent of the customers currently \n",
    "leasing sports cars will switch to sedans. In addition, 5 percent of the customers\n",
    "driving minivans or sport utility vehicles will also switch to sedans. These results are\n",
    "summarized in the first row of Table 1. The second row indicates the percentages of\n",
    "customers that will lease sports cars the next time, and the final two rows give the\n",
    "percentages that will lease minivans and SUV, respectively.\n",
    "\n",
    "<center>\n",
    "   <img src=\"lecture18-fig3.png\" style=\"width: 400px;\"/> \n",
    "</center>\n",
    "    \n",
    "Suppose that initially there are 200 sedans leased and 100 of each of the other three\n",
    "types of vehicles. If we set\n",
    "$$ \n",
    "  A = \\begin{bmatrix} \n",
    "        0.80 & 0.10 & 0.05 & 0.05 \\\\ \n",
    "        0.10 & 0.80 & 0.05 & 0.05 \\\\\n",
    "        0.05 & 0.05 & 0.80 & 0.10 \\\\\n",
    "        0.05 & 0.05 & 0.10 & 0.80\n",
    "       \\end{bmatrix}\n",
    "       \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n",
    "   \\vec{x}_0 = \n",
    "    \\begin{bmatrix}\n",
    "       200\\\\ 100 \\\\ 100 \\\\ 100\n",
    "    \\end{bmatrix}\n",
    "$$\n",
    "then we can determine how many people will lease each type of vehicle two years later\n",
    "by setting\n",
    "$$\n",
    " \\vec{x}_1 = A\\vec{x}_0 = \\begin{bmatrix} \n",
    "        0.80 & 0.10 & 0.05 & 0.05 \\\\ \n",
    "        0.10 & 0.80 & 0.05 & 0.05 \\\\\n",
    "        0.05 & 0.05 & 0.80 & 0.10 \\\\\n",
    "        0.05 & 0.05 & 0.10 & 0.80\n",
    "       \\end{bmatrix}\n",
    "    \\begin{bmatrix}\n",
    "       200\\\\ 100 \\\\ 100 \\\\ 100\n",
    "    \\end{bmatrix}\n",
    "     =\n",
    "     \\begin{bmatrix}\n",
    "       180\\\\ 110\\\\ 105\\\\ 105\n",
    "     \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "We can predict the numbers for future leases by setting\n",
    "$$\n",
    " \\vec{x}_{k+1} = A\\vec{x}_k \\ \\ \\mbox{for}\\ n = 1, 2, ...\n",
    "$$\n",
    "The vectors $\\vec{x}_i$ produced in this manner are referred to as **state vectors**, and \n",
    "the sequence\n",
    "of state vectors is called a **Markov chain**. The matrix $A$ is referred to as a *transition\n",
    "matrix*. The entries of each column of $A$ are nonnegative numbers that add up to 1.\n",
    "Each column can be viewed as a probability vector. For example, the first column of\n",
    "$A$ corresponds to individuals currently leasing sedans. The entries in this column are\n",
    "the probabilities of choosing each type of vehicle when the lease is renewed.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24519a12",
   "metadata": {},
   "source": [
    "If we divide the entries of the initial vector by 500 (the total number of customers),\n",
    "then the entries of the new initial state vector\n",
    "$$\n",
    " \\vec{x}_0 = (0.40, 0.20, 0.20, 0.20)^T\n",
    "$$\n",
    "represent the proportions of the population that rent each type of vehicle. The entries\n",
    "of $\\vec{x}_1$  will represent the proportions for the next lease. Thus, $\\vec{x}_0$ and $\\vec{x}_1$ are probability\n",
    "vectors, and it is easily seen that the succeeding state vectors in the chain will all be\n",
    "probability vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d4a9198a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k =   1  0.36  0.22  0.21 0.21\n",
      "k =   2  0.33  0.23  0.22 0.22\n",
      "k =   3  0.31  0.24  0.22 0.22\n",
      "k =   4  0.29  0.25  0.23 0.23\n",
      "k =   5  0.28  0.25  0.23 0.23\n",
      "k =   6  0.27  0.25  0.24 0.24\n",
      "k =   7  0.27  0.25  0.24 0.24\n",
      "k =   8  0.26  0.25  0.24 0.24\n",
      "k =   9  0.26  0.25  0.24 0.24\n",
      "k =  10  0.26  0.25  0.24 0.24\n",
      "k =  11  0.26  0.25  0.25 0.25\n",
      "k =  12  0.25  0.25  0.25 0.25\n",
      "k =  13  0.25  0.25  0.25 0.25\n",
      "k =  14  0.25  0.25  0.25 0.25\n",
      "k =  15  0.25  0.25  0.25 0.25\n",
      "k =  16  0.25  0.25  0.25 0.25\n",
      "k =  17  0.25  0.25  0.25 0.25\n",
      "k =  18  0.25  0.25  0.25 0.25\n",
      "k =  19  0.25  0.25  0.25 0.25\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "B = np.array([[0.80, 0.10, 0.05, 0.05], [00.10, 0.80, 0.05, 0.05],\n",
    "              [0.05, 0.05, 0.80, 0.10], [0.05, 0.05, 0.10, 0.80]])\n",
    "\n",
    "p = np.array([0.4, 0.2, 0.2, 0.2])\n",
    "N = 20\n",
    "pn = np.zeros((N, len(p)))\n",
    "pn[0,:] = p.copy()\n",
    "\n",
    "for i in range(1,N):\n",
    "    pn[i,:] = B.dot(pn[i-1,:])\n",
    "    print('k = %3d  %3.2f  %3.2f  %3.2f %3.2f'  % (i, pn[i,0], pn[i,1], pn[i,2], pn[i,3]) )\n",
    "    \n",
    "plt.figure(figsize=(12,4))\n",
    "plt.plot(pn[:,0], label='Sedan')\n",
    "plt.plot(pn[:,1], label='Sports Car')\n",
    "plt.plot(pn[:,2], label='Minivans')\n",
    "plt.plot(pn[:,3], label='SUV')\n",
    "plt.legend()\n",
    "#plt.grid()\n",
    "plt.xlabel('Leasing Period')\n",
    "plt.title('Leasing Proportions');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a52e042a",
   "metadata": {},
   "source": [
    "As $k$ increases, $\\vec{x}_k$ approaches the **steady-state** vector\n",
    "$$\n",
    " \\vec{x}_{ST} = (0.25, 0.25, 0.25, 0.25)^T\n",
    "$$\n",
    "Thus, the Markov chain model predicts that, in the long run, the leases will be divided\n",
    "equally among the four types of vehicles."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "986272ae",
   "metadata": {},
   "source": [
    "Theoretically, the long-term behavior of the stochastic process can be determined by the eigenvalues and eigenvectors of the transition matrix $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "7e035b82",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9999999999999997 [-0.5 -0.5 -0.5 -0.5]\n",
      "0.7 [-7.07106781e-01  7.07106781e-01 -7.44188381e-16 -8.13575913e-16]\n",
      "0.7999999999999996 [ 0.5  0.5 -0.5 -0.5]\n",
      "0.7 [-0.26314552  0.26314552 -0.65631885  0.65631885]\n"
     ]
    }
   ],
   "source": [
    "eigenvalues, eigenvectors = np.linalg.eig(B)\n",
    "\n",
    "for eigenvalue, eigenvector in zip(eigenvalues, eigenvectors.T):\n",
    "    print(eigenvalue, eigenvector)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66d600ed",
   "metadata": {},
   "source": [
    "The eigenvalues of $A$ are $\\lambda_1 = 1, \\lambda_2 = 0.8$, and $\\lambda_3 = \\lambda_4 = 0.7$. \n",
    "Even though $A$ has multiple eigenvalues, it does have *four linearly\n",
    "independent eigenvectors* and hence it can be diagonalized, $A = PDP^{-1}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "e757a265",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1.  0.  0.  0. ]\n",
      " [0.  0.7 0.  0. ]\n",
      " [0.  0.  0.8 0. ]\n",
      " [0.  0.  0.  0.7]]\n"
     ]
    }
   ],
   "source": [
    "D = np.diag(eigenvalues)\n",
    "print(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "bf2fd835",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-5.00000000e-01 -7.07106781e-01  5.00000000e-01 -2.63145518e-01]\n",
      " [-5.00000000e-01  7.07106781e-01  5.00000000e-01  2.63145518e-01]\n",
      " [-5.00000000e-01 -7.44188381e-16 -5.00000000e-01 -6.56318853e-01]\n",
      " [-5.00000000e-01 -8.13575913e-16 -5.00000000e-01  6.56318853e-01]]\n"
     ]
    }
   ],
   "source": [
    "P = np.array(eigenvectors)\n",
    "print(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "a0fa388b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-5.00000000e-01 -5.00000000e-01 -5.00000000e-01 -5.00000000e-01]\n",
      " [-7.07106781e-01  7.07106781e-01  2.83508510e-01 -2.83508510e-01]\n",
      " [ 5.00000000e-01  5.00000000e-01 -5.00000000e-01 -5.00000000e-01]\n",
      " [ 4.22897735e-17  4.22897735e-17 -7.61824832e-01  7.61824832e-01]]\n"
     ]
    }
   ],
   "source": [
    "Pinv = np.linalg.inv(P) #The inverse of P\n",
    "print(Pinv)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e193880e",
   "metadata": {},
   "source": [
    "Python 3.5 onwards also has an explicit operator for dot product, `@`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "d2f46773",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.8  0.1  0.05 0.05]\n",
      " [0.1  0.8  0.05 0.05]\n",
      " [0.05 0.05 0.8  0.1 ]\n",
      " [0.05 0.05 0.1  0.8 ]]\n"
     ]
    }
   ],
   "source": [
    "L = P@D@Pinv\n",
    "print(L)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "052998ba",
   "metadata": {},
   "source": [
    "Indeed, we see that $L = A$. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f1f331d6",
   "metadata": {},
   "source": [
    "**Matrix Powers.** Let's compute $A^{20}$ in two different way; one is directly with $A$ itself, the other uses its diagonalization \n",
    "$A^{20} = PD^{20}P^{-1}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "6cf8719d",
   "metadata": {},
   "outputs": [],
   "source": [
    "k = 20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "39357d23",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "33.5 µs ± 3.2 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit  # Time execution of a Python statement or expression in the cell. This magic command should come in the first line.\n",
    "result = B.copy()\n",
    "for _ in range(1,k):\n",
    "    result = result @ B"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "74e96c02",
   "metadata": {},
   "source": [
    "Let's use diagonalization to do the same computation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "82796ea3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.29 µs ± 316 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "P @ D**k @ Pinv"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc675330",
   "metadata": {},
   "source": [
    "Diagonalization computes $A^{k}$ much faster!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f2edf60",
   "metadata": {},
   "source": [
    "Now, ley's try to find the steady-state vector by computing $A^{k}$ with $k$ big enough, say $k=100$,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "0fcace5a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.25, 0.25, 0.25, 0.25])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = p\n",
    "x100 = P @ D**(100) @ Pinv @x0\n",
    "x100"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d2e9ee14",
   "metadata": {},
   "source": [
    "In fact, \n",
    "$$\\vec{x}_k = A^k\\vec{x}_0 = PD^kP^{-1}\\vec{x}_0 = 0.25*(1,1,1,1)^T - 0.05 (0.8)^k * (-1,-1,1,1)^T + 0.10 (0.7)^k* (1,-1,0,0)^T$$\n",
    "Thus \n",
    "$$ \\lim_{k\\rightarrow\\infty}\\vec{x}_k = (0.25, 0.25, 0.25, 0.25)^T\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d66f972",
   "metadata": {},
   "source": [
    "In general, we will assume that the initial vector $\\vec{x}_0$  in a Markov chain is a probability vector, \n",
    "and this in turn implies that all of the state vectors are probability vectors.\n",
    "One would expect, then, that if the chain converges to a steady-state vector $x$, then the\n",
    "steady-state vector must also be a probability vector. This is indeed the case, as we see\n",
    "in the next theorem.\n",
    "\n",
    "**Theorem 18.7.** If a Markov chain with an $n\\times n$ transition matrix $A$ converges to a steady-state\n",
    "vector $\\vec{x}_{ST}$, then \n",
    "  1. $\\vec{x}_{ST}$ is a probability vector.\n",
    "  2. $\\lambda_1 = 1$ is an eigenvalue of $A$ and $\\vec{x}_{ST}$ is an eigenvector corresponding to $\\lambda_1$;\n",
    "     $A\\vec{x}_{ST} = \\vec{x}_{ST}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "773dab59-2a63-4bf6-b39d-077a1e88f5b1",
   "metadata": {},
   "source": [
    "**Remark.** In Example 18.6, python code yielded the eigenvector \n",
    "$(-0.5, -0.5, -0.5, -0.5)$ corresponding to $\\lambda_1$. But \n",
    "$\\vec{x}_{ST} = (0.25, 0.25, 0.25, 0.25)$. What happened? Remember that\n",
    "$\\vec{x}_{ST}$ should be a probability vectore and the sum of its entries\n",
    "must be equal to $1$. So, we have to scale the vector $(-0.5, -0.5, -0.5, -0.5)$\n",
    "by multiplying $\\frac{1}{(-0.5) + (-0.5) + (-0.5) + (-0.5)} = -\\frac{1}{2}$\n",
    "to get $\\vec{x}_{ST} = (0.25, 0.25, 0.25, 0.25)$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3c02c21",
   "metadata": {},
   "source": [
    "### Exercises "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22c20c68",
   "metadata": {},
   "source": [
    "1. Each year, employees at a company are given the option of donating to a local charity as part of a payroll deduction plan. In general, 80 percent of the employees enrolled in the plan in any one year will choose to sign up again the following year, and 30 percent of the unenrolled will choose to enroll the following year. Determine the transition matrix for the Markov process and find the steady-state vector. What percentage of employees would you expect to find enrolled in the program in the long run?\n",
    "2. A system consists of three servers, A, B and C. The specifications are as follows: “If server A is down, then the propability that server B is up 90% and the probability that  server C is up is 10%. Likewise, the diagram illustrates the probability of each server up when a server is down.  <center><img src=\"lecture18-fig4.png\" style=\"width: 200px;\"/>  </center> Determine the transition matrix for the Markov process and find the steady-state vector. What is the chance that server C is up when server A is down in the long run?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e825c2a7",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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