{
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  {
   "cell_type": "markdown",
   "metadata": {
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    "slide_helper": "slide_end",
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     "slide_type": "slide"
    }
   },
   "source": [
    "# 参加者間のネットワークについて"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end",
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   "source": [
    "例2の結論でも述べたように、すべての発言者が$[0,1]$の間の一様な乱数を発言として発するという設定では、参加者の違いや距離などは全く寄与しない。\n",
    "したがって、次に考えることは個人間のリンクにおいて成り立つ関係はどのようになっているかということである。\n",
    "ここでも、まずは先行研究としてなされていることも多いと思われるので、そのまとめと一般化、それから簡単なオリジナルの場合を考えることにする。\n",
    "最終的な結論に行き着くために、これら個人間のリンクについての性質をまとめておき、その後にそれを反映した評価関数や閾値などを設定することにする。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "時刻$k$に$i$が発言を行い、その後時刻$k+1$に$j$が発言$x_{k+1}^{j}$を行う確率$p_{k}(i,j)$を\n",
    "\n",
    "$$p_{k}(i,j) = \\frac{g_{k}(d_{ij})}{\\sum_{j} g_{k}(d_{ij})}$$\n",
    "\n",
    "として考えていた。\n",
    "\n",
    "これは適応度モデルと違い、(初期のやりとりを新たなノードの追加と見なさなければ)新たなノードを追加することはない。\n",
    "また、はじめにスケールフリー性と呼ばれる性質を導くネットワークを作るものとなったWSモデルとも、そのモデルでは枝のつなぎかえの操作であるのに対して、こちらは最初からリンクの数を増やしていく点で異なる。修正WSモデルでは、少数の新たな枝を追加するようにしているが、そのオーダーなどから考えても、現在考えているモデルとは異なることに留意すべきである。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "subslide_end",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "~~さて、現段階(2014/12/09)で良さそうな先行研究なども見つかりそうになかったので~~\n",
    "(複雑ネットワークは基本的にノードを増やしていく操作を行っている。これはむしろ意見の作るネットワークに対して適用すべき事柄である。)\n",
    "、上のような場合を考えるために、設定を別の視点から見直して、シミュレーションしやすい設定に変える。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$n$人の参加者がいるときに、発言者$i$を、アルファベット$A=\\{0, \\dots , n-1\\}$の一点と考え、確率変数$x\\in A$で表すことにする。次にこうしてラベリングされた参加者を円周上に並べる(図)。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end"
    },
    "slide_helper": "subslide_end",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "<img src=circle_walk1.jpg width=200 />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "このとき点$i$から時計回りに$a$目盛進む確率を$p_{i}(a)$と書くことにすると、進む目盛を確率変数$X$で表した時、$X$に対する確率密度関数は$p_{i}(X)$と書ける。\n",
    "また、ここで$X\\in \\{ 0, \\dots , n\\} = A$である。\n",
    "\n",
    "ここで、時刻$k$の粒子の位置を$x_{k}$として、その確率過程を式で表すと\n",
    "\n",
    "$$x_{k+1} = x_{k} + X \\mod n$$\n",
    "\n",
    "となる。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import operator\n",
    "\n",
    "def accumulate(iterable, func=operator.add):\n",
    "    \"\"\"Return running totals\n",
    "    \n",
    "    Usage:\n",
    "    accumulate([1,2,3,4,5]) --> 1 3 6 10 15\n",
    "    accumulate([1,2,3,4,5], operator.mul) --> 1 2 6 24 120\n",
    "    \"\"\"\n",
    "    it = iter(iterable)\n",
    "    total = next(it)\n",
    "    yield total\n",
    "    for element in it:\n",
    "        total = func(total, element)\n",
    "        yield total\n",
    "\n",
    "def circle_walk(P, n=6, i=0, K=100):\n",
    "    '''\n",
    "    alternative view of malkov chain\n",
    "    \n",
    "    [Keyword arguments] = [default value]:\n",
    "    n = 6   -- a number of participants\n",
    "    i = 0   -- initial condition\n",
    "    K = 100 -- trial\n",
    "    '''\n",
    "    k = 0\n",
    "    speaker = [] # time series of participants\n",
    "    speaker.append(i)\n",
    "\n",
    "    def gonext(i):\n",
    "        w = list(accumulate(P[i])) # create weight list\n",
    "        rn = np.random.rand()\n",
    "        for j, _w in enumerate(w):\n",
    "            if rn <= _w:\n",
    "                return j\n",
    "\n",
    "    while True:\n",
    "        if k+1 > K:\n",
    "            break\n",
    "        k += 1\n",
    "        i = (i + gonext(i))%n\n",
    "        speaker.append(i)\n",
    "    return speaker"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
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2xBmFiy7K9+B2u+44TFGlUJb7KOa8+sUoxCgF1z3iKsesi6t+/RpoBjEKtaA7OHt55pBR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\njEL+44pRiNteB2VTlEJRo1C1+wjUeT3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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f73d8209490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 6 # a number of participants\n",
    "K = 100 # trial\n",
    "\n",
    "p = np.random.rand(n) # random probability P_{X}: prob X\n",
    "s = np.sum(p)\n",
    "pp = p/s\n",
    "\n",
    "P = []\n",
    "for i in range(n):\n",
    "    P.append(np.roll(pp, i))\n",
    "P = np.reshape(P, (n,n))\n",
    "\n",
    "ans = circle_walk(P, n=n, i=0, K=K)\n",
    "plt.plot(range(K+1), ans)\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel('speaker')\n",
    "plt.ylim(-1, n)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "-----\n",
    "(2014/12/10)上に述べたようなモデルはマルコフ連鎖で表すことができた。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "(引用)マルコフ連鎖のシミュレーション(山本浩・森隆一・藤曲哲郎, シミュレーションによる確率論, 日本評論社, 137-143, 1993.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "## マルコフ連鎖のシミュレーション"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "時間と共にランダムな変化をする現象を考える。このような例として、バクテリアの増殖、天候の変化、人口の増減などが挙げられる。これらの現象をモデル化するときには、適当な単位時間を設定し、この何倍かの時刻$n$での状態$X_{n}$を調べる。単位時間としては1時間、1日、1年、1世代などが考えられ、もんっ第二より適切なものを選ぶ。確率論論としては確率変数列$X_{0}, X_{1}, \\cdots ,X_{n}, \\cdots$を取り扱うことになる。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "各確率変数$X_{n}$は問題により人数、位置などをあらわす。$X_{n}$のとり得る値全体の集合を状態空間といい、$S$であらわす。6章では$S$は自然数全体、または、その有限の部分集合として定式化できる問題を取り扱う。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "$\\{X_{n}\\}=\\{X_{n}: n=0, 1, \\cdots\\}$に対して、時刻$n+1$の状態$X_{n+1}$の状態$X_{n+1}$は時刻$n-1$までの状態には関係せず、時刻$n$での状態$X_{n}$だけで決まると仮定する。この仮定はマルコフ性とよばれ、現在が与えられたとき過去と現愛は独立であると表現できる。これにより、現在の状態がわかれば未来(次の時刻)の状態を過去の状態に依存せずに確率的に求めることができる。したがって、計算機を用い、乱数の値により一定の操作を繰り返すことにより$\\{X_{n}\\}$のシミュレーションが容易に実行できる。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "一般に確率変数列$\\{X_{n}\\}$に対して、$X_{n}$または$\\displaystyle M_{n} = \\frac{X_{0} + X_{1} + \\cdots + X_{n-1}}{n}$のどちらかの確率変数の$n\\rightarrow \\infty$としたときの状態が問題となる。$X_{n}$が収束すれば$M_{n}$は同じ極限に収束するが、逆は成り立たない。したがって、モデルによりどちらかの確率変数を取り上げる。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "6.1節ではマルコフ連鎖$X_{n}$の説明と記号の準備を行う。6.2節では、現金引き出し機の順番を待つ人の数を扱ったモデルを取り上げ、$M_{n}$の収束を問題とする。6.3節ではランダムウォークを取り扱い、$X_{n}$の$n\\rightarrow \\infty$の状態を考える。6.4節では個体が出生死滅していく集団の個体数を扱った人口モデルを取り上げ、集団が死滅するか個体数が爆発するかを問題とする。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### 6．1 マルコフ連鎖"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "この章の所で述べた$\\{X_{n}\\}$に対するマルコフ性という仮定を、条件付き確率を用いて数学的に定式化しよう。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定義 1** 確率変数$\\{X_{n} :n=0,1,2, \\cdots\\}$が次の性質(マルコフ性)をもつとき、**マルコフ連鎖**とよぶ:\n",
    "\n",
    ">任意の$n$個の状態$i_{0}, i_{1}, \\cdots ,i_{n-1}$と2個の状態$i,j$に対して、\n",
    ">$$P(X_{n+1}=j|X_{0}=i_{0}, \\cdots, X_{n-1} = i_{n-1}, X_{n} = i) = P(X_{n+1}=j|X_{n}=i)$$\n",
    ">このとき、上の条件付き確率を$p(i,j;n)$とあらわし、**時刻$n$での遷移確率**とよぶ。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "定義より$p(i,j;n)$は時刻$n$で状態$i$にいるという条件の下で、次の時刻$n+1$に状態$j$へと移る確率である。$p(i,j;n)$を$(i,j)$成分とする行列$P(n) = (p(i,j;n))_{i,j}$が定義でき、この行列$P(n)$を**と呼ぶ時刻$n$での遷移行列**という。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "$P(n)=(p(i,j;n))_{i,j}$が$n$に依存しないとき、$\\{X_{n}\\}$を時間的に一様なマルコフ連鎖という。この場合はパラメータ$$nの必要性がないので単に遷移行列、遷移確率とよび、$P,p(i,j)$とあらわすことにする。6章では、この時間的に一様なマルコフ連鎖を取り扱う。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "$\\pi(j) = P(X_{0} = j), j\\in S,$で与えられる状態空間$S$上の確率分布$\\pi$をマルコフ連鎖の初期分布という。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "時刻$1$での分布$P(X_{1}=j)$を考えよう。任意の$j$に対して、\n",
    "$$\\{X_{1} = j\\} = \\bigcup_{k}\\{X_{0} = k, X_{1}=j\\}$$\n",
    "右辺の各集合は$k$に対して互いに背反であるから、\n",
    "$$\\begin{align}\n",
    "P(X_{1} = j) &= \\sum_{k}P(X_{0}=k, X_{1}=j)\\\\\n",
    "&= \\sum_{k} P(X_{0} = k)P(X_{1} =j|X_{0}=k)\\\\\n",
    "&= \\sum_{k} \\pi(k)p(k,j).\n",
    "\\end{align}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "$\\pi$を行ベクトルと考えれば、最後の行は行ベクトル$\\pi P$の$j$成分であり、次の補題が成立する。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**補題 1** $P(X_{1} = j) = (\\pi P)(j) = \\sum_{k}\\pi(k)p(k,j)\\ \\ (j\\in S).$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**例 1** 3.3節例題6は時間的に一様なマルコフ連鎖である。この例では単位時間として親から子供への1世代をとる。親から2ステップ後の結果とは、孫が大学に入るとき理工系か文科系かをあらわす。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "状態空間$S$は$\\{0=\\text{理工系}, 1=\\text{文科系}\\}$、遷移確率は\n",
    "\n",
    "$$p(0,0) = 0.5,\\ p(0,1) = 0.5,\\ p(1,0) = 0.2,\\ p(1,1) = 0.8$$\n",
    "\n",
    "遷移行列は、\n",
    "\n",
    "$$P = \\left(\\begin{array}{cc} 0.5 & 0.5 \\\\ 0.2 & 0.8 \\end{array}\\right) .$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "各世代が大学で専攻する割合を次のように行ベクトルで表示する。\n",
    "$$(\\text{理工系の確率}\\ \\ \\text{文科系の確率})$$\n",
    "親が理工系であることは、初期分布が行ベクトル$(1\\ 0)$であると言い換えられ、補題1より、子供が専攻する確率は$(1\\ 0)P = (0.5\\ 0.5)$である。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "定義より$p(i,j)$は非負であり、任意の$i$に対して次の等式が成り立つ。\n",
    "$$\\sum_{j}p(i,j) = 1.$$\n",
    "ここで、遷移行列という概念を行列の性質として(再)定義しよう。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定義 2** すべての成分が非負で、任意の行列に関する和が1となる行列を**確率行列**という。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "マルコフ連鎖が与えられたとき、遷移行列と初期分布が定まる。1．3節で適当な標本空間$\\Omega$での事象に対して加算加法性をもつ集合関数として確率$P(\\cdot)$を定義した。マルコフ連鎖を取り扱う場合、$X_{n}$の各時刻で$n$での値$x_{n}(n=0,1,\\cdots)$を順に並べた$(x_{0},x_{1},\\cdots , x_{n}, \\cdots)$を標本点と考える。したがって、標本空間$\\Omega$を状態空間$S$の元の無限列全体とし、$\\omega\\in \\Omega$に対して$X_{n}(\\omega)$は$\\omega$のf第$n$座標とする($n$は$0$より始める)。この場合、$\\Omega$は一定のものであるから、定義1でのマルコフ性は確率$P$に対する仮定となる。このとき、次の定理が成り立つ。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定理 1** 確率行列$P$と分布$\\pi$が与えられたとき、任意の$n$と状態空間の$n+1$個の点$j_{0},j_{1},\\cdots ,j_{n}$に対して、\n",
    "$$P(X_{0} = j_{0}, X_{1} = j_{1}, \\cdots , X_{n} =j_{n}) = \\pi(j_{0})p(j_{0},j_{1})\\cdots p(j_{n-1}, j_{n})$$\n",
    "をみたす$\\Omega$上の確率$P$がただ1つ存在する。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "この定理はマルコフ連鎖を数学的に扱う場合に基本となる定理である。証明には測度論的な考えが必要となり、ここでは省略する。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**系** 定理1で定まる$P$に対して、$\\{X_{n}\\}$は$\\pi$を初期分布、$P$を遷移行列とするマルコフ連鎖である。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**証明** 定理1の条件式で$n=0$とすることにより、$\\pi$は初期分布となる。遷移行列を計算する。\n",
    "$$\\begin{align}P(X_{n+1} &= j | X_{0} = i_{0}, \\cdots, X_{n-1} = i_{n-1}, X_{n}=i)\\\\\n",
    "&= \\frac{P(X_{0} = i_{0}, \\cdots , X_{n-1} = i_{n-1}, X_{n}=i, X_{n+1} = j)}{P(X_{0} = i_{0}, \\cdots, X_{n-1} = i_{n-1}, X_{n}=i)}\\\\\n",
    "&= \\frac{\\pi(i_{0})p(i_{0},i_{1})\\cdots p(i_{n-1},i_{n})p(i,j)}{\\pi(i_{0})p(i_{0},i_{1})\\cdots p(i_{n-1}, i)}\\\\\n",
    "&= p(i,j)\\ \\ \\ \\blacksquare\n",
    "\\end{align}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**補題 2** $P = (p(i,j))$を遷移行列とするとき、$P^{n}$も遷移行列となる。 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**証明** 行列$P^{n}$の$(i,j)$成分を$p^{(n)}(i,j)$とすれば、示すべきことは、すべての$n$に対し$\\sum_{j}p^{(n)}(i,j) = 1$であり、これを$n$についての数学的帰納法で証明する。$n=1$の場合は、遷移行列の定義式であり、明らかである。\n",
    "\n",
    "いま$n$まで等式が成立していると仮定する。$P^{n+1} = P\\cdot P^{n}$より、\n",
    "\n",
    "$$\\begin{align}\\sum_{j} p^{(n+1)}(i,j) &= \\sum_{j}\\sum_{k}p(i,k)p^{(n)}(k,j)\\\\\n",
    "&= \\sum_{k}p(i,k)\\sum_{j}p^{(n)}(k,j) = \\sum_{k}p(i,k) = 1.\\ \\ \\ \\blacksquare \\end{align}$$\n",
    "\n",
    "状態空間の$n+1$個の点$j_{0}, \\cdots ,j_{n}$に対して、事象$A_{j_{0}, \\cdots ,j_{n}}$を次で定義する。\n",
    "\n",
    "$$A_{j_{0}\\cdots ,j_{n}} = \\{ X_{0} = j_{0}, \\cdots , X_{n}=j_{n}\\}.$$\n",
    "\n",
    "$\\pi(i) = P(X_{0} = i)>0$と仮定する。条件付き確率の定義より、\n",
    "\n",
    "$$P(A_{j_{0},\\cdots , j_{n}} | X_{0} = i) = \\frac{P(A_{j_{0}, \\cdots , j_{n}} \\cap \\{ X_{0} = i \\} )}{ P(X_{0} = i) }$$\n",
    "\n",
    "このとき、"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "(1) $j_{0} = i$の場合、\n",
    "\n",
    "$$\\begin{align}\n",
    "A_{ j_{0}, \\cdots , j_{n} } \\cap \\{ X_{0} = i \\} &= A_{ j_{0}, \\cdots , j_{n} },\\\\\n",
    "P(A_{ j_{0}, \\cdots , j_{n} } | X_{0} = i) &= p(j_{0}, j_{1}) \\cdots p(j_{n-1}, j_{n}).\n",
    "\\end{align}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "(2) $j_{0} \\neq i$の場合、$A_{j_{o}\\cdots, j_{n}} \\cap \\{X_{0} = i\\} = \\phi,\\ P(A_{j_{0},\\cdots ,j_{n}} | X_{0} =i) = 0.$上の結果を定理としてまとめる。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定理 2** $\\pi(i) >0$とするとき、\n",
    "$$P(A_{j_{0}\\cdots ,j_{n}} | X_{0} = i) = \\left\\{ \\begin{array}{ll}\n",
    "p(j_{0},j_{1})\\cdots p(j_{n-1},j_{n}) & i=j_{0}\\text{の場合},\\\\\n",
    "0 & i\\neq j_{0}\\text{の場合}.\n",
    "\\end{array}\\right.$$\n",
    "左辺の条件付き確率は状態$i$より出発した場合の条件付き確率であるから、$\\pi(i) = P(X_{0} =i) =0$の場合には左辺は意味をもたないが、右辺は$\\pi(i)$の値に関係なく意味をもつ。右辺を条件付き確率$P(A_{j_{0}\\cdots ,j_{n}} | X_{0} = i) = P_{\\{X_{0} =i\\}}(A_{j_{0}\\cdots ,j_{n}})$と区別するために$P_{i}$と書きあらわす。これを定義としてまとめよう。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定義 3**\n",
    "$$P_{i}(A_{j_{0}\\cdots ,j_{n}}) = \\left\\{ \\begin{array}{ll}\n",
    "p(j_{0},j_{1})\\cdots p(j_{n-1},j_{n}) & i=j_{0}\\text{の場合},\\\\\n",
    "0 & i\\neq j_{0}\\text{の場合}.\n",
    "\\end{array}\\right.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**系** 事象$A$に対して、$P(A) = \\sum_{i}\\pi(i)P_{i}(A).$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "上で定義した$P_{i}$に関して成り立つことのいくつかの性質を挙げる。\n",
    "\n",
    "(1) $\\pi(i)> 0$の場合、$P_{i}(A) = P(A|X_{0}=i).$\n",
    "\n",
    "事象$A$に対して確率$P_{i}(A)$を$i$より出発した場合のAの確率と言いあらわす。実際の用いる場合、系のように$\\pi(i)$をかける場合が多い。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "(2) $P_{i}(X_{1} = j) = \\sum_{k}P_{i}(X_{0}=k,X_{1} = j) = \\sum_{k}P_{i}(A_{k,j}) = P_{i}(A_{i,j}) = p(i,j).$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "(3) 状態空間の$n$個の点$j_{1},\\cdots ,j_{n}$に対して$B_{j_{1},\\cdots ,j_{n}} = \\{X_{1} = j_{1}, \\cdots , X_{n} =j_{n}\\}$とするとき、\n",
    "$$\\begin{align}P_{i}(B_{j_{1},\\cdots ,j_{n}}) &= \\sum_{j}P_{i}(\\{X_{0} = j\\} \\cap B_{j_{1},\\cdots ,j_{n}})\\\\\n",
    "&= \\sum_{j}P_{i}(A_{j, j_{1},\\cdots ,j_{n}}) = P_{i}(A_{i, j_{1},\\cdots ,j_{n}})\\\\\n",
    "&= p(i,j_{1}) \\cdots p(j_{n-1},j_{n}).\n",
    "\\end{align}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "(4) $p(i,j)>0$と仮定する。\n",
    "$$\\begin{align}P_{i}(X_{2}= k|X_{1} = j) &= \\frac{P_{i}(X_{1} = j, X_{2} = k)}{P_{i}(X_{1} = j)} = \\frac{P_{i}(B_{j,k})}{P_{i}(B_{j})}\\\\\n",
    "&= \\frac{p(i,j)p(j,k)}{p(i,j)} = p(j,k)\\\\\n",
    "&= P_{i}(X_{1=k})\\end{align}$$\n",
    "\n",
    "定理2を一般化した次の定理も同様にして証明できる。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定理 3** 任意の$m,n$と$n+1$個の状態$j_{0},j_{1},\\cdots ,j_{n}$に対して、\n",
    "$$P(X_{m} = j_{0}, \\cdots ,X_{m+n} = j_{n}|X_{m} = i) = P_{i}(A_{j_{0},\\cdots ,j_{n}}).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "定理3の応用として2ステップの変化を考えてみよう。時刻$m$から時刻$m+2$への変化の確率$P(X_{m+2}=j|X_{m}=i)$を計算する。\n",
    "$$\\begin{align}P(X_{m+2} =j|X_{m}=i) &= \\sum_{k}P(X_{m+2} = j, X_{m+1} = k, X_{m}=i) | X_{m}=i)\\\\\n",
    "&= \\sum_{k}P_{i}(A_{i,k,j}) = \\sum_{k}p(i,k)p(k,j).\n",
    "\\end{align}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "最後の和は$P^{2} = P\\times P$の$(i,j)$成分をあらわし、$m$に無関係である。同様にして、次の定理も証明できる。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定理 4** $n$ステップの変化は遷移行列の$n$乗であらわされる。任意の$m\\ge 0, n\\ge 1$と$i,j$に対して、\n",
    "$$P(X_{n+m} = j|X_{m} = i) = p^{(n)}(i,j)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "補題1と同様にして、次の系が一般に成り立つ。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**系** $P(X_{n} = j) = (\\pi P^{n})(j) = \\sum_{k}p^{(n)}(k,j).$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**例2** この系を用いれば3.3節、例題6は簡単に解くことができる。孫が専攻する確率は$(1\\ 0)\\cdot P^{2} = (0.35\\ 0.65)$となるから、孫が理工系である確率は0.35である。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "補題2と定理4より、マルコフ連鎖の変化を$n$ステップごとにみることは$P^{n}$を1つの遷移行列とするマルコフ連鎖をみることと同じである。状態空間$S$上の関数$f_{j}(s)(j\\in S)$を\n",
    "\n",
    "$$f_{j}(s) = \\left\\{ \\begin{array}{ll}1 & s=j\\text{の場合}\\\\\n",
    "0 & s \\neq j\\text{の場合}\\end{array}\\right.$$\n",
    "\n",
    "と定義し、任意の状態$j$に対して、確率変数列$\\{V_{jn}\\}$を\n",
    "\n",
    "$$\\begin{align}V_{j0} &= 0\\\\\n",
    "V_{jn} &= f_{j}(X_{0}) + \\cdots + f_{j}(X_{n-1})\\ \\ \\ \\ (n\\ge 1)\\end{align}$$\n",
    "\n",
    "と定義すると、$V_{jn}$は時刻$n-1$までに$j$を訪れた回数になる。(条件付き)確率$P_{i}(A)$による確率変数$X$の期待値を$E_{i}[X]$とあらわし、条件付き期待値とよぶ。これは4．1節での期待値の定義で、確率$P(A)$を(条件付き)確率$P_{i}(A)$に置き換えたものである。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定義 4**\n",
    "$$E_{i}[X] = \\sum_{k}P_{i}(X=k).$$\n",
    "\n",
    "これを用いれば、\n",
    "$$E_{i}[f_{i}(X_{k})] = P_{i}(X_{k} = j) = p^{(k)}(i,j).$$\n",
    "\n",
    "$k$に関する和をとれば、$E_{i}[V_{jn}]$は行列の和$I + P + \\cdots + P^{n-1}$の$(i,j)$成分である(ただし、$I$は単位行列とし、$P^{0} = i$とする)。この結果をまとめると次の定理5となる。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**定理 5** 状態$i$より出て、時刻$n-1$までの間に状態$j$を訪れた回数の平均は行列$I + P + \\cdots + P^{n-1}$の$(i,j)$成分で与えられる。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**例 3** ここで、3.3節例題6をもう一度考える。この場合、$V_{jn}$は$n$世代までの$n$人のうち理工系の人数の平均をあらわす。\n",
    "\n",
    "\n",
    "$$\\begin{array}{ll}\n",
    "P = \\left( \\begin{array}{cc}0.5 & 0.5 \\\\ 0.2 & 0.8\\end{array}\\right) &\n",
    "P^{2} =\\left( \\begin{array}{cc}0.35 & 0.65 \\\\ 0.26 & 0.74\\end{array}\\right) \\\\\n",
    "P^{3} =\\left( \\begin{array}{cc}0.305 & 0.695 \\\\ 0.278 & 0.722\\end{array}\\right) &\n",
    "P^{4} =\\left( \\begin{array}{cc}0.2915 & 0.7085 \\\\ 0.2924 & 0.7166\\end{array}\\right) \\\\\n",
    "I + P + \\cdots + P^{4} =\\left( \\begin{array}{cc}2.4465 & 2.5535\\\\ 1.0214 & 3.9786\\end{array}\\right) & \\ \\\\\n",
    "\\end{array}$$\n",
    "\n",
    "これより、親が理工系ならば5世代のうち理工系は約半分、親が文系ならば5世代のうち約1人が理工系であることが期待される。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "subslide_end",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### 適用"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "以上の一般のマルコフ連鎖についての議論を踏まえて、現在考えている系に適用すると、確率変数は図でラベル付けされた数字であり、遷移行列はその時刻に依存しない。また、各状態$i$において、遷移確率はその点からの位置関係のみで決まっている。\n",
    "\n",
    "点$i$から時計回りに$a$目盛進む確率を$p_{i}(a)$と書くことにしており、しかもこれが$i$に依らないとき($p_{i}(a) = p(a),\\  ^{\\forall} i$)を考えていたので、このマルコフ連鎖の遷移行列$P$は\n",
    "\n",
    "$$P(n) = P = \\left( \\begin{array}{cccccc}\n",
    "p(0) & p(1) & p(2) & p(3) & p(4) & p(5) \\\\\n",
    "p(5) & p(0) & p(1) & p(2) & p(3) & p(4) \\\\\n",
    "p(4) & p(5) & p(0) & p(1) & p(2) & p(3) \\\\\n",
    "p(3) & p(4) & p(5) & p(0) & p(1) & p(2) \\\\\n",
    "p(2) & p(3) & p(4) & p(5) & p(0) & p(1) \\\\\n",
    "p(1) & p(2) & p(3) & p(4) & p(5) & p(0) \n",
    "\\end{array} \\right)$$\n",
    "\n",
    "のようにかける。\n",
    "\n",
    "定理5から、状態$i$より出て、時刻$n-1$までの間に状態$j$を訪れた回数の平均は行列$I+P + P^{n-1}$の$(i,j)$成分で与えられるから、\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "subslide_end",
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "def cal_f(n, P, K):  \n",
    "    f = np.identity(n) + P\n",
    "    if K == 1:\n",
    "        return f\n",
    "    for k in range(2, K+1):\n",
    "        _k = 2\n",
    "        tmp = P\n",
    "        while _k < k+1:\n",
    "            tmp = np.dot(tmp, P)\n",
    "            _k += 1\n",
    "        f += tmp\n",
    "        \n",
    "    return f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "internals": {
     "slide_type": "subslide"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td> 17.767557</td>\n",
       "      <td> 16.464353</td>\n",
       "      <td> 16.534830</td>\n",
       "      <td> 16.594450</td>\n",
       "      <td> 16.918806</td>\n",
       "      <td> 16.720004</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td> 16.720004</td>\n",
       "      <td> 17.767557</td>\n",
       "      <td> 16.464353</td>\n",
       "      <td> 16.534830</td>\n",
       "      <td> 16.594450</td>\n",
       "      <td> 16.918806</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td> 16.918806</td>\n",
       "      <td> 16.720004</td>\n",
       "      <td> 17.767557</td>\n",
       "      <td> 16.464353</td>\n",
       "      <td> 16.534830</td>\n",
       "      <td> 16.594450</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td> 16.594450</td>\n",
       "      <td> 16.918806</td>\n",
       "      <td> 16.720004</td>\n",
       "      <td> 17.767557</td>\n",
       "      <td> 16.464353</td>\n",
       "      <td> 16.534830</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td> 16.534830</td>\n",
       "      <td> 16.594450</td>\n",
       "      <td> 16.918806</td>\n",
       "      <td> 16.720004</td>\n",
       "      <td> 17.767557</td>\n",
       "      <td> 16.464353</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td> 16.464353</td>\n",
       "      <td> 16.534830</td>\n",
       "      <td> 16.594450</td>\n",
       "      <td> 16.918806</td>\n",
       "      <td> 16.720004</td>\n",
       "      <td> 17.767557</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>6 rows × 6 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "           0          1          2          3          4          5\n",
       "0  17.767557  16.464353  16.534830  16.594450  16.918806  16.720004\n",
       "1  16.720004  17.767557  16.464353  16.534830  16.594450  16.918806\n",
       "2  16.918806  16.720004  17.767557  16.464353  16.534830  16.594450\n",
       "3  16.594450  16.918806  16.720004  17.767557  16.464353  16.534830\n",
       "4  16.534830  16.594450  16.918806  16.720004  17.767557  16.464353\n",
       "5  16.464353  16.534830  16.594450  16.918806  16.720004  17.767557\n",
       "\n",
       "[6 rows x 6 columns]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = cal_f(n,P,K)\n",
    "from pandas import DataFrame\n",
    "DataFrame(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "検算"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "internals": {
     "slide_helper": "subslide_end"
    },
    "slide_helper": "subslide_end",
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td> 2.4465</td>\n",
       "      <td> 2.5535</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td> 1.0214</td>\n",
       "      <td> 3.9786</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>2 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "        0       1\n",
       "0  2.4465  2.5535\n",
       "1  1.0214  3.9786\n",
       "\n",
       "[2 rows x 2 columns]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DataFrame(cal_f(2, np.array([[0.5, 0.5], [0.2, 0.8]]), 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "subslide_end",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Counter({4: 23, 2: 21, 0: 20, 3: 15, 1: 13, 5: 9})\n",
      "[ 17.76755711  16.46435264  16.53482981  16.59445001  16.9188062\n",
      "  16.72000422]\n"
     ]
    }
   ],
   "source": [
    "import collections\n",
    "count = collections.Counter(ans)\n",
    "print count\n",
    "print f[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "subslide_end",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "平均してみて、あっているかどうか。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "internals": {
     "slide_helper": "subslide_end",
     "slide_type": "subslide"
    },
    "slide_helper": "subslide_end",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 167.74596838  166.67534215  166.64948632  166.72541605  166.70481025\n",
      "  166.49897685]\n",
      "1001.0\n",
      "[195.96666666666667, 137.98, 194.79666666666665, 137.38666666666666, 194.93333333333334, 139.93666666666667]\n",
      "1001.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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kYCc23MJzwDvOewusVV8KdAT+A/TEcvyxvgn2r74Kr7xi77nuk08s2A0ebP+hTj/dLuw2\na2ansSUlFhRzzeHDls9fvNha+2edZT15Wra06wwHDtgFtl277Kzg3nvTXePcFw5byu2JJ6wF3rq1\n7YvWrS0HXqeOHYxHj7YeWJIZqhPs3Zw4TQGGYIF8EzAeaATc4Hz+GhboAQYD9wMlQBlwHRUDfTn7\n9qX3Riovde1qLacTT7T0Rt++dvMFWKspV09j69e3Pv1g+dklS6yHRLNmVhYOW5fVpk0tB9uund15\nOGeOzZ93XvTf6dNP7SARmZeqOXLE+nQ//DC8+6612keNyvxeKFJ9ab+p6tFH7YLLY4+lqSaSUd5/\n3wacC4ctFfTSS9bjYuFCC/6/+IVd23j9dXvfts1uHFuyBF57ze5tOHzYXmPGWPfUVMT2WKoN4bAd\nuFauhHfesTOdhg3t+kbknoyTTrIGQf36la8rkgtPZu9eu1nwxRdtvb162bRfGlq5orZb9rXKTy17\nqVzfvnZTS8TPfmb3KfTta7nixYvt+s7w4XYj2dNPw4UXWlAPhWyMow4dLB02dCjcf79dVGzQwLrt\nVWbBAru2sGGDpZ6uuebYd1OHQnYQuvLKaI+jyM12kyfbRemvvrLus5EeVwcP2nWMoiJLkfTvb7nx\nkhK7meeZZ+zi5c6dNn/xxdYza9kyS3m1amUHs379LIhPm2bLde1qPaKuuspSNNOm2dlTmza2zpEj\nraeMusb6U9pb9rffbqfld9yRpppIxguHrYU/alQ08P7f/1mwvPXWxN37wK6LPPywXRTfuDHayj98\n2ILz++/bPQbbtkW7/73xhq27Wze772P+fAueV19taaZdu6Kv2bOty/AFF1j9GjWyFvmuXXbj2x//\naOmRVassXdKtm10I3bTJutM++2zlqbuNG61OX3xhB7zGja2+eXmWhmnWzIL4GWfYzUQ33GC/X1Bg\nw10cOmS594suio7hJNkrqwdCu/Zau9Kf7T1QJLPt3WtBs0kTG5DqmWesRT1tml2UvOwya4UPGWJd\nByO++MLOLGbPthZ68+aWBmne3NJIt91mAf7AAWuJf/WVXZNp1658GmjNGhtyonlzOxCcdFLtbOdX\nX0VHblXPmNyT1cH+iivsNHX06DTVREQkS1Qn2Kf9SVXK2YuI1D4FexERH1CwFxHxAQV7EREfULAX\nEfGBtPbGCYetH/LBg/4aplREJBVZ2xvn0CEL9gr0IiK1K63BXikcERFvKNiLiPiAgr2IiA8o2IuI\n+EDag33jxumsgYiIP6Q12B8+rCcOiYh4Ia3BvrTUhpUVEZHapWAvIuIDboL9RGA7sDym7HTgPeAj\nYAYQm3m/G1gLrAbOP9aKFexFRLzhJthPAkbElf0VuBM4DXgdiDxUsDtwufM+AnjyWL+hYC8i4g03\nwX4esDuurLNTDvA2cKkzfTEwBSgBNgLrgP7JVqxgLyLijVRz9iuxwA5wGZDvTLcBNscstxlom2wl\nCvYiIt6o5Nn2Sf0E+DPwayxnf+QYy4YTFRYWFrJ4sT2kORQKEgwGU6yKiEhuCoVChEKhGlmX26Ey\nC4CZQK8En3UBJgMDgHFO2YPO+yxgPLAo7jvhcDjMY49BcTH8+c9VqrOIiC+lY4jjk2O+fw/wlDM/\nA7gCqAd0wHL7i5OtRGkcERFvuEnjTAGGAC2ATVhLvRFwg/P5a8BzzvQqYKrzfhS4niRpHFCwFxHx\niptgPzpJebLkywPOq1IK9iIi3tAdtCIiPqBgLyLiAwr2IiI+oGAvIuIDaQ32R48q2IuIeEEtexER\nH0h7sM9LdcAGERFxLe3BXi17EZHap2AvIuIDCvYiIj6gYC8i4gMK9iIiPqBgLyLiAwr2IiI+oGAv\nIuIDCvYiIj6gYC8i4gMK9iIiPqBgLyLiA26C/URgO7A8pqw/sBgoApYA/ZzyAuCQU14EPHmsFSvY\ni4h4w02wnwSMiCubAPwa6A3c68xHrHPKewPXH2vFCvYiIt5wE+znAbvjyj4HTnSmmwJbUvlxBXsR\nEW+kOpr8OGA+8BB2wPhOzGcdsBTOXuAeZ7mEFOxFRLyRarB/FrgJeB24DMvrnwdsBfKxM4E+wHSg\nB7A/fgWFhYUUF8PkyVC3bpBgMJhiVUREclMoFCIUCtXIugIulysAZgK9nPl9QJOYdewhmtaJNRe4\nDVgaVx4Oh8MMGgS//z2cc06V6iwi4kuBQADcx+1yUu16uQ4Y4kwPA9Y40y2ASGKmI9AZWJ9sJUrj\niIh4w00aZwoW2FsAm7DeN9cCTwD1sa6W1zrLDgbuB0qAMuA6rNWfkIK9iIg33AT70UnKByQom+a8\nXFGwFxHxhu6gFRHxAQV7EREfULAXEfEBBXsRER9QsBcR8YG0BvujRxXsRUS8oJa9iIgPpD3Y56U6\nOo+IiLiW9mCvlr2ISO1TsBcR8QEFexERH1CwFxHxAQV7EREfULAXEfEBBXsRER9QsBcR8YG0Bftw\n2F510nq4ERHxh7SF2tJSC/SBlB6dKyIiVZHWYK8UjoiIN9wE+4nAdmB5TFl/YDFQBCwB+sV8djew\nFlgNnJ9spQr2IiLecRPsJwEj4somAL8GegP3OvMA3YHLnfcRwJPJfkPBXkTEO26C/Txgd1zZ58CJ\nznRTYIszfTEwBSgBNgLrsLOAChTsRUS8k+oAw+OA+cBD2AHjO055G2BhzHKbgbaJVqBgLyLinVSD\n/bPATcDrwGVYXv+8JMuGExU++GAhX38NhYUQDAYJBoMpVkVEJDeFQiFCoVCNrMttx8cCYCbQy5nf\nBzSJWcceLK0zzil70HmfBYwHFsWtL7x1a5jevWHbthRqLSLiQwHrq55Sh/VUu16uA4Y408OANc70\nDOAKoB7QAeiM9dqpQGkcERHvuEnjTMECewtgE9b75lrgCaA+cMiZB1gFTHXejwLXkySNo2AvIuKd\ndN2/Gi4uDjN8OGzYkKYaiIhkmXSkcapNLXsREe8o2IuI+ICCvYiIDyjYi4j4gIK9iIgPKNiLiPiA\ngr2IiA+kLdgfPapgLyLilbS27PNSHYZNRESqRGkcEREfULAXEfEBBXsRER9QsBcR8QEFexERH1Cw\nFxHxAQV7EREfULAXEfEBBXsRER9QsBcR8QE3wX4isB1YHlP2ClDkvDY47wAF2APII589mWylCvYi\nIt5xMzrNJOBx4IWYsitiph8C9sTMrwN6V7ZSBXsREe+4CfbzsBZ7IgHgR8DQqv6wgr2IiHeqm7M/\nB0vxFMeUdcBSOCFgULIvKtiLiHinuoMMjwZejpnfCuQDu4E+wHSgB7A//oszZhSycycUFkIwGCQY\nDFazKiIiuSUUChEKhWpkXQGXyxUAM4FeMWV5wGYsqG9N8r25wG3A0rjy8GOPhVm7Fh5/3H1lRUT8\nLBAIgPu4XU510jjnAh9TPtC3ACLJmY5AZ2B9oi8rjSMi4h03wX4KsADoAmwCxjjllzufxRoMLMNy\n9n8HrqN8T51vHDkC9eunUGMREakyNzn70UnKxyQom+a8KvX113D88W6WFBGR6krbHbQK9iIi3lGw\nFxHxAQV7EREfSGuw1wVaERFvqGUvIuIDaQv2hw8r2IuIeEUtexERH1CwFxHxAQV7EREfULAXEfEB\nBXsRER9QP3sRER9Qy15ExAcU7EVEfEA3VYmI+IBy9iIiPpC2YF+nDuRV93HnIiLiStqCvVI4IiLe\ncRPsJwLbgeUxZa9gz5ktAjY47xF3A2uB1cD5yVaqYC8i4h03iZRJwOPACzFlV8RMP0T0oeLdsQeR\ndwfaAm9jDyovi1+pgr2IiHfctOznAbuTfBYAfgRMceYvdqZLgI3AOqB/oi/q4qyIiHeqm7M/B0vx\nFDvzbYDNMZ9vxlr4FahlLyLineoG+9HAy5UsE05UqGAvIuKd6nR+zAN+APSJKdsC5MfMt3PKKtix\no5DCQpsOBoMEg8FqVEVEJPeEQiFCoVCNrCvgcrkCYCbQK6ZsBHAXMDSmrDvW0u9P9ALtKVRs3YeH\nDw/z9tsp1FhExKcCgQC4j9vluEnjTAEWYL1qNgFjnPLLiV6YjVgFTHXe/wVcj9I4IiJpl9IRogaE\nL700zKuvpunXRUSyUG237GuFWvYiIt5RsBcR8YG0BXvdVCUi4h217EVEfEDBXkTEBxTsRUR8QMFe\nRMQH0hbs+yccC1NERGpD2h4MODdcSPC++yqUjx8ynsJgYYXywlAh9/1by2t5La/ltXwq0nYHbTic\ncBQFERFJIivvoBUREe8o2IuI+ICCvYiIDyjYi4j4gIK9iIgPKNiLiPiAgr2IiA8o2IuI+ICCvYiI\nD7gJ9hOB7cDyuPKfAx8DK4A/OGUFwCGgyHk9WSO1FBGRanET7CcBI+LKhgIXAacBPYGHYj5bB/R2\nXtfXQB2zTigUSncVak0ubxto+7Jdrm9fdbgJ9vOA3XFlPwN+D5Q48ztrslLZLpf/4HJ520Dbl+1y\nffuqI9WcfWdgMLAQCAF9Yz7rgKVwQsCgatRNRERqSKpDHOcBzYCBQD9gKtAR2ArkY2cCfYDpQA9g\nf7VrKiIita6A8hdo/wUMiZlfB5yU4HtzsaAfbx0Q1ksvvfTSq0qvdaQo1Zb9dGAY8G+gC1AP+BJo\ngbXqS7GWfmdgfYLvn5Li74qISC2ZgqVnDgObgDHAccBkrLX/ARB0lr0E64pZ5JSP9LiuIiIiIiLi\nhRHAamAtcFea61JTNgIfYWczi52y5sBbwBpgNtA0LTVLTaKb6I61PXdj+3M1cL5HdayORNtXCGwm\nejPgBTGfZdP25WPXyVZiZ9g3OeW5sv+SbV8hubH/jgcWAR8Cq7Du7ZCF+68udnGhAEsDfQh0S2eF\nasgGbGfEmgDc6UzfBTzoaY2q5xzshrjYYJhse7pj+/E4bL+uI/OH4Ei0feOBWxMsm23b1wo4w5lu\nBHyC/R/Llf2XbPtyZf8BnOC852Fd2wdRQ/vPyw3vj1VmI3Yz1ivAxR7+fm2KfwDwRcDzzvTzwChv\nq1MtiW6iS7Y9F2PXdEqw/boO28+ZLNH2QeKHOGfb9m3D/vMDHMCGM2lL7uy/ZNsHubH/AA467/Ww\nBvJuamj/eRns22IXeCM2E91R2SwMvA28D4x1ylpiqQKc95ZpqFdNSrY9bbD9GJHN+/TnwDLgWaKn\nydm8fQXYGcwicnP/FWDbt9CZz5X9Vwc7oG0nmrKqkf3nZbAPe/hbXjob+6O7ALgBSxPEivSPzRWV\nbU82butT2J3fZwCfAw8fY9ls2L5GwGvAzVS8oTEX9l8j4FVs+w6QW/uvDNuOdtgoBUPjPk95/3kZ\n7LdgF1gi8il/VMpWnzvvO4HXsdOo7Vh+EaA1sCMN9apJybYnfp+2c8qyzQ6i/4n+SvRUOBu37zgs\n0E/G7oeB3Np/ke17kej25dL+i9gLvAGcSRbuvzygGDv9qkduXKA9AWjsTDcE3sWuiE8g2ttoHNl1\ngRYq3jGdbHsiF4jqYS2rYhLnTjNNAeW3r3XM9C3Ay850tm1fAHgBeCSuPFf2X7Lty5X914JoCqoB\n8B9gOFm6/y7ArqCvw7oMZbsO2D/2h1hXsMg2Ncfy+NnY9TJyE90RojfRHWt7fontz9XAdz2taWri\nt+8nWAD5CMv5Tqf8NZZs2r5BWBrgQ6LdEEeQO/sv0fZdQO7sv17AUmz7PgLucMpzZf+JiIiIiIiI\niIiIiIiIiIiIiIiIiIiIiIiIiIhIVZ0I/KySZd6t5PMDKfzuLc73Irfsn409kvPqFNYlIiKVKKD8\neDcRAdyPFxI/QqQbQ7CxSm6JKftRCusREREXXsEe/FAETMXGYHoeG7MoMjJgpOU+HXv+wAqizyCA\naLBviI04+CF2ADlW8L4Ma9VHHknZGBuPRkREakF7oi379kApFZ/aEwnmzZz3Bs53msV9finwl5jv\nNTnG70YOBG8Bp2It/ZOrUnERL2T68xhF3ArETX9KtLUd72as1f4e1urvHPf5R8B5WHpmELDPxe+/\nBFyFjbi403WtRTyiYC+56qsk5UOwMcIHYk8EKgKOj1tmLdGHkv8W+HWSdbXChksGe6DGJWTQeOIi\nsfLSXQGRGrKf6INkjuVE7CHOX2Npl4EJlmntLPMS9sSga5Ksqx8wJ+b3V6AUjmQoBXvJFV9iXSuX\nAx+T+Fl4fPjUAAAATUlEQVScYWAW8FNgFXYR9724zwFOw54OVIY95CRRl85hQCFQH3seKtij8r6s\nxjaIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIhITfh/07B7xt4HqesAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f73b9461610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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NMycBGzvx/bcBHyZ6W3yx07mWepDBXtVoczvppxB9hB9PPBFoLjCoYJ6ltD6U\n/OvA19pZ1kFEd8kQD9N4P7tYf+JSvv69nQGpAhppfYhMR/YlHuL8KtHscnyReYYn89xGPDHoE+0s\n61hgVt73L8QmHO3CDPaqBi8Tl1YuAJ6k+LM4c8C9wKeBxcRJ3D8VTAc4gng6UDPxkJNil3S+A6gH\nBhLPQoV4TN7L3SiDJEmSJEmSJEmSJEmSJEmSJEmSJEmqlP8HFGTWuOfvGV4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f73b7f47c10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f73c19f7790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f73b7f09850>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ptn77sB8/QDl2O5OOZM72i7Z+kBnbD+CE+9wEayCXkaDtl8xg3xE7wRuwi+CG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\nhP8HHXirIULKEScAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f73c19eb210>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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7/SqmMDpoW1E/JcShwD+wXvEHqB9t9Gvyp4MWvnvFdLSyhDqIirFa1UoabjfN\nNWUcWL5OYa9vxab5gPwsXxEwhfopTEIKZR9GK18h7MP21Dc/HQK8B5xFHu+787Ae9Eps2FC+64b9\nwRdhQ8FCZToSa8fPt6GXkRfRXc3By3IXti+XAedmNKeJaegiwSnY0NnFWE0xvH8l38o3CGsKWET9\nMMThFM4+bKh851EY+7APsAAr28fAHV56oew7ERERERERERERERERERERERERERERERERERERiVcb\n4IYY6/wjxvJvEvjeW73PhS7XPx27JecvEtiWiIjEUMaB892EFNH4+UIiZ4dsjKHYXCW3hqX9PIHt\niIhII7yI3fhhIfASNv/Ss9h8RaGZAUM191exew8sof7+A1Af7A/FZhxchB1ADha8L8Zq9aHbUbbG\n5qIREZE0OIb6mv0xwH6+e9eeUDBv6z0f4n2mbcTyi4Anwz53+EG+N3Qg+DtwLFbT7xBPxkUyJZfv\nxyjSWEURrz+nvrYd6Ras1j4Hq/X3ilj+MXA21jwzCNjRiO+fClyBzba4qdG5FskgBXspRNVR0odi\nc4QPxO4ItBBoGbHOCupvSv7/gd9G2dbR2HTJYDfT+Ck5Np+4SLhm2c6ASArspP4mMgfTBruJ8x6s\n2WVgA+t08taZit0x6Noo2zoFeDvs+5egJhzJYQr2Ugi2YEMrK4BPafhenA6YAfxfYCnWiTsnYjnA\nSdjdgeqwm5w0NKTzTKAcaIHdCxXsNnlbkiiDiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIpMr/\nAlOuIOMuGhg0AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f73b7f509d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 6 # a number of participants\n",
    "K = 1000 # max time\n",
    "\n",
    "\n",
    "N = 300 # trial for averaging\n",
    "\n",
    "p = np.random.rand(n) # random probability P_{X}: prob X\n",
    "s = np.sum(p)\n",
    "pp = p/s\n",
    "P = []\n",
    "for i in range(n):\n",
    "    P.append(np.roll(pp, i))\n",
    "P = np.reshape(P, (n,n))\n",
    "\n",
    "i = 0 # initial value\n",
    "\n",
    "f = cal_f(n, P, K)\n",
    "print f[i]\n",
    "print reduce(operator.add, f[i])\n",
    "\n",
    "_N = 0\n",
    "_ave = [[] for _i in range(n)]\n",
    "c = collections.Counter()\n",
    "while _N < N:\n",
    "    c = c + collections.Counter(circle_walk(P, n, i, K))\n",
    "    _N += 1\n",
    "    for k, v in dict(c).items():\n",
    "        _ave[k].append(float(v)/_N)\n",
    "last = [_ave[a][-1] for a in range(n)]\n",
    "print last\n",
    "print reduce(operator.add, last)\n",
    "        \n",
    "def plot_for_j(j):\n",
    "    plt.plot(range(1, N+1), _ave[j])\n",
    "    plt.plot([1, N], [f[i][j], f[i][j]], '--')\n",
    "    plt.xlabel(r'trials $N$')\n",
    "    plt.ylabel(r'')\n",
    "    plt.show()\n",
    "\n",
    "for j in range(n):\n",
    "    plot_for_j(j)"
   ]
  },
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    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "どう見ても別の値で収束している???"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {
     "slide_type": "subslide"
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   "source": [
    "ともかく(?)、このようにマルコフ連鎖としてこの状況をとらえたとき、距離の効果を入れるなら、遷移確率にその効果を埋め込むことが出来る。このとき確率$p(i,j)$と$p(j,i)$は一般には異なっていても良い。\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_helper": "subslide_end"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "参考になる?[計算機実習 問題12.16 拡散が支配する1次元化学反応系のモンテカルロ・シミュレーション](http://nbviewer.ipython.org/github/ssh0/12-16_diffusion-dominant_chemical_reaction/blob/master/12-16_diffusion-dominant_chemical_reaction.ipynb)"
   ]
  }
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