{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 多次元配列の計算"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "行列のおはなし。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ニューラルネットワークのための行列の積\n",
    "\n",
    "入力 $x_1, x_2$ を $X$、重み $w_1, w_2, w_3 ... w_6$ を $W$ とすると、その出力結果 $Y$ はどうなるのか…？というプログラムを書く。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X.shape = (2,)\n",
      "W = [[1 3 5]\n",
      " [2 4 6]]\n",
      "W.shape = (2, 3)\n",
      "Y = [ 5 11 17]\n",
      "Y.shape = (3,)\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "X = np.array([1, 2])                 # 1 x 2の行列\n",
    "print( f'X.shape = {X.shape}')\n",
    "\n",
    "W = np.array([[1, 3, 5], [2, 4, 6]]) # 2 x 3の行列\n",
    "print( f'W = {W}')\n",
    "print( f'W.shape = {W.shape}')\n",
    "\n",
    "Y = np.dot(X, W)                    # 1 x 3 の行列\n",
    "print( f'Y = {Y}')\n",
    "print( f'Y.shape = {Y.shape}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 🤔コピペしたけどよくわからぬ\n",
    "\n",
    "一旦式にしてみよう。\n",
    "\n",
    "入力は2つ、数列は$x_1, x_2$\n",
    "\n",
    "出力結果$Y$は…今回は2つの入力から、3つの出力を得るので\n",
    "$$\n",
    "   Y = y_1, y_2, y_3\n",
    "$$\n",
    "\n",
    "重み$W$は…2つの入力に対して出力を3つにしたいので…重みは $2 \\times 3 = 6$ 個必要だ。\n",
    "$$\n",
    "   w_1, w_2, w_3, w_4, w_5, w_6\n",
    "$$\n",
    "\n",
    "出力結果$Y$を得るための計算は\n",
    "$$\n",
    "    y_1 = x_1 w_1 + x_2 w_2\n",
    "$$\n",
    "$$\n",
    "    y_2 = x_1 w_3 + x_2 w_4\n",
    "$$\n",
    "$$\n",
    "    y_3 = x_1 w_5 + x_2 w_6\n",
    "$$\n",
    "\n",
    "つまり、\n",
    "$$\n",
    "    Y = y_1, y_2, y_3 \\\\\n",
    "    = (x_1 w_1 + x_2 w_2), (x_1 w_3 + x_2 w_4), (x_1 w_5 + x_2 w_6)\n",
    "$$\n",
    "\n",
    "\n",
    "うお、見難いな！…しかしなんとなく行列の積になりそうになってきた。\n",
    "\n",
    "仮に、$Y' = \\begin{bmatrix} y_1 & y_2 & y_3 \\end{bmatrix}$とすると\n",
    "\n",
    "$$\n",
    "    Y' = \\begin{bmatrix}\n",
    "        x_1 w_1 + x_2 w_2 & x_1 w_3 + x_2 w_4 & x_1 w_5 + x_2 w_6\n",
    "    \\end{bmatrix}\\\\\n",
    "    = \\begin{bmatrix}\n",
    "          x_1 & x_2\n",
    "      \\end{bmatrix}\n",
    "      \\begin{bmatrix}\n",
    "          w_1 & w_3 & w_4 \\\\\n",
    "          w_2 & w_5 & w_6\n",
    "      \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "それっぽくなってきた。\n",
    "\n",
    "$\\begin{bmatrix}\n",
    "          x_1 & x_2\n",
    "      \\end{bmatrix}$を$X$、$\\begin{bmatrix}\n",
    "          w_1 & w_3 & w_4 \\\\\n",
    "          w_2 & w_5 & w_6\n",
    "      \\end{bmatrix}$を$W$として、置き換えると\n",
    "      \n",
    "$$\n",
    "    Y' = X \\times W\n",
    "$$\n",
    "\n",
    "つまり、ニューラルネットワークの計算は、行列に置き換えると計算しやすい、ということ。おー、納得した！"
   ]
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