diff --git a/BP Algorithm.ipynb b/BP Algorithm.ipynb
new file mode 100644
index 0000000..de98784
--- /dev/null
+++ b/BP Algorithm.ipynb	
@@ -0,0 +1,545 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# BP Algorithm\n",
+    "谢昊君 1200015556 元培学院"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# k层BP 算法推导\n",
+    "$W^{(k)}$表示k-1层到第k层的权重矩阵,$z^i=f(W^i\\cdot a^{i-1}+b)$,或每次将$a$加入单位向量，使原式变为$z^i =[f(W^i\\cdot a^{i-1});1]$\n",
+    "\n",
+    "$$a^{(i+1)}=f(z^{(i+1)})$$\n",
+    "\n",
+    "激活函数满足$f'(z)=f(z)(1-f(z))$\n",
+    "\n",
+    "$h_{W,b}(x)$表示该神经网络对x特征的预测\n",
+    "\n",
+    "损失函数为：\n",
+    "$$J(W,b; x,y) = \\frac{1}{2} \\left\\| h_{W,b}(x) - y \\right\\|^2.$$\n",
+    "对m个输入样本而言，总损失函数为:\n",
+    "$$J(W,b)=\\frac{1}{m}\\sum_i ^m J(W,b;x^{(i)},y^{(i)})$$\n",
+    "\n",
+    "对该损失函数按梯度最优化更新权重矩阵$W$，即\n",
+    "\\begin{align}\n",
+    "W_{ij}^{(l)} &= W_{ij}^{(l)} - \\alpha \\frac{\\partial}{\\partial W_{ij}^{(l)}} J(W,b) \\\\\n",
+    "b_{i}^{(l)} &= b_{i}^{(l)} - \\alpha \\frac{\\partial}{\\partial b_{i}^{(l)}} J(W,b)\n",
+    "\\end{align}\n",
+    "\n",
+    "接下来求$\\frac{\\partial}{\\partial W_{ij}^{(l)}} J(W,b) $.\n",
+    "\n",
+    "首先考虑损失函数对神经元的导数,每一个神经元对loss function的影响/贡献,定义$\\delta_i ^k$ 为第k层第i个神经元对loss function的贡献\n",
+    "\n",
+    "于是有\n",
+    "$$\\delta_i ^k = \\frac{\\partial}{\\partial z_i ^k} \\frac{1}{2} \\left\\| h_{W,b}(x) - y \\right\\|^2 = - (y_i - a^{(n_l)}_i) \\cdot f'(z^{(n_l)}_i)\n",
+    "$$\n",
+    "\n",
+    "继续向内层求导则有：\n",
+    "\n",
+    "$$\\delta^{(l)}_i = \\left( \\sum_{j=1}^{s_{l+1}} W^{(l)}_{ji} \\delta^{(l+1)}_j \\right) f'(z^{(l)}_i)$$\n",
+    "\n",
+    "于是根据链式法则对权重$W_ij$和$b$有\n",
+    "$$ \\frac{\\partial}{\\partial W_{ij}^{(l)}} J(W,b; x, y) = a^{(l)}_j \\delta_i^{(l+1)}$$\n",
+    "$$ \\frac{\\partial}{\\partial b_{i}^{(l)}} J(W,b; x, y) = \\delta_i^{(l+1)}$$\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import numpy.random as rd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "# def activation function\n",
+    "def sigmoid(x):\n",
+    "    return 1.0/(1.0 + np.exp(-x))\n",
+    "\n",
+    "def sigmoid_prime(x):\n",
+    "    return sigmoid(x)*(1.0-sigmoid(x))\n",
+    "\n",
+    "def tanh(x):\n",
+    "    return np.tanh(x)\n",
+    "\n",
+    "def tanh_prime(x):\n",
+    "    return 1.0 - x**2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 186,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "# 双层的前馈神经网络的BP算法。\n",
+    "class NeuralNetwork:\n",
+    "    # 默认激活函数为S型函数\n",
+    "    def __init__(self, layers, activation='sigmoid'):\n",
+    "        # 精度设置\n",
+    "        self.error_criterion = []\n",
+    "        # 初始化error数(随机设置一个比criterion大的数),假如小于criterion，即停止迭代\n",
+    "        self.error = 1\n",
+    "        if activation == 'sigmoid':\n",
+    "            self.activation = sigmoid\n",
+    "            self.activation_prime = sigmoid_prime\n",
+    "        elif activation == 'tanh':\n",
+    "            self.activation = tanh\n",
+    "            self.activation_prime = tanh_prime\n",
+    "\n",
+    "        # Set weights\n",
+    "        self.weights = []\n",
+    "        # Layer 设置\n",
+    "        for i in range(1, len(layers) - 1):\n",
+    "            # 初始权重介于-1,1之间，生成一特征数量*神经元数量的权重矩阵，+1表示常数项对应的权重。\n",
+    "            r = 2*np.random.random((layers[i-1] + 1, layers[i] )) -1\n",
+    "            self.weights.append(r)\n",
+    "        # output layer - random((2+1, 1)) : 3 x 1\n",
+    "        r = 2*np.random.random( (layers[i] + 1, layers[i+1])) - 1\n",
+    "        self.weights.append(r)\n",
+    "\n",
+    "    def fit(self, X, y, learning_rate=1, epochs=10000,error_criterion=0.01):\n",
+    "        self.error_criterion = error_criterion\n",
+    "        # epochs为迭代次数上限，最终选择在误差小于criterion后或者迭代100000次后，停止。         \n",
+    "        for k in range(epochs):\n",
+    "            if k % 1000 == 0: \n",
+    "                print('epochs:', k)\n",
+    "            \n",
+    "            # 计算mean squared error，若小于criterion即停止更新。\n",
+    "            prediction = self.predict(X)\n",
+    "            mean_square_error = ((y-prediction)**2).mean()\n",
+    "            self.error = mean_square_error\n",
+    "            if mean_square_error < error_criterion:\n",
+    "                break\n",
+    "            # 随机选取元素做更新\n",
+    "            i = np.random.randint(X.shape[0])\n",
+    "            a = [X[i]]\n",
+    "\n",
+    "            for l in range(len(self.weights)):\n",
+    "                    dot_value = np.dot(np.hstack((1,a[l])), self.weights[l])\n",
+    "                    activation = self.activation(dot_value)\n",
+    "                    a.append(activation)\n",
+    "\n",
+    "            errors = y[i] - a[-1]\n",
+    "            \n",
+    "            deltas = [errors * self.activation_prime(a[-1])]\n",
+    "            \n",
+    "            #生成倒序delta\n",
+    "            for l in range(len(a) - 2, 0, -1): \n",
+    "                deltas.append(np.dot(self.weights[l][1:],deltas[-1]).dot(self.activation_prime(a[l])))\n",
+    "\n",
+    "            deltas.reverse()\n",
+    "\n",
+    "            # 更新权重\n",
+    "            for i in range(len(self.weights)):\n",
+    "                layer = np.atleast_2d(np.hstack((1,a[i])))\n",
+    "                delta = np.atleast_2d(deltas[i])\n",
+    "                self.weights[i] += learning_rate * layer.T.dot(delta)\n",
+    "            \n",
+    "    # 由输入和参数计算神经网络输出值\n",
+    "    def predict(self, X): \n",
+    "        tmp = X\n",
+    "        for l in range(0, len(self.weights)):\n",
+    "            ones = np.ones((tmp.shape[0],1))\n",
+    "            dots =np.dot(np.hstack((ones, tmp)), self.weights[l])\n",
+    "            tmp = self.activation(dots)\n",
+    "        return tmp\n",
+    "       \n",
+    "    def mean_square_error(self,X,y):\n",
+    "        return ((y-self.predict(X))**2).mean()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 第二题\n",
+    "对XOR运算进行训练,但结果不是很理想.\n",
+    "即使在迭代9000次,(之前尝试国100000次的情形),最终结果仍然停留在0.5处.不明白原因."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 187,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "epochs: 0\n",
+      "epochs: 1000\n",
+      "epochs: 2000\n",
+      "epochs: 3000\n",
+      "epochs: 4000\n",
+      "epochs: 5000\n",
+      "epochs: 6000\n",
+      "epochs: 7000\n",
+      "epochs: 8000\n",
+      "epochs: 9000\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Test for XOR\n",
+    "nn = NeuralNetwork([2, 3, 1])\n",
+    "\n",
+    "X = np.array([[0, 0],[0, 1],[1, 0],[1, 1]])\n",
+    "\n",
+    "y = np.array([0, 1, 1, 0])\n",
+    "\n",
+    "nn.fit(X, y)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 188,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 0.3429196 ],\n",
+       "       [ 0.35613153],\n",
+       "       [ 0.33280655],\n",
+       "       [ 0.33445151]])"
+      ]
+     },
+     "execution_count": 188,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "nn.predict(X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 189,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.26937145976643928"
+      ]
+     },
+     "execution_count": 189,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "nn.error"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Test for Question 3\n",
+    "使用2-m-1结构的前馈网络逼近$[0,1]^2$上的连续函数$y=f(x)=1/(1+\\sqrt(x_1 ^2+x_2 ^2))$\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "# train_fun for numpy array\n",
+    "def train_fun(x):\n",
+    "    return 1/(1+(x*x).sum(1))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "# 1*1 内的均匀格点\n",
+    "data=[]\n",
+    "for i in range(100):\n",
+    "    for j in range(100):\n",
+    "        data.append([i,j])\n",
+    "data = np.array(data)\n",
+    "data = 0.005 + data/100"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "# 随机选取5000个\n",
+    "train_loc = rd.choice(10000,5000,replace = False)\n",
+    "tmp = set(train_loc)\n",
+    "tmp2 = set(range(10000))\n",
+    "tmp3 = tmp2.difference_update(tmp)\n",
+    "training_data = data[train_loc]\n",
+    "testing_data = data[list(tmp3)]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "此处不知道为什么，速度很慢。始终难以收敛到要求。（以及达到了，但没有正常break）"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 148,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "epochs: 0\n"
+     ]
+    },
+    {
+     "ename": "KeyboardInterrupt",
+     "evalue": "",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[1;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
+      "\u001b[1;32m<ipython-input-148-28a25798e9a2>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[0mtestnn\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mNeuralNetwork\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m5\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      2\u001b[0m \u001b[0my\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mtrain_fun\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mtraining_data\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 3\u001b[1;33m \u001b[0mtestnn\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mfit\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mtraining_data\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[1;32m<ipython-input-147-141028aa2cca>\u001b[0m in \u001b[0;36mfit\u001b[1;34m(self, X, y, learning_rate, epochs, error_criterion)\u001b[0m\n\u001b[0;32m     34\u001b[0m             \u001b[1;31m# 计算mean squared error，若小于criterion即停止更新。\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     35\u001b[0m             \u001b[0mprediction\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpredict\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mX\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 36\u001b[1;33m             \u001b[0mmean_square_error\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m(\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m-\u001b[0m\u001b[0mprediction\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m**\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmean\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     37\u001b[0m             \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0merror\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mmean_square_error\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     38\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0mmean_square_error\u001b[0m \u001b[1;33m<\u001b[0m \u001b[0merror_criterion\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
+      "\u001b[1;31mKeyboardInterrupt\u001b[0m: "
+     ]
+    }
+   ],
+   "source": [
+    "testnn = NeuralNetwork([2,5,1])\n",
+    "y = train_fun(training_data)\n",
+    "testnn.fit(training_data,y)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 152,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[array([[-0.50325027, -0.80837581, -0.71209436,  0.54072039,  0.15305065],\n",
+       "        [-1.76344064, -1.44920066, -0.89819535, -1.91194376, -1.52794614],\n",
+       "        [-2.01661244, -1.75081518, -1.8705454 , -0.86569561, -1.70634533]]),\n",
+       " array([[-0.05322174],\n",
+       "        [ 0.52131733],\n",
+       "        [ 0.6899994 ],\n",
+       "        [ 0.60008547],\n",
+       "        [ 0.0413903 ],\n",
+       "        [ 1.7299185 ]])]"
+      ]
+     },
+     "execution_count": 152,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "testnn.weights"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "此处显示对训练样本的mean squared error已经小于error criterion"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 53,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.0057749077749055504"
+      ]
+     },
+     "execution_count": 53,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "testnn.error"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 153,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 0.5693499 ],\n",
+       "       [ 0.60821122],\n",
+       "       [ 0.65583173],\n",
+       "       ..., \n",
+       "       [ 0.57624142],\n",
+       "       [ 0.59523835],\n",
+       "       [ 0.57949126]])"
+      ]
+     },
+     "execution_count": 153,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "testnn.predict(training_data)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 178,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "predition = testnn.predict(testing_data)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 179,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "tmp = predition - train_fun(testing_data)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "对样本的预测mean squared error远远大于训练样本，显示由过拟合的现象。这也与损失函数中并没有添加正则项有关。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 181,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.030990427107420407"
+      ]
+     },
+     "execution_count": 181,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "tmp**2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "def mean_squared_error()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "# 对不同的m值分别测试，计算predicting error并记录。\n",
+    "error =[]\n",
+    "for m in range(3,10):\n",
+    "    nn = NeuralNetwork[[2,m,1]]\n",
+    "    nn.fit(training_data,y)\n",
+    "    error.append(nn.mean_sqaured_error(testing_data,train_fun(testing_data)))\n",
+    "    "
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.5.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}