diff --git a/5_nn/1-Perceptron.ipynb b/5_nn/1-Perceptron.ipynb
index c908e87..24e2692 100644
--- a/5_nn/1-Perceptron.ipynb
+++ b/5_nn/1-Perceptron.ipynb
@@ -197,19 +197,24 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[2 2 2 2 2 1 1 2 1 1 1 1 2 1 1 1 1 2 2 2]\n"
+     ]
+    },
+    {
      "data": {
-      "image/png": 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psOK+ldyy9noioub1SlI1JnyBX/SPH+aNhy5gw5pNnH7+nzF1+lQu+OKH+KuTP8vGtZs587LTWXDQ61/xnt6eXlbc+ySVytD00M/v+Le6ZFnz1Iu//UjWbRtfor+3n/Zp7XVZtySN1YQv8PapUzjzsve8YtmiQxbw9ae+uPf3TGvn9b/7OtY+vY6W1uCwow+uS5b3XHgyt1xzBxHB2057i+UtqVAT/iBmtTZ3b+Hb//B9ps6YynsuOJnpM6fVZb1PPbSK3p4+Djv6YKdPJDXE3g5iTtoCl6TJorRnoYyn/r5+Hv35E6x7trvoKJI0ZhN+Dny89PX2c/5bPzV0YLJS4b/d8kne8mdHFh1LkvZZ047An+x6ijW/eZGel3bSu6OPb37u20VHkqQxadoC71h4AJWBoVMCp0ydwhsPW1BwIkkam6Yt8Plv6uDyWz7BEccdxn/4wDv48DXnFB1JksakaefAAY4++SiOPvmoomNIUlWqHoFHxKKI+ElEPBYRj0bEhfUMJkl6dbWMwAeAT2TmAxExG1geEXdm5mN1yiZJehVVj8Azc01mPjD8eBuwAvBIoCQ1SF0OYkbEYuBI4L49fG9JRHRFRFd3txfMSFK91FzgETEL+BfgoszcOvL7mbk0Mzszs7Ojo6PWzUmShtVU4BExhaHyvjkzb6tPJEnSvqjlLJQAvgKsyMzP1y+SJGlf1DICPxZ4H3B8RDw4/OfkOuWSJI2i6tMIM/MewA/ElqSCNO2l9JJUdhZ4FTZ3b2HHtp6iY0hqchb4GP3vi7/KWYs+wnvnn8v/+9f7i44jqYlZ4GOwZf1WvvOlHzHQN0Dfzn6+9PGv1WW9v37waf761L/jcx+4ji3rdzuVXpL2qKk/jXCs2qdNoaVl6LhtBMyeO6vmdfb19vPJP72C7Vt20DallfXPrefqu66oeb2SJj9H4GMwfdZ0/vqfP87rFr+Wg448kE8vu7jmdW7fvJ3enj4ABvoHee6JF2pep6Tm0LQj8J7tO5nS3kbblLH9FRxzaifHnLrbzaGrNue1+3Pk8UfwyD0rqFQqnPHxU+u2bkmTW1MW+NJLvs5t136PKVPbuPKOy/ijPz28sCwRwZXfuZQV9z7JzP1ncOARbyosi6RyaboplA1rNvGv/+v7DA4MsnN7L/9w3vVFR6K1tZXD/+Qwy1vSmDRdgU9pb+PlC0gjYPqsacUGkqQqNV2B73fAbC76xyXM6diPRYcu5NKbzi86kiRVJTKzYRvr7OzMrq6uhm1PkiaDiFiembudPdF0I3BJmiwscEkqKQtckkrKApekkrLAJamkLHBJKikLXJJKqqHngUdEN/BMwzZYH/OA9UWHKEgz7zu4/828/xNt39+UmR0jFza0wMsoIrr2dAJ9M2jmfQf3v5n3vyz77hSKJJWUBS5JJWWBj25p0QEK1Mz7Du5/M+9/KfbdOXBJKilH4JJUUha4JJWUBb4HEbEoIn4SEY9FxKMRcWHRmYoQEa0R8cuI+G7RWRotIuZExK0R8XhErIiIY4rO1CgRcfHwz/0jEbEsIib1basi4oaIWBcRj+yybG5E3BkRK4e/vqbIjHtjge/ZAPCJzHwz8FbgvIh4c8GZinAhsKLoEAX5AvDDzDwU+EOa5O8hIhYAFwCdmXk40AqcWWyqcfc14KQRyy4D7srMg4G7hp9POBb4HmTmmsx8YPjxNob+8S4oNlVjRcRC4BSg+Ls+N1hE7A8cB3wFIDP7MnNzoaEaqw2YHhFtwAzghYLzjKvM/BmwccTi04Abhx/fCJzeyEz7ygIfRUQsBo4E7is4SqNdC1wCVArOUYQDgW7gq8NTSNdHxMyiQzVCZj4PXAM8C6wBtmTmj4pNVYj5mblm+PFaYH6RYfbGAn8VETEL+BfgoszcWnSeRomIdwHrMnN50VkK0gYcBXwpM48EtjNBf4Wut+G53tMY+k/sDcDMiDi72FTFyqFzrSfk+dYW+F5ExBSGyvvmzLyt6DwNdizw7ohYBXwTOD4ivlFspIZaDazOzJd/67qVoUJvBicCT2dmd2b2A7cBbys4UxFejIjXAwx/XVdwnj2ywPcgIoKh+c8Vmfn5ovM0WmZ+KjMXZuZihg5g/Tgzm2YUlplrgeci4pDhRScAjxUYqZGeBd4aETOG/x2cQJMcwB3hDuCc4cfnALcXmGWvLPA9OxZ4H0MjzweH/5xcdCg11PnAzRHxMPBHwGeLjdMYw7913Ao8APyKoY4oxWXl1YqIZcC9wCERsToizgWuAt4ZESsZ+q3kqiIz7o2X0ktSSTkCl6SSssAlqaQscEkqKQtckkrKApekkrLAJamkLHBJKqn/D+qhvYKqDzFRAAAAAElFTkSuQmCC\n",
+      "image/png": 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jiZui/hQdHc3hwoKT7lNUnEd0dPnz3EVEgkGBfpyePXtSUHiAfQe2ldmeX5DD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\nAElFTkSuQmCC\n",
       "text/plain": [
-       "<Figure size 432x288 with 1 Axes>"
+       "<matplotlib.figure.Figure at 0x7f2187fecc88>"
       ]
      },
-     "metadata": {
-      "needs_background": "light"
-     },
+     "metadata": {},
      "output_type": "display_data"
     }
    ],
@@ -238,12 +243,16 @@
     "x = x[shuffled_index]\n",
     "y = y[shuffled_index]\n",
     "\n",
-    "\n",
     "train_data = np.concatenate((x, y[:, np.newaxis]), axis=1)\n",
     "\n",
+    "label_y = y.copy()\n",
+    "label_y[label_y==-1] = 2;\n",
+    "print(label_y)\n",
     "# plot data\n",
-    "plt.scatter(train_data[:,0], train_data[:,1], c=train_data[:,2], marker='.')\n",
+    "plt.scatter(train_data[:,0], train_data[:,1], marker='.',  s = 300,\n",
+    "            c=label_y, cmap=plt.cm.Spectral)\n",
     "plt.title(\"Data\")\n",
+    "plt.savefig(\"perceptron_sample_data.pdf\")\n",
     "plt.show()\n"
    ]
   },
@@ -256,7 +265,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": 21,
    "metadata": {
     "lines_to_end_of_cell_marker": 2
    },
@@ -265,37 +274,47 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "update weight/bias:  3.662519607024163 3.038628576040485 0.5\n",
-      "update weight/bias:  3.1283482021588505 1.595298136548129 0.0\n",
-      "update weight/bias:  1.7022056555578562 0.9488336160189257 -0.5\n",
-      "update weight/bias:  0.6064534385664728 0.25821553124912766 -1.0\n",
-      "update weight/bias:  -0.0072588404447664345 -1.1732780618880432 -1.5\n",
-      "update weight/bias:  3.56979581700504 4.157268901515593 -1.0\n",
-      "update weight/bias:  2.7104258301108155 3.6232316160976543 -1.5\n",
-      "update weight/bias:  1.6146736131194321 2.9326135313278563 -2.0\n",
-      "update weight/bias:  1.0805022082541196 1.4892830918355 -2.5\n",
-      "update weight/bias:  -0.3456403383468747 0.8428185713062968 -3.0\n",
-      "update weight/bias:  3.4230241861565407 3.6955935657768997 -2.5\n",
-      "update weight/bias:  2.5636541992623156 3.161556280358961 -3.0\n",
-      "update weight/bias:  1.1375116526613214 2.5150917598297577 -3.5\n",
-      "update weight/bias:  0.38678027777241897 1.1329290169473543 -4.0\n",
-      "update weight/bias:  -1.446079265832824 -0.7361650837497964 -4.5\n",
-      "update weight/bias:  1.7930043074867144 5.9278879714909145 -4.0\n",
-      "update weight/bias:  1.2588329026214018 4.484557531998558 -4.5\n",
-      "update weight/bias:  0.3697277316535954 3.2936957431536 -5.0\n",
-      "update weight/bias:  -0.519377439314211 2.1028339543086423 -5.5\n",
-      "update weight/bias:  -2.352236982919454 0.23373985361149163 -6.0\n",
-      "update weight/bias:  1.8159337720901148 4.415105700242464 -5.5\n",
-      "update weight/bias:  0.7328910527198487 3.024123458986641 -6.0\n",
-      "update weight/bias:  -0.3501516666504174 1.6331412177308182 -6.5\n",
-      "w =  [-0.3501516666504174, 1.6331412177308182]\n",
-      "b =  -6.5\n",
+      "update weight/bias:  2.9024433699190153 3.129619118339762 0.5\n",
+      "update weight/bias:  2.013338198951209 1.9387573294948042 0.0\n",
+      "update weight/bias:  0.4222250668142189 0.37906946899829386 -0.5\n",
+      "update weight/bias:  -0.4668801041535875 -0.8117923198466639 -1.0\n",
+      "update weight/bias:  1.6670650562137663 2.551793701566567 -0.5\n",
+      "update weight/bias:  0.8076950693195415 2.0177564161486283 -1.0\n",
+      "update weight/bias:  -0.05167491757468334 1.4837191307306896 -1.5\n",
+      "update weight/bias:  -0.8024062924635857 0.10155638784828613 -2.0\n",
+      "update weight/bias:  2.4366772808559527 6.765609443088997 -1.5\n",
+      "update weight/bias:  1.6859459059670503 5.383446700206593 -2.0\n",
+      "update weight/bias:  0.25980335936605603 4.7369821796773905 -2.5\n",
+      "update weight/bias:  -0.6293018116017504 3.546120390832433 -3.0\n",
+      "update weight/bias:  -1.7123445309720164 2.1551381495766098 -3.5\n",
+      "update weight/bias:  -2.246515935837329 0.7118077100842535 -4.0\n",
+      "update weight/bias:  0.9925676374822094 7.375860765324964 -3.5\n",
+      "update weight/bias:  0.1331976505879846 6.841823479907025 -4.0\n",
+      "update weight/bias:  -0.48051462842325465 5.410329886769854 -4.5\n",
+      "update weight/bias:  -1.906657175024249 4.763865366240651 -5.0\n",
+      "update weight/bias:  -2.6573885499131515 3.3817026233582475 -5.5\n",
+      "update weight/bias:  1.0051310571110115 6.420331199398733 -5.0\n",
+      "update weight/bias:  -0.07791166225925461 5.0293489581429105 -5.5\n",
+      "update weight/bias:  -1.1609543816295207 3.638366716887088 -6.0\n",
+      "update weight/bias:  -1.6951257864948333 2.1950362773947316 -6.5\n",
+      "update weight/bias:  3.626233232216561 6.632855560776491 -6.0\n",
+      "update weight/bias:  2.5304810152251775 5.942237476006693 -6.5\n",
+      "update weight/bias:  1.1043384686241833 5.29577295547749 -7.0\n",
+      "update weight/bias:  -0.7285210749810598 3.4266788547803397 -7.5\n",
+      "update weight/bias:  -1.342233353992299 1.995185261643169 -8.0\n",
+      "update weight/bias:  1.5602100159267163 5.124804379982931 -7.5\n",
+      "update weight/bias:  1.0260386110614037 3.6814739404905743 -8.0\n",
+      "update weight/bias:  0.13693344009359731 2.4906121516456166 -8.5\n",
+      "w =  [0.13693344009359731, 2.4906121516456166]\n",
+      "b =  -8.5\n",
+      "\n",
       "\n",
       "\n",
       "ground_truth:  [-1. -1. -1. -1. -1.  1.  1. -1.  1.  1.  1.  1. -1.  1.  1.  1.  1. -1.\n",
       " -1. -1.]\n",
-      "predicted:     [-1. -1. -1. -1. -1. -1.  1. -1.  1.  1.  1.  1. -1.  1.  1.  1.  1. -1.\n",
-      " -1. -1.]\n"
+      "predicted:     [-1. -1. -1. -1.  1.  1.  1. -1.  1.  1.  1.  1. -1.  1.  1.  1.  1. -1.\n",
+      " -1. -1.]\n",
+      "accuracy:      0.95\n"
      ]
     }
    ],
@@ -303,12 +322,14 @@
     "import random\n",
     "import numpy as np\n",
     "\n",
-    "# 符号函数\n",
+    "\n",
     "def sign(v):\n",
+    "    \"\"\"符号函数\"\"\"\n",
     "    if v > 0:  return 1\n",
     "    else:      return -1\n",
     "    \n",
     "def perceptron_train(train_data, eta=0.5, n_iter=100):\n",
+    "    \"\"\"对感知机模型进行训练\"\"\"\n",
     "    weight = [0, 0]      # 权重\n",
     "    bias = 0             # 偏置量\n",
     "    learning_rate = eta  # 学习速率\n",
@@ -331,6 +352,7 @@
     "    return weight, bias\n",
     "\n",
     "def perceptron_pred(data, w, b):\n",
+    "    \"\"\"输入数据，模型，对数据进行分类\"\"\"\n",
     "    y_pred = []\n",
     "    for d in data:\n",
     "        x1, x2, y = d\n",
@@ -340,17 +362,24 @@
     "    return np.array(y_pred, dtype=float)\n",
     "\n",
     "\n",
-    "# do training\n",
+    "# 训练感知机\n",
     "w, b = perceptron_train(train_data)\n",
     "print(\"w = \", w)\n",
     "print(\"b = \", b)\n",
     "\n",
-    "# predict \n",
+    "# 预测 \n",
     "y_pred = perceptron_pred(train_data, w, b)\n",
     "\n",
+    "# 计算分类精度\n",
+    "c = y_pred == y\n",
+    "cn = np.sum(c == True)\n",
+    "acc = cn / len(y_pred)\n",
+    "print()\n",
+    "\n",
     "print(\"\\n\")\n",
     "print(\"ground_truth: \", train_data[:, 2])\n",
-    "print(\"predicted:    \", y_pred)"
+    "print(\"predicted:    \", y_pred)\n",
+    "print(\"accuracy:     \", acc)"
    ]
   },
   {
@@ -380,7 +409,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.9"
+   "version": "3.5.4"
   }
  },
  "nbformat": 4,
diff --git a/5_nn/2-mlp_bp.ipynb b/5_nn/2-mlp_bp.ipynb
index b7ef799..7a79ea5 100644
--- a/5_nn/2-mlp_bp.ipynb
+++ b/5_nn/2-mlp_bp.ipynb
@@ -719,7 +719,9 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "import numpy as np\n",
@@ -1024,7 +1026,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.9"
+   "version": "3.5.4"
   }
  },
  "nbformat": 4,
diff --git a/5_nn/3-softmax_ce.ipynb b/5_nn/3-softmax_ce.ipynb
index 651302d..f222841 100644
--- a/5_nn/3-softmax_ce.ipynb
+++ b/5_nn/3-softmax_ce.ipynb
@@ -11,13 +11,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "softmax经常被添加在分类任务的神经网络中的输出层，神经网络的反向传播中关键的步骤就是求导，从这个过程也可以更深刻地理解反向传播的过程，还可以对梯度传播的问题有更多的思考。\n",
+    "`Softmax`经常被添加在分类任务的神经网络中的输出层，神经网络的反向传播中关键的步骤就是求导，从这个过程也可以更深刻地理解反向传播的过程，还可以对梯度传播的问题有更多的思考。\n",
     "\n",
     "## 1. softmax 函数\n",
     "\n",
-    "softmax(柔性最大值)函数，一般在神经网络中， softmax可以作为分类任务的输出层。其实可以认为softmax输出的是几个类别选择的概率，比如我有一个分类任务，要分为三个类，softmax函数可以根据它们相对的大小，输出三个类别选取的概率，并且概率和为1。\n",
+    "`softmax`(柔性最大值)函数，一般在神经网络中， `softmax`可以作为分类任务的输出层。其实可以认为`softmax`输出的是几个类别选择的概率，比如有一个分类任务，要分为三个类，softmax函数可以根据它们相对的大小，输出三个类别选取的概率，并且概率和为1。\n",
     "\n",
-    "Softmax从字面上来说，可以分成`soft`和`max`两个部分。`max`故名思议就是最大值的意思。Softmax的核心在于`soft`，而`soft`有软的含义，与之相对的是`hard`硬。很多场景中需要我们找出数组所有元素中值最大的元素，实质上都是求的`hardmax`。下面使用`Numpy`模块实现hardmax。"
+    "Softmax从字面上来说，可以分成`soft`和`max`两个部分。`max`故名思议就是最大值的意思。Softmax的核心在于`soft`，而`soft`有软的含义，与之相对的是`hard`硬。很多场景中需要找出数组所有元素中值最大的元素，实质上都是求的`hardmax`。下面使用`Numpy`模块实现hardmax。"
    ]
   },
   {
@@ -62,7 +62,7 @@
     "\n",
     "![softmax_demo](images/softmax_demo.png)\n",
     "\n",
-    "softmax直白来说就是将原来输出是$[3,1,-3]$通过softmax函数作用，就映射成为(0,1)的值，而这些值的累和为1（满足概率的性质），那么我们就可以将它理解成概率，在最后选取输出结点的时候，我们就可以选取概率最大（也就是值对应最大的）结点，作为我们的预测目标！\n"
+    "softmax直白来说就是将原来输出是$[3,1,-3]$通过softmax函数作用，就映射成为(0,1)的值，而这些值的累和为1（满足概率的性质），那么我们就可以将它理解成概率，在最后选取输出结点的时候，选取概率最大（也就是值对应最大的）结点，作为预测目标！\n"
    ]
   },
   {
@@ -108,7 +108,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "以一个神经元的二类分类训练为例，进行两次实验（神经网络常用的激活函数为`sigmoid`函数，该实验也采用该函数）：输入一个相同的样本数据x=1.0（该样本对应的实际分类y=0）；两次实验各自随机初始化参数，从而在各自的第一次前向传播后得到不同的输出值，形成不同的代价（误差）：\n",
+    "以一个神经元的二类分类训练为例，进行两次实验（神经网络常用的激活函数为`sigmoid`函数，该实验也采用该函数）：输入一个相同的样本数据$x=1.0$（该样本对应的实际分类$y=0$）；两次实验各自随机初始化参数，从而在各自的第一次前向传播后得到不同的输出值，形成不同的代价（误差）：\n",
     "\n",
     "![cross_entropy_loss_1](images/cross_entropy_loss_1.png)\n",
     "实验1：第一次输出值为0.82\n",
@@ -267,7 +267,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.9"
+   "version": "3.5.4"
   }
  },
  "nbformat": 4,
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diff --git a/5_nn/perceptron_sample_data.pdf b/5_nn/perceptron_sample_data.pdf
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