一、問題
OCR（光學字元識别）是機器學習重要的應用之一，一般要經過二值化、去噪、傾斜校正、特征抽取、字元切割、字元識别、後處理等過程。其中難度最大的是字元切割，最關鍵的步驟是字元識别。一般進行字元識别的方法有kNN，SVM，CNN等方式，其中比較好用的是SVM。作者在這裡實作的是相對較為簡單的kNN（k近鄰）算法，用以完成經典的MNIST資料集的字元識别工作。該資料集的訓練集共計60000條資料，測試集共計10000條資料。
二、原理
kNN的核心思想：如果一個樣本在特征空間中的k個最相鄰的樣本中的大多數屬于某一個類别，則該樣本也屬于這個類别，并具有這個類别上樣本的特性。一般來講，對于類域的交叉或重疊較多的待分樣本集來說，kNN相比其它方法更為适合。
三、解決
①導入資料，觀察示例圖确定基本解決方案
②定義kNN函數
③調用kNN函數
④比較不同數量的訓練集、不同的距離度量函數對結果準确率、時間開銷的影響
首先是對資料集内的資料進行導入、劃分資料集、觀察示例圖等正常操作，讀者不必在這些代碼上花費大量時間，可以跳讀下一代碼段。
%matplotlib inline
import numpy as np
from PIL import Image
from matplotlib import pyplot as plt
import copy
import scipy.ndimage
import sys, os
DATASET_PATH = r'D:\檔案路徑'
DATASET_FILE = os.path.join(DATASET_PATH, 'mnist.npz')
f = np.load(DATASET_FILE)
x_train, y_train = f['x_train'], f['y_train']
x_test, y_test = f['x_test'], f['y_test']
#unit8（無符号的整數，unit8是0～255
def img_show(img):
plt.imshow(Image.fromarray(np.uint8(img)))
plt.axis('on') # 關掉坐标軸為 off
plt.title('image') # 圖像題目
plt.show()
img = x_train[0]#訓練圖像賦給img
print(img.shape) # (784,)
img = img.reshape(28, 28) # 把圖像的形狀變為原來的尺寸
img_show(img)
實際操作中，我們可以隻定義一種距離度量方法，最常用的就是歐式距離。這裡定義了歐式距離、曼哈頓距離、切比雪夫距離和闵科夫斯基距離四種距離度量方法，目的是對四種方法進行比較。可以在後面的叙述中看到，歐式距離效果最好。
def euclidean_dist(x,y):
return np.linalg.norm(x-y)
def manhattan_dist(x,y):
return np.sum(np.abs(x-y))
def chebyshev_dist(x,y):
return np.max(np.abs(x-y))
def minkowski_dist(x,y):
return np.sqrt(np.sum(np.square(x-y)))
接下來就是對kNN函數的定義。函數包括四個參數：
x表示向量清單，是拟進行标注的圖檔的特征矩陣拉伸成向量後的向量；
M表示樣本矩陣，用來訓練分類器；
k表示目标點的鄰居個數；
dtype表示度量方法，共有0，1，2，3四個選項，分别對應上面的四種距離度量方法。
函數輸出的是M中與x最近的k個樣本的下标。
值得注意的是，這裡對x、M都做了拉伸變換。而這二者都必須是array類型才能利用numpy的方法進行距離度量。是以無論傳進來是什麼類型（實際上是list類型），都先轉換為array。
注意，下面M[:10000]代表使用傳入的訓練集中前10000條資料進行訓練，目的是加快訓練速度，但也犧牲了一部分精度。實際操作時，這部分可以進行更改。
def KNN(x, M, k, dtype):
x = np.array(x)
M = np.array(M)
orin_dist = []
dist = []
dist0 = 0
idx = []
for a in M[:10000]:#這裡可以調整訓練集大小
if dtype == 0:
dist0 = euclidean_dist(x,a)
elif dtype == 1:
dist0 = manhattan_dist(x,a)
elif dtype == 2:
dist0 = chebyshev_dist(x,a)
elif dtype == 3:
dist0 = minkowski_dist(x,a)
dist.append(dist0)
for i in dist:
orin_dist.append(i)
dist.sort()
for i in range(k):
idx.append(orin_dist.index(dist[i]))
return idx
定義查找結果函數。傳入的是目标點的k個鄰居共同組成的向量，傳出的是這k個鄰居确定的數字結果。
def find(x_result):
y_result = y_train[x_result]
from collections import Counter
res0 = Counter(y_result).most_common(1)
res = res0[0][0]
print("這個數字是"+str(res))
return res
定義驗證函數。若比對成功，說明結果正确；正确率等于正确驗證數量除以總驗證數量。
def vali(result, r_result):
r_con = 0
c_con = 0
for i in range(len(result)):
if(result[i] == r_result[i]):
r_con += 1
c_con += 1
print("共"+str(r_con)+"個結果正确，正确率為"+str(r_con/c_con))
這一步完成了對x_train資料集的矩陣拉伸工作。
new_x_train = []
for i in x_train:
new_x_train.append(np.ravel(i))
這一步完成了kNN函數調用，并輸出結果。
注意，這裡dtype的位置傳的參數是3，代表使用了闵科夫斯基距離。實際操作時，這一參數可以修改。
y_result = []
for i in x_test[:500]:
x_result = KNN(np.ravel(i),new_x_train,11,3)
y_result.append(find(x_result))
print(y_result)
在上面的介紹中提到了兩處參數的修改。分别調整這兩處參數，得到使用1000條、10000條、60000條訓練集中的資料進行訓練得到的y_train結果（使用的是歐式距離）；以及在1000條訓練資料下分别使用四種距離度量方法進行度量的所得結果。
将上面得到的結果手動複制到下面，進行正确率比較。
y_r1000 = [7, 2, 1, 0, 0, 1, 4, 9, 8, 9, 0, 6, 9, 0, 1, 1, 9, 7, 3, 4, 9, 6, 6, 5, 1, 0, 7, 9, 0, 1, 3, 1, 3, 6, 7, 2, 7, 1, 1, 1, 1, 7, 9, 1, 1, 5, 1, 1, 9, 4, 6, 3, 5, 1, 0, 0, 4, 1, 9, 1, 7, 8, 9, 1, 7, 1, 7, 4, 3, 0, 7, 0, 2, 7, 1, 7, 1, 7, 1, 7, 9, 6, 2, 7, 8, 4, 7, 5, 6, 1, 3, 6, 1, 3, 1, 9, 1, 7, 6, 9, 1, 0, 5, 4, 9, 9, 2, 1, 9, 4, 8, 1, 1, 9, 1, 1, 9, 4, 7, 7, 5, 6, 7, 6, 7, 9, 0, 5, 8, 5, 6, 6, 5, 7, 8, 1, 0, 1, 6, 7, 6, 7, 3, 1, 9, 1, 8, 2, 0, 1, 9, 9, 9, 5, 1, 5, 6, 0, 3, 1, 4, 6, 5, 4, 6, 5, 4, 1, 1, 9, 9, 7, 3, 1, 1, 1, 1, 8, 1, 8, 1, 8, 1, 0, 1, 6, 2, 5, 0, 1, 1, 1, 0, 7, 0, 1, 1, 6, 9, 2, 3, 6, 1, 1, 1, 1, 9, 3, 2, 9, 7, 1, 9, 1, 9, 0, 3, 8, 7, 5, 9, 0, 2, 7, 1, 3, 8, 1, 1, 1, 5, 1, 8, 7, 7, 9, 2, 1, 4, 1, 5, 3, 8, 7, 1, 1, 0, 6, 4, 1, 9, 1, 9, 1, 7, 1, 1, 1, 2, 0, 8, 1, 7, 7, 1, 1, 0, 1, 3, 0, 3, 0, 1, 9, 9, 9, 1, 8, 2, 1, 2, 9, 1, 1, 9, 2, 6, 4, 1, 1, 8, 2, 9, 1, 0, 9, 0, 0, 2, 8, 1, 7, 1, 1, 9, 0, 1, 1, 4, 1, 3, 0, 0, 5, 1, 9, 1, 5, 0, 6, 1, 1, 9, 1, 6, 9, 6, 0, 7, 1, 1, 1, 1, 3, 3, 1, 9, 7, 0, 6, 5, 1, 1, 3, 8, 1, 0, 5, 1, 1, 1, 5, 0, 6, 1, 8, 5, 1, 1, 9, 4, 6, 7, 1, 5, 0, 6, 5, 6, 1, 7, 2, 0, 8, 8, 5, 9, 1, 1, 4, 0, 7, 3, 7, 6, 1, 6, 1, 1, 7, 2, 8, 6, 1, 7, 5, 1, 5, 4, 4, 2, 1, 1, 1, 1, 4, 5, 0, 5, 1, 7, 7, 8, 7, 9, 9, 1, 9, 2, 1, 1, 2, 9, 2, 0, 9, 9, 1, 4, 1, 1, 1, 6, 4, 9, 8, 9, 3, 7, 6, 0, 0, 3, 1, 8, 0, 6, 1, 9, 5, 3, 3, 1, 3, 9, 1, 1, 6, 9, 0, 9, 6, 6, 6, 7, 8, 8, 2, 8, 8, 8, 7, 6, 1, 8, 4, 1, 2, 1, 3, 1, 9, 7, 1, 9, 0, 8, 9, 9, 1, 6, 5, 2, 3, 7, 6, 9, 1, 0, 1]
y_r10000 = [7, 2, 1, 0, 4, 1, 9, 9, 4, 9, 0, 6, 9, 0, 1, 5, 9, 7, 3, 4, 9, 6, 6, 5, 4, 0, 7, 4, 0, 1, 3, 1, 3, 6, 7, 2, 7, 1, 1, 1, 1, 7, 4, 2, 1, 5, 1, 1, 9, 4, 6, 3, 5, 1, 6, 0, 4, 1, 9, 1, 7, 8, 1, 1, 7, 1, 6, 4, 3, 0, 7, 0, 2, 9, 1, 7, 1, 7, 9, 7, 9, 6, 2, 7, 8, 4, 7, 3, 6, 1, 3, 6, 1, 3, 1, 4, 1, 7, 6, 9, 6, 0, 5, 4, 9, 9, 2, 1, 9, 9, 8, 1, 1, 9, 7, 1, 1, 4, 9, 7, 8, 6, 1, 6, 7, 9, 0, 5, 8, 5, 6, 6, 8, 7, 8, 1, 0, 1, 6, 9, 6, 7, 3, 1, 7, 1, 8, 2, 0, 1, 9, 8, 5, 8, 1, 5, 6, 0, 3, 1, 4, 6, 5, 4, 6, 5, 4, 1, 1, 4, 9, 7, 3, 1, 2, 1, 1, 8, 1, 8, 1, 8, 1, 0, 1, 9, 2, 3, 0, 1, 1, 1, 0, 9, 0, 1, 1, 6, 9, 2, 3, 6, 1, 1, 1, 3, 9, 8, 2, 9, 7, 5, 9, 1, 9, 0, 3, 6, 5, 5, 7, 2, 2, 7, 1, 3, 8, 1, 1, 1, 3, 1, 8, 7, 1, 9, 2, 1, 4, 1, 5, 8, 8, 7, 1, 6, 0, 6, 4, 1, 9, 1, 9, 5, 7, 1, 1, 1, 2, 6, 8, 1, 7, 7, 1, 1, 8, 1, 3, 0, 3, 0, 1, 9, 9, 9, 1, 8, 2, 1, 2, 9, 1, 1, 9, 2, 6, 9, 1, 5, 9, 2, 9, 2, 0, 9, 0, 0, 2, 8, 1, 7, 1, 1, 9, 0, 2, 9, 4, 1, 3, 0, 0, 3, 1, 9, 1, 5, 3, 5, 1, 7, 9, 1, 6, 9, 6, 0, 7, 1, 1, 2, 1, 5, 3, 1, 9, 7, 8, 6, 6, 1, 1, 3, 8, 1, 0, 5, 1, 3, 1, 8, 0, 6, 1, 8, 5, 1, 9, 9, 4, 6, 7, 2, 8, 0, 6, 5, 6, 1, 7, 2, 0, 8, 8, 5, 4, 1, 1, 4, 0, 7, 3, 7, 6, 1, 6, 1, 1, 9, 2, 8, 6, 1, 7, 5, 2, 5, 4, 4, 2, 1, 3, 9, 2, 4, 5, 0, 3, 1, 7, 7, 8, 7, 9, 7, 1, 9, 2, 1, 9, 2, 9, 2, 0, 4, 9, 1, 8, 8, 1, 1, 6, 5, 9, 1, 9, 3, 7, 6, 0, 0, 3, 1, 8, 0, 6, 9, 8, 3, 3, 8, 1, 3, 9, 1, 1, 6, 8, 0, 9, 6, 6, 6, 7, 8, 8, 2, 8, 5, 8, 9, 6, 1, 8, 4, 1, 2, 6, 9, 1, 9, 7, 1, 9, 0, 8, 9, 9, 1, 0, 5, 2, 3, 7, 6, 9, 1, 8, 1]
y_r60000 = [7, 2, 1, 0, 4, 1, 9, 9, 0, 9, 0, 6, 9, 0, 1, 8, 9, 7, 3, 4, 9, 6, 6, 5, 4, 0, 7, 4, 0, 1, 3, 1, 3, 0, 7, 2, 7, 1, 2, 1, 1, 7, 4, 2, 1, 5, 1, 1, 9, 4, 6, 3, 5, 0, 6, 0, 4, 1, 9, 1, 7, 8, 4, 3, 7, 1, 6, 4, 3, 0, 7, 0, 2, 9, 1, 7, 1, 7, 9, 7, 9, 6, 2, 7, 8, 4, 7, 8, 6, 1, 3, 6, 1, 3, 1, 4, 1, 7, 6, 9, 6, 0, 5, 4, 9, 9, 2, 1, 9, 9, 8, 1, 1, 9, 1, 9, 9, 4, 9, 8, 8, 6, 7, 6, 7, 4, 0, 5, 8, 5, 6, 6, 3, 7, 8, 1, 0, 1, 6, 9, 6, 7, 3, 1, 7, 1, 8, 2, 0, 1, 9, 8, 5, 3, 1, 5, 6, 0, 3, 1, 8, 6, 5, 4, 6, 5, 4, 5, 1, 4, 9, 7, 2, 1, 2, 1, 1, 8, 1, 8, 1, 8, 1, 0, 1, 9, 2, 3, 0, 1, 1, 1, 0, 9, 0, 1, 1, 6, 4, 2, 3, 6, 1, 1, 1, 1, 9, 5, 2, 9, 4, 5, 9, 1, 9, 0, 3, 6, 5, 5, 7, 2, 2, 7, 1, 2, 8, 1, 1, 7, 3, 1, 8, 8, 7, 9, 2, 2, 4, 1, 5, 8, 8, 7, 1, 2, 0, 2, 4, 1, 9, 1, 9, 5, 7, 1, 2, 1, 2, 6, 8, 5, 7, 7, 1, 1, 8, 1, 8, 0, 3, 0, 1, 9, 9, 9, 1, 8, 2, 1, 2, 9, 1, 5, 9, 2, 6, 4, 1, 8, 9, 2, 9, 1, 0, 4, 0, 0, 2, 8, 1, 7, 1, 7, 9, 0, 2, 1, 8, 1, 3, 0, 0, 3, 1, 9, 1, 5, 2, 8, 1, 7, 9, 3, 0, 9, 2, 0, 7, 1, 1, 2, 1, 8, 3, 1, 9, 7, 8, 6, 6, 1, 1, 3, 8, 1, 0, 5, 1, 3, 1, 5, 0, 6, 1, 8, 5, 1, 8, 4, 4, 6, 8, 2, 5, 0, 6, 5, 6, 3, 7, 2, 0, 8, 8, 5, 4, 1, 1, 4, 0, 7, 3, 7, 6, 1, 6, 1, 1, 9, 2, 8, 6, 1, 9, 5, 2, 5, 4, 4, 2, 1, 3, 8, 7, 4, 5, 0, 3, 1, 7, 7, 8, 7, 9, 7, 1, 9, 2, 1, 1, 2, 9, 2, 0, 4, 9, 1, 4, 8, 1, 8, 1, 5, 9, 8, 8, 3, 7, 6, 0, 0, 3, 1, 8, 0, 6, 9, 8, 3, 3, 3, 2, 3, 9, 1, 1, 6, 8, 0, 9, 6, 6, 6, 7, 8, 8, 2, 7, 8, 8, 9, 6, 1, 8, 4, 1, 2, 1, 8, 1, 9, 7, 1, 4, 0, 8, 9, 9, 1, 0, 5, 2, 3, 7, 6, 9, 4, 0, 1]
y_r1000_eu = [7, 2, 1, 0, 0, 1, 4, 9, 8, 9, 0, 6, 9, 0, 1, 1, 9, 7, 3, 4, 9, 6, 6, 5, 1, 0, 7, 9, 0, 1, 3, 1, 3, 6, 7, 2, 7, 1, 1, 1, 1, 7, 9, 1, 1, 5, 1, 1, 9, 4, 6, 3, 5, 1, 0, 0, 4, 1, 9, 1, 7, 8, 9, 1, 7, 1, 7, 4, 3, 0, 7, 0, 2, 7, 1, 7, 1, 7, 1, 7, 9, 6, 2, 7, 8, 4, 7, 5, 6, 1, 3, 6, 1, 3, 1, 9, 1, 7, 6, 9, 1, 0, 5, 4, 9, 9, 2, 1, 9, 4, 8, 1, 1, 9, 1, 1, 9, 4, 7, 7, 5, 6, 7, 6, 7, 9, 0, 5, 8, 5, 6, 6, 5, 7, 8, 1, 0, 1, 6, 7, 6, 7, 3, 1, 9, 1, 8, 2, 0, 1, 9, 9, 9, 5, 1, 5, 6, 0, 3, 1, 4, 6, 5, 4, 6, 5, 4, 1, 1, 9, 9, 7, 3, 1, 1, 1, 1, 8, 1, 8, 1, 8, 1, 0, 1, 6, 2, 5, 0, 1, 1, 1, 0, 7, 0, 1, 1, 6, 9, 2, 3, 6, 1, 1, 1, 1, 9, 3, 2, 9, 7, 1, 9, 1, 9, 0, 3, 8, 7, 5, 9, 0, 2, 7, 1, 3, 8, 1, 1, 1, 5, 1, 8, 7, 7, 9, 2, 1, 4, 1, 5, 3, 8, 7, 1, 1, 0, 6, 4, 1, 9, 1, 9, 1, 7, 1, 1, 1, 2, 0, 8, 1, 7, 7, 1, 1, 0, 1, 3, 0, 3, 0, 1, 9, 9, 9, 1, 8, 2, 1, 2, 9, 1, 1, 9, 2, 6, 4, 1, 1, 8, 2, 9, 1, 0, 9, 0, 0, 2, 8, 1, 7, 1, 1, 9, 0, 1, 1, 4, 1, 3, 0, 0, 5, 1, 9, 1, 5, 0, 6, 1, 1, 9, 1, 6, 9, 6, 0, 7, 1, 1, 1, 1, 3, 3, 1, 9, 7, 0, 6, 5, 1, 1, 3, 8, 1, 0, 5, 1, 1, 1, 5, 0, 6, 1, 8, 5, 1, 1, 9, 4, 6, 7, 1, 5, 0, 6, 5, 6, 1, 7, 2, 0, 8, 8, 5, 9, 1, 1, 4, 0, 7, 3, 7, 6, 1, 6, 1, 1, 7, 2, 8, 6, 1, 7, 5, 1, 5, 4, 4, 2, 1, 1, 1, 1, 4, 5, 0, 5, 1, 7, 7, 8, 7, 9, 9, 1, 9, 2, 1, 1, 2, 9, 2, 0, 9, 9, 1, 4, 1, 1, 1, 6, 4, 9, 8, 9, 3, 7, 6, 0, 0, 3, 1, 8, 0, 6, 1, 9, 5, 3, 3, 1, 3, 9, 1, 1, 6, 9, 0, 9, 6, 6, 6, 7, 8, 8, 2, 8, 8, 8, 7, 6, 1, 8, 4, 1, 2, 1, 3, 1, 9, 7, 1, 9, 0, 8, 9, 9, 1, 6, 5, 2, 3, 7, 6, 9, 1, 0, 1]
y_r1000_ma = [7, 2, 1, 0, 9, 1, 4, 9, 9, 7, 0, 6, 9, 0, 1, 1, 9, 7, 3, 4, 9, 6, 6, 1, 1, 0, 7, 9, 0, 1, 3, 1, 3, 6, 7, 2, 7, 1, 1, 1, 1, 7, 9, 1, 1, 6, 1, 1, 9, 4, 6, 3, 5, 1, 0, 0, 4, 1, 9, 1, 7, 8, 1, 1, 7, 1, 1, 4, 3, 0, 7, 0, 3, 7, 1, 7, 1, 7, 1, 7, 9, 6, 2, 7, 1, 4, 7, 3, 6, 1, 3, 6, 1, 3, 1, 9, 1, 1, 6, 9, 1, 0, 5, 4, 9, 9, 2, 1, 9, 9, 1, 1, 1, 9, 1, 1, 1, 4, 7, 7, 5, 6, 1, 6, 7, 1, 0, 5, 8, 1, 6, 6, 5, 7, 8, 1, 0, 1, 6, 7, 1, 7, 3, 1, 7, 1, 9, 2, 0, 1, 9, 9, 1, 5, 1, 5, 6, 0, 3, 1, 4, 6, 5, 4, 6, 5, 4, 1, 1, 9, 9, 7, 3, 1, 1, 1, 1, 6, 1, 8, 1, 1, 1, 0, 1, 9, 2, 5, 0, 1, 1, 1, 0, 1, 0, 1, 1, 6, 9, 2, 0, 6, 1, 1, 1, 1, 9, 3, 1, 9, 7, 1, 9, 1, 9, 0, 3, 1, 7, 5, 9, 0, 2, 7, 1, 3, 8, 1, 1, 1, 5, 1, 6, 7, 1, 9, 2, 1, 4, 1, 5, 3, 8, 7, 1, 1, 0, 6, 4, 1, 9, 1, 9, 1, 7, 1, 1, 1, 2, 0, 8, 1, 7, 7, 1, 1, 0, 1, 3, 0, 3, 0, 1, 9, 9, 9, 1, 8, 2, 1, 2, 9, 1, 1, 9, 2, 6, 4, 1, 1, 9, 2, 9, 1, 0, 9, 0, 0, 2, 8, 1, 7, 1, 1, 7, 0, 1, 1, 4, 1, 1, 0, 0, 1, 1, 9, 1, 1, 0, 6, 1, 1, 9, 1, 6, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 9, 7, 5, 6, 1, 1, 1, 3, 8, 1, 0, 5, 1, 1, 1, 5, 0, 6, 1, 1, 5, 1, 1, 9, 4, 6, 7, 1, 5, 0, 1, 1, 6, 1, 7, 1, 1, 8, 1, 5, 9, 1, 1, 4, 0, 1, 3, 7, 6, 1, 6, 1, 1, 7, 2, 8, 6, 1, 7, 5, 1, 1, 4, 4, 2, 1, 1, 1, 1, 4, 5, 0, 5, 1, 7, 1, 8, 7, 9, 9, 1, 9, 1, 1, 1, 2, 9, 2, 0, 4, 9, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 3, 7, 6, 0, 0, 3, 1, 1, 0, 6, 1, 9, 5, 3, 3, 1, 1, 9, 1, 1, 6, 9, 0, 9, 6, 6, 6, 7, 8, 8, 2, 7, 8, 1, 9, 6, 1, 8, 4, 1, 2, 1, 3, 1, 9, 7, 1, 4, 0, 7, 9, 9, 1, 6, 6, 2, 3, 7, 6, 9, 1, 0, 1]
y_r1000_ch = [4, 2, 1, 2, 9, 0, 7, 1, 4, 5, 0, 6, 6, 0, 5, 0, 0, 3, 4, 1, 5, 6, 5, 4, 4, 4, 2, 5, 3, 1, 0, 9, 2, 1, 1, 6, 5, 3, 1, 3, 1, 5, 3, 3, 9, 0, 9, 5, 5, 7, 6, 0, 0, 0, 4, 1, 2, 1, 4, 8, 5, 2, 3, 7, 4, 5, 7, 4, 5, 0, 2, 0, 0, 5, 0, 5, 3, 4, 0, 2, 9, 8, 9, 5, 4, 2, 0, 0, 5, 1, 9, 6, 1, 3, 3, 5, 9, 3, 9, 6, 1, 0, 9, 0, 1, 5, 4, 2, 5, 5, 4, 0, 2, 0, 9, 4, 4, 1, 4, 5, 0, 2, 0, 4, 3, 5, 9, 4, 6, 0, 5, 5, 5, 3, 0, 1, 6, 9, 5, 4, 6, 3, 3, 1, 3, 6, 6, 6, 0, 8, 5, 1, 4, 5, 1, 4, 5, 8, 0, 9, 5, 2, 0, 0, 5, 6, 5, 3, 1, 9, 4, 6, 3, 0, 0, 0, 2, 0, 2, 5, 0, 8, 6, 5, 0, 5, 0, 1, 6, 5, 4, 1, 2, 2, 0, 3, 1, 5, 5, 6, 5, 5, 9, 3, 1, 0, 0, 2, 2, 9, 4, 1, 5, 3, 0, 0, 9, 0, 3, 4, 7, 0, 0, 9, 8, 0, 4, 8, 1, 2, 0, 9, 5, 1, 0, 2, 5, 6, 1, 0, 5, 1, 8, 1, 2, 0, 6, 1, 9, 9, 0, 3, 6, 4, 1, 0, 0, 4, 5, 2, 2, 0, 7, 7, 1, 8, 5, 2, 4, 0, 0, 5, 1, 6, 4, 2, 0, 0, 5, 0, 0, 5, 2, 0, 5, 0, 6, 5, 5, 0, 6, 6, 4, 2, 1, 5, 0, 0, 0, 0, 1, 3, 8, 2, 4, 0, 2, 0, 1, 3, 5, 5, 6, 2, 5, 8, 1, 0, 3, 1, 0, 2, 5, 9, 5, 5, 4, 5, 6, 4, 0, 3, 1, 0, 6, 0, 1, 2, 2, 6, 6, 0, 2, 0, 1, 2, 0, 8, 3, 0, 0, 0, 1, 2, 8, 6, 7, 1, 5, 1, 1, 5, 4, 0, 0, 1, 5, 5, 6, 2, 5, 5, 0, 4, 5, 0, 0, 8, 3, 4, 4, 2, 5, 7, 5, 1, 6, 2, 2, 3, 3, 3, 6, 1, 9, 3, 1, 4, 0, 7, 6, 3, 0, 5, 9, 5, 8, 5, 1, 9, 1, 5, 5, 9, 5, 1, 3, 2, 6, 6, 8, 5, 3, 3, 2, 5, 0, 1, 9, 0, 1, 2, 3, 2, 1, 0, 0, 3, 7, 3, 3, 5, 3, 3, 9, 0, 5, 1, 3, 0, 5, 5, 6, 1, 2, 1, 5, 5, 1, 5, 0, 5, 5, 0, 1, 5, 2, 0, 5, 7, 3, 0, 9, 1, 5, 3, 1, 5, 1, 1, 3, 3, 4, 8, 0, 1, 0, 7, 0, 1, 0, 3, 5, 6, 3, 5, 2, 7, 1, 4]
y_r1000_mi = [1, 1, 1, 1, 6, 1, 5, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 5, 8, 6, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 6, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 8, 5, 8, 8, 1, 1, 5, 1, 1, 1, 1, 8, 1, 8, 1, 1, 6, 1, 1, 1, 1, 0, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 8, 0, 8, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 8, 1, 1, 1, 8, 5, 1, 1, 8, 1, 1, 0, 1, 1, 1, 8, 1, 1, 5, 1, 1, 8, 6, 1, 7, 1, 1, 1, 1, 1, 1, 6, 1, 1, 8, 6, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 6, 1, 1, 1, 8, 1, 0, 1, 1, 7, 1, 5, 1, 1, 1, 1, 1, 1, 1, 6, 1, 5, 1, 1, 0, 8, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 6, 5, 1, 1, 1, 2, 5, 6, 1, 1, 8, 5, 1, 0, 1, 1, 1, 1, 1, 1, 1, 6, 1, 6, 1, 1, 1, 9, 6, 1, 1, 0, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 5, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 6, 1, 1, 8, 5, 8, 1, 8, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 8, 1, 5, 1, 1, 6, 1, 8, 1, 6, 1, 6, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 5, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 1, 1, 1, 5, 1, 5, 8, 1, 1, 1, 5, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 6, 8, 1, 1, 1, 1, 1, 1, 5, 1, 1, 4, 0, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 8, 8, 1, 1, 4, 1, 1, 1, 1, 1, 1, 5, 1, 8, 5, 1, 6, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 5, 1, 1, 1, 8, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 8, 6, 1, 8, 8, 1, 1, 5, 1, 6, 8, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1, 5, 0, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1]
調用驗證函數，比較不同方法結果的準确率。
四、反思
1000訓練資料，歐氏距離
10000訓練資料，歐氏距離
60000訓練資料，歐氏距離
1000訓練資料，歐氏距離
1000訓練資料，曼哈頓距離
1000訓練資料，切比雪夫距離
1000訓練資料，闵科夫斯基距離
資料表明，歐式距離是最适合kNN進行文字識别的距離度量方法；同時筆者估算了一下，以筆者的電腦性能，如果用60000條訓練資料，大概要1.4個小時才能跑完10000條測試集資料，時間原因沒有進行驗證。不過這個時間也是可以接受的。
方案還有可改進的地方，歡迎留言交流。