Finished homework_03_kmeans.

homework_03_kmeans/homework/3-1.py

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+'''
+Author: SJ2050
+Date: 2021-11-05 23:21:46
+LastEditTime: 2021-11-07 15:14:50
+Version: v0.0.1
+Description: Homework3.1.
+Copyright © 2021 SJ2050
+'''
+import numpy as np
+from basic_kMeans import read_csv_data, show_figure, KMeans
+
+if __name__ == '__main__':
+    data_file = 'dataset_circles.csv'
+    original_data = read_csv_data(data_file)
+
+    distance_func = lambda p1, p2: np.linalg.norm(p1 - p2)
+    def compute_center_point_func(points):
+        n = len(points)
+        return sum(points)/n
+
+    clustered_points, center_points, iter_num, cost = KMeans(original_data[:, 0:2], 2, \
+                                                            distance_func, \
+                                                            compute_center_point_func, \
+                                                            1000, 1e-3).run()
+    print(f'迭代次数: {iter_num}, 代价函数值为: {cost:.3f}')
+
+    classified_points = np.array([(clustered_points[i][j][0], clustered_points[i][j][1], i) \
+                                        for i in range(2) \
+                                            for j in range(len(clustered_points[i]))])
+    center_points = [(center_points[i][0], center_points[i][1], i) for i in range(2)]
+    show_figure(classified_points, center_points)

homework_03_kmeans/homework/3-2.py

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+'''
+Author: SJ2050
+Date: 2021-11-07 13:49:04
+LastEditTime: 2021-11-07 15:05:48
+Version: v0.0.1
+Description: Homework3.2.
+Copyright © 2021 SJ2050
+'''
+import numpy as np
+from basic_kMeans import read_csv_data, show_figure, KMeans
+
+if __name__ == '__main__':
+    data_file = 'dataset_circles.csv'
+    original_data = read_csv_data(data_file)
+
+    distance_func = lambda p1, p2: abs(np.linalg.norm(p1)-np.linalg.norm(p2))
+    def compute_center_point_func(points):
+        n = len(points)
+        return np.array([0 ,sum(np.linalg.norm(points, axis=1))/n])
+
+    clustered_points, center_points, iter_num, cost = KMeans(original_data[:, 0:2], 2, \
+                                                            distance_func, \
+                                                            compute_center_point_func, \
+                                                            1000, 1e-3).run()
+    print(f'迭代次数: {iter_num}, 代价函数值为: {cost:.3f}')
+
+    classified_points = np.array([(clustered_points[i][j][0], clustered_points[i][j][1], i) \
+                                        for i in range(2) \
+                                            for j in range(len(clustered_points[i]))])
+    center_points = [(center_points[i][0], center_points[i][1], i) for i in range(2)]
+    show_figure(classified_points, center_points)
+

homework_03_kmeans/homework/3-3.py

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+'''
+Author: SJ2050
+Date: 2021-11-07 15:13:40
+LastEditTime: 2021-11-07 21:14:18
+Version: v0.0.1
+Description: Use a density-based clustering algorithm.
+Copyright © 2021 SJ2050
+'''
+import numpy as np
+import random
+from basic_kMeans import read_csv_data, show_figure
+
+def compute_distances(points):
+    """Compute the distances between every two points.
+
+    Args：
+        points: All points.
+
+    Returns:
+        distances: The distances between every two points.
+
+    """
+    points = np.array(points, dtype=np.float64)
+    n = len(points)
+    distances = np.zeros((n, n))
+
+    for i in range(n):
+        for j in range(n):
+            if i <= j:
+                distances[i, j] = np.linalg.norm(points[i]-points[j])
+            else:
+                distances[i, j] = distances[j, i]
+
+    return distances
+
+def find_neighbors(distances, p_ind, eps):
+    """Find neighbors of point p.
+
+    When the distance between two points is less than eps, they are considered as neighbors.
+
+    Args:
+        distances: The distances between every two points (numpy 2d array).
+        p_ind： Reference point's index.
+        eps: The distance between two neighbors should be less than eps.
+
+    Returns:
+        neighbors: A set of the reference point's neighbors' indices.
+    """
+    neighbors = {i for i in range(distances.shape[0]) if distances[p_ind][i] < eps and p_ind != i}
+
+    return neighbors
+
+def DBSCAN(points, eps, min_Pts):
+    """A density-based clustering algorithm
+
+    Args:
+        points: All points.
+        eps: The distance between two neighbors should be less than eps.
+        min_Pts: Each core object should at least have `min_Pts` neighbors.
+
+    Returns:
+        clusters: Clusters of given points.
+    """
+    # initialization
+    distances = compute_distances(points)
+    each_point_neighbors = [find_neighbors(distances, i, eps) for i in range(len(points))]
+    core_objects = {i for i in range(len(points)) \
+                        if len(each_point_neighbors[i]) >= min_Pts}
+    unvisited_indices = {i for i in range(len(points))}
+
+    clusters = []
+    while len(core_objects) > 0:
+        unvisited_indices_old = unvisited_indices.copy()
+        core_obj = random.choice(list(core_objects))
+        Q = [core_obj]
+        unvisited_indices = unvisited_indices - {core_obj}
+
+        while len(Q) > 0:
+            q = Q.pop(0)
+            Nq = each_point_neighbors[q]
+            if len(Nq) >= min_Pts:
+                delta = Nq & unvisited_indices
+                Q.extend(list(delta))
+                unvisited_indices = unvisited_indices - delta
+
+        Ck = unvisited_indices_old - unvisited_indices
+        clusters.append(Ck)
+        core_objects = core_objects - Ck
+
+    clusters.append(unvisited_indices)
+    return clusters
+
+if __name__ == '__main__':
+    data_file = 'dataset_circles.csv'
+    original_data = read_csv_data(data_file)
+    points = original_data[:, :2]
+    eps = 5
+    min_Pts = 5
+
+    clusters = DBSCAN(points, eps, min_Pts)
+    classified_points = [(points[j][0], points[j][1], i) \
+                            for i in range(len(clusters)) \
+                                for j in clusters[i]]
+    show_figure(classified_points, [])

homework_03_kmeans/homework/basic_kMeans.py

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+'''
+Author: SJ2050
+Date: 2021-11-05 23:21:46
+LastEditTime: 2021-11-07 15:10:24
+Version: v0.0.1
+Description: Basic K-Means algorithm.
+Copyright © 2021 SJ2050
+'''
+import csv
+import random
+import numpy as np
+from matplotlib import pyplot as plt
+
+def read_csv_data(data_file):
+    """Read data from file(.csv format).
+
+    Args:
+        data_file: Raw data file(.csv).
+
+    Returns:
+        A 2d numpy array of data from input file.
+    """
+    with open(data_file, 'r') as fp_inp_data:
+        reader = csv.reader(fp_inp_data)
+        result = [(item[0], item[1], item[2]) for item in reader]
+
+    return np.array(result, dtype=np.float64)
+
+def show_figure(points, center_points):
+    """Show result through figure.
+
+    Args:
+        points: Points (size: N*3). For each point, first and second components represent location
+                and third compoint represents class type.
+        center_points: Center points to be highlighted.
+
+    Returns:
+        None.
+    """
+    points = np.array(points)
+    x = points[:, 0]
+    y = points[:, 1]
+    c = points[:, 2]
+    plt.scatter(x, y, s=20, c=c)
+
+    if len(center_points) > 0:
+        center_points = np.array(center_points)
+        center_x = center_points[:, 0]
+        center_y = center_points[:, 1]
+        center_c = center_points[:, 2]
+        plt.scatter(center_x, center_y, s=100, c=center_c, marker='x')
+
+    plt.show()
+
+class KMeans():
+    """Basic KMeans class.
+    """
+    def __init__(self, points, class_num, distance_func, compute_center_point_func, \
+                                                                    max_iter_num, atol):
+        self.points = np.array(points, dtype=np.float64)
+        self.class_num = class_num
+        self.distance_func = distance_func
+        self.compute_center_point_func = compute_center_point_func
+        self.max_iter_num = max_iter_num
+        self.atol = atol
+        self.clustered_points = []
+        self.center_points = []
+
+    @property
+    def cost(self):
+        assert len(self.clustered_points) == self.class_num, "点簇数目与分类数不一致"
+        assert len(self.center_points) == self.class_num, "中点的个数与分类数不一致"
+
+        cost = 0
+        for k in range(self.class_num):
+            for i in range(len(self.clustered_points[k])):
+                cost += self.distance_func(self.clustered_points[k][i], self.center_points[k])**2
+
+        return cost
+
+    def choose_which_class_belonging_to(self, point):
+        distances = [self.distance_func(point, self.center_points[i]) for i in range(len(self.center_points))]
+        return np.argmin(distances)
+
+    def initialize(self):
+        n = len(self.points)
+        center_points_indices = random.sample([i for i in range(n)], self.class_num)
+        self.center_points = self.points[center_points_indices]
+        self.clustered_points = [[] for i in range(self.class_num)]
+
+        for p in self.points:
+            belonging_class = self.choose_which_class_belonging_to(p)
+            self.clustered_points[belonging_class].append(p)
+
+    def cluster(self):
+        self.center_points = [self.compute_center_point_func(self.clustered_points[i]) for i in range(self.class_num)]
+
+        self.clustered_points = [[] for i in range(self.class_num)]
+
+        for p in self.points:
+            belonging_class = self.choose_which_class_belonging_to(p)
+            self.clustered_points[belonging_class].append(p)
+
+    def run(self):
+        self.initialize()
+        prev_cost = None
+        curr_cost = self.cost
+        for i in range(self.max_iter_num):
+            if prev_cost and abs(curr_cost - prev_cost) < self.atol:
+                break
+            self.cluster()
+            prev_cost = curr_cost
+            curr_cost = self.cost
+
+        return self.clustered_points, self.center_points, i, curr_cost
+
+
+
+if __name__ == '__main__':
+    # test
+    def compute_center_point(points):
+        n = len(points)
+        return sum(points)/n
+
+    data_file = 'dataset_circles.csv'
+    original_data = read_csv_data(data_file)
+
+    distance_func = lambda p1, p2: np.linalg.norm(p1 - p2)
+    clustered_points, center_points, iter_num, cost = KMeans(original_data[:, 0:2], 2, \
+                                                            distance_func, \
+                                                            compute_center_point, \
+                                                            1000, 1e-3).run()
+    print(f'迭代次数: {iter_num}, 代价函数值为: {cost:.3f}')
+
+    classified_points = np.array([(clustered_points[i][j][0], clustered_points[i][j][1], i) \
+                                        for i in range(2) \
+                                            for j in range(len(clustered_points[i]))])
+    center_points = [(center_points[i][0], center_points[i][1], i) for i in range(2)]
+    show_figure(classified_points, center_points)

homework_03_kmeans/homework/dataset_circles.csv

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homework_03_kmeans/homework/作业报告.md

@@ -0,0 +1,40 @@
+# 作业报告
+
+## 1. 基本的kMeans方法
+
+​		首先我编写了一个最基础的kMeans类，聚类方法采用简单的离最近重心进行归类，并且不进行任何映射。原始数据点如下所示：
+
+![](images/original_points.png)
+
+<center>图1. 原始数据点</center>
+
+采用基础的kMeans方法分成两类后的结果如下所示：
+
+![](images/basic_kMeans.png)
+
+<center>图2. 采用基本的kMeans方法进行分类</center>
+
+## 2. 对原始数据进行变换
+
+​		对原始数据点进行分析后，觉得可以将原始点映射到一个离`(0, 0)`点距离的空间中去，同样对数据归为两类，最终运行结果如下所示：
+
+![](images/map.png)
+
+<center>图3. 对原始数据点进行变换后再使用kMeans归类</center>
+
+可以看到，通过对原始点进行变换后，归类后的效果更好。
+
+
+
+## 3. 其他聚类方法
+
+​		除了上述简单的聚类方法后，我还通过查找资料学习了到了一种根据密度进行聚类的方法，其核心思想是认为同一类别的样本，它们之间应该是紧密相连的，也就是说，在该类别任意样本周围不远处一定有同类别的样本存在，通过将紧密相连的样本划为一类，这样就得到了一个聚类类别。其伪代码如下：
+
+![](images/pseudocode.png)
+
+密度聚类不需要提前给定分类数，但其对参数`eps`和`minPts`比较敏感，如果选取不当，可以不能将数据点归类开，或者归类质量较低。对于给定的数据，在对参数多次尝试后，最后发现当`eps=5`，`minPts=5`时聚类效果不错，运行结果如下所示：
+
+![](images/density_cluster.png)
+
+<center>图4. 采用密度聚类</center>
+