From b39d143adc757fa84926ba305f469eefb95b0f8b Mon Sep 17 00:00:00 2001 From: SJ2050cn Date: Sun, 7 Nov 2021 21:18:11 +0800 Subject: [PATCH] Finished homework_03_kmeans. --- homework_03_kmeans/homework/3-1.py | 31 ++ homework_03_kmeans/homework/3-2.py | 32 ++ homework_03_kmeans/homework/3-3.py | 104 +++++ homework_03_kmeans/homework/basic_kMeans.py | 139 ++++++ .../homework/dataset_circles.csv | 400 ++++++++++++++++++ .../homework/images/basic_kMeans.png | Bin 0 -> 63721 bytes .../homework/images/density_cluster.png | Bin 0 -> 62530 bytes homework_03_kmeans/homework/images/map.png | Bin 0 -> 62845 bytes .../homework/images/original_points.png | Bin 0 -> 68484 bytes .../homework/images/pseudocode.png | Bin 0 -> 347832 bytes homework_03_kmeans/homework/作业报告.md | 40 ++ 11 files changed, 746 insertions(+) create mode 100644 homework_03_kmeans/homework/3-1.py create mode 100644 homework_03_kmeans/homework/3-2.py create mode 100644 homework_03_kmeans/homework/3-3.py create mode 100644 homework_03_kmeans/homework/basic_kMeans.py create mode 100644 homework_03_kmeans/homework/dataset_circles.csv create mode 100644 homework_03_kmeans/homework/images/basic_kMeans.png create mode 100644 homework_03_kmeans/homework/images/density_cluster.png create mode 100644 homework_03_kmeans/homework/images/map.png create mode 100644 homework_03_kmeans/homework/images/original_points.png create mode 100644 homework_03_kmeans/homework/images/pseudocode.png create mode 100644 homework_03_kmeans/homework/作业报告.md diff --git a/homework_03_kmeans/homework/3-1.py b/homework_03_kmeans/homework/3-1.py new file mode 100644 index 0000000..80572fe --- /dev/null +++ b/homework_03_kmeans/homework/3-1.py @@ -0,0 +1,31 @@ +''' +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) diff --git a/homework_03_kmeans/homework/3-2.py b/homework_03_kmeans/homework/3-2.py new file mode 100644 index 0000000..5e74bc3 --- /dev/null +++ b/homework_03_kmeans/homework/3-2.py @@ -0,0 +1,32 @@ +''' +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) + diff --git a/homework_03_kmeans/homework/3-3.py b/homework_03_kmeans/homework/3-3.py new file mode 100644 index 0000000..bacc092 --- /dev/null +++ b/homework_03_kmeans/homework/3-3.py @@ -0,0 +1,104 @@ +''' +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, []) diff --git a/homework_03_kmeans/homework/basic_kMeans.py b/homework_03_kmeans/homework/basic_kMeans.py new file mode 100644 index 0000000..61fb59f --- /dev/null +++ b/homework_03_kmeans/homework/basic_kMeans.py @@ -0,0 +1,139 @@ +''' +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, \ 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图1. 原始数据点
+ +采用基础的kMeans方法分成两类后的结果如下所示: + +![](images/basic_kMeans.png) + +
图2. 采用基本的kMeans方法进行分类
+ +## 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) + +
图3. 对原始数据点进行变换后再使用kMeans归类
+ +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) + +
图4. 采用密度聚类
+