From ee3ae3c28ac1c19237e62f09ada1f75c43833d96 Mon Sep 17 00:00:00 2001 From: xhs7700 Date: Sun, 28 Mar 2021 16:08:43 +0800 Subject: [PATCH] assignment-1 --- assignment-1/submission/18307130090/README.md | 132 +++++++++++++++ .../18307130090/img/readme/acc_cov.png | Bin 0 -> 26052 bytes .../18307130090/img/readme/acc_mean.png | Bin 0 -> 26837 bytes .../18307130090/img/readme/acc_nums.png | Bin 0 -> 25034 bytes .../18307130090/img/readme/test.png | Bin 0 -> 40051 bytes .../18307130090/img/readme/time.png | Bin 0 -> 24702 bytes .../18307130090/img/readme/train.png | Bin 0 -> 44228 bytes .../submission/18307130090/img/test.png | Bin 0 -> 44544 bytes .../submission/18307130090/img/train.png | Bin 0 -> 45151 bytes assignment-1/submission/18307130090/source.py | 160 ++++++++++++++++++ 10 files changed, 292 insertions(+) create mode 100644 assignment-1/submission/18307130090/README.md create mode 100644 assignment-1/submission/18307130090/img/readme/acc_cov.png create mode 100644 assignment-1/submission/18307130090/img/readme/acc_mean.png create mode 100644 assignment-1/submission/18307130090/img/readme/acc_nums.png create mode 100644 assignment-1/submission/18307130090/img/readme/test.png create mode 100644 assignment-1/submission/18307130090/img/readme/time.png create mode 100644 assignment-1/submission/18307130090/img/readme/train.png create mode 100644 assignment-1/submission/18307130090/img/test.png create mode 100644 assignment-1/submission/18307130090/img/train.png create mode 100644 assignment-1/submission/18307130090/source.py diff --git a/assignment-1/submission/18307130090/README.md b/assignment-1/submission/18307130090/README.md new file mode 100644 index 0000000..7f12492 --- /dev/null +++ b/assignment-1/submission/18307130090/README.md @@ -0,0 +1,132 @@ +# PRML-2021 Assignment1 + +姓名:夏海淞 + +学号:18307130090 + +## 简述 + +在本次实验中,我实现了K近邻算法,在自己生成的数据集上进行测试,将其在不同数据集上的性能与作为对照的随机算法进行了比较。同时,尝试对性能瓶颈进行了一定优化,效果较好。 + +## 算法介绍 + +### 数据集生成 + +`source.py`中的`generate_data`函数用于生成数据集。该函数接收`mean` `cov` `nums`三个数组参数,每个数组的第`i`个元素分别表示第`i`类点的均值、协方差和数量。`generate_data`函数通过调用`numpy.random.multivariate_normal`方法生成数据。 + +`source.py`中的`show_plot`函数用于绘制数据集的散点图。该函数接收两个`ndarray`参数和一个字符串参数,分别表示样本点、样本标签和图片名称。`show_plot`函数通过调用`matplotlib`库完成绘制,将图存放至项目的`\img`路径中。 + +按照实验要求,将数据打乱后,取80%作为训练集,20%作为测试集。 + +训练集和测试集的散点图如下所示: + +
+ train + test +
+ +### 模型训练与预测 + +#### `fit` + +`fit`方法首先将训练数据进一步按照80%/20%的比例划分为训练集和验证集,随后枚举K近邻算法中的参数`k`,最大值设为20,进行训练。 + +对于每个`k`,枚举验证集中的点,在训练集中计算出距离该点最近的`k`个点,在`k`个点对应的标签中选取数量最多的作为验证点的预测标签。最后计算出验证集的平均预测准确率acc,选取acc最大的k作为模型的参数。 + +#### `predict` + +`predict`方法与`fit`方法类似:将`fit`方法中训练出的`k`作为参数,枚举测试集的点,计算出距离该点最近的`k`个点。选取数量最多的标签作为预测标签。因为该过程与`fit`方法中枚举`k`进行训练的过程完全一致,因此将其封装成函数进行复用,同时也方便后续对其进行性能优化。 + +## 实验结果与探究 + +### 数据集参数 + +$$ +\mu_x=[5,5] ,\Sigma_x=\begin{bmatrix}34&1 \\\\ 5&10\end{bmatrix},|x|=1600\\\\ +\mu_y=[10,15],\Sigma_y=\begin{bmatrix}20&4 \\\\ 3&24\end{bmatrix},|y|=400\\\\ +\mu_z=[20,5],\Sigma_z=\begin{bmatrix}30&2 \\\\ 1&10\end{bmatrix},|z|=2000 +$$ + +### 实验结果 + +重复10次实验,得到每次实验的模型参数`k`、训练准确率`train_acc`、测试准确率`test_acc`和对照准确率(按照权重随机选择)`control`: + +| k | train_acc | test_acc | control | +| ---- | --------- | -------- | ------- | +| 8 | 0.9047 | 0.8988 | 0.4075 | +| 4 | 0.8984 | 0.8775 | 0.4438 | +| 10 | 0.8875 | 0.8888 | 0.43 | +| 18 | 0.8813 | 0.8888 | 0.43 | +| 15 | 0.8922 | 0.8688 | 0.4075 | +| 14 | 0.9016 | 0.88 | 0.41 | +| 9 | 0.8891 | 0.8863 | 0.405 | +| 19 | 0.8969 | 0.8863 | 0.405 | +| 18 | 0.8875 | 0.8613 | 0.4038 | +| 14 | 0.8875 | 0.8925 | 0.445 | + +得到训练准确率、测试准确率和对照准确率的平均值分别为0.8927,0.8829和0.4188。 + +由此可知该模型的泛化能力较好,性能相较随机选择算法有明显提高。 + +### 实验探究 + +该部分主要探究不同数据集对K近邻算法性能的影响因素。 + +#### 数据量大小 + +固定上述数据集的均值和协方差,改变样本容量为原先的$k$倍$(k\in[0.1,2])$,作出$k$关于`acc`的图如下: + +acc_nums + +由图可知,数据集容量的大小对于模型性能没有明显的影响。 + +推测:出现这种现象的原因可能是K近邻模型过于简单,样本容量较小时也能得到较好的性能。 + +#### 数据集距离 + +固定上述数据集的协方差和容量,改变均值为原先的$k$倍$(k\in[0.1,2])$,作出$k$关于`acc`的图如下: + +acc_mean + +由图可知,数据集的均值越大(即不同数据集的距离越大),模型效果越好。 + +推测:出现这种现象的原因可能是不同数据集之间距离越远,测试点周围属于自己的类的训练点比例越高,因此性能越好。 + +#### 数据集方差 + +固定上述数据集的均值和容量,改变协方差为原先的$k$倍$(k\in[0.1,2])$,作出$k$关于`acc`的图如下: + +acc_cov + +由图可知,数据集的协方差越大,模型效果越好。 + +推测:出现这种现象的原因和数据集距离的原因类似。数据集协方差增大时,不同数据集越倾向于“混合”在一起,使得测试点周围属于自己的类的点比例不断减小,性能随之降低。 + +#### 总结 + +综上,数据集的容量对K近邻模型的影响较小,而不同数据集的距离和协方差分别对K近邻模型性能产生了正面和负面的影响。 + +## 性能优化 + +### 问题归纳 + +注意到K近邻算法的核心操作是:给定$n$个点$x_1,x_2,\cdots,x_n$和一个查询点$y$,要求返回距离$y$最近的$k$个点。常见的算法为计算出$x_i(i=1,2,\cdots,n)$与$y$的距离后按照距离进行排序,取出前$k$个点。这样做的时间复杂度为$O(n\log n)$。 + +### 优化1 + +考虑到$k$一般很小(具体实现中$k\le25$),因此尝试使用堆进行优化。算法步骤为: + +1. 计算出$x_1,\cdots,x_k$与$y$的距离,插入大根堆中; +2. 计算出$x_i(i=k+1,k+2,\cdots,n)$与$y$的距离,与堆顶元素比较。如果$dist(x_i,y)$比堆顶元素小,则弹出堆顶元素,将$dist(x_i,y)$插入堆中。 + +算法结束时堆中的元素即为$x_i$与$y$的距离中最近的$k$个。时间复杂度$O(n\log k)$。由于$k\le25$,因此算法近似为$O(n)$。 + +然而在具体实现中发现优化后速度反而慢于优化前。可能原因是暴力排序调用的是`numpy.argsort`,其底层调用了C的库,常数较小;而堆优化调用的是`heapq`库,完全由Python实现,常数较大,因此性能反而不如优化前。 + +### 优化2 + +后来发现`numpy`库提供了一个名为`numpy.argpartition`的方法。该方法与快速排序的`partition`操作类似,接收一个`ndarray`参数和一个`int`参数并返回一个数组,其中前$k$个为前$k$小的元素参数。由主定理可知,该方法的时间复杂度为$O(n)$。因此改用`numpy.argpartition`计算距离$y$最近的$k$个点,提高了运算速度。 + +`fit`方法分别在使用`numpy.argsort`和`numpy.argpartition`的情况下的运行时间比较图: + +time \ No newline at end of file diff --git a/assignment-1/submission/18307130090/img/readme/acc_cov.png b/assignment-1/submission/18307130090/img/readme/acc_cov.png new file mode 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self.train_data = None + self.train_labels = None + self.k = None + + # func=bf表示使用O(nlogn)的算法 func=opt表示使用O(n)的算法 + def get_predict_labels(self, k, train_data, train_labels, valid_data, func='bf'): + + # 确保func字段只能取bf或者opt + assert func in {'bf', 'opt'} + + predict_labels = np.array([]) + for valid_dot in valid_data: + + # 计算每个train_dot与valid_dot之间的距离 + dist = np.linalg.norm(train_data - valid_dot, axis=1) + + # 计算距离最小的k个train_dot的下标 + dist_index = np.argsort(dist)[:k] if func == 'bf' else np.argpartition(dist, k)[:k] + + # 计算数量最多的标签 + count_dict = {} + max_count = 0 + for index in dist_index: + index = int(index) + train_label = train_labels[index] + count_dict[train_label] = count_dict.get(train_label, 0) + 1 + max_count = max(max_count, count_dict[train_label]) + predict_label = np.array([]) + for train_label, count in count_dict.items(): + if max_count != count: continue + predict_label = np.append(predict_label, train_label) + predict_labels = np.append(predict_labels, np.random.choice(predict_label)) + + return predict_labels + + def fit(self, input_data, input_labels): + self.train_data = input_data + self.train_labels = input_labels + + # 将数据打乱 + shuffled_data, shuffled_labels = shuffle(input_data, input_labels) + + # 划分为训练集和验证集 + ratio, data_size = 0.2, shuffled_data.shape[0] + valid_size = int(data_size * ratio) + train_size = data_size - valid_size + valid_data, valid_labels = shuffled_data[:valid_size], shuffled_labels[:valid_size] + train_data, train_labels = shuffled_data[valid_size:], shuffled_labels[valid_size:] + + # 枚举k,求出最佳参数 + k_size = min(25, train_size) + max_acc, best_k = -1, 0 + for k in range(1, k_size): + predict_labels = self.get_predict_labels(k, train_data, train_labels, valid_data, func='opt') + acc = np.mean(np.equal(predict_labels, valid_labels)) + # print(f'k={k} acc={acc}') + if acc > max_acc: + max_acc = acc + best_k = k + print(f'k={best_k} train_acc={max_acc}') + self.k = best_k + + def predict(self, test_data): + return self.get_predict_labels(self.k, self.train_data, self.train_labels, test_data, func='opt') + + +def generate_data(mean, cov, nums): + n = len(mean) + assert n == len(cov) and n == len(nums) + data = np.concatenate([np.random.multivariate_normal(mean[i], cov[i], int(nums[i])) for i in range(n)]) + labels = np.concatenate([np.ones(int(nums[i]), dtype=int) * i for i in range(n)]) + + data, labels = shuffle(data, labels) + + ratio, data_size = 0.2, len(data) + test_size = int(ratio * data_size) + test_data, test_label = data[:test_size], labels[:test_size] + train_data, train_label = data[test_size:], labels[test_size:] + np.save('data.npy', (train_data, train_label, test_data, test_label)) + + +def shuffle(data, labels): + data_size = len(data) + assert data_size == len(labels) + + indices = np.random.permutation(data_size) + return data[indices], labels[indices] + + +def save_plot(data, labels, name): + data_size = len(data) + assert data_size == len(labels) + total = {} + for i in range(data_size): + label = labels[i] + if label not in total: + total[label] = [] + else: + total[label].append(data[i]) + for category in total.values(): + if category == []: continue + category = np.array(category) + plt.scatter(category[:, 0], category[:, 1]) + plt.title(name) + plt.savefig(f'./img/{name}') + plt.close() + + +def read(): + return np.load('data.npy', allow_pickle=True) + + +def generate_control(nums, length): + n = len(nums) + labels = [i for i in range(n)] + return random.choices(labels, nums, k=length) + + +def train(mean, cov, nums, generate, ratio=(1, 1, 1)): + if generate: + generate_data(mean * ratio[0], cov * ratio[1], nums * ratio[2]) + train_data, train_label, test_data, test_label = read() + save_plot(train_data, train_label, 'train') + save_plot(test_data, test_label, 'test') + knn = KNN() + start_time = time.time() + knn.fit(train_data, train_label) + end_time = time.time() + training_time = end_time - start_time + # print(f'training stime={training_time} s') + ans = knn.predict(test_data) + control_group = generate_control(nums, len(test_label)) + test_acc = np.mean(np.equal(ans, test_label)) + control = np.mean(np.equal(control_group, test_label)) + return test_acc, control + + +if __name__ == '__main__': + nums = np.array([1600, 400, 2000], dtype=int) + mean = np.array([[5, 5], [10, 15], [20, 5]]) + cov = np.array([ + [[34, 5], [5, 10]], + [[20, 5], [5, 24]], + [[30, 5], [5, 10]] + ]) + generate = True if len(sys.argv) > 1 and sys.argv[1] == 'g' else False + acc, control = train(mean, cov, nums, generate) + print(f'acc={acc} control={control}') + pass -- Gitee