离群点检测——局部离群因子(Local Outlier Factor，LOF)算法

1 概述

离群点是观察的数据集中明显异常的数据点，或者说，离群点的数据分布与数据集的整体分布不同。离群点检测的目的是检测出那些与正常数据差别较大的数据点，然后根据具体的问题作进一步处理。

离群点检测算法主要有基于统计、聚类、分类、信息论、距离、密度等相关的方法，列表如下

检测方法	描述	优缺点
基于统计	根据数据的分布特点，选择一个概率分布模型对数据进行匹配，将不能匹配的数据点识别为离群点。	优点：统计方法广泛。缺点：在高维数据上的应用效果不够理想；实际数据分布规律无法预估，难以用单一的分布模型来刻画。
基于聚类	应用聚类算法对数据进行聚类操作，将不归属于任何一个类簇的点识别为离群点。	优点：聚类算法理论完善。缺点：主要做聚类，附带检测离群点，检测效果不够理想；时间复杂度较高。
基于分类	应用分类算法，对数据点做是否离群的类别判定。	优点：分类算法理论完善。缺点：对训练集的数据质量要求较高。
基于信息论	将信息论的理论应用到离群点检测中。	优点：仅依赖于数据对象的本身属性特性；数据属性类型适应性强，既可以是数值型，也可以是标称属性。缺点：计算和度量复杂数据的信息熵或Kolomogorov复杂度较为困难。
基于距离	对某一个数据点，超过一定部分的数据与它的距离都大于一定值，那么将它识别为离群点。	优点：方法简单，易于操作。缺点：对参数敏感；时间复杂度偏高；在高维稀疏数据集上效果不理想。
基于密度	根据数据的密集情况，计算每个数据对象的局部离群因子，用以标识数据的离群程度。选出top(n)个离群程度最大的点作为离群点。	优点：方法简洁，不受数据分布影响。缺点：对近邻参数较为敏感；时间复杂度较高；在高维大数据集上效率较低。

【注】

1)离群点不同于噪声，非噪声点也可能离群，噪声应该在离群点检测前完成去除。

2)离群点检测算法的评价指标同二分类，可使用正确率(Accuracy)、查准率(Precision)、查全率(Recall)、F值(F1-scores)等指标进行评估。

本文介绍一种基于密度的离群点检测方法——局部离群因子算法。

2 局部离群因子(Local Outlier Factor，LOF)算法

2.1 算法思想

局部离群因子(LOF，又叫局部异常因子)算法是Breunig于2000年提出的一种基于密度的局部离群点检测算法，该方法适用于不同类簇密度分散情况迥异的数据。如下图中，集合C1是低密度区域，集合C2是高密度区域，依据传统的基于密度的离群点检测算法，点p与C2中邻近点的距离小于C1中任何一个数据点与其邻近点的距离，点p会被看作是正常的点，而在局部来看，点p却是事实上的孤立点，LOF算法即可以有效地实现对该种情形的离群点检测。

LOF算法的基本思想是，根据数据点周围的数据密集情况，首先计算每个数据点的一个局部可达密度，然后通过局部可达密度进一步计算得到每个数据点的一个离群因子，该离群因子即标识了一个数据点的离群程度，因子值越大，表示离群程度越高，因子值越小，表示离群程度越低。最后，输出离群程度最大的top(n)个点。

2.2 概念定义

（1）点到点的距离：

$d(p,o)$ ,数据点p到数据点o的距离。

（2）第k距离：

数据点p的第k距离 $d_{k}(p)$ ，定义为： $d_k(p)=d(p,o)$ ，满足

a)在集合中至少有不包括p在内的k个点o’，使得 $d(p,o')\leqslant d(p,o)$ ;

b)在集合中至多有不包括p在内的k-1个点o’，使得 $d(p,o')< d(p,o)$ .

通俗地讲，就是以p为圆心向外辐射，直至涵盖了第k个邻近点。下图中示意了p的第5距离

(3)第k距离邻域：

数据点p的第k距离邻域 $N_k(p)$ ，指点p的第k距离内的所有点的集合，包括第k距离上的点。

易知，有 $\left | N_k(p) \right |\geqslant k$ .

(4)第k可达距离：

$reach\_dist_k(o,p)=max\left \{ d_k(o),d(o,p) \right \}$

数据点o到数据点p的第k可达距离，定义为点o的第k距离和点o到点p的距离中的较大者。如下图中，o1到p的第5可达距离为 $d(o_1,p)$ ，o2到p的第5可达距离为 $d_5(o_2)$

易知，点o到点o的第k邻域内所有点的第k可达距离均为 $d_k(o)$ .

(5)局部可达密度(local reachability density)：

$lrd_k(p)=1/\left ( \frac{\sum_{o\in N_k(p) }^{}reach\_dist_k(o,p)}{\left | N_k(p) \right |} \right )$

数据点p的第k局部可达密度，即点p的第k距离邻域内的所有点到点p的平均第k可达距离的倒数。它表征了点p的密度情况，点p与周围点密集度越高，各点的可达距离越可能是较小的各自的第k距离，lrd值越大；点p与周围点的密集度越低，各点的可达距离越可能是较大的两点间的实际距离，lrd值越小。

6)局部离群因子：

$LOF_k(p)=\frac{\sum_{o\in N_k(p)}^{}\frac{lrd_k(o)}{lrd_k(p)}}{\left | N_k(p) \right |}=\frac{\sum_{o\in N_k(p)}^{}lrd_k(o)}{\left | N_k(p) \right |}/lrd_k(p)$

数据点p的第k局部离群因子，意为将点p的 $N_k(p)$ 邻域内所有点的平均局部可达密度与点p的局部可达密度作比较，这个比值越大于1，表明p点的密度越小于其周围点的密度，p点越可能是离群点；这个比值越小于1，表明p点的密度越大于其周围点的密度，p点越可能是正常点。

2.3 算法描述

输入：数据点集合D;

输出：离群点集合O.

计算每个点的局部可达密度，进而计算得到每个点的局部离群因子，选取输出离群程度最高的n个点：

(1)计算每个点的第k距离邻域内各点的第k可达距离：

$reach\_dist_k(o,p)=max\left \{ d_k(o),d(o,p) \right \}$

其中， $d_k(o)$ 为领域点o的第k距离， $d(o,p)$ 为邻域点o到点p的距离.

(2)计算每个点的局部第k局部可达密度：

$lrd_k(p)=1/\left ( \frac{\sum_{o\in N_k(p)}^{}reach\_dist_k(o,p)}{\left | N_k(p) \right |} \right )$

其中， $N_k(p)$ 为p点的第k距离邻域.

(3)计算每个点的第k局部离群因子：

$LOF_k(p)=\frac{\sum_{o\in N_k(p)}^{}\frac{lrd_k(o)}{lrd_k(p)}}{\left | N_k(p) \right |}=\frac{\sum_{o\in N_k(p)}^{}lrd_k(o)}{\left | N_k(p) \right |}/lrd_k(p)$

其中， $N_k(p)$ 为p点的第k距离邻域.

(4)对最大的n个局部离群因子所属的数据点，输出离群点集合：

$O=\left \{ o_1,o_2,...,o_n \right \}$

3 python实现

算法实现，lof.py文件：

#!/usr/bin/python
# -*- coding: utf8 -*-
from __future__ import divisiondef distance_euclidean(instance1, instance2):"""Computes the distance between two instances. Instances should be tuples of equal length.Returns: Euclidean distanceSignature: ((attr_1_1, attr_1_2, ...), (attr_2_1, attr_2_2, ...)) -> float"""def detect_value_type(attribute):"""Detects the value type (number or non-number).Returns: (value type, value casted as detected type)Signature: value -> (str or float type, str or float value)"""from numbers import Numberattribute_type = Noneif isinstance(attribute, Number):attribute_type = floatattribute = float(attribute)else:attribute_type = strattribute = str(attribute)return attribute_type, attribute# check if instances are of same lengthif len(instance1) != len(instance2):raise AttributeError("Instances have different number of arguments.")# init differences vectordifferences = [0] * len(instance1)# compute difference for each attribute and store it to differences vectorfor i, (attr1, attr2) in enumerate(zip(instance1, instance2)):type1, attr1 = detect_value_type(attr1)type2, attr2 = detect_value_type(attr2)# raise error is attributes are not of same data type.if type1 != type2:raise AttributeError("Instances have different data types.")if type1 is float:# compute difference for floatdifferences[i] = attr1 - attr2else:# compute difference for stringif attr1 == attr2:differences[i] = 0else:differences[i] = 1# compute RMSE (root mean squared error)rmse = (sum(map(lambda x: x ** 2, differences)) / len(differences)) ** 0.5return rmseclass LOF:"""Helper class for performing LOF computations and instances normalization."""def __init__(self, instances, normalize=True, distance_function=distance_euclidean):self.instances = instancesself.normalize = normalizeself.distance_function = distance_functionif normalize:self.normalize_instances()def compute_instance_attribute_bounds(self):min_values = [float("inf")] * len(self.instances[0])  # n.ones(len(self.instances[0])) * n.infmax_values = [float("-inf")] * len(self.instances[0])  # n.ones(len(self.instances[0])) * -1 * n.inffor instance in self.instances:min_values = tuple(map(lambda x, y: min(x, y), min_values, instance))  # n.minimum(min_values, instance)max_values = tuple(map(lambda x, y: max(x, y), max_values, instance))  # n.maximum(max_values, instance)self.max_attribute_values = max_valuesself.min_attribute_values = min_valuesdef normalize_instances(self):"""Normalizes the instances and stores the infromation for rescaling new instances."""if not hasattr(self, "max_attribute_values"):self.compute_instance_attribute_bounds()new_instances = []for instance in self.instances:new_instances.append(self.normalize_instance(instance))  # (instance - min_values) / (max_values - min_values)self.instances = new_instancesdef normalize_instance(self, instance):return tuple(map(lambda value, max, min: (value - min) / (max - min) if max - min > 0 else 0,instance, self.max_attribute_values, self.min_attribute_values))def local_outlier_factor(self, min_pts, instance):"""The (local) outlier factor of instance captures the degree to which we call instance an outlier.min_pts is a parameter that is specifying a minimum number of instances to consider for computing LOF value.Returns: local outlier factorSignature: (int, (attr1, attr2, ...), ((attr_1_1, ...),(attr_2_1, ...), ...)) -> float"""if self.normalize:instance = self.normalize_instance(instance)return local_outlier_factor(min_pts, instance, self.instances, distance_function=self.distance_function)def k_distance(k, instance, instances, distance_function=distance_euclidean):# TODO: implement caching"""Computes the k-distance of instance as defined in paper. It also gatheres the set of k-distance neighbours.Returns: (k-distance, k-distance neighbours)Signature: (int, (attr1, attr2, ...), ((attr_1_1, ...),(attr_2_1, ...), ...)) -> (float, ((attr_j_1, ...),(attr_k_1, ...), ...))"""distances = {}for instance2 in instances:distance_value = distance_function(instance, instance2)if distance_value in distances:distances[distance_value].append(instance2)else:distances[distance_value] = [instance2]distances = sorted(distances.items())neighbours = []k_sero = 0k_dist = Nonefor dist in distances:k_sero += len(dist[1])neighbours.extend(dist[1])k_dist = dist[0]if k_sero >= k:breakreturn k_dist, neighboursdef reachability_distance(k, instance1, instance2, instances, distance_function=distance_euclidean):"""The reachability distance of instance1 with respect to instance2.Returns: reachability distanceSignature: (int, (attr_1_1, ...),(attr_2_1, ...)) -> float"""(k_distance_value, neighbours) = k_distance(k, instance2, instances, distance_function=distance_function)return max([k_distance_value, distance_function(instance1, instance2)])def local_reachability_density(min_pts, instance, instances, **kwargs):"""Local reachability density of instance is the inverse of the average reachabilitydistance based on the min_pts-nearest neighbors of instance.Returns: local reachability densitySignature: (int, (attr1, attr2, ...), ((attr_1_1, ...),(attr_2_1, ...), ...)) -> float"""(k_distance_value, neighbours) = k_distance(min_pts, instance, instances, **kwargs)reachability_distances_array = [0] * len(neighbours)  # n.zeros(len(neighbours))for i, neighbour in enumerate(neighbours):reachability_distances_array[i] = reachability_distance(min_pts, instance, neighbour, instances, **kwargs)sum_reach_dist = sum(reachability_distances_array)if sum_reach_dist == 0:return float('inf')return len(neighbours) / sum_reach_distdef local_outlier_factor(min_pts, instance, instances, **kwargs):"""The (local) outlier factor of instance captures the degree to which we call instance an outlier.min_pts is a parameter that is specifying a minimum number of instances to consider for computing LOF value.Returns: local outlier factorSignature: (int, (attr1, attr2, ...), ((attr_1_1, ...),(attr_2_1, ...), ...)) -> float"""(k_distance_value, neighbours) = k_distance(min_pts, instance, instances, **kwargs)instance_lrd = local_reachability_density(min_pts, instance, instances, **kwargs)lrd_ratios_array = [0] * len(neighbours)for i, neighbour in enumerate(neighbours):instances_without_instance = set(instances)instances_without_instance.discard(neighbour)neighbour_lrd = local_reachability_density(min_pts, neighbour, instances_without_instance, **kwargs)lrd_ratios_array[i] = neighbour_lrd / instance_lrdreturn sum(lrd_ratios_array) / len(neighbours)def outliers(k, instances, **kwargs):"""Simple procedure to identify outliers in the dataset."""instances_value_backup = instancesoutliers = []for i, instance in enumerate(instances_value_backup):instances = list(instances_value_backup)instances.remove(instance)l = LOF(instances, **kwargs)value = l.local_outlier_factor(k, instance)if value > 1:outliers.append({"lof": value, "instance": instance, "index": i})outliers.sort(key=lambda o: o["lof"], reverse=True)return outliers

测试程序，test_lof.py文件：

# -*- coding: utf8 -*-
instances = [(-4.8447532242074978, -5.6869538132901658),(1.7265577109364076, -2.5446963280374302),(-1.9885982441038819, 1.705719643962865),(-1.999050026772494, -4.0367551415711844),(-2.0550860126898964, -3.6247409893236426),(-1.4456945632547327, -3.7669258809535102),(-4.6676062022635554, 1.4925324371089148),(-3.6526420667796877, -3.5582661345085662),(6.4551493172954029, -0.45434966683144573),(-0.56730591589443669, -5.5859532963153349),(-5.1400897823762239, -1.3359248994019064),(5.2586932439960243, 0.032431285797532586),(6.3610915734502838, -0.99059648246991894),(-0.31086913190231447, -2.8352818694180644),(1.2288582719783967, -1.1362795178325829),(-0.17986204466346614, -0.32813130288006365),(2.2532002509929216, -0.5142311840491649),(-0.75397166138399296, 2.2465141276038754),(1.9382517648161239, -1.7276112460593251),(1.6809250808549676, -2.3433636210337503),(0.68466572523884783, 1.4374914487477481),(2.0032364431791514, -2.9191062023123635),(-1.7565895138024741, 0.96995712544043267),(3.3809644295064505, 6.7497121359292684),(-4.2764152718650896, 5.6551328734397766),(-3.6347215445083019, -0.85149861984875741),(-5.6249411288060385, -3.9251965527768755),(4.6033708001912093, 1.3375110154658127),(-0.685421751407983, -0.73115552984211407),(-2.3744241805625044, 1.3443896265777866)]from lof import outliers
lof = outliers(5, instances)for outlier in lof:print (outlier["lof"],outlier["instance"])from matplotlib import pyplot as px,y = zip(*instances)
p.scatter(x,y, 20, color="#0000FF")for outlier in lof:value = outlier["lof"]instance = outlier["instance"]color = "#FF0000" if value > 1 else "#00FF00"p.scatter(instance[0], instance[1], color=color, s=(value-1)**2*10+20)p.show()

运行结果：

输出离群点的lof值及坐标信息

可视化，其中红色的点为检测出的离群点

参考

陈瑜. 离群点检测算法研究[D].兰州大学,2018.
https://blog.csdn.net/wangyibo0201/article/details/51705966
https://blog.csdn.net/ilike_program/article/details/85009618