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Appendix: Coefficient of Variation (CV)
Z-score method
If data is distributed normally then we can use z-score method (or the three-sigma rule) to detect outliers (values exceeding 3 std deviations in both directions are considered as anomalies).
The following python snippet demonstrates how to detect anomalies in data:
import numpy as np# outliers exceed 3 std deviation in both directionsdata=[2,3,3,12,3,9,12,4,340,5,2,2,6,1,8,6,1,300,7,5,4,3,9]outliers=[]threshold=3 # three-sigma rule
m=np.mean(data)std=np.std(data)for val in data: z_score=(val-m)/std if np.abs(z_score)> threshold: outliers.append(val)print("data: ",data)print("outliers: ",outliers)data: [2, 3, 3, 12, 3, 9, 12, 4, 340, 5, 2, 2, 6, 1, 8, 6, 1, 300, 7, 5, 4, 3, 9]
outliers: [340, 300]
Tukey’s Method
If data is not distributed normally it’s common to use Tukey’s Method to detect outliers/anomalies. In Tukey’s method, we define a lower limit and upper limit.
Data within these limits, is considered ‘clean’ or normal. The lower and upper limits are determined in a robust way. That means, that the upper and lower limits do not get influenced by the presence of the outliers. This is a distinction from some other methods like the z-score method (described above), where the lower and upper limits are influenced by the outliers.In general, it is better to use robust methods.
The Upper and Lower limits are defined as follows:
Lower Limit = 25th Percentile — k*IQR
Upper Limit = 75th Percentile + k*IQR
where, k is generally 1.5 but must be adjusted if required. IQR is the Inter-Quartile Range (IQR = 75th Percentile — 25th Percentile of data) of the variable.
Example [ detection of anomalies with panda module, python ]:
import pandas as pddf=pd.read_csv(r'C:\Tool\StudentsPerformance.csv')Q1 = df['score'].quantile(0.25)Q3 = df['score'].quantile(0.75)df['outliers'] = df['score'].apply(lambda x: 'Outlier' if x > Q3+1.5*IQR or x<Q1-1.5*IQR else 'Normal')
The Outlier Ratio
Appendix: Coefficient of Variation (CV)
The Coefficient of Variation (CV), known also as the relative standard deviation, is a normalized measure of dispersion and computed as the ratio of the standard deviation (STD) to the mean:
The standard deviation (STD) shows the average distance between data points from the mean. The STD on one dataset and the STD on other dataset can significantly differ in magnitudes while the variability is similar. Therefore we need normalize the standard deviation to gauge variability as a dimensionless index.
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