PyNomaly is a Python 3 implementation of LoOP (Local Outlier Probabilities). LoOP is a local density based outlier detection method by Kriegel, Kröger, Schubert, and Zimek which provides outlier scores in the range of [0,1] that are directly interpretable as the probability of a sample being an outlier.

The outlier score of each sample is called the Local Outlier Probability. It measures the local deviation of density of a given sample with respect to its neighbors as Local Outlier Factor (LOF), but provides normalized outlier scores in the range [0,1]. These outlier scores are directly interpretable as a probability of an object being an outlier. Since Local Outlier Probabilities provides scores in the range [0,1], practitioners are free to interpret the results according to the application.

Like LOF, it is local in that the anomaly score depends on how isolated the sample is with respect to the surrounding neighborhood. Locality is given by k-nearest neighbors, whose distance is used to estimate the local density. By comparing the local density of a sample to the local densities of its neighbors, one can identify samples that lie in regions of lower density compared to their neighbors and thus identify samples that may be outliers according to their Local Outlier Probability.

The authors' 2009 paper detailing LoOP's theory, formulation, and application is provided by Ludwig-Maximilians University Munich - Institute for Informatics; LoOP: Local Outlier Probabilities.

This Python 3 implementation uses Numpy and the formulas outlined in LoOP: Local Outlier Probabilities to calculate the Local Outlier Probability of each sample.

- Python 3.5 - 3.8
- numpy >= 1.16.3
- python-utils >= 2.3.0
- (optional) numba >= 0.45.1

Numba just-in-time (JIT) compiles the function with calculates the Euclidean distance between observations, providing a reduction in computation time (significantly when a large number of observations are scored). Numba is not a requirement and PyNomaly may still be used solely with numpy if desired (details below).

First install the package from the Python Package Index:

```
pip install PyNomaly # or pip3 install ... if you're using both Python 3 and 2.
```

Then you can do something like this:

```
from PyNomaly import loop
m = loop.LocalOutlierProbability(data).fit()
scores = m.local_outlier_probabilities
print(scores)
```

where *data* is a NxM (N rows, M columns; 2-dimensional) set of data as either a Pandas DataFrame or Numpy array.

LocalOutlierProbability sets the *extent* (in integer in value of 1, 2, or 3) and *n_neighbors* (must be greater than 0) parameters with the default
values of 3 and 10, respectively. You're free to set these parameters on your own as below:

```
from PyNomaly import loop
m = loop.LocalOutlierProbability(data, extent=2, n_neighbors=20).fit()
scores = m.local_outlier_probabilities
print(scores)
```

This implementation of LoOP also includes an optional *cluster_labels* parameter. This is useful in cases where regions
of varying density occur within the same set of data. When using *cluster_labels*, the Local Outlier Probability of a
sample is calculated with respect to its cluster assignment.

```
from PyNomaly import loop
from sklearn.cluster import DBSCAN
db = DBSCAN(eps=0.6, min_samples=50).fit(data)
m = loop.LocalOutlierProbability(data, extent=2, n_neighbors=20, cluster_labels=list(db.labels_)).fit()
scores = m.local_outlier_probabilities
print(scores)
```

**NOTE**: Unless your data is all the same scale, it may be a good idea to normalize your data with z-scores or another
normalization scheme prior to using LoOP, especially when working with multiple dimensions of varying scale.
Users must also appropriately handle missing values prior to using LoOP, as LoOP does not support Pandas
DataFrames or Numpy arrays with missing values.

It may be helpful to use just-in-time (JIT) compilation in the cases where a lot of
observations are scored. Numba, a JIT compiler for Python, may be used
with PyNomaly by setting `use_numba=True`

:

```
from PyNomaly import loop
m = loop.LocalOutlierProbability(data, extent=2, n_neighbors=20, use_numba=True, progress_bar=True).fit()
scores = m.local_outlier_probabilities
print(scores)
```

Numba must be installed if the above to use JIT compilation and improve the
speed of multiple calls to `LocalOutlierProbability()`

, and PyNomaly has been
tested with Numba version 0.45.1. An example of the speed difference that can
be realized with using Numba is avaialble in `examples/numba_speed_diff.py`

.

You may also choose to print progress bars *with our without* the use of numba
by passing `progress_bar=True`

to the `LocalOutlierProbability()`

method as above.

The *extent* parameter controls the sensitivity of the scoring in practice. The parameter corresponds to
the statistical notion of an outlier defined as an object deviating more than a given lambda (*extent*)
times the standard deviation from the mean. A value of 2 implies outliers deviating more than 2 standard deviations
from the mean, and corresponds to 95.0% in the empirical "three-sigma" rule. The appropriate parameter should be selected
according to the level of sensitivity needed for the input data and application. The question to ask is whether it is
more reasonable to assume outliers in your data are 1, 2, or 3 standard deviations from the mean, and select the value
likely most appropriate to your data and application.

The *n_neighbors* parameter defines the number of neighbors to consider about
each sample (neighborhood size) when determining its Local Outlier Probability with respect to the density
of the sample's defined neighborhood. The idea number of neighbors to consider is dependent on the
input data. However, the notion of an outlier implies it would be considered as such regardless of the number
of neighbors considered. One potential approach is to use a number of different neighborhood sizes and average
the results for reach observation. Those observations which rank highly with varying neighborhood sizes are
more than likely outliers. This is one potential approach of selecting the neighborhood size. Another is to
select a value proportional to the number of observations, such an odd-valued integer close to the square root
of the number of observations in your data (*sqrt(n_observations*).

We'll be using the well-known Iris dataset to show LoOP's capabilities. There's a few things you'll need for this example beyond the standard prerequisites listed above:

- matplotlib 2.0.0 or greater
- PyDataset 0.2.0 or greater
- scikit-learn 0.18.1 or greater

First, let's import the packages and libraries we will need for this example.

```
from PyNomaly import loop
import pandas as pd
from pydataset import data
import numpy as np
from sklearn.cluster import DBSCAN
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
```

Now let's create two sets of Iris data for scoring; one with clustering and the other without.

```
# import the data and remove any non-numeric columns
iris = pd.DataFrame(data('iris'))
iris = pd.DataFrame(iris.drop('Species', 1))
```

Next, let's cluster the data using DBSCAN and generate two sets of scores. On both cases, we will use the default
values for both *extent* (0.997) and *n_neighbors* (10).

```
db = DBSCAN(eps=0.9, min_samples=10).fit(iris)
m = loop.LocalOutlierProbability(iris).fit()
scores_noclust = m.local_outlier_probabilities
m_clust = loop.LocalOutlierProbability(iris, cluster_labels=list(db.labels_)).fit()
scores_clust = m_clust.local_outlier_probabilities
```

Organize the data into two separate Pandas DataFrames.

```
iris_clust = pd.DataFrame(iris.copy())
iris_clust['scores'] = scores_clust
iris_clust['labels'] = db.labels_
iris['scores'] = scores_noclust
```

And finally, let's visualize the scores provided by LoOP in both cases (with and without clustering).

```
fig = plt.figure(figsize=(7, 7))
ax = fig.add_subplot(111, projection='3d')
ax.scatter(iris['Sepal.Width'], iris['Petal.Width'], iris['Sepal.Length'],
c=iris['scores'], cmap='seismic', s=50)
ax.set_xlabel('Sepal.Width')
ax.set_ylabel('Petal.Width')
ax.set_zlabel('Sepal.Length')
plt.show()
plt.clf()
plt.cla()
plt.close()
fig = plt.figure(figsize=(7, 7))
ax = fig.add_subplot(111, projection='3d')
ax.scatter(iris_clust['Sepal.Width'], iris_clust['Petal.Width'], iris_clust['Sepal.Length'],
c=iris_clust['scores'], cmap='seismic', s=50)
ax.set_xlabel('Sepal.Width')
ax.set_ylabel('Petal.Width')
ax.set_zlabel('Sepal.Length')
plt.show()
plt.clf()
plt.cla()
plt.close()
fig = plt.figure(figsize=(7, 7))
ax = fig.add_subplot(111, projection='3d')
ax.scatter(iris_clust['Sepal.Width'], iris_clust['Petal.Width'], iris_clust['Sepal.Length'],
c=iris_clust['labels'], cmap='Set1', s=50)
ax.set_xlabel('Sepal.Width')
ax.set_ylabel('Petal.Width')
ax.set_zlabel('Sepal.Length')
plt.show()
plt.clf()
plt.cla()
plt.close()
```

Your results should look like the following:

**LoOP Scores without Clustering**

Note the differences between using LocalOutlierProbability with and without clustering. In the example without clustering, samples are scored according to the distribution of the entire data set. In the example with clustering, each sample is scored according to the distribution of each cluster. Which approach is suitable depends on the use case.

**NOTE**: Data was not normalized in this example, but it's probably a good idea to do so in practice.

When using numpy, make sure to use 2-dimensional arrays in tabular format:

```
data = np.array([
[43.3, 30.2, 90.2],
[62.9, 58.3, 49.3],
[55.2, 56.2, 134.2],
[48.6, 80.3, 50.3],
[67.1, 60.0, 55.9],
[421.5, 90.3, 50.0]
])
scores = loop.LocalOutlierProbability(data, n_neighbors=3).fit().local_outlier_probabilities
print(scores)
```

The shape of the input array shape corresponds to the rows (observations) and columns (features) in the data:

```
print(data.shape)
# (6,3), which matches number of observations and features in the above example
```

Similar to the above:

```
data = np.random.rand(100, 5)
scores = loop.LocalOutlierProbability(data).fit().local_outlier_probabilities
print(scores)
```

PyNomaly provides the ability to specify a distance matrix so that any distance metric can be used (a neighbor index matrix must also be provided). This can be useful when wanting to use a distance other than the euclidean.

```
from sklearn.neighbors import NearestNeighbors
data = np.array([
[43.3, 30.2, 90.2],
[62.9, 58.3, 49.3],
[55.2, 56.2, 134.2],
[48.6, 80.3, 50.3],
[67.1, 60.0, 55.9],
[421.5, 90.3, 50.0]
])
neigh = NearestNeighbors(n_neighbors=3, metric='hamming')
neigh.fit(data)
d, idx = neigh.kneighbors(data, return_distance=True)
m = loop.LocalOutlierProbability(distance_matrix=d, neighbor_matrix=idx, n_neighbors=3).fit()
scores = m.local_outlier_probabilities
```

The below visualization shows the results by a few known distance metrics:

**LoOP Scores by Distance Metric**

PyNomaly also contains an implementation of Hamlet et. al.'s modifications to the original LoOP approach [4], which may be used for applications involving streaming data or where rapid calculations may be necessary. First, the standard LoOP algorithm is used on "training" data, with certain attributes of the fitted data stored from the original LoOP approach. Then, as new points are considered, these fitted attributes are called when calculating the score of the incoming streaming data due to the use of averages from the initial fit, such as the use of a global value for the expected value of the probabilistic distance. Despite the potential for increased error when compared to the standard approach, it may be effective in streaming applications where refitting the standard approach over all points could be computationally expensive.

While the iris dataset is not streaming data, we'll use it in this example by taking the first 120 observations as training data and take the remaining 30 observations as a stream, scoring each observation individually.

Split the data.

```
iris = iris.sample(frac=1) # shuffle data
iris_train = iris.iloc[:, 0:4].head(120)
iris_test = iris.iloc[:, 0:4].tail(30)
```

Fit to each set.

```
m = loop.LocalOutlierProbability(iris).fit()
scores_noclust = m.local_outlier_probabilities
iris['scores'] = scores_noclust
m_train = loop.LocalOutlierProbability(iris_train, n_neighbors=10)
m_train.fit()
iris_train_scores = m_train.local_outlier_probabilities
```

```
iris_test_scores = []
for index, row in iris_test.iterrows():
array = np.array([row['Sepal.Length'], row['Sepal.Width'], row['Petal.Length'], row['Petal.Width']])
iris_test_scores.append(m_train.stream(array))
iris_test_scores = np.array(iris_test_scores)
```

Concatenate the scores and assess.

```
iris['stream_scores'] = np.hstack((iris_train_scores, iris_test_scores))
# iris['scores'] from earlier example
rmse = np.sqrt(((iris['scores'] - iris['stream_scores']) ** 2).mean(axis=None))
print(rmse)
```

The root mean squared error (RMSE) between the two approaches is approximately 0.199 (your scores will vary depending on the data and specification). The plot below shows the scores from the stream approach.

```
fig = plt.figure(figsize=(7, 7))
ax = fig.add_subplot(111, projection='3d')
ax.scatter(iris['Sepal.Width'], iris['Petal.Width'], iris['Sepal.Length'],
c=iris['stream_scores'], cmap='seismic', s=50)
ax.set_xlabel('Sepal.Width')
ax.set_ylabel('Petal.Width')
ax.set_zlabel('Sepal.Length')
plt.show()
plt.clf()
plt.cla()
plt.close()
```

**LoOP Scores using Stream Approach with n=10**

When calculating the LoOP score of incoming data, the original fitted scores are not updated. In some applications, it may be beneficial to refit the data periodically. The stream functionality also assumes that either data or a distance matrix (or value) will be used across in both fitting and streaming, with no changes in specification between steps.

Please use the issue tracker to report any erroneous behavior or desired feature requests.

If you would like to contribute to development, please fork the repository and make
any changes to a branch which corresponds to an open issue. Hot fixes
and bug fixes can be represented by branches with the prefix `fix/`

versus
`feature/`

for new capabilities or code improvements. Pull requests will
then be made from these branches into the repository's `dev`

branch
prior to being pulled into `main`

. Pull requests which are works in
progress or ready for merging should be indicated by their respective
prefixes ([WIP] and [MRG]). Pull requests with the [MRG] prefix will be
reviewed prior to being pulled into the `main`

branch.

When contributing, please ensure to run unit tests and add additional tests as
necessary if adding new functionality. To run the unit tests, use `pytest`

:

```
python3 -m pytest --cov=PyNomaly
```

To run the tests with Numba enabled, simply set the flag `NUMBA`

in `test_loop.py`

to `True`

. Note that a drop in coverage is expected due to portions of the code
being compiled upon code execution.

Semantic versioning is used for this project. If contributing, please conform to semantic versioning guidelines when submitting a pull request.

This project is licensed under the Apache 2.0 license.

If citing PyNomaly, use the following:

```
@article{Constantinou2018,
doi = {10.21105/joss.00845},
url = {https://doi.org/10.21105/joss.00845},
year = {2018},
month = {oct},
publisher = {The Open Journal},
volume = {3},
number = {30},
pages = {845},
author = {Valentino Constantinou},
title = {{PyNomaly}: Anomaly detection using Local Outlier Probabilities ({LoOP}).},
journal = {Journal of Open Source Software}
}
```

- Breunig M., Kriegel H.-P., Ng R., Sander, J. LOF: Identifying Density-based Local Outliers. ACM SIGMOD International Conference on Management of Data (2000). PDF.
- Kriegel H., Kröger P., Schubert E., Zimek A. LoOP: Local Outlier Probabilities. 18th ACM conference on Information and knowledge management, CIKM (2009). PDF.
- Goldstein M., Uchida S. A Comparative Evaluation of Unsupervised Anomaly Detection Algorithms for Multivariate Data. PLoS ONE 11(4): e0152173 (2016).
- Hamlet C., Straub J., Russell M., Kerlin S. An incremental and approximate local outlier probability algorithm for intrusion detection and its evaluation. Journal of Cyber Security Technology (2016). DOI.

- The authors of LoOP (Local Outlier Probabilities)
- Hans-Peter Kriegel
- Peer Kröger
- Erich Schubert
- Arthur Zimek

- NASA Jet Propulsion Laboratory

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