PyNomaly 0.3.5
A Python 3 implementation of LoOP: Local Outlier Probabilities, a local density based outlier detection method providing an outlier score in the range of [0,1].
License: Apache License, Version 2.0
Keywords: outlier, anomaly, detection, machine, learning, probability

# PyNomaly

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. 

PyNomaly is a core library of [deepchecks](https://github.com/deepchecks/deepchecks), [OmniDocBench](https://github.com/opendatalab/OmniDocBench) and [pysad](https://github.com/selimfirat/pysad). 

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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](http://www.dbs.ifi.lmu.de/Publikationen/Papers/LoOP1649.pdf).

## Implementation

This Python 3 implementation uses Numpy and the formulas outlined in
[LoOP: Local Outlier Probabilities](http://www.dbs.ifi.lmu.de/Publikationen/Papers/LoOP1649.pdf)
to calculate the Local Outlier Probability of each sample.

## Dependencies
- Python 3.8 - 3.13
- 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). 

## Quick Start

First install the package from the Python Package Index:

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

Alternatively, you can use conda to install the package from conda-forge:

```shell
conda install conda-forge::pynomaly
```
Then you can do something like this:

```python
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:

```python
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.

```python
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.

### Utilizing Numba and Progress Bars

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`:

```python
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.

### Choosing Parameters

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*).

## Iris Data Example

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.

```python
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.

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

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).

```python
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.

```python
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).

```python
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**
![LoOP Scores without Clustering](https://github.com/vc1492a/PyNomaly/blob/main/images/scores.png)

**LoOP Scores with Clustering**
![LoOP Scores with Clustering](https://github.com/vc1492a/PyNomaly/blob/main/images/scores_clust.png)

**DBSCAN Cluster Assignments**
![DBSCAN Cluster Assignments](https://github.com/vc1492a/PyNomaly/blob/main/images/cluster_assignments.png)


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.

## Using Numpy

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

```python
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:

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

Similar to the above:

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

## Specifying a Distance Matrix

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.

Note that in order to maintain alignment with the LoOP definition of closest neighbors, 
an additional neighbor is added when using [scikit-learn's NearestNeighbors](https://scikit-learn.org/1.5/modules/neighbors.html) since `NearestNeighbors` 
includes the point itself when calculating the cloest neighbors (whereas the LoOP method does not include distances to point itself). 

```python
import numpy as np
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]
])

# Generate distance and neighbor matrices
n_neighbors = 3 # the number of neighbors according to the LoOP definition 
neigh = NearestNeighbors(n_neighbors=n_neighbors+1, metric='hamming')
neigh.fit(data)
d, idx = neigh.kneighbors(data, return_distance=True)

# Remove self-distances - you MUST do this to preserve the same results as intended by the definition of LoOP
indices = np.delete(indices, 0, 1)
distances = np.delete(distances, 0, 1)

# Fit and return scores
m = loop.LocalOutlierProbability(distance_matrix=d, neighbor_matrix=idx, n_neighbors=n_neighbors+1).fit()
scores = m.local_outlier_probabilities
```

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

**LoOP Scores by Distance Metric**
![DBSCAN Cluster Assignments](https://github.com/vc1492a/PyNomaly/blob/main/images/scores_by_distance_metric.png)

## Streaming Data

PyNomaly also contains an implementation of Hamlet et. al.'s modifications
to the original LoOP approach [[4](http://www.tandfonline.com/doi/abs/10.1080/23742917.2016.1226651?journalCode=tsec20)],
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.
```python
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.
```python
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
```

```python
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.

```python
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.

```python
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**
![LoOP Scores using Stream Approach with n=10](https://github.com/vc1492a/PyNomaly/blob/main/images/scores_stream.png)

### Notes
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.

## Contributing

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`. 

### Commit Messages and Releases

**Your commit messages are important** - here's why. 

PyNomaly leverages [release-please](https://github.com/googleapis/release-please-action) to help automate the release process using the [Conventional Commits](https://www.conventionalcommits.org/) specification. When pull requests are opened to the `main` branch, release-please will collate the git commit messages and prepare an organized changelog and release notes. This process can be completed because of the Conventional Commits specification. 

Conventional Commits provides an easy set of rules for creating an explicit commit history; which makes it easier to write automated tools on top of. This convention dovetails with SemVer, by describing the features, fixes, and breaking changes made in commit messages. You can check out examples [here](https://www.conventionalcommits.org/en/v1.0.0/#examples). Make a best effort to use the specification when contributing to Infactory code as it dramatically eases the documentation around releases and their features, breaking changes, bug fixes and documentation updates. 

### Tests
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 -s -v
```

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. 

## Versioning
[Semantic versioning](http://semver.org/) is used for this project. If contributing, please conform to semantic
versioning guidelines when submitting a pull request.

## License
This project is licensed under the Apache 2.0 license.

## Research
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}
}
```


## References
1. 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](http://www.dbs.ifi.lmu.de/Publikationen/Papers/LOF.pdf).
2. Kriegel H., Kröger P., Schubert E., Zimek A. LoOP: Local Outlier Probabilities. 18th ACM conference on Information and knowledge management, CIKM (2009). [PDF](http://www.dbs.ifi.lmu.de/Publikationen/Papers/LoOP1649.pdf).
3. Goldstein M., Uchida S. A Comparative Evaluation of Unsupervised Anomaly Detection Algorithms for Multivariate Data. PLoS ONE 11(4): e0152173 (2016).
4. 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](http://www.tandfonline.com/doi/abs/10.1080/23742917.2016.1226651?journalCode=tsec20).

## Acknowledgements
- The authors of LoOP (Local Outlier Probabilities)
    - Hans-Peter Kriegel
    - Peer Kröger
    - Erich Schubert
    - Arthur Zimek
- [NASA Jet Propulsion Laboratory](https://jpl.nasa.gov/)
    - [Kyle Hundman](https://github.com/khundman)
    - [Ian Colwell](https://github.com/iancolwell)