# Mahalanobis distance anomaly detection

Mahalanobis distance is an effective multivariate distance metric that measures the distance between a point and a distribution. It is an extremely useful metric having, excellent applications in multivariate anomaly detection, classification on highly imbalanced datasets and one-class classification.

This post explains the intuition and the math with practical examples on three machine learning use cases. Mahalanobis distance is an effective multivariate distance metric that measures the distance between a point vector and a distribution. It has excellent applications in multivariate anomaly detection, classification on highly imbalanced datasets and one-class classification and more untapped use cases. Considering its extremely useful applications, this metric is seldom discussed or used in stats or ML workflows.

This post explains the why and the when to use Mahalanobis distance and then explains the intuition and the math with useful applications. Euclidean distance is the commonly used straight line distance between two points.

If the two points are in a two-dimensional plane meaning, you have two numeric columns p and q in your datasetthen the Euclidean distance between the two points p1, q1 and p2, q2 is:. Well, Euclidean distance will work fine as long as the dimensions are equally weighted and are independent of each other.

Only the units of the variables change. Since both tables represent the same entities, the distance between any two rows, point A and point B should be the same. But Euclidean distance gives a different value even though the distances are technically the same in physical space.

That is, if the dimensions columns in your dataset are correlated to one another, which is typically the case in real-world datasets, the Euclidean distance between a point and the center of the points distribution can give little or misleading information about how close a point really is to the cluster.

The above image on the right is a simple scatterplot of two variables that are positively correlated with each other. That is, as the value of one variable x-axis increases, so does the value of the other variable y-axis.

The two points above are equally distant Euclidean from the center. But only one of them blue is actually more close to the cluster, even though, technically the Euclidean distance between the two points are equal. This is because, Euclidean distance is a distance between two points only. It does not consider how the rest of the points in the dataset vary. So, it cannot be used to really judge how close a point actually is to a distribution of points.

What we need here is a more robust distance metric that is an accurate representation of how distant a point is from a distribution. Mahalonobis distance is the distance between a point and a distribution.

And not between two distinct points. It is effectively a multivariate equivalent of the Euclidean distance. It was introduced by Prof.Detecting outliers in a set of data is always a tricky business.

How do we know a data point is an outlier?

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How do we make sure we are detecting and discarding only true outliers and not cherry-picking from the data? We can however work out a few good methods to help us make sensible judgements. Today we are going to discuss one of these good methods, namely the Mahalanobis distance for outlier detection. The aficionados of this blog may remember that we already discussed a fairly involved method to detect outliers using Partial Least Squares. What we are going to work out today is instead a simpler method, very useful for classification problems.

Therefore we can use PCA as a stepping stone for outliers detection in classification. For a couple of our previous posts on PCA check out the links below:.

For this tutorial, we are going to use NIR reflectance data of fresh plums acquired from to nm with steps of 2 nm. The data is available for download at our Github repository.

So far so good. We are now going to use the score plot to detect outliers. More precisely, we are going to define a specific metric that will enable to identify potential outliers objectively. This metric is the Mahalanobis distance. But before I can tell you all about the Mahalanobis distance however, I need to tell you about another, more conventional distance metric, called the Euclidean distance.

That is the conventional geometrical distance between two points. Consider the score plot above. Pick any two points. The distance between the two according to the score plot units is the Euclidean distance. This is the whole business about outliers detection. Again, look at the score plot above. I bet you can approximately pinpoint the location of the average or centroid of the cloud of points, and therefore easily identify the points which are closer to the centre and those sitting closer to the edges.

This concept can be made mathematically precise. If we drew the score plot using the correct aspect ratio, the cloud of point would squash to an ellipsoidal shape.An anomaly detection method for virtual machines in a cloud system, in which an HsMM is trained by searching the state information of the normal virtual machines in the cloud system, and a corresponding algorithm is designed to detect and calculate the probabilistic logarithm probability and the Mahalanobis distance of the dynamic changing behaviors of the resources in the cloud system when each virtual machine is online.

If the Mahalanobis distance value of an online virtual machine is detected being higher than the preset threshold value of the cloud system, it is suggested that the virtual machine is operating anomalously.

This application is a National Stage entry under 35 U. CN This invention relates to the field of network technologies, and more particularly, an anomaly detection method for the virtual machines in a cloud system.

More and more companies and enterprises have been increasingly reducing their costs by transferring some of their infrastructures of information technology to providers of cloud system services, such as data centers with distributed storage infrastructures and other types of cloud computing systems. Such providers of cloud system services have established various types of virtual infrastructures, including private and public cloud systems with commercially virtualized software such as Vmware and vSphere.

The data of such cloud systems can be distributed to hundreds of interconnecting computers, storage devices and other physical machines. In a public or private cloud system, enterprises that rent computing and storage resources from providers of cloud system services are called cloud renter. When they store their data in a cloud system, the security of their data is at the risk of being attacked outside.

For instance, a provider of the cloud system services may establish multiple virtual machines for different renter in one physical host; if one of the virtual machines malfunctions because of internal viruses or external attacks, the data security of the other virtual machines in the same physical host of the cloud system is threatened.

The anomalous virtual machine can enormously threaten the normal operations of the other virtual machines sharing the same physical host and hinder the cloud system from providing services to the other normal virtual machines, endangering the security of the cloud system.

Currently, there are a few anomaly detection methods for detecting anomalies of the virtual machines in a cloud system, and the existing defensive technologies have failed to monitor the dynamic changes of the virtual machines in a cloud system, which are deficient in performance to a certain degree.

Review Questions 3 - Mahalanobis distance and rank of covariance matrix

It is of great significance to guarantee the availability of a cloud system for its normal virtual machines, specifically in the following two aspects: the first is to provide reasonable resource distribution services for the normal use of the virtual machines; the second is to check the availability of the other normal virtual machines when one virtual machine in the cloud system becomes anomalous, i.

This invention provides a real-time reliable anomaly detection method for the virtual machines in a cloud system, which can reduce the impact of anomaly detection on the global operation of the cloud system and ensure the availability of the normal virtual machines in the cloud system for its users. To fulfill that purpose, the invention provides an anomaly detection method for the virtual machines in a cloud system:.

A search module for Searching the virtual machine's state attribute information can search for the state attribute information of each virtual machine in a cloud system and then send the information to an HsMM Hidden semi-Markov Model online detection module in real time for anomaly detection.

The above-mentioned HsMM online detection module can detect any anomaly of the virtual machines and send the state attribute information of the anomalous virtual machines to a detection and treatment system in the cloud system to remove the anomalies. The above-mentioned anomaly detection and treatment system will detect any anomalous virtual machine in the cloud system, remove the anomalies and send warnings to its corresponding cloud renter if the anomaly degree has not reached the preset index of the cloud system.

And if the anomaly degree has reached the preset index, the anomaly detection and treatment system will send warnings to its corresponding cloud renter and shut down each anomalous virtual machine.

The aforementioned anomaly detection method for the virtual machines in a cloud system comprises the following steps:.

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Step 1: Search the state attribute value of each virtual machine in the normal state in a cloud system through a module searching the virtual machine's attribute information; the normal state herein refers to the state of a virtual machine without any internal virus or external attack.

Step 2: Take the state attribute value of each virtual machine in the normal state as an observation sequence, train an HsMM and design an HsMM online detection algorithm. Step 3: Search the online state information of each virtual machine in the cloud with a module searching the virtual machine's state attribute information in accordance with a preset time interval of the cloud system, and send the information to the HsMM online detection module in real time.

Step 4: The aforementioned HsMM online detection module, which is based on an algorithm obtained from Step 2, can detect the state and behavior of each virtual machine online and calculate its probabilistic logarithm probability and the Mahalanobis distance so as to judge whether the virtual machine is anomalous or not.

Step 5: Make a comparison between the Mahalanobis distance calculated by the online behavior of each virtual machine with the preset threshold value Q of the cloud system, and judge whether the Mahalanobis distance value of the online behavior of each virtual machine is higher than the threshold value Q.

If the former is higher than the latter, turn to Step 6; if not, turn to Step 3. Step 6: Start the anomaly detection and treatment system of the cloud system, and detect each virtual machine whose detection result is higher than the preset threshold value Q of the cloud system. Step 7: Judge whether the anomaly index of each virtual machine detected in Step 6 is higher than the maximum preset threshold value E max of the anomaly detection and treatment system.

If the anomaly index is equal to or higher than the maximum threshold value E maxturn to Step 8. If the anomaly index is lower than E maxthe anomaly detection and treatment system will remove the anomaly and send warnings to the corresponding cloud renter and then turn to Step 3.

Step 8: The anomaly detection and treatment system will send warnings to the corresponding cloud renter when the anomaly index of one virtual machine in the cloud system is equal to or higher than the maximum threshold value E max and shut down that virtual machine.R's mahalanobis function provides a simple means of detecting outliers in multidimensional data.

When plotting these data generated for this example using an interactive plotyou could mark as outliers those points that are, for instance, more than two sample standard deviations from the mean height or mean weight:.

Note that the point with height equal to cm in the bottom-right corner of the graph has not been marked as an outlier, as it's less than 2 standard deviations from the mean height and mean weight. And yet that is the point that most clearly does not follow the linear relationship between height and weight that we see in this data.

It is—arguably—the real outlier here.

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By the way, the choice of scales for the above graph is somewhat misleading. A clearer picture of the effect of height on weight would have been obtained by at least letting the y scale start at zero. But I'm using this data merely to illustrate outlier detection; I hope you'll overlook this bad practice! The above code marks as outliers the two most extreme points according to their Mahalanobis distance also known as the generalised squared distance. This is, very roughly speaking, the distance of each point the rows of the dataframe from the centre of the data that the dataframe comprises, normalised by the standard deviation of each of the variables the columns of the dataframe and adjusted for the covariances of those variables.

For details, visit Wikipedia's page on Mahalanobis distance. As you can see, this time the point in the bottom-right corner of the graph has been caught:. And this technique works in higher dimensions too. This code produces a 3-dimensional spinnable scatterplot:. As you can see from the above code, the mahalanobis function calculates the Mahalanobis distance of a dataframe using a supplied vector of means and a supplied covariance matrix. You'll typically want to use it as in the examples above, passing in a vector of means and a covariance matrix that have been calculated from the dataframe under consideration.

For example:. The resulting vector of distances can be used to weed out the most extreme rows of a dataframe. This is often useful when you want to quickly check whether an analysis you're running is overly affected by extreme points.

First run the analysis on the full dataset, then remove the most extreme points using the above technique… and then run your analysis again. If there's a big difference in the results, you may want to consider using an analysis that is more robust against outliers. Be wary of mahalanobis when your data exhibit nonlinear relationships, as the Mahalanobis distance equation only accounts for linear relationships.

For example, try running the following code:. Note that the most obvious outlier has not been detected because the relationship between the variables in the dataset under consideration is nonlinear. Toggle navigation. Last revised 30 Nov For example, suppose you have a dataframe of heights and weights: hw data.An implementation of a density based outlier detection method - the Local Outlier Factor Technique, to find frauds in credit card transactions.

For detecting both local and global outliers. Compute the Mahalanobis distance from a centroid for a given set of training points. Implement a k-nearest neighbor kNN classifier. Graph-based image anomaly detection algorithm leveraging on the Graph Fourier Transform. A collection of interesting, memorable, and well Contains the codes for Extended Histogram of Gradients for object recognition developed by me during my PhD studies.

Structure informed clustering based population structure correction strategy. An example of a minimum distance classificator doing a comparison between using Mahalanobis distance and Euclidean distance. Add a description, image, and links to the mahalanobis-distance topic page so that developers can more easily learn about it. Curate this topic. To associate your repository with the mahalanobis-distance topic, visit your repo's landing page and select "manage topics.

Star Code Issues Pull requests. Multi-target tracker based on cost computation. Updated Feb 15, Python. Updated Mar 25, CSS. Star 3. Updated Apr 30, Python. Star 2. Updated Oct 19, Python. Star 1. Updated Jul 29, Jupyter Notebook.

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Updated Jan 17, Python. A credit card fraud detection algorithm. Updated May 8, Jupyter Notebook. Tools for quantifying latent space class separations. Updated Mar 24, C. Plugins to Phy1 - additional features to Phy. Updated Nov 23, Python. Updated Nov 12, Python.In this article, I will introduce a couple of different techniques and applications of machine learning and statistical analysis, and then show how to apply these approaches to solve a specific use case for anomaly detection and condition monitoring.

These are all terms you have probably heard or read about before. However, behind all of these buzz words, the main goal is the use of technology and data to increase productivity and efficiency.

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The connectivity and flow of information and data between devices and sensors allows for an abundance of available data. The key enabler is then being able to use these vast amounts of available data and actually extract useful information, making it possible to reduce costs, optimize capacity, and keep downtime to a minimum.

This is where the recent buzz around machine learning and data analytics comes into play. Anomaly detection or outlier detection is the identification of rare items, events or observations which raise suspicions by differing significantly from the majority of the data.

Typically, anomalous data can be connected to some kind of problem or rare event such as e. This connection makes it very interesting to be able to pick out which data points can be considered anomalies, as identifying these events are typically very interesting from a business perspective. This brings us to one of the key objectives: How do we identify whether data points are normal or anomalous? In some simple cases, as in the example figure below, data visualization can give us important information.

In this case of two-dimensional data X and Yit becomes quite easy to visually identify anomalies through data points located outside the typical distribution. However, looking at the figures to the right, it is not possible to identify the outlier directly from investigating one variable at the time: It is the combination of the X and Y variable that allows us to easily identify the anomaly.

This complicates the matter substantially when we scale up from two variables to 10—s of variables, which is often the case in practical applications of anomaly detection.

## mahalanobis-distance

Any machine, whether it is a rotating machine pump, compressor, gas or steam turbine, etc. That point might not be that of an actual failure or shutdown, but one at which the equipment is no longer acting in its optimal state.

This signals that there might be need of some maintenance activity to restore the full operating potential. The most common way to perform condition monitoring is to look at each sensor measurement from the machine and to impose a minimum and maximum value limit on it. If the current value is within the bounds, then the machine is healthy.

If the current value is outside the bounds, then the machine is unhealthy and an alarm is sent. This procedure of imposing hard coded alarm limits is known to send a large number of false alarms, that is alarms for situations that are actually healthy states for the machine. There are also missing alarms, that is situations that are problematic but are not alarmed. The first problem not only wastes time and effort but also availability of the equipment. The second problem is more crucial as it leads to real damage with the associated repair cost and lost production.

Both problems result from the same cause: The health of a complex piece of equipment cannot be reliably judged based on the analysis of each measurement on its own as also illustrated in figure 1 in the above section on anomaly detection. We must rather consider a combination of the various measurements to get a true indication of the situation. It is hard to cover the topics of machine learning and statistical analysis for anomaly detection without also going into some of the more technical aspects.

I will still avoid going too deep into the theoretical background but provide some links to more detailed descriptions.

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If you are more interested in the practical applications of machine learning and statistical analysis when it comes to e. As dealing with high dimensional data is often challenging, there are several techniques to reduce the number of variables dimensionality reduction.

One of the main techniques is principal component analysis PCAwhich performs a linear mapping of the data to a lower-dimensional space in such a way that the variance of the data in the low-dimensional representation is maximized. In practice, the covariance matrix of the data is constructed and the eigenvectors of this matrix are computed.

The eigenvectors that correspond to the largest eigenvalues the principal components can now be used to reconstruct a large fraction of the variance of the original data. The original feature space has now been reduced with some data loss, but hopefully retaining the most important variance to the space spanned by a few eigenvectors. As we have noted above, for identifying anomalies when dealing with one or two variables, data visualization can often be a good starting point.

However, when scaling this up to high-dimensional data which is often the case in practical applicationsthis approach becomes increasingly difficult.My previous article on anomaly detection and condition monitoring has received a lot of feedback. Many of the questions I receive, concern the technical aspects and how to set up the models etc. Due to this, I decided to write a follow-up article covering all the necessary steps in detail, from pre-processing data to building models and visualizing results.

For an introduction to anomaly detection and condition monitoring, I recommend first reading my original article on the topic. This provides the neccesary background information on how machine learning and data driven analytics can be utilized to extract valuable information from sensor data.

The current article focuses mostly on the technical aspects, and includes all the code needed to set up anomaly detection models based on multivariate statistical analysis and autoencoder neural networks. To replicate the results in the original article, you first need to download the dataset from the NASA Acoustics and Vibration Database.

Each data set consists of individual files that are 1-second vibration signal snapshots recorded at specific intervals.

Each file consists of The file name indicates when the data was collected. Each record row in the data file is a data point. Larger intervals of time stamps showed in file names indicate resumption of the experiment in the next working day. The first step is to import some useful packages and libraries for the analysis:. An assumption is that gear degradation occur gradually over time, so we use one datapoint every 10 minutes in the following analysis.

Each 10 minute datapoint is aggregated by using the mean absolute value of the vibration recordings over the We then merge together everything in a single dataframe. In the following example, I use the data from the 2nd Gear failure test see readme document for further info on that experiment.

After loading the vibration data, we transform the index to datetime format using the following conventionand then sort the data by index in chronological order before saving the merged dataset as a.

To do this, we perform a simple split where we train on the first part of the dataset which should represent normal operating conditionsand test on the remaining parts of the dataset leading up to the bearing failure. I then use preprocessing tools from Scikit-learn to scale the input variables of the model.

As dealing with high dimensional sensor data is often challenging, there are several techniques to reduce the number of variables dimensionality reduction. One of the main techniques is principal component analysis PCA.