Many machine learning techniques make use of distance calculations as a measure of similarity between two points. For example, in k-means clustering, we assign data points to clusters by calculating and comparing the distances to each of the cluster centers.
What happens, though, when the components have different variances, or there are correlations between components? First, a note on terminology.
Consider the following cluster, which has a multivariate distribution. This cluster was generated from a normal distribution with a horizontal variance of 1 and a vertical variance of 10, and no covariance. In order to assign a point to this cluster, we know intuitively that the distance in the horizontal dimension should be given a different weight than the distance in the vertical direction. We can account for the differences in variance by simply dividing the component differences by their variances.
The general equation for the Mahalanobis distance uses the full covariance matrix, which includes the covariances between the vector components. With that in mind, below is the general equation for the Mahalanobis distance between two vectors, x and y, where S is the covariance matrix. Side note: As you might expect, the probability density function for a multivariate Gaussian distribution uses the Mahalanobis distance instead of the Euclidean.
See the equation here. Now we can work through the Mahalanobis equation to see how we arrive at our earlier variance-normalized distance equation.
But when happens when the components are correlated in some way? Correlation is computed as part of the covariance matrix, S. Subtracting the means causes the dataset to be centered around 0, 0. Your original dataset could be all positive values, but after moving the mean to 0, 0roughly half the component values should now be negative.
Similarly, the bottom-right corner is the variance in the vertical dimension.
Mahalonobis Distance – Understanding the math with examples (python)
The bottom-left and top-right corners are identical. If the data is evenly dispersed in all four quadrants, then the positive and negative products will cancel out, and the covariance will be roughly zero. As another example, imagine two pixels taken from different places in a black and white image.
If the pixels tend to have the same value, then there is a positive correlation between them. If the pixels tend to have opposite brightnesses e. If the pixel values are entirely independent, then there is no correlation. The cluster of blue points exhibits positive correlation. Looking at this plot, we know intuitively the red X is less likely to belong to the cluster than the green X. Even taking the horizontal and vertical variance into account, these points are still nearly equidistant form the center.
The Mahalanobis distance takes correlation into account; the covariance matrix contains this information. We can gain some insight into it, though, by taking a different approach. The two eigenvectors are the principal components.Multivariate outliers can be a tricky statistical concept for many students.
Multivariate outliers are typically examined when running statistical analyses with two or more independent or dependent variables. Here we outline the steps you can take to test for the presence of multivariate outliers in SPSS. This could be, for example, a group of independent variables used in a multiple linear regression or a group of dependent variables used in a MANOVA.
Move the variables that you want to examine multivariate outliers for into the independent s box. Then click OK to run the linear regression. Sort this column in descending order so the larger values appear first. The degrees of freedom will correspond to the number of variables you have grouped together to calculate the Mahalanobis Distances in this care three: Age, TestScoreA, and TestScoreB.
Now write the expression: 1 — CDF. For X1, substitute the Mahalanobis Distance variable that was created from the regression menu Step 4 above. For X2, substitute the degrees of freedom — which corresponds to the number of variables being examined in this case 3.
By using this formula, we are calculating the p -value of the right-tail of the chi-square distribution. Click OK to compute the variable. This new variable will appear at the end of your spreadsheet. Multivariate outliers will be present wherever the values of the new probability variable are less than.
In this case, there were three multivariate outliers. Prior to running inferential analyses, it would be advisable to remove these cases. Call Us: Blog About Us. Pin It on Pinterest.The Mahalanobis distance is a measure of the distance between a point P and a distribution D, introduced by P. Mahalanobis in This distance is zero if P is at the mean of D, and grows as P moves away from the mean along each principal component axis. If each of these axes is re-scaled to have unit variance, then the Mahalanobis distance corresponds to standard Euclidean distance in the transformed space.
The Mahalanobis distance is thus unitless and scale-invariantand takes into account the correlations of the data set. If the covariance matrix is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance. If the covariance matrix is diagonalthen the resulting distance measure is called a standardized Euclidean distance :.
Mahalanobis distance is preserved under full-rank linear transformations of the space spanned by the data. This means that if the data has a nontrivial nullspace, Mahalanobis distance can be computed after projecting the data non-degenerately down onto any space of the appropriate dimension for the data.
We can find useful decompositions of the squared Mahalanobis distance that help to explain some reasons for the outlyingness of multivariate observations and also provide a graphical tool for identifying outliers.
Consider the problem of estimating the probability that a test point in N -dimensional Euclidean space belongs to a set, where we are given sample points that definitely belong to that set. Our first step would be to find the centroid or center of mass of the sample points. Intuitively, the closer the point in question is to this center of mass, the more likely it is to belong to the set.
However, we also need to know if the set is spread out over a large range or a small range, so that we can decide whether a given distance from the center is noteworthy or not. The simplistic approach is to estimate the standard deviation of the distances of the sample points from the center of mass.
If the distance between the test point and the center of mass is less than one standard deviation, then we might conclude that it is highly probable that the test point belongs to the set. The further away it is, the more likely that the test point should not be classified as belonging to the set.
By plugging this into the normal distribution we can derive the probability of the test point belonging to the set. The drawback of the above approach was that we assumed that the sample points are distributed about the center of mass in a spherical manner.
Were the distribution to be decidedly non-spherical, for instance ellipsoidal, then we would expect the probability of the test point belonging to the set to depend not only on the distance from the center of mass, but also on the direction.
In those directions where the ellipsoid has a short axis the test point must be closer, while in those where the axis is long the test point can be further away from the center. Putting this on a mathematical basis, the ellipsoid that best represents the set's probability distribution can be estimated by building the covariance matrix of the samples. The Mahalanobis distance is the distance of the test point from the center of mass divided by the width of the ellipsoid in the direction of the test point.
For a normal distribution in any number of dimensions, the probability density of an observation is uniquely determined by the Mahalanobis distance d. For number of dimensions other than 2, the cumulative chi-squared distribution should be consulted.
In a normal distribution, the region where the Mahalanobis distance is less than one i. Mahalanobis distance is proportional, for a normal distribution, to the square root of the negative log likelihood after adding a constant so the minimum is at zero. If we square both sides, and take the square-root, we will get an equation for a metric that looks a lot like the Mahalanobis distance:.
The resulting magnitude is always non-negative and varies with the distance of the data from the mean, attributes that are convenient when trying to define a model for the data. Mahalanobis's definition was prompted by the problem of identifying the similarities of skulls based on measurements in Mahalanobis distance is widely used in cluster analysis and classification techniques. It is closely related to Hotelling's T-square distribution used for multivariate statistical testing and Fisher's Linear Discriminant Analysis that is used for supervised classification.
In order to use the Mahalanobis distance to classify a test point as belonging to one of N classes, one first estimates the covariance matrix of each class, usually based on samples known to belong to each class.The Mahalanobis Distance, widely used in cluster and classification algorithms, can be quite useful to detect outliers in multivariate data.
For each taco, we have the following ingredients:. Now, envision that the quantity of our ingredients were the coordinates in our axes. A small increase in taco meat would not alter the recipe or desirability of the taco on a large scale. Nor would a small decrease in cheese impact the taste test. By doing so, we can identify outliers easier. The idea is to calculate the covariance matrix of each class to help identify the relative distance between the two attributes from their centroid, a base or central point that is the overall mean for multivariate data.
Section 2: Means and Sample Size We then calculate the means of each variable. Section 3: Input Data and Find Difference We input data we want to find the distance from the mean v and then we calculate the difference between the new vector and the mean vector. S function, we calculate the individual variances for each variable. Section 6: Create Covariance Matrix We then create a covariance matrix labeled covar matrix by inserting the outputs from sections 4 and 5. Here are the steps:. Try and figure out how to do so by applying the steps above!
Section 9: Calculate Mahalanobis Distance The last step! We just created a covariance matrix, took the inverse of the covariance matrix, multiplied that inverse covariance matrix with the difference of the target vector to the mean, multiplied that output with the transposed difference, and then took the square root of the output.
Please let me know if there are any questions or concerns down in the comments section. I am fairly new at calculating the Mahalanobis Distance, so please do let me know if there are any errors! You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account.
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. Say I work out the mahalanobis distance 'D' to measure the separation between two objects which aren't normally distributed. I've read that using Chi-Square Distribution is one way, using N-1 degree of freedom and converting the distance to Chi-square p values. However, it states that because isn't normally distributed some conversion is recommended.
In my case, where I have one distance 'D' and I can't re-scale it, is using the above still advisable? If if it advisable, could one briefly explain the conversion where e. You need to have some assumed distribution, normal or not, in which to compare your 'D' to. If you only have one D and no other information, then there's no way to tell whether it's an outlier or not. If you have information about the population distances then your first step would probably be to plot the distances in a boxplot, then overlay your one D and see where it lies in terms of a percentile.
Then you could use that information to deem whether it's an outlier in the practical and contextual sense. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.
Mahalanobis Distance for Spotting Outliers
Ask Question. Asked 5 years ago. Active 3 years, 1 month ago. Viewed 5k times. Say I now want to use 'D' against some critical values to decide if it's an outlier or not. If it isn't, what alternative method could I use to generate some critical values?
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Related 6. Hot Network Questions. Question feed.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. Mahalanobis in and has been used in various statistical applications ever since. The above three steps are meant to address the problems with Euclidean distance we just talked about.
But how? We then divide this by the covariance matrix or multiply by the inverse of the covariance matrix. If the variables in your dataset are strongly correlated, then, the covariance will be high.
Dividing by a large covariance will effectively reduce the distance. So effectively, it addresses both the problems of scale as well as the correlation of the variables that we talked about in the introduction.I previously described how to use Mahalanobis distance to find outliers in multivariate data.
This article takes a closer look at Mahalanobis distance. A subsequent article will describe how you can compute Mahalanobis distance. In statistics, we sometimes measure "nearness" or "farness" in terms of the scale of the data. Often "scale" means "standard deviation. You can also specify the distance between two observations by specifying how many standard deviations apart they are. For many distributions, such as the normal distribution, this choice of scale also makes a statement about probability.
Specifically, it is more likely to observe an observation that is about one standard deviation from the mean than it is to observe one that is several standard deviations away.
Because the probability density function is higher near the mean and nearly zero as you move many standard deviations away. For normally distributed data, you can specify the distance from the mean by computing the so-called z-score. This is a dimensionless quantity that you can interpret as the number of standard deviations that x is from the mean.
You can generalize these ideas to the multivariate normal distribution. The following graph shows simulated bivariate normal data that is overlaid with prediction ellipses. The prediction ellipses are contours of the bivariate normal density function. In the graph, two observations are displayed by using red stars as markers. The first observation is at the coordinates 4,0whereas the second is at 0,2. The question is: which marker is closer to the origin? The origin is the multivariate center of this distribution.
The answer is, "It depends how you measure distance. However, for this distribution, the variance in the Y direction is less than the variance in the X direction, so in some sense the point 0,2 is "more standard deviations" away from the origin than 4,0 is. Notice the position of the two observations relative to the ellipses.
What does this mean? It means that the point at 4,0 is "closer" to the origin in the sense that you are more likely to observe an observation near 4,0 than to observe one near 0,2. The probability density is higher near 4,0 than it is near 0,2. In this sense, prediction ellipses are a multivariate generalization of "units of standard deviation. A point p is closer than a point q if the contour that contains p is nested within the contour that contains q.
You can use the probability contours to define the Mahalanobis distance. The Mahalanobis distance has the following properties:. For univariate normal data, the univariate z-score standardizes the distribution so that it has mean 0 and unit variance and gives a dimensionless quantity that specifies the distance from an observation to the mean in terms of the scale of the data. After transforming the data, you can compute the standard Euclidian distance from the point z to the origin.
This measures how far from the origin a point is, and it is the multivariate generalization of a z-score. You can rewrite z T z in terms of the original correlated variables. The Mahalanobis distance accounts for the variance of each variable and the covariance between variables. Geometrically, it does this by transforming the data into standardized uncorrelated data and computing the ordinary Euclidean distance for the transformed data.
In this way, the Mahalanobis distance is like a univariate z-score: it provides a way to measure distances that takes into account the scale of the data. His areas of expertise include computational statistics, simulation, statistical graphics, and modern methods in statistical data analysis. Pingback: The curse of dimensionality: How to define outliers in high-dimensional data?Finding Multivariate Outliers with the Mahalanobis Distance Test in SPSS
I got 20 values of MD [2.