For an observation of multidimensional variable $$x_i$$ and the set of observations $$X$$, the Mahalanobis distance tells how far $$x_i$$ is from the center of the data with the shape of the dataset considered (i.e., far from center but with multiple samples in the proximity of Euclidean space is fine, but if it is alone in the proximity is an outlier). The Mahalanobis distance is defined as

$\text{MD}_i = \sqrt{(x_i-\bar{x})^\top V^{-1} (x_i - \bar{x})}$

where $$V$$ is the sample covariance matrix and $$\bar{x}$$ is the sample mean. Euclidean distance will consider only $$\bar{x}$$, but it is the covariance matrix $$V$$ to tell how much variation you have on each dimension. The Mahalanobis distance is scale-invariant. It roughly tells how many standard deviations the point $$x_i$$ is from the center $$\bar{x}$$. For $$p$$-dimensional observations, we can set the cutoff for outlier to be $$\text{MD}_i^2 > \chi_{p,1-\alpha/2}^2$$ where $$\chi_{p,v}^2$$ is the inverse CDF at value $$v$$ of $$\chi^2$$ distribution with dof $$p$$.

The Mahalanobis distance will suffer from masking effect, i.e. multiple outliers may not produce large distance. To solve this, we need sample mean and sample covariance matrix to be robust in case of adding outliers. Rousseeuw and Zomeren (1990) proposed minimum volume ellipsoid (MVE), which replaced $$\bar{x}$$ and $$V$$ with a vector $$T$$ and positive semidefinite matrix $$C$$ and define the robust distnace,

$\text{RD}_i = \sqrt{(x_i-T)^\top C^{-1} (x_i - T)}$

The vector $$T$$ and matrix $$C$$ is found such that

\begin{aligned} \min\quad & \text{det} C \\ \text{subject to } & \big\lvert \big\{i : (x_i - T)^\top C^{-1} (x_i - T) \le a^2 \big\}\big\rvert \ge \frac{n+p+1}{2} \end{aligned}

where $$a^2$$ is a constant chosen to be $$\chi_{p,0.5}^2$$ if assuming majority of the data are from normal distribution. The observation is $$p$$ dimensional and the dataset has $$n$$ observations. The MVE has breakdown point approximately at 50%, as the above optimization problem is constrained by half of the data points are with short robust distance.

Solving the above optimization problem can be computationally expensive. Hence Gao et al (2005) proposed Max-Eigen difference (MED). The algorithm is as follows:

1. For dataset $$X$$, find the sample mean $$\bar{x}$$ and covariance matrix $$V$$, then define $$\lambda_1\ge \cdots \ge \lambda_p$$ as the eigenvalues of $$V$$ and their corresponding eigenvectors are $$v_1,\cdots,v_p$$
2. Define $$X^{(i)}$$ as the dataset with observation $$x_i$$ removed. We find covariance matrix $$V^{(i)}$$ of $$X^{(i)}$$ its eigenvalues $$\lambda_1^{(i)}\ge \cdots \ge\lambda_p^{(i)}$$ and eigenvectors $$v_1^{(i)},\cdots,v_p^{(i)}$$
3. Define distance $d_i = \lVert \lambda_1^{(i)} v_1^{(i)} - \lambda_1 v_1\rVert \Big(1-\prod_{j=1}^p \mathbb{I}[ y_{ij}^2 \lt \lambda_j]\Big)$ where $$y_{ij} = (x_i - \bar{x})^\top v_j$$ and $$\mathbb{I}[\cdot]$$ is the indicator function. Then the MED is $$d_i$$ normalized,
$\text{MED}_{i} = \frac{d_i}{ \sum_{j=1}^n d_j}$

This works because the outliers will deeply influence the first eigenvalue and eigenvector.

These are not the only way to find outliers. Kannan and Manoj (2015) provided a survey of some other method as well. For example, the Cook’s distance is to consider a linear regression setting $$y_i \sim x_i$$ and asks how one point can influence the least square estimate. This distance can be written in different ways:

\begin{aligned} D_i &= \frac{1}{pMSE} \sum_{j=1}^n (\hat{y}_i - \hat{y}_{j(i)} \\ D_i &= \frac{e^2}{pMSE}\left(\frac{h_{ii}}{(1-h_{ii})^2}\right) \\ D_i &= \frac{(\hat{\beta}-\hat{\beta}^{-i})^\top (X^\top X) (\hat\beta - \hat\beta^{-i})}{(1+p)s^2} \end{aligned}

where the model is $$y = \beta^\top x$$ and $$\hat\beta$$ is the least square estimate of $$\beta$$. The $$\hat\beta^{-i}$$ is the same estimate with observation $$x_i$$ removed. Similarly $$\hat{y}_j = \hat\beta^\top x_j$$ and $$\hat{y}_{j(i)} = \hat\beta^{-i\top} x_j$$. The quantity $$h_{ii}$$ is the $$i$$-th diagonal element of the hat matrix

$H = X(X^\top X)^{-1} X^\top$

In fact, $$h_{ii}$$ is the leverage score in which it is high if $$x_i$$ is an outlier. $$h_{ii}\in[0,1]$$ and we can use the cutoff $$h_{ii}>3p/n$$ to determine an outlier.

DFFITS is another measure similar to Cook’s distance. Consider the same linear regression setting, it tells how much the regression function changes if observation $$x_i$$ is removed. The metric is

$\text{DFFITS}_i = \frac{\hat{y}_i - \hat{y}_{i(i)}}{\sigma_{(i)}\sqrt{h_ii}}$

This metric should less than 1 for small samples and it should be less than $$2\sqrt{p/n}$$ for large samples. Otherwise the points should be check for outlier.

### Univariate case

The case of identifying outliers in univariate samples are much easier as there is only one dimension to consider. Manoj and Kannan (2013) has another survey.

The simplest one is the quantile method, which defines the outlier as those with value below $$Q_1 - 1.5 \text{IQR}$$ and those above $$Q_3 + 1.5\text{IQR}$$.

Grubbs test is for the hypothesis ($$H_1$$) that the dataset has at least one outlier against the null hypothesis ($$H_0$$) that there are no outlier. The statistic depends on the sample mean $$\bar{x}$$ and sample standard deviation $$s$$:

$G = \frac{\max_i \vert x_i - \bar{x}_i\vert}{s}$

Bernard Rosner (1983) take this one step further to define the generalized extreme studentized deviate (ESD) test. It is a test comparing the null hypothesis of no outlier against the hypothesis of up to $$r$$ outliers. Define the statistic similar to Grubbs’:

$R_i = \frac{\max_i \vert x_i - \bar{x}_i\vert}{s}$

and find the observation $$x_i$$ that maximized $$R_i$$ and remove it from the dataset. Repeat this process for $$r$$ times to remove the top $$r$$ observations. The corresponding statistics are denoted as $$R_1,\cdots,R_r$$. The test is then defining the critical region:

$\lambda_i = \frac{(n-i)t_{p,n-i-1}}{\sqrt{n-i-1+t^2_{p,n-i-1}}(n-i+1)}$

where $$i=1,2,\cdots,r$$ and $$t_{p,v}$$ is the inverse CDF at value $$p$$ of the $$t$$ distribution with $$v$$ dof. We set

$$p = 1-\frac{\alpha}{2(n-i-1)}$$. The removed observation $$x_i$$ is an outlier if $$R_i > \lambda_i$$.

For an easier test, Doxin (1950) proposed the following: Arrage the observations in order $$x_1\le x_2 \le \cdots \le x_n$$, and depends on the size $$n$$, define

\begin{aligned} R_{10} &= \frac{x_n - x_{n-1}}{x_n - x_1} & \text{for }&3\le n\le 7 \\ R_{11} &= \frac{x_n - x_{n-1}}{x_n-x_2} && 8\le n\le 10\\ R_{21} &= \frac{x_n-x_{n-1}}{x_n-x_2} && 11\le n\le 13 \\ R_{22} &= \frac{x_n-x_{n-2}}{x_n-x_3} && 14\le n\le 30 \end{aligned}

and this value will exceed some critical value if $$x_n$$ is an outlier (too large). Apply the same formula with the order $$x_1\ge x_2 \ge \cdots \ge x_n$$ will find if $$x_n$$ is an outlier at the lower end (too small).

Finally, the Hampel method uses the median instead of mean. Denote the median of the dataset as $$M_x$$ and the deviation from median as $$r_i = x_i - M_x$$, then the median of the absolute deviation is denoted as $$M_{\vert r\vert}$$. The observation $$x_i$$ is an outlier if we found $$\vert r_i\vert \ge 4.5 M_{\vert r\vert}$$.

### References

• Peter J. Rousseeuw and Bert C. van Zomeren (1990) Unmasking Multivariate Outliers and Leverage Points. Journal of the American Statistical Association, 85(411):633-639
• Shaogen Gao, Guoying Li, and Dongqian Wang (2005) A New Approach for Detecting Multivariate Outliers. Communications in Statistics — Theory and Models, 34:1857-1865
• K. Senthamarai Kannan and K. Manoj (2015) Outlier Detection in Multivariate Data. Applied Mathematical Sciences, 9(47):2317-2324
• K. Senthamarai Kannan and K. Manoj (2013) Comparison of methods for detecting outliers. International Journal of Scientific & Engineering Research, 4(9):709-714
• F. E. Grubbs (1969) Procedures for detecting outlying observations in samples. American Statistical Association and American Society for Quality. Technometrics, 11:1-21
• Bernard Rosner (1983) Percentage Points for a Generalized ESD many-outlier procedure. Technometrics 25(2):165-172
• W. J. Doxin (1950) Analysis of extreme values. The Annals of Mathematical Statistics, 21(4):488-506