Definition of True (TPR) and False (FPR) Positive Rate.
1. Introduction
Data science is an evolutionary step in interdisciplinary fields incorporating computer science, statistics, engineering, and mathematics. At its core, data science involves using automated and robust approaches to analyze massive amounts of data and to extract informative knowledge from them. Data science transforms traditional ways of analyzing problems and creates powerful new solutions. Diverse computational and analytical techniques contribute to data science. In this chapter, we review and also propose one type of data mining and pattern recognition strategy that has been under development in multiple disciplines (e.g. statistics and machine learning) with important applications  outlier or novelty detection [14].
In biomedical engineering, data science can make healthcare and medical imaging science not only more efficient but also more effective for better outcomes and earlier detection. Outlier and novelty detection in these domains plays an essential role, though it may be underappreciated and not well understood by many biomedical practitioners. From the healthcare point of view, an outlier probably reflects the need for heightened vigilance, if not fullfledged intervention. For example, an abnormally high glucose reading for a diabetic patient is an outlier which may require action. In highdimensional medical imaging, developing automated and robust outlier detection methods is a critical preprocessing step for any subsequent statistical analysis or medical research.
An exact definition of an outlier or novelty typically depends on hidden assumptions regarding the data structure and the associated detection method, though some definitions are general enough to cope with varieties of data and methods. For example, outliers can be considered as patterns in data that do not conform to a welldefined notion of “normal” behavior, or as observations in a data set which appear to be inconsistent with the remainder of that set of data. Figure 1 shows outliers in a 2dimensional dataset. Since most of the observations fall into clusters N1 and N2, they are two “normal” regions; while points in region O1 as well as points o2 and o3 are outliers (in red), due to their sufficiently far distance from the “normal” regions. Identifying observations inconsistent with the “normal” data, or detecting previously unobserved emergent or novel patterns is commonly referred to as
Outlier and novelty detection methods can be divided into
Another related topic is robust statistics for estimation that can handle outliers or at least is less sensitive to the influence of outliers. Robust statistics perform well with data drawn from a wide range of probability distributions, especially for distributions that are not normally distributed. Robust statistical methods have been developed for many common problems, such as estimating data properties including location and scatter or estimating model parameters as in regression analysis [10, 11]. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a parametric distribution. A typical procedure or example of the former case is for multivariate estimation of location and covariance as well as for multivariate outlier detection. In this case, as a first step, the approaches often try to search for a minimum number of observations with a certain degree of confidence being outlierfree. Based on this starting subset, location and covariance can be estimated robustly. In a second step, outliers can be identified through computing the observations’ distances with respect to these initial estimates.
In this chapter, we review and also propose statistical and machine learning approaches for outlier and novelty detection, as well as robust methods that can handle outliers in data and imaging sciences. In particular, robust statistical techniques based on the Minimum Covariance Determinant (MCD) are introduced in Section 2, which include a classical and fast computation scheme of MCD and a few robust regression strategies. We present our newly developed multivariate Voronoi outlier detection (MVOD) method for time series data and some preliminary results in Section 3. This approach copes with outliers in a multivariate framework via designing and extracting effective attributes or features from the data; Voronoi diagrams allow for automatic configuration of the neighborhood relationship of the data points, facilitating the differentiation of outliers and nonoutliers. Section 4 reviews varieties of machine learning methods for novelty detection, with a focus on probabilistic approaches. In Section 5, we present some existing and new technologies related to outliers and novelty in the area of imaging sciences. Section 6 provides concluding remarks of the chapter.
2. Robust statistical methods using Minimum Covariance Determinant (MCD)
The Minimum Covariance Determinant (MCD) estimator is a highly robust estimator of multivariate location and scatter. Since estimating the covariance matrix is the cornerstone of many multivariate statistical methods, the MCD has also been used to develop robust and computationally efficient multivariate techniques.
2.1. MCD and its fast computing algorithm
Given a dataset consisting of
A point with a larger Mahalanobis distance will lie further away from the center of the data cloud than a point with a smaller Mahalanobis distance. A robust distance (
The classical estimates can be sensitive to outliers, while the MCD estimate is robust [8, 12, 13]. The MCD relies on a subset of the total observations. Choosing this subset makes the algorithm robust because it is less sensitive to the influence of outlying points. Figure 3 illustrates the difference between these two estimates; it is a scatterplot of the distances for an example dataset with 70 observations and 2 variables (i.e.
The MCD has a userdetermined parameter,
Computing the exact MCD is possible but computationally difficult, as it requires the evaluation of all
Going from
2.2. Robust multivariate regression and Multivariate LeastTrimmed Squares (MLTS) estimator
Section 2.1 introduced the robust MCD estimator and showed how the MCD can be computed efficiently. In this section, we review different frameworks for applying the MCD estimator to multivariate regression. These methods offer robust alternatives to standard multiple regression analysis.
We first look at robust multivariate regression in [19]. Suppose we have a full dataset of predictors and responses containing no outliers; computing the regression parameter estimates from the full dataset using a least squares procedure will yield accurate results. With outliers present in the dataset, the MCD is used to search for a subset of size
Different from the above robust multivariate regression, the multivariate least trimmed squares (MLTS) estimator in [20] first fits a regression model to the subset of data and then calculates the covariance matrix of the residuals. The estimator is defined by minimizing a trimmed sum of squared Mahalanobis distances, and can be computed by a fast algorithm. Let us consider the classical multivariate regression framework. Assume we have a sample of data defined as
The classical least squares estimator for the regression parameter is given by:
and the classical estimator of the scatter matrix is:
These classical least squares estimators are sensitive to outliers. A robust alternative to these estimators based on the residuals is achieved as below. For any
where
where
where
Using the MLTS as a means to estimate the parameters of the Vector Autoregressive (VAR) Model was presented in [21]. The VAR model is popular for modeling multiple time series. Estimation of its parameters based a typical least squares method is unreliable when outliers are present in the data. Development of robust procedures for multiple time series analysis is more crucial than for univariate time series analysis due to the data correlation structure. Experimental results in [21] show that applying the reweighted MLTS procedure to the VAR model leads to robust multivariate regression estimators with improved performance.
3. Multivariate Voronoi Outlier Detection (MVOD) for time series
In order to better analyze multivariate time series data, we have recently proposed a general outlier detection method based on the mathematical principles of Voronoi diagrams. It is general because different attributes or features can be extracted from the data for Voronoi diagram construction. These attributes or features can be designed based on the nature of the data and the outliers. This has the potential to increase the accuracy and precision of outlier detection for specific application problems.
3.1. Background on Voronoi diagram
Our new method requires a Voronoi diagram, which is composed of Voronoi cells [22]. A Voronoi diagram is a way of dividing space into regions. Assume we have a set
where
Figure 4 shows part of a Voronoi diagram, assuming Euclidean distance between the points. If one used a different distance metric, the Voronoi diagram would be configured differently. The plane is decomposed into
3.2. Our proposed MVOD method
The Voronoi Outlier Index (VOInd) used in our Multivariate Voronoi Outlier Detection (MVOD) method is based upon the Voronoi notion of nearest neighbors. For a point
Our method is based upon the geometric principles of Voronoi diagrams for defining the neighborhood relationship of the data points and this facilitates the assignment of group or data membership (i.e. outliers and nonoutliers). Construction of a two dimensional Voronoi diagram requires two coordinates for each data point. Based on the nature of the data and the nature of the outliers to be identified, we can embed their attributes into the coordinates via extracting different valid features from the data. Here, we present one such case of the MVOD framework for feature extraction; but many others are also possible, including nonparametric forms. Figure 5 overviews the process and the rest of this subsection explains the steps in more detail.
Although a regression model is used here in Step 2 to extract the feature value, in fact, our method does not require this model. With either Step 1 or Step 2 alone, we will have a corresponding nonparametric or parametric basis, both of which could be suitable for different applications or datasets.
Note that a Voronoi outlier factor is used in [24] as the index which, however, was completely univariate in nature, since the xand ycoordinates were based on a univariate time series. One of our primary motivations for this study is to create a novel and general MVOD method, which can detect outliers in time series data in a multivariate framework with multiple, interlocking sets of variables.
3.3. Experimental evaluation and results
Definition of TPR and FPR  
= TP / (TP + FN) 






TP  FP  = FP / (FP +TN) 


FN  TN 
The alpha parameter in the MLTS method determines both the size of the subset to use as well as a critical value in a chisquare distribution. If an observation is greater than this threshold in the chisquare distribution, then the MLTS method flags the observation as an outlier. However, it is critical to note that a onetoone correspondence does not exist between the alpha value chosen, and the number of outliers flagged. For instance, one could set alpha at 0.10 but only have 2 out of 100 observations flagged as outliers. Partly for this reason we considered a range of alpha values and then averaged across this range to fairly compare with the MVOD method. For all simulated time series, we considered alpha between 0.01 and 0.20.















MVOD  0.52  0.52  0.54  0.037  0.065  0.094 
MLTS  0.21  0.37  0.32  0.012  0.028  0.047  

MVOD  0.91  0.79  0.78  0.025  0.041  0.056  
MLTS  0.63  0.61  0.73  0.002  0.012  0.011  

MVOD  0.96  0.83  0.86  0.023  0.037  0.045  
MLTS  0.93  0.78  0.80  0.004  0.006  0.005  

MVOD  0.96  0.86  0.88  0.023  0.034  0.042  
MLTS  0.97  0.87  0.85  0.002  0.002  0.003  

MVOD  0.96  0.86  0.90  0.023  0.034  0.039  
MLTS  0.96  0.90  0.87  0.002  0.002  0.002 
In the results presented next, we obtained the TPR and FPR for the two methods in the following way. For a given number of outliers with a specific outlier magnitude, we averaged a total of five cases. The five cases averaged always included the threshold (MVOD) or alpha value (MLTS) corresponding with the number of outliers, but also contained the preceding four cases as well. For instance, in the 10 outlier case, we took the results for threshold=10 (MVOD), as well as thresholds of 9, 8, 7 and 6. In the corresponding MLTS case, we would have taken alpha=0.10, 0.09, 0.08, 0.07 and 0.06. The TPR and FPR for each of these five cases for each method were averaged to obtain the values shown in Table 2.
4. Machine learning methods for novelty detection
Novelty detection can be considered as the task of classifying test data that differ in some respect from the data that are available during training. This may be approached within the framework of “oneclass classification” [3], in which a model is built to describe “normal” training data. Novelty detection methods can be categorized into several areas such as probabilistic, distancebased, reconstructionbased, domainbased, and informationtheoretic techniques. In this section, we mainly introduce the first category of probabilistic approaches, and briefly summarize the others.
4.1. Probabilistic approaches
Probabilistic approaches to novelty detection are based on estimation of the generative probability density function of the data, which may then be used to define thresholds for the boundaries of “normality” in the data space and test whether or not a test sample is from the same distribution. Statistical hypothesis tests are the simplest statistical techniques for novelty detection [5]. Among the different statistical tests for novelty detection, here we concentrate on more advanced statistical modeling methods involving complex, multivariate data distributions. Techniques for estimating the underlying data density from multivariate training data broadly fall into parametric and nonparametric methods. The former imposes a restrictive model on the data, leading to a large bias when the model does not fit the data; the later builds up a very flexible model with fewer assumptions but requires a large sample size for a reliable fit of all free parameters when the model size becomes large.
In parametric approaches, the widely used distribution form for continuous variables is Gaussian. The involved parameters are estimated from the training data via
Nonparametric methods do not assume a fixed structure of a model; the model grows in size as necessary to fit the data and accommodate the data complexity. A common nonparametric approach for probabilistic density estimation is the kernel density estimator [26], which estimates the probability density function with a large number of kernels over the data space. The kernel density estimator places a kernel (e.g. Gaussian) on each data point and then sums the contributions from a localized neighborhood of the kernel. This is the socalled
4.2. Other categories
Distancebased approaches, such as clustering or nearestneighbor methods [3537], are another types of techniques that can be used for classification or for estimating the probability density function of data. The underlying assumption is that “normal” data are tightly clustered, while novel data occur far from their nearest neighbors. These methods use welldefined distance metrics to compute the distance (e.g. similarity measure) between two data points.
Reconstructionbased methods involve training a regression model with the training data [3, 38, 39]. The distance between the test vector and the output of the system (i.e. the reconstruction error) can be related to the novelty score, which would be high when “abnormal” data occurs. For instance, neural networks can be used in this way and show many of the same advantages for novelty detection as they do for typical classification applications. Another type of reconstructionbased novelty detection is subspacebased techniques. They assume that data can be projected or embedded into a lower dimensional subspace, which makes better discrimination of “normal” and “abnormal” data easier.
Domainbased methods often aim to describe a domain that contains “normal” data through a boundary around the “normal” class following the distribution of the data without explicitly providing a distribution [40, 41]. These techniques are usually insensitive to the specific sampling and density of the interested class. The location to the boundary is the criterion for determining the class membership of unknown data. Novelty detection support vector machines (SVMs) are the “oneclass SVMs”, which set the location of the novelty boundary only based on the data lying closest to it in the transformed feature space. That is, the novelty boundary is determined without considering the data that are not support vectors.
Informationtheoretic methods calculate the information content of a dataset with measures such as entropy, relative entropy, and Kolmogorov complexity, etc. [42, 43]. The key idea is that novel data alter the information content in a dataset significantly. A common procedure is: metrics are computed using the entire dataset and then the subset of points whose elimination from the dataset causes the largest difference in the metric are identified. The data contained in this subset is then assumed to be novel data.
5. Robust estimator and outlier detection in highdimensional medical imaging
The statistical analysis of medical images is challenging, not only because of the highdimensionality and low signaltonoise ratio of the data, but also due to varieties of errors in the image acquisition processes, such as scanner instabilities, acquisition artifacts, and issues associated with the experimental protocol [44]. Furthermore, populations under study typically present high variability [45, 46], and therefore the corresponding imaging data may have uncommon though technically correct observations. Such outliers deviating from normality could be numerous. With emergence of large medical imaging databases, developing automated outlier detection methods turns out to be a critical preprocessing step for any subsequent statistical analysis or group study. In addition, medical imaging data are usually strongly correlated [47]; outlier detection approaches based on multivariate models are thus crucial and desirable. Procedures using the classical MCD estimator are not wellsuited for such highdimensional data.
In [48], several extensions to the classical outlier detection framework are proposed to handle highdimensional imaging data. Specifically, the MCD robust estimator were modified so that it can be used for detecting outliers when the number of observations is small compared to the number of available features. This is achieved through introducing regularization in the definition and estimation of the MCD. Three regularization procedures were presented and compared:
Outlier detection methods described above can serve as a statistical control on subject inclusion in neuroimaging. However, sometimes it is controversial regarding whether or not outliers should be discarded, and, if so, what tolerance to use. An alternative strategy is to utilize outlierresistant techniques for statistical inference, which would also compensate for inexact hypotheses including data normality and homogeneous dataset. Robust techniques are especially useful when a large number of regressions are tested and assumptions cannot be evaluated for each individual regression, as with neuroimaging data.
Both individual subject and group analyses are required in neuroimaging. At a typical single subject level, a multiple regression model is used for the time series data at each voxel [49, 50], and outliers (or other assumption violations) in the time series would impact the model fitness. Robust regression can minimize the influence of these outliers. At the group level, after spatial normalization, a common strategy is to first save the regression parameters for each subject at each voxel and then perform a test on the parameter values. Robust regression used at this level can minimize the influence of outlying subjects. Wager et al. [51] used simulations to evaluate several robust techniques against ordinary least squares regression, and apply robust regression to secondlevel group analyses in three real fMRI datasets. Experimental results demonstrate that robust Iteratively Reweighted Least Squares (IRLS) at the second level is computationally efficient; it increases statistical power and decreases false positive rates when outliers are present. Without the presence of outliers, IRLS controls false positive rates at an appropriate level. In summary, IRLS shows significant advantages in group data analysis and in the hemodynamic response shape estimation for fMRI time series data.
6. Conclusions
Outlier and novelty detection is a primary step in many data mining and analysis applications, including healthcare and medical research. In this chapter, statistical and machine learning methods for outlier and novelty detection, and robust approaches for handling outliers in data and imaging sciences were introduced and reviewed. Particularly, we also presented our new method for outlier detection in time series data based on the Voronoi diagram (i.e. MVOD). There are several key advantages of our method. First, it copes with outliers in a multivariate framework by accounting for multivariate structure in the data. Second, it is flexible in extracting valid features for differentiating outliers from nonoutliers, in the sense that we have the option of using or not using a parametric model. Lastly, Voronoi diagrams capture the geometric relationship embedded in the data points. Initial experimental results show that our MVOD method can lead to accurate, sensitive, and robust identification of outliers in multivariate time series.
It is often difficult to reach a precise definition of outlier or novelty, and suggesting an optimal approach for outlier or novelty detection is even more challenging. The variety of practical and theoretical considerations arising in realworld datasets lead to the variety of techniques utilized [52]. Therefore, there is no single universally applicable detection method due to the large variety of considerations, which could include the application domain, the type of data such as dimension, and the availability of training data, etc. Based on the application and the nature of the associated data, developing suitable computational methods that can robustly and efficiently extract useful quantitative information from big data is still a current challenge and gaining increasing interest in data and imaging sciences.
Acknowledgments
This work is supported in part by a grant from the National Institute of Health, K25AG033725.
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