1 Introduction
Anomalous pattern detection is the task of identifying subsets of data points that systematically differ from the underlying model. Identifying anomalous patterns in realworld data is critical for understanding how people and systems deviate from expected behavior. In the spatiotemporal domain, timely identification of such patterns can allow for effective interventions. For example, detecting anomalous increases in opioid deaths can enable health care workers to effectively target overdose prevention programs. Similarly, patterns of increased 311 calls can help cities to better target services and allocate resources.
To detect these anomalous patterns, we will address three key challenges. First, realworld data is extremely complex with nontrivial correlations across space, time, and other features. Treating data points as iid ignores important covariance structure and will substantially overestimate the anomalousness of detected patterns. Second, an event of interest often affects multiple nearby points. Simply considering how anomalous is each individual point loses power to detect subtle anomalies. Third, anomalous patterns are often irregularly shaped or discontiguous due to latent demographic or geographic features. Searching for these complex patterns is important for precision and detection power, yet exhaustive methods are computationally intractable and may result in overfitting.
A sensible approach to this problem is modelbased anomaly detection, where a distribution is fit to model “regular” data. Points with a low likelihood under this distribution are identified as anomalous
(Chandola et al., 2009; Hodge and Austin, 2004). To address the complex correlations in realworld systems, Gaussian processes (GPs) provide a natural means of learning covariance structure from data. However, GP anomaly detection has been typically used to classify
individualpoints as outliers
(Smith et al., 2014; Kowalska and Peel, 2012; Stegle et al., 2008). Such approaches have difficulty when confronted with subtle anomalies, where each individual data point may seem to conform to the underlying distribution, yet when taken as a group, they form a collectively anomalous pattern. Thus anomalous pattern detection is a conceptually and statistically different problem than anomaly or outlier detection.
A few recent GP models consider anomalous intervals (Reece et al., 2015) and sophisticated change points (Saatçi et al., 2010; Herlands et al., 2016) to detect intervals of anomalous points. However, these methods (the first two of which are applied exclusively to onedimensional data) are limited to contiguous intervals in the input domain and cannot model the irregularly shaped anomalies we expect in complex data. Cheng et al. (2015) recently developed an anomalous pattern detection technique for spatiotemporal data. However, this approach requires a corpus of anomalyfree training data, can only detect contiguous anomalous patterns, and is specific to video data.
In the statistics literature, spatial and subset scanning methods are commonly used to identify collectively anomalous subsets of data (Kulldorff, 1997; Neill, 2012). By combining information across a subset of data elements, they generate a strong signal of anomalous behavior. These approaches compute a loglikelihood ratio (LLR) of subsets being drawn from a null or anomalous distribution. The LLR is a powerful statistic that measures how much evidence exists in the data to conclude if the subset exhibits abnormal behavior (Kulldorff, 1997; Neill et al., 2005). A core challenge of subset scanning is searching through the possible subsets of data elements (Neill and Moore, 2004; Agarwal et al., 2006; Duczmal et al., 2007; Wu et al., 2009). Neill (2012) shows that certain LLR statistics satisfying a lineartime subset scanning (LTSS) property can be optimized in by ordering points according to a particular “priority function” and evaluating only of the subsets. However, LTSS assumes that we can compute the contributions of individual points to the LLR. This is possible only when assuming that data is uncorrelated under the null (Neill, 2009), yet when applied to noniid data this independence assumption would result in substantial false positive rates, as correlated fluctuations will be mistaken for anomalous movements.
1.1 Contributions
In this paper we introduce novel techniques for identifying anomalous patterns in noniid data. Our methods are powerful and interpretable. By combining naturally interpretable GPs with localized anomalous patterns we can describe the “regular” data dynamics as well as quantify and corroborate anomalous regions with domain experts. Our main contributions are:

Combining GP modeling with subset scanning for powerful and interpretable detection of anomalous patterns in highly correlated data.

Proposing a new likelihood ratio statistic and subset scan technique for correlated data that do not assume conditional independence.

Performing holdout GP inference while computing our new likelihood ratio statistic conditioned on GP hyperparameters, to avoid corrupting the null model with anomalies.

Developing two novel, principled approaches to the NPhard problem of searching for the most anomalous subset, through a new iterative method and an application of the Generalized Rayleigh Quotient respectively.

We demonstrate our methods on numeric simulations, opioidrelated deaths, 311 calls for service data, and multiple streams of sewer flooding reports and tree damage reports, illustrating interpretable and policyrelevant results.
The paper proceeds as follows: §2 provides background on GPs. §3 introduces a novel loglikelihood ratio statistic for noniid data. §4 details the Gaussian Process Neighborhood Scan (GPNS) and the Gaussian Process Subset Scan (GPSS). Experimental results on numerical and real data are presented in §5.
2 Gaussian Processes
Consider data, , where , are inputs or covariates, and ,
are outputs or response variables indexed by
. We assume that is generated from by a latent function with a GP prior. In particular, ,A GP is a nonparametric prior over functions completely specified by mean and covariance functions. The mean function, , is the prior expectation of . The covariance function is given by (Rasmussen, 2006).
In this paper we use three important properties of GPs. First, we can draw samples from a GP prior since conditional on GP hyperparameters any finite collection of function values is distributed
Second, if a function has a Gaussian noise model, , then conditional on hyperparameters and data , we can derive a closed form expression for the predictive distribution of ,
(1)  
Third, GP hyperparameters, , can be learned by maximum likelihood optimization. While naively this requires computations, we use scalable GP learning in the structured kernel inference (SKI) framework (Wilson and Nickisch, 2015) for scalability.
3 LLR statistic for noniid data
Considering as defined in §2, we are interested in anomalous patterns that systematically differ from the underlying data distribution. We frame this search as an LLR comparison between a null model of “regular” behavior and an alternative model of “anomalous” behavior. A single latent GP defines both models. Subsets of data with the highest LLR scores are identified as the most anomalous.
Using a GP as the foundational modeling technique enables us to learn complex covariance structure and seamlessly extend to high dimensions as well as missing data. GPs are also naturally interpretable, which can provide insight about the “regular” data dynamics.
Consider a given subset of data points defined by the binary weighting vector
, where if is included in the subset and if excluded. Our null model, , assumes that all points (regardless of ) are drawn from a function with a GP prior: , where and . Our alternative model, , assumes that for , and for , where is any function of the latent GP.Here we focus on the case of a mean shift, , . The covariance structure remains the same in the null and alternative models. This allows us to efficiently compute the posterior mean vector and covariance matrix through GP inference, where under , and under . For posterior and we condition on all data outside the subset of points represented by
, ensuring that null model estimates are not corrupted by anomalous observations. However, since anomalies are assumed to be rare, their influence on parameter estimation is minimal. Therefore we use all
for GP learning of the parameters of the null model .We concentrate on mean changes since many real world cases concern anomalous levels of a quantity. Increases in localized drug overdoses, crime, and calls for city service are all mean shifts of great importance. Methods for identifying arbitrary changes in distribution – while able to detect other sorts of patterns – have reduced power to detect such mean shifts, due to more diffuse inductive biases. Persistent changes in covariance structure are typically considered changepoints and require substantial data in both regimes as opposed to the localized anomalous patterns we detect.
To measure how anomalous is a subset defined by , we compute the generalized loglikelihood ratio, , where:
(2) 
Here MNPDF
is the multivariate normal probability density function. The most anomalous subset,
, is(3)  
where for notational brevity. Conditional on , the MLE can be calculated in closed form, (see Appendix A). Nevertheless, maximizing is an NPcomplete Integer Quadratic Program (Del Pia et al., 2014), so an optimal solution requires exponentialtime computation. Note that the LTSS condition for a log lineartime subset search described in Neill (2012) does not apply, since it requires independent data with a diagonal covariance matrix.
3.1 Randomization testing
Given a method for finding anomalous subsets, the following randomization testing procedure determines an level significance threshold for conditional on the parameters of the null model:

Repeatedly draw , at the same covariates, , as the real data for .

Scan over with the chosen subset searching method. For each randomization save the most anomalous LLR value, .

Determine an level threshold for significance based on the quantile of the maximum LLR values, above which any from the original scan is considered statistically significant.
4 Efficient subset scanning
Having defined the LLR scan statistic to evaluate how anomalous is a given subset, we must now decide over which subsets to scan. Unconstrained optimization over subsets is computationally infeasible for an exhaustive search. Additionally, an unconstrained search may return an unrelated set of points, reducing interpretability and increasing the potential for overfitting. Anomalous events in human data, such as drug usage and requests for government services, often affect multiple nearby points. Thus we assume that anomalous points are near one another. For example, in spatiotemporal data we assume that anomalous points are clustered in space and time. Following Neill (2012), we define the local “neighborhood” of each data point, consisting of that point and its nearest neighbors, for some . We propose two approaches for using these neighborhoods to identify anomalous patterns: Gaussian Process Neighborhood Scan (GPNS) and Gaussian Process Subset Scan (GPSS).
4.1 GP Neighborhood Scan (GPNS)
Given a maximum neighborhood size , GPNS searches over the local neighborhoods consisting of the neighborhood for each point where . Where neighborhoods are defined by Euclidean distance, the set of search regions are circular in shape. For each neighborhood, , we obtain posterior and conditional on and points . We then compute for the neighborhood where , i.e., we evaluate the alternative hypothesis of the entire neighborhood being anomalous. GPNS pseudocode is presented in Alg. 1.
4.2 GP Subset Scan (GPSS)
While GPNS simplifies the exponential search, it requires constraining assumptions about the shape of neighborhoods and is only able to discover contiguous, spherical anomalous patterns. While there are approaches to increase the variety of neighborhood shapes without substantially degrading computational efficiency (Kulldorff et al., 2006; Neill and Moore, 2004; Kulldorff, 2001), these methods still require strict specification of potential anomalies. Such foreknowledge is unrealistic in realworld applications where natural boundaries, demographics, and stochastic effects lead to irregularlyshaped patterns. In such cases GPNS has reduced detection and explanatory power.
To flexibly detect irregularlyshaped patterns, GPSS conducts an unconstrained search for the most anomalous subset within neighborhoods of fixed size . Specifically, we identify the subset of points that maximize the LLR within each neighborhood. This allows us to identify highly irregular and even noncontiguous anomalous patterns. By restricting the search within a local neighborhood, we ensure that the identified patterns are coherent and interpretable. GPSS requires evaluating neighborhoods, as presented in Alg. 2.
Unfortunately, this procedure requires finding that maximizes the LLR of a subset within the neighborhood, . This is still an Integer Quadratic Program, whose optimal solution is intractable even for moderately sized neighborhoods. Instead, below we formulate three approaches for finding approximate solutions.
4.2.1 for conditionally optimal subset
Due to the full rank covariance matrix, we are unable to disentangle the individual contributions from each point to the LLR. However, if we condition on some subset of points, , we are able to compute the conditional contribution of each point. First, note that conditional on we can decompose from Eq. 3 into a sum over each of the points in the neighborhood
(4)  
The contribution of point to the LLR is the difference in LLR between and . Due to the outer and inner sums, the change in the LLR is:
(5) 
To maximize the LLR a point is only added to the subset if its contribution is positive. By setting Eq. 5 to zero we can compute , the maximum value for which to include point .
(6) 
As proved in Speakman et al. (2016), we obtain the conditional optimal subset by using as a priority function, ranking each data point by , and iteratively compute the score function for subsets including each additional point. This yields a log linear search over data points. Such an approach identifies the most anomalous subset with a positive mean shift. To find the most anomalous subset with a negative mean shift we simply rank data points by
Since the derivation of is conditional on a subset , we obtain the conditional optimal subset. In order to approximate an optimal solution we iteratively compute the conditional optimal subset beginning with a null subset, . This requires computation for some number of iterations. Pseudocode for this algorithm can be found in Appendix B.
For a diagonal , orders points according to , which is equivalent to the LTSS priority function for an independent Gaussian subset scan (Speakman et al., 2016). Thus approach identifies the optimal subset in the independent case and is conditionally optimal in the dependent case.
4.2.2 Generalized Rayleigh Quotient method
We consider an alternative optimization approach to obtain an approximately optimal subset. Consider plugging the MLE solution, , into from Eq. 3,
(7)  
If we relax such that , this can be rewritten as the generalized Rayleigh quotient, , where , and . Note that is a symmetric matrix and is a Hermitian positivedefinite matrix. Taking the Cholesky decomposition , the generalized Rayleigh quotient can be written as a Rayleigh quotient (Yu et al., 2013), , where and . The maximum of the Rayleigh quotient,
, is the largest eigenvector of
. Since we defined , then the maximum is the relaxed solution to our original optimization problem from Eq. 7.Although has noninteger elements, the ordering of the elements of this eigenvector corresponds to the importance of the data points in the neighborhood. Thus we scan over the ordered elements of , iteratively adding each to the subset. Maximizing over this linear number of subsets provides an approximate solution to the constrained integer program.
4.2.3 Forward stepwise optimization
A third approximation approach uses a greedy forward stepwise algorithm that iteratively sets one element such that the objective is minimized in each iteration. Once the objective cannot be further minimized the optimization is terminated, thereby providing a greedy optimal solution. For a neighborhood of size , the stepwise approach may require up to iterations, evaluating subsets at each iteration for a total of computations.
4.3 Efficient MultiStream Search
Often we are interested in searching for anomalous patterns across multiple dimensions, or streams, of data. For example, anomalous patterns of damaged trees and sewer flooding can help localize severe storm damage. Multistream search can enhance the signal of subtle anomalies that affect multiple streams, and reduce false positive detections when perturbations in a single stream are not important to the application.
In principle, GPNS and GPSS can handle multiple streams by stacking the data from each stream and adding a final dimension to indicate from which stream the data came. Yet naive GP inference requires complexity, so repeatedly concatenating data from multiple streams quickly leads to scalability issues. On the other hand, Kroneckerbased scalability require a kernel that is multiplicatively decomposable over the input dimensions (Saatçi, 2012). This implies that the prior correlation structure is the same over all data dimensions except for the stream indicator. For example, Kronecker structure in spatiotemporal settings constrains streams to have the same prior spatiotemporal correlations. This assumption is overly restrictive for the complex data in which we are interested.
Instead, we learn independent GPs for each stream of data and then scan over neighborhoods in the data jointly for all streams. Posteriors for each stream are independently inferred from the associated GP. Thus for streams , the posterior distribution for subset scanning contains a block diagonal covariance,
In this manner each stream can flexibly learn different prior covariance structures while still ensuring scalability equivalent to singlestream GPNS and GPSS. The one drawback of this approach is that interstream covariance information is not exploited for GP inference.
5 Experiments
We evaluate GPNS and GPSS using numeric simulations and three urban spatiotemporal datasets. We compare the methods against a number of competitive baseline algorithms from contemporary literature. First, we compare to an independent Gaussian subset scan, a state of the art anomalous pattern detection algorithm (Neill, 2009, 2012). Additionally, we compare against a standard GP anomaly detection approach (Kowalska and Peel, 2012; Stegle et al., 2008), in which we use the posterior distribution of the null GP model regressed over the entire dataset to classify points beyond a given level significance threshold as anomalies. While all GP methods in this paper are agnostic to kernel choice, an RBF kernel and linear mean function were used for all experiments.
Although anomalous pattern detection is a distinct problem from outlier or anomalous point detection, we also compare against two commonly used outlier detection techniques: a oneclass SVM (Schölkopf et al., 2001) and robust multivariate outlier detection using the Mahalanobis distance (Rousseeuw and Leroy, 2005; Rousseeuw and Van Zomeren, 1990).
An additional real data analysis is in Appendix D.
5.1 Numeric experiments
For each numeric test, baseline data is drawn from a 2D GP (Rasmussen and Nickisch, 2016). Multiplicative anomalies of arbitrary shape are injected by scaling randomly sampled points, within a randomly chosen neighborhood, by a factor of . (Note that this simulation does not correspond to our method’s assumption of an additive mean shift.) The most anomalous subset is computed using GPSS methods and baseline approaches. For the baseline GP approach and oneclass SVM we provide additional information (the true percentage of the anomalous data) in order to determine their threshold levels.
Varying the multiplicative factor between 1 and 2 we compute the average precision and recall in Fig.
1 over 50 tests in a 400 point grid for each multiplicative factor. Randomization testing () is performed for each synthetic test to determine the score threshold for significance. For precision and recall, truly anomalous points are “positive” and all other data is “negative.” The GPSS approaches dominate all other methods for nearly the entire test range, with performing best overall.Additionally, for each test we use an exhaustive search to find the subset with the highest LLR. The ratios of the LLR of approximate GPSS solutions to are shown in Fig. 2. Note that all approximation methods are relatively close to the optimal value. While the approach dominates at large magnitudes, the GRQ dominates at small magnitudes and achieves a relatively stable ratio across all tests.
To test the methods’ scalability we vary the maximum neighborhood size and measure run time. In Fig. 2 we compare GPSS, GPNS, and an exhaustive search for the optimal subset. The exhaustive search quickly becomes computationally intractable. Despite the added flexibility, GPSS is faster than GPNS because GP posterior inference is performed for fewer neighborhoods.
We consider the effect of the density of anomalies on GPSS and GPNS where “density” is defined by the proportion of anomalous points in the true subset (Fig. 3). While the stepwise method is competitive with the and GRQ approaches at low densities, its precision and recall drop off steeply at high densities. Additionally, in relatively low density anomalies, where the anomalous shapes may be highly irregular, GPNS has substantially reduced precision and recall.
5.2 Urban opioid overdose deaths
A recent United States opioid epidemic has garnered national attention (US Department of Health and Human Services, 2016). We study monthly opioid overdose deaths in New York from 19992015 (US CDC, 2017). Data is provided at a county level for Manhattan, Brooklyn, Queens, the Bronx, Nassau County, and Suffolk County. Data is missing for some months in different counties. We apply GPSS and baseline approaches jointly to data across all time, latitude, and longitude, with randomization testing at .
All three GPSS approaches identify two statistically significant anomalous patterns. While precise points
selected by the methods differ slightly, Fig. 4 depicts the two anomalous regions discovered by in blue circles and red crosses. With the exception of the independent subset scan, the baseline methods failed to discover a coherent anomalous pattern. Instead they selected individual points across space and time. For example, see results from the oneclass SVM in Fig. 4.
The anomalies detected by GPSS correspond to important public health events. The blue circles at the end of 2015 indicate a surge in opioid deaths corresponding to a well known plague of fentanylrelated deaths in NYC (City of New York Office of the Mayor, 2017). The anomaly denoted by red crosses in 2006 is particularly interesting since it indicates a spike in opioid deaths immediately preceding the introduction of community training programs to administer a lifesaving naloxone drug. This may indicate a surge in fatalities that was cut short by making naloxone more widely available and educating communities in its use.
5.3 Manhattan 311 requests
New York City’s 311 system enables residents to request government services. We consider a local public health event that occurred on 01/22/16 in upper Manhattan. On that day, local news reported that residents were concerned due to brown tap water (Pichardo, 2016; CBS New York, 2016b). Detecting the extent of the residents’ concerns is important to help identify and mitigate public health risks.
We consider daily 311 requests in Manhattan for the month of January 2016, aggregated over a 0.08 mile grid (City of New York, 2017). We apply GPSS methods and baseline approaches with neighborhoods of up to 15 points. All GPSS methods identified an anomalous pattern around the locations and time of the water discoloration event. Baseline methods tended to substantially overestimate the anomaly’s extent in both space and time. These results from January 22 are represented by the GRQ and the Robust baselines in Fig. 5. Blue and yellow squares indicate low and high volume of reports, respectively. Red squares indicate the top anomalous regions discovered by each method.
Ground truth does not exist for these hyperlocal events so we cannot compute precision and recall. However, 311 requests have labeled types, although we used aggregated 311 calls as our data inputs. For each method we compute the ratio of waterrelated 311 calls to nonwaterrelated calls in the detected anomalies. This “water signaltonoise” ratio, listed in Table 1, indicates how precisely each method identified regions associated with many waterrelated requests. The entire dataset has a water signaltonoise of 0.07.
Model  SignaltoNoise 

GRQ  7.22 
Stepwise  7.22 
7.22  
Independent SS  7.06 
Baseline GP  0.44 
Oneclass SVM  0.23 
RobustCov  0.12 
5.4 Multistream: trees and sewers
Using the multistream procedure from §4.3, we consider 311 reports of damaged trees and sewer issues. Both streams indicate weatherrelated issues: damaged trees indicate high winds while sewer calls indicate substantial precipitation. Together, these data identify areas with dangerous poststorm conditions. Each complaint type is fit with an independent GP and the entire data is scanned jointly for anomalies.
We analyze data in Brooklyn aggregated weekly over a 0.08 mile grid (City of New York, 2017). We conduct analyses for 2016 and 2010 with results depicted in Figs. 6 and 7. The number of sewer reports (per week, per cell) are plotted on top, and damaged tree reports on bottom. Red squares indicate the top anomalous regions discovered using the approach.
The most anomalous regions in 2016 were all concentrated during the week of July 20th when a significant summer storm felled trees and flooded sewers, thus jointly affecting both data streams (CBS New York, 2016a). Conversely, although the week of July 13th experienced elevated reports of felled trees no anomalous region is detected since there is no corresponding increase in sewer flooding. This demonstrates how multistream search may help to regulate GPSS.
The most anomalous regions in 2010 were all concentrated during the week of September 15th when an urban tornado cut through Brooklyn (AccuWeather, 2013). Unlike the 2016 results, these anomalies only occurred in reports of damaged trees. Also note the lone yellow square in the sewer data of September 22. Though the square indicates elevated number of calls, GPSS does not consider it anomalous since it does not represent a systematic shift in space and time.
6 Conclusions
We develop two GPbased subset scanning approaches to accurately and efficiently detect anomalous patterns in complex, highly correlated data. The results of GPNS and GPSS are coherent, powerful, and interpretable. While the simpler GPNS method may be sufficient for circular clusters, GPSS provides additional flexibility to accurately identify irregular cluster shapes. Unlike individual anomaly detection methods, the spatial locality enforced by our methods ensures coherent explanation of detected anomalous patterns. As the 311 and opioid applications demonstrate, our approaches can be used for studying and informing policy decisions. Future work could integrate categorical variables or extend to Studentt processes
(Shah et al., 2014) for dealing with heavytailed noise.This work was supported by NSF awards GRFP DGE1252522 and IIS0953330, the RK Mellon Foundation, and NCSU Laboratory for Analytical Sciences.
Appendix A Alternative model MLE
Given data, , we can determine the optimal mean shift, through maximum likelihood estimation as shown below. Let be the posterior mean and covariance of the null model in the domain of , and denote for brevity.
(8)  
We take the derivative with respect to and set it to zero
(9)  
Appendix B Iterative algorithm to approximate optimal subset
Since the derivation of is conditional on a subset , we obtain the conditional optimal subset. In order to approximate an optimal solution we use iteratively compute the conditional optimal subset beginning with a null subset, . This is an algorithm for some number of iterations, where is the size of the neighborhood. Pseudocode is depicted in Alg. 3.
Appendix C Constrained optimization over blocks
Although we focus on unconstrained subsets searching within neighborhoods, real world applications sometimes require a more constrained optimization. For example, in spatiotemporal phenomena it is often useful to consider anomalous patterns that are nearby in space and contiguous over time. We can enforce such constraints by predefining mutually exclusive blocks of points, where points in a block must all either be included in, or excluded from, a subset.
When considering blocks of points we can compute the total contribution from all points in the block, though we must also account for additional offdiagonal terms in due to the blocking of data points. Following the derivation in Section 4.2.1 of the main paper, we can derive the for each block,
(10) 
This can be used in a lightly modified version of Algorithm 3 where the of blocks, not individual points, is iteratively computed.
Appendix D School Absenteeism
Public schools in New York City record and publish daily student attendance (NYC Department of Education, 2017). Given the importance of education on future outcomes there is tremendous interest in understanding patterns of school absenteeism. We consider public school attendance data in Manhattan for the 20152016 school year. The data is messy, with missing entries and nonuniform placement of school locations. We aggregate data at weekly level and remove the last four weeks of the school year since they contain known high absenteeism rates that are not of interest to Department of Education officials.
We apply GPSS methods and baseline approaches with neighborhoods of up to ten local schools. All GPSS methods identified an anomaly around January to February 2016 concentrated on West Side of Manhattan. The results from GRQ around the time of the detected anomaly are presented in Fig. 8. Each dot represents a school location, with yellow dots indicating high attendance and blue dots indicating low attendance. The spacetime locations of schools in the top ten anomalous subsets are bordered in red.
The detected anomalies correspond to a category five blizzard which may have disrupted teachers and students from attending school even though no snow day closings were reported at the time. Further research is required to understand why the West Side of Manhattan differed systematically from the rest of the borough. Baseline anomaly detection methods did not identify a coherent anomaly and instead detected anomalies throughout the year.
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