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By Katie Day (A)theologies, Guest Bloggers, OMFG, Politics/Law, RD Blog July 8, 2015
Black Churches are Burning: Is It the 1990s All Over Again?
For many Americans, news reporting that at least seven predominantly black churches have been destroyed by fire since the horrific murders at Emanuel AME Church in Charleston last month feels like "déjà vu all over again."
We remember all too well the daily images of burning churches on the nightly news in the late 1990s. I was involved in extensive research on the phenomenon at the time, so in many ways I feel as if I'm watching reruns of old news reels.
Still, can this be the beginning of another wave of racist violence targeting the spiritual homes of African-American Christians?
On June 30, a seventh church fire, at 100-year-old Mt. Zion AME Church in Greeleyville, S.C., was a devastating repeat of history. In 1995, two members of the Ku Klux Klan burnt Mt. Zion to the ground almost exactly 20 years ago to the day of last month's blaze.
Once again Mt. Zion has been reduced to ashes, but the FBI says the likely cause of last month's fire there was a lightning strike, not hatred. Intuitively, the smoke smells like a hate crime, a violent act meant to intimidate the African-American population. But is it?
At the outset, the obvious needs to be stated: not all church fires are crimes motivated by racial hatred. Many are accidental, whether faulty wiring or a lightning strike. It takes a lot to move a fire from "suspicious origins" to "intentional," and therefore arson.
It is even harder to prove that an arson is a hate crime.
There is a lot at stake here: hate crimes are prosecuted as civil rights violations and carry heavier penalties than arsons. Under the Church Arson Prevention Act, passed in 1996, penalties for both were increased when a house of worship is the target. Sometimes racist language is spray-painted on a church, or the perpetrators themselves reveal their motives. But often acts of arson have been depicted in court as just drunken mischief, allowing defendants to avoid the heavier sentences.
So, how does this latest wave of church burnings compare to the phenomenon of the 1990s? There are some striking, and familiar, similarities as well as some significant differences.
An obvious difference is that of scale. The Church Arson Prevention Act established the Church Arson Task Force which began documenting the burning of houses of worship from 1995. In their first report in 1997 they reported that 429 worship sites had been destroyed by arson or bombing in two and a half years. One third were predominantly African American churches in the south. They were able to get civil rights convictions for racial hate crimes in 60 percent of the cases in five Southern states and Nevada.
In subsequent years, the numbers fell: there were 209 church arsons in 1997 and 166 the following year, according to their annual reports. The last report was issued in 2001. Altogether there were almost 900 reported cases of church arson in the late 1990s—a huge number compared to what we've seen in the last two weeks.
A second difference is that of public response. In 1995-96, there was a unique alliance of actors creating "perfect storm" conditions. Black churches had long been the targets of racial hatred. But it was when Green Bay Packers player, Reggie White, spoke in an interview during the Super Bowl of 1996 about the arson and bombing of a church he pastored in Knoxville, Tenn., that the issue burst into mainstream media.
Within weeks, the White House, news media, and National Council of Christian Churches began working together to document and formulate responses to this wave of attacks on Black churches. USA Today and the New York Times provided daily coverage, as did major television network news programs. They deployed many reporters on the story, who were in communication with the Justice Department and ATF officials, who were themselves scrambling to keep up with events.
President Clinton, facing re-election, met directly with pastors of burned churches and with the National Council of Churches. He then moved quickly to pass the Church Arson Prevention Act, which established the Church Arson Prevention Task Force, an unprecedented collaborative effort between the Civil Rights Division of Department of Justice, the Bureau of Alcohol, Tobacco and Firearms, FEMA, the FBI, and others.
The Church Arson Prevention Act also made the penalties for church arsons more severe, designated the cancellation of church insurance policies by insurance companies to be a crime, designated loan funds for rebuilding, and developed strategies for arson prevention and community building.
Meanwhile, the NCC was raising unprecedented amounts of funding from organizations, celebrities, and individual congregations. The burning of black churches was an outrage that transcended theological differences. Millions of dollars were directed to herculean rebuilding efforts, channeling hundreds of volunteers to church sites for one-to-two-week stints.
Mt. Zion A.M.E. was among the churches rebuilt by those efforts.
The wave of church burnings had hit a national nerve. The public conversation around racism had been simmering in the early 1990s, but had spiked in intensity after the nation saw video of the beating of Rodney King in 1991 and the LA riots triggered by the acquittal of the officers who'd beaten him, as well as the trial and aquittal of O.J. Simpson in 1995. These racially divisive events painfully exposed the different perspectives of African American and white communities writ large.
The attacks on black churches, therefore, provided a respite of cohesion—an issue that seemingly most Americans could agree was wrong and want to do something about. I interviewed many of the white volunteers who helped rebuild black churches and who felt that, by helping to restore a church destroyed by hate, they were playing a part in dismantling racism and restoring justice.
The current spate of church burnings in the wake of the Charleston massacre has drawn much earlier public attention to what might be a trend. No doubt social media—see the #BlackLivesMatter and #WhoIsBurningBlackChurches on Twitter and Facebook—has accelerated public awareness.
So far there no concerted response by the federal government, such as re-instituting the Task Force from the 1990s, or a assembling a central faith-based organization of volunteers to rebuild the churches. But both, or similar efforts, still could happen.
This ripple-that-might-be-a-wave of church burnings comes at a time when nerves once again are raw: this time, the outrageous murder of innocent church folk in the familiar activity of gathering for a bible study. Once again, this racist act shocked and offended all Americans, of whatever political, religious or ethnic stripe. Presidential leadership has again helped frame the public's response: in his eloquent eulogy for the Rev. Clementa Pinckney, President Obama called us back to being the United States.
Hate crimes against black churches (and other religious groups as well) continue to occur. Many of the burned churches in the 1990s were surprised by all the media attention and that white Americans were just becoming aware of what had been part of the African American experience for generations. (A number of the sites, in fact, had been victims of arson in the past.)
Perhaps going forward after Charleston, we no longer will be able to plead ignorance of the unacceptable targeting of faith communities by those filled with racial hatred. There are certainly preventative measures that churches can take. But the deeper work of prevention is the responsibility of all of us.
black church burningschurch arson prevention actku klux klanracist violence
Katie Day is the Charles A. Schieren Professor of Church and Society at The Lutheran Theological Seminary at Philadelphia. Prof. Day was a founding member of the Global Network for Public Theology. Currently she is co-editing the Brill Companion on Public Theology, to be published in 2016. In addition to her research on urban religion, Prof. Day has conducted major studies of faith-based community organizing, Black Church arsons and the volunteer effort to rebuild them, religious responses to HIV/AIDS in South Africa, and religious social service agencies. Currently she is conducting research on "God and Guns," looking at the role of religion in the construction of meaning and public discourse on guns.
Latest Posts By Katie Day
Previous articleAsk the Dust: Unbelievers, Bunker-Dwellers, Anti-Natalists
Next article'My Name is Legion': Sources and Forces of White Supremacy in the U.S.
'My Name is Legion': Sources and Forces of White Supremacy in the U.S.
#WhoIsBurningBlackChurches: Is It Freedom Summer Again?
What Would Jesus Chew?, Bishops Bet on B-Ball, The Incredibly Shrinking Brain


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A list of research papers on face recognition
By Josh Davis
Davis, J. P., Maigut, A., & Forrest, C. L. D. (in press). The wisdom of the crowd: A case of post- to ante-mortem face matching by police super-recognisers. Forensic Science International, accepted 30 July 2019.
This case report describes novel methodology used to identify a 43-year-old post-mortem photo of a drowned male recovered from a London river in the 1970s. Embedded in an array of foils, police super-recognisers (n = 25) possessing superior simultaneous face matching ability, and police controls (n = 139) provided confidence ratings as to the similarity of the post-mortem photo to an ante-mortem photo of a man who went missing at about the same time. Indicative of a match, compared to controls, super-recognisers provided higher ratings to the target than the foils. Effects were enhanced when drawing on the combined wisdom of super-recogniser crowds, but not control crowds. These findings supported additional case evidence allowing the coroner to rule that the deceased male and missing male were likely one and the same person. A description of how similar super-recogniser wisdom of the crowd procedures could be applied to other visual image identification cases when no other method is feasible is provided.
Davis, J. P. (2019). The worldwide public impact of identifying super-recognisers for police and business. The Cognitive Psychology Bulletin, 4, Spring 2019, 17-21.
Super-recognisers occupy the extreme top end of a wide spectrum of human face recognition ability. Although test scores provide evidence of super-recognisers' quantitative superiority, their abilities may be driven by qualitatively different cognitive or neurological mechanisms. Some super-recognisers scoring exceptionally highly on multiple short-term face memory tests do not achieve superior performances on measures of simultaneous face matching, long term face memory and/or spotting faces in a crowd. Heterogeneous performance patterns have implications for police, security or business aiming to utilise super-recognisers' superior skills. Drawing on a global participant base (n ≈ 6,000,000), as well as theory and empirical research, this paper describes the background, development, and employment of tests designed to measure four components of superior face processing to assist in recruitment and deployment decisions.
Satchell, L. P., Davis, J. P., Julle-Danière, E., Tupper, N., & Marshman, P. (2019). Recognising faces but not traits: Accurate personality judgment from faces is unrelated to superior face memory. Journal of Research in Personality.
It is suggested that accurate personality judgments of faces are driven by a morphological 'kernel of truth' from face shape. We hypothesised that this relationship could lead to those with better face identification ability being better at personality judgments. We investigated the relationship between face memory, face matching, Big Five personality traits, and accuracy in recognising Big Five personality traits from 50 photographs of unknown faces. In our sample (n = 792) there was overall good (but varying) face memory and personality judgment accuracy. However, there was convincing evidence that these two skills do not correlate (all r < 0.06). We also replicate the known relationship between extraversion and face memory ability in the largest sample to date.
Belanova, E., Davis, J. P., & Thompson, T. (2018). Cognitive and neural markers of super-recognisers' face processing superiority and enhanced cross-age effect. Cortex, 98, 91-101.
Super-recognisers inhabit the extreme high end of an adult face processing ability spectrum in the population. While almost all research in this area has evaluated those with poor or mid-range abilities, evaluating whether super-recognisers' superiority generates distinct electrophysiological brain activity, and transcends to different age group faces (i.e., children's) is important for enhancing theoretical understanding of normal and impaired face processing. It may also be crucial for policing, as super-recognisers may be deployed to operations involving child identification and protection. In Experiment 1, super-recognisers (n = 315) outperformed controls (n = 499) at adult and infant face recognition, while also displaying larger cross-age effects. These findings were replicated in Experiment 2 (super-recognisers, n = 19; controls, n = 28), although one SR with frequent infant exposure showed no cross-age effect. Compared to controls, super-recognisers also generated significantly greater electrophysiological activity in event-related potentials associated with pictorial processing (P1) and explicit recognition (P600). Experiment 3, employing an upright and inverted sequential matching design found super-recognisers (n = 24) outperformed controls (n = 20) at adult and infant face matching, but showed no upright cross-age matching effects. Instead, they displayed larger inversion effects, and cross-age inversion effects, implicating the role of holistic processing in their perceptual superiority. Larger cross-age effects in recognition, but not matching suggests that super-recognisers' adult face recognition is partly driven by experience. However, their enhanced infant face recognition suggest super-recognisers' superiority is also experience-independent, results that have implications for policing and for models of face recognition.
Davis, J. P., Treml. F., Forrest, C., & Jansari, A (2018). Identification from CCTV: Assessing police super-recognisers ability to spot faces in a crowd and susceptibility to change blindness. Applied Cognitive Psychology, 32(3), 337-353.
Police worldwide regularly review closed‐circuit television (CCTV) evidence in investigations. This research found that London police experts who work in a full‐time "Super‐Recogniser Unit" and front line police identifiers regularly making suspect identifications from CCTV possessed superior unfamiliar face recognition ability and, with higher levels of confidence, outperformed controls at locating actors in a bespoke Spot the Face in a Crowd Test. Police were also less susceptible to change blindness errors and possessed higher levels of conscientiousness and lower levels of neuroticism and openness. Controls who took part in Spot the Face in a Crowd Test actor familiarisation training outperformed untrained controls, suggesting this exercise might enhance identification of persons of interest in real investigations. This research supports an accumulating body of evidence demonstrating that international police forces may benefit from deploying officers with superior face recognition ability to roles such as CCTV review, as these officers may be the most likely to identify persons of interest.
Durova, M. D., Dimou, A., Litos, G., Daras, P., & Davis, J. P. (2017). TooManyEyes: Super-recogniser directed identification of target individuals on CCTV. Proceedings of the 8th IET International Conference on Imaging for Crime Detection and Prevention (ICDP-17), IET Digital Library, 43-48. DOI.org/10.5281/zenodo.1071986
For the current research, a `Spot the Face in a Crowd Test' (SFCT) comprising six video clips depicting target-actors and multiple bystanders was loaded on TooManyEyes, a bespoke multi-media platform adapted here for the human-directed identification of individuals in CCTV footage. To test the utility of TooManyEyes, police `super-recognisers' (SRs) who may possess exceptional face recognition ability, and police controls attempted to identify the target-actors from the SFCT. As expected, SRs correctly identified more target-actors; with higher confidence than controls. As such, the TooManyEyes system provides a useful platform for uploading tests for selecting police or security staff for CCTV review deployment.
Davis, J. P., & Tamonytė, D. (2017). Masters of disguise: Super-recognisers' superior memory for concealed unfamiliar faces. Proceedings of the 2017 Seventh International Conference on Emerging Security Technologies (EST), 6-8 September 2017, Canterbury, UK. DOI: 10.1109/EST.2017.8090397
The deployment of police super-recognisers (SRs) with exceptional face recognition ability, has transformed the manner in which some forces manage CCTV evidence. In London, SRs make high numbers of sometimes disguised suspect identifications from CCTV. In two experiments measuring immediate and one-week memory of unfamiliar faces in disguise, SRs were more accurate and confident than controls at correctly identifying targets, and ruling out faces not seen before. Accuracy and confidence were highest when targets wore no disguise, followed by hat and plaster, sunglasses, and balaclavas respectively. Even in the balaclava condition, SRs were more accurate than chance levels. These findings add to an accumulating body of empirical evidence demonstrating that SRs possess wide-ranging enhanced face processing abilities, and their deployment should complement ever advancing computerised face recognition systems.
Davis, J. P., Lander, K., Evans, R., & Jansari, A. (2016). Investigating predictors of superior face recognition ability in police super-recognisers. Applied Cognitive Psychology, 30(6), 827–840. DOI: 10.1002/acp.3260
There are large individual differences in the ability to recognise faces. Super‐recognisersare exceptionally good at face memory tasks. In London, a small specialist pool of police officers (also labelled 'super‐recognisers' by the Metropolitan Police Service) annually makes 1000's of suspect identifications from closed‐circuit television footage. Some suspects are disguised, have not been encountered recently or are depicted in poor quality images. Across tests measuring familiar face recognition, unfamiliar face memory and unfamiliar face matching, the accuracy of members of this specialist police pool was approximately equal to a group of non‐police super‐recognisers. Both groups were more accurate than matched control members of the public. No reliable relationships were found between the face processing tests and object recognition. Within each group, however, there were large performance variations across tests, and this research has implications for the deployment of police worldwide in operations requiring officers with superior face processing ability.
Davis, J. P., Thorniley, S., Gibson, S., & Solomon, C. (2016). Holistic facial composite construction and subsequent lineup identification accuracy: Comparing adults and children. Journal of Psychology: Interdisciplinary and Applied, 150(1), 102-118. doi: 10.1080/00223980.2015.1009867
When the police have no suspect, they may ask an eyewitness to construct a facial composite of that suspect from memory. Faces are primarily processed holistically, and recently developed computerized holistic facial composite systems (e.g., EFIT-V) have been designed to match these processes. The reported research compared children aged 6–11 years with adults on their ability to construct a recognizable EFIT-V composite. Adult constructor's EFIT-Vs received significantly higher composite-suspect likeness ratings from assessors than children's, although there were some notable exceptions. In comparison to adults, the child constructors also overestimated the composite-suspect likeness of their own EFIT-Vs. In a second phase, there were no differences between adult controls and constructors in correct identification rates from video lineups. However, correct suspect identification rates by child constructors were lower than those of child controls, suggesting that a child's memory for the suspect can be adversely influenced by composite construction. Nevertheless, all child constructors coped with the demands of the EFIT-V system, and the implications for research, theory, and the criminal justice system practice are discussed.
Davis, J. P., Maigut, A. C., Jolliffe, D., Gibson, S., & Solomon, C. (2015). Holistic facial composite creation and subsequent video line-up eyewitness identification paradigm. Journal of Visualized Experiments, 106, e53298. DOI:10.3791/53298
The paradigm detailed in this manuscript describes an applied experimental method based on real police investigations during which an eyewitness or victim to a crime may create from memory a holistic facial composite of the culprit with the assistance of a police operator. The aim is that the composite is recognized by someone who believes that they know the culprit. For this paradigm, participants view a culprit actor on video and following a delay, participant-witnesses construct a holistic system facial composite. Controls do not construct a composite. From a series of arrays of computer-generated, but realistic faces, the holistic system construction method primarily requires participant-witnesses to select the facial images most closely meeting their memory of the culprit. Variation between faces in successive arrays is reduced until ideally the final image possesses a close likeness to the culprit. Participant-witness directed tools can also alter facial features, configurations between features and holistic properties (e.g., age, distinctiveness, skin tone), all within a whole face context. The procedure is designed to closely match the holistic manner by which humans' process faces. On completion, based on their memory of the culprit, ratings of composite-culprit similarity are collected from the participant-witnesses. Similar ratings are collected from culprit-acquaintance assessors, as a marker of composite recognition likelihood. Following a further delay, all participants — including the controls — attempt to identify the culprit in either a culprit-present or culprit-absent video line-up, to replicate circumstances in which the police have located the correct culprit, or an innocent suspect. Data of control and participant-witness line-up outcomes are presented, demonstrating the positive influence of holistic composite construction on identification accuracy. Correlational analyses are conducted to measure the relationship between assessor and participant-witness composite-culprit similarity ratings, delay, identification accuracy, and confidence to examine which factors influence video line-up outcomes.
Davis, J. P., Simmons, S., Sulley, L., Solomon, C. J., & Gibson, S. J. (2015). An evaluation of post-production facial composite enhancement techniques. Journal of Forensic Practice, 17(4), 1-12. DOI 10.1108/JFP-08-2015-0042
– The purpose of this paper is to describe four experiments evaluating post-production enhancement techniques with facial composites mainly created using the EFIT-V holistic system.
– Experiments 1-4 were conducted in two stages. In Stage 1, constructors created between one and four individual composites of unfamiliar targets. These were merged to create morphs. Additionally in Experiment 3, composites were vertically stretched. In Stage 2, participants familiar with the targets named or provided target-similarity ratings to the images.
– In Experiments 1-3, correct naming rates were significantly higher to between-witness 4-morphs, within-witness 4-morphs and vertically stretched composites than to individual composites. In Experiment 4, there was a positive relationship between composite-target similarity ratings and between-witness morph-size (2-, 4-, 8-, 16-morphs).
– The likelihood of a facial composite being recognised can be improved by morphing and vertical stretch.
– This paper improves knowledge of the theoretical underpinnings of these facial composite post-production enhancement techniques. This should encourage acceptance by the criminal justice system, and lead to better detection outcomes.
Davis, J. P., Valentine, T., Memon, A., & Roberts, A. J. (2015). Identification on the street: a field comparison of police street identifications and video line-ups in England. Psychology, Crime and Law, 1, 9-27. DOI.org/10.1080/1068316X.2014.915322
A street identification or live show-up provides an eyewitness with an opportunity to identify a suspect shortly after a crime. In England, the majority of suspects identified are subsequently included in a video line-up for the same witness to view. In Study 1, robbery squad data from three English police forces recorded 696 crimes, the identification procedures employed and prosecution decisions. A street identification was the most frequent identification procedure, being attempted in 22.7% of investigations, followed by mugshot albums (11.2%) and video line-ups (3.4%). In Study 2, data of 59 crimes were collected in which suspects, identified in a street identification, were subsequently filmed for a video line-up. Across both studies, most (84%) suspects identified in the street were subsequently identified in a video line-up, indicative of a commitment effect, in which a witness conforms to their first identification decision. All suspects identified in two procedures were eventually cautioned or charged to appear in court. The ground truth of suspect guilt in these field data cannot be determined. However, suggestions are made for reducing the likelihood of a mistaken identification of an innocent suspect caught up in an investigation; all possible steps should be taken to reduce the inherent suggestiveness of the street identification procedure.
Davis, J. P., Gibson, S, & Solomon, C. (2014). The positive influence of creating a holistic facial composite on video lineup identification. Applied Cognitive Psychology, 28, 634–639. DOI: 10.1002/acp.3045
Witnesses to a crime may be asked to create a facial composite of the offender from memory. They may then view a suspect in a police line‐up. Previous research on this topic has found both recognition impairment and enhancement following composite construction. In Experiment 1, creator‐participants employed the holistic facial composite system system EFIT‐V or the feature‐based E‐FIT system to create a single composite, and in Experiment 2, creators constructed up to three EFIT‐Vs. In both experiments, facial composite creators were one‐and‐a‐half times more likely than non‐composite creating controls to make correct target identifications from a video line‐up. No between condition effects were found in target‐absent trials in Experiment 1. The development of holistic facial composite systems has enhanced suspect identification rates in police investigations, and these results suggest that the use of such a system can also have a positive influence on a composite‐creating witness' later recognition of the suspect.
Roberts, A. J., Davis, J. P., Valentine, T., & Memon, A. (2014). Should we be concerned about street identifications? Criminal Law Review, 9, 633-653.
The use of street identification procedures—informal procedures in which witnesses attempt to identify an offender, usually soon after the commission of a crime and close to where it occurred—has attracted significant concern. These procedures are generally thought to give rise to a greater risk of mistaken identification because they lack the safeguards of formal procedures conducted under controlled conditions. This article describes the findings of empirical research undertaken by the authors. The research had three broad objectives. The first was to collect data which would provide some indication of the extent to which street identifications are used by police in England and Wales. The second was to compare the reliability of street identifications and video identification procedures involving the use of foils. The final objective was to investigate the influence that a street identification procedure would have on a subsequent video identification procedure involving the same witness and suspect. The findings suggest that substantial numbers of street identifications are conducted but, perhaps counter-intuitively, in terms of the risk of mistaken identification of innocent suspects, such procedures may be no less reliable than video identification procedures. Following identification of a suspect in a street identification, there is a very high likelihood that a formal procedure involving the same suspect and witness will result in the suspect being identified again, notwithstanding that the suspect is innocent.
Valentine, T., Davis, J. P., Memon, A., & Roberts, A. (2012). Live showups and their influence on a subsequent video lineup. Applied Cognitive Psychology, 26(1), 1-23. DOI: 10.1002/acp.1796
A live showup (known as a street identification in the UK) allows the perpetrator to be identified shortly after a street crime. If the suspect disputes the identification, a video line‐up often ensues. Four experiments examined the reliability of live showups and their influence on a subsequent video line‐up using realistic procedures and conditions. Similar proportions of culprits and innocent suspects were identified from live showups and video line‐ups. Both culprits and innocent suspects previously identified were likely to be identified again in a subsequent line‐up, with delays from a few minutes to a month. Only a weak effect of clothing bias was observed. There was strong evidence of commitment to a previous identification but no reliable evidence of source monitoring errors. The results suggest that a live showup is not less fair than a line‐up, but the use of repeated identification procedures introduces an unfair bias against innocent suspects.
Davis, J. P., Valentine, T., & Davis, R. E. (2010). Computer assisted facial comparison of photographic images taken from full-face and profile viewpoints. Forensic Science International, 200, 165-176. DOI: 10.1016/j.forsciint.2010.04.012
Expert witnesses using facial comparison techniques are regularly required to disambiguate cases of disputed identification in CCTV images and other photographic evidence in court. This paper describes a novel software-assisted photo-anthropometric facial landmark identification system, DigitalFace tested against a database of 70 full-face and profile images of young males meeting a similar description. The system produces 37 linear and 25 angular measurements across the two viewpoints. A series of 64 analyses were conducted to examine whether separate novel probe facial images of target individuals whose face dimensions were already stored within the database would be correctly identified as the same person. Identification verification was found to be unreliable unless multiple distance and angular measurements from both profile and full-face images were included in an analysis.
Valentine, T., Davis, J. P., Thorner, K., Solomon, C., & Gibson, S. (2010). Evolving and combining facial composites: Between-witness and within-witness morphs compared. Journal of Experimental Psychology: Applied, 16(1), 72-86. DOI: 10.1037/a0018801.
Student participant-witnesses produced 4 composites of unfamiliar faces with a system that uses a genetic algorithm to evolve appearance of artificial faces. Morphs of 4 composites produced by different witnesses (between-witness morphs) were judged better likenesses (Experiment 1) and were more frequently named (Experiment 2) by participants who were familiar with the target actors than were morphs of 4 composites produced by a single witness (within-witness morphs). Within-witness morphs were judged better likenesses and more frequently named than the best or the first-produced individual composites. The same results for likeness judgments were observed after possible artifacts in the comparison of between- and within-witness morphs were eliminated (Experiment 3). Experiment 4 showed that both internal and external features were better represented in morphs than in the original composites, although the representation of internal features improved more. The results suggest that morphing improves the representation of faces by reducing random error. Between-witness morphs yield more benefit than within-witness morphs by reducing consistent but idiosyncratic errors of individual witnesses. The experiments provide the first demonstration of an advantage for within-witness morphs produced using a single system. Experiment 2 provides the first demonstration of a reliable advantage for between-witness morphs in the most forensically relevant task: naming a composite of a familiar person produced by a witness who was unfamiliar with the target. Morphing would enhance the recognition of facial composites of criminals. Within-witness morphing provides a methodology for use in crimes in which the victim is the only witness. (Contains 2 footnotes, 2 tables, and 5 figures.)
Davis, J. P., Sulley, L. Solomon, C., & Gibson, S. (2010). A comparison of individual and morphed facial composites created using different systems. In G. Howells, K. Sirlantzis, A. Stoica, T. Huntsberger and A.T. Arslan (Eds.), 2010 IEEE International Conference on Emerging Security Technologies (pp. 56–60). Canterbury: IEEE. DOI: 10.1109/EST.2010.29
An evaluation of individual and morphed composites created using the E-FIT and EFIT-V production systems was conducted. With the assistance of trained police staff, composites of unfamiliar targets were constructed from memory following a Cognitive Interview. EFIT-V composite production followed either a two-day delay, or on the same day as viewing a video of the target. E-FIT composites were created on the same day as viewing the target video. Morphs were produced from merging either two, or three composites created by the same witness, but with the assistance of a different operator. Participants familiar with the targets supplied similarity-to-target photograph ratings. No differences were found in the rated quality of composites created using E-FIT or EFIT-V, although a two-day delay in production resulted in inferior images. Morphs were rated as better likenesses than individual composites, although the benefits were greater with EFIT-Vs. Encouraging witnesses to create more than one composite image for subsequent morphing might enhance the likelihood of recognition of facial composites of criminals.
Davis, J. P., & Valentine, T. (2009). CCTV on trial: Matching video images with the defendant in the dock. Applied Cognitive Psychology, 23, 482-505. DOI: 10.1002/acp.1490
The experiments reported in this paper investigated simultaneous identity matching of unfamiliar people physically present in person with moving video images typical of that captured by closed circuit television (CCTV). This simulates the decision faced by a jury in court when the identity of somebody caught on CCTV is disputed. Namely, 'is the defendant in the dock the person depicted in video'? In Experiment 1, the videos depicted medium‐range views of a number of actor 'culprits'. Experiment 2 used similar quality images taken a year previously, some of which showed the culprits in disguise. Experiment 3 utilised high‐quality close‐up video images. It was consistently found that in both culprit‐present and culprit‐absent videos and in optimal conditions, matching the identity of a person in video can be highly susceptible to error.
Scott, S. K., Rosen, S., Beaman, C. P., Davis, J. P., & Wise, R. J. S. (2009). The neural processing of masked speech: Evidence for different mechanisms in the left and right temporal lobes. Journal of the Acoustical Society of America, 125, 1737-1743. DOI: 10.1121/1.3050255
It has been previously demonstrated that extensive activation in the dorsolateral temporal lobes associated with masking a speech target with a speech masker, consistent with the hypothesis that competition for central auditory processes is an important factor in informational masking. Here, masking from speech and two additional maskers derived from the original speech were investigated. One of these is spectrally rotated speech, which is unintelligible and has a similar (inverted) spectrotemporal profile to speech. The authors also controlled for the possibility of "glimpsing" of the target signal during modulated masking sounds by using speech-modulated noise as a masker in a baseline condition. Functional imaging results reveal that masking speech with speech leads to bilateral superior temporal gyrus (STG) activation relative to a speech-in-noise baseline, while masking speech with spectrally rotated speech leads solely to right STG activation relative to the baseline. This result is discussed in terms of hemispheric asymmetries for speechperception, and interpreted as showing that masking effects can arise through two parallel neural systems, in the left and right temporal lobes. This has implications for the competition for resources caused by speech and rotated speech maskers, and may illuminate some of the mechanisms involved in informational masking.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

Most competitors in ML jump straight to XGBoost,
\citet{DBLP:journals/corr/ChenG16}, an implementation of
the gradient boosting algorithm, because of its speed and accurate predictive
power. Part of XGBoost amazing speed must be ascribed to the fact that
predictions are piecewise constant. From a modeling perspective, this might certainly
be the right thing to do in high-dimensional input spaces, but if the number of features
is low, a piecewise linear model is likely to yield a better predictive performance.

This is best seen on a one-dimensional function. Consider the synthetic
\emph{HeavySine} function, \citet{donoho1995adapting},
a sinusoid of period 1 with two jumps, at $t_1=.3$ and $t_2=.72$:
\begin{equation}
f(t)=4 \sin(4 \pi t) - \text{sign}(t-0.3)-\text{sign}(0.72-t)
\end{equation}
Given enough trees, XGBoost can adequately fit the noisy data set,
Figure \ref{fig: many trees vs. single tree}, left.
If we constrain XGBoost to a single tree and further regularize the model
to prevent over-fitting, the piecewise constant nature of the
predictions is clearly revealed, Figure \ref{fig: many trees vs. single tree},
middle. A single tree with a linear model at the leaves produces visually
far better results, Figure \ref{fig: many trees vs. single tree}, right.
To get better results in terms of the NMSE performance metric,
more trees are needed though.
By adding quadratic terms, we can even get superior results.

\begin{figure}[h]
  \centering
  \includegraphics[width=1.0\linewidth]{./fig1.png}
  \caption{Regression of the \emph{HeavySine} function based on 201 noisy
  samples (the noise is Gaussian with zero mean and variance $\sigma^2 = 0.05$).
  Left: XGBoost with default setting except that the learning rate is reduced
  to 0.1 and the number of trees boosted to 50.
  Middle: XGBoost using a single tree.
  Right: LinXGBoost using also a single tree.
  In the middle and right plots, both XGBoost and LinXGBoost share the same
  tree maximum depth, 30, and regularization term on the number
  of leaves $\gamma=3$ and L2 regularization term on weights $\alpha=0$.}
  \vspace{-25pt}
  \label{fig: many trees vs. single tree}
\end{figure}

A piecewise (constant or linear) model is particularly suited
for the regression of functions
that exhibit jumps or discontinuities.
Finding jumps in otherwise smooth functions is a notoriously hard challenge.
It usually involves the derivation of criteria for choosing the number
and placement of the jumps, \citet{lee2002automatic}.
Even Bayesian models treat exclusively jumps in one-dimensional signals,
i.e. time series, \citet{adams2007bayesian}.

A piecewise linear model is appropriate for functions whose smoothness
is input-dependent. There are well established techniques
to regress functions whose smoothness does
not vary (e.g. the Nadaraya-Watson kernel estimator or Gaussian processes,
both with a fixed bandwidth). While convenient, the assumption that the
smoothness is input-independent is rarely realistic. Extensions to model
functions with varying length-scales are not trivial (e.g. \emph{adaptive}
Nadaraya-Watson kernel estimator or \emph{non-stationary} Gaussian processes).
Nevertheless, this is essential in many fundamental problems.
Modeling terrain surfaces, for instance, given sets of noisy elevation
measurements is even more challenging since it requires the ability
to balance smoothing against the preservation of discontinuities
(see e.g. \citet{plagemann2008nonstationary}).

At the core of gradient boosting are regression trees. A regression tree
decomposes the input space
into subdomains. This is reminiscent of the \emph{domain decomposition} approach
to regress non-stationary functions, \citet{park2011domain}.
In each input subdomain, the function can be approximately regarded
as stationary. Therefore, local regression is applied in each subdomain with
a fixed length-scale or bandwidth. But this results in discontinuities in
prediction on boundaries of the subdomains, \citet{JMLR:v17:15-327}.
To mitigate this drawback,
the assignment of data points to models can also be soft:
The assignments are treated as unobserved random variables,
\citet{rasmussen2002infinite}.
Because of the uncertainty in the assignments,
discontinuities in predictions are smoothed out.

Domain decomposition is however rarely done in a principled way: In
\citet{kim2005analyzing}, the uncertainty in the number of
disjoint regions, their shapes and the model within regions was dealt with in
a fully Bayesian fashion. However, this feat has a price: It is fairly
involved and slow (because of Reversible jump MCMC) and limited to small
feature space dimensions (because of the Voronoi tessellation).

Finding the right size of the local region for linearization is a problem faced
in Locally Weighted Learning, \citet{englert2012locally}.
\citet{meier2014local} developed a Bayesian
localized regression algorithm. The local models were added incrementally
but not in a fully consistent manner.

Though XGBoost is not Bayesian, the trees are grown and the scores at the
leaves are chosen in a principled way: To minimize an objective function.
Consequently, XGBoost can automatically captured jumps and discontinuities,
Figure \ref{fig: pc vs. pl}, left.
Through the extension to local \emph{linear} models, we make it able to
additionally better model smooth functions with varying length-scales
using fewer trees, Figure \ref{fig: pc vs. pl}, middle and right.

\begin{figure}[h]
  \centering
  \includegraphics[width=1.0\linewidth]{./fig2.png}
  \caption{Left: Regression of a noisy data set from a piecewise constant function.
  XGBoost can automatically infer the right number of partitions using a single
  tree.
  Middle: Regression of a noisy data set from a piecewise linear function.
  Because XGBoost predictions are piecewise constant,
  we are forced to boost the number of trees (and so reduce the learning rate)
  to fit the data. The tree was not regularized: A strong regularization would
  lead to piecewise constant predictions.
  Right: LinXGBoost using a single tree provides a much better match.
  Notice that the tree was regularized.}
  \label{fig: pc vs. pl}
\end{figure}

It is not possible to natively handle categorical features;
as in XGBoost, they must be encoded into numerical vectors
using e.g. one-hot encoding.

Notice that plugging-in higher-order models at the tree leaves
was advocated by \citet{torgo1997functional}
to produce \emph{local regression trees}. He acknowledged that
this strategy brings significant gains in terms of prediction accuracy
at the cost of an increase of computation time.

\section{The vanilla XGBoost}
\label{bosting}

We follow and borrow material from the clear-cut presentation in \citet{chen2014introduction}.
We refer to this presentation for further details.

XGBoost implements a Boosting algorithm.
Boosting algorithms belongs to \emph{ensemble machine learning methods}.
Specifically, they iteratively add predictors that focus on improving
the current model, and this is achieved by modifying the learning problem
(the objective in XGBoost) between iterations, see e.g.
\citet{elith2008working}.
Hence, boosting algorithms greedily approximate a function.

The predictors XGBoost builds are \emph{regression trees}.
A regression tree has
decision rules and scores at the leaves.
It is a function since it maps features (the attributes) to values (the scores).
The prediction at $\boldsymbol{x}_*$ is given by
\begin{equation}
\hat{y} = \sum_{k=1}^K{ f_k(\boldsymbol{x}_*) }
\end{equation}
assuming we have $K$ trees.
$f_k \in \mathcal{F}$ where $\mathcal{F}$ is the space of functions
containing all regression trees.
The set of parameters is thus $\boldsymbol{\theta} = \{ f_1, \cdots, f_K \}$.
Instead of learning weights as in linear regression or logistic regression,
we are learning functions.

In XGBoost, the learning of functions is done by defining
an objective to minimize:
\begin{equation}
Obj = \sum_{i=1}^n{ \ell(y_i, \hat{y}_i) } + \sum_{k=1}^K{ \Omega_k(f_k) }
\end{equation}
where $\hat{y}_i$ is the prediction and $y_i$ the observation
at $\boldsymbol{x}_i$ for $i=1,\cdots,n$.
$\ell(\cdot,\cdot)$ designates a loss function.
The first term is the training loss,
the second penalizes the complexity of trees.

In XGBoost, the objective at step \(t\) is defined as
\begin{eqnarray}
Obj^{(t)} &=& \sum_{i=1}^n{ \ell(y_i, \hat{y}_i^{(t)}) } + \sum_{k=1}^t{ \Omega_k(f_k) } \\
              &=& \sum_{i=1}^n{ \ell(y_i, \hat{y}_i^{(t-1)}+f_t(\boldsymbol{x}_i)) } + \sum_{k=1}^t{ \Omega_k(f_k) }
\end{eqnarray}
$f_t(\boldsymbol{x}_i)$ can be thought of as a perturbation (the residual),
and so the loss is, to second-order accuracy:
\begin{equation}
\ell(y_i, \hat{y}_i^{(t-1)}+f_t(\boldsymbol{x}_i))
\approx
\ell(y_i, \hat{y}_i^{(t-1)})
+ g_i f_t(\boldsymbol{x}_i)
+ \frac{1}{2} h_i  f_t(\boldsymbol{x}_i)^2
\end{equation}
where $g_i$ (resp. $h_i$) is the first (resp. second) derivative of
the loss w.r.t. its second argument evaluated at $(y_i, \hat{y}_i^{(t-1)})$.
For the square loss, $\ell(y_i, \hat{y})=(y_i- \hat{y})^2$,
we have $g_i=2 (\hat{y}_i^{(t-1)}-y_i)$ and $h_i=2$.
In that particular case, the approximation is exact.

Removing the constant terms, we get
\begin{equation}
\label{obj at round t}
Obj^{(t)}
=
\sum_{i=1}^n{
                    [ g_i f_t(\boldsymbol{x}_i)) + \frac{1}{2} h_i  f_t(\boldsymbol{x}_i)^2 ]
                    }
+ \Omega_k(f_t)
\end{equation}
with
\begin{equation}
\Omega_k(f_t) = \gamma T + \frac{1}{2} \lambda \sum_{k=1}^T{ w_k^2 }
\end{equation}
to control the complexity of the tree.
Notice that the number of leaves $T$ and
the scores  of the tree at the leaves $w_k$ ought to be indexed
by $t$ to refer to the $t$-th iteration.

Assuming that the tree structure is known, it is possible to derivative
the optimal weights. We have $f_t(\boldsymbol{x}) = w_{q(\boldsymbol{x})}$
where $q(\boldsymbol{x})$ is the assignment function
which assigns every data point to the $q(\boldsymbol{x})$-th leaf.
Then define $I_j = \{ i | q(\boldsymbol{x}_i) = j \}$,
the indices of all points that end up in the $j$-leaf.
All data points in the same leaf share the same prediction.
Consequently, the objective function can be recast as
\begin{equation}
Obj^{(t)}
=
\sum_{j=1}^T{
                    \left[
                    \left( \sum_{i \in I_j}{ g_i } \right) w_j
                    +
                     \frac{1}{2} \left( \lambda + \sum_{i \in I_j}{ h_i } \right) w_j^2
                   \right]
                   }
+ \gamma T
\end{equation}
We can solve for the optimal weights $w_j^*$: Setting $G_i = \sum_{i \in I_j}{ g_i }$
and $H_i = \sum_{i \in I_j}{ h_i }$, we have
\begin{equation}
w_j^* = - \frac{ G_j }{ H_j + \lambda }
\end{equation}
and so
\begin{equation}
Obj^{(t)}
=
-\frac{1}{2}
\sum_{j=1}^T{
                    \frac{ G_j^2 }{ H_j + \lambda }
                   }
+ \gamma T
\end{equation}

The tree is grown in a top-down recursive fashion:
For each feature in turn, XGBoost sorts the numbers
and scans the best splitting point.
The change of objective after a split is the \emph{gain}:
\begin{equation}
gain
=
\frac{1}{2}
\left[
- \frac{ (G_L+G_R)^2}{H_L+H_R+\lambda}
+
\left(
  \frac{ G_L^2}{H_L+\lambda}
+ \frac{ G_R^2}{H_R+\lambda}
\right)
\right]
- \gamma
\end{equation}
The tree is grown
until the maximum depth is reached.
Then nodes with a negative gain are pruned out in a bottom-up fashion.

\section{Linear models at the leaves}

Instead of a score $w$, a leaf stores the weights of a linear model,
$\boldsymbol{w} \in \mathbb{R}^{d+1}$ where $d$ is the dimension of the
feature space, such that the local prediction
at an unseen input $\boldsymbol{x}_*$ at the leaf is given by
$\boldsymbol{w}^T \tilde{\boldsymbol{x}}_*$
where $\tilde{\boldsymbol{x}}_* = [ \boldsymbol{x}_* \ , \ 1]^T$.

This extension is natural:
\begin{itemize}
\item The simple model $\boldsymbol{w}^T \tilde{\boldsymbol{x}}$ captures
both the linear and constant models.
\item It is a form of \emph{locally weighted polynomial regression}.
However, instead of having a bandwidth controlling how much of the data is
used to fit each local polynomial, all data points inside a specific
hypervolume (the volume of feature space assigned to a leaf) are
considered for model building. As in \emph{mixture of experts},
the input space is divided into regions within which specific separate experts
make predictions.
\end{itemize}

The objective at round $t$  changes from \ref{obj at round t} to
\begin{equation}
Obj^{t)}
=
\sum_{i=1}^n{
  \left[ g_i \tilde{\boldsymbol{x}}_i^T \boldsymbol{w}_{q(\boldsymbol{x}_i)} + \frac{1}{2} \boldsymbol{w}_{q(\boldsymbol{x}_i)}^T h_i \tilde{\boldsymbol{x}}_i \tilde{\boldsymbol{x}}^T_i  \boldsymbol{w}_{q(\boldsymbol{x}_i)} \right]
  }
+ \gamma T + \frac{1}{2} \lambda \sum_{j=1}^T{ \boldsymbol{w}_j^T \boldsymbol{w}_j  }
\end{equation}
Regrouping by leaf, we get
\begin{equation}
Obj^{t)}
=
\sum_{j=1}^T{
  \left[
   \left(
   \sum_{i \in I_j} { g_i \tilde{\boldsymbol{x}}_i^T }
    \right)
   \boldsymbol{w}_j
 +
   \frac{1}{2} \boldsymbol{w}_j^T
     \left(
     \lambda \boldsymbol{I}_{d+1}
     + \sum_{i \in I_j}{  h_i \tilde{\boldsymbol{x}}_i \tilde{\boldsymbol{x}}^T_i }
      \right)
     \boldsymbol{w}_j
  \right] }
+ \gamma T
\end{equation}
By defining
\begin{equation}
\tilde{\boldsymbol{g}}^T_j = \sum_{i \in I_j}{ g_i \tilde{\boldsymbol{x}}_i^T }
\ , \ \
\tilde{\boldsymbol{H}}_j = \sum_{i \in I_j}{  h_i \tilde{\boldsymbol{x}}_i \tilde{\boldsymbol{x}}^T_i  }
\end{equation}
we have:
\begin{equation}
Obj^{t)}
=
\sum_{j=1}^T{
  \left[
   \tilde{\boldsymbol{g}}^T_j \boldsymbol{w}_j +
   \frac{1}{2} \boldsymbol{w}_j^T
               ( \lambda \boldsymbol{I}_{d+1} + \tilde{\boldsymbol{H}}_j )
     \boldsymbol{w}_j
  \right]
+ \gamma T
  }
\end{equation}
Assuming the structure of the tree fixed, the optimal weights are given by
\begin{equation}
\boldsymbol{w}_j^*
= - ( \lambda \boldsymbol{I}_{d+1} + \tilde{\boldsymbol{H}}_j )^{-1} \tilde{\boldsymbol{g}}_j
\end{equation}
We have $\tilde{\boldsymbol{g}}_j = \tilde{\boldsymbol{X}}_j^T \boldsymbol{g}_j$
where $\tilde{\boldsymbol{X}}_j^T$ is the $(d+1)$-x-$|I_j|$ matrix
whose columns are the $\tilde{\boldsymbol{x}}_i$.
From the dyadic expansion, we can see  that
$\tilde{\boldsymbol{H}}_j = \tilde{\boldsymbol{X}}_j^T \boldsymbol{H}_j \tilde{\boldsymbol{X}}_j$
where $\boldsymbol{H}_j$ is a diagonal matrix with elements
the $h_i$, $i \in I_j$.
Because $h_i$ is the second derivative of a loss function, it is positive.
Hence $\tilde{\boldsymbol{H}}_j$ is symmetric positive semi-definite and
$\tilde{\boldsymbol{C}}_j = \lambda \boldsymbol{I} + \tilde{\boldsymbol{H}}_j$
is symmetric positive definite as long as $\lambda > 0$.
If so, $ \tilde{\boldsymbol{C}}_j$ is Cholesky decomposed to solve
for the optimal weights $\boldsymbol{w}_j^*$.
If $\lambda = 0$, then $\tilde{\boldsymbol{C}}_j$ is rank at most $d$,
and so degenerate, if $|I_j|<d+1$
(less than $d+1$ input points fall into the leaf).
In that particular situation, we fall back to the piecewise constant model.

$\tilde{\boldsymbol{C}}_j$ is a $(d+1)$-x-$(d+1)$ matrix
where $d$ is the number of features: $\boldsymbol{x} \in \mathbb{R}^d$.
If $d$ is much larger than the cardinality of $I_j$,
then it might be convenient to use the Woodbury formula. Indeed, we have
\begin{equation}
\tilde{\boldsymbol{H}}_j
=
(\tilde{\boldsymbol{X}}_j^T \boldsymbol{H}_j^{\frac{1}{2}})
(\tilde{\boldsymbol{X}}_j ^T \boldsymbol{H}_j^{\frac{1}{2}})^T
=\tilde{\boldsymbol{X}}_j^h \tilde{\boldsymbol{X}}_j^{hT}
\end{equation}
Using the Woodbury formula, we get:
\begin{equation}
( \lambda \boldsymbol{I}_{d+1} + \tilde{\boldsymbol{H}}_j )^{-1}
=
\frac{1}{\lambda} \boldsymbol{I}_{|I_j|}
-
\tilde{\boldsymbol{X}}^h \left( \boldsymbol{I}_{|I_j|}
+
\frac{1}{\lambda} \tilde{\boldsymbol{X}}^{hT} \tilde{\boldsymbol{X}}^h \right)^{-1} \tilde{\boldsymbol{X}}^{hT}
\end{equation}
Now the matrix to invert has size $|I_j|$-x-$|I_j|$.

For the square loss, the optimal weight is
the regularized least-square solution of the residual at a given leaf.
Indeed, we have
\begin{equation}
\boldsymbol{w}_j^*
= \left( \frac{\lambda}{2} \boldsymbol{I}_{d+1} + \tilde{\boldsymbol{X}}_j^T \tilde{\boldsymbol{X}}_j \right)^{-1} \tilde{\boldsymbol{X}}_j^T ( \boldsymbol{y}_j-\hat{\boldsymbol{y}}_j^{(t-1)})
\end{equation}

The objective to minimize is now
\begin{equation}
Obj^{(t)}
=
-\frac{1}{2}
\sum_{j=1}^T{
   \tilde{\boldsymbol{g}}_j^T
   ( \lambda \boldsymbol{I}_{d+1} + \tilde{\boldsymbol{H}}_j )^{-1}
  \tilde{\boldsymbol{g}}_j }
+ \gamma T
\end{equation}

As in XGBoost, we exhaustively search at each node
for the best axis-aligned split, namely the
split associated with the maximum gain:
\begin{eqnarray}
gain
&=&
\frac{1}{2}
\left[
\tilde{\boldsymbol{g}}_{L,R}^T
\boldsymbol{w}_{L,R}^*
-
\left(
\tilde{\boldsymbol{g}}_{L}^T
\boldsymbol{w}_{L}^*
+
\tilde{\boldsymbol{g}}_{R}^T
\boldsymbol{w}_{R}^*
\right)
\right] \nonumber - \gamma
\end{eqnarray}

This extension is computationally intensive
even though the matrices to invert are small
provided that $d$ is small.
One possibility to drastically reduce the compute at
the expense of accuracy is to use the vanilla XGBoost
at the beginning and switch to linear models at the nodes
once the number of elements in nodes is lower than a user-defined threshold.
Notice though that it is expected that fewer trees will be needed.

In XGBoost, a tree is grown until the maximum depth is reached.
Then nodes with a negative gain are pruned out in a bottom-up fashion.
Why do we accept negative gains?
In the middle of the tree construction,
the gain might be negative, but then the following gains might be significant.
This is reminiscent of the exploitation vs. exploration
in many disciplines, e.g. Reinforcement Learning:
The best long-term strategy may involve short-term sacrifices.
However, all sacrifices are unlikely to be worth it.
Thus, in LinXGBoost, we investigate all subtrees
starting from nodes with a negative gain
in a top-to-bottom fashion
and the subtrees that do not lead to a decrease of the objective
are pruned out.
Eventually, if a tree has a single leaf and it does not lead to
a decrease of the objective,
then the tree is removed and the tree building process is stopped.

As a result, the maximum depth of the trees is set to a very large value.
The tree depth, and so the model complexity,
is limited by the minimum number of samples per leaf and the
minimum number of samples for a split.
For large datasets, the depth could easily exceed the default maximum depth,
so that the default plays nevertheless a role in combating overfitting.

There is another major difference with XGBoost: The bias term
is not regularized, as is usual in e.g. Ridge Regression,
see \citet{friedman2001elements}.
Consequently, the matrix
$\tilde{\boldsymbol{C}}_j = \lambda \boldsymbol{I} + \tilde{\boldsymbol{H}}_j$
is re-written as
$\tilde{\boldsymbol{C}}_j = \boldsymbol{\Lambda} + \tilde{\boldsymbol{H}}_j$
where $\boldsymbol{\Lambda}=\text{diag}(\lambda, \cdots, \lambda,0)$
is a diagonal matrix with $d+1$ elements.

\section{Experiments}

First of all, we checked that LinXGBoost yields the same results as XGBoost.
This was gauged visually for one-dimensional problems,
and by comparing performance metrics in higher dimensions.

Because models at the leaves in XGBoost are linear,
they capture the constant model, and so
we expect LinXGBoost to produce no worse results that XGBoost
except that LinXGBoost is prone to overfit.
In the limit, we could use in LinXGBoost as much trees as in XGBoost
and drastically regularize LinXGBoost but this would be pointless.
In our experiments, we observed that we could in general
get as good results as XGBoost using at most two trees,
and that past five trees, results were either worse or the gain
in accuracy was negligibly small so that the additional compute
by adding more trees was not compensated.
Hence we searched for the best number of LinXGBoost trees
using up to five trees for a random run of an experiment,
and thereafter the number of LinXGBoost trees was fixed
for all runs of the experiment except when mentioned.

In the experiments,
we compare LinXGBoost to the vanilla XGBoost and to Random Forest,
often used as a starting point in ML competitions for regression.

\textbf{Preprocessing}
Data re-scaling is not necessary for the vanilla XGBoost and Random Forest.
Since our extension is based on decision trees, it ought to work straight
out of the-box. But since at the leaves we have a linear model, it is better
to have zero mean features. Indeed, leaving the regularization term by side,
the matrix $\tilde{\boldsymbol{X}}_j^T \tilde{\boldsymbol{X}}_j$
must be inverted ($\boldsymbol{H}_j$ is constant for the square loss).
If the features have zero mean, then
$\tilde{\boldsymbol{X}}_j^T \tilde{\boldsymbol{X}}_j$
is nothing else than $\text{cov}(\tilde{\boldsymbol{X}}_j,\tilde{\boldsymbol{X}}_j)$,
and so, if the covariates $\tilde{\boldsymbol{X}}_j$ are almost
linearly independent, then
$\tilde{\boldsymbol{X}}_j^T \tilde{\boldsymbol{X}}_j$
is approximately diagonal.
If the features do not have zero mean, depending on the problem,
$\tilde{\boldsymbol{X}}_j^T \tilde{\boldsymbol{X}}_j$
might be severely ill-conditioned.

\textbf{Performance metrics}
The performance on all experiments is assessed with the Normalized Mean Square Error (NMSE)
\begin{equation}
\text{NMSE}=\frac{\sum_{i=1}^n{ (y_i - \hat{y}_i)^2}}{\sum_{i=1}^n{ (y_i - \overline{y})^2}}
\end{equation}
The method of guessing the mean of the test points
has a NMSE of approximately 1.

\textbf{The Jakeman test functions}
We exercise our model on two synthetic datasets
from \citet{2011arXiv1110.0010J}, both defined on
the unit square. The \emph{Jakeman1} function, Figure \ref{Jakeman1 function},
\begin{equation}
f_1(x_1,x_2)=\frac{1}{|0.3-x_1^2-x_2^2|+0.1}
\label{jakeman1 eq}
\end{equation}
is discontinuous at $x_1^2+x_2^2 = 0.3$, and the \emph{Jakeman4} function,
Figure \ref{Jakeman4 function},
\begin{equation}
f_4(x_1,x_2)
=
\begin{cases} 0 & \text{ if } x_1>0.5 \text{ or } x_2>0.5 \\
\exp(0.5 x_1 + 3 x_2) & \text{otherwise}\end{cases}
\label{jakeman4 eq}
\end{equation}
exhibits a jump at $x_1=0.5, x_2 \in [0,0.5]$ and $x_2=0.5, x_1 \in [0,0.5]$.

To make things interesting, i.i.d. Gaussian noise is added to the training samples.
XGBoost can cope with the noise through subsampling: Only a random fraction
of the training samples pass down a tree. We observed that subsampling plays
a key role when the function exhibits a discontinuity,
otherwise its benefit is minor if the function has a jump.
For subsampling to be effective, a large number of trees must be built.
Hence subsampling degrades the performance of LinXGBoost.
It is better to increase the minimum number of samples at a leaf:
The linear fit to the underlying function is more robust (less susceptible
to be altered by the noise) if we consider more samples.
A positive side-effect is that the trees are shallower and thus
the compute gets faster.

\begin{table}[!ht]
  \caption{NMSE results on the \emph{Jakeman} synthetic test datasets.
  Smaller values are better.}
  \label{sample-table-Jakeman}
  \centering
  \small
  \begin{tabular}{l|c|c|c|c|}
                   & \emph{Jakeman1}       & \emph{Jakeman1}               & \emph{Jakeman4}                 & \emph{Jakeman4}          \\
    Method         & 11-x-11               &  41-x-41                      & 11-x-11                         &  41-x-41                 \\
    \midrule
    XGBoost        &  0.0858 $\pm$ 0.0026  & 0.0083 $\pm$ 0.0002           &  \textbf{0.5778} $\pm$ 0.0214   & 0.1335 $\pm$ 0.0021      \\
    LinXGBoost     &  0.0724 $\pm$ 0.0094  & \textbf{0.0046} $\pm$ 0.0004  &  0.6480 $\pm$ 0.0294            & 0.1385 $\pm$ 0.0024      \\
    Random Forest  &  0.0838 $\pm$ 0.0020  & 0.0113 $\pm$ 0.0002           &  0.5907 $\pm$ 0.0194            & 0.1376 $\pm$ 0.0022      \\
    \bottomrule
  \end{tabular}
  \vspace{-10pt}
\end{table}

Datasets are gridded: Training is carried out on
the small 11-x-11 and medium 41-x-41 configurations,
and testing is done on a 1001-x-1001 configuration.
All model parameters are tuned for each run of an experiment
by conducting an exhaustive grid search
(10-fold cross-validation on the training data set).
An experiment consists of 20 runs.
The number of LinXGBoost trees is fixed, whereas the best number of XGBoost trees
is found by cross-validation. The XGBoost regularizer $\lambda$ was set to 0,
the best setting in all circumstances.
Results are presented in table \ref{sample-table-Jakeman}.

\begin{figure}[!tbp]
  \centering
  \hspace*{\fill}%
  \subfloat[3D view.]{\includegraphics[width=0.25\linewidth]{func_Jakeman_f1_2D_cropped.pdf}\label{fig:j1 1}}\hfill%
  \subfloat[Projection onto the XY-plane.]{\includegraphics[width=0.25\textwidth]{ground_truth_jakeman1.png}\label{fig:j1 2}}
\hspace*{\fill}%
  \caption{The \emph{Jakeman1} test function.}
  \label{Jakeman1 function}

  \centering
  \subfloat[Training dataset: 11-x-11 noisy samples]{\includegraphics[width=0.48\linewidth]{jakeman1_11.png}\label{fig:j1 3}}
  \hfill
  \subfloat[Training dataset: 41-x-41 noisy samples]{\includegraphics[width=0.48\linewidth]{jakeman1_41.png}\label{fig:j1 4}}
  \caption{Results for the noisy \emph{Jakeman1} test function ($\sigma^2=0.05$) for a random run
           (see eq. \ref{jakeman1 eq}).
           Training sample data is gridded. Next to the contours, we plot the results
           on the diagonal $x_1=x_2$.}
  \vspace{-20pt}
  \label{Jakeman1 function: results for a random run}
\end{figure}

For the \emph{Jakeman1} function, on the small training dataset,
LinXGBoost has 3 trees whereas XGBoost has around 50 trees
(the exact number is run-dependent).
There is no clear-cut winner but we observe that
the variance of LinXGBoost results is fairly high.
On the medium training dataset, LinXGBoost has 5 trees
and XGBoost needs circa 150 trees. Nevertheless, LinXGBoost
beats XGBoost by a large margin. Random Forest performs
slightly worse than XGBoost.
A random run is shown in Figure \ref{Jakeman1 function: results for a random run}.

Again our expectations, LinXGBoost performs slightly worse than XGBoost and Random Forest
on the \emph{Jakeman4} function on the small training dataset.
Results of all methods
are close to each other in terms of NMSE on the medium dataset.
LinXGBoost is limited to 3 trees. Including more trees in LinXGBoost model
does not improve performance.
The main contribution to the error comes from the
jump being at the wrong position: All methods put the jump
far to the right\footnote{The reason therefor is that the training data is gridded.
Indeed, if we repeat the previous experiment with
41x41=1681 input data points randomly distributed in the unit square,
then the error is one order of magnitude lower, and LinXGBoost with 5 estimators
performs as good as XGBoost with 100 estimators.}
The improvement of LinXGBoost in smoother regions is
marginal though visually noticeable.
Both assertions can be verified on the
random run shown in Figure \ref{Jakeman4 function: results for a random run}.

\begin{figure}[!tbp]
  \centering
  \hspace*{\fill}%
  \subfloat[3D view.]{\includegraphics[width=0.30\linewidth]{func_Jakeman_f4_2D_cropped.pdf}\label{fig:j4 1}}\hfill%
  \subfloat[Projection onto the XY-plane.]{\includegraphics[width=0.25\textwidth]{ground_truth_jakeman4.png}\label{fig:j4 2}}
\hspace*{\fill}%
  \caption{The \emph{Jakeman4} test function.}
  \label{Jakeman4 function}

  \centering
  \subfloat[Training dataset: 11-x-11 noisy samples]{\includegraphics[width=0.48\linewidth]{jakeman4_11.png}\label{fig:j4 3}}
  \hfill
  \subfloat[Training dataset: 41-x-41 noisy samples]{\includegraphics[width=0.48\linewidth]{jakeman4_41.png}\label{fig:j4 4}}
  \caption{Results for the noisy \emph{Jakeman4} test function ($\sigma^2=0.05$) for a random run
           (see eq. \ref{jakeman4 eq}) .
           Training sample data is gridded. Next to the contours, we plot the results
           on the line $x_1=0.1$.}

  \label{Jakeman4 function: results for a random run}
\end{figure}

\iffalse
Irritated by the poor performance of LinXGBoost
on the \emph{Jakeman4} function,
we devise a variation of it: The unit square is partitioned
into 4 distinct regions  and sample randomly

LinXGBoost 2 estimators
XGBoost       :      0.08774 +/-      0.02807
LinXGBoost    :      0.14103 +/-      0.16492
Random Forests:      0.08725 +/-      0.01597

XGBoost       :      0.00492 +/-      0.00112
LinXGBoost    :      0.00379 +/-      0.00122
Random Forests:      0.00672 +/-      0.00128

\begin{figure}[!tbp]
  \centering
  \hspace*{\fill}%
  \subfloat[3D view.]{\includegraphics[width=0.25\linewidth]{func_Jakeman_f4_2D_cropped.pdf}\label{fig:f1}}\hfill%
  \subfloat[Projection onto the XY-plane.]{\includegraphics[width=0.25\textwidth]{ground_truth_jakeman4.png}\label{fig:f2}}
\hspace*{\fill}%
  \caption{The \emph{Jakeman4} test function.}
  \label{Jakeman4 function}

  \centering
  \subfloat[Training dataset: 11-x-11 noisy samples]{\includegraphics[width=0.48\linewidth]{jakeman4_11.png}\label{fig:f1}}
  \hfill
  \subfloat[Training dataset: 41-x-41 noisy samples]{\includegraphics[width=0.48\linewidth]{jakeman4_41.png}\label{fig:f2}}
  \caption{Results for the noisy \emph{Jakeman4} test function ($\sigma^2=0.05$) for a random run
           (see eq. \ref{jakeman4 eq}) .
           Training sample data is gridded. Next to the contours, we plot the results
           on the line $x_1=0.1$.}
  \label{Jakeman4 function: results for a random run}

\end{figure}

\fi

\textbf{The Friedman1 test function}
The \emph{Friedman1} data set is a synthetic dataset.
It has been previously employed in evaluations of MARS,
\citet{friedman1991multivariate},
and bagging, \citet{breiman1996bagging}.
According to Friedman,
it is particularly suited to examine the ability of methods
to uncover interaction effects that are present in the data.
In its basic version, there are 10 independent covariates,
uniformly distributed on (0,1)
and only five of these are related to the target via
\begin{equation}
y=10 \sin(\pi x_1 x_2) + 20 (x_3-0.5)^2 + 10 x_4 + 5 x_5 + \epsilon
\end{equation}
where $\epsilon$ is Gaussian noise with $\sigma=1$.

We consider the dataset from the LIACC repository
that contains 40768 data points.
Results are averaged over 20 randomly splits of this data set.
Each split has 200 training and 40568 testing points.
Model parameters are tuned for each split
by a 10-fold cross-validation on the training data set.

In a random run, we found that LinXGBoost needs 2 to 3 trees
and so for the experiment the number of trees in LinXGBoost
is either 2 or 3 (determined by cross-validation),
whereas XGBoost has approximately 50 estimators.
Results are presented in table \ref{sample-table-Friedman}.
The error of Random Forest is twice as high as the error of XGBoost,
and LinXGBoost does even better than XGBoost on average, but
LinXGBoost results are a bit more volatile,
presumably because LinXGBoost uses very few trees.

\begin{table}[ht]
  \caption{NMSE results on the \emph{Friedman1} synthetic test dataset.
  Smaller values are better.}
  \label{sample-table-Friedman}
  \centering
  \small
  \begin{tabular}{l|c|}
    Method         &  \emph{Friedman1} \\
    \midrule
    XGBoost        &  0.1303 $\pm$  0.0138 \\
    LinXGBoost     &  \textbf{0.1133} $\pm$  0.0199 \\
    Random Forest  &  0.2278 $\pm$  0.0197 \\
    \bottomrule
  \end{tabular}
\end{table}

\textbf{The Combined Cycle Power Plant Data Set (CCPP)}
The dataset, available at the UCI Machine Learning Repository,
contains 9568 data points collected from a Combined Cycle Power Plant
over 6 years (2006-2011),
when the power plant was set to work with full load.
Features consist of
hourly average ambient variables Temperature (T),
Ambient Pressure (AP),
Relative Humidity (RH) and exhaust Vacuum (V)
to predict the net hourly electrical energy output (PE)
of the plant.

Among all datasets used for the benchmark,
this dataset is the only real-world dataset
that was not purposely generated for the validation of algorithms.
As such, we cannot rule out the presence of outliers.
Outliers have a detrimental impact on the results of XGBoost and LinXGBoost
because both make use of the square loss function.
Nevertheless, this impact is attenuated by the fact that
XGBoost and LinXGBoost
are based on decision trees and so outliers are isolated
into small clusters.
Extreme values do not affect the entire model because of local model fitting.

The summary of the data set per variable is:
\begin{verbatim}
                AT            V           AP           RH           PE
count  9568.000000  9568.000000  9568.000000  9568.000000  9568.000000
mean     19.651231    54.305804  1013.259078    73.308978   454.365009
std       7.452473    12.707893     5.938784    14.600269    17.066995
min       1.810000    25.360000   992.890000    25.560000   420.260000
2
5
7
max      37.110000    81.560000  1033.300000   100.160000   495.760000
\end{verbatim}

All features are real-valued (no categorical features).
Furthermore, there are no missing values.

Are there \emph{extreme outliers} ?
Any point beyond 2*step where step=1.5(Q3-Q1)
is considered an extreme outlier, whereas a point beyond
a step is considered a
\emph{mild outlier} \footnote{See
\href{http://www.itl.nist.gov/div898/handbook/prc/section1/prc16.htm}{What are outliers in the data?}
from NIST/SEMATECH e-Handbook of Statistical Methods.}.
Recall that Q1 and Q3 are the 25th and 75th percentile of the data
for a given variable respectively.
Answer: No.

Are there \emph{multivariate outliers} ?
A scatter-plot matrix is an appropriate tool to rapidly
scrutinize many variables for patters (for linear trends,
the correlation coefficients are printed) and outliers.
Figure \ref{fig: scatter-plot matrix} clearly reveals some abnormalities:
\begin{itemize}
\item There are 13 points at the lower end of variable V (colored red) that
are away from the bulk of points (colored blue): 7 points for which V=25.36
and 6 points for which V=25.88.
\item There are 2 isolated points (colored red) at the lower end of variable PE.
\item There is a strip of 15 points (colored green) for which V=71.14,
which indicates that the data is likely to have been thresholded.
\end{itemize}
All those points can all the more safely be removed from the data set
since we have plenty of data points.

\begin{figure}[h]
  \centering
  \includegraphics[width=15.0cm]{scatter_plot_matrix.png}
  \caption{Scatter-plot matrix for the CCPP dataset. Points that are not colored blue are considered outliers.}
  \label{fig: scatter-plot matrix}
\end{figure}

Another key property of the data set depicted
in Figure \ref{fig: scatter-plot matrix} is that the target PE seems to be
a (highly noisy) linear function of variable AT: The coefficient of correlation tops
at -0.95. Hence a linear regression is a worthy contender to XGBoost, LinXGBoost
and Random Forest for this particular problem.

The 9568 records set is split into training and test sets in a 70-30 proportion.
The split is repeated 20 times independently. For each split,
a 3-fold cross-validation  is carried out to find the best hyperparameters.

Results are presented in table \ref{real-world-datasets}.
XGBoost wins by a large margin and Random Forest performs slightly better
than LinXGBoost with 4 trees. Results from the linear regression
are fairly good and define the base line,
but it is no surprise since the relationship
between target and features is almost linear.

\textbf{Pumadyn-8nm} The data set was generated using a robot-arm simulation.
It is highly non-linear and has very low noise. It contains 8192 data samples
with 8 attributes. We follow exactly the same procedure as with the
CCPP data set.
Table \ref{real-world-datasets} shows that XGBoost with 150 estimators (set variable)
provides slightly better results than Random Forest with 100 estimators (also set variable)
and LinXGBoost with 3 estimators.

\begin{table}[ht]
  \caption{NMSE results on the CCPP (Combined Cycle Power Plant) and pumadyn-8nm datasets.
  Smaller values are better.}
  \label{real-world-datasets}
  \centering
  \small
  \begin{tabular}{l|c|c|}
    Method         &  CCPP                   & pumadyn-8nm \\
    \midrule
    Linear reg.    &  0.07133 $\pm$  0.00283   & 0.48262 $\pm$ 0.01152              \\
    XGBoost        &  \textbf{0.03371} $\pm$  0.00233   &  \textbf{0.03737} $\pm$  0.00145  \\
    LinXGBoost     &  0.03567 $\pm$  0.00289   &  0.03876 $\pm$  0.00177  \\
    Random Forest  &  0.03988 $\pm$  0.00260   &  0.03898 $\pm$  0.00138  \\
    \bottomrule
  \end{tabular}
\end{table}

\section{Conclusion}

The gradient boosting algorithm XGBoost was extended into LinXGBoost
so that linear models
are stored at the leaves. Computations are still analytically tractable.

This strategy is supposed to rip benefits in terms of accuracy
in low dimensional input space but it is computationally intensive.
As expected, experiments demonstrate that far fewer trees (in general less than
five) are needed
to get the accuracy XGBoost yields with hundreds of trees.
LinXGBoost seems to shine in one-dimensional problems, but
in higher input dimensions, it does not clearly beat XGBoost or Random Forest.

In a future work, we will investigate whether LinXGBoost can offer
substantial improvements in classification.

\subsubsection*{Code}

The LinXGBoost code is written in Python.
It is not an extension of XGBoost.
Why ?
At the first sight, the XGBoost code can appear arcane because of precisely
what makes XGBoost awesome: It is written in C++
and ported to R, Python, Java, Scala, and more,
it runs on a single machine, Hadoop, Spark, Flink and DataFlow,
and it is highly optimized.
All of this bloats the code and makes it harder to pinpoint the core routines
to change (and to apply the changes without breaking any feature of XGBoost!).
We felt it is far more easier to implement
LinXGBoost from scratch.
LinXGBoost is a naive implementation
in Python in less than 350 human-readable lines based on \citet{chen2014introduction}.
Hence the interpretation of the parameters is obvious:
It complies with \citet{chen2014introduction}.

The LinXGBoost implementation does not strive for speed.
The current version runs only on a single machine and is not multithreaded.
Therefore, we do not mention runtime performance.

The software implementation is made available
at \url{https://github.com/ldv1}.
It is envisaged to extend the XGBoost code in a future work.

\iffalse
\subsubsection*{Acknowledgments}

Use unnumbered third level headings for the acknowledgments. All
acknowledgments go at the end of the paper. Do not include
acknowledgments in the anonymized submission, only in the final paper.
\fi

\medskip

\bibliographystyle{dinat}

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text:
Q: Using awk, how do I match pattern and variants? I've been struggling with this for a while in regex testers but what came up as a correct regex pattern actually failed. I've got a large file, tab delimited, with numerous types of data. I want to print a specific column, with the characters XYZ, and it's subsequent values.
In the specific column I'm interested in I have values like:
XYZ

ABCDE

XYZ/WORDS

XYZ/ABCDE

ABFE

XYZ

regex tester that was successful was something like:
XYZ(.....)*

It obviously fails when implemented as:
awk '{if ($1=="XYZ(......)*") print$0}'

What regex character do I use to denote that I want everything after the backslash(/), including the original pattern (XYZ)?
Specifically, I want to be able to capture all instances of XYZ, and print the other columns that go along with them (hence the print$0). Specifically, capture these values:
XYZ

XYZ/WORDS

XYZ/ABCDE

Thank you

A: Setup: (assuming actual data file does not include blank lines)
$ cat x
XYZ 1 2 3 4
ABCDE 1 2 3 4
XYZ/WORDS 1 2 3 4
XYZ/ABCDE 1 2 3 4
ABFE 1 2 3 4
XYZ 1 2 3 4

If you merely want to print all rows where the first field starts with XYZ:
$ awk '$1 ~ /^XYZ/' x
XYZ 1 2 3 4
XYZ/WORDS 1 2 3 4
XYZ/ABCDE 1 2 3 4
XYZ 1 2 3 4

If this doesn't provide the expected results then please update the question with more details (to include a more representative set of input data and the expected output).

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