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# Does high-rise residential building design shape antisocial behaviour?

Yung Yau (Department of Public Policy, City University of Hong Kong, Kowloon, Hong Kong)

ISSN: 0263-7472

Publication date: 20 August 2018

## Abstract

### Purpose

The purpose of this paper is to examine links between environmental design of high-rise housing communities and residents’ perceptions about antisocial behaviour (ASB).

### Design/methodology/approach

A conceptual framework was proposed to investigate correlations between architectural design parameters and perceived severity of ASB activity. A questionnaire was administered to test the relationships. Residents of 14 public rental housing estates in Hong Kong participated, and 422 complete responses were analysed.

### Findings

Strong correlation was discovered between elements of residential design and residents’ perceptions of ASB severity. Block layout, building height and number of flats per floor affected residents’ feelings about ASB threat. Access to outside air in communal corridors also significantly reduced residents’ complaints about ASB.

### Practical implications

This study offers insights into how architectural design of high-rise residences might reduce residents’ perception of ASB severity. Findings impact current ASB research, but also architects’ and developers’ designs. Better planned built environments will enhance residents’ security and satisfaction, reinforcing communities.

### Originality/value

Previous studies have ignored whether architectural design of high-rises could directly influence residents’ perception of ASB severity. This study is the first to focus on the relationship.

## Keywords

#### Citation

Yau, Y. (2018), "Does high-rise residential building design shape antisocial behaviour?", Property Management, Vol. 36 No. 4, pp. 483-503. https://doi.org/10.1108/PM-10-2017-0057

### Publisher

:

Emerald Publishing Limited

## Introduction

As scholars have suggested, aspects of housing quality shape inhabitants’ lives (Dunn, 2000; Newcombe et al., 2005; Raw and Hamilton, 1995). On the one hand, design factors like layout, location, orientation and size significantly affect the intrinsic quality of housing. On the other hand, operational and adjustable factors like effective management, the ease of forming neighbourhood relationships, and residents’ general behaviour have significant effects on housing quality and, thus, residents’ quality of life (Bonaiuto et al., 2003; Wong et al., 2006). One key influence on residents’ lives is vulnerability to antisocial behaviour (ASB). ASB is “[any] behaviour that causes harassment to a community” (Local Government Information Unit, 1997, p. 5). Some kinds of ASB are criminal-like activities that injure other residents or vandalise properties while most other types of ASB are really minor offences that disrupt the peace and quietness of a neighbourhood but not prohibited by the criminal law (Local Government Information Unit, 1997; Millie et al., 2005).

Individuals who indulge in ASB disturb the peace, comfort, or privacy of other residents in the community. ASB can also affect other nearby residential areas. For example, residents who have been subjected to protracted ASB locally grow irritable and lose confidence in their neighbourhoods. High rates of ASB affect entire residential areas. Proliferation of antisocial acts signals that a neighbourhood’s social controls and behavioural norms are breaking down (Perkins and Taylor, 1996; Skogan and Maxfield, 1981). Vandalism and housing disrepair make chaotic environments which increase residents’ perception that they are vulnerable to crime (McCrea et al., 2005; Wood et al., 2008). Similarly, empirical studies have identified direct correlation between a neighbourhood’s levels of physical and social disorders (e.g. litter, graffiti and vandalism) and enhanced fear of crime (LaGrange et al., 1992; Perkins and Taylor, 1996). Neighbours’ sense that the community is decaying has been indexed to heightened symptoms of anxiety and depression (Latkin and Curry, 2003). Latkin et al. (2005) have shown in one controlled empirical study that physical and social disorders in one neighbourhood contributed to residents’ extreme distress and, ultimately, was a contributing factor to incipient drug addiction. ASB is both physically and mentally harmful. Its influences on individuals are both direct and indirect (Curtis et al., 2004; Molnar et al., 2004; Putrik et al., 2015; Stansfeld et al., 2000). Social psychologists have found that ASB causes social disruption and reduces residential satisfaction (Agyemang et al., 2007, Jacobson et al., 2008; Weidemann and Anderson, 1982). High rates of ASB and widespread incivilities among residents drive down a neighbourhood’s property values (Brown, 2011).

Across the world, ASB is increasingly attracting scholars’ attention. Besides, recognising the catastrophic consequences of allowing ASB to proliferate, public authorities in different countries have implemented policies to fight it. Cities have enacted laws enforcing stricter tenancy regulations and have employed lease termination as a deterrent against and final remedy for recurrent ASB. Similar legal restrictions, such as forcing newcomers to complete probationary tenancies, courts summarily issuing ASB orders against repeat offenders, and civil orders that penalise parents who say they cannot control their children’s behaviour, have been widely used (Chartered Institute of Housing, 2001; Flint and Nixon, 2006; Hunter et al., 2005). In Hong Kong, a penalty-point system has been institutionalized since 2003 to expel public housing tenants who repeatedly commit proscribed ASB (Yau, 2011). From a communitarian perspective, key scholars have argued that more effective approaches to reducing ASB would focus on enhancing the entire community’s social identity and heightening solidarity among neighbours (Yau, 2014; Hopkins-Burke and Hodgson, 2015).

In the literature, it is often argued that social environments of housing communities have a significant bearing on the proliferation of ASB in the neighbourhoods. However, for this research project, it is particularly intriguing that Yau (2014) has also suggested public housing-related ASB could be mitigated through better residential design. This is a remarkably different approach from legal and community initiatives directed towards changing residents’ behaviours. Yet empirical research assessing the influence of housing design on ASB has been exceptionally rare (Cozens et al., 2001a, b, 2002). With this deficiency in mind, the current Hong Kong-based research aims to investigate whether differences in housing design contribute to changing levels of ASB activity. This study of potential correlations between housing features and the prevalence of ASB in Hong Kong’s high-rise public rental housing estates is organised as follows. First, the most relevant scholarship on ASB will be assessed. Puzzlingly, major research has rarely considered whether the design parameters of a large multi-unit residential edifice could exacerbate or limit ASB activities. An analytic framework for assessing evidence of these factors will then be detailed and its criteria justified. Applications of the investigative framework to survey data collected from residents of high-rise public rental housing in Hong Kong will offer patterns that suggest strong contributors to ASB. The analysis results will be interpreted. Last, a nuanced discussion of the results’ implications for housing design, planning and management and future research will be offered.

## Literature review: ASB and housing design

### Epidemic housing-related ASB around the world

Residential communities are where ASB occurs frequently and produces its damaging consequences (Bullock and Jones, 2004; Flint, 2004). Increased ASB activities in residential communities preoccupy policy-makers in many countries. ASB has attracted extensive attention from academics, architects, urban planners, housing authorities, law-enforcement authorities, residents’ organisations, and other stakeholders. Take for example the fact that over 87 per cent of respondents to a recent study of social housing in New South Wales, Australia, reported experiencing verbal abuse and threats from neighbours (Stein and Marjoribanks, 2014). Approximately 10 per cent of Australian households have reported illegal drug use in their residential communities, while 25 per cent complained that their neighbourhoods had been vandalized (Australian Bureau of Statistics, 2006). In the American Housing Survey 2009, approximately 25.4mn respondents complained of excessive noise in their communities, and 9.8mn identified reckless littering as a chronic problem (United States Census Bureau, 2010). In the study conducted in the early 1990s, 20 per cent of Scotland’s public housing tenants complained that neighbours repeatedly indulged in nuisance behaviours (Clapham et al., 1995). In England and Wales between April 1999 and December 2013, 24,427 ASB Orders were issued (Home Office, 2014). Meanwhile, in Singapore’s public housing estates, residents reported that neighbours’ noisiness had significantly diminished their quality of life (Housing and Development Board, 2014a). In Singaporean public housing, nearly half of all households suffered from nuisance behaviours (Housing and Development Board, 2014b). Yau (2012b) documented that, when questioned, nearly 25 per cent of Hong Kong public housing tenants characterised the neighbourhood problems they endured as intolerable.

### Causes and determinants of ASB proliferation

ASB proliferation has many plausible causes. Seen as a product of poor moral principle and weak willpower, ASB becomes an issue of individual and agency. Intrapersonal factors like poor education, chronic unemployment, and substance abuse affect people’s tendency to engage in ASB. If ASB is a problem of individual self-control, then the root problem is residents’ decisions to engage in ASB. Poor decision-making has been linked to interpersonal issues such as family problems and poor peer relationships (Home Office, 2003). Research in the UK has found that personality disorders and mental health issues frequently prompt individuals to act-out antisocially (Cale, 2006). In Australia, McGee et al. (2009) revealed that ASB was perpetrated at significantly higher rates by children who struggled with family problems. However, ASB authorities have also linked widespread behavioural disorders to socio-structural conditions endemic to large urban populations. Ipsos Mori (2006) and Jacobs and Arthurson (2003) have suggested that profoundly diminished job opportunities for young people and the consequences of poor parenting are crucial structural determinants. Millie (2009) has contended that the growth of ASB-related community problems adheres to the statistical evidence of deteriorating family values and moral standards in modern societies (Millie, 2009).

Another stream of ASB research has focussed on how social systems can influence individual behaviours and minimise broad social circumstances that spark ASB activities in marginalised groups (Burney, 2009; Flint and Rowlands, 2003). These studies have paid considerable attention to the circumstances that shape how community feelings grow sensitised to the apprehending threats and vigilantly detecting communal conflicts that might not be clearly related to ASB. ASB activities can invite more destructive behaviours from disaffected residents. But residents who are keenly alert for evidence of suspected ASB are likely to find it, and confirmed anxieties increase. Divisive communities colour residents’ perceptions of how ASB affects their quality of life. Studies have shown that socio-economic factors such as gender, marital status, and length of residence clearly influenced residents’ sensitivity to possible ASB activity (Taylor et al., 2010). Moley (2008) concluded that, in the UK, perceptions that ASB-related problems required more attention from authorities were exhibited most often by residents in neighbourhoods that were “hard-pressed” or of “moderated means” (p. 125). Another British study found that residents in social housing, as well as those living in flats and maisonettes, were predisposed to show high levels of sensitivity to ASB (Taylor et al., 2010). Furr-Holden et al. (2011) concluded that the presence of abandoned buildings in neighbourhoods powerfully contributed to juvenile ASB activities. These crack properties can be easily accessible sites of gang, prostitution and drug-related undertakings. However, in an Australian study by McGee et al. (2009), data analysis failed to support the theory that more abandoned buildings necessarily elicited higher rates of ASB. McGee et al. (2009) questioned whether the socialized expectation that derelict buildings invited delinquency was empirically valid or whether studies that supported the claim suffered from confirmation bias.

### Housing design: the missing piece to the puzzle of ASB research

Factors related to individual circumstances and structural social trends do clearly affect ASB incidence rates. Along with these issues of individual dysfunction and group conflict, another set of conditions — the effective or poor design of housing and of residential communities — has been linked both to documented rates of antisocial activity and to residents’ enhanced sensitivity to ASB activities. Poor housing design has been connected to neighbourhood problems in Australia (Jacobs and Arthurson, 2003). However, though the poor residential design’s contribution to disruptive behaviours has been observed, the apparent relationship has not been systematically tested. By contrast, empirical evidence that poor housing design increases local crime rates has been well documented (e.g. Newman and Franck, 1980, 1982). According to the rational offender perspective, criminals prefer to commit crimes that “require the least effort, provide the highest benefits and pose the lowest risks” (Taylor and Harrell, 1996, p. 2). If these are the “rational” criminal’s primary considerations, then crime prevention must take account of a wide range of circumstances that make criminal acts more tempting or less profitable.

Design features of residential environments can contribute both to inviting criminality and to deterring it. Consequently, “crime prevention through environmental design” deserves recognition as one of the most influential paradigms in forensic sociological studies. Experts agreed that different physical features impact whether homes would attract criminals (Colquhoun, 2004; Newman, 1973; Robinson, 1996; Stollard, 1991). It makes intuitive sense that criminal opportunities would be reduced by design features that offer residents better surveillance, that segment outdoor spaces into locations more easily controlled by small groups of neighbours, that clearly demarcate private from public spaces, and that locate home entryways close to well-used public thoroughfares. Architectural design features that reduce crime ought to, one anticipates, also diminish residents’ fear of imminent crime. Effective crime-deterrent architectural designs strengthen community-based informal control of residences (Newman, 1972; Taylor and Harrell, 1996). Building layouts that minimise the number of apartments sharing access to one outer threshold, which provide a single means of entry and exit from a few homes, enhance residents’ ability to monitor their environment and reduces strangers’ casual access. Reinforcing gates, doors, and windows with physical barriers also dissuades criminals (Newman and Franck, 1982). Better lighting in communal areas enhances communal surveillance and deters would-be criminals (Welsh and Farrington, 2008). Manzi and Smith-Bowers (2005) provided a wealth of empirical proof that incorporating visible gating in common entrances made homes more secure. There is a wealth of studies on environmental criminology and public health that document the impact of built environmental designs on crime incidence. Moreover, Wood et al. (2008) have found that the visible security of effectively designed residences enhances residents’ apprehension of their own domestic safety. Foster et al. (2010) have also documented links between security-enhanced neighbourhood designs and residents’ confidence.

The majority of studies on ASB activity have been conducted in the West, specifically in Australia, the UK and the USA. Research focussing on how ASB affects residential communities in the East has lagged. The paucity of research in Eastern countries on ASB is particularly glaring when one considers how prevalent ASB activities are in many of the high-rise residential communities that predominate in cities like Hong Kong, Taipei and Singapore. A separate factor that requires consideration is whether the logical tenets and empirically proven crime-prevention methods of environmental crime deterrence translate unequivocally to ASB prevention. Take for instance, Newman’s (1973) concept of defensible space: it works well for deterring burglars but does not necessarily reduce conflictual behaviours among neighbours.

To give credit where it is due, Yau (2014) argued that the problem of ASB in Hong Kong’s housing estates could be mitigated through better housing design. His analysis has not elicited a wave of new empirical studies on how residential designs could diminish Eastern cities’ endemic ASB problems and enhance community solidarity. The need to better understand how built environments affect ASB occurrence is no less imperative for the lack of research. Responding to this pronounced gap, this research directly examines the effects of housing design parameters on ASB activity in Eastern urban residential communities. Spurred by this study, the researcher anticipates other ASB experts will explore the ways in which design details in residential construction can influence psycho-social responses to the problem of individuals seeking to live comfortably and privately while always in close proximity to many more people. How might better housing design influence residents’ sense of being vulnerable to ASB?

## Research method

### Analytic framework of this research

To explore possible relationships between housing design parameters and rates of ASB, the researcher adopted a multiple regression analysis approach, as it allows for demographic and socio-economic variables to be controlled. For this study, an analytic framework was developed for empirical testing. The framework is graphically depicted in Figure 1. In the framework, the severity of ASB problems in a housing estate is hypothesised to be dependent on two sets of factors. The first set comprises housing design parameters, including block layout, corridor design, the number of storeys in each block and the number of flats per storey. These design parameters are the central foci of this empirical study. The second set of potentially significant circumstances consists of demographic and socio-economic characteristics of a specific high-rise community’s residents. Characteristics include residents’ gender, age and education level. These factors serve as control variables in the exploratory analysis. Definitions and measurements of the primary factors being considered in the framework are explained in the following section.

### Measurement of constructs

This study treated the incidence level of ASB in the particular residential neighbourhoods being examined as the dependent variable for the empirical analysis. Rather than using objective measurements, like annual records of the number of ASB incidents reported or the number of punitive orders served against perpetrators, this study adopted residents’ subjective perceptions of ASB as the principal index of ASB manifestation for several reasons. First, the statistics about ASB incidents and resident complaints disseminated by Hong Kong’s public authorities do not accurately reflect the number of ASB incidents reported in residential communities (Yau, 2016). Public statistics are undermined by victims’ reluctance to report ASB incidents (Home Office, 2008). Second, the number of cease-and-desist orders served against ASB perpetrators is a poor indicator of ASB activities’ frequency in a community because the frequency with which statutory orders are issued depends on unknown limitations, like whether public resources have been allocated for combatting ASB. Accuracy of the records of ASB orders served is also contingent on the bureaucratic effectiveness of the enforcement authorities. Third and most importantly, statistics about ASB incidents and citizens’ complaints publicly available from Hong Kong’s government only reflect aggregated totals of ASB activities at the level of the whole territory. No discrete information is available for street-block and estate-level activities.

Without reliable empirical data on ASB activities, this study has had to utilise residents’ self-reported experiences of ASB activities, which we have identified with residents’ self-assessment of past and future ASB threat. To gauge the incidence level of ASB witnessed by a local resident in a public rental housing estate, the resident was asked to rate the following seven aspects of ASB using a five-point Likert scale (5=very serious; 4=serious; 3=quite serious; 2=a little serious; 1=not a problem at all). The classes of ASB respondents assessed were as follows:

• neighbour-generated noise;

• pet nuisances;

• neglected water seepage or dripping;

• littering;

• abusiveness of neighbours in common areas;

• graffiti, unlawful flyer-posting and vandalism; and

• harassment and intimidation.

The overall level of intensity of ASB activity perceived by the resident was taken as the simple average of the seven responses.

As for the exogenous variables, block layout refers to the type of building plan configuration implemented in an individual residential block. In Hong Kong, a rental housing block is commonly configured in a cruci, cross, trident, H-shaped, linear or slab form. As shown in Figures 2–4, habitable rooms (e.g. living rooms, dining rooms or bedrooms) of a flat face those of adjacent flat in the first three configurations. It is expected that noise, odour and other nuisances can be more easily spread from one flat to another. On the contrary, the H-shaped, linear and slab forms minimise the mutual disturbance among residents. Figures 5 and 6 demonstrate that in these two layout designs, the living or dining rooms of adjacent flats do not face each other. Block layout is indicated by a set of dummy variables. The variable LINEAR equalled 1 if the housing block was a linear or slab form and zero if otherwise. The variable HSHAPE equalled 1 if the housing block was in an H-shaped form and zero if otherwise. The variable TRIDENT equalled 1 if the housing block was a trident-shaped block and zero if otherwise. The reference categories are the cross and cruci forms.

The category corresponding to the design of a block’s corridors (CDES), reflects the extent of the outer boundary walls of communal corridors that is exposed to external air. The variable is taken as the percentage of the enclosing walls of the communal corridors on a typical floor of a housing block, measured in length on a building plan, that have openable windows or other portals that permit access to external space and air. Figure 7 illustrates the measurement of CDES. It was anticipated that the effects of noise and odour originating from one flat and permeating other adjacent flats would be diminished if the communal corridor linking the flats were more “open”. The openings on the corridor’s containment walls can serve as points to allow sound and foul air to discharge from the interior to the exterior of a housing block. Airborne sound cannot be reflected where there is an opening in the walls. With a higher degree of “openness”, the communal corridor can allow undesirable noise and odour generated in one flat to attenuate more effectively before they reach other flats. Besides, two factors relate to building scale. The first is the total number of domestic storeys in a residential block (STOREY). The second is the average number of flats per floor in a residential block (FLAT).

The key control variables picked for the study characterised demographic and socio-economic factors: gender, age, educational attainment and personal income and household size are employed. A resident’s gender is indicated by the dummy variable MALE, equalling 1 for males and zero for women. For each resident’s age group, the variable AGE is indicated with a six-category scale (6=65 years old or above; 5=55–64 years old; 4=45–54 years old; 3=35–44 years old; 2=25–34 years old; and 1=18–24 years old). The highest educational attainment of the resident (EDU) is measured using a six-category scale (6=postgraduate degree or above; 5=bachelor degree; 4=sub-degree post-secondary education; 3=upper secondary school or matriculation; 2=lower secondary school; and 1=primary school or below). Income level (PINC) was indicated with a seven-point measure: 7=HK$30,000 or above; 6=HK$25,000–HK$29,999; 5=HK$20,000–HK$24,999; 4=HK$15,000–HK$19,999; 3=HK$10,000–HK$14,999; 2=HK$4,000–HK$9,999; and 1=below HK$4,000. Income brackets identified each resident’s average monthly personal income. Household size (HSIZE) was measured as the number of household members living in a resident’s flat. In addition to the five demographic and socio-economic factors mentioned above, the length in years of a resident’s occupation of the same housing estate (LOR) is included in the model to indicate the resident’s attachment to the estate.

### Collection of data

Empirical testing of the analytic model elaborated in the previous section required that relevant data be collected from Hong Kong’s public rental housing estates. Both the primary data sourced directly from residents and secondary data on the housing blocks’ construction were employed for the analyses. For the primary data, an online structured questionnaire survey was conducted in Hong Kong in the period between November 2016 and March 2017. A questionnaire was designed to solicit the information required to empirically test the analytic model. Before the survey officially started, the questionnaire had been pretested and modified in response to testers’ feedback. When the questionnaire was revised to its final version, a letter was sent to the sampled households in selected public rental housing estates. Letters invited the respective household heads to complete an online survey. As for the secondary data, estate-based information such as development scale and occupant density were obtained from various governmental sources such as the website of the Housing Department and government’s census data.

In total, 14 public rental housing estates in Hong Kong were sampled. These housing estates are scattered throughout Hong Kong, and their housing blocks had been constructed in different design configurations and to different scales of square-footage and occupant-density. Selective sampling allowed for a more meaningful research analysis. In all, five of these 14 housing estates were located on Hong Kong Island, three in Kowloon and six in the New Territories. The scales of the estates ranged from 900 residential units to 7,900 housing units. In the 14 sampled estates, there was a total of 106 housing blocks. In each of the sampled housing estates, approximately 20 per cent of the housing blocks were randomly selected, resulting in a total of 23 housing blocks that contained tenants who were invited to participate in the study. As Table I shows, ten housing blocks (43.5 per cent) came from the New Territories, five (21.7 per cent) from Kowloon and eight (34.8 per cent) from Hong Kong Island.

Of the sampled housing blocks, the basic design formats were distributed in these proportions. Eight housing blocks (34.8 per cent) were designed with a linear or slab layout and another eight (34.8 per cent) with cruci or cross form layouts. The number of storeys in these housing blocks in these estates ranged from 14 to 40, with a mean of 23.3. On average, there were 14.5 residential units on each floor. The average age of the 23 housing blocks was 30.1 years. The percentage of an enclosure walls’ perimeter for a typical communal corridor that was open to external air ranged from 3.2 to 33.6 per cent, with a mean of 12.1 per cent.

In each of these 23 housing blocks, approximately 25 per cent of the residential units were randomly sampled — for a total of 1,950 residential units. Invitations were sent to the household heads of these residential units. Household heads were requested to answer an online survey, which included the study’s questionnaire. The researcher contacted households once, waited for responses, and then repeated the invitation to households that had not replied. After the two appeals, a total of 451 replies were received, representing an overall response rate of 23.1 per cent. Among these returned questionnaires, 422 were deemed valid for the analysis. The remaining 29 replies were incomplete so they were discarded. Multiple regression technique was employed to analyse the survey data.

Table II summarises the characteristics of the 422 respondents. Over half of the respondents were male. The respondents’ ages were distributed quite evenly across adult age groups. Data on education level revealed that approximately 15 per cent of the respondents had attained bachelor degrees or higher. Comparative data for income showed that the group earning HK$15,000–HK$19,999 contributed the largest share (35.5 per cent) of any one income bracket for average monthly income. Regarding household income, the sample looks comparable with the population in public rental housing in Hong Kong. According to the 2016 Population By-census, the median monthly domestic household income in all public rental housing was HK$16,000 (Census and Statistics Department, 2017). The average household size was 3.4 people, and the respondents’ mean length of residence in the same estate was 18.1 years. As Table III shows, residents considered neighbours’ noise to be the single-most serious type of ASB. Harassment and intimidation was the ASB category respondents complained of least. The correlation matrix of continuous independent variables is presented in Table IV. Inspection of the correlation matrix does not reveal any significant problems of multicollinearity in the data set. ### Results of regression analyses Two models were used for empirically testing relationships between housing design parameters and ASB perceptions. Model 1 included housing design parameters (e.g. block layout, building height and number of flats per storey) only as independent variables. In addition to the housing design parameters, Model 2 incorporated the control variables (e.g. the respondents’ demographic and socio-economic characteristics and length of residence). Table V shows the results of the model estimation using the ordinary least squared method. The estimation returned adjusted R2 of 0.42 and 0.49 for Models 1 and 2, respectively, implying around 42–49 per cent of the variation in the dependent variable (i.e. ASB) could be explained by differences in the independent variables incorporated in the respective models. As far as the control variables are concerned, only age showed significant (at 10 per cent level at least) effects on ASB perceptions. Older respondents tended to perceive a less serious level of ASB in their residential neighbourhoods. A ten-year increase in age resulted in a 0.5-point reduction in the overall severity level of ASB. On the other hand, gender, education level, personal income and household size did not correlate with any pronounced trend in the respondents’ ASB perceptions (even at the 10 per cent level). Similarly, length of residence had no significant correlation with ASB perception. As for the housing design parameters which are the foci of the current study, the estimation results of Models 1 and 2 were similar. The estimated coefficients of the dummy variables LINEAR (−0.1062 and −0.0956) and HSPAPE (−0.0808 and −0.0776) were found to be negative, and they were statistically significant at the 10 per cent level at least in the two estimation models. Thus, compared with residents in housing blocks with cruci- or cross-form layouts, respondents living in housing blocks with a linear, slab or H-shaped form perceived less serious ASB problems, ceteris paribus. In both models, there was no significant difference between the estimated coefficients of LINEAR and HSPAPE (even at the 10 per cent level). In other words, when other factors were controlled, residents living in housing blocks in a linear or slab-form layout and with an H-shaped form perceived similar levels of ASB severity. On the other hand, in both models, the dummy variable TRIDENT was found to have no insignificant impact on ASB (even at the 10 per cent level), indicating that residents living in housing blocks in a trident form and in blocks with cruci and cross forms did not have significantly dissimilar ASB perceptions, keeping other things constant. The estimated coefficient of the variable CDES was found negative and statistically significant at the 1 per cent level in both models. These results suggested that residents living in a housing block where the communal corridor had a greater proportion of perimeter opening to external air perceived a lower level of ASB severity in their housing estates. Say according to the estimation results of Model 2, when the “openness” of the communal corridor was increased by 10 per cent (or 0.1), the overall ASB score would drop by 0.46. If communal corridors of a housing block were more “open” to external air, the residents’ perception of threatening or nuisance ASB activity was diminished. The building height factor, as proxied by the number of storeys in a building block, was found to have significant (at the 10 per cent level in Model 1 and 5 per cent level in Model 2) positive impacts on ASB perceptions. Yet, the impacts brought by building heights were rather marginal. A rise in the number of storeys in a housing block by 10 would contribute to a 0.05-point increment only in the overall ASB level. As for the variable FLAT, it also had a positive and statistically significant (at the 1 per cent level in both models) estimated coefficient, signifying a positive relationship between number of flats per storey and ASB perception. ## Discussion and implications The estimation results of the empirical model suggested that block layout and corridor design were the factors that powerfully influenced residents’ perceptions of ASB level in their housing estates. The analysis results confirmed our expectations. By carefully arranging the housing units on a floor and making openings on enclosure walls of communal corridors, dampen the spread of noise, odour and other nuisances among residents. These enhancements can be incorporated in the design stage — long before residents and housing managers struggle to find solutions to ASB problems during the operation stage that will likely last decades. As the analysis results suggest, linear, slab and H-shape forms are preferable block layouts, as they can reduce residents’ perceptions of ASB environmental effects. Trident, cruci and cross layouts are less preferable from the perspective of ASB perception minimisation. Apart from block layout, corridor design was also found to correlate strongly with perceived ASB level. Residents living in a housing block with communal corridors that were more open to external air complained less of ASB effects. The findings of this research should encourage architects to reduce future residents’ perception of ASB distractions by investing communal corridors with more openable windows and other safe means of accessing “outside” air. Alternatively, balconies could be integrated in design of communal corridors. Furthermore, as the analysis results implied, adding more storeys to a housing block will likely increase ASB effects, and it will certainly increase residents’ perceptions that their peace and safety is threatened by neighbours’ ASB activities. The same problems are exacerbated by plans that add additional units to each storey. These relationships echo arguments by Makinde et al. (2016) that every crowded living environment promotes ASB activity. When there are too many storeys in a building and too many flats compacted on each floor, the boundaries between “private” flats and public, shared areas (including communal corridors and environmental cleanliness) become uncomfortably thin. Tenants who grow sensitive to the fact that, even when they are alone in their rental units, they are sharing space with others in the building will feel vulnerable to neighbours’ constant activities. Inclined to interpret neighbours’ unwanted intrusions on their privacy as ASB, residents are more likely to feel like victims. If their resentments increase, they might become intentional sources of retaliatory ASB. The exploratory analyses produced provocative results that should be explored in future research. Previous scholarship on ASB has argued that residents’ age was a primary determinant in sensitivity to ASB; however, this research has indicated that a more imposing source of influences that decreased or elevated tenants’ sensitivity to ASB activity were key housing design parameters. These aspects of high-rise buildings’ physical construction connected to residents’ sense that their privacy was secure from or vulnerable to neighbours’ intrusive proximity. Building features that also proved significant correlated to tenants’ sense that public corridors had access to open air. Renters’ perceptions about these conditions powerfully influenced their sense of the quality of their living environments. In order to curb ASB activities in high-rise residences and reduce disputes between neighbours, the Hong Kong government would be wise to consider these discoveries when they seek to integrate ASB-dampening features in designs for new public and private housing developments. This study’s findings should also be cautionary for governmental and professional organisations that certify sustainable building projects (e.g. Building Environmental Assessment Method Plus). With this study’s main recommendations in mind, architects and real estate developers should remember that a new building’s incorporation of renewable materials and implementation of new energy-saving and waste-reducing technologies is very important. But the ultimate purpose of all residential projects is to create an enhanced sustainable built environment for individuals and communities to coexist in peacefully and happily. Second, to make a housing estate pleasant place for living and visiting, housing managers should make sure that those ASB-dampening design features will not be “disabled” in the building operation stage. Over years, building managers will need to oversee repairs, renovations, and alterations to guarantee that ASB-reducing features are not sabotaged by poorly conceived retrofitting. Indeed, tenants must contribute to community health by ensuring that ASB-suppressing features continue to work. Openable windows in communal corridors should not be kept closed. Closed windows retain heat, but they also trap noise and odour within communal corridors. Housing managers who oversee maintenance of new cost-efficient but crowded government-subsidised high-rise residences must be aware that tenants recognise when increasing the number of flats per storey has occurred at the expense of privacy. Too many tenants inhabiting each storey will become sensitised to ASB activities. Housing managers might need to proactively institute more stringent house rules (like “no-pet” policies) or patrol common areas where friction between neighbours often occurs. This research’s implications are likely not limited to enhancing ASB governance in Hong Kong’s residential high-rises. If additional research confirms these findings, the architectural features we discovered to be most effective in securing residents’ contentment could prove advantageous in other countries high-rise and large-scale residential structures, especially those which are being implemented in Asia. As Yau (2012a) contended, ASB activities have proliferated in Hong Kong’s private residential housing developments in a fashion comparable to its spread in crowded public rental housing communities. This research has investigated residents’ quality of life in public housing, but the ASB-suppressing architectural features we have identified might have equally enduring benefits for multi-unit private housing in Asian cities. ### Conclusion and agenda for future research It has been widely accepted that ASB activities are a matter of burgeoning concern, as much for community government as for city and district law enforcement. The possible effects of design and construction features on large multi-unit structures have not been seriously investigated until now. Scholarship on ASB’s disruptive influence on communities has focussed on individual behaviours and structural social trends in contemporary home life. Therefore, the researcher finds it exciting that this study’s conclusions have pointed to a more powerful set of environmental influences correlating to effective building design. This contributes to the existing body of ASB research by providing new, statistically supported theories of how building features could both reduce the conditions that contribute to disruptive ASB and, by increasing residents’ sense of well-being, diminish their perceived sensitivity to problems of close cohabitation that could be identified as ASB. The findings of this research have direct relevance to current public housing policy. But their implications extend to the potential benefit of many modern housing stakeholders: real-estate developers, tenants and owner-occupiers, housing managers, architects and governments. The relationship between housing design and residents’ perceived sensitivity to ASB disruption is complex. Nevertheless, because unhappy residents’ reactions to perceived ASB-related conflict affect whole communities’ well-being, identifying likely environmental contributors to ASB potentially impacts community health, residential satisfaction and social capital. This avenue of exploration warrants further theoretical consideration and empirical study. The current study focusses on public housing only. The housing design-ASB relationship in private housing should be examined. Moreover, only several factors such as block layout, openness of communal corridor, building height and development density were incorporated in the exploratory analysis in the current study. Some design parameters that could have effects on residents’ ASB perceptions were ignored. These parameters include thicknesses of internal walls and floor slabs, and types of window systems. Future studies should be carried out to find out the effects of these design parameters. Furthermore, it is necessary to develop a more complex modelling of the linkages between the housing design, neighbourhood cohesion and ASB perception in the future. While Yau (2014) argued that solidarity with one’s community could be nurtured through better housing design, it seems logical that some housing design features might enhance a neighbourhood’s solidarity but simultaneously exacerbate residents’ sensitivity to ASB influences by outsiders and residents who resist conformity. Future studies need to look at an expanded range of housing design parameters. The spectrum of ASB activities researchers observe should be expanded. Further, future studies are warranted to explore the effects of housing design parameters or features on ASB and neighbourhood cohesion and the dynamic relationships between ASB and neighbourhood cohesion after controlling the structural and contextual factors, as shown in Figure 8. Such research will facilitate better understanding of how institutionalized environmental features affect residents’ perceptions of their security and well-being in their living places. And yet this study also signals a profound shift in ASB studies that, by drawing attention to designed residential communities, opens the door of disciplinary opportunity to architects, engineers, ecologists, and urban planners to contribute to the study of environment’s influence on individual and communal perception of social disruptive interactions by neighbours. The theoretical implications are challenging and intriguing. But the practical results of future research into the influence of built environments on large, complexly engineered residences promise concrete rewards from better designed homes and better organised communities that will help engender sustainable communities for all our future cities. ## Figures #### Figure 1 Analytic framework contrasting variables’ influence on ASB #### Figure 2 Cruci form of block layout #### Figure 3 Cross form of block layout #### Figure 4 Trident form of block layout #### Figure 5 Linear form of block layout #### Figure 6 H-shaped form of block layout #### Figure 7 Graphical illustration of measurement of CDES #### Figure 8 Proposed conceptual framework for subsequent research ## Table I Characteristics of sampled housing blocks Characteristic Count Percentage Location Hong Kong Island 8 34.8 Kowloon 5 21.7 New territories 10 43.5 Block Layout Linear or slab form 8 34.8 H-shaped form 4 17.4 Trident form 3 13.0 Cruci or cross form 8 34.8 Note: n=23 ## Table II Characteristics of the survey respondents Characteristic Count Percentagea Gender Male 268 63.5 Female 154 36.5 Age 18–24 years old 15 3.6 25–34 years old 60 14.2 35–44 years old 138 32.7 45–54 years old 143 33.9 55–64 years old 58 13.7 65 years old or above 8 1.9 Education level Primary school or below 68 16.1 Lower secondary school 109 25.8 Upper secondary school or matriculation 108 25.6 Sub-degree post-secondary education 73 17.3 Bachelor degree 58 13.7 Postgraduate degree or above 6 1.4 Average monthly personal income Below HK$4,000 42 10.0
HK$4,000-HK$9,999 71 16.8
HK$10,000-HK$14,999 83 19.7
HK$15,000-HK$19,999 150 35.5
HK$20,000-HK$24,999 51 12.1
HK$25,000-HK$29,999 16 3.8
HK\$30,000 or above 9 2.1
Layout of housing block
Linear or slab form 131 31.0
H-shaped form 68 16.1
Trident form 63 14.9
Cruci or cross form 160 37.9

Notes: n=422. aThe percentages of the categories for each characteristic may not total 100% because of rounding

## Table III

Respondents’ perceptions of ASB severity

Type of ASB Very serious Serious Quite serious A little serious Not a problem at all
Noise 120 (28.4%) 107 (25.4%) 86 (20.4%) 80 (19.0%) 29 (6.9%)
Pet nuisances 90 (21.3%) 89 (21.1%) 116 (27.5%) 86 (20.4%) 41 (9.7%)
Neglected water seepage or dripping 86 (20.4%) 100 (23.7%) 110 (26.1%) 79 (18.7%) 47 (11.1%)
Littering 117 (27.7%) 109 (25.8%) 86 (20.4%) 70 (16.6%) 40 (9.5%)
Abusiveness in common areas 86 (20.4%) 110 (26.1%) 104 (24.6%) 81 (19.2%) 41 (9.7%)
Graffiti, flyer-posting, and vandalism 87 (20.6%) 90 (21.8%) 112 (26.5%) 90 (21.3%) 41 (9.7%)
Harassment and intimidation 83 (19.7%) 94 (22.3%) 98 (23.2%) 103 (24.4%) 44 (10.4%)

Notes: n=422. The figures in parentheses represent proportions of respondents

## Table IV

Correlation coefficient matrix of continuous independent variables

Variable AGE EDU PINC HSIZE LOR CDES STOREY FLAT
AGE 1
EDU −0.015 1
PINC −0.010 −0.331 1
HSIZE 0.001 −0.087 0.082 1
LOR 0.049 0.147 −0.223 −0.029 1
CDES −0.057 −0.005 −0.020 −0.064 −0.014 1
STOREY −0.017 −0.036 0.009 0.067 −0.014 0.017 1
FLAT −0.009 −0.075 0.222 0.118 0.013 0.052 −0.059 1

## Table V

Results of the ordinary least squared estimation of the empirical models

Model 1 Model 2
Variable Coefficient t-statistic Coefficient t-statistic
Intercept 3.6874 30.5300*** 3.8236 25.1482***
LINEAR −0.1062 −2.0255** −0.0956 −1.8260*
HSHAPE −0.0808 −2.1079** −0.0776 −2.0303**
TRIDENT 0.0927 1.5095 0.0730 1.1855
CDES −4.7635 −18.4959*** −4.6472 −17.7415***
STOREY 0.0051 1.9030* 0.0054 2.0504**
FLAT 0.0247 8.9628*** 0.0242 8.8510***
MALE 0.0277 0.8731
AGE −0.0516 −3.6734***
EDU 0.0157 1.4183
PINC 0.0013 0.1158
HSIZE 0.0021 0.1964
LOR −0.0020 −0.8040
R2 0.4346 0.4959
F-statistics 6.0409*** 6.8132***
Dependent variable ASB ASB
Number of observations 442 442

Notes: *,**,***Denote the estimated coefficients of the variables or test statistics to be significant at the 10, 5 and 1 per cent levels, respectively

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## Acknowledgements

The final form of this article owes much to the discussion at the Workshop on Property Management and Land Management: New Challenges in Declining and Growing High-rise Cities held in Hong Kong on 13 June 2017. The work described in this paper was supported by the Research Grant Writing Fund offered by the College of Liberal Arts and Social Sciences, City University of Hong Kong.

## Corresponding author

Dr Yung Yau can be contacted at: y.yau@cityu.edu.hk


text:
Lemma 38.42.1. Let $X$ be a scheme. Let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$. Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_ X$-modules such that $\mathcal{F}^ i$ is $\mathcal{I}$-torsion free for all $i$. Then $\eta _\mathcal {I}\mathcal{F}^\bullet$ is a complex of quasi-coherent $\mathcal{O}_ X$-modules. Moreover, if $U = \mathop{\mathrm{Spec}}(A) \subset X$ is affine open and $D \cap U = V(f)$, then $\eta _ f(\mathcal{F}^\bullet (U))$ is canonically isomorphic to $(\eta _\mathcal {I}\mathcal{F}^\bullet )(U)$.

Proof. Omitted. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

format: plain


text:
## 11.1 Complex seasonality

So far, we have considered relatively simple seasonal patterns such as quarterly and monthly data. However, higher frequency time series often exhibit more complicated seasonal patterns. For example, daily data may have a weekly pattern as well as an annual pattern. Hourly data usually has three types of seasonality: a daily pattern, a weekly pattern, and an annual pattern. Even weekly data can be challenging to forecast as it typically has an annual pattern with seasonal period of $$365.25/7\approx 52.179$$ on average.

Such multiple seasonal patterns are becoming more common with high frequency data recording. Further examples where multiple seasonal patterns can occur include call volume in call centres, daily hospital admissions, requests for cash at ATMs, electricity and water usage, and access to computer web sites.

Most of the methods we have considered so far are unable to deal with these seasonal complexities. Even the ts class in R can only handle one type of seasonality, which is usually assumed to take integer values.

To deal with such series, we will use the msts class which handles multiple seasonality time series. This allows you to specify all of the frequencies that might be relevant. It is also flexible enough to handle non-integer frequencies.

Despite this flexibility, we don’t necessarily want to include all of these frequencies — just the ones that are likely to be present in the data. For example, if we have only 180 days of data, we may ignore the annual seasonality. If the data are measurements of a natural phenomenon (e.g., temperature), we can probably safely ignore any weekly seasonality.

The top panel of Figure 11.1 shows the number of retail banking call arrivals per 5-minute interval between 7:00am and 9:05pm each weekday over a 33 week period. The bottom panel shows the first three weeks of the same time series. There is a strong daily seasonal pattern with frequency 169 (there are 169 5-minute intervals per day), and a weak weekly seasonal pattern with frequency $$169 \times 5=845$$. (Call volumes on Mondays tend to be higher than the rest of the week.) If a longer series of data were available, we may also have observed an annual seasonal pattern.

p1 <- autoplot(calls) +
ylab("Call volume") + xlab("Weeks") +
scale_x_continuous(breaks=seq(1,33,by=2))
p2 <- autoplot(window(calls, end=4)) +
ylab("Call volume") + xlab("Weeks") +
scale_x_continuous(minor_breaks = seq(1,4,by=0.2))
gridExtra::grid.arrange(p1,p2)

### STL with multiple seasonal periods

The mstl() function is a variation on stl() designed to deal with multiple seasonality. It will return multiple seasonal components, as well as a trend and remainder component.

calls %>% mstl() %>%
autoplot() + xlab("Week")

There are two seasonal patterns shown, one for the time of day (the third panel), and one for the time of week (the fourth panel). To properly interpret this graph, it is important to notice the vertical scales. In this case, the trend and the weekly seasonality have relatively narrow ranges compared to the other components, because there is little trend seen in the data, and the weekly seasonality is weak.

The decomposition can also be used in forecasting, with each of the seasonal components forecast using a seasonal naïve method, and the seasonally adjusted data forecasting using ETS (or some other user-specified method). The stlf() function will do this automatically.

calls %>%  stlf() %>%
autoplot() + xlab("Week")

### Dynamic harmonic regression with multiple seasonal periods

With multiple seasonalities, we can use Fourier terms as we did in earlier chapters (see Sections 5.4 and 9.5). Because there are multiple seasonalities, we need to add Fourier terms for each seasonal period. In this case, the seasonal periods are 169 and 845, so the Fourier terms are of the form $\sin\left(\frac{2\pi kt}{169}\right), \quad \cos\left(\frac{2\pi kt}{169}\right), \quad \sin\left(\frac{2\pi kt}{845}\right), \quad \text{and} \quad \cos\left(\frac{2\pi kt}{845}\right),$ for $$k=1,2,\dots$$. The fourier() function can generate these for you.

We will fit a dynamic harmonic regression model with an ARMA error structure. The total number of Fourier terms for each seasonal period have been chosen to minimise the AICc. We will use a log transformation (lambda=0) to ensure the forecasts and prediction intervals remain positive.

fit <- auto.arima(calls, seasonal=FALSE, lambda=0,
xreg=fourier(calls, K=c(10,10)))
fit %>%
forecast(xreg=fourier(calls, K=c(10,10), h=2*169)) %>%
autoplot(include=5*169) +
ylab("Call volume") + xlab("Weeks")

This is a large model, containing 40 parameters: 4 ARMA coefficients, 20 Fourier coefficients for frequency 169, and 16 Fourier coefficients for frequency 845. We don’t use all the Fourier terms for frequency 845 because there is some overlap with the terms of frequency 169 (since $$845=5\times169$$).

### TBATS models

An alternative approach developed by uses a combination of Fourier terms with an exponential smoothing state space model and a Box-Cox transformation, in a completely automated manner. As with any automated modelling framework, there may be cases where it gives poor results, but it can be a useful approach in some circumstances.

A TBATS model differs from dynamic harmonic regression in that the seasonality is allowed to change slowly over time in a TBATS model, while harmonic regression terms force the seasonal patterns to repeat periodically without changing. One drawback of TBATS models, however, is that they can be slow to estimate, especially with long time series. Hence, we will consider a subset of the calls data to save time.

calls %>%
subset(start=length(calls)-2000) %>%
tbats() -> fit2
fc2 <- forecast(fit2, h=2*169)
autoplot(fc2, include=5*169) +
ylab("Call volume") + xlab("Weeks")

Here the prediction intervals appear to be much too wide – something that seems to happen quite often with TBATS models unfortunately.

### Complex seasonality with covariates

TBATS models do not allow for covariates, although they can be included in dynamic harmonic regression models. One common application of such models is electricity demand modelling.

Figure 11.6 shows half-hourly electricity demand in Victoria, Australia, during 2014, along with temperatures for the same period for Melbourne (the largest city in Victoria).

autoplot(elecdemand[,c("Demand","Temperature")],
facet=TRUE) +
scale_x_continuous(minor_breaks=NULL,
breaks=2014+
cumsum(c(0,31,28,31,30,31,30,31,31,30,31,30))/365,
labels=month.abb) +
xlab("Time") + ylab("")

Plotting electricity demand against temperature (Figure 11.7) shows that there is a nonlinear relationship between the two, with demand increasing for low temperatures (due to heating) and increasing for high temperatures (due to cooling).

elecdemand %>%
as.data.frame() %>%
ggplot(aes(x=Temperature, y=Demand)) + geom_point() +
xlab("Temperature (degrees Celsius)") +
ylab("Demand (GW)")

We will fit a regression model with a piecewise linear function of temperature (containing a knot at 18 degrees), and harmonic regression terms to allow for the daily seasonal pattern.

cooling <- pmax(elecdemand[,"Temperature"], 18)
fit <- auto.arima(elecdemand[,"Demand"],
xreg = cbind(fourier(elecdemand, c(10,10,0)),
heating=elecdemand[,"Temperature"],
cooling=cooling))

Forecasting with such models is difficult because we require future values of the predictor variables. Future values of the Fourier terms are easy to compute, but future temperatures are, of course, unknown. If we are only interested in forecasting up to a week ahead, we could use temperature forecasts obtain from a meteorological model. Alternatively, we could use scenario forecasting (Section 4.5) and plug in possible temperature patterns. In the following example, we have used a repeat of the last two days of temperatures to generate future possible demand values.

temps <- subset(elecdemand[,"Temperature"],
start=NROW(elecdemand)-2*48+1)
fc <- forecast(fit,
xreg=cbind(fourier(temps, c(10,10,0)),
heating=temps, cooling=pmax(temps,18)))
autoplot(fc, include=14*48)

Although the short-term forecasts look reasonable, this is a crude model for a complicated process. The residuals demonstrate that there is a lot of information that has not been captured with this model.

checkresiduals(fc)
#>
#>  Ljung-Box test
#>
#> data:  Residuals from Regression with ARIMA(5,1,4) errors
#> Q* = 738412, df = 3455, p-value <2e-16
#>
#> Model df: 49.   Total lags used: 3504

More sophisticated versions of this model which provide much better forecasts are described in and .


text:
# Determining BETA for an MFB filter

Discussion in 'Homework Help' started by kdillinger, Nov 6, 2009.

1. ### kdillinger Thread Starter Active Member

Jul 26, 2009
141
3
Hello all,

I need some advice on an approach to determine β for a multiple feedback filter. My scanner is broke, but I have pages and pages of lengthy derivations and I am stuck.

My first approach was to consider two β's and them add them together using superposition. β1 would be at the R3/C1/R2 node (see attached) and β2 would be at the classic summing junction.

I was oh so close to Equation 16 from TI's application note (also in the attached).

Rather than ask for someone to help me with the math at this point, I would like to know if my approach makes sense. β1 + β2 = β(total) or as shown in the attached WORD doc.

The more I think about this, I believe I cannot do this but I am lost. Is there a better approach of how I can view the circuit? I can trod through the math, but I need some insight on how I can setup my nodes in these two feedback loops.

One is easy, two is driving me to drink.

-Ken

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2. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,558
452
Please post in some other format than .doc; .pdf preferably.

I viewed your .doc attachment on another computer. I don't see anything that looks like it came from a TI app note.

Can you provide a link to the TI app note, or the title of the note?

Last edited: Nov 6, 2009
3. ### Ron H AAC Fanatic!

Apr 14, 2005
7,049
673
I don't see any equations in the attachment.

Jan 28, 2005
9,030
218

hgmjr

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5. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
789
Re the attachment - capacitors CT & C3 seem pointless. If the op-amp is functioning 'normally' then the negative input should be driven to ground potential - thereby shunting those caps to ground.

6. ### kdillinger Thread Starter Active Member

Jul 26, 2009
141
3
I figured I would post the solution since I have been away due to personal errands. I finished this over the weekend if everyone is still interested.
I am posting only the first and last page of my derivation (total of 6).

And, yes, TNK, the capacitor will be shunted to ground when considering the signal gain, not the closed loop again or noise-gain as it is referred to in literature.

This exercise is to find β and 1/β and the capacitor from the inverting node to ground absolutely comes into play.

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7. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,558
452
I don't think you explained yourself very well in your very first post.

Are you assuming that the general feedback formula:

$Vout=\frac{A}{1-A\beta}Vin$

is to be taken as applicable to your filter, and then you are to find β as it appears in the general formula?

Or is the β you are trying to find something else?

8. ### kdillinger Thread Starter Active Member

Jul 26, 2009
141
3
Yes, β, as it appears in the general feedback formula. The feedback attenuation at the summing junction.

9. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,558
452
I had to go dig out one of my old textbooks to be sure of what is meant by the β network in a circuit like this.

It looks to me like this is a shunt-shunt feedback network, in which case the β network is determined by shorting the input, applying a test voltage (1 volt) to the part of the network connected to Vout, and calculating the current into a short to ground at the Vfb node. This is different from what you calculated.

Are you still working on this problem, and do my comments make any sense to you?


text:
# How can I interpret a moderately negative correlation?

I've conducted a Spearman's rank-order correlation test on a set of data. The Score which a participant got on a test is the independent variable, and their Responses to a Likert-type question are the dependent variable. I wanted to see if there was a main effect between the Score, and Responses to any of the questions. There were 19 questions. I only found one correlation, between the Score and Question 5, and it looked like this:

r = -.408, p = .003

So it's showing a moderate negative correlation, implying that as participants' score in the test goes up, their responses to Question 5 go down. As Question 5 is Likert-type, with 5 being the best and 1 being the worst, this means that that participants who scored higher tend to rate Question 5 worse.

This bit is fine. What I don't know how to interpret is the moderate correlation. Because of the low p-value, I guess that means that the test is reliable, but what does it mean to have a reliable moderate correlation? Perhaps I'm overthinking this, but I just don't know how to make sense of this result.

A couple of caveats before moving on to the actual question you asked -

First, with 19 tests (really, more than one test) you should be adjusting for multiple comparisons. If you perform 20 independent tests of null hypotheses that are in fact true, you'd expect to get, on average, one p-value $\leq 0.05$, with the distinct possibility of getting more... which implies that your overall probability of rejecting a true null hypothesis is actually quite high.

One well-respected procedure you can use is the Benjamini-Hochberg procedure. You would need to choose a false discovery rate (FDR), that is, a desired maximum expected proportion of "discoveries" (i.e., rejections of the null hypothesis) that are false (i.e., the null hypothesis in those cases is actually true.) In your case, the criterion for your single "significant" correlation becomes $FDR/19$, which equals 0.0026 if you set the FDR at 0.05. You would conclude that you could not reject any of the null hypotheses at an FDR of 0.05.

Having written that, let's assume you set the FDR at 0.1 and consequently did reject the null hypothesis for the correlation above. Now, why is that correlation relatively large (in absolute value)? When you're comparing several (in this case) correlations, the largest ones probably got that way through some combination of underlying value and randomness, writing very loosely. This brings us to the second caveat - you can't trust that the largest effect size is an unbiased estimate of its true effect. There are ways of correcting for this, too, but I won't go into them here - they are more complex than FDR, and, depending on your data, may not make much difference.

On to the question! What does moderate correlation mean? I suspect you are overthinking the issue to some extent. It simply means that there is some relationship between the two variables in question, but that there's also a lot of randomness affecting one or both variables, or perhaps other variables affect the two variables in question, so the direct relationship isn't strong, but it's certainly noticeable. Plots help:

These two variables have a correlation of -0.44, about the same as yours. You can see there is a relationship, but it's certainly not a strong one, nor is it so weak it can be ignored. Hence, "moderate".

• Thanks for the answer, that's really helpful, insightful and informative! I'm sure you're right about overthinking it, I just don't have the head for stats - and more importantly, I wasn't sure what to write in my analysis section for my dissertation. I guess what I was really looking for was whether I could say anything more than the fact that there is a moderate negative correlation. But I almost certainly don't have the information to make inferences about what this correlation means - especially given that it's the only result with a >.3 or <-.3 r-value. – Lou May 19 '18 at 19:48
• Well, a lot about having a head for stats is really about just doing stuff, thereby developing the intuition that helps a lot with this sort of thing. If the answer's good enough, please accept it - that lets future readers know that it addresses the question without them being forced to read it to find out. – jbowman May 19 '18 at 19:53
• Heaven forbid people have to read things :'). – Lou May 19 '18 at 20:02


text:
### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

### Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

### Plum Tree

Label this plum tree graph to make it totally magic!

# Pair Squares

##### Stage: 5 Challenge Level:

Solutions are given here for finding sets of three numbers such that the sum of any two of them is a perfect square. Can you generalize the methods to find sets of four numbers with this property?

Hyeyoun Chung from St Paul's Girls' School noticed that if the square numbers chosen are: $x^2$, $(x+1)^2$ and $(x+3)^2$ and the corresponding set of three integers are $a,\ b$ and $c$, with $a< b< c$ and: \eqalign{ a + b &= x^2 \cr a + c &= (x + 1)^2 \cr b + c &= (x + 3)^2 }. then these these three equations can be solved to give: \eqalign { c &= {x^2 + 8x + 10 \over 2} \cr b &= {x^2 + 4x + 8 \over 2} \cr a &= {x^2 -4x - 8 \over 2} }.

Therefore, from a single square number, $x^2$, three integers $a$, $b$ and $c$ can be found with values such that the sum of any pair of these numbers is a perfect square. The only problem with this system is, however, that the square number must be even, otherwise $a$, $b$ and $c$ will not be integers.

Yatir Halevi tackled the problem as follows:

We are given 3 square numbers: $a^2,\ b^{2}$ and $c^{2}$, and we have to find a general formula for finding 3 integers that are 'Pair Squares' such that the sum of any pair is a square number, that is: $x+y=a^2$, $y+z=b^2$ and $x+z=c^2$.

Solving for $x,\ y$ and $z$ we get:

\eqalign{ a^2-b^2 &= x-z \cr c^2 &= x+z \cr 2x &= a^2 + c^2 - b^2 \cr x &= (a^2- b^2 + c^2)/2}

By the same methods we get: \eqalign{ y &=(a^2+b^2-c^2)/2 \cr z &=(-a^2+b^2+c^2)/2.}

In order for $x,\ y$ and $z$ to be integers; either $a,\ b$ and $c$ are all even or two are odd and the other is even because if an integer $n$ is even or odd so is its square. This is proved as follows: if $n$ is odd then it can be written $n=2m+1$, so $n^2=(2m+1)^2=4m^2+4m+1=2(2m^2+2m)+1=2q+1$ which is odd. If $n$ is even then $n=2m$, so $n^2=(2m)^2=4m^2=2(m^2)$ which is even.

For the first case we can generate Pair Squares by choosing any three even squares, for example 4, 16 and 36 and the formula gives:

\eqalign { x &= (4 -16 + 36)/2 = 12 \cr y&= (4 + 16 -36)/2 = -8 \cr z&= -4 +16 +36)/2 = 24. } This is a set of pair squares because $12-8=4$, $-8+24=16$ and $24+12=36$.

For the second case we can generate Pair Squares by choosing any two odd squares together with one even square, for example 9, 1 and 4 and the formula gives: \eqalign { x &= (9 - 1 + 4)/2 = 6 \cr y&= (9 + 1 - 4)/2 = 3 \cr z&= -9 +1 + 4)/2 = -2. }

Checking that this is a set of pair squares we have $6+3=9,\ 3-2=1,\ 6-2=4$.

As for four integers that are pair squares: take an even square $a^2$ and halve it and you have $b=a^2/2$ giving $\{b,\ b,\ b,\ b\}$ for example $\{8, 8, 8, 8\}$ or $\{8,8,8,17\}$.

Saul Foresta used a similar method and gave the following example: $x=54$, $y=10$ and $z=90$


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# Seven family members are seated around their circular dinner

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Seven family members are seated around their circular dinner  [#permalink]

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Updated on: 09 Jul 2013, 07:43
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Question Stats:

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### HideShow timer Statistics

Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Originally posted by eladshush on 04 Oct 2010, 05:30.
Last edited by Bunuel on 09 Jul 2013, 07:43, edited 1 time in total.
Edited the OA.
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Re: Arrangements around the table  [#permalink]

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04 Oct 2010, 08:06
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Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Glue Michael and Bobby so that they create one unit, so we would have total of 6 units: {1}{2}{3}{4}{5}{MB} --> # of different arrangements of $$n$$ objects around the table (circular arrangements) is is $$(n-1)!$$, so our 6 objects can be arranged in $$(6-1)!=5!$$.

On the other hand Michael and Bobby in 2! ways --> total $$5!*2!=240$$.

Similar question also posted by you: ways-to-sit-around-the-table-102187.html
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Re: Arrangements around the table  [#permalink]

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04 Oct 2010, 08:01
2
Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Are you sure you got the right OA on this one ?

As I see it : 7 members, and Michael & Bobby sit together. So treat those two like a single person. Now you have to arrange 6 people around a table. This may be done in 5! or 120 ways

The number of ways to arrange Michael & Bobby is 2 (MB or BM).

So net there are 240 ways or B

I don't think this is a GMAT level question, may be a tad too hard
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Re: Seven family members are seated around their circular dinner  [#permalink]

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29 Jun 2016, 01:25
1
Seven family members are seated around their circular dinner table. If only arrangements that are considered distinct are those where family members are seated in different locations relative to each other, and Michael and Bobby insist on sitting next to one another, then how many distinct arrangements around the table are possible?

A. 120
B. 240
C. 360
D. 480
E. 720

Lets consider Michael and Bobby as one individual and fix their position so that all the members do NOT move together while they remain in same order relatively

Now after fixing Michael and Bobby we have 5 other member left to change their positions among themselves which can change positions in 5! ways

but Michael and Bobby and exchange positions between the two in 2! ways'

Hence, Total ways of different arrangements = 5!*2! = 120*2 = 240

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Re: Seven family members are seated around their circular dinner   [#permalink] 25 Nov 2017, 07:33
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