title: Construction of lattice rules for multiple integration based on a weighted discrepancy

authors: Sinescu, Vasile

subjects: Lattice rules

abstract: High-dimensional integrals arise in a variety of areas, including quantum physics, the physics and chemistry of molecules, statistical mechanics and more recently, in financial applications.
In order to approximate multidimensional integrals, one may use Monte Carlo methods in which the quadrature points are generated randomly or quasi-Monte Carlo methods, in which points are generated deterministically. One particular class of quasi-Monte Carlo methods for multivariate integration is represented by lattice rules. Lattice rules constructed throughout this thesis allow good approximations to integrals of functions belonging to certain
weighted function spaces. These function spaces were proposed as an explanation as to why integrals in many variables appear to be successfully approximated although the standard theory indicates that the number of quadrature
points required for reasonable accuracy would be astronomical because of the large number of variables.
The purpose of this thesis is to contribute to theoretical results regarding the construction of lattice rules for multiple integration. We consider both lattice rules for integrals over the unit cube and lattice rules suitable for integrals over Euclidean space. The research reported throughout the thesis is devoted to finding the generating vector required to produce lattice rules that have what is termed a  low weighted discrepancy . In simple terms, the  discrepancy  is a measure of the uniformity of the distribution of the quadrature points or in other settings, a worst-case error. One of the assumptions used in these weighted function spaces is that variables are arranged in the decreasing order of their importance and the assignment of weights in this situation results in so-called  product weights . In other applications it is rather the importance of group of variables that matters. This situation is modelled by using function spaces in which the weights are  general . In the weighted
settings mentioned above, the quality of the lattice rules is assessed by the  weighted discrepancy  mentioned earlier. Under appropriate conditions on the weights, the lattice rules constructed here produce a convergence rate of the error that ranges 
from O(n−1/2) to the (believed) optimal O(n−1+δ) for any δ gt 0, with the involved constant independent of the dimension.

text: Construction of lattice rules for multiple integration based on a weighted discrepancy High-dimensional integrals arise in a variety of areas, including quantum physics, the physics and chemistry of molecules, statistical mechanics and more recently, in financial applications.
In order to approximate multidimensional integrals, one may use Monte Carlo methods in which the quadrature points are generated randomly or quasi-Monte Carlo methods, in which points are generated deterministically. One particular class of quasi-Monte Carlo methods for multivariate integration is represented by lattice rules. Lattice rules constructed throughout this thesis allow good approximations to integrals of functions belonging to certain
weighted function spaces. These function spaces were proposed as an explanation as to why integrals in many variables appear to be successfully approximated although the standard theory indicates that the number of quadrature
points required for reasonable accuracy would be astronomical because of the large number of variables.
The purpose of this thesis is to contribute to theoretical results regarding the construction of lattice rules for multiple integration. We consider both lattice rules for integrals over the unit cube and lattice rules suitable for integrals over Euclidean space. The research reported throughout the thesis is devoted to finding the generating vector required to produce lattice rules that have what is termed a  low weighted discrepancy . In simple terms, the  discrepancy  is a measure of the uniformity of the distribution of the quadrature points or in other settings, a worst-case error. One of the assumptions used in these weighted function spaces is that variables are arranged in the decreasing order of their importance and the assignment of weights in this situation results in so-called  product weights . In other applications it is rather the importance of group of variables that matters. This situation is modelled by using function spaces in which the weights are  general . In the weighted
settings mentioned above, the quality of the lattice rules is assessed by the  weighted discrepancy  mentioned earlier. Under appropriate conditions on the weights, the lattice rules constructed here produce a convergence rate of the error that ranges 
from O(n−1/2) to the (believed) optimal O(n−1+δ) for any δ gt 0, with the involved constant independent of the dimension.

year: 2008

title: Female suicidal behaviour: Initiation, cessation and prevention

authors: Curtis, Cate

abstract: This thesis explores non-fatal suicidal behaviour in young women. The approach taken is qualitative in orientation. The central form of data collection was the semi-structured interview. Interviews were conducted with key informants, including counsellors, clinicians and social workers and with women who had engaged in suicidal behaviour. The primary focus of the research was to explore the narratives of women who had engaged in suicidal behaviour, to contextualise their insights, understand their experiences and to examine the meanings of events leading to, and implicated in the recovery from, suicidal behaviour. This material demonstrates the importance of gender in suicidal behaviour and cessation as well as highlighting the limitations of many available therapeutic settings. A key finding of the research is the confirmation of sexual abuse as a common precursor to suicidal behaviour, especially when in conjunction with other, mainly familial, risk factors and an absence of protective factors. More significantly, the women noted that the effects of sexual abuse were exacerbated by problems with disclosure. Issues of control emerged as key to both suicidal behaviour and cessation, relating to family circumstance, abuse, disclosure of abuse and efficacy of forms of intervention. In contrast to the literature which suggests that any movement of self-harming behaviours is along a continuum from the less to more severe, the data gathered in this research suggests an episodic, or punctuated, pattern of behaviour, in which the protagonist moves between self-mutilation and suicidal behaviour. These findings problematise the notion of self-mutilation and suicidal behaviour as somewhat distinct behaviours. More generally, the material gathered from the women emphasizes the multi-causal and complex aspects of suicidal behaviour. The complexity of lived experience has implications for effective intervention and prevention strategies. Many of the women stressed the problematic nature of the forms of intervention they were able to access. Indeed, some regarded their experiences of intervention as reinforcing their feelings of lack of control This was largely confirmed by analysis of the interviews with key informants.

text: Female suicidal behaviour: Initiation, cessation and prevention This thesis explores non-fatal suicidal behaviour in young women. The approach taken is qualitative in orientation. The central form of data collection was the semi-structured interview. Interviews were conducted with key informants, including counsellors, clinicians and social workers and with women who had engaged in suicidal behaviour. The primary focus of the research was to explore the narratives of women who had engaged in suicidal behaviour, to contextualise their insights, understand their experiences and to examine the meanings of events leading to, and implicated in the recovery from, suicidal behaviour. This material demonstrates the importance of gender in suicidal behaviour and cessation as well as highlighting the limitations of many available therapeutic settings. A key finding of the research is the confirmation of sexual abuse as a common precursor to suicidal behaviour, especially when in conjunction with other, mainly familial, risk factors and an absence of protective factors. More significantly, the women noted that the effects of sexual abuse were exacerbated by problems with disclosure. Issues of control emerged as key to both suicidal behaviour and cessation, relating to family circumstance, abuse, disclosure of abuse and efficacy of forms of intervention. In contrast to the literature which suggests that any movement of self-harming behaviours is along a continuum from the less to more severe, the data gathered in this research suggests an episodic, or punctuated, pattern of behaviour, in which the protagonist moves between self-mutilation and suicidal behaviour. These findings problematise the notion of self-mutilation and suicidal behaviour as somewhat distinct behaviours. More generally, the material gathered from the women emphasizes the multi-causal and complex aspects of suicidal behaviour. The complexity of lived experience has implications for effective intervention and prevention strategies. Many of the women stressed the problematic nature of the forms of intervention they were able to access. Indeed, some regarded their experiences of intervention as reinforcing their feelings of lack of control This was largely confirmed by analysis of the interviews with key informants.

year: 2003

title: Social consequences of Tūhoe migration: Voices from home in Te Urewera

authors: Nikora, Linda Waimarie

subjects: age effects

abstract: A previous study by Nikora, Guerin, Rua & Te Awekotuku (2004) of the social consequences of Tūhoe moving to the Waikato region found employment and a tertiary education to be the primary motivators. Tūhoe ‘movers’ remained in contact with those who remain in Te Urewera and retained a wish to return to their tribal homelands if presented with work opportunities or upon retirement. Through 29 intensive conversational interviews conducted by Tūhoe community researchers with Tūhoe people we explored the social consequences of migration upon those who had remained ‘at home’. Most participants recognised and strongly supported ‘movers’ to better themselves and find successful futures but hoped that they would one day return. An emerging theme was an apparent age-distribution gap of people between about 20 and 50 years of age in the Tuhoe homelands. This is concerning for a number of reasons.  The ‘age gap’ may potentially bring about a) a lack of good role-models for younger children; b) a lack of people to help out with heavy work; c) a gap in people to sustain traditional teaching models for children, and d) a lack of qualified people in trades and professions. The age gap also means that a whole middle-aged group that in other communities might be contributing funds for capital expenditures are not present in the community.

text: Social consequences of Tūhoe migration: Voices from home in Te Urewera A previous study by Nikora, Guerin, Rua & Te Awekotuku (2004) of the social consequences of Tūhoe moving to the Waikato region found employment and a tertiary education to be the primary motivators. Tūhoe ‘movers’ remained in contact with those who remain in Te Urewera and retained a wish to return to their tribal homelands if presented with work opportunities or upon retirement. Through 29 intensive conversational interviews conducted by Tūhoe community researchers with Tūhoe people we explored the social consequences of migration upon those who had remained ‘at home’. Most participants recognised and strongly supported ‘movers’ to better themselves and find successful futures but hoped that they would one day return. An emerging theme was an apparent age-distribution gap of people between about 20 and 50 years of age in the Tuhoe homelands. This is concerning for a number of reasons.  The ‘age gap’ may potentially bring about a) a lack of good role-models for younger children; b) a lack of people to help out with heavy work; c) a gap in people to sustain traditional teaching models for children, and d) a lack of qualified people in trades and professions. The age gap also means that a whole middle-aged group that in other communities might be contributing funds for capital expenditures are not present in the community.

year: 2008