On Nonlinear Functionals of Random Spherical Eigenfunctions

We prove Central Limit Theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combine asymptotic analysis of higher order moments for Legendre polynomials and, in addition, recent results on Malliavin calculus and Total Variation bounds for Gaussian subordinated fields. We discuss application to geometric functionals like the Defect and invariant statistics, e.g. polyspectra of isotropic spherical random fields. Both of these have relevance for applications, especially in an astrophysical environment.Comment: 24 page

The defect variance of random spherical harmonics

The defect of a function $f:M\rightarrow \mathbb{R}$ is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is obtaining the asymptotic shape of the defect variance, in the high frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern Cosmological data analysis.Comment: 19 page

On the number of excursion sets of planar Gaussian fields

37 pages, 14 figures37 pages, 14 figures37 pages, 14 figure

The structure of paranoia in the general population

Background Psychotic phenomena appear to form a continuum with normal experience and beliefs, and may build on common emotional interpersonal concerns. Aims We tested predictions that paranoid ideation is exponentially distributed and hierarchically arranged in the general population, and that persecutory ideas build on more common cognitions of mistrust, interpersonal sensitivity and ideas of reference. Method Items were chosen from the Structured Clinical Interview for DSM-IV Axis II Disorders (SCID-II) questionnaire and the Psychosis Screening Questionnaire in the second British National Survey of Psychiatric Morbidity (n = 8580), to test a putative hierarchy of paranoid development using confirmatory factor analysis, latent class analysis and factor mixture modelling analysis. Results Different types of paranoid ideation ranged in frequency from less than 2% to nearly 30%. Total scores on these items followed an almost perfect exponential distribution (r = 0.99). Our four a priori first-order factors were corroborated (interpersonal sensitivity; mistrust; ideas of reference; ideas of persecution). These mapped onto four classes of individual respondents: a rare, severe, persecutory class with high endorsement of all item factors, including persecutory ideation; a quasi-normal class with infrequent endorsement of interpersonal sensitivity, mistrust and ideas of reference, and no ideas of persecution; and two intermediate classes, characterised respectively by relatively high endorsement of items relating to mistrust and to ideas of reference. Conclusions The paranoia continuum has implications for the aetiology, mechanisms and treatment of psychotic disorders, while confirming the lack of a clear distinction from normal experiences and processes

Are genetic risk factors for psychosis also associated with dimension-specific psychotic experiences in adolescence?

Psychosis has been hypothesised to be a continuously distributed quantitative phenotype and disorders such as schizophrenia and bipolar disorder represent its extreme manifestations. Evidence suggests that common genetic variants play an important role in liability to both schizophrenia and bipolar disorder. Here we tested the hypothesis that these common variants would also influence psychotic experiences measured dimensionally in adolescents in the general population. Our aim was to test whether schizophrenia and bipolar disorder polygenic risk scores (PRS), as well as specific single nucleotide polymorphisms (SNPs) previously identified as risk variants for schizophrenia, were associated with adolescent dimension-specific psychotic experiences. Self-reported Paranoia, Hallucinations, Cognitive Disorganisation, Grandiosity, Anhedonia, and Parent-rated Negative Symptoms, as measured by the Specific Psychotic Experiences Questionnaire (SPEQ), were assessed in a community sample of 2,152 16-year-olds. Polygenic risk scores were calculated using estimates of the log of odds ratios from the Psychiatric Genomics Consortium GWAS stage-1 mega-analysis of schizophrenia and bipolar disorder. The polygenic risk analyses yielded no significant associations between schizophrenia and bipolar disorder PRS and the SPEQ measures. The analyses on the 28 individual SNPs previously associated with schizophrenia found that two SNPs in TCF4 returned a significant association with the SPEQ Paranoia dimension, rs17512836 (p-value=2.57x10-4) and rs9960767 (p-value=6.23x10-4). Replication in an independent sample of 16-year-olds (N=3,427) assessed using the Psychotic-Like Symptoms Questionnaire (PLIKS-Q), a composite measure of multiple positive psychotic experiences, failed to yield significant results. Future research with PRS derived from larger samples, as well as larger adolescent validation samples, would improve the predictive power to test these hypotheses further. The challenges of relating adult clinical diagnostic constructs such as schizophrenia to adolescent psychotic experiences at a genetic level are discussed

Reduced P300 amplitude during retrieval on a spatial working memory task in a community sample of adolescents who report psychotic symptoms.

BACKGROUND: Deficits in working memory are widely reported in schizophrenia and are considered a trait marker for the disorder. Event-related potentials (ERPs) and imaging data suggest that these differences in working memory performance may be due to aberrant functioning in the prefrontal and parietal cortices. Research suggests that many of the same risk factors for schizophrenia are shared with individuals from the general population who report psychotic symptoms. METHODS: Forty-two participants (age range 11--13 years) were divided into those who reported psychotic symptoms (N = 17) and those who reported no psychotic symptoms, i.e. the control group (N = 25). Behavioural differences in accuracy and reaction time were explored between the groups as well as electrophysiological correlates of working memory using a Spatial Working Memory Task, which was a variant of the Sternberg paradigm. Specifically, differences in the P300 component were explored across load level (low load and high load), location (positive probe i.e. in the same location as shown in the study stimulus and negative probe i.e. in a different location to the study stimulus) and between groups for the overall P300 timeframe. The effect of load was also explored at early and late timeframes of the P300 component (250-430 ms and 430-750 ms respectively). RESULTS: No between-group differences in the behavioural data were observed. Reduced amplitude of the P300 component was observed in the psychotic symptoms group relative to the control group at posterior electrode sites. Amplitude of the P300 component was reduced at high load for the late P300 timeframe at electrode sites Pz and POz. CONCLUSIONS: These results identify neural correlates of neurocognitive dysfunction associated with population level psychotic symptoms and provide insights into ERP abnormalities associated with the extended psychosis phenotype