We present an approximated maximum likelihood method for the multifractal random walk processes of [E. Bacry et al., Phys. Rev. E 64, 026103 (2001)]. The likelihood is computed using a Laplace approximation and a truncation in the dependency structure for the latent volatility. The procedure is implemented as a package in the r computer language. Its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments. The method is applied to estimate parameters for various financial stock indices.

ObjectivesTo describe the prevalence of general (body mass index (BMI) ≥30 kg/m2) and abdominal (waist circumference women >88 cm, men >102 cm) obesity in Tromsø 7 (2015–2016), and the secular change from Tromsø 6 (2007–2008). Furthermore, to study longitudinal changes in body weight and waist circumference from Tromsø 6 to Tromsø 7.SettingA population study in Tromsø, Norway.ParticipantsThe cross-sectional analyses included 20 855 participants in Tromsø 7 (aged ≥40 years) and 12 868 in Tromsø 6 (aged ≥30 years). The longitudinal analyses included 8592 participants with repeated measurements, aged 35–79 in Tromsø 6.Outcome measuresMean age-specific and sex-specific BMI, waist circumference, prevalence of general and abdominal overweight and obesity, as well as longitudinal changes in body weight and waist circumference according to sex and birth cohort.ResultsOver 8 years, the age-adjusted prevalence of general obesity increased (p<0.0001) from 20.1% to 23.0% in women and from 20.7% to 25.2% in men. The age-adjusted prevalence of abdominal obesity did not increase in women (from 54.7% to 53.4%), and the increase in men was modest (from 36.8% to 38.6%, p=0.003). Longitudinal analyses showed an increase in body weight, by 1.1 kg (95% CI 0.9 to 1.2) in women and 0.7 kg (95% CI 0.6 to 0.9) in men, and also waist circumference, by 1.3 cm (95% CI 1.0 to 1.5) in women and 1.4 cm (95% CI 1.2 to 1.6) in men. There were inverse relationships (p<0.001) between age at baseline and change in weight and waist circumference.ConclusionsRepeated cross-sectional analyses showed that the prevalence of general obesity increased, whereas the increase in abdominal obesity was less marked. Longitudinal analyses showed increases in both body weight and waist circumference. The youngest age groups have the largest increase.

We discuss stochastic modeling of volatility persistence and anti-correlations in electricity spot prices, and for this purpose we present two mean-reverting versions of the multifractal random walk (MRW). In the first model the anti-correlations are modeled in the same way as in an Ornstein-Uhlenbeck process, i.e. via a drift (damping) term, and in the second model the anti-correlations are included by letting the innovations in the MRW model be fractional Gaussian noise with H < 1/2. For both models we present approximate maximum likelihood methods, and we apply these methods to estimate the parameters for the spot prices in the Nordic electricity market. The maximum likelihood estimates show that electricity spot prices are characterized by scaling exponents that are significantly different from the corresponding exponents in stock markets, confirming the exceptional nature of the electricity market. In order to compare the damped MRW model with the fractional MRW model we use ensemble simulations and wavelet-based variograms, and we observe that certain features of the spot prices are better described by the damped MRW model. The characteristic correlation time is estimated to approximately half a year.

The scaling function F(s) in detrended fluctuation analysis (DFA) scales as F(s)∼s^{H} for stochastic processes with Hurst exponent H. This scaling law is proven for stationary stochastic processes with 0

This paper studies regional climate variability for the time period 1900–2013 using parsimonious stochastic models. Instrumental data records on 5° × 5°, 2° × 2°, and equal-area grids are examined. A long-range dependent (LRD) stochastic process is used as a simplified description of the multitude of response times in the climate system. Fitting a linear trend to the global mean surface temperature (GMST) implies a warming of 0.08 decade−1, which is highly significant under an LRD null hypothesis (p &lt; 10−4). The regional trends are distributed around the global mean trend, while the fluctuation levels increases when going from global to regional scale. The temperature fluctuations of the tropical oceans are observed to be strongly influenced by El Niño–Southern Oscillation (ENSO) and, therefore, more consistent with autoregressive processes of order 1 [AR(1)]. A likelihood-ratio test is used to systematically determine the best null model [AR(1) or LRD]. About 80% of the regional warming trends are found to be significant (with a 5% significance level).