Statistical properties of complex systems -lognormal and related distributions-

Precise valuation of the economic efficiency of risky investment projects in the mineral sector has a direct impact on the range of future investments. Since the mid-90s, a number of enterprises have also been giving increased attention to the valuation of managerial flexibility that cannot normally be estimated with classical discounted cash flow (DCF) analysis. This has been the result of a development in the real options analysis (ROA) and the simplification of its algorithms, most of which have been achieved through: ♦ incorporating lattice models, ♦ introducing a single uncertain project parameter (gross present value, PV) as an underlying instrument, ♦ assuming that the underlying asset follows the multiplicative stochastic process, ♦ introducing the ‘marketed asset disclaimer’ (MAD) assumption. Unfortunately, in most cases, models constructed on the abovementioned assumptions and modifications are not consistent with real projects. Some analysts recognize that project PVs might not follow the multiplicative process, which could have a direct impact on the project’s value. In order to improve the MAD approach, the paper proposes a modified model where the multiplicative tree is replaced with an additive one. In addition, methods of ‘additive volatility’ calculation and ‘dividend’ adjustments were suggested.

Some approaches in statistical physics have been applied to analyzing economic and social events. Recently, Kobayashi, Kuninaka et al. reviewed the statistical properties of several social and biological phenomena as ‘‘complex systems’’ from the viewpoint of stochastic processes. In their review, the population size distribution of municipalities in Japan was treated as one example. They stated that the log-normal distribution is basic for complex systems, which are generally described as a multiplicative random growth given by the equation xðt þ 1Þ xðtÞ 1⁄4 ðtÞxðtÞ. Here, xðtÞ is the population size at time t for example, and ðtÞ is the growth rate, which is a random variable. This property is well known as Gibrat’s law. For the population size distribution, the following feature has been recognized: the major part of the distribution obeys the log-normality, but the tail part corresponding to a large size exhibits the power law. Sasaki, Kuninaka et al. classified Japanese municipalities into three types (i.e., village, town, and city) and found that villages and cities are fitted with the log-normal and power-law distributions, respectively. Their conclusion was that the difference in these distributions is due to the existence of a lower population threshold in cities, but not in villages. The power-law distribution is reproducible by an additional effect to the above multiplicative random growth. Particularly, the following models have been applied to the Japanese case: (i) Random growth model: This model treats multiplicative random growth with an additive random noise, namely, the Kesten process, which possesses a powerlaw property. (ii) Migration model: There are two approaches focusing on population migration. (ii-a) Tomita and Hayashi adopted an urn model where preferential attachment in complex networks was applied. (ii-b) Sasaki and coworkers proposed another model and took population thresholds into account. Depending on the existence of such thresholds, the size distributions of villages, towns, and cities have been separately reproduced. It is noted that the emergence of the power law due to a threshold has already been reported in fragmentation process. In this short note, we review model (ii-b) with modification by focusing on the preferential migration effect, which is a different viewpoint from those in Refs. 9 and 15. Now, N sites representing each municipality are prepared. The time evolution of xjðtÞ, the size of the j-th site, is described by xjðt þ 1Þ xjðtÞ 1⁄4 XN

The fluctuation-spectrum analysis for long-time statistical characteristics of the multiplicative stochastic process described by the Langevin equation of motion drt / dt =( Δ -β rt2 + ft ) rt , (β>0, Δ >0), ft being the Gaussian-white noise, is performed by rigorously solving the eigenvalue problem of the extended Fokker-Planck operator Hq . This is done by observing how the fluctuation of the coarse-grained average α t = t-1 \int 0trskds reduces as t →∞. Furthermore for Δ →0 the scaling behaviors of characteristic functions relevant to the fluctuation-spectrum approach are discussed.

Content Propagation in Online Social Networks

Empirical analysis of Zipf’s law, power law, and lognormal distributions in medical discharge reports

of catching r individuals of any given species, m being the expected number. Here m may be called the abundance of the species concerned, but it must be recognized that this means abundance relative to both the intensity and the method of sampling. As in the paper of Fisher, Corbet & Williams (1943), the structure of the population may be investigated by studying the numbers of species represented in the sample by various numbers of individuals. The expected number of species with r individuals in the sample is nr = S(e-mmr/r!), (1) where S denotes summation over all species in the population. In practice the number of species is large, and for theoretical purposes the frequency distribution of their abundances is taken to be continuous. Three such distributions which have been suggested are: the Eulerian or Type III distribution a e-mf me-i dm, the limit of this as e -* 0, and the log-normal distribution. The first of these alternatives leads to a negative binomial series of nr, and the second to the logarithmic series investigated by Fisher et al. (1943); on the properties of these and related distributions, cf. Anscombe (1950). Observational evidence in favour of the normal distribution of log n has been adduced by Preston (1948); in the same paper he has worked out some examples of expected sample frequencies, using a discrete approximation to the normal distribution. His results indicate that the expected sample frequencies, grouped along the scale of r by octaves, conform well to the population curve after the first few octaves. The object of this paper is to give tables and formulae for calculation of the n, when m has a given log-normal distribution. Information of the kind thus provided, especially regarding the frequencies of rare species, is becoming increasingly necessary for the comparison of theory with the observed results as these accumulate. It is convenient to work with the

Long-tailed distributions in biological systems have been studied. First, we found that lognormal distributions show excellent fit with various data for the duration distribution of disability in aged people, irrespective of their severity and gender. The robust lognormal distribution of disability implies that the incidence of diseases can be completed by many independent subprocesses in succession. Next, we studied food fragmentation by human mastication. A lognormal distribution well fits to the entire region for masticated food fragments for a small number of chewing strokes. Furthermore, the tail of the fragmentsize distribution changes from the lognormal distribution to a power-law one as the chewing stroke number increases. The good data fitting by the lognormal and power-law distribution implies that two functions of mastication, a sequential fragmentation with cascade and randomness and a lower threshold for fragment size, may affect the size distribution of masticated food fragments.

Is the cross-sectional distribution of house prices close to a (log) normal distribution, as is often assumed in empirical studies on house price indexes? How does the distribution evolve over time? To address these questions, we investigate the cross-sectional distribution of house prices in the Greater Tokyo Area for the period 1986 to 2009. We find that size-adjusted house prices follow a lognormal distribution except for the period of the housing bubble and its collapse in Tokyo, for which the price distribution has a substantially heavier right tail than that of a lognormal distribution. In addition, we find that, during the bubble era, the sharp price movements were concentrated in particular areas, and this spatial heterogeneity is the source of the fat upper tail. These findings suggest that the shape of the size-adjusted price distribution, especially the shape of the tail part, may contain information useful for the detection of housing bubbles. Specifically, the presence of a bubble can be safely ruled out if recent price observations are found to follow a lognormal distribution. On the other hand, if there are many outliers, especially near the upper tail, this may indicate the presence of a bubble, since such price observations are unlikely to occur if they follow a lognormal distribution. This method of identifying bubbles is quite different from conventional ones based on aggregate measures of housing prices, and therefore should be a useful tool to supplement existing methods.

Power Laws in Real Estate Prices? Some Evidence

Power Laws in Real Estate Prices? Some Evidence

We present event distributions for the polymer translocation obtained by extensive Langevin dynamics simulations. Such distributions have not been reported previously and they provide new understanding of the stochastic characteristics of the process. We extract at a high length scale resolution distributions of polymer segments that continuously traverse through a nanoscale pore. The obtained log-normal distributions together with the characteristics of polymer translocation suggest that it is describable as a multiplicative stochastic process. In spite of its clear out-of-equilibrium nature the forced translocation is surprisingly similar to the unforced case. We find forms for the distributions almost unaltered with a common cut-off length. We show that the individual short-segment and short-time movements inside the pore give the scaling relations τ∼Nα and τ∼f-β for the polymer translocation.

The author presents a study on the development of approximate theoretical forms of the distributions of wave steepness and surf parameter. The approach is based on assuming the random wave process to be long crested and narrow banded. The results are compared with data from measurements at sea representing two severe storms. Both parameters are lognormal distributed. The resulting statistics for the surf parameter are applied to breakers at normal incidence on sloping beaches. As stated by the author, the joint statistics of wave steepness with wave heights or crest heights, or the wave steepness with wave heights or crest heights above a specified threshold may be more appropriate in formulating risks of capsizing of vessels, overtopping, and slamming forces on seawalls, etc. However, the wave steepness is of interest by itself, particularly in relation with many of the surf-zone processes. This discussion points out that other data sets may result in other distributions. This is exemplified by using the data referred to by the author—the data used by Myrhaug and Kjeldsen 1984, 1987 , Myrhaug and Rue 1993 , and Myrhaug and Kvalsvold 1995 . These papers discuss various aspects of wave steepness statistics using data from a large population of waves obtained by waverider buoys at three different deep water locations on the Norwegian continental shelf. Fig. 1 here shows the cdf of the normalized wave steepness s=S /Srms in Weibull scale, where Srms is as defined in Eq. 20 of the original paper. The pluses represent the Myrhaug and Kjeldsen 1984 data. The full line represents the combined lognormal and Weibull distribution obtained as a best fit by eye to the data; the pdf of the lognormal distribution is given by Eq. 17 of the original paper with , = −0.257,0.523 for s 1.2; the pdf of the Weibull distribution is given by Eq. 14 with , = 1.40,0.84 for s 1.2. It appears that this model gives a good representation of the data. Myrhaug and Rue 1993 used the Weibull model to study the statistics of two successive wave steepness parameters with the focus on steep waves S 0.10 . Myrhaug and Kjeldsen 1984 , Myrhaug and Kvalsvold 1995 , and Myrhaug 1994 discussed the joint distribution of wave height and wave steepness by taking f h ,s = f s h f h where f s h is given by the lognormal distribution in Myrhaug and Kjeldsen 1984, Eq. 4 , with x=s, Eq. 18 , and Eq. 19 , and f h is given by the Weibull distribution in Myrhaug and Kjeldsen

A Class of Flow Bifurcation Models with Lognormal Distribution and Fractal Dispersion

In this contribution, I address the function of multiplicative stochastic processes in modelling the occurrence of power-law city size distributions. As an explanation of the result of Zipf’s rank analysis, Simon’s model is presented in a mathematically elementary way, with a thorough discussion of the involved hypotheses. Emphasis is put on the flexibility of the model, as to its possible extensions and the relaxation of some strong assumptions. I point out some open problems regarding the prediction of the detailed shape of Zipf’s rank plots, which may be tackled by means of such extensions.KeywordsWord FrequencyUrban SystemEvolution StepUrban SettlementCity SizeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Galactic center black hole Sagittarius A* is a variable near-infrared (NIR) source that exhibits bright flux excursions called flares. When flux from Sgr A* is detected, the light curve has been shown to exhibit red noise characteristics and the distribution of flux densities is non-linear, non-Gaussian, and skewed to higher flux densities. However, the low-flux density turnover of the flux distribution is below the sensitivity of current single-aperture telescopes. For this reason, the median NIR flux has only been inferred indirectly from model fitting, but it has not been directly measured. In order to explore the lowest flux ranges, to measure the median flux density, and to test if the previously proposed flux distributions fit the data, we use the unprecedented resolution of the GRAVITY instrument at the VLTI. We obtain light curves using interferometric model fitting and coherent flux measurements. Our light curves are unconfused, overcoming the confusion limit of previous photometric studies. We analyze the light curves using standard statistical methods and obtain the flux distribution. We find that the flux distribution of Sgr A* turns over at a median flux density of (1.1 ± 0.3) mJy. We measure the percentiles of the flux distribution and use them to constrain the NIR K-band spectral energy distribution. Furthermore, we find that the flux distribution is intrinsically right-skewed to higher flux density in log space. Flux densities below 0.1 mJy are hardly ever observed. In consequence, a single powerlaw or lognormal distribution does not suffice to describe the observed flux distribution in its entirety. However, if one takes into account a power law component at high flux densities, a lognormal distribution can describe the lower end of the observed flux distribution. We confirm the rms–flux relation for Sgr A* and find it to be linear for all flux densities in our observation. We conclude that Sgr A* has two states: the bulk of the emission is generated in a lognormal process with a well-defined median flux density and this quiescent emission is supplemented by sporadic flares that create the observed power law extension of the flux distribution.