scholarly journals Estimating Conditional Power for Sequential Monitoring of Covariate Adaptive Randomized Designs: The Fractional Brownian Motion Approach

2021 ◽  
Vol 5 (3) ◽  
pp. 114
Author(s):  
Yiping Yang ◽  
Hongjian Zhu ◽  
Dejian Lai
Keyword(s):  
Clinical Trials ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Independent Increment ◽  
Conditional Power ◽  
Test Statistics ◽  
Test Statistic ◽  
Adaptive Randomization ◽  
Error Terms

Conditional power based on classical Brownian motion (BM) has been widely used in sequential monitoring of clinical trials, including those with the covariate adaptive randomization design (CAR). Due to some uncontrollable factors, the sequential test statistics under CAR procedures may not satisfy the independent increment property of BM. We confirm the invalidation of BM when the error terms in the linear model with CAR design are not independent and identically distributed. To incorporate the possible correlation structure of the increment of the test statistic, we utilize the fractional Brownian motion (FBM). We conducted a comparative study of the conditional power under BM and FBM. It was found that the conditional power under FBM assumption was mostly higher than that under BM assumption when the Hurst exponent was greater than 0.5.

INTERIM ANALYSIS OF CLINICAL TRIALS: SIMULATION STUDIES OF CONDITIONAL POWER UNDER FRACTIONAL BROWNIAN MOTION

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950133
Author(s):  
VIVIAN J. HUANG ◽  
DEJIAN LAI
Keyword(s):  
Clinical Trials ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Interim Analysis ◽  
Treatment Effects ◽  
Statistical Hypothesis ◽  
Boundary Crossing ◽  
Conditional Power ◽  
Statistical Hypothesis Testing ◽  
Wide Range

Classical Brownian motion (BM) techniques for statistical monitoring of clinical trials have been widely used. The conditional power (CP) and [Formula: see text]-spending function-based boundary crossing probabilities are popular procedures for statistical hypothesis testing under the assumption of BM. However, in some clinical trials, the assumptions of BM may not be fully met for the design and data analysis. Therefore, a more general class of stochastic processes, fractional Brownian motion (FBM), was proposed in the literature to model the test statistics derived from interim analysis of clinical trials. To investigate the properties of FBM, in this paper, we simulated a wide range of FBM data, e.g. [Formula: see text] (BM) versus [Formula: see text] (FBM), with treatment effects versus without treatment effects. Then the performance of CP-based interim analysis was compared by assuming that the data follow BM or FBM. Our simulation study suggested that CP under the FBM assumptions was generally higher than that under the BM assumptions when [Formula: see text] and also matched well with the empirical results.

scholarly journals GENERALIZED (m, k)-Zipf LAW FOR FRACTIONAL BROWNIAN MOTION-LIKE TIME SERIES WITH OR WITHOUT EFFECT OF AN ADDITIONAL LINEAR TREND

2003 ◽  
Vol 14 (03) ◽  
pp. 351-365 ◽  
Author(s):  
PH. BRONLET ◽  
M. AUSLOOS
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Size Effects ◽  
Hurst Exponent ◽  
Linear Trend ◽  
Finite Size ◽  
Time Dependent ◽  
Finite Size Effects ◽  
The Relationship ◽  
Zipf Law

We have translated fractional Brownian motion (FBM) signals into a text based on two "letters", as if the signal fluctuations correspond to a constant stepsize random walk. We have applied the Zipf method to extract the ζ′ exponent relating the word frequency and its rank on a log–log plot. We have studied the variation of the Zipf exponent(s) giving the relationship between the frequency of occurrence of words of length m < 8 made of such two letters: ζ′ is varying as a power law in terms of m. We have also searched how the ζ′ exponent of the Zipf law is influenced by a linear trend and the resulting effect of its slope. We can distinguish finite size effects, and results depending whether the starting FBM is persistent or not, i.e., depending on the FBM Hurst exponent H. It seems then numerically proven that the Zipf exponent of a persistent signal is more influenced by the trend than that of an antipersistent signal. It appears that the conjectured law ζ′ = |2H - 1| only holds near H = 0.5. We have also introduced considerations based on the notion of a time dependent Zipf law along the signal.

scholarly journals Spectral characteristics of high latitude raw 40 MHz cosmic noise signals

2015 ◽  
Vol 2 (4) ◽  
pp. 969-987
Author(s):  
C. M. Hall
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Spectral Characteristics ◽  
Noise Signal ◽  
Density Profiles ◽  
Generalized Hurst Exponent ◽  
Electron Density Profiles ◽  
Cosmic Noise ◽  
Non Gaussian

Abstract. Cosmic noise at 40 MHz is measured at Ny-Ålesund (79° N, 12° E) using a relative ionospheric opacity meter ("riometer"). A riometer is normally used to determine the degree to which cosmic noise is absorbed by the intervening ionosphere, giving an indication of ionization of the atmosphere at altitudes lower than generally monitored by other instruments. The usual course is to determine a "quiet-day" variation, this representing the galactic noise signal itself in the absence of absorption; the current signal is then subtracted from this to arrive at absorption expressed in dB. By a variety of means and assumptions, it is thereafter possible to estimate electron density profiles in the very lowest reaches of the ionosphere. Here however, the entire signal, i.e. including the cosmic noise itself will be examined and spectral characteristics identified. It will be seen that distinct spectral subranges are evident which can, in turn be identified with non-Gaussian processes characterized by generalized Hurst exponents, α. Considering all periods greater than 1 h, α &amp;approx; 1.24 – an indication of fractional Brownian motion, whereas for periods greater than 1 day α &amp;approx; 0.9 – approximately pink noise and just in the domain of fractional Gaussian noise. The results are compared with other physical processes suggesting that absorption of cosmic noise is characterized by a generalized Hurst exponent &amp;approx; 1.24 and thus non-persistent fractional Brownian motion, whereas generation of cosmic noise is characterized by a generalized Hurst exponent &amp;approx; 1.

scholarly journals On some statistics of fractional Brownian motion

2021 ◽  
pp. 131-138
Author(s):  
Viktor Bondarenko
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Limit Theorems ◽  
Goodness Of Fit ◽  
Mean Square ◽  
One Step ◽  
Optimal Forecasting

Fractional Brownian motion as a method for estimating the parameters of a stochastic process by variance and one-step increment covariance is proposed and substantiated. The root-mean-square consistency of the constructed estimates has been proven. The obtained results complement and generalize the consequences of limit theorems for fractional Brownian motion, that have been proved in the number of articles. The necessity to estimate the variance is caused by the absence of a base unit of time and the estimation of the covariance allows one to determine the Hurst exponent. The established results let the known limit theorems to be used to construct goodness-of-fit criteria for the hypothesis “the observed time series is a transformation of fractional Brownian motion” and to estimate the error of optimal forecasting for time series.

Effects of Noise on the Approximate Entropy of Fractional Brownian Motion Sequence

2013 ◽  
Vol 444-445 ◽  
pp. 698-702
Author(s):  
Xu Yi Hu ◽  
Li Wan ◽  
Dan Ying Xie
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Approximate Entropy ◽  
Poisson Noise ◽  
System Complexity ◽  
Sequence Change ◽  
Noise Type ◽  
Motion Sequence

Approximate entropy is a widely used technique to measure system complexity or regularity. In this paper, the effects of noise on the approximate entropy of fractional Brownian motion were investigated by some factors including the value of Hurst exponent, different noise type and coefficients. The results show that the values of approximate entropy of fractional Brownian motion decrease with the increase of Hurst exponent. The values increase in different degree after adding white noise in the sequence of fractional Brownian motion, and tend to be stable with the data lengthened. Meanwhile, the values of approximate entropy of mixed sequence change obviously by adding Poisson noise, while multiplying the coefficients of Poisson noise, the effects on the approximate entropy become greater.

scholarly journals On the Solution of Fractional Option Pricing Model by Convolution Theorem

2019 ◽  
pp. 143-157
Author(s):  
A. I. Chukwunezu ◽  
B. O. Osu ◽  
C. Olunkwa ◽  
C. N. Obi
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Long Memory ◽  
Hurst Exponent ◽  
Convolution Theorem ◽  
Pricing Model ◽  
One Dimensional ◽  
Black Scholes ◽  
Option Pricing Model

The classical Black-Scholes equation driven by Brownian motion has no memory, therefore it is proper to replace the Brownian motion with fractional Brownian motion (FBM) which has long-memory due to the presence of the Hurst exponent. In this paper, the option pricing equation modeled by fractional Brownian motion is obtained. It is further reduced to a one-dimensional heat equation using Fourier transform and then a solution is obtained by applying the convolution theorem.

scholarly journals Complexity signatures in the geomagnetic H component recorded by the Tromsø magnetometer (70° N, 19° E) over the last ¼ century

2014 ◽  
Vol 1 (1) ◽  
pp. 895-915
Author(s):  
C. M. Hall
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Geomagnetic Latitude ◽  
Horizontal Component ◽  
Fluctuation Analysis ◽  
Gaussian Distributions ◽  
Generalized Hurst Exponent ◽  
Solar Disturbances ◽  
Detrended Fluctuation

Abstract. Solar disturbances, depending on the orientation of the interplanetary magnetic field, typically result in perturbations of the geomagnetic field as observed by magnetometers on the ground. Here, the geomagnetic field's horizontal component, as measured by the ground-based observatory-standard magnetometer at Tromsø (70° N, 19° E) is examined for signatures of complexity. 25 year-long 10 s resolution datasets are analysed, but for fluctuations with timescales less than 1 day. Quantile-quantile (Q-Q) plots are employed first, revealing the fluctuations are better represented by Cauchy rather than Gaussian distributions. Thereafter, both spectral density and detrended fluctuation analysis methods are used to estimate values of the generalized Hurst exponent, α. The results are then compared with independent findings. Inspection and comparison of the spectral and detrended fluctuation analyses reveals that timescales between 1 h and 1 d are characterized by fractional Brownian motion with a generalized Hurst exponent of ~1.4 whereas including timescales as short as 1 min suggests fractional Brownian motion with a generalized Hurst exponent of ~1.6. This is consistent with changes in the position of the auroral electrojet that can be considered rapid during the course of an evening, whereas from minute-to-minute the electrojet moves more persistently in geomagnetic latitude.

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