1,960 research outputs found
A statistical analysis of particle trajectories in living cells
Recent advances in molecular biology and fluorescence microscopy imaging have
made possible the inference of the dynamics of single molecules in living
cells. Such inference allows to determine the organization and function of the
cell. The trajectories of particles in the cells, computed with tracking
algorithms, can be modelled with diffusion processes. Three types of diffusion
are considered : (i) free diffusion; (ii) subdiffusion or (iii) superdiffusion.
The Mean Square Displacement (MSD) is generally used to determine the different
types of dynamics of the particles in living cells (Qian, Sheetz and Elson
1991). We propose here a non-parametric three-decision test as an alternative
to the MSD method. The rejection of the null hypothesis -- free diffusion -- is
accompanied by claims of the direction of the alternative (subdiffusion or a
superdiffusion). We study the asymptotic behaviour of the test statistic under
the null hypothesis, and under parametric alternatives which are currently
considered in the biophysics literature, (Monnier et al,2012) for example. In
addition, we adapt the procedure of Benjamini and Hochberg (2000) to fit with
the three-decision test setting, in order to apply the test procedure to a
collection of independent trajectories. The performance of our procedure is
much better than the MSD method as confirmed by Monte Carlo experiments. The
method is demonstrated on real data sets corresponding to protein dynamics
observed in fluorescence microscopy.Comment: Revised introduction. A clearer and shorter description of the model
(section 2
Quickest detection in coupled systems
This work considers the problem of quickest detection of signals in a coupled
system of sensors, which receive continuous sequential observations from
the environment. It is assumed that the signals, which are modeled by general
It\^{o} processes, are coupled across sensors, but that their onset times may
differ from sensor to sensor. Two main cases are considered; in the first one
signal strengths are the same across sensors while in the second one they
differ by a constant. The objective is the optimal detection of the first time
at which any sensor in the system receives a signal. The problem is formulated
as a stochastic optimization problem in which an extended minimal
Kullback-Leibler divergence criterion is used as a measure of detection delay,
with a constraint on the mean time to the first false alarm. The case in which
the sensors employ cumulative sum (CUSUM) strategies is considered, and it is
proved that the minimum of CUSUMs is asymptotically optimal as the mean
time to the first false alarm increases without bound. In particular, in the
case of equal signal strengths across sensors, it is seen that the difference
in detection delay of the -CUSUM stopping rule and the unknown optimal
stopping scheme tends to a constant related to the number of sensors as the
mean time to the first false alarm increases without bound. Alternatively, in
the case of unequal signal strengths, it is seen that this difference tends to
zero.Comment: 29 pages. SIAM Journal on Control and Optimization, forthcomin
Sequential Testing with Uniformly Distributed Size
Sequential procedures of testing for structural stability do not provide enough guidance on the shape of boundaries that are used to decide on acceptance or rejection, requiring only that the overall size of the test is asymptotically controlled. We introduce and motivate a reasonable criterion for a shape of boundaries which requires that the test size be uniformly distributed over the testing period. Under this criterion, we numerically construct boundaries for most popular sequential tests that are characterized by a test statistic behaving asymptotically either as a Wiener process or Brownian bridge. We handle this problem both in a context of retrospecting a historical sample and in a context of monitoring newly arriving data. We tabulate the boundaries by Â
tting them to certain exible but parsimonious functional forms. Interesting patterns emerge in an illustrative application of sequential tests to the Phillips curve model.Structural stability; sequential tests; CUSUM; retrospection; monitoring; boundaries; asymptotic size
A range unit root test
Since the seminal paper by Dickey and Fuller in 1979, unit-root tests have conditioned the standard approaches to analyse time series with strong serial dependence, the focus being placed in the detection of eventual unit roots in an autorregresive model fitted to the series. In this paper we propose a completely different method to test for the type of long-wave patterns observed not only in unit root time series but also in series following more complex data generating mechanisms. To this end, our testing device analyses the trend exhibit by the data, without imposing any constraint on the generating mechanism. We call our device the Range Unit Root (RUR) Test since it is constructed from running ranges of the series. These statistics allow a more general characterization of a strong serial dependence in the mean behavior, thus endowing our test with a number of desirable properties, among which its error-model-free asymptotic distribution, the invariance to nonlinear monotonic transformations of the series and the robustness to the presence of level shifts and additive outliers. In addition, the RUR test outperforms the power of standard unit root tests on near-unit-root stationary time series and is asymptotically immune to noise
Power of Change-Point Tests for Long-Range Dependent Data
We investigate the power of the CUSUM test and the Wilcoxon change-point test
for a shift in the mean of a process with long-range dependent noise. We derive
analytiv formulas for the power of these tests under local alternatives. These
results enable us to calculate the asymptotic relative efficiency (ARE) of the
CUSUM test and the Wilcoxon change point test. We obtain the surprising result
that for Gaussian data, the ARE of these two tests equals 1, in contrast to the
case of i.i.d. noise when the ARE is known to be
A Binary Control Chart to Detect Small Jumps
The classic N p chart gives a signal if the number of successes in a sequence
of inde- pendent binary variables exceeds a control limit. Motivated by
engineering applications in industrial image processing and, to some extent,
financial statistics, we study a simple modification of this chart, which uses
only the most recent observations. Our aim is to construct a control chart for
detecting a shift of an unknown size, allowing for an unknown distribution of
the error terms. Simulation studies indicate that the proposed chart is su-
perior in terms of out-of-control average run length, when one is interest in
the detection of very small shifts. We provide a (functional) central limit
theorem under a change-point model with local alternatives which explains that
unexpected and interesting behavior. Since real observations are often not
independent, the question arises whether these re- sults still hold true for
the dependent case. Indeed, our asymptotic results work under the fairly
general condition that the observations form a martingale difference array.
This enlarges the applicability of our results considerably, firstly, to a
large class time series models, and, secondly, to locally dependent image data,
as we demonstrate by an example
Power of change-point tests for long-range dependent data
We investigate the power of the CUSUM test and the Wilcoxon change-point tests for a shift in the mean of a process with long-range dependent noise. We derive analytic formulas for the power of these tests under local alternatives. These results enable us to calculate the asymptotic relative efficiency (ARE) of the CUSUM test and the Wilcoxon change point test. We obtain the surprising result that for Gaussian data, the ARE of these two tests equals 1, in contrast to the case of i.i.d. noise when the ARE is known to be 3/Ï.Herold Dehling and Aeneas Rooch were supported in part by the German Research Foundation (DFG) through the Collaborative Research Center SFB 823 Statistical Modelling of Nonlinear Dynamic Processes. Murad S. Taqqu was supported in part by NSF grant DMS-1309009 at Boston University. (SFB 823 - German Research Foundation (DFG); DMS-1309009 - NSF at Boston University)Published versio
- âŠ