The expectation-maximization (EM) algorithm uses incomplete data to estimate the parameters of probabilistic models, and it has been widely used in machine learning. In this paper, EM techniques are applied to Time-domain fluorescence lifetime imaging microscopy (FLIM) systems for estimating fluorescence lifetimes without measuring the instrument response functions (IRF). The results of Monte Carlo simulations indicate that the proposed approach can obtain comparable or better accuracy and precision performances than the previously reported method.
Introduction: Time-correlated single-photon counting (TCSPC) has excellent timing performances, and it is routinely used for fluorescence lifetime imaging microscopy (FLIM) system [1, 2]. FLIM
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This method is widely used for parameter estimation with incomplete or missing data [10, 11, 12]. In this paper, a new EM-based Lifetime Estimation (EMLE) algorithm is proposed to simultaneously estimate the IRF and lifetime, and it shows better photon efficiency compared with EKF in [9].
Theory: According to the EM theory [10, 11, 12], we assume that {xi, i = 0, 1, …, N-1} are observation values of a random variable x whose density function shown in Fig.1(a) is , (1) where f(x|ξi) is the density function of ith component with a parameter ξi , λj is the component weight satisfying Σλj= 1, j = 0,1, …, L-1, and ϕ = [ξ0, λ0, ξ1, λ1,, ξL-1, λL-1]. ξi and λj can be estimated using {x0, x0, …, xN-1}. An expectation step (E-step) and a maximization step (M-step) are performed iteratively [10, 11], shown in Fig. 1(b).
In E-step, the posterior probability pij is . (2)
In M-step, ϕ that maximizes the expected log-likelihood in E-step can be calculated from , (3) where ϕi = [ξi, λi] is the parameter for the ith component.
a b
Fig. 1 Overview of EMLE a Mixture of the density function b EMLE processing flow.
In a TCSPC-FLIM experiment, we assume that the fluorescence density function is g(t), the IRF is IRF(t), and the measured fluorescence decay is y(t). y(t) is the sum of an additive Poisson noise, v(t) and the convolution of g(t) and
"This course is intended to highlight mathematical principles, concepts, and techniques that are often used in scientific applications and illustrate how these techniques are employed in the context of specific problems in physics, chemistry, and biology. Topics include mathematical
MLE can sometimes result in parameter estimates of zero, if the data does not happen to contain any training samples satisfying the condition in the numerator. To avoid this, it is common to use a “smoothed” estimate which effectively adds in a number of additional “hallucinated” samples, and which assumes these hallucinated examples are spread evenly over the possible values of $X^j$, or equivalently a MAP estimate
The trace statistics ʎ trace and the maximum Eigen statistics ʎ max were used and the results are presented in table 3 and 4 below.
The model parameters are estimated from the EP and therefore the AR can be calculated within the TP (Strong, 1992). Explicitly, the AR which
The remaining results used to obtain the graph in the next section can be obtained by, iteratively substituting the parameters shown in table 3 below for the various architectures and various population sizes. The system parameters given in [16] is shown in table 3.
The objective in Lab 8 is to measure wavelength of five emission lines of light and a laser beam through the principle of light interference.
The expected value approach is used to determine the expected value with perfect information (EVPI). The EVPI is obtained by subtracting the expected value without perfect information (EMV) from the EVPI.
C. An unknown, rectangular substance measures 3.6 cm high, 4.21 cm long, and 1.17 cm wide.
BSP, S. (2010). How is EM different from light microscopy? Retrieved April 25, 2015, from http://bsp.med.harvard.edu/node/222
The data sets for problems 5 and 6 can be found through the Pearson Materials in the Student Textbook Resource Access link, listed under Academic Resources. The data is listed in the data file named Lesson 20 Exercise File 1. Answer Exercises 5 and 6 based on the following research problem:
Following that, the expected values for decision nodes 6 and 7 should also be calculated. The following results were obtained:
Using a statistical model they created (See Appendix), Entine and Small ended up with the following results:
We select alpha1 and alpha2 that make the largest progress towards the global maximum value on each side of the hyper plane according to the heuristic function. The heuristic function is as follow:
Estimating the mixing density of a mixture distribution remains an interesting problem in the statistics literature. Stochastic approximation (SA) provides a fast recursive way for numerically maximizing a function under measurement error. Using suitably chosen weight/step-size the stochastic approximation algorithm converges to the true solution, which can be adapted to estimate the components of the mixing distribution from a mixture, in the form of recursively learning, predictive recursion method. The convergence depends on a martingale construction and convergence of related series and heavily depends on the independence of the data. The general algorithm may not hold if dependence is present. We have proposed a novel martingale decomposition to address the case of dependent data.
Estimating the mixing density of a mixture distribution remains an interesting problem in the statistics literature. Stochastic approximation (SA) provides a fast recursive way for numerically maximizing a function under measurement error. Using suitably chosen weight/step-size the stochastic approximation algorithm converges to the true solution, which can be adapted to estimate the components of the mixing distribution from a mixture, in the form of recursively learning, predictive recursion method. The convergence depends on a martingale construction and convergence of related series and heavily depends on the independence of the data. The general algorithm may not hold if dependence is present. We have proposed a novel martingale decomposition to address the case of dependent data.