## Statistical properties of amplitude and decay parameter estimators for fluorescence lifetime imaging |

Optics Express, Vol. 21, Issue 5, pp. 6061-6075 (2013)

http://dx.doi.org/10.1364/OE.21.006061

Acrobat PDF (906 KB)

### Abstract

We analyze the statistical properties of the maximum likelihood estimator, least squares estimator, and Pearson’s *χ*^{2}-based and Neyman’s *χ*^{2}-based estimators for the estimation of decay constants and amplitudes for fluorescence lifetime imaging. Our analysis is based on the linearization of the gradient of the objective functions around true parameters. The analysis shows that only the maximum likelihood estimator based on the Poisson likelihood function yields unbiased and efficient estimation. All other estimators yield either biased or inefficient estimations. We validate our analysis by using simulations.

© 2013 OSA

## 1. Introduction

*χ*

^{2}and Neyman’s

*χ*

^{2}, that originate from approximations of the Poisson likelihood function to the Gaussian likelihood function [1

1. J. R. Lakowicz, *Principles of Fluorescence Spectroscopy* (Kluwer Academic/Plenum, 1999) [CrossRef] .

9. P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J. **82**, 2244–2255 (2002) [CrossRef] [PubMed] .

*χ*

^{2}-based measures (including the sum of square errors, which is

*χ*

^{2}for data having equal variances) have been frequently used to measure the goodness of fit in data analysis [10], and because it may look natural to measure the goodness of fit by a function that includes the sum of square errors between the model and data, as in the case with

*χ*

^{2}-based measures. However, since

*χ*

^{2}-based measures assume that each data value is a Gaussian random variable, estimation using such measures is based on an incorrect probability model. For the case of high photon counts, it is known that the Gaussian likelihood function might be an alternative to the Poisson likelihood function, because the Poisson probability distribution can be approximated by the Gaussian probability distribution [11

11. T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A **457**, 384–401 (2001) [CrossRef] .

*χ*

^{2}-based estimators because these estimators are either biased or inefficient, as demonstrated herein. Nevertheless, the least squares estimator (LSE) was used in software packages for FLIM [6], and

*χ*

^{2}-based estimators were used in FLIM imaging [7

7. S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J. **87**, 2807–17 (2004) [CrossRef] [PubMed] .

9. P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J. **82**, 2244–2255 (2002) [CrossRef] [PubMed] .

*χ*

^{2}-based investigations that were reported earlier [12

12. M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. **73**, 2078–2086 (2001) [CrossRef] [PubMed] .

11. T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A **457**, 384–401 (2001) [CrossRef] .

12. M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. **73**, 2078–2086 (2001) [CrossRef] [PubMed] .

13. K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst. **7**, 322–336 (2011) [CrossRef] [PubMed] .

14. J. Tellinghuisen and C. W. Wilkerson, “Bias and precision in the estimation of exponential decay parameters from sparse data,” Anal. Chem. **65**, 1240–1246 (1993) [CrossRef] .

*χ*

^{2}-based estimator for the asymptotic case of infinite photon counts are identical in their analysis [14

14. J. Tellinghuisen and C. W. Wilkerson, “Bias and precision in the estimation of exponential decay parameters from sparse data,” Anal. Chem. **65**, 1240–1246 (1993) [CrossRef] .

*χ*

^{2}-based estimator. In fact, the Pearson’s

*χ*

^{2}-based estimator should not be used since, as demonstrated herein, this estimator is biased for a finite number of photon counts.

*χ*

^{2}, Pearson’s

*χ*

^{2}, and Neyman’s

*χ*

^{2}methods by linearizing the estimators. Such analysis was used effectively in some previous works to analyze the properties of nonlinear estimators [15

15. J. Fessler, “Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography,” IEEE Trans. Image Process. **5**, 493 –506 (1996) [CrossRef] [PubMed] .

16. J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag. **23**, 1430 –1444 (2004) [CrossRef] .

*χ*

^{2}and Neyman’s

*χ*

^{2}-based estimation yield a biased estimator. We use simulations to validate our analysis.

## 2. Theory

### 2.1. Problem formulation

*y*(

*t*), that is measured at time

_{i}*t*from a mixture of fluorophores, as a Poisson random variable [1

_{i}1. J. R. Lakowicz, *Principles of Fluorescence Spectroscopy* (Kluwer Academic/Plenum, 1999) [CrossRef] .

*λ*(

*t*;

_{i}*θ*), is defined as follows: where, * is the convolution operator,

*h*(

*t*) is the impulse response function (IRF) of a measurement system, and

*g*(

*t*;

*θ*) is an exponential decay model that is defined as follows: where, the 2

*M*× 1 long vector of

*unknown*parameters,

*θ*= [

*τ*

_{0},

*τ*

_{1},...,

*τ*

_{M}_{−1},

*A*

_{0},

*A*

_{1},...,

*A*

_{M−1}], consists of the decay constants

*τ*and the amplitudes

_{i}*A*of all fluorophores [1

_{i}1. J. R. Lakowicz, *Principles of Fluorescence Spectroscopy* (Kluwer Academic/Plenum, 1999) [CrossRef] .

*λ*(

*t*;

_{i}*θ*) by

*λ*(

_{i}*θ*) and

*y*(

*t*) by

_{i}*y*. Because FLIM should be based on accurate decay properties, the parameter vector

_{i}*θ*should be accurately estimated from the noisy measurement

*Y*= [

*y*

_{0},

*y*

_{1},...,

*y*

_{N}_{−1}]. To achieve this goal, the estimation is usually carried out by determining the minimizer of an objective function, in the following way: Since it is desired that the mean of the estimator be the same as the true parameter (

*i.e., unbiased*) and that the variance of the estimator be as small as possible, one should design the objective function based on the underlying probabilistic nature of the measured photon counts. One of the most useful estimators is the MLE, which is defined by the maximizer of the likelihood function of

*Y*. The MLE for the measurement that is modeled in Eq. (1) is determined by the minimizer of the negative Poisson log-likelihood function, defined as follows: If one assumes that the measured data is acquired under additive Gaussian noises with equal variances, then the LSE would be the MLE for such measured data. The objective function for the LSE is defined as follows: Since each measurement

*y*has different variance, one may attempt to estimate the vector

_{i}*θ*by considering different variances for different

*y*. Because the mean and the variance of the Poisson random variable

_{i}*y*are the same as

_{i}*λ*(

_{i}*θ*), assuming each measurement as a Gaussian random variable with equal mean and variance yields the well known Pearson’s

*χ*

^{2}objective function defined, as follows [11

11. T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A **457**, 384–401 (2001) [CrossRef] .

*χ*

^{2}is the modified Neyman’s

*χ*

^{2}, which is obtained by substituting the variance in the denominator with max{

*y*, 1}, as follows: The denominator of the original Neyman’s

_{i}*χ*

^{2}is

*y*, which is based on the intuition that

_{i}*E*[

*y*] is

_{i}*λ*(

_{i}*θ*). However, since the original Neyman’s

*χ*

^{2}function is not defined when

*y*is zero, we study the modified Neyman’s

_{i}*χ*

^{2}-based method as suggested previously [11

**457**, 384–401 (2001) [CrossRef] .

*y*are non-zero, the modified Neyman’s

_{i}*χ*

^{2}is the same as the original [11

**457**, 384–401 (2001) [CrossRef] .

### 2.2. Mean and variance approximation

16. J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag. **23**, 1430 –1444 (2004) [CrossRef] .

*true*parameter as reported previously [15

15. J. Fessler, “Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography,” IEEE Trans. Image Process. **5**, 493 –506 (1996) [CrossRef] [PubMed] .

16. J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag. **23**, 1430 –1444 (2004) [CrossRef] .

**23**, 1430 –1444 (2004) [CrossRef] .

*operator is defined by the column gradient using*

_{θ}**23**, 1430 –1444 (2004) [CrossRef] .

*Φ(*

_{θ}*θ*̂,

*Y*) by the first-order Taylor series expansion that is performed around the

*true*parameter

*θ*, as follows [16

**23**, 1430 –1444 (2004) [CrossRef] .

*θ*, and when

*θ*̂ belongs to the local region. For such cases, the Hessian matrix ∇

*[∇*

_{θ}*Φ(*

_{θ}*θ*;

*Y*)] is a positive definite matrix. Since ∇

*Φ(*

_{θ}*θ*̂;

*Y*) = 0, the estimator

*θ*̂ can be approximated as follows [16

**23**, 1430 –1444 (2004) [CrossRef] .

**23**, 1430 –1444 (2004) [CrossRef] .

**H**is the expectation of Hessian matrix at

*θ*and is defined as follows: In addition, the covariance of the estimator may be approximated as follows [16

**23**, 1430 –1444 (2004) [CrossRef] .

*i.e*., high signal to noise scenario).

### 2.3. Cramer-Rao bound

*j*-th component of any unbiased estimator

*θ*̂

*is bounded by*

_{j}*J*, as follows [17]: where, the matrix

_{jj}**J**is the inverse of the Fisher information matrix

**F**, which is defined as follows: We compute the Cramer-Rao bound for the estimation of

*θ*from

*Y*(defined in Eq. (1)). By using the negative log-likelihood function defined in Eq. (5), it is straightforward to show that the (

*j*,

*k*)-th element of the Fisher information matrix

*F*can be defined as follows: Note that it is possible to simplify the Fisher information matrix, using the relation

*E*[

*y*] = Var[

_{i}*y*] =

_{i}*λ*(

_{i}*θ*).

### 2.4. Maximum likelihood estimation

### 2.5. Least squares estimation

**H**=

*E*[∇

*[∇*

_{θ}*Φ*

_{θ}*(*

_{LS}*θ*;

*Y*)]

*] and*

^{T}**K**=

*E*[∇

*Φ*

_{θ}*(*

_{LS}*θ*;

*Y*)∇

*Φ*

_{θ}*(*

_{LS}*θ*;

*Y*)

*]. Then the (*

^{T}*j*,

*k*)-th elements of

**H**and

**K**matrices are defined as follows: The approximated covariance function of the LSE,

*θ*̂

*, satisfies the following inequality due to the matrix Cauchy-Schwarz inequality [18*

_{LS}18. T. G., “A matrix extension of the cauchy-schwarz inequality,” Econ. Lett. **63**, 1–3 (1999) [CrossRef] .

**A**≥

**B**is understood in the sense that

**A**–

**B**matrix is positive semidefinite. The proof of this inequality is included in Appendix

**A**. Therefore, we conclude that the variance of the LSE is larger than the MLE, although the estimator is unbiased. This is because the LSE is based on an incorrect probability model.

### 2.6. Pearson’s χ^{2}

### 2.7. Neyman’s χ^{2}

### 2.8. Numerical optimization

*χ*

^{2}-based estimations, it is required to solve the minimization problem defined in Eq. (4) with the objective functions defined in Eq. (5) through Eq. (8). Note that the minimization problem is a constrained optimization problem with non-negativity constraints since both decay and amplitude parameters have positive values. We solve the constrained minimization problem using the trust region reflective method [21

21. J. Moré and D. Sorensen, “Computing a trust region step,” SIAM J. Sci. Comput. **4**, 553–572 (1983) [CrossRef] .

23. R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program. **40**, 247–263 (1988) [CrossRef] .

**H**

*of the negative log-likelihood function, which is the objective function for the MLE, is defined as follows: For the quadratic approximation of the negative log-likelihood function, we ignore the first term in Eq. (29) since inclusion of the second order derivative term can be destabilizing if the model fits badly [10, p. 688]. The Hessian matrices for the*

^{M}*χ*

^{2}-based estimators are also computed using only first order derivative terms for the same reason.

*χ*

^{2}-based estimators are non-convex functions, numerical optimization may find only a local minimum depending on initial parameter values. Because it is very challenging to analyze the behaviors of local minima theoretically, we study the effects of local minima in decay and amplitude parameter estimation using numerical simulations in the following section.

## 3. Simulation results

*σ*= 1 and 5 nonzero points. Note that the Gaussian function was often used to model an IRF [2

2. H. E. Grecco, P. Roda-Navarro, and P. J. Verveer, “Global analysis of time correlated single photon counting FRET-FLIM data,” Opt. Express **17**, 6493–6508 (2009) [CrossRef] [PubMed] .

*y*whose mean is

_{i}*λ*(

_{i}*θ*) with uniformly spaced

*t*from

_{i}*t*

_{0}= 0 to

*t*

_{N}_{−1}= 12.788

*ns*. We set true values as

*τ*

_{1}= 1

*ns*,

*τ*

_{2}= 4

*ns*. To simulate a different number of photon counts, we changed

*A*

_{1}from 23 to 125 while we set

*A*

_{2}=

*A*

_{1}− 10. By using the generated photon counts (

*i.e.*, the Poisson random variables), we estimated the parameters using the MLE, LSE, and Pearson’s

*χ*

^{2}and Neyman’s

*χ*

^{2}-based estimators. We set initial parameter values by two times the true values. We repeated the estimation 5000 times, using different Poisson random variable realizations.

*τ*

_{1},

*τ*

_{2},

*A*

_{1},

*A*

_{2}] from the 5000 estimations with the approximated mean values that were obtained using the formula developed in the preceding section for the Pearson’s and Neyman’s

*χ*

^{2}-based estimators. Notably, because there is no bias in our approximation, the approximated means of the MLE and the LSE are the same as true ones. As shown in Fig. 1, the sample means of the MLE and the LSE are very close to the true parameters for the entire range of photon counts. However, the Pearson’s

*χ*

^{2}-based estimator shows positive biases in estimating

*τ*

_{1}and

*τ*

_{2}. There also exists a small positive bias in estimating

*A*

_{1}and a small negative bias in estimating

*A*

_{2}. For higher photon counts, the approximated mean formula is in better agreement with the sample mean. The approximated mean values are included only in the range for which the approximated Hessian is a positive definite matrix. Compared with the Pearson’s

*χ*

^{2}-based estimator, the Neyman’s

*χ*

^{2}-based estimator exhibits more complicated behaviors. The Neyman’s

*χ*

^{2}-based estimator shows positive biases in estimating

*τ*

_{1}and

*τ*

_{2}when photon counts are very low. However, in the region wherein photon counts are relatively high, the biases are negative. The approximated mean formula agrees with the sample mean only in the regime of high photon counts. As a matter of fact, the approximated Hessian matrix of the Neyman’s

*χ*

^{2}is positive definite only for sufficiently high photon counts, indicating that the Neyman’s

*χ*

^{2}-based estimator might be more nonlinear than the Pearson’s

*χ*

^{2}-based estimator.

**A**, the approximated variance of the LSE is always greater than the Cramer-Rao bound. The variances of the Pearson’s

*χ*

^{2}-based estimator are larger than those of the MLE for low photon counts, although the variances become close to the Cramer-Rao bound as the number of photons increases. Note that even for high photon counts, the Pearson’s

*χ*

^{2}-based estimator is biased in estimating

*τ*

_{1}and

*τ*

_{2}. For Neyman’s

*χ*

^{2}-based estimator, the variances are greater than those of the MLE for relatively low photon counts. Although the variances for estimating

*τ*

_{1}and

*τ*

_{2}approach the Cramer-Rao bound as photon counts increase, variances in estimated amplitudes do not approach the Cramer-Rao bound even in the case of very high photon counts, as is shown in Fig. 2(c) and Fig. 2(d). We believe that the Neyman’s

*χ*

^{2}-based estimator should not be used because it is more biased than the Pearson’s

*χ*

^{2}-based method, in addition to showing larger variances. Note that the Neyman’s

*χ*

^{2}-based method was a frequently used method in FLIM imaging [7

7. S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J. **87**, 2807–17 (2004) [CrossRef] [PubMed] .

12. M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. **73**, 2078–2086 (2001) [CrossRef] [PubMed] .

*χ*

^{2}-based methods even for the case of more than 45000 photon counts. In addition, since the LSE shows larger variances than the MLE, there is no reason to favor the LSE over to the MLE.

*χ*

^{2}-based estimators are non-convex functions. To study the effects of local minimizers in estimating decay and amplitude parameters, we conducted another simulation study. We generated 1024 Poisson random variables

*y*whose mean is

_{i}*λ*(

_{i}*θ*) with true parameter values as

*τ*

_{1}= 1

*ns*,

*τ*

_{2}= 4

*ns*,

*A*

_{1}= 72, and

*A*

_{2}= 62 (number of average total photon counts is 24852). Then, we estimated the decay and the amplitude parameters using the MLE and the

*χ*

^{2}-based methods while increasing initial parameter values from the true values to five times the true values. We repeated these estimations 5000 times with different Poisson random variable realizations. Figure 3 shows the sample mean of each estimator from 5000 realizations for different initial parameters. To our surprise, all the four estimation methods are very robust to the change of initial parameters. As shown in Fig. 3, the sample mean values of estimated parameters using the four methods are almost the same for the entire range of initial parameters. Similar results are found in the sample variance of each estimator which is shown in Fig. 4. One can see that the sample variances of the MLE and the

*χ*

^{2}-based estimations do not change much for the entire range of initial parameters. In addition, the sample variance of the MLE and the LSE agree with the Cramer-Rao bound and the variance approximation formula for the LSE, respectively. These simulation results suggest that the objective functions of the four estimation methods might be locally convex functions for the cases of high photon counts.

*λ*(

_{i}*θ*) to

*τ*

_{1}= 1

*ns*,

*τ*

_{2}= 2

*ns*,

*A*

_{1}= 28 and

*A*

_{2}= 18 (number of average total photon counts is 5138). Then, we conducted the same simulation study as the previous one. Figure 5 shows the sample mean values of estimated decay and amplitude parameters for different initial parameters. One can see that the sample mean values of the MLE exhibit small biases. In addition, the sample mean values are different for different initial parameter values. We suspect that this dependency on initial parameter is due to the effects of local minimizers. Compared with the MLE, the Pearson’s

*χ*

^{2}-based estimation was more robust to the change of initial parameters. However, the Pearson’s

*χ*

^{2}-based method shows larger biases than the MLE as shown in Fig. 5(a) through Fig. 5(d). The Neyman’s

*χ*

^{2}and the LSE exhibit very large bias in estimating

*τ*

_{2}for some initial parameters as shown in Fig. 5(b). Although the amounts of the biases of the MLE are different depending on initial parameters, the sample mean values of the MLE are the closest ones to the true parameter values for the entire range of initial parameters.

*χ*

^{2}were smaller than those of the MLE for

*τ*

_{1},

*A*

_{1}and

*A*

_{2}. However, in estimating

*τ*

_{2}, Pearson’s and Neyman’s

*χ*

^{2}showed larger variances than the MLE. In addition, the variances of the LSE were always larger than the MLE. Note that although the Pearson’s

*χ*

^{2}-based estimator shows smaller variances than the MLE in estimating some parameters, it has larger biases than the MLE. Since unbiased estimation of decay and amplitude parameters is important, we believe that one should choose the MLE for the cases of low photon counts with similar decay parameters. Note that a previous investigation also reported that the MLE should be used for the cases of low photon counts based on empirical comparisons [12

**73**, 2078–2086 (2001) [CrossRef] [PubMed] .

## 4. Conclusion

*χ*

^{2}-based estimators showed biases, it can be concluded that they are not suitable for even high photon counts. We believe that the analysis presented in this work can also be used to predict the performance of the estimators for FLIM.

## Appendix A: Inequality for the variance approximations of MLE and LSE

18. T. G., “A matrix extension of the cauchy-schwarz inequality,” Econ. Lett. **63**, 1–3 (1999) [CrossRef] .

**C**≥

**0**means that the matrix

**C**is a positive semidefinite matrix. We define

*N*× 2

*M*matrices

*A*and

*B*as follows: Note that the Fisher information matrix is defined as follows: In addition, since the

**H**matrix, defined in Eq. (21) is

**A**, and the

^{T}B**K**matrix, defined in Eq. (22), is

**A**, the covariance approximation of the LSE satisfies the following inequality: where

^{T}A*θ*̂

*denotes MLE.*

_{MLE}## Appendix B: Mean approximation for the estimation of a single time constant using the Pearson’s *χ*^{2}

*A*is very large, since the first term is proportional to

*A*but others are not. Therefore, we approximate the second derivative of the objective function as follows: Therefore, the bias is approximated as follows: Note that the bias in estimating

*τ*approaches zero as

*A*approaches infinity. However, note that the convergence rate is rather slow, since the bias is proportional to the reciprocal value of the amplitude

*A*.

## Acknowledgment

## References and links

1. | J. R. Lakowicz, |

2. | H. E. Grecco, P. Roda-Navarro, and P. J. Verveer, “Global analysis of time correlated single photon counting FRET-FLIM data,” Opt. Express |

3. | S. Kumar, C. Dunsby, P. A. A. D. Beule, D. M. Owen, U. Anand, P. M. P. Lanigan, R. K. P. Benninger, D. M. Davis, M. A. A. Neil, P. Anand, C. Benham, A. Naylor, and P. M. W. French, “Multifocal multiphoton excitation and time correlated single photon counting detection for 3-D fluorescence lifetime imaging,” Opt. Express |

4. | B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem. |

5. | H. Cramer, |

6. | S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw. |

7. | S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J. |

8. | N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem. |

9. | P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J. |

10. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

11. | T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A |

12. | M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem. |

13. | K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst. |

14. | J. Tellinghuisen and C. W. Wilkerson, “Bias and precision in the estimation of exponential decay parameters from sparse data,” Anal. Chem. |

15. | J. Fessler, “Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography,” IEEE Trans. Image Process. |

16. | J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag. |

17. | H. Van Trees, |

18. | T. G., “A matrix extension of the cauchy-schwarz inequality,” Econ. Lett. |

19. | P. Hall and B. Selinger, “Better estimates of exponential decay parameters,” J. Phys. Chem. |

20. | K. J. Mighell, “Parameter estimation in astronomy with poisson-distributed data. I. the |

21. | J. Moré and D. Sorensen, “Computing a trust region step,” SIAM J. Sci. Comput. |

22. | M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput. |

23. | R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(180.2520) Microscopy : Fluorescence microscopy

(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence

**ToC Category:**

Microscopy

**History**

Original Manuscript: January 3, 2013

Revised Manuscript: February 23, 2013

Manuscript Accepted: February 24, 2013

Published: March 4, 2013

**Virtual Issues**

Vol. 8, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Jeongtae Kim and Jiyeong Seok, "Statistical properties of amplitude and decay parameter estimators for fluorescence lifetime imaging," Opt. Express **21**, 6061-6075 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6061

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### References

- J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Kluwer Academic/Plenum, 1999). [CrossRef]
- H. E. Grecco, P. Roda-Navarro, and P. J. Verveer, “Global analysis of time correlated single photon counting FRET-FLIM data,” Opt. Express17, 6493–6508 (2009). [CrossRef] [PubMed]
- S. Kumar, C. Dunsby, P. A. A. D. Beule, D. M. Owen, U. Anand, P. M. P. Lanigan, R. K. P. Benninger, D. M. Davis, M. A. A. Neil, P. Anand, C. Benham, A. Naylor, and P. M. W. French, “Multifocal multiphoton excitation and time correlated single photon counting detection for 3-D fluorescence lifetime imaging,” Opt. Express15, 12548–12561 (2007). [CrossRef] [PubMed]
- B. B. Collier and M. J. McShane, “Dynamic windowing algorithm for the fast and accurate determination of luminescence lifetimes,” Anal. Chem.84, 4725–4731 (2012). [CrossRef] [PubMed]
- H. Cramer, Mathematical Methods of Statistics (Princeton University, 1999).
- S. Laptenok, K. M. Mullen, J. W. Borst, I. H. M. van Stokkum, V. V. Apanasovich, and A. J. W. G. Visser, “Fluorescence lifetime imaging microscopy (FLIM) data analysis with TIMP,” J. Stat. Softw.18, 1–20 (2007).
- S. Pelet, M. J. R. Previte, L. H. Laiho, and P. T. C. So, “A fast global fitting algorithm for fluorescence lifetime imaging microscopy based on image segmentation.” Biophys. J.87, 2807–17 (2004). [CrossRef] [PubMed]
- N. Boens, W. Qin, N. Basarić, J. Hofkens, M. Ameloot, J. Pouget, J.-P. Lefèvre, B. Valeur, E. Gratton, M. vandeVen, N. D. Silva, Y. Engelborghs, K. Willaert, A. Sillen, G. Rumbles, D. Phillips, A. J. W. G. Visser, A. van Hoek, J. R. Lakowicz, H. Malak, I. Gryczynski, A. G. Szabo, D. T. Krajcarski, N. Tamai, and A. Miura, “Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy,” Anal. Chem.79, 2137–2149 (2007). [CrossRef] [PubMed]
- P. J. Steinbach, R. Ionescu, and C. R. Matthews, “Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding,” Biophys. J.82, 2244–2255 (2002). [CrossRef] [PubMed]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++ - The Art of Scientific Computing (Cambridge University, 2002).
- T. Hauschild and M. Jentschel, “Comparison of maximum likelihood estimation and chi-square statistics applied to counting experiments,” Nucl. Instrum. Meth. A457, 384–401 (2001). [CrossRef]
- M. Maus, M. Cotlet, J. Hofkens, T. Gensch, F. C. De Schryver, J. Schaffer, and C. A. M. Seidel, “An experimental comparison of the maximum likelihood estimation and nonlinear least-squares fluorescence lifetime analysis of single molecules,” Anal. Chem.73, 2078–2086 (2001). [CrossRef] [PubMed]
- K. A. Walther, B. Papke, M. B. Sinn, K. Michel, and A. Kinkhabwala, “Precise measurement of protein interacting fractions with fluorescence lifetime imaging microscopy,” Mol. BioSyst.7, 322–336 (2011). [CrossRef] [PubMed]
- J. Tellinghuisen and C. W. Wilkerson, “Bias and precision in the estimation of exponential decay parameters from sparse data,” Anal. Chem.65, 1240–1246 (1993). [CrossRef]
- J. Fessler, “Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography,” IEEE Trans. Image Process.5, 493 –506 (1996). [CrossRef] [PubMed]
- J. Kim and J. Fessler, “Intensity-based image registration using robust correlation coefficients,” IEEE Trans. Med. Imag.23, 1430 –1444 (2004). [CrossRef]
- H. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (John Wiley & Sons, 2001).
- T. G., “A matrix extension of the cauchy-schwarz inequality,” Econ. Lett.63, 1–3 (1999). [CrossRef]
- P. Hall and B. Selinger, “Better estimates of exponential decay parameters,” J. Phys. Chem.85, 2941–2946 (1981). [CrossRef]
- K. J. Mighell, “Parameter estimation in astronomy with poisson-distributed data. I. the χγ2 statistic,” The Astrophys. J.518, 380 (1999). [CrossRef]
- J. Moré and D. Sorensen, “Computing a trust region step,” SIAM J. Sci. Comput.4, 553–572 (1983). [CrossRef]
- M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput.21, 1–23 (1999). [CrossRef]
- R. Byrd, R. Schnabel, and G. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program.40, 247–263 (1988). [CrossRef]

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