Time-Resolved Fluorescence Wiki

bootstrap

Anonymous (anonymous@undisclosed.example.com) — 2014-01-21T16:41:23+00:00

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	Specifically, the bootstrap analysis takes a subset of N randomly chosen points out of the experimental data set, which itself consists of N data points. Since the points are chosen randomly, some of the points will be selected more than once; others will not be selected at all. Thus the 'simulated' data set is not identical to the original one, while the statistical properties of the original data set are maintained (since the number of data points remains the same).		Specifically, the bootstrap analysis takes a subset of N randomly chosen points out of the experimental data set, which itself consists of N data points. Since the points are chosen randomly, some of the points will be selected more than once; others will not be selected at all. Thus the 'simulated' data set is not identical to the original one, while the statistical properties of the original data set are maintained (since the number of data points remains the same).
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-	At PQ the bootstrap method is exclusively used in [[FluoFit]]. For ensemble fitting as in [[FLIM]] it is -alas!- way too slow.
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	===== Advantages and disadvantages =====		===== Advantages and disadvantages =====

marquardt-levenberg

Anonymous (anonymous@undisclosed.example.com) — 2015-02-10T16:34:33+00:00

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	====== Marquardt-Levenberg ======		====== Marquardt-Levenberg ======

-	The Marquardt-Levenberg method is a means of optimising parameters in a least squares fit. It is used in our [[products:FluoFit]], [[products:SymPhoTime]] and a variety of other products (online-Fitting, FluoLib...).	+	The Marquardt-Levenberg method is a means of optimising parameters in a least squares fit. It is used in our [[software:FluoFit]], [[software:SymPhoTime]] and a variety of other products (online-Fitting, FluoLib...).

	Basically it works by examining how changes of a certain parameter affect the residuals trace, that means, it calculates a pointwise gradient of the residuals trace for each parameter. Marquard-Levenberg optimisation is fairly efficient and is regarded as state-of-the-art in least squares fitting.		Basically it works by examining how changes of a certain parameter affect the residuals trace, that means, it calculates a pointwise gradient of the residuals trace for each parameter. Marquard-Levenberg optimisation is fairly efficient and is regarded as state-of-the-art in least squares fitting.

support_plane_analysis

Anonymous (anonymous@undisclosed.example.com) — 2014-01-21T16:46:58+00:00

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	===== References =====		===== References =====

-	Lakowicz JR (1999)// Principles of Fluorescence Spectroscopy//, 2nd edn. Kluver Academic/Plenum Publishers, New York (available in room C202)*	+	*Lakowicz JR (1999)// Principles of Fluorescence Spectroscopy//, 2nd edn. Kluver Academic/Plenum Publishers, New York

chi_square_management

Anonymous (anonymous@undisclosed.example.com) — 2014-01-21T16:22:05+00:00

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	====== Chi square ======		====== Chi square ======

-	The (normalised) χ<sup>2</sup> (read: chi square) is the optimisation parameter for [[least squares]] fitting (for a definition see there). For a perfect fit it should be near 1. As a measure of the goodness-of-fit it is insufficient. Other methods have to be used in addition, like examining the weighted [[residuals]] trace and the [[autocorrelation function]] of the weighted residuals trace.	+	The (normalized) χ<sup>2</sup> (read: chi square) is the optimization parameter for [[least squares]] fitting (for a definition see there). For a perfect fit it should be near 1. As a measure of the goodness-of-fit it is insufficient. Other methods have to be used in addition, like examining the weighted [[residuals]] trace and the [[autocorrelation function]] of the weighted residuals trace.

asymptotic_standard_errors

Anonymous (anonymous@undisclosed.example.com) — 2015-02-10T16:33:41+00:00

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	The asymptotic standard errors (ASE) are used for analyzing fitting parameter error intervals. For illustration let's start at the best fit parameter set, which can be regarded as a single point in the parameter space. Now we remove the parameter for which we want to calculate the error intervals from this location, that is, we take a single step parallel to its parameter axis. We then calculate the reduced X<sup>2</sup> for this new paramter set. By iterating this procedure we get X<sup>2</sup> as a function of the parameter of interest. The intersection points of this function with a given X<sup>2</sup> confidence limit define the boundaries of the confidence interval of the parameter.		The asymptotic standard errors (ASE) are used for analyzing fitting parameter error intervals. For illustration let's start at the best fit parameter set, which can be regarded as a single point in the parameter space. Now we remove the parameter for which we want to calculate the error intervals from this location, that is, we take a single step parallel to its parameter axis. We then calculate the reduced X<sup>2</sup> for this new paramter set. By iterating this procedure we get X<sup>2</sup> as a function of the parameter of interest. The intersection points of this function with a given X<sup>2</sup> confidence limit define the boundaries of the confidence interval of the parameter.

-	At PQ the asymptotic standard errors are supported by [[products:FluoFit]] and the [[products:SymPhoTime]] software.	+	At PQ the asymptotic standard errors are supported by [[software:FluoFit]] and the [[software:SymPhoTime]] software.

least_squares

Anonymous (anonymous@undisclosed.example.com) — 2014-04-09T20:43:37+00:00

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	====== Least squares ======		====== Least squares ======

-	Least squares is an optimization paradigm for matching data ('fitting') with a parametrised model equation. A famous example is the linear regression used for finding the linear equation that best matches a given set of data points.	+	Least squares is an optimization paradigm for matching data ('{{wiki>Regression_analysis\|fitting}}') with a parametrised model equation. A famous example is the linear regression used for finding the linear equation that best matches a given set of data points.

	The least squares measure for the goodness-of-fit is		The least squares measure for the goodness-of-fit is

poisson_distribution

Anonymous (anonymous@undisclosed.example.com) — 2014-04-09T20:47:01+00:00

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	$$P_{\nu}(n)={{\nu^n~e^{-\nu}}\over{n!}}$$		$$P_{\nu}(n)={{\nu^n~e^{-\nu}}\over{n!}}$$

-	The Poisson distribution is of interest especially for [[TCSPC]]: The expected number of photons in any TCSPC channel is given by the 'real' decay (including convolution with the IRF etc.), while the stochastic nature of the measurement process (either a photon is detected or it is not) introduces noise, which follows a Poisson distribution. In the limit of large ${\nu}^{}_{}$ the Poisson distribution approaches a [[Gaussian distribution]] with a width of $\sqrt{\nu}$ centred around ${\nu}^{}_{}$.	+	The Poisson distribution is of interest especially for [[TCSPC]]: The expected number of photons in any TCSPC channel is given by the 'real' decay (including convolution with the IRF etc.), while the stochastic nature of the measurement process (either a photon is detected or it is not) introduces noise, which follows a Poisson distribution. In the limit of large ${\nu}^{}_{}$ the Poisson distribution approaches a [[wp>Normal_distribution\|Gaussian distribution]] with a width of $\sqrt{\nu}$ centred around ${\nu}^{}_{}$.

	In the Gaussian limit [[least squares]] [[wp>Regression_analysis\|fitting]] may be applied, otherwise [[MLE]] [[wp>Regression_analysis\|fitting]] is preferable.		In the Gaussian limit [[least squares]] [[wp>Regression_analysis\|fitting]] may be applied, otherwise [[MLE]] [[wp>Regression_analysis\|fitting]] is preferable.

residuals

Anonymous (anonymous@undisclosed.example.com) — 2015-02-10T16:35:23+00:00

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	$$w_i=\sqrt{D_i^{exp}}$$		$$w_i=\sqrt{D_i^{exp}}$$

-	The resudials trace is of importance within any framework concerned with fitting, as the [[products:SymPhoTime]] software or [[products:FluoFit]].	+	The resudials trace is of importance within any framework concerned with fitting, as the [[software:SymPhoTime]] software or [[software:FluoFit]].

fast_lifetime

Anonymous (anonymous@undisclosed.example.com) — 2013-08-06T14:43:32+00:00

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	A method for estimating the average lifetime, calculated non-distinctive on different exponentials in a simple single path calculation, compared to a complete fitting operation against a more complex, non-linear model. Often used with [[FLIM]] go get a first idea of the lifetime distribution in the image.		A method for estimating the average lifetime, calculated non-distinctive on different exponentials in a simple single path calculation, compared to a complete fitting operation against a more complex, non-linear model. Often used with [[FLIM]] go get a first idea of the lifetime distribution in the image.

-	In detail, FastLT is calculating the barycentre of the (pseudo–)pixel's decay. The time span from the barycentre of the IRF to the barycentre of the decay equals the average lifetime. This estimate is very fast and does not suffer as much from low statistics. If an [[IRF]] is not available, the "time zero" $t_0_$ has to be estimated differently, for example by using the rising flank of the decay or the entrance of the [[FWHM]] interval. However, this may introduce a systematic shift in the estimated average lifetimes.	+	In detail, FastLT is calculating the barycentre of the (pseudo–)pixel's decay. The time span from the barycentre of the IRF to the barycentre of the decay equals the average lifetime. This estimate is very fast and does not suffer as much from low statistics. If an [[IRF]] is not available, the "time zero" $t_\theta$ has to be estimated differently, for example by using the rising flank of the decay or the entrance of the [[FWHM]] interval. However, this may introduce a systematic shift in the estimated average lifetimes.

monte_carlo

Anonymous (anonymous@undisclosed.example.com) — 2014-04-09T20:45:42+00:00

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-	PicoQuant Software makes use of Monte Carlo methods mainly as a means of finding initial values for [[fitting]] parameters before optimisation by a [[Marquardt-Levenberg]] algorithm. The principle is simple: A large number of random parameter sets is generated, for each one a [[chi square]] is calculated and the best one is taken as the initial parameter set.	+	PicoQuant Software makes use of Monte Carlo methods mainly as a means of finding initial values for [[wp>Regression_analysis\|fitting]] parameters before optimisation by a [[Marquardt-Levenberg]] algorithm. The principle is simple: A large number of random parameter sets is generated, for each one a [[chi square]] is calculated and the best one is taken as the initial parameter set.

	The second application is [[error analysis]], namely the [[bootstrap method]].		The second application is [[error analysis]], namely the [[bootstrap method]].