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z045z!R9raJ9btL9{hh^asVnODJ>w|Meg#i5^)NPKqBY>|@ghEn4dZP=Pn>Pa^iuq3 zaQ6(gMb#gctmRVL9HFKO4o~UwZ{wYTBrWrCR9M>;>SLcC-%5Y%Y+T zY$O;n~-A2Gq$>x1+Q%#r)cC3@UsTJA1 zbWf}%;^C`IB5cf;3eJ(&)p=h%!rODg2R%cq<0GJZA))$lqpI;=vD#i5^rVhElA=H znPPX=tN$A-D@h;~(ki1dWPAIC4!5dJkUM($92<;CXe0^_7dL5Kd~H(n|;C$I(rqGVlcw3)ul{J81KB8 z(vsG*CcG^ycuxv>!b}%DbTaUsGV){zD(`Czk}I%^eoBqE`@Rb|DlchLEkUSmkZAj9 z$nS|8#H8fBO@kK?^#-*41Y2g5bFPEA!_Cf&MH8{CWsd7j7;cKLv>NNldm+iOeAxr6 z=^wOvn_)OVHhG~l_rilHX4f8BXlaD+nWp;q=tQ{HR1o=WMG^r`NJ@ zdUAS-cm!YE+^H8mJaHseWh|h~T7SjV4rIalTC!CsyJ^#M#$nUmZtuVX6ZytvFXRog zkLZ~%-yY7Q&`b&^0O@?|g$l0abF`yLk85orpJyD8V5nVnv1Q^!Ut|D>F@ zuArf?%AJ+W)R(jNa^#qzJvvHE8TML_S^}v=eiQE-qM^>eai!9f!o{q`Lngg#e|2XT z0@-bTY4N3DBWtjnox-vHyr{kG=LN6;jFyqNGDIt1Ri52Sl*g*za+g;S$pE%TpFC7b zWaEFqBzXVmW`KnzWN 0$ is the +frame interval (we'll neglect gaps in the trajectory for now). +Let $\Delta B_{i} = B_{i \Delta t} - B_{(i-1) \Delta t}$ be the +$i^{\text{th}}$ increment (``jump''). Then the joint distribution over +these jumps is +\begin{align*} + p_{\Delta B_{1}, ..., \Delta B_{m}} \left( \Delta b_{1}, ..., \Delta b_{m} \right) &= \prod\limits_{i=1}^{m} \frac{ \exp \left( - \Delta b_{i}^{2} / 4 D \Delta t \right) }{ \sqrt{ 4 \pi D \Delta t } } \\ + &= \frac{1}{\left( 4 \pi D \Delta t \right)^{m/2}} \exp \left( - \frac{1}{4 D \Delta t} \left[ \Delta b_{1}^{2} + ... + \Delta b_{m}^{2} \right] \right) +\end{align*} + +This distribution does not depend on the individual jumps, +only on the sum of their squares. It is straightforward to +show that the sum of squares $X = \Delta B_{1}^{2} + ... + \Delta B_{m}^{2}$ +has a gamma distribution. First, the cdf for $\Delta B_{i}^{2}$ is +\begin{align*} + F_{\Delta B^{2}} \left( x \right) &:= \Pr \left( \Delta B^{2} \leq x \right) \\ + &= \Pr \left( \Delta B \leq \sqrt{x} \right) - \Pr \left( \Delta B < -\sqrt{x} \right) \\ + &= F_{\Delta B} \left( \sqrt{x} \right) - F_{\Delta B} \left( - \sqrt{x} \right) +\end{align*} + +Differentiating both sides, we obtain the pdf +\begin{align*} + f_{\Delta B^{2}}(x) &= \frac{\partial F_{\Delta B^{2}}(x)}{\partial x} \\ + &= \frac{1}{2 \sqrt{x}} \left( + f_{\Delta B} \left( \sqrt{x} \right) + f_{\Delta B} \left( - \sqrt{x} \right) + \right) +\end{align*} + +Substituting the normal distribution from equation \ref{eq:rule1} as $f_{\Delta B}$, +we have +\[ + f_{\Delta B^{2}}(x) = \frac{ e^{-x / 4 D \Delta t} }{ \sqrt{4 \pi D \Delta t x} } +\] + +which we recognize as a gamma distribution with parameters $\alpha = 1/2$ and +$\beta = 1 / 4 D \Delta t$: +\begin{align*} + \Delta B^{2} &\sim \text{Gamma} \left( \frac{1}{2}, \frac{1}{4 D \Delta t} \right) +\end{align*} + +Now, the sum of $m$ independent and identically distributed gamma random variables +with the distribution $\text{Gamma} \left( \alpha, \beta \right)$ has the distribution +$\text{Gamma} \left( m \alpha, \beta \right)$. From this, we have +\begin{equation} + X := \Delta B_{1}^{2} + ... + \Delta B_{m}^{2} \sim \text{Gamma} \left( + \frac{m}{2}, \frac{1}{4 D \Delta t} + \right) +\end{equation} + +which corresponds to the pdf +\begin{equation} + f_{X}(x) = \frac{x^{\frac{m}{2}-1} e^{-x / 4 D \Delta t}}{\Gamma \left( \frac{m}{2} \right) \left( 4 D \Delta t \right)^{m/2}}, \qquad x \geq 0 +\end{equation} + +Now, suppose that a particle's motion in $d$ spatial dimensions can be +modeled as independent Brownian motions in each dimension. Due to the Pythagorean +theorem, the sum of its squared displacement is now +$X = X_{\text{dimension 1}} + X_{\text{dimension 2}} + ...$, where +the terms on the right-hand side are the sums of its squared displacements along +each spatial dimension. Insofar as these sums are iid, we have +\begin{equation}\label{eq:brownian_likelihood} + X \sim \text{Gamma} \left( \frac{md}{2}, \frac{1}{4 D \Delta t} \right) +\end{equation} + +Frequently, we will find it convenient to parametrize the distribution by +the \emph{scale} parameter $\phi = 4 D \Delta t$ instead of $D$, +in which case the pdf becomes +\begin{equation}\label{eq:brownian_likelihood_2} + f_{X}(x | \phi) = \frac{x^{\frac{md}{2} - 1} e^{-x / \phi}}{\Gamma \left( \frac{md}{2} \right) \phi^{\frac{md}{2}}}, \quad x \geq 0 +\end{equation} + +Finally, at the time of writing we are only interested in 2D tracking ($d=2$), +and so the likelihood simplifies to +\begin{align}\begin{split}\label{eq:brownian_likelihood_3} + X &\sim \text{Gamma} \left( m, \phi \right) \\ + \Rightarrow f_{X}(x | \phi) &= \frac{x^{m-1} e^{-x/\phi}}{\Gamma (m) \phi^{m}}, \quad x \geq 0 +\end{split}\end{align} + +where as before $\phi = 4 D \Delta t$. +For the sake of readability we use equation \ref{eq:brownian_likelihood_3} +throughout this text, with the understanding that generalizing to $d \neq 2$ +requires substituting $m$ with $md / 2$. + +\section{Bayesian inference on single-state Brownian motion} + +As a building block to mixture models, we first consider a simple +Bayesian model for one-state Brownian motion. Suppose we have a +2D Brownian trajectory with $m$ jumps and sum of squared displacements $x$. +Our goal is to infer a distribution over the scale parameter (diffusion +coefficient) $\phi$ given the observed $x$. Bayes' theorem prescribes +\[ + f_{\phi | X}(\phi | x) \propto f_{X}(x | \phi) f_{\phi}(\phi) +\] + +where $f_{X}(x | \phi)$ is given by equation \ref{eq:brownian_likelihood_3}. +For the prior $f_{\phi}(\phi)$, we assume an inverse gamma distribution: +\begin{align*} + \phi &\sim \text{InvGamma} \left( \alpha_{0}, \beta_{0} \right) \\ + \Rightarrow f_{\phi} \left( \phi \right) &= \frac{\beta_{0}^{\alpha_{0}} e^{-\beta_{0} / \phi}}{\Gamma \left( \alpha_{0} \right) \phi^{\alpha_{0} + 1} }, \quad \phi \geq 0 +\end{align*} + +Substituting into Bayes' theorem and retaining only factors that depend +directly on $\phi$, +\[ + f_{\phi | X} (\phi | x) \propto \frac{ + e^{-(\beta_{0} + x) / \phi} + }{ + \phi^{\alpha_{0} + m + 1} + } +\] + +After renormalization, we obtain +\[ + f_{\phi | X} (\phi | x) = \frac{ (\beta_{0} + x)^{\alpha_{0}+m-1} e^{-(\beta_{0} + x) / \phi}}{\Gamma (\alpha_{0} + m) \phi^{\alpha_{0} + m + 1}} +\] + +which we recognize as another inverse gamma distribution +\[ + \phi \ | \ X \sim \text{InvGamma} \left( \alpha_{0} + m, \beta_{0} + x \right) +\] + +So - due to the conjugacy of the Brownian likelihood \ref{eq:browian_likelihood_3} +and the inverse gamma prior - inference boils down to simply adding +shape and scale parameters. We can obtain a point estimate of the +diffusion coefficient by taking the mean of the posterior distribution: +\[ + \mathbb{E} \left[ D | x \right] = \frac{1}{4 \Delta t} \mathbb{E} \left[ \phi | x \right] = \frac{\beta_{0} + x}{4 \Delta t \left( \alpha_{0} + m - 1 \right)} +\] + +Notice that the $-1$ in the denominator is analogous to the Bessel +correction for the sample variance, as the diffusion coefficient is essentially +a variance. + +\section{Brownian mixtures} + +Most samples of interest in SPT contain a mixture of particles in different +states characterized by distinct diffusion coefficients. Suppose we have a dataset +with $N$ trajectories, each of which originates from one of $K$ distinct states +characterized by scale parameters $\phi_{1}, ..., \phi_{K}$. There are multiple +unknowns here: we don't know the scale parameters for each state, the relative +abundances (``occupations'') of each state, nor which state each trajectory comes +from. To represent this uncertainty, we construct the following mixture model: + +\begin{align}\label{eq:brownian_mixture} + \boldsymbol{\tau} &\sim \text{Dirichlet} \left( \alpha_{0}, ..., \alpha_{0} \right), &\quad \boldsymbol{\tau} \in C^{K-1} \\ + \phi_{j} &\sim \text{InvGamma} \left( \alpha_{0}, \beta_{0} \right), &\quad j \in \left\{ 1, 2, ..., K \right\} \\ + \vec{Z}_{i} \ | \ \boldsymbol{\tau} &\sim \boldsymbol{\tau}, &\quad i \in \left\{ 1, 2, ..., N \right\} \\ + X_{i} \ | \ Z_{ij} = 1, \phi_{j} &\sim \text{Gamma} \left( m_{i}, \phi_{j} \right), &\quad i \in \left\{ 1, 2, ..., N \right\} +\end{align} + +Breaking this down: +\begin{itemize} + \item $\boldsymbol{\tau}$ is an unknown distribution over $K$ states (in other words, it belongs to the simplex $C^{K-1}$). + \item Each state $j$ is characterized by a scale parameter $\phi_{j}$. + \item The priors over each $\phi_{j}$ are independent and given by $\text{InvGamma} (\alpha_{0}, \beta_{0})$. + \item To each observation $i$, we associate a one-hot vector $\vec{Z}_{i} \in \left\{ 0, 1 \right\}^{K}$ so that $Z_{ik} = 1$ if observation $i$ originates from state $j$ and $Z_{ik} = 0$ otherwise. Given knowledge of $\boldsymbol{\tau}$, the event $Z_{ij} = 1$ has probability $\tau_{j}$. + \item $m_{i}$ is the number of jumps in trajectory $i$. + \item $X_{i}$ is the sum of squared displacements for trajectory $i$. +\end{itemize} + +We will use $\vec{X} = \left( X_{1}, ..., X_{N} \right)$ and +$\vec{Z} = \left( Z_{1}, ..., Z_{N} \right)$ to represent the sequence of +all $X_{i}$ and $\vec{Z}_{i}$. +Likewise, we'll use $\boldsymbol{\phi} = (\phi_{1}, ..., \phi_{K})$ to represent the sequence of scale parameters for all $K$ states. +Then the total probability function for this model factors as +\[ + p \left( \vec{X}, \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) = p \left( \vec{X} | \vec{Z}, \boldsymbol{\phi} \right) p \left( \vec{Z} | \boldsymbol{\tau} \right) p \left( \boldsymbol{\tau} \right) p \left( \boldsymbol{\phi} \right) +\] + +For later parts, it's useful to write out each of these terms explicitly: +\begin{equation}\label{eq:eqA} + p \left( \vec{X} | \vec{Z}, \boldsymbol{\phi} \right) = \prod\limits_{i=1}^{N} \prod\limits_{j=1}^{K} \left[ + \frac{ x_{i}^{m_{i}-1} e^{-x_{i} / \phi_{j}}}{\Gamma (m_{i}) \phi_{j}^{m_{i}}} + \right]^{Z_{ij}} +\end{equation} + +\begin{equation}\label{eq:eqB} + p \left( \vec{Z} | \boldsymbol{\tau} \right) = \prod\limits_{i=1}^{N} \prod\limits_{j=1}^{K} \tau_{j}^{Z_{ij}} +\end{equation} + +\begin{equation}\label{eq:eqC} + p \left( \boldsymbol{\tau} \right) = \frac{1}{B(\alpha_{0}, ..., \alpha_{0})} \prod\limits_{j=1}^{K} \tau_{j}^{\alpha_{0} - 1} +\end{equation} + +\begin{equation}\label{eq:eqD} + p \left( \boldsymbol{\phi} \right) = \prod\limits_{j=1}^{K} \frac{\beta_{0}^{\alpha_{0}} e^{-\beta_{0} / \phi_{j}}}{\Gamma (\alpha_{0}) \phi_{j}^{\alpha_{0} + 1}} +\end{equation} + +(Here, $B(\alpha_{0}, ..., \alpha_{0})$ is the multivariate beta function.) \newline + +Of all of these parameters, the only one we actually observe is $\vec{X}$. +Our goal is to infer a posterior distribution over the remaining variables +given an observed set of trajectories. + +\section{Approximative inference on Brownian mixtures} + +Given an observed set of trajectories $\vec{X} = (X_{1}, ..., X_{N})$, we +want to infer the probability of the unobserved (latent) variables +$\vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi}$ in the mixture \ref{eq:brownian_mixture}. +In other words, we want to infer the posterior distribution +$p \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} | \vec{X} \right)$. +Solving this problem exactly is intractable. There are two fallbacks: +(a) approximate the distribution with a finite sequence of samples using +MCMC methods or (b) construct a tractable approximation to the posterior. +In \verb|vbdiff|, we take the latter approach. We follow +a classic variational Bayesian approach and paraphrase certain steps; +for more detail see Bishop's book \cite{bishop2006}. \newline + +We will approximate the posterior with a tractable distribution $q$: +\[ + q \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) \approx p \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} | \vec{X} \right) +\] + +We will use the evidence lower bound (ELBO) as the objective function for $q$: +\begin{equation}\label{eq:elbo} + \mathcal{L} \left[ q \right] := \mathbb{E}_{\vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \sim q} \left[ + \log \left( + \frac{ p \left( \vec{X}, \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right)}{ q \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) } + \right) + \right] +\end{equation} + +By factoring $p \left( \vec{X}, \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) = p \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} | \vec{X} \right) p \left( \vec{X} \right)$ in equation \ref{eq:elbo}, it straightforward to show that maximizing $\mathcal{L} [q]$ corresponds to minimizing the Kullback-Leibler divergence between the true and approximative posteriors, and that $\mathcal{L}[q]$ forms a lower bound on the marginal log likelihood $\log p (\vec{X})$: +\begin{align*} + \underset{q}{\text{argmax}} \: \mathcal{L}[q] &= \underset{q}{\text{argmin}} \: \text{KL} \left( q(\vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi}) || p (\vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} | \vec{X} ) \right) \\ + \mathcal{L}[q] &\leq \log p (\vec{X}) +\end{align*} + +This last point is crucial, as it will form the basis for model selection considered later on. For now, notice that $p(\vec{X})$ can be seen as the average probability of the observed data over all possible model parameters $\vec{X}, \boldsymbol{\tau}, \boldsymbol{\phi}$ with some $K$. \newline + +What form will we choose for $q$? As it turns out, we only need to make a +single constraint to find a tractable $q$: assume that $q$ factors into +separate distributions over $\vec{Z}$ and $\boldsymbol{\tau}, \boldsymbol{\phi}$: +\[ + q \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) = q \left( \vec{Z} \right) q \left( \boldsymbol{\tau}, \boldsymbol{\phi} \right) +\] + +This factorability criterion - a kind of mean field approximation - is sufficient +to completely specify the form of the optimal $q$. It can be shown \cite{bishop2006} +that $\underset{q}{\text{argmax}} \: \mathcal{L}[q]$ satisfies +\begin{align}\begin{split}\label{eq:em_equations} + \log q \left( \vec{Z} \right) &= \mathbb{E}_{\boldsymbol{\tau}, \boldsymbol{\phi} \sim q (\boldsymbol{\tau}, \boldsymbol{\phi})} \left[ \log p \left( \vec{X}, \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) \right] \\ + \log q \left( \boldsymbol{\tau}, \boldsymbol{\phi} \right) &= \mathbb{E}_{\vec{Z} \sim q(\vec{Z})} \left[ \log p \left( \vec{X}, \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) \right] +\end{split}\end{align} + +We solve these equations in Appendix \ref{appendix:derivation_approx_posterior}. +Skipping to the punchline, the optimal approximation $q$ is given by +\begin{align}\begin{split} + q \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) &= q \left( \vec{Z} \right) q \left( \boldsymbol{\tau} \right) q \left( \boldsymbol{\phi} \right) \\ + q \left( \vec{Z} \right) &= \prod\limits_{i=1}^{N} \prod\limits_{j=1}^{K} r_{ij}^{Z_{ij}} \\ + q \left( \boldsymbol{\tau} \right) &= \text{Dirichlet} \left( + \boldsymbol{\tau} | \alpha_{0} + \alpha_{1}, ..., \alpha_{0} + \alpha_{K} + \right) \\ + q \left( \boldsymbol{\phi} \right) &= \prod\limits_{j=1}^{K} \text{InvGamma} \left( + \phi_{j} | \alpha_{0} + \alpha_{j}, \beta_{0} + \beta_{j} + \right) +\end{split}\end{align} + +where +\begin{align}\begin{split} + r_{ij} &= \frac{ + \exp \left( + \mathbb{E}_{q} \left[ \log \tau_{j} \right] - x_{i} \mathbb{E}_{q} \left[ \phi_{j}^{-1} \right] - m_{i} \mathbb{E}_{q} \left[ \log \phi_{j} \right] + \right) + }{ + \sum\limits_{k=1}^{K} \exp \left( + \mathbb{E}_{q} \left[ \log \tau_{k} \right] - x_{i} \mathbb{E}_{q} \left[ \phi_{k}^{-1} \right] - m_{i} \mathbb{E}_{q} \left[ \log \phi_{k} \right] + \right) + } \\ + \alpha_{j} &= \sum\limits_{i=1}^{N} \mathbb{E}_{q} \left[ Z_{ij} \right] m_{i} \\ + \beta_{j} &= \sum\limits_{i=1}^{N} \mathbb{E}_{q} \left[ Z_{ij} \right] x_{i} +\end{split}\end{align} + +Inference consists of cyclically updating $r_{ij}$, $\alpha_{j}$, and $\beta_{j}$ +until convergence. This can be done by substituting the following expectations at each update: +\begin{align*} + \mathbb{E}_{q} \left[ Z_{ij} \right] &= r_{ij} \\ + \mathbb{E}_{q} \left[ \log \tau_{j} \right] &= \psi \left( \alpha_{0} + \alpha_{j}) - \psi (K \alpha_{0} + \alpha_{1} + ... + \alpha_{K} \right) \\ + \mathbb{E}_{q} \left[ \phi_{j}^{-1} \right] &= \frac{\alpha_{0} + \alpha_{j}}{\beta_{0} + \beta_{j}} \\ + \mathbb{E}_{q} \left[ \log \phi_{j} \right] &= \log (\beta_{0} + \beta_{j}) - \psi (\alpha_{0} + \alpha_{j}) +\end{align*} + +\subsection{Choice of hyperparameters} + +There are three hyperparameters in the mixture model \ref{eq:brownian_mixture}: +\begin{itemize} + \item $\alpha_{0}$, the pseudocounts accorded to each state in the prior; + \item $\beta_{0}$, the prior scale parameter for each state; + \item $K$, the number of states. +\end{itemize} + +The last ($K$) is treated in the next section \ref{section:elbo}. The number of pseudocounts is often set to $\alpha_{0} = 2$ in \verb|vbdiff|, to avoid division by zero when evaluating prior means. For $\beta_{0}$, suppose we have some prior guess for the diffusion coefficient $D_{0}$. Since $\phi = 4 D \Delta t$ and the prior mean value of $\phi$ is $\beta_{0} / (\alpha_{0} - 1)$, a sensible choice is to set $\beta_{0} = 4 \Delta t (\alpha_{0} - 1) D_{0}$. This corresponds to the average sum of squared displacements for a trajectory with $\alpha_{0}$ jumps and diffusion coefficient $D_{0}$, sampled with frame interval $\Delta t$. + +\section{Evidence lower bound (ELBO)}\label{section:elbo} + +A central hyperparameter in the mixture model \ref{eq:brownian_mixture} is the number of states $K$. How do we choose $K$ in a principled way? \newline + +Besides providing an objective function for variational Bayesian inference, +the evidence lower bound \ref{eq:elbo} can also be used for \emph{model selection}. Because $\mathcal{L}[q] \leq \log p \left( \vec{X} \right)$, the ELBO provides an approximation to the probability of the observed data $\vec{X}$ averaged over all model parameters $\vec{X}, \boldsymbol{\tau}, \boldsymbol{\phi}$ corresponding to a particular $K$. If $K$ is too small, then there may not exist model parameters that adequately describe the data. If $K$ is too high, then even if model parameters exist that adequately describe the data, most do not. In short, we should take the $K$ with the highest $\mathcal{L}[q]$. \newline + +Expanding the ELBO equation \ref{eq:elbo}, we have +\begin{align} + \mathcal{L} [q] &= \mathbb{E}_{q} \left[ \log p \left( \vec{X} | \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) \right] \quad &\text{(term A)} \\ + &\quad + \mathbb{E}_{q} \left[ \log p \left( \vec{Z} | \boldsymbol{\tau} \right) \right] \quad &\text{(term B)} \\ + &\quad + \mathbb{E}_{q} \left[ \log p \left( \boldsymbol{\tau} \right) \right] \quad &\text{(term C)} \\ + &\quad + \mathbb{E}_{q} \left[ \log p \left( \boldsymbol{\phi} \right) \right] \quad &\text{(term D)} \\ + &\quad - \mathbb{E}_{q} \left[ \log q \left( \vec{Z} \right) \right] \quad &\text{(term E)} \\ + &\quad - \mathbb{E}_{q} \left[ \log q \left( \boldsymbol{\tau} \right) \right] \quad &\text{(term F)} \\ + &\quad - \mathbb{E}_{q} \left[ \log q \left( \boldsymbol{\phi} \right) \right] \quad &\text{(term G)} +\end{align} + +Separately, these terms are: +\begin{align*} + \mathbb{E}_{q} \left[ \log p \left( \vec{X} | \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) \right] &= \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{K} \mathbb{E}_{q} \left[ Z_{ij} \right] \Big( + (m_{i} - 1) \log x_{i} - \log \Gamma (m_{i}) \\ + &\qquad - m_{i} \mathbb{E}_{q} \left[ \log \phi_{j} \right] - x_{i} \mathbb{E}_{q} \left[ \phi_{j}^{-1} \right] \Big) \\ + \mathbb{E}_{q} \left[ \log p \left( \vec{Z} | \boldsymbol{\tau} \right) \right] &= \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{K} \mathbb{E}_{q} \left[ Z_{ij} \right] \mathbb{E}_{z} \left[ \log \tau_{j} \right] \\ + \mathbb{E}_{q} \left[ \log p \left( \boldsymbol{\tau} \right) \right] &= - \log B (\alpha_{0}, ..., \alpha_{0}) + \sum\limits_{j=1}^{K} (\alpha_{0} - 1) \mathbb{E}_{q} \left[ \log \tau_{j} \right] \\ + \mathbb{E}_{q} \left[ \log p \left( \boldsymbol{\phi} \right) \right] &= \sum\limits_{j=1}^{K} \left( + \alpha_{0} \log \beta_{0} - \log \Gamma (\alpha_{0}) - \beta_{0} \phi_{j}^{-1} - (\alpha_{0} + 1) \log \phi_{j} + \right) \\ + \mathbb{E}_{q} \left[ \log q \left( \vec{Z} \right) \right] &= \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{K} \mathbb{E}_{q} \left[ Z_{ij} \right] \log r_{ij} \\ + \mathbb{E}_{q} \left[ \log q \left( \boldsymbol{\tau} \right) \right] &= - \log B (\alpha_{0} + \alpha_{1}, ..., \alpha_{0} + \alpha_{K} ) + \sum\limits_{j=1}^{K} (\alpha_{0} + \alpha_{j} - 1) \mathbb{E}_{q} \left[ \log \tau_{j} \right] \\ + \mathbb{E}_{q} \left[ \log q \left( \boldsymbol{\phi} \right) \right] &= \sum\limits_{j=1}^{K} \Big( + (\alpha_{0} + \alpha_{j}) \log (\beta_{0} + \beta_{j}) - \log \Gamma (\alpha_{0} + \alpha_{j}) \\ + &\qquad - (\beta_{0} + \beta_{j}) \mathbb{E}_{q} \left[ \phi_{j}^{-1} \right] - (\alpha_{0} + \alpha_{j} + 1) \mathbb{E}_{q} \left[ \log \phi_{j} \right] + \Big) +\end{align*} + +where, as before, $B(x_{1}, ..., x_{K})$ is the multivariate beta function and the inner expectations are given by +\begin{align*} + \mathbb{E}_{q} \left[ Z_{ij} \right] &= r_{ij} \\ + \mathbb{E}_{q} \left[ \phi_{j}^{-1} \right] &= \frac{\alpha_{0} + \alpha_{j}}{\beta_{0} + \beta_{j}} \\ + \mathbb{E}_{q} \left[ \log \phi_{j} \right] &= \log (\beta_{0} + \beta_{j}) - \psi (\alpha_{0} + \alpha_{j}) \\ + \mathbb{E}_{q} \left[ \log \tau_{j} \right] &= \psi (\alpha_{0} + \alpha_{j}) - \psi (K \alpha_{0} + \alpha_{1} + ... + \alpha_{k} ) +\end{align*} + +where $\psi$ is the digamma function. + +\appendix + +\section{Appendix A: Derivation of approximate posterior for Brownian mixtures}\label{appendix:derivation_approx_posterior} + +To identify the exact form of the approximative posterior $q(\vec{X}, \boldsymbol{\tau}, \boldsymbol{\phi})$ for Brownian mixtures, we will solve the system of equations \ref{eq:em_equations} given the model specified by \ref{eq:eqA}, \ref{eq:eqB}, \ref{eq:eqC}, and \ref{eq:eqD}. \newline + +First we consider the second equation in \ref{eq:em_equations}. Writing this out, +\begin{align}\begin{split}\label{eq:int1} + \log q \left( \boldsymbol{\tau}, \boldsymbol{\phi} \right) &= \mathbb{E}_{\vec{Z} \sim q(\vec{Z})} \left[ \log p \left( \vec{X}, \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) \right] \\ + &= \sum\limits_{j=1}^{K} \left( \alpha_{0} - 1 + \sum\limits_{i=1}^{N} \mathbb{E}_{q} \left[ Z_{ij} \right] m_{i} \right) \log \tau_{j} \\ + &\quad - \sum\limits_{j=1}^{K} \left( \beta_{0} + \sum\limits_{i=1}^{N} \mathbb{E}_{q} \left[ Z_{ij} \right] x_{i} \right) \phi_{j}^{-1} \\ + &\quad - \sum\limits_{j=1}^{K} \left( + \alpha_{0} + 1 + \sum\limits_{i=1}^{N} \mathbb{E}_{q} \left[ Z_{ij} \right] m_{i} + \right) \log \phi_{j} \\ + &\qquad + \text{constant} +\end{split}\end{align} + +where the constant collects all terms that do not directly depend +on $\boldsymbol{\tau}$ or $\boldsymbol{\phi}$. We have made one +arbitrary modification in equation \ref{eq:int1}, weighting the statistical +weight of trajectory $i$ toward state $j$ by its number of jumps $m_{i}$ +rather than the same for all tracks. This accounts for a bias unmodeled +by the naive mixture \ref{eq:brownian_mixture} - fast states tend to produce +many short trajectories while slow states tend to produce a few long trajectories. Weighting statistical weight by the number of jumps in each trajectory removes some of this bias; the remaining bias can be addressed via defocalization corrections. \newline + +Examining the form of equation \ref{eq:int1}, we see that it is a product +of a Dirichlet distribution over $\boldsymbol{\tau}$ and independent inverse +gamma distributions over each scale parameter $\phi_{j}$: +\begin{align}\begin{split}\label{eq:int2} + q \left( \boldsymbol{\tau}, \boldsymbol{\phi} \right) &= q \left( \boldsymbol{\tau} \right) \prod\limits_{j=1}^{K} q \left( \phi_{j} \right) \\ + q \left( \boldsymbol{\tau} \right) &= \text{Dirichlet} \left( \boldsymbol{\tau} \ | \ \alpha_{0} + \alpha_{1}, ..., \alpha_{0} + \alpha_{K} \right) \\ + q \left( \phi_{j} \right) &= \text{InvGamma} \left( + \alpha_{0} + \alpha_{j}, \beta_{0} + \beta_{j} + \right) \\ + \alpha_{j} &= \sum\limits_{i=1}^{N} \mathbb{E}_{q} \left[ Z_{ij} \right] m_{i} \\ + \beta_{j} &= \sum\limits_{i=1}^{N} \mathbb{E}_{q} \left[ Z_{ij} \right] x_{i} +\end{split}\end{align} + +We still need to solve $q(\vec{Z})$. Expanding the first equation +in \ref{eq:em_equations}, we have +\begin{align*} + \log q \left( \vec{Z} \right) &= \mathbb{E}_{\boldsymbol{\tau}, \boldsymbol{\phi} \sim q (\boldsymbol{\tau}, \boldsymbol{\phi})} \left[ \log p \left( \vec{X}, \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} \right) \right] \\ + &= \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{K} Z_{ij} \Big( + \mathbb{E}_{q} \left[ \log \tau_{j} \right] + (m_{i}-1) \log x_{i} - x_{i} \mathbb{E}_{q} \left[ \phi_{j}^{-1} \right] \\ + &\quad - \log \Gamma (m_{i}) - m_{i} \mathbb{E}_{q} \left[ \log \phi_{j} \right] \Big) + \text{constant} +\end{align*} + +where, as before, we've aggregated terms that do not depend on $\vec{Z}$ +into the constant. +Normalizing over all states for each trajectory $i$, we have +\begin{align}\begin{split}\label{eq:int3} + q \left( \vec{Z} \right) &= \prod\limits_{i=1}^{N} \prod\limits_{j=1}^{K} r_{ij}^{Z_{ij}} \\ + r_{ij} &= \frac{ + \exp \left( + \mathbb{E}_{q} \left[ \log \tau_{j} \right] - x_{i} \mathbb{E}_{q} \left[ \phi_{j}^{-1} \right] - m_{i} \mathbb{E}_{q} \left[ \log \phi_{j} \right] + \right) + }{ + \sum\limits_{k=1}^{K} \exp \left( + \mathbb{E}_{q} \left[ \log \tau_{k} \right] - x_{i} \mathbb{E}_{q} \left[ \phi_{k}^{-1} \right] - m_{i} \mathbb{E}_{q} \left[ \log \phi_{k} \right] + \right) + } +\end{split}\end{align} + +Together, equations \ref{eq:int2} and \ref{eq:int3} specify the approximative posterior. The expectation-maximization routine cyclically updates the expectations in these equations, obtaining a progressively better estimate of the posterior. Convergence is guaranteed because the ELBO is convex with respect to each factor in $q$ \cite{boyd2004}. Since we know the forms of $q(\vec{Z})$, $q(\boldsymbol{\tau})$, and $q(\boldsymbol{\phi})$, the updates for each expectation can be evaluated in closed form: +\begin{align*} + \mathbb{E}_{q} \left[ Z_{ij} \right] &= r_{ij} \\ + \mathbb{E}_{q} \left[ \tau_{j} \right] &= \psi \left( \alpha_{0} + \alpha_{j} \right) - \psi \left( K \alpha_{0} + \alpha_{1} + ... + \alpha_{K} \right) \\ + \mathbb{E}_{q} \left[ \phi_{j}^{-1} \right] &= \frac{\alpha_{0} + \alpha_{j}}{\beta_{0} + \beta_{j}} \\ + \mathbb{E}_{q} \left[ \log \phi_{j} \right] &= \log (\beta_{0} + \beta_{j}) - \psi ( \alpha_{0} + \alpha_{j} ) +\end{align*} + +where $\alpha_{j}, \beta_{j}$ are defined as in equation \ref{eq:int2} and +$\psi(\tau) = \frac{\partial}{\partial \tau} \log \Gamma (\tau)$ is the +digamma function. + +\begin{thebibliography}{9} + +\bibitem{bishop2001} +Corduneanu A. \& Bishop C. M. Variational Bayesian model selection for mixture distributions. \emph{Artificial Intelligence and Statistics}, 27-34 (2001). + +\bibitem{bishop2006} +Bishop C. M. \emph{Pattern Recognition and Machine Learning}. Springer (2006). + +\bibitem{boyd2004} +Boyd S. \& Vandenberghe L. \emph{Convex Optimization}. Cambridge University Press (2004). + +\end{thebibliography} + +\end{document} diff --git a/tests/test_vbdiff.py b/tests/test_vbdiff.py index c105474..d103492 100644 --- a/tests/test_vbdiff.py +++ b/tests/test_vbdiff.py @@ -29,3 +29,23 @@ def test_end_to_end(self): assert (occs >= 0.0).all() assert (diff_coefs >= 0.0).all() assert abs(occs.sum() - 1.0) < 1e-5 + + def test_empty(self): + spots = pd.DataFrame( + { + "x": np.zeros(0, dtype=np.float64), + "y": np.zeros(0, dtype=np.float64), + "trajectory": np.zeros(0, dtype=np.int64), + "frame": np.zeros(0, dtype=np.int64), + } + ) + n_states = 2 + with self.assertRaises(RuntimeError): + occs, diff_coefs = vbdiff( + spots, + n_states=n_states, + pixel_size_um=0.16, + frame_interval=0.00748, + loc_error=0.035, + dz=0.7, + ) From 19611cf2d01af0d6b75f0845f47af95e3c4e91bb Mon Sep 17 00:00:00 2001 From: Alec Heckert Date: Sat, 3 Feb 2024 17:16:20 -0500 Subject: [PATCH 02/11] small corrections to ELBO calculation --- src/emdiff/vb.py | 48 +++++++++++++++++++++++++++++++++--------------- 1 file changed, 33 insertions(+), 15 deletions(-) diff --git a/src/emdiff/vb.py b/src/emdiff/vb.py index a81f2e2..bf49105 100644 --- a/src/emdiff/vb.py +++ b/src/emdiff/vb.py @@ -227,7 +227,8 @@ def corr_defoc(A, B, n): # beta factor for the inverse gamma posterior over phi. # Initially, we set this to the beta parameter corresponding # to the mean prior value of phi - B0 = phi * (A - 1) + # B0 = phi * (A - 1) + B0 = phi * A B = B0.copy() ## INITIALIZATION @@ -317,7 +318,7 @@ def corr_defoc(A, B, n): exp_log_occs = digamma(n) - digamma(n.sum()) exp_log_phi = np.log(B) - digamma(A) exp_inv_phi = A / B - elbo, model_likelihood = calc_elbo( + elbo, evA, evB, evC, evD, evE, evF, evG = calc_elbo( sum_r2, n_jumps, r, n, A, B, n0, A0, B0, exp_log_occs, exp_inv_phi, exp_log_phi ) @@ -345,7 +346,7 @@ def corr_defoc(A, B, n): # Return the parameters for the mean field approximation to the posterior # distribution if return_posterior: - return r, n, A, B, occs, D_mean, elbo, model_likelihood + return r, n, A, B, occs, D_mean, elbo, evA, evB, evC, evD, evE, evF, evG else: return occs, D_mean @@ -364,6 +365,7 @@ def calc_elbo( exp_inv_phi, exp_log_phi, identifiability_corr=True, + mode: int = 1, ): """ Calculate the variational evidence lower bound for @@ -398,10 +400,16 @@ def calc_elbo( - loggamma(n_jumps) - exp_log_phi[j] * n_jumps ) - evA = ((r * n_jumps) * _evA).sum() + if mode == 0: + evA = ((r * n_jumps) * _evA).sum() + elif mode == 1: + evA = (r * _evA).sum() # Due to the prior over state assignments (decreases with higher K; large magnitude) - evB = (exp_log_occs * (r * n_jumps).sum(axis=1)).sum() + if mode == 0: + evB = (exp_log_occs * (r * n_jumps).sum(axis=1)).sum() + elif mode == 1: + evB = (exp_log_occs * r.sum(axis=1)).sum() # Due to the prior over mixing coefficients (decreases with higher K; low magnitude) evC = (n0[0] - 1) * exp_log_occs.sum() - logbeta(*n0) @@ -410,8 +418,17 @@ def calc_elbo( evD = (A0 * np.log(B0) - loggamma(A0) - B0 * A / B - (A0 + 1) * exp_log_phi).sum() # Due to the posterior over state assignments (decreases with higher K; high magnitude) - nonzero = r > 0 - evE = (r[nonzero] * np.log(r[nonzero])).sum() + if mode == 0: + r_adj = r + 1e-8 + evE = ((r_adj * n_jumps) * np.log(r_adj)).sum() + elif mode == 1: + # nonzero = (r > 0).all(axis=0) + # evE = ((r[:, nonzero] * n_jumps[nonzero]) * np.log(r[:,nonzero])).sum() + + # print(f"(r<=0).sum() = {(r<=0).sum()}") + # print(f"(r>0).sum() = {(r>0).sum()}") + r_adj = r + 1e-8 + evE = (r_adj * np.log(r_adj)).sum() # Due to the posterior over mixing coefficients (increases with higher K; low magnitude) evF = ((n - 1) * exp_log_occs).sum() - logbeta(*n) @@ -424,17 +441,18 @@ def calc_elbo( # For checking numerical accuracy if False: - print("evA: ", evA) - print("evB: ", evB) - print("evC: ", evC) - print("evD: ", evD) - print("evE: ", evE) - print("evF: ", evF) - print("evG: ", evG) + print(f"K = {K}") + print(f"evA:\t{evA:.3f}") + print(f"evB:\t{evB:.3f}") + print(f"evC:\t{evC:.3f}") + print(f"evD:\t{evD:.3f}") + print(f"evE:\t{evE:.3f}") + print(f"evF:\t{evF:.3f}") + print(f"evG:\t{evG:.3f}") # Apply an identifiability correction (usually small in magnitude # compare to the other terms) if identifiability_corr: elbo += loggamma(K + 1) - return elbo, evA + return elbo, evA, evB, evC, evD, evE, evF, evG From db6a1c9e9023478db6289dfd8504fe5304157fdb Mon Sep 17 00:00:00 2001 From: Alec Heckert Date: Sat, 3 Feb 2024 14:17:41 -0800 Subject: [PATCH 03/11] bump emdiff version to 3 --- pyproject.toml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/pyproject.toml b/pyproject.toml index dbb3c4d..5ffb253 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -4,7 +4,7 @@ build-backend = "setuptools.build_meta" [project] name = "emdiff" -version = "2.0.0" +version = "3.0.0" requires-python = ">=3.7" readme = "README.md" authors = [ From 8e7b6c1dcb09c59c78967d24bbac3e4f2d0f898b Mon Sep 17 00:00:00 2001 From: Alec Heckert Date: Sun, 4 Feb 2024 13:25:54 -0500 Subject: [PATCH 04/11] add vbdiff.pdf --- doc/sim/README.md | 3 + doc/sim/plot_results.py | 141 ++++++++++++++ doc/sim/simulate.py | 333 +++++++++++++++++++++++++++++++++ doc/static/elbo_comparison.png | Bin 0 -> 444757 bytes doc/vbdiff.pdf | Bin 188628 -> 648963 bytes doc/vbdiff.tex | 135 ++++++++++++- 6 files changed, 611 insertions(+), 1 deletion(-) create mode 100644 doc/sim/README.md create mode 100644 doc/sim/plot_results.py create mode 100644 doc/sim/simulate.py create mode 100644 doc/static/elbo_comparison.png diff --git a/doc/sim/README.md b/doc/sim/README.md new file mode 100644 index 0000000..6ee1e27 --- /dev/null +++ b/doc/sim/README.md @@ -0,0 +1,3 @@ +This directory contains code to reproduce +the analysis in Figure 1 of the `vbdiff.pdf` +document. It requires the [strobesim](https://github.com/alecheckert/strobesim) package. diff --git a/doc/sim/plot_results.py b/doc/sim/plot_results.py new file mode 100644 index 0000000..6240b52 --- /dev/null +++ b/doc/sim/plot_results.py @@ -0,0 +1,141 @@ +#!/usr/bin/env python +import matplotlib +import matplotlib.pyplot as plt +import numpy as np +import os +import pandas as pd + + +matplotlib.rcParams["font.family"] = "sans-serif" +matplotlib.rcParams["font.sans-serif"] = "Arial" + + +def savefig(out_png: str): + plt.tight_layout() + plt.savefig(out_png, dpi=600) + plt.close() + os.system(f"open {out_png} -a Preview") + + +def open_spine(ax: matplotlib.axes.Axes): + for s in ["top", "right"]: + ax.spines[s].set_visible(False) + + +def add_log_ticks(ax: matplotlib.axes.Axes, side="x"): + """Force log-spaced major and minor ticks, regardless of + figure size.""" + if side == "x": + lo, hi = ax.get_xlim() + else: + lo, hi = ax.get_ylim() + log_lo = int(np.floor(np.log10(lo))) + log_hi = int(np.ceil(np.log10(hi))) + major_ticks = [] + minor_ticks = [] + for major in range(log_lo, log_hi + 1): + x = 10**major + major_ticks.append(x) + for i in range(10): + minor_ticks.append(i * x) + major_ticks = np.array(major_ticks) + minor_ticks = np.array(minor_ticks) + major_ticks = major_ticks[np.logical_and(major_ticks >= lo, major_ticks <= hi)] + minor_ticks = minor_ticks[np.logical_and(minor_ticks >= lo, minor_ticks <= hi)] + if side == "x": + ax.set_xticks(major_ticks, minor=False) + ax.set_xticks(minor_ticks, minor=True) + else: + ax.set_yticks(major_ticks, minor=False) + ax.set_yticks(minor_ticks, minor=True) + + +def figure1(): + """Show distribution of the max-ELBO state for each + simulation.""" + # 30000 simulated particles ~= 7000 observed tracks, since most + # simulated particles do not pass through the simulated focus. + conditions = [ + ("Mixture 1: $K_{true} = 1$", "results_1states_ntracks30000.csv"), + ("Mixture 2: $K_{true} = 2$", "results_2states_ntracks30000.csv"), + ("Mixture 3: $K_{true} = 3$", "results_3states_ntracks30000.csv"), + ("Mixture 4: $K_{true} = 4$", "results_4states_ntracks30000.csv"), + ("Mixture 5: $K_{true} = 5$", "results_5states_ntracks30000.csv"), + ("Mixture 6: $K_{true} = 6$", "results_6states_ntracks30000.csv"), + ] + + n_conditions = len(conditions) + fig, axes = plt.subplots( + n_conditions, 1, figsize=(2.5, n_conditions * 1), sharex=True + ) + fontsize = 12 + ind = np.arange(1, 8) + + for i, (title, csv) in enumerate(conditions): + ax = axes[i] + f = pd.read_csv(csv) + f["replicate"] = np.repeat(np.arange(len(f) // 7), 7) + by_state = pd.Series(np.zeros(len(ind), dtype=np.float64), index=ind) + obs = f.loc[f.groupby("replicate")["elbo"].idxmax()].groupby("n_states").size() + by_state.loc[obs.index] = obs + ax.bar(by_state.index, by_state, color="w", edgecolor="k", width=0.8) + open_spine(ax) + ax.tick_params(labelsize=fontsize) + # ax.set_ylabel("Frequency", fontsize=fontsize) + ax.set_title(title, fontsize=fontsize) + + axes[-1].set_xlabel("# states in model ($K$)", fontsize=fontsize) + axes[-1].set_xticks(by_state.index) + + savefig("figure1.png") + + +def figure2(): + """Show the ground truth models (diffusion coefficients + and occupations) for each of the simulations considered + in figure1.""" + models = [ + # 1 states + (np.array([5.0]), np.array([1.0])), + # 2 states + (np.array([5.0, 20.0]), np.array([0.5, 0.5])), + # 3 states + (np.array([0.1, 1.0, 5.0]), np.array([0.2, 0.4, 0.4])), + # 4 states + (np.array([0.02, 0.3, 2.0, 8.0]), np.array([0.1, 0.3, 0.2, 0.4])), + # 5 states + (np.array([0.04, 0.43, 1.2, 4.9, 11.0]), np.array([0.1, 0.15, 0.3, 0.25, 0.2])), + # 6 states + ( + np.array([0.01, 0.2, 0.9, 2.3, 6.6, 15.0]), + np.array( + [0.01278217, 0.31094062, 0.0178494, 0.44588117, 0.11149864, 0.101048] + ), + ), + ] + + n_models = len(models) + fig, axes = plt.subplots(n_models, 1, figsize=(2.5, 1 * n_models), sharex=True) + fontsize = 12 + xlim = (10 ** (-2.5), 10**2.5) + + for i, (diff_coefs, occs) in enumerate(models): + ax = axes[i] + for d, o in zip(diff_coefs, occs): + ax.plot([d, d], [0, o], color="r") + ax.set_xscale("log") + ax.set_xlim(xlim) + ax.set_ylim((0, ax.get_ylim()[1])) + ax.tick_params(labelsize=fontsize) + open_spine(ax) + ax.set_title("Mixture %d: $K_{true} = %d$" % (i + 1, i + 1), fontsize=fontsize) + add_log_ticks(ax, side="x") + + axes[-1].set_xlabel("Diff. coef. ($\mu$m$^{2}$/s)", fontsize=fontsize) + + savefig("figure2.png") + + +if __name__ == "__main__": + figure1() + figure2() diff --git a/doc/sim/simulate.py b/doc/sim/simulate.py new file mode 100644 index 0000000..d5474e2 --- /dev/null +++ b/doc/sim/simulate.py @@ -0,0 +1,333 @@ +#!/usr/bin/env python +import dask +import numpy as np +import pandas as pd +from dask.diagnostics import ProgressBar +from itertools import chain +from typing import List, Tuple +from emdiff.vb import vbdiff +from strobesim import strobe_multistate + + +FRAME_INTERVAL = 0.005 +FOCAL_DEPTH_UM = 0.7 +LOC_ERROR = 0.02 + + +def fit_vbdiff( + spots: pd.DataFrame, n_states: int +) -> Tuple[np.ndarray, np.ndarray, float, dict]: + """Fit an observed set of trajectories to a multi-state + Brownian mixture model using vbdiff. We run the algorithm + 5 times with randomly initialized starting diffusion + coefficients and take the resulting model with the highest + ELBO. + + Parameters + ---------- + spots : pandas.DataFrame with columns "frame", + "trajectory", "x", and "y" + n_states : number of states in the mixture model + + Returns + ------- + 1D ndarray of shape *n_states*, diffusion coefficients in + squared microns per sec; + 1D ndarray of shape *n_states*, state occupations; + float, evidence lower bound (ELBO); + dict with keys "A", "B", "C", "D", "E", "F", and "G", + individual terms contributing to ELBO + """ + diff_coefs = None + occs = None + elbo = -np.inf + evidence_terms = None + for _ in range(5): + guess_diff_coefs = np.random.gamma(1.0, 8.0, size=n_states) + ( + r, + n, + A, + B, + occs_, + diff_coefs_, + elbo_, + evA, + evB, + evC, + evD, + evE, + evF, + evG, + ) = vbdiff( + spots.copy(), + n_states=n_states, + pixel_size_um=1.0, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + pseudocounts=2.0, + return_posterior=True, + dz=FOCAL_DEPTH_UM, + guess=guess_diff_coefs, + ) + if elbo_ > elbo: + diff_coefs = diff_coefs_ + occs = occs_ + elbo = elbo_ + evidence_terms = { + "A": evA, + "B": evB, + "C": evC, + "D": evD, + "E": evE, + "F": evF, + "G": evG, + "n_states": n_states, + "elbo": elbo, + } + + return diff_coefs, occs, elbo, evidence_terms + + +def simulate_and_fit_1states(): + n_replicates = 64 + n_tracks = 10000 + results = [] + + @dask.delayed + def run_replicate(rep_idx: int) -> List[dict]: + spots = strobe_multistate( + n_tracks, + np.array([5.0]), + np.array([1.0]), + motion="brownian", + geometry="sphere", + radius=5.0, + dz=FOCAL_DEPTH_UM, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + track_len=100, + bleach_prob=0.1, + ) + results = [] + for n_states in [1, 2, 3, 4, 5, 6, 7]: + print(f"\nfitting replicate {rep_idx} to K={n_states}...") + _, _, _, evidence_terms = fit_vbdiff(spots, n_states=n_states) + results.append(evidence_terms) + return results + + tasks = map(run_replicate, range(n_replicates)) + with ProgressBar(): + results = dask.compute(*tasks, num_workers=16, scheduler="processes") + + results = pd.DataFrame(list(chain(*results))) + print(results) + results.to_csv(f"results_1states_ntracks{n_tracks}.csv", index=False) + + +def simulate_and_fit_2states(): + n_replicates = 64 + n_tracks = 10000 + results = [] + + @dask.delayed + def run_replicate(rep_idx: int) -> List[dict]: + spots = strobe_multistate( + n_tracks, + np.array([5.0, 20.0]), + np.array([0.5, 0.5]), + motion="brownian", + geometry="sphere", + radius=5.0, + dz=FOCAL_DEPTH_UM, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + track_len=100, + bleach_prob=0.1, + ) + results = [] + for n_states in [1, 2, 3, 4, 5, 6, 7]: + print(f"\nfitting replicate {rep_idx} to K={n_states}...") + _, _, _, evidence_terms = fit_vbdiff(spots, n_states=n_states) + results.append(evidence_terms) + return results + + tasks = map(run_replicate, range(n_replicates)) + with ProgressBar(): + results = dask.compute(*tasks, num_workers=16, scheduler="processes") + + results = pd.DataFrame(list(chain(*results))) + print(results) + results.to_csv(f"results_2states_ntracks{n_tracks}.csv", index=False) + + +def simulate_and_fit_3states(): + n_replicates = 64 + n_tracks = 10000 + results = [] + + @dask.delayed + def run_replicate(rep_idx: int) -> List[dict]: + spots = strobe_multistate( + n_tracks, + np.array([0.1, 1.0, 5.0]), + np.array([0.2, 0.4, 0.4]), + motion="brownian", + geometry="sphere", + radius=5.0, + dz=FOCAL_DEPTH_UM, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + track_len=100, + bleach_prob=0.1, + ) + results = [] + for n_states in [1, 2, 3, 4, 5, 6, 7]: + print(f"\nfitting replicate {rep_idx} to K={n_states}...") + _, _, _, evidence_terms = fit_vbdiff(spots, n_states=n_states) + results.append(evidence_terms) + return results + + tasks = map(run_replicate, range(n_replicates)) + with ProgressBar(): + results = dask.compute(*tasks, num_workers=16, scheduler="processes") + + results = pd.DataFrame(list(chain(*results))) + print(results) + results.to_csv(f"results_3states_ntracks{n_tracks}.csv", index=False) + + +def simulate_and_fit_4states(): + n_replicates = 64 + n_tracks = 10000 + results = [] + + @dask.delayed + def run_replicate(rep_idx: int) -> List[dict]: + spots = strobe_multistate( + n_tracks, + np.array([0.02, 0.3, 2.0, 8.0]), + np.array([0.1, 0.3, 0.2, 0.4]), + motion="brownian", + geometry="sphere", + radius=5.0, + dz=FOCAL_DEPTH_UM, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + track_len=100, + bleach_prob=0.1, + ) + results = [] + for n_states in [1, 2, 3, 4, 5, 6, 7]: + print(f"\nfitting replicate {rep_idx} to K={n_states}...") + _, _, _, evidence_terms = fit_vbdiff(spots, n_states=n_states) + results.append(evidence_terms) + return results + + tasks = map(run_replicate, range(n_replicates)) + with ProgressBar(): + results = dask.compute(*tasks, num_workers=16, scheduler="processes") + + results = pd.DataFrame(list(chain(*results))) + print(results) + results.to_csv(f"results_4states_ntracks{n_tracks}.csv", index=False) + + +def simulate_and_fit_5states(): + n_replicates = 64 + n_tracks = 10000 + results = [] + + @dask.delayed + def run_replicate(rep_idx: int) -> List[dict]: + spots = strobe_multistate( + n_tracks, + np.array([0.04, 0.43, 1.2, 4.9, 11.0]), + np.array([0.1, 0.15, 0.3, 0.25, 0.2]), + motion="brownian", + geometry="sphere", + radius=5.0, + dz=FOCAL_DEPTH_UM, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + track_len=100, + bleach_prob=0.1, + ) + results = [] + for n_states in [1, 2, 3, 4, 5, 6, 7]: + print(f"\nfitting replicate {rep_idx} to K={n_states}...") + _, _, _, evidence_terms = fit_vbdiff(spots, n_states=n_states) + results.append(evidence_terms) + return results + + tasks = map(run_replicate, range(n_replicates)) + with ProgressBar(): + results = dask.compute(*tasks, num_workers=16, scheduler="processes") + + results = pd.DataFrame(list(chain(*results))) + print(results) + results.to_csv(f"results_5states_ntracks{n_tracks}.csv", index=False) + + +def simulate_and_fit_6states(): + n_replicates = 64 + n_tracks = 10000 + results = [] + + @dask.delayed + def run_replicate(rep_idx: int) -> List[dict]: + spots = strobe_multistate( + n_tracks, + np.array([0.01, 0.2, 0.9, 2.3, 6.6, 15.0]), + np.array( + [0.01278217, 0.31094062, 0.0178494, 0.44588117, 0.11149864, 0.101048] + ), + motion="brownian", + geometry="sphere", + radius=5.0, + dz=FOCAL_DEPTH_UM, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + track_len=100, + bleach_prob=0.1, + ) + results = [] + for n_states in [1, 2, 3, 4, 5, 6, 7]: + print(f"\nfitting replicate {rep_idx} to K={n_states}...") + _, _, _, evidence_terms = fit_vbdiff(spots, n_states=n_states) + results.append(evidence_terms) + return results + + tasks = map(run_replicate, range(n_replicates)) + with ProgressBar(): + results = dask.compute(*tasks, num_workers=16, scheduler="processes") + + results = pd.DataFrame(list(chain(*results))) + print(results) + results.to_csv(f"results_6states_ntracks{n_tracks}.csv", index=False) + + +if __name__ == "__main__": + simulate_and_fit_1states() + simulate_and_fit_2states() + simulate_and_fit_3states() + simulate_and_fit_4states() + simulate_and_fit_5states() + simulate_and_fit_6states() + if False: + spots = strobe_multistate( + 10000, + np.array([0.02, 0.3, 2.0, 8.0]), + np.array([0.1, 0.3, 0.2, 0.4]), + motion="brownian", + geometry="sphere", + radius=5.0, + dz=FOCAL_DEPTH_UM, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + track_len=100, + bleach_prob=0.1, + ) + print(spots) + print(spots["trajectory"].nunique()) + print(spots.groupby("trajectory").size().mean()) diff --git a/doc/static/elbo_comparison.png b/doc/static/elbo_comparison.png new file mode 100644 index 0000000000000000000000000000000000000000..cd0e26208c4dba69b870949a5b4996c3f6db676c GIT binary patch literal 444757 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fast, simple tool for Brownian mixture modeling} \date{February 2024} \author{Alec Heckert} \maketitle @@ -23,18 +23,33 @@ over three factors: (1) the diffusion coefficients for each state, (2) the mixing coefficients, and (3) the origin state for each individual trajectory. Unlike more sophisticated frameworks like vbSPT, it does not consider state transitions. -As a consequence, it is much faster than these more general models. \newline +As a consequence, it is much faster than these more general models, +and was intended for inference on large-scale single particle tracking +(SPT) datasets. \newline -Here, we derive \verb|vbdiff|, starting from basic one-state Brownian motion. Many of these results are standard but scattered over different texts; the goal +Here, we derive \verb|vbdiff| starting from basic one-state Brownian motion. Many of these results are standard but scattered over different texts; the goal of this document is provide a unified, accessible introduction to the variational Brownian mixture model. We also give the evidence lower bound (ELBO) for \verb|vbdiff| in closed form. This can be used, for example, to choose the -appropriate number of states when modeling a Brownian mixture \cite{bishop2001}. +appropriate number of states when modeling a Brownian mixture \cite{bishop2001}. \newline + +This document was written rather quickly and paraphrases some +sections of the author's thesis. If you find errors, I would be +grateful (and will buy you a coffee) - let me know at +alecheckert@gmail.com. \tableofcontents \section{Brownian motion}\label{section:brownian_motion} +This section reviews Brownian mixture models, starting from +simple one-state Brownian motion. It defines several terms +relevant for the next section on the \verb|vbdiff| algorithm. +The central result is the mixture model represented by relations +\ref{eq:brownian_mixture} - \ref{eq:brownian_mixture_last}. + +\subsection{Single state Brownian motion} + A continuous-time stochastic process $B_{t} \in \mathbb{R}$ ($t \in \mathbb{R}$) is Brownian if \begin{align} B_{t} - B_{s} &\sim \mathcal{N} \left( 0, 2 D \left| t - s \right| \right) \qquad &\text{for any $t, s$} \label{eq:rule1} \\ @@ -129,7 +144,7 @@ \section{Brownian motion}\label{section:brownian_motion} throughout this text, with the understanding that generalizing to $d \neq 2$ requires substituting $m$ with $md / 2$. -\section{Bayesian inference on single-state Brownian motion} +\subsection{Bayesian inference on single-state Brownian motion} As a building block to mixture models, we first consider a simple Bayesian model for one-state Brownian motion. Suppose we have a @@ -142,10 +157,10 @@ \section{Bayesian inference on single-state Brownian motion} where $f_{X}(x | \phi)$ is given by equation \ref{eq:brownian_likelihood_3}. For the prior $f_{\phi}(\phi)$, we assume an inverse gamma distribution: -\begin{align*} +\begin{align}\begin{split}\label{eq:inverse_gamma_prior} \phi &\sim \text{InvGamma} \left( \alpha_{0}, \beta_{0} \right) \\ \Rightarrow f_{\phi} \left( \phi \right) &= \frac{\beta_{0}^{\alpha_{0}} e^{-\beta_{0} / \phi}}{\Gamma \left( \alpha_{0} \right) \phi^{\alpha_{0} + 1} }, \quad \phi \geq 0 -\end{align*} +\end{split}\end{align} Substituting into Bayes' theorem and retaining only factors that depend directly on $\phi$, @@ -167,8 +182,8 @@ \section{Bayesian inference on single-state Brownian motion} \phi \ | \ X \sim \text{InvGamma} \left( \alpha_{0} + m, \beta_{0} + x \right) \] -So - due to the conjugacy of the Brownian likelihood \ref{eq:browian_likelihood_3} -and the inverse gamma prior - inference boils down to simply adding +So - because the Brownian likelihood \ref{eq:brownian_likelihood_3} is conjugate to the inverse gamma prior \ref{eq:inverse_gamma_prior} - +inference boils down to simply adding shape and scale parameters. We can obtain a point estimate of the diffusion coefficient by taking the mean of the posterior distribution: \[ @@ -176,10 +191,9 @@ \section{Bayesian inference on single-state Brownian motion} \] Notice that the $-1$ in the denominator is analogous to the Bessel -correction for the sample variance, as the diffusion coefficient is essentially -a variance. +correction for the sample variance, as the diffusion coefficient is essentially a variance over position. -\section{Brownian mixtures} +\subsection{Brownian mixtures} Most samples of interest in SPT contain a mixture of particles in different states characterized by distinct diffusion coefficients. Suppose we have a dataset @@ -193,7 +207,7 @@ \section{Brownian mixtures} \boldsymbol{\tau} &\sim \text{Dirichlet} \left( \alpha_{0}, ..., \alpha_{0} \right), &\quad \boldsymbol{\tau} \in C^{K-1} \\ \phi_{j} &\sim \text{InvGamma} \left( \alpha_{0}, \beta_{0} \right), &\quad j \in \left\{ 1, 2, ..., K \right\} \\ \vec{Z}_{i} \ | \ \boldsymbol{\tau} &\sim \boldsymbol{\tau}, &\quad i \in \left\{ 1, 2, ..., N \right\} \\ - X_{i} \ | \ Z_{ij} = 1, \phi_{j} &\sim \text{Gamma} \left( m_{i}, \phi_{j} \right), &\quad i \in \left\{ 1, 2, ..., N \right\} + X_{i} \ | \ Z_{ij} = 1, \phi_{j} &\sim \text{Gamma} \left( m_{i}, \phi_{j} \right), &\quad i \in \left\{ 1, 2, ..., N \right\} \label{eq:brownian_mixture_last} \end{align} Breaking this down: @@ -240,13 +254,16 @@ \section{Brownian mixtures} Our goal is to infer a posterior distribution over the remaining variables given an observed set of trajectories. -\section{Approximative inference on Brownian mixtures} +\section{Variational Bayes inference on Brownian mixtures} + +\subsection{vbdiff} Given an observed set of trajectories $\vec{X} = (X_{1}, ..., X_{N})$, we want to infer the probability of the unobserved (latent) variables -$\vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi}$ in the mixture \ref{eq:brownian_mixture}. +$\vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi}$ in the mixture \ref{eq:brownian_mixture} - \ref{eq:brownian_mixture_last}. In other words, we want to infer the posterior distribution -$p \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} | \vec{X} \right)$. +$p \left( \vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi} | \vec{X} \right)$. \newline + Solving this problem exactly is intractable. There are two fallbacks: (a) approximate the distribution with a finite sequence of samples using MCMC methods or (b) construct a tractable approximation to the posterior. @@ -274,7 +291,7 @@ \section{Approximative inference on Brownian mixtures} \mathcal{L}[q] &\leq \log p (\vec{X}) \end{align*} -This last point is crucial, as it will form the basis for model selection considered later on. For now, notice that $p(\vec{X})$ can be seen as the average probability of the observed data over all possible model parameters $\vec{X}, \boldsymbol{\tau}, \boldsymbol{\phi}$ with some $K$. \newline +This last point is crucial, as it will form the basis for model selection considered later on. For now, notice that $p(\vec{X})$ can be seen as the average probability of the observed data over all possible latent parameters $\vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi}$ with some $K$. \newline What form will we choose for $q$? As it turns out, we only need to make a single constraint to find a tractable $q$: assume that $q$ factors into @@ -328,6 +345,8 @@ \section{Approximative inference on Brownian mixtures} \mathbb{E}_{q} \left[ \log \phi_{j} \right] &= \log (\beta_{0} + \beta_{j}) - \psi (\alpha_{0} + \alpha_{j}) \end{align*} +where $\psi(\alpha) = \frac{\D}{\D \alpha} \log \Gamma (\alpha)$ is the digamma function. + \subsection{Choice of hyperparameters} There are three hyperparameters in the mixture model \ref{eq:brownian_mixture}: @@ -337,14 +356,14 @@ \subsection{Choice of hyperparameters} \item $K$, the number of states. \end{itemize} -The last ($K$) is treated in the next section \ref{section:elbo}. The number of pseudocounts is often set to $\alpha_{0} = 2$ in \verb|vbdiff|, to avoid division by zero when evaluating prior means. For $\beta_{0}$, suppose we have some prior guess for the diffusion coefficient $D_{0}$. Since $\phi = 4 D \Delta t$ and the prior mean value of $\phi$ is $\beta_{0} / (\alpha_{0} - 1)$, a sensible choice is to set $\beta_{0} = 4 \Delta t (\alpha_{0} - 1) D_{0}$. This corresponds to the average sum of squared displacements for a trajectory with $\alpha_{0}$ jumps and diffusion coefficient $D_{0}$, sampled with frame interval $\Delta t$. +The last ($K$) is treated in the next section \ref{subsection:elbo}. The number of pseudocounts is often set to $\alpha_{0} = 2$ in \verb|vbdiff|, to avoid division by zero when evaluating prior means. For $\beta_{0}$, suppose we have some prior guess for the diffusion coefficient $D_{0}$. Since $\phi = 4 D \Delta t$ and the prior mean value of $\phi$ is $\beta_{0} / (\alpha_{0} - 1)$, a sensible choice is to set $\beta_{0} = 4 \Delta t (\alpha_{0} - 1) D_{0}$. This corresponds to the average sum of squared displacements for a trajectory with $\alpha_{0}$ jumps and diffusion coefficient $D_{0}$, sampled with frame interval $\Delta t$. -\section{Evidence lower bound (ELBO)}\label{section:elbo} +\subsection{Evidence lower bound (ELBO)}\label{subsection:elbo} A central hyperparameter in the mixture model \ref{eq:brownian_mixture} is the number of states $K$. How do we choose $K$ in a principled way? \newline Besides providing an objective function for variational Bayesian inference, -the evidence lower bound \ref{eq:elbo} can also be used for \emph{model selection}. Because $\mathcal{L}[q] \leq \log p \left( \vec{X} \right)$, the ELBO provides an approximation to the probability of the observed data $\vec{X}$ averaged over all model parameters $\vec{X}, \boldsymbol{\tau}, \boldsymbol{\phi}$ corresponding to a particular $K$. If $K$ is too small, then there may not exist model parameters that adequately describe the data. If $K$ is too high, then even if model parameters exist that adequately describe the data, most do not. In short, we should take the $K$ with the highest $\mathcal{L}[q]$. \newline +the evidence lower bound \ref{eq:elbo} can also be used for \emph{model selection}. Because $\mathcal{L}[q] \leq \log p \left( \vec{X} \right)$, the ELBO provides an approximation to the probability of the observed data $\vec{X}$ averaged over all model parameters $\vec{Z}, \boldsymbol{\tau}, \boldsymbol{\phi}$ corresponding to a particular $K$. If $K$ is too small, then there may not exist model parameters that adequately describe the data. If $K$ is too high, then even if model parameters exist that adequately describe the data, most do not. In short, we should take the $K$ with the highest $\mathcal{L}[q]$. This means we have both model parameters that adequately describe the data and do not have vast tracts of parameter space that do not describe the data. \newline Expanding the ELBO equation \ref{eq:elbo}, we have \begin{align} @@ -390,12 +409,12 @@ \section{Evidence lower bound (ELBO)}\label{section:elbo} \caption{Using the ELBO to determine the number of states $K$ in a Brownian mixture. We simulated Brownian mixtures with different numbers of states ($K_{\text{true}}$), then attempted to recover the correct $K$ using vbdiff. To do this, we fit each simulated mixture to models with $K=1$, 2, 3, 4, 5, 6, or 7 and took the $K$ with the highest ELBO. Brownian mixture simulations were performed with the \href{https://github.com/alecheckert/strobesim}{strobesim} package, which simulates SPT-like conditions including 20 nm localization error, 700 nm depth of field, bleaching rate 10 Hz, frame rate 200 Hz, and approximately 7000 observed trajectories per simulation. Mean trajectory length is $\sim 3-4$ frames under these conditions (depending on the simulation). (A) Each of the six Brownian mixtures simulated. Red bars indicate ground truth states and the height of the bars indicates each state's \emph{occupation} - the fraction of particles originating from that state. (B) Histogram of recovered $K$ over 64 replicates of each simulated mixture. Notice that uncertainty in $K$ rises as $K_{\text{true}}$ rises.} \end{figure} -\section{Addressing experimental biases} +\subsection{Experimental biases} \verb|vbdiff| also incorporates two experimental biases common in SPT experiments: localization error and defocalization. -\subsection{Localization error} +\subsubsection{Localization error} In real SPT experiments, the position of a particle isn't known exactly, but is always associated with some measurement error (known as @@ -432,7 +451,7 @@ \subsection{Localization error} $\phi_{j} = 4 (D \Delta t + \sigma_{\text{loc}}^{2})$ as the parameter for each state. -\subsection{Defocalization} +\subsubsection{Defocalization} In real SPT experiments, particles are only observed when their positions coincide with the imaging system's depth of field. @@ -517,7 +536,7 @@ \subsection{Defocalization} \appendix -\section{Appendix A: Derivation of approximate posterior for Brownian mixtures}\label{appendix:derivation_approx_posterior} +\section{Appendix A: Derivation of vbdiff approximate posterior}\label{appendix:derivation_approx_posterior} To identify the exact form of the approximative posterior $q(\vec{X}, \boldsymbol{\tau}, \boldsymbol{\phi})$ for Brownian mixtures, we will solve the system of equations \ref{eq:em_equations} given the model specified by \ref{eq:eqA}, \ref{eq:eqB}, \ref{eq:eqC}, and \ref{eq:eqD}. \newline From 89970df557b3c83ae3650fa76d2d8c0ca42bdabb Mon Sep 17 00:00:00 2001 From: Alec Heckert Date: Sun, 4 Feb 2024 14:07:07 -0500 Subject: [PATCH 06/11] clean up ELBO calculation --- doc/sim/simulate.py | 39 ++++++------------------------ src/emdiff/vb.py | 58 ++++++++++++++++++++++----------------------- 2 files changed, 35 insertions(+), 62 deletions(-) diff --git a/doc/sim/simulate.py b/doc/sim/simulate.py index d5474e2..dd33fe4 100644 --- a/doc/sim/simulate.py +++ b/doc/sim/simulate.py @@ -41,25 +41,10 @@ def fit_vbdiff( diff_coefs = None occs = None elbo = -np.inf - evidence_terms = None + elbo_terms = None for _ in range(5): guess_diff_coefs = np.random.gamma(1.0, 8.0, size=n_states) - ( - r, - n, - A, - B, - occs_, - diff_coefs_, - elbo_, - evA, - evB, - evC, - evD, - evE, - evF, - evG, - ) = vbdiff( + (r, n, A, B, occs_, diff_coefs_, elbo_terms) = vbdiff( spots.copy(), n_states=n_states, pixel_size_um=1.0, @@ -70,23 +55,13 @@ def fit_vbdiff( dz=FOCAL_DEPTH_UM, guess=guess_diff_coefs, ) - if elbo_ > elbo: + if elbo_terms["elbo"] > elbo: diff_coefs = diff_coefs_ occs = occs_ - elbo = elbo_ - evidence_terms = { - "A": evA, - "B": evB, - "C": evC, - "D": evD, - "E": evE, - "F": evF, - "G": evG, - "n_states": n_states, - "elbo": elbo, - } - - return diff_coefs, occs, elbo, evidence_terms + elbo = elbo_terms["elbo"] + elbo_terms["n_states"] = n_states + + return diff_coefs, occs, elbo, elbo_terms def simulate_and_fit_1states(): diff --git a/src/emdiff/vb.py b/src/emdiff/vb.py index bf49105..885484f 100644 --- a/src/emdiff/vb.py +++ b/src/emdiff/vb.py @@ -318,7 +318,7 @@ def corr_defoc(A, B, n): exp_log_occs = digamma(n) - digamma(n.sum()) exp_log_phi = np.log(B) - digamma(A) exp_inv_phi = A / B - elbo, evA, evB, evC, evD, evE, evF, evG = calc_elbo( + elbo = calc_elbo( sum_r2, n_jumps, r, n, A, B, n0, A0, B0, exp_log_occs, exp_inv_phi, exp_log_phi ) @@ -346,7 +346,7 @@ def corr_defoc(A, B, n): # Return the parameters for the mean field approximation to the posterior # distribution if return_posterior: - return r, n, A, B, occs, D_mean, elbo, evA, evB, evC, evD, evE, evF, evG + return r, n, A, B, occs, D_mean, elbo else: return occs, D_mean @@ -365,11 +365,11 @@ def calc_elbo( exp_inv_phi, exp_log_phi, identifiability_corr=True, - mode: int = 1, -): +) -> dict: """ Calculate the variational evidence lower bound for - a particular mixture model. + a particular mixture model. For more details on this + calculation, see `emdiff/doc/vbdiff.pdf`. args ---- @@ -383,11 +383,15 @@ def calc_elbo( returns ------- - ( - float, evidence lower bound; - float, model likelihood - ) - + dict with keys + "elbo": actual evidence lower bound, + "A": E[log p(X|Z,tau,phi)] + "B": E[log p(Z|tau)] + "C": E[log p(tau)] + "D": E[log p(phi)] + "E": E[log q(Z)] + "F": E[log q(tau)] + "G": E[log q(phi)] """ K, N = r.shape @@ -400,16 +404,10 @@ def calc_elbo( - loggamma(n_jumps) - exp_log_phi[j] * n_jumps ) - if mode == 0: - evA = ((r * n_jumps) * _evA).sum() - elif mode == 1: - evA = (r * _evA).sum() + evA = (r * _evA).sum() # Due to the prior over state assignments (decreases with higher K; large magnitude) - if mode == 0: - evB = (exp_log_occs * (r * n_jumps).sum(axis=1)).sum() - elif mode == 1: - evB = (exp_log_occs * r.sum(axis=1)).sum() + evB = (exp_log_occs * r.sum(axis=1)).sum() # Due to the prior over mixing coefficients (decreases with higher K; low magnitude) evC = (n0[0] - 1) * exp_log_occs.sum() - logbeta(*n0) @@ -418,17 +416,8 @@ def calc_elbo( evD = (A0 * np.log(B0) - loggamma(A0) - B0 * A / B - (A0 + 1) * exp_log_phi).sum() # Due to the posterior over state assignments (decreases with higher K; high magnitude) - if mode == 0: - r_adj = r + 1e-8 - evE = ((r_adj * n_jumps) * np.log(r_adj)).sum() - elif mode == 1: - # nonzero = (r > 0).all(axis=0) - # evE = ((r[:, nonzero] * n_jumps[nonzero]) * np.log(r[:,nonzero])).sum() - - # print(f"(r<=0).sum() = {(r<=0).sum()}") - # print(f"(r>0).sum() = {(r>0).sum()}") - r_adj = r + 1e-8 - evE = (r_adj * np.log(r_adj)).sum() + r_adj = r + 1e-8 + evE = (r_adj * np.log(r_adj)).sum() # Due to the posterior over mixing coefficients (increases with higher K; low magnitude) evF = ((n - 1) * exp_log_occs).sum() - logbeta(*n) @@ -455,4 +444,13 @@ def calc_elbo( if identifiability_corr: elbo += loggamma(K + 1) - return elbo, evA, evB, evC, evD, evE, evF, evG + return { + "elbo": elbo, + "A": evA, + "B": evB, + "C": evC, + "D": evD, + "E": evE, + "F": evF, + "G": evG, + } From 64553e61f1a4b765df6fc5cdad26c8d17796d301 Mon Sep 17 00:00:00 2001 From: Alec Heckert Date: Sun, 4 Feb 2024 14:18:52 -0500 Subject: [PATCH 07/11] change title --- doc/vbdiff.pdf | Bin 649839 -> 649527 bytes doc/vbdiff.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/vbdiff.pdf b/doc/vbdiff.pdf index 183ae32ddca6a5ac44005ee819240a5966b4af10..fdf6c0180be13317519cf408cf49cc0a8ec96a4f 100644 GIT binary patch delta 9874 zcmai(V{j%6tf*_--P*RTt!>+HZTGEh8(Z6UyR~h1Yumlwx##}BlX+&6nN0p9nF(%| zsM;2&5`@4wc@jmifYWh!TnN1DGsj`og zN@2h@O`{HkgciDbfbUn=2g4NLjGjQRpNr(Kwf9;59K{jQj^Ee_Snb_Ees71v8)h|- z-Tq-(LjBu1$WQw=en0=-*L8`LY#g7K{TeuRN(){>ccC>6fexQjCl{8@vA-hD;R>B1 zG!NpRv>^(K-<@ieTO2Z}R9iOjJml^xoa|U3R(Zf5P%=NS-cLSm#H^(TLXMaU3+|p; z=&sso%PbiVVmsX5euV=@U7co9P|oqpC8d;z3`8F(%`oEZO$mNMuT?fz-)wq(pw3G`+Gog4vZ zsG+HB!IbioCP1j{n3{h4|*p-;LR#(#tfO`EfC8*Rctyn!LP_Q zTnpvrRpoF>vf+G=!Q&?)b^8yIA89t5&t>my+PL4 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zf8tmox#R(+=}u4j8Ef3;z4;)YHmG$yf4`Ci!lgiF+#Dc!$*}YBNb>V?@$&P16X)g< z=NISTkz$wT{7*`7uuDtui;xTb|5uDMdY$0`(0uId|MNM(1(K$XYpDJp7~VeC$#pNH zKF!yZph??cz-`;6lKNh=qWcU@Ky04L&Vx5(?uarrR?j^jorlR&ZCJ|g71z-7_R{h^ z@f4WvzUA&K(yf8;V%&^|&MuXgQD0f|)BN6(%{2}pHOk_o}CWi}s0{*Ssk&cEO zmTI9Ti9-!S23X-p!vcX!6(9m;VMZ3m1WPaQG#SD~S7?1-A8=!*FVt3cq#CCwNY&>A zqcUfNaM%eiEHyF1;}&3Ph7zeErw0qvlPrY7%^+z1q^-wl4#q<+)Vy}a)tw9<6GNnj z@^jlv18~C&lxw-#!^_)?3(iQMUm36zn>4$_TXO|#0Si!@L-E>;inZ$_4*js+UcTAV z%*0jsBap!Q?RS}i!pFAT6KO$7w&^IVs)@xvH)bHbLX1oyF5m<4vdL^RHci=1;OTlH zkqtOJ(cL7gHG%oe-{u1fQdd`9T zBm)I}Um&R?S?=uX17EHoA(HlYF7@S^R2Jrg@hQl=t2e-(VA@T9kz<9iP`2U%y;x6>+rUSdysYGIeUA_Y& z2*TXcH^5J*serESv z(+3qx4{0zx>lTcl6nfPa&&$j~g;YhCRqclbeRJ_ Date: Sun, 4 Feb 2024 14:38:07 -0500 Subject: [PATCH 08/11] use max a priori guess for beta, to avoid instability when pseudocounts==1 --- src/emdiff/vb.py | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/src/emdiff/vb.py b/src/emdiff/vb.py index 885484f..77b6c51 100644 --- a/src/emdiff/vb.py +++ b/src/emdiff/vb.py @@ -226,8 +226,7 @@ def corr_defoc(A, B, n): # beta factor for the inverse gamma posterior over phi. # Initially, we set this to the beta parameter corresponding - # to the mean prior value of phi - # B0 = phi * (A - 1) + # to the max a priori value of phi B0 = phi * A B = B0.copy() From 54b620ababf1e5c5e731550eae09a39f55f01b99 Mon Sep 17 00:00:00 2001 From: Alec Heckert Date: Mon, 5 Feb 2024 10:48:06 -0500 Subject: [PATCH 09/11] add retries arg to vbdiff to make it easier to rerun with different initial parameters --- src/emdiff/vb.py | 70 ++++++++++++++++++++++++++++++++++---------- tests/test_vbdiff.py | 34 +++++++++++++++++++++ 2 files changed, 88 insertions(+), 16 deletions(-) diff --git a/src/emdiff/vb.py b/src/emdiff/vb.py index 77b6c51..2108a62 100644 --- a/src/emdiff/vb.py +++ b/src/emdiff/vb.py @@ -24,21 +24,22 @@ def vbdiff( - tracks, - n_states=2, - pixel_size_um=0.16, - frame_interval=0.01, - pos_cols=["y", "x"], - loc_error=0.035, - max_jumps_per_track=None, - start_frame=0, - max_iter=10000, - convergence=1.0e-8, - dz=np.inf, - guess=None, - pseudocounts=2.0, - return_posterior=False, - allow_neg_diff_coef=False, + tracks: pd.DataFrame, + n_states: int = 2, + pixel_size_um: float = 0.16, + frame_interval: float = 0.01, + pos_cols: list = ["y", "x"], + loc_error: float = 0.035, + max_jumps_per_track: int = None, + start_frame: int = 0, + max_iter: int = 10000, + convergence: float = 1e-8, + dz: float = np.inf, + guess: np.ndarray = None, + pseudocounts: float = 2.0, + return_posterior: bool = False, + allow_neg_diff_coef: bool = False, + retries: int = 0, ): """ Evaluate a variational Bayesian approximation to the posterior @@ -100,6 +101,10 @@ def vbdiff( negative values if the observed motion is actually slower than the user-provided localization error + retries : run multiple iterations of vbdiff with + different (random) initial diffusion + coefficients and return the one with the + highest ELBO returns ------- @@ -140,7 +145,6 @@ def vbdiff( D_mean, 1D ndarray of shape (n_states); the mean diffusion coefficients under the posterior model ) - """ # For internal convenience le2 = loc_error**2 @@ -342,6 +346,40 @@ def corr_defoc(A, B, n): if not allow_neg_diff_coef: D_mean[D_mean < 0] = 0 + # Rerun algorithm multiple times with different initial guesses, + # if desired. Note that this may return different answers for + # multiple runs to vbdiff. If this is not desired, call + # np.random.seed prior to running vbdiff. + if retries > 0: + for _ in range(retries): + guess_diff_coefs = np.random.gamma(1, 8, size=n_states) + r_, n_, A_, B_, occs_, D_mean_, elbo_ = vbdiff( + tracks=tracks, + n_states=n_states, + pixel_size_um=pixel_size_um, + frame_interval=frame_interval, + pos_cols=pos_cols, + loc_error=loc_error, + max_jumps_per_track=max_jumps_per_track, + start_frame=start_frame, + max_iter=max_iter, + convergence=convergence, + dz=dz, + guess=guess_diff_coefs, + pseudocounts=pseudocounts, + return_posterior=True, + allow_neg_diff_coef=allow_neg_diff_coef, + retries=0, + ) + if elbo_["elbo"] > elbo["elbo"]: + r = r_ + n = n_ + A = A_ + B = B_ + occs = occs_ + D_mean = D_mean_ + elbo = elbo_ + # Return the parameters for the mean field approximation to the posterior # distribution if return_posterior: diff --git a/tests/test_vbdiff.py b/tests/test_vbdiff.py index d103492..4cb04fe 100644 --- a/tests/test_vbdiff.py +++ b/tests/test_vbdiff.py @@ -49,3 +49,37 @@ def test_empty(self): loc_error=0.035, dz=0.7, ) + + def test_retries(self): + """Multiple runs with different initial guesses + should return the same answer if we call np.random.seed + in advance.""" + target = FIXTURES / "sample_tracks.zip" + assert target.is_file() + spots = pd.read_csv(str(target)) + n_states = 7 + retries = 5 + np.random.seed(1) + occs1, diffcoefs1 = vbdiff( + spots, + n_states=n_states, # number of diffusive states + pixel_size_um=0.16, # microns + frame_interval=0.00748, # seconds + loc_error=0.035, # microns + dz=0.7, # focal depth, microns + max_iter=10, + retries=retries, + ) + np.random.seed(1) + occs2, diffcoefs2 = vbdiff( + spots, + n_states=n_states, # number of diffusive states + pixel_size_um=0.16, # microns + frame_interval=0.00748, # seconds + loc_error=0.035, # microns + dz=0.7, # focal depth, microns + max_iter=10, + retries=retries, + ) + np.testing.assert_allclose(occs1, occs2, atol=1e-6, rtol=1e-6) + 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@@ -57,7 +57,7 @@ \subsection{Single state Brownian motion} \end{align} where $\mathcal{N} \left( 0, 2 D \left| t - s \right| \right)$ is a normal distribution -with mean 0 and variance $2 D \left| t - s \right|$. \newline +with mean 0 and variance $2 D \left| t - s \right|$. $D \geq 0$ is a parameter called the \emph{diffusion coefficient}, parametrizing the scale of the motion. \newline Suppose we measure a Brownian trajectory in a single particle tracking (SPT) experiment. We sample the particle's position at $m+1$ timepoints @@ -92,7 +92,7 @@ \subsection{Single state Brownian motion} Substituting the normal distribution from equation \ref{eq:rule1} as $f_{\Delta B}$, we have \[ - f_{\Delta B^{2}}(x) = \frac{ e^{-x / 4 D \Delta t} }{ \sqrt{4 \pi D \Delta t x} } + f_{\Delta B^{2}}(x) = \frac{ e^{-x / 4 D \Delta t} }{ \sqrt{4 \pi D \Delta t x} }, \quad x \geq 0 \] which we recognize as a gamma distribution with parameters $\alpha = 1/2$ and From e5e671141e6f500e4b847cd1984c7cfd94cb81f7 Mon Sep 17 00:00:00 2001 From: Alec Heckert Date: Mon, 5 Feb 2024 13:48:09 -0500 Subject: [PATCH 11/11] use new retries arg in simulate.py --- doc/sim/simulate.py | 34 +++++++++++----------------------- 1 file changed, 11 insertions(+), 23 deletions(-) diff --git a/doc/sim/simulate.py b/doc/sim/simulate.py index dd33fe4..a4aa3e1 100644 --- a/doc/sim/simulate.py +++ b/doc/sim/simulate.py @@ -38,29 +38,17 @@ def fit_vbdiff( dict with keys "A", "B", "C", "D", "E", "F", and "G", individual terms contributing to ELBO """ - diff_coefs = None - occs = None - elbo = -np.inf - elbo_terms = None - for _ in range(5): - guess_diff_coefs = np.random.gamma(1.0, 8.0, size=n_states) - (r, n, A, B, occs_, diff_coefs_, elbo_terms) = vbdiff( - spots.copy(), - n_states=n_states, - pixel_size_um=1.0, - frame_interval=FRAME_INTERVAL, - loc_error=LOC_ERROR, - pseudocounts=2.0, - return_posterior=True, - dz=FOCAL_DEPTH_UM, - guess=guess_diff_coefs, - ) - if elbo_terms["elbo"] > elbo: - diff_coefs = diff_coefs_ - occs = occs_ - elbo = elbo_terms["elbo"] - elbo_terms["n_states"] = n_states - + r, n, A, B, occs, diff_coefs, elbo_terms = vbdiff( + spots.copy(), + n_states=n_states, + pixel_size_um=1.0, + frame_interval=FRAME_INTERVAL, + loc_error=LOC_ERROR, + pseudocounts=2.0, + return_posterior=True, + dz=FOCAL_DEPTH_UM, + retries=6, + ) return diff_coefs, occs, elbo, elbo_terms

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