From 53a91fd05dfe9f999e86455e74dcdf403e882109 Mon Sep 17 00:00:00 2001 From: John Mason Date: Wed, 17 Oct 2018 16:36:01 -0700 Subject: [PATCH 1/2] Graphical demonstration of a simple parameter estimation problem. --- .gitignore | 7 + .../simple_parameter_estimation/README.md | 30 ++ .../description.pdf | Bin 0 -> 352174 bytes .../description.tex | 223 ++++++++++ .../simple_parameter_estimation/main.py | 395 ++++++++++++++++++ .../out/figure1-raw_objective_values.pdf | Bin 0 -> 13879 bytes .../out/figure2-log_objective_values.pdf | Bin 0 -> 21260 bytes .../out/figure3-derivative_alpha.pdf | Bin 0 -> 18842 bytes .../out/figure4-derivative_beta.pdf | Bin 0 -> 22664 bytes .../figure5-best_local_search_direction.pdf | Bin 0 -> 10574 bytes .../out/figure6-gradient_descent_flow.pdf | Bin 0 -> 103279 bytes 11 files changed, 655 insertions(+) create mode 100644 prototypes/simple_parameter_estimation/README.md create mode 100644 prototypes/simple_parameter_estimation/description.pdf create mode 100644 prototypes/simple_parameter_estimation/description.tex create mode 100755 prototypes/simple_parameter_estimation/main.py create mode 100644 prototypes/simple_parameter_estimation/out/figure1-raw_objective_values.pdf create mode 100644 prototypes/simple_parameter_estimation/out/figure2-log_objective_values.pdf create mode 100644 prototypes/simple_parameter_estimation/out/figure3-derivative_alpha.pdf create mode 100644 prototypes/simple_parameter_estimation/out/figure4-derivative_beta.pdf create mode 100644 prototypes/simple_parameter_estimation/out/figure5-best_local_search_direction.pdf create mode 100644 prototypes/simple_parameter_estimation/out/figure6-gradient_descent_flow.pdf diff --git a/.gitignore b/.gitignore index bc7ed1967e..5ad4d8b757 100644 --- a/.gitignore +++ b/.gitignore @@ -142,3 +142,10 @@ reconstruction/ecoli/dataclasses/process/two_component_system_odes_fitter.py # Mongo Explorer plugin: .idea/mongoSettings.xml + +# LaTeX compilation files # +########################### +*.aux +*.fdb_latexmk +*.fls +*.synctex.gz diff --git a/prototypes/simple_parameter_estimation/README.md b/prototypes/simple_parameter_estimation/README.md new file mode 100644 index 0000000000..f39889e3c2 --- /dev/null +++ b/prototypes/simple_parameter_estimation/README.md @@ -0,0 +1,30 @@ + +A demonstration of a simple parameter estimation procedure on a standard +Michaelis-Menten rate law, with a desired rate of reaction and no prior +observed parameter values. The system has been reduced and transformed into +two parameters for better visualization (but without loss of generality). + +Dependencies +------------ + +Requires `numpy` and `matplotlib`. Designed for Python 2 but should be readily portable to Python 3. + +Execution +--------- + +Call `.\main.py` or `python main.py`. + +Output +------ + +Six single-panel figures (as PDFs) will be output into the `out` directory. If no `out` directory exists, it will be created. + +Extension +--------- + +The constants in `main.py` should be well documented. Modifying the underlying model or optimization problem would require an extensive rewrite. + +Long description +---------------- + +See `description.pdf`. diff --git a/prototypes/simple_parameter_estimation/description.pdf b/prototypes/simple_parameter_estimation/description.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0a0bef144310112f0bbe4ec88099674544967e53 GIT binary patch literal 352174 zcmeFZby#G}k}rz8JB_)_ugk_pV|A) zoOy4)ci$hkzK>dKRYhh-Mr1}rM*Ip2B{2yG03$mb#q|8p5*#ZrGqIzI4ICdI9Fv^6 zgQc4lF@TGMo%pvG9Fv5#otwD}F_VOyv75P=xv8U>Ih=q1oU5CQxv@Q**K(JZtm6*} zwC*F#^Je_2(gFX^S%4rAB)ckf>S0o_Ta^9jtwuII*D&-%Q|-w-MgB~(IC1T@aRas& zY7I?IzZ>-?1AYDXcM{8TJo!cJ-@nOI%9Fs+HK^$1)1~O!*RNl#XRhYQRyHxJ`NQ}v z4FPWXVh5Iv`{lGtJ-5hntp`7`=y-spVG(!WhPG>twvN*4t8X+w9L#p74az*4#mMYc1(_i|8 z?j%C3m9*IFa?7rp`zfDY2>diR{;Nz7Ar9m_G^NH5QXhWSy!F;1NdgQV&X%Tj6akZ!C z(d@ z2B>!vCtlO)FyXbj;LxjExIUj&CeeV7U)qnyqBaWzV%hFkIbhpkVa%0IN z-*GPqAuG_3P;zgx+-@UaF})GyQI>sW@cFXQ!Wk>8AkmulTdDy$NgW3CIA+Py1`7LK 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kinetic rate law} +\author{John Mason} +\date{October 2018} + +\begin{document} + +\maketitle + +\section{Background} + +This is a highly reduced basis to serve as an introduction to my broader parameter estimation work (forthcoming publication). + +\section{Model} + +For this demonstration, I will use a basic Michaelis-Menten kinetic rate law: + +\begin{align} +v = k E \frac{C}{C + K} \label{equation:MM} +\end{align} + +Note the simplified notation -- for this level of complexity, I want to avoid superscripts and subscripts as much as possible. Here \(k\) is the catalytic rate constant, \(E\) is the enzyme concentration, \(C\) is the metabolite concentration, and \(K\) is the so-called Michaelis-Menten constant (an effective saturation constant). + +\section{Objective} + +Our goal is to match some target rate of reaction, \(v^\star\). We are not considering any prior information about parameter values, so our system is not necessarily an optimization problem -- we simply wish to solve \(v = v^\star\). For the sake of generality, however, I will express this as an optimization problem. + +In this context we want the residual \(v - v^\star\) to be as small (in the sense of closeness to zero) as possible. Thus we need to apply some metric to this residual. For purposes that I won't expand here, minimizing the square of this residual is a practical choice: + +\begin{align} +\min_{k, E, C, K} \left(v - v^\star\right)^2 +\end{align} + +For expressive convenience I will define \(f = \left(v - v^\star\right)^2\). + +\section{The objective value space} + +As previously mentioned, this problem can be solved directly via analytic approaches. For demonstrative purposes I will instead be using graphical approaches. + +Graphical approaches only work in low dimensionality space. Unfortunately we have a function of four variables that we wish to illustrate. This isn't an issue if we lump together parameters. + +Choosing \(m = k E\) as the maximum velocity, and \(s = c/K\) as the saturation ratio, we can instead write + +\begin{align} +v = m \frac{s}{s + 1} +\end{align} + +This is a problem in two variables. Thus we can now sweep \(m\) and \(s\), and plot a heat or height map of \(f\). + +Another observation. Both our original and lumped parameters are strictly non-negative values, and do tend to vary over order of magnitude. Thus it is possible and convenient to work in logarithmically rescaled parameters\footnote{There is some weirdness about zero, since the logarithm is not finite at this limit. In practice, this isn't terribly important, although it can lead to some numerical weirdness.}. I defined \(\alpha = \ln m\) and \(\beta = \ln s\). I give these no particular names -- just think of them as the magnitudes (exponents) of their corresponding lumped variables. Now, + +\begin{align} +v &= \frac{ e^{\alpha + \beta}}{e^\beta + 1} +\\ &= e^\alpha \frac{1}{1 + e^{-\beta}} +\end{align} + +This new form is very nice for exploring the parameter space, as we simply need to vary \(\alpha\) and \(\beta\) linearly. This is demonstrated in Figure \ref{figure:objective}. + +\begin{figure}[!ht] +\centering +\includegraphics{figure1-raw_objective_values} +\caption{\textbf{Raw objective value sweep.}} +\label{figure:objective} +\end{figure} + +Unfortunately the objective grows rapidly for large \(\alpha\) and \(\beta\). It's easier to see the overall trend by plotting \(\ln f\), as shown in Figure \ref{figure:log-objective}. Since \(\ln x\) is a monotonic function, this does not change the optimization problem. + +\begin{figure}[!ht] +\centering +\includegraphics{figure2-log_objective_values} +\caption{\textbf{Logarithmically scaled objective value sweep.}} +\label{figure:log-objective} +\end{figure} + +Now there is an obvious `valley' where the objective is minimized, corresponding to the 1D space of values for which combinations of the parameters solve our optimization problem. Already it should be apparent that our choice of lumped parameters was more than a convenience -- for large \(\beta\), the correct choice approaches \(\alpha \approx 0\); on the other end, we have \(\beta \approx -\alpha\). + +This is consistent with our knowledge about the problem. When \(\beta\) is large, the system is highly saturated, and then all that matters is the maximum rate of reaction \(\alpha\). In the opposite regime, near the limit of mass-action kinetics, a decrease in saturation requires a commensurate increase in the maximum rate of reaction to achieve the target rate. + +\section{The gradient value space} + +Most optimization algorithms are, at some level, performing a gradient descent. Even if gradient descent can't result in a global solution, the gradient can still provide useful local information. + +In our case, the gradient is just the two partial derivatives \(\partial f/\partial \alpha\) and \(\partial f / \partial \beta\). Since \(f\) is hard to visualize, I again instead look at the gradient on the logarithm of \(f\), which is just + +\begin{align} +\frac{\partial \ln f}{\partial \alpha} = \frac{1}{f} \frac{\partial f}{\partial \alpha} +\end{align} + +and + +\begin{align} +\frac{\partial \ln f}{\partial \beta} = \frac{1}{f} \frac{\partial f}{\partial \beta} +\end{align} + +These terms are depicted in Figures \ref{figure:derivative-alpha} and \ref{figure:derivative-beta}. + +\begin{figure}[!ht] +\centering +\includegraphics{figure3-derivative_alpha} +\caption{\textbf{Derivative of objective with respect to \(\alpha\).}} +\label{figure:derivative-alpha} +\end{figure} + +\begin{figure}[!ht] +\centering +\includegraphics{figure4-derivative_beta} +\caption{\textbf{Derivative of objective with respect to \(\beta\).}} +\label{figure:derivative-beta} +\end{figure} + +What we see is consistent with the analysis of Figure \ref{figure:log-objective}. Gradients, unfortunately, are somewhat hard to interpret. Another way to look at this is by just computing the direction of the optimal local\footnote{Note that this is not the \textit{global} best search direction, which would corrspond to snapping directly to the curved line of solutions. In gradient descent, you're only allowed to look at the slope of the ground at your feet, so to speak.} gradient descent, as shown in Figure \ref{figure:search-direction}. + +\begin{figure}[!ht] +\centering +\includegraphics{figure5-best_local_search_direction} +\caption{\textbf{Best local search direction.} The color at each point indicates the direction (in radians, with 0 corresponding to `right' and moving counter-clockwise) of the direction of gradient descent. Roughly, `red' corresponds to `left', `cyan' corresponds to `right', `yellow-green' to `down', and `blue-violet' to `up'.} +\label{figure:search-direction} +\end{figure} + +From this depiction we can clearly see that there are four regimes. At the top these correspond to moving `right' or `left' in the parameter space, while at the bottom, the best local search direction is diagonal (up-right or down-left). This is easier to see by looking at a flow diagram, as in Figure \ref{figure:flow}. + +\begin{figure}[!ht] +\centering +\includegraphics{figure6-gradient_descent_flow} +\caption{\textbf{Gradient descent flow diagram.}} +\label{figure:flow} +\end{figure} + +The purposes of this exercise are as follows: + +\begin{itemize} + \item To develop an intuition about the parameter space, and how that corresponds to the objective space. + \item To show how a transformation can simplify both the visualization and interpretation of the problem. + \item To demonstrate that \(\alpha\) and \(\beta\) are a logical parameter basis for these problems. +\end{itemize} + +\section{Parameter space exploration} + +\(\alpha\) and \(\beta\) are not the only possible choices for lumped parameters in this model, but they probably are the best. Remember, \(\alpha\) corresponds to the maximum rate of reaction, while \(\beta\) corresponds to the degree of saturation. A third parameter, \(\gamma = \alpha + \beta\), corresponds to the rate of reaction in the limit of mass-action kinetics. When divided by the enzme concentration, this term is often called the `catalytic efficiency'\footnote{For reasons, I'll add, that I don't really understand.}. Any two of \(\alpha\), \(\beta\), and \(\gamma\) are sufficient to define this system. Which is the `right' choice? + +Let's define a few things. + +\begin{align} +a &= \ln k +\\ b &= \ln E +\\ c &= \ln C +\\ d &= \ln K +\end{align} + +These four new variables are just logarithms of our original independent parameters, and thus themselves are an independent parameter basis. Now, + +\begin{align} +\alpha &= a + b +\\ \beta &= c - d +\\ \gamma &= a + b + c -d +\end{align} + +Already we see something critical -- \(\alpha\) and \(\beta\) are independent with respect to the underlying independent model parameters, while \(\gamma\) collides with both. In practical terms, we can modify \(a\) or \(b\) to modify \(\alpha\) without disrupting \(\beta\), and vice-versa for \(c\) and \(d\). Thus, even though they represent lumped quantities, we can safely claim to modify \(\alpha\) and \(\beta\) as if they were independent, as it's trivial to resolve that into changes into the underlying model parameters. This isn't the case with \(\gamma\) and \(\alpha\) or \(\beta\). + +Regardless, let's bully ahead with \(\gamma\) and see what happens. I'm going to choose \(\alpha\) and \(\gamma\) as my lumped model parameters to explore, since they both correspond to rates (but in different model regimes). This gives us the model equation + +\begin{align} +v = \frac{1}{e^{-\alpha} + e^{-\gamma}} +\end{align} + +The derivation of this form, and its graphical exploration, are left as exercises to the reader\footnote{Meaning that I've lost interest in writing any more code. However I promise you that it has an interesting shape.}. I do think this form is quite suggestive, given its symmetric nature. + +Now, we can claim to modify these two values independently, but how do we resolve this into real parameter values if their underlying definitions are in fact interdependent? This relation is expressed in matrix-vector form as + +\begin{align} +\underbrace{\begin{bmatrix} \alpha \\ \gamma \end{bmatrix}}_{=y} += \underbrace{\begin{bmatrix} +1 & +1 & \phantom{\pm}0 & \phantom{\pm}0 \\ +1 & +1 & +1 & -1 \end{bmatrix}}_{=\begin{bmatrix}a_1 & a_2\end{bmatrix}^\text{T} = A} +\underbrace{\begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix}}_{=x} +\end{align} + +This new matrix form \(y = A x\) will simplify our analysis. Our question, more explicitly, is this: given values of \(y\), how can we find \(x\)? This is an underdetermined problem -- which is good, but means we need some way to manage the degrees of freedom. Taking the Moore-Penrose pseudoinverse of \(A\), we can find an \(x\) for any \(y\) as + +\begin{align} +x = A^\dagger y +\end{align} + +where + +\begin{align} +A^\dagger = \frac{1}{2} \begin{bmatrix} ++1 & \phantom{\pm}0 \\ ++1 & \phantom{\pm}0 \\ +-1 & +1 \\ ++1 & -1 +\end{bmatrix} +\end{align} + +Now, what exactly is going on here? First off, let's note that the second column in \(A^\dagger\) corresponds to \(\beta\). That's pretty interesting, considering that we haven't brought up \(\beta\) at all in this problem. It fell out naturally. + +How about that first vector? Interestingly, the dot product of this vector with the vector of coefficients defing \(\gamma\) is zero -- they're orthogonal! So, on a per-column basis, these vector represent independent means of modifying the underlying independent parameters to obtain a change in one of \(\alpha\) or \(\beta\) without changing the value of the other. + +If you follow that -- and I know it's difficult -- then it's worth reexamining the case where we choose \(\alpha\) and \(\beta\) as our lumped parameters, as in the previous section. Then + +\begin{align} +A^\dagger = \frac{1}{2} \begin{bmatrix} ++1 & \phantom{\pm}0 \\ ++1 & \phantom{\pm}0 \\ +\phantom{\pm}0 & +1 \\ +\phantom{\pm}0 & -1 +\end{bmatrix} +\end{align} + +These columns are just our definitions of \(\alpha\) and \(\beta\), reemphasizing their intrinsically independent nature\footnote{Perhaps it would be more appropriate to start calling these orthogonal.}. + +This begins to get at the parsimonious perturabtion approach I take in my larger parameter estimtion system. In short -- regardless of the basis taken, it's possible to explore the parameter space in a fashion that perturbs combined parameters of interest (like a maximum rate of reaction) while holding others as close to constant as possible. + +What I haven't answered -- and what is, in truth, a lingering question, is which parameter space is the most natural. The ideal parameter space would reorient the system such that perturbation to the parameters would most reliably and rapidly yield net movement towards the global (true) solution. Apart from empirically testing alternatives, I'm not sure which choice is best. For my current parameter estimation system, I use the equivalents of \(\alpha\) and \(\beta\), though \(\gamma\)-like parameters in place of \(\beta\)-like would be interesting to test. + +\end{document} diff --git a/prototypes/simple_parameter_estimation/main.py b/prototypes/simple_parameter_estimation/main.py new file mode 100755 index 0000000000..6ac3a06bdb --- /dev/null +++ b/prototypes/simple_parameter_estimation/main.py @@ -0,0 +1,395 @@ +#!/usr/bin/python + +''' + +A demonstration of a simple parameter estimation procedure on a standard +Michaelis-Menten rate law, with a desired rate of reaction and no prior +observed parameter values. The system has been reduced and transformed into +two parameters for better visualization (but without loss of generality). + +TODO: shrink color bars, better ticks + +''' + +# Imports + +# Future features +from __future__ import absolute_import, division, print_function + +# Built-ins +import os + +# Third-party +import numpy as np + +import matplotlib.pyplot as plt + + +# Constants (independent and derived) + +# Optimization constants + +TARGET_RATE_OF_REACTION = 1.0 + +# Parameter sweep constants + +# The parameter search window (up and down). Should be small and positive, no +# less than 1e-15 (float64 precision limit) and no greater than 1. +SPREAD = 1e-3 + +ALPHA_MIN = np.log(TARGET_RATE_OF_REACTION * SPREAD) +ALPHA_MAX = np.log(TARGET_RATE_OF_REACTION * SPREAD**-1) + +BETA_MIN = np.log(SPREAD) +BETA_MAX = np.log(SPREAD**-1) + +N_POINTS = 1000 # Needs to be large to see minima + +# Plotting constants +IMSHOW_STANDARD_KWARGS = dict( + interpolation = 'nearest', # "nearest" = no interpolation (nearest neighbor) + origin = 'lower', # place origin in bottom-left (standard for plotting) + ) + +# Color maps +CMAP_GENERAL = 'viridis' # arbitrary values, "zero" has no particular meaning +CMAP_DIVERGENT = 'RdBu' # positive/negative values, "zero" has a specific meaning (must be centered!) +CMAP_ONESIDED = 'bone' # minimum value has a specific meaning +CMAP_ROTATIONAL = 'hsv' # circularly permuted values (min/max must be chosen appropriately) + +EXTENT = ( + ALPHA_MIN, ALPHA_MAX, + BETA_MIN, BETA_MAX + ) + +N_TICKS = 5 # should be a small integer, preferably odd + +TICKS_FRACTIONAL_MARGIN = 0.9 # prevent ticks from being too close to the edge + +FIGSIZE = (4, 4) + +OUTDIR = 'out' +EXTENSION = ( # Choose one + # 'png' + 'pdf' + ) + +# 'Raw' string needed to avoid clashes between escape characters for Python and +# for the LaTeX renderer. +ALPHA_STRING = r'$\alpha$' +BETA_STRING = r'$\beta$' + +# Functions + +# Model functions + +def rate_of_reaction(alpha, beta): + ''' + Compute the rate of reaction subject to two lumped parameters. + + Parameters: + alpha: float + Natural logarithm of the maximum velocity (AKA rate of reaction). + + beta: float + Natural logarithm of the saturation ratio (metabolite concetration + divided by corresponding Michaelis-Menten saturation constant). + + Returns: + The "velocity" or rate of reaction. (float) + + Formulation: + + Starting from standard MM kinetics, + + v = k E C / (C + K) + + v: rate of reaction + k: catalytic rate constant + E: enzyme concentration + C: metabolite concentration + K: Michaelis-Menten saturation constant + + Lump the parameters: + + vmax = k E: maximum "velocity" or rate of reaction + s = C/K: saturation ratio + + Now, rewrite the system, + + v = vmax * s / (s + 1) + + Without loss of generality, rewrite in terms of logarithmic parameters: + + alpha = ln(vmax) + beta = ln(s) + + Now, finally, + + v = exp(alpha + beta) / (exp(beta) + 1) + + Finally, for numerical purposes, we rewrite as + + v = exp(alpha) / (1 + exp(-beta)) + + Apart from their physical interpretation, alpha can be thought of as the + height of the curve, and beta can be thought of as the horizontal shift. + ''' + return np.exp(alpha) / (1 + np.exp(-beta)) + + +# Optimization functions + +def objective(alpha, beta): + ''' + Returns the square of the difference between the target rate of reaction + and the calculated rate of reaction based on the two parameters alpha and + beta. See the documentation for 'rate_of_reaction' for more information. + ''' + return (rate_of_reaction(alpha, beta) - TARGET_RATE_OF_REACTION)**2 + +def grad(alpha, beta): + ''' + Computes the gradient of the objective with respect to the model + parameters. + + Note that these derivatives were computed by hand. As such I'm not + perfectly confident in them. + ''' + # TODO: validate + + ror = rate_of_reaction(alpha, beta) + dobj_dalpha = 2 * ror * (ror - TARGET_RATE_OF_REACTION) + dobj_dbeta = dobj_dalpha / (np.exp(beta) + 1) + + return dobj_dalpha, dobj_dbeta + + +# Plotting functions + +def imshow_percentile(array, bottom, top, **kwargs): + ''' + Plot an array as an image, choosing upper and lower limits using + pecentiles. + ''' + (vmin, vmax) = np.percentile(array, [bottom, top]) + + plt.imshow(array, vmin = vmin, vmax = vmax, **kwargs) + +def imshow_percentile_centered(array, top, **kwargs): + ''' + Plot an array as an image, choosing upper and lower limits using + a percentile. Data will be centered about zero. + ''' + vmax = np.percentile(np.abs(array), top) + vmin = -vmax + + plt.imshow(array, vmin = vmin, vmax = vmax, **kwargs) + +def choose_ticks(min_value, max_value, n_ticks, fractional_margin): + ''' + Select plotted ticks. + ''' + return np.linspace( + np.ceil(min_value * fractional_margin), + np.floor(max_value * fractional_margin), + n_ticks + ) + +def ax(ticks, label, tick_function, label_function, ticks_only = False): + ''' + Axis formatting function. + ''' + if ticks_only: + tick_function(ticks, ['']*ticks.size) + + else: + tick_function(ticks) + label_function(label) + +def finalize(figure_number, title): + plt.tight_layout() + plt.savefig(os.path.join(OUTDIR, + 'figure{}-{}.{}'.format(figure_number, title, EXTENSION) + )) + plt.clf() + +# Execution + +def main(): + # Sweep the parameter space, computing the objective and gradient + + alpha_range = np.linspace(ALPHA_MIN, ALPHA_MAX, N_POINTS) + beta_range = np.linspace(BETA_MIN, BETA_MAX, N_POINTS) + + (alpha, beta) = np.meshgrid(alpha_range, beta_range) + + obj = objective(alpha, beta) + + (grad_alpha, grad_beta) = grad(alpha, beta) + + # Plotting + + if not os.path.exists(OUTDIR): + os.mkdir(OUTDIR) + + # The derivative of a logarithm of a function is the derivative of the function + # divided by the function itself. + log_grad_alpha = grad_alpha / obj + log_grad_beta = grad_beta / obj + + xticks = choose_ticks( + ALPHA_MIN, ALPHA_MAX, + N_TICKS, TICKS_FRACTIONAL_MARGIN + ) + yticks = choose_ticks( + BETA_MIN, BETA_MAX, + N_TICKS, TICKS_FRACTIONAL_MARGIN + ) + + # Conveience functions - defined here because they refer to values that + # strictly exist in this scope. + + def xax(ticks_only = False): + ax(xticks, ALPHA_STRING, plt.xticks, plt.xlabel, ticks_only) + + def yax(ticks_only = False): + ax(yticks, BETA_STRING, plt.yticks, plt.ylabel, ticks_only) + + + # Figure 1: Raw objective value + + # The raw objective value is hard to see, but is provided for context. + + plt.figure(figsize = FIGSIZE) + + imshow_percentile( + obj, + 0, 95, # Value grows very rapidly - use 95th percentile as max + **dict( + IMSHOW_STANDARD_KWARGS, + cmap = CMAP_ONESIDED, + extent = EXTENT + ) + ) + + plt.colorbar() + + xax() + yax() + + finalize(1, 'raw_objective_values') + + + # Figure 2: Log-scaled objective value + + # Log-scaling the objective value makes the features of the objective value + # space clear. + + plt.figure(figsize = FIGSIZE) + + imshow_percentile( + np.log(obj), + 1, 100, # objective falls off to zero at minima + **dict( + IMSHOW_STANDARD_KWARGS, + cmap = CMAP_ONESIDED, + extent = EXTENT + ) + ) + + plt.colorbar() + + xax() + yax() + + finalize(2, 'log_objective_values') + + + # Figure 3: Derivative of log objective w.r.t alpha + + plt.figure(figsize = FIGSIZE) + + imshow_percentile_centered( + log_grad_alpha, + 95, # log gradient near minima is very steep since the objective approaches zero + **dict( + IMSHOW_STANDARD_KWARGS, + cmap = CMAP_DIVERGENT, + extent = EXTENT + ) + ) + + plt.colorbar() + + xax() + yax() + + finalize(3, 'derivative_alpha') + + + # Figure 4: Derivative of log objective w.r.t beta + + plt.figure(figsize = FIGSIZE) + + imshow_percentile_centered( + log_grad_beta, + 95, # log gradient near minima is very steep since the objective approaches zero + **dict( + IMSHOW_STANDARD_KWARGS, + cmap = CMAP_DIVERGENT, + extent = EXTENT + ) + ) + + plt.colorbar() + + xax() + yax() + + finalize(4, 'derivative_beta') + + + # Figure 5: Optimal search direction + + plt.figure(figsize = FIGSIZE) + + # Note that the best search angle doesn't change if we are minimizing the + # objective or the logarithmic objective, since they differ only by a scaling + # factor. + best_search_angle = np.arctan2(-grad_beta, -grad_alpha) # Negative because we are minimizing + + plt.imshow( + best_search_angle, + vmin = -np.pi, vmax = +np.pi, # Range of arctan2 + cmap = CMAP_ROTATIONAL, + extent = EXTENT, + **IMSHOW_STANDARD_KWARGS + ) + + plt.colorbar() + + xax() + yax() + + finalize(5, 'best_local_search_direction') + + + # Figure 6: Flow plot + + plt.figure(figsize = FIGSIZE) + + plt.streamplot(alpha, beta, -grad_alpha, -grad_beta, density = [0.5, 0.5]) + plt.axis('square') + + xax() + yax() + + # Need to set limits explicitly, unlike with plt.imshow + plt.xlim(ALPHA_MIN, ALPHA_MAX) + plt.ylim(BETA_MIN, BETA_MAX) + + finalize(6, 'gradient_descent_flow') + +if __name__ == '__main__': + main() diff --git a/prototypes/simple_parameter_estimation/out/figure1-raw_objective_values.pdf b/prototypes/simple_parameter_estimation/out/figure1-raw_objective_values.pdf new file mode 100644 index 0000000000000000000000000000000000000000..16baf07d3207538a794ca0791853f682d660de63 GIT binary patch literal 13879 zcmeHuc|4Te+jz>>M3J>f6A9U7H3lvA?7NUK7(1ge_DI?HlszhGQ6gKiFWDmLp+YH3 ziAt7ImTd32M|!k;pLhHIKEHoneNN}T&wZV9u5(@2xwdmgP*+Js94Uc;3EqAQ&#izV z;Rra%+6g8n2ba_(*b_bADA1x0mo)TsC&DEa2^4}0$sT_2AdKi{3js>6>!9pIQ8l0t 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.../simple_parameter_estimation/main.py | 5 ++++- .../out/figure1-raw_objective_values.pdf | Bin 13879 -> 13879 bytes .../out/figure2-log_objective_values.pdf | Bin 21260 -> 21260 bytes .../out/figure3-derivative_alpha.pdf | Bin 18842 -> 18842 bytes .../out/figure4-derivative_beta.pdf | Bin 22664 -> 22664 bytes .../figure5-best_local_search_direction.pdf | Bin 10574 -> 10574 bytes .../out/figure6-gradient_descent_flow.pdf | Bin 103279 -> 103279 bytes 7 files changed, 4 insertions(+), 1 deletion(-) diff --git a/prototypes/simple_parameter_estimation/main.py b/prototypes/simple_parameter_estimation/main.py index 6ac3a06bdb..50a98b36c9 100755 --- a/prototypes/simple_parameter_estimation/main.py +++ b/prototypes/simple_parameter_estimation/main.py @@ -68,7 +68,10 @@ FIGSIZE = (4, 4) -OUTDIR = 'out' +OUTDIR = os.path.join( + os.path.split(__file__)[0], + 'out' + ) EXTENSION = ( # Choose one # 'png' 'pdf' diff --git 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delta 22 dcmeC!$k?%waf4ncySbr>fw`&SX4BAMW&l=T2CD!7 diff --git a/prototypes/simple_parameter_estimation/out/figure5-best_local_search_direction.pdf b/prototypes/simple_parameter_estimation/out/figure5-best_local_search_direction.pdf index e3affee3d9ae7256bfb2aff18d1009264f3393c9..c7f177b8ab7224511733533b73ab93211ce5b1c2 100644 GIT binary patch delta 20 bcmX>XbS`MaS5XbS`MaS5^?43AO+L delta 25 hcmaFAjP3n0wuUW?Jrmf?4NVNpO^vrtp1^pA2>^>q39|qI