Map making in Matlab

Map making in Matlab

Greetings,

This weeks post will cover the basics of generating maps in Matlab.  Julie’s recent post showed how to do some of this in Python, but, Matlab is also widely used by the community.  You can get a lot done with Matlab, but in this post we’ll just cover a few of the basics.

We’ll start off by plotting a map of the continental United States, with the states.  We used three  this with three commands: usamap, shaperead, and geoshow.  usamap creates an empty map axes having the Lambert Projection covering the area of the US, or any state or collection of states.  shaperead reads shapefiles (duh) and returns a Matlab geographic data structure, composed of both geographic data and attributes.  This Matlab data structure then interfaces really well with various Matlab functions (duh).  Finally, geoshow plots geographic data, in our case on the map axes we defined.  Here’s some code putting it all together.

hold on
figure1 = figure;
ax = usamap('conus');

set(ax, 'Visible', 'off')
latlim = getm(ax, 'MapLatLimit');
lonlim = getm(ax, 'MapLonLimit');
states = shaperead('usastatehi',...
 'UseGeoCoords', true, 'BoundingBox', [lonlim', latlim']);
geoshow(ax, states, 'FaceColor', [0.5 0.5 0.5])
tightmap
hold off

Note that ‘usastatehi’ is a shapefile containing the US states (duh) that’s distributed with Matlab. The above code generates this figure:

conus_blank

Now, suppose we wanted to plot some data, say a precipitation forecast, on our CONUS map.  Let’s assume our forecast is being made at many points (lat,long).  To interpolate between the points for plotting we’ll use Matlab’s griddata function.  Once we’ve done this, we use the Matlab’s contourm command.  This works exactly like the normal contour function, but the ‘m’ indicates it plots map data.

xi = min(x):0.5:max(x);
yi = min(y):0.5:max(y);
[XI, YI] = meshgrid(xi,yi);
ZI = griddata(x,y,V,XI,YI);

hold on
figure2 = figure;
ax = usamap('conus');

set(ax, 'Visible', 'off')
latlim = getm(ax, 'MapLatLimit');
lonlim = getm(ax, 'MapLonLimit');
states = shaperead('usastatehi',...
 'UseGeoCoords', true, 'BoundingBox', [lonlim', latlim']);
geoshow(ax, states, 'FaceColor', [0.5 0.5 0.5])

contourm(YI,-1*XI,ZI)
tightmap
hold off

Here x, y, and V are vectors of long, lat, and foretasted precipitation respectively.  This code generates the following figure:

conus_contour

Wow!  Louisiana is really getting hammered!  Let’s take a closer look.  We can do this by changing the entry to usamap to indicate we want to consider only Louisiana.  Note, usamap accepts US postal code abbreviations.

ax = usamap('LA');

Making that change results in this figure:

LA_contour

Neat!  We can also look at two states and add annotations.  Suppose, for no reason in particular, you’re interested in the location of Tufts University relative to Cornell.  We can make a map to look at this with the textm and scatterm functions.  As before, the ‘m’ indicates the functions  plot on a map axes.

hold on
figure4 = figure;
ax = usamap({'MA','NY'});

set(ax, 'Visible', 'off')
latlim = getm(ax, 'MapLatLimit');
lonlim = getm(ax, 'MapLonLimit');
states = shaperead('usastatehi',...
 'UseGeoCoords', true, 'BoundingBox', [lonlim', latlim']);
geoshow(ax, states, 'FaceColor', [0.5 0.5 0.5])
scatterm(42.4075,-71.1190,100,'k','filled')
textm(42.4075+0.2,-71.1190+0.2,'Tufts','FontSize',30)

scatterm(42.4491,-76.4842,100,'k','filled')
textm(42.4491+0.2,-76.4842+0.2,'Cornell','FontSize',30)
tightmap
hold off

This code generates the following figure.

Cornell_Tufts

Cool! Now back to forecasts.  NOAA distributes short term Quantitative Precipitation Forecasts (QPFs) for different durations every six hours.  You can download these forecasts in the form of shapefiles from a NOAA server.  Here’s an example of a 24-hour rainfall forecast made at 8:22 AM UTC on April 29.

fill_94qwbg

Wow, that’s a lot of rain!  Can we plot our own version of this map using Matlab!  You bet!  Again we’ll use usamap, shaperead, and geoshow.  The for loop, (0,1) scaling, and log transform are simply to make the color map more visually appealing for the post.  There’s probably a cleaner way to do this, but this got the job done!

figure5 = figure;
ax = usamap('conus');
S=shaperead('94q2912','UseGeoCoords',true);

set(ax, 'Visible', 'off')
latlim = getm(ax, 'MapLatLimit');
lonlim = getm(ax, 'MapLonLimit');
states = shaperead('usastatehi',...
 'UseGeoCoords', true, 'BoundingBox', [lonlim', latlim']);
geoshow(ax, states, 'FaceColor', [0.5 0.5 0.5])
p = colormap(jet);

N = max(size(S));
d = zeros(N,1);
for i = 1:N
 d(i) = log(S(i).QPF);
end

y=floor(((d-min(d))/range(d))*63)+1;
col = p(y,:);
for i = 1:N
 geoshow(S(i),'FaceColor',col(i,:),'FaceAlpha',0.5)%,'SymbolSpec', faceColors)
end

This code generates the following figure:

conus_shape

If you are not plotting in the US, Matlab also has a worldmap command.  This works exactly the same as usamap, but now for the world (duh).  Matlab is distibuted with a shapefile ‘landareas.shp’ which contains all of the land areas in the world (duh).  Generating a global map is then trivial:

figure6 = figure;

worldmap('World')
land = shaperead('landareas.shp', 'UseGeoCoords', true);
geoshow(land, 'FaceColor', [0.15 0.5 0.15])

Which generates this figure.

globe

 

Matlab also comes with a number of other included that might be of interest.  For instance, shapefiles detailing the locations of major world cities, lakes, and rivers.  We can plot those with the following code:

figure7 = figure;

worldmap('World')
land = shaperead('landareas.shp', 'UseGeoCoords', true);
geoshow(land, 'FaceColor', [0.15 0.5 0.15])
lakes = shaperead('worldlakes', 'UseGeoCoords', true);
geoshow(lakes, 'FaceColor', 'blue')
rivers = shaperead('worldrivers', 'UseGeoCoords', true);
geoshow(rivers, 'Color', 'blue')
cities = shaperead('worldcities', 'UseGeoCoords', true);
geoshow(cities, 'Marker', '.', 'Color', 'red')

Which generates the figure:

globe_river

But suppose we’re interested in one country or a group of countries.  worldmap works in the same usamap does.  Also, you can plot continents, for instance Europe.

worldmap('Europe')

Europe.png

Those are the basics, but there are many other capabilities, including 3-D projections. I can cover this in a later post if there is interest.

contour

That’s it for now!

A visual introduction to data compression through Principle Component Analysis

Principle Component Analysis (PCA) is a powerful tool that can be used to create parsimonious representations of a multivariate data set. In this post I’ll code up an example from Dan Wilks’ book Statistical Methods in the Atmospheric Sciences to visually illustrate the PCA process. All code can be found at the bottom of this post.

As with many of the examples in Dr. Wilks’ excellent textbook, we’ll be looking at minimum temperature data from Ithaca and Canandaigua, New York  (if anyone is interested, here is the distance between the two towns).  Figure 1 is a scatter plot of the minimum temperature anomalies at each location for the month of January 1987.

raw_data

Figure 1: Minimum temperature anomalies in Ithaca and Canandaigua, New York in January 1987

As you can observe from Figure 1, the two data sets are highly correlated, in fact, they have a Pearson correlation coefficient of 0.924. Through PCA, we can identify the primary mode of variability within this data set (its largest “principle component”) and use it to create a single variable which describes the majority of variation in both locations. Let x define the matrix of our minimum temperature anomalies in both locations. The eigenvectors (E) of the covariance matrix of x describe the primary modes variability within the data set. PCA uses these eigenvectors to  create a new matrix, u,  whose columns contain the principle components of the variability in x.

u = xE

Each element in u is a linear combination of the original data, with eigenvectors in E serving as a kind of weighting for each data point. The first column of u corresponds to the eigenvector associated with the largest eigenvalue of the covariance matrix. Each successive column of u represents a different level of variability within the data set, with u1 describing the direction of highest variability, u2 describing the direction of the second highest variability and so on and so forth. The transformation resulting from PCA can be visualized as a rotation of the coordinate system (or change of basis) for the data set, this rotation is shown in Figure 2.

PCA visualization

Figure 2: Geometric interpretation of PCA

As can be observed in Figure 2, each data point can now be described by its location along the newly rotated axes which correspond to its corresponding value in the newly created matrix u. The point (16, 17.8), highlighted in Figure 2, can now be described as (23, 6.6) meaning that it is 23 units away from the origin in the direction of highest variability and 6.6 in the direction of second highest variability. As shown in Figure 2, the question of “how different from the mean” each data point is can mostly be answered by looking at its  corresponding u1 value.

Once transformed, the original data can be recovered through a process known as synthesis. Synthesis  can be thought of as PCA in reverse. The elements in the original data set x, can be approximated using the eigenvalues of the covariance matrix and the first principle component, u1.

\tilde{x} = \tilde{u}\tilde{E}^T

Where:

\tilde{x}\hspace{.1cm} is\hspace{.1cm} the\hspace{.1cm} reconstructed\hspace{.1cm} data\hspace{.1cm} set

\tilde{u}\hspace{.1cm} is\hspace{.1cm} the\hspace{.1cm} PCs\hspace{.1cm} used \hspace{.1cm} for \hspace{.1cm} reconstruction\hspace{.1cm} (in\hspace{.1cm} our\hspace{.1cm} case\hspace{.1cm} the\hspace{.1cm} first\hspace{.1cm} column)

\tilde{E}\hspace{.1cm} is \hspace{.1cm} the \hspace{.1cm} eigenvector\hspace{.1cm} of \hspace{.1cm} the \hspace{.1cm} PCs \hspace{.1cm} used

For our data set, these reconstructions seem to work quite well, as can be observed in Figure 3.

final_synth.png

 

 

Data compression through PCA can be a useful alternative tolerant methods for dealing with multicollinearity, which I discussed in my previous post. Rather than running a constrained regression, one can simply compress the data set to eliminate sources of multicollinearity. PCA can also be a helpful tool for identifying patterns within your data set or simply creating more parsimonious representations of a complex set of data. Matlab code used to create the above plots can be found below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ithaca_Canandagua_PCA
% By: D. Gold
% Created: 3/20/17
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% This script will perform Principle Component analysis on minimum
% temperature data from Ithaca and Canadaigua in January, 1987 provided in 
% Appendix A of Wilks (2011). It will then estimate minimum temperature
% values of both locations using the first principle component.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% create data sets
clear all

% data from appendix Wilks (2011) Appendix A.1
Ith = [19, 25, 22, -1, 4, 14, 21, 22, 23, 27, 29, 25, 29, 15, 29, 24, 0,... 
 2, 26, 17, 19, 9, 20, -6, -13, -13, -11, -4, -4, 11, 23]';

Can = [28, 28, 26, 19, 16, 24, 26, 24, 24, 29, 29, 27, 31, 26, 38, 23,... 
 13, 14, 28, 19, 19, 17, 22, 2, 4, 5, 7, 8, 14, 14, 23]';

%% center the data, plot temperature anomalies, calculate covariance and eigs

% center the data
x(:,1) = Ith - mean(Ith);
x(:,2) = Can - mean(Can);

% plot anomalies
figure
scatter(x(:,1),x(:,2),'Filled')
xlabel('Ithaca min temp anomaly ({\circ}F)')
ylabel('Canandagua min temp anomaly ({\circ}F)')

% calculate covariance matrix and it's corresponding Eigenvalues & vectors
S = cov(x(:,1),x(:,2));
[E, Lambda]=eigs(S);

% Identify maximum eigenvalue, it's column will be the first eigenvector
max_lambda = find(max(Lambda)); % index of maximum eigenvalue in Lambda
idx = max_lambda(2); % column of max eigenvalue

%% PCA
U = x*E(:,idx);

%% synthesis
x_syn = E(:,idx)*U'; % reconstructed values of x

% plot the reconstructed values against the original data
figure
subplot(2,1,1)
plot(1:31,x(:,1) ,1:31, x_syn(1,:),'--')
xlim([1 31])
xlabel('Time (days)')
ylabel('Ithaca min. temp. anomalies ({\circ}F)')
legend('Original', 'Reconstruction')
subplot(2,1,2)
plot(1:31, x(:,2),1:31, x_syn(2,:)','--')
xlim([1 31])
xlabel('Time (days)')
ylabel('Canadaigua min. temp. anomalies ({\circ}F)')
legend('Original', 'Reconstruction')

 

Sources:

Wilks, D. S. (2011). Statistical methods in the atmospheric sciences. Amsterdam: Elsevier Academic Press.

Plotting Probability Ellipses for Bivariate Normal Distributions

Plotting probability ellipses can be a useful way to visualize bivariate normal distributions. A probability ellipse represents a contour of constant probability within which a certain percentage of the distribution lies. The width and orientation of probability ellipses can yield information about the correlation between the two data points of interest.

To plot probability ellipses of a bivariate normal distribution, you need to have a vector containing the means of both data sets of interest as well as the covariance matrix for the two data sets. Each probability ellipse is centered around the means of the two data sets and oriented in the direction of the first eigenvector of the covariance matrix. The length of the primary axis of each ellipse is proportional to the value of the percentile of the Chi Squared distribution for the given percentile the ellipse represents.

Below, I’ve coded an example (using Matlab) that is presented in the very informative textbook Statistical Methods in the Atmospheric Sciences by Dan Wilks. The example uses one month of daily minimum temperature data from the towns of Ithaca and Canandaigua, New York. Notice that the 50% ellipse in the center of the plot encloses roughly half the data points, indicating that the contours were plotted correctly.

blog_80

Probability Ellipses for Ithaca and Canandaigua Minimum Temperatures

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Plotting  probability ellipses of the bivariate normal distribution
% By: D. Gold
% Created: 10/28/16
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This script will plot the multivariate cumulative probability contours 
% of two data sets that are fit to a multivariate normal distribution
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%
clear all; close all; clc;

%%% Create Data Sets %%%
Ith_MinT = [19, 25, 22, -1, 4, 14, 21, 22, 23, 27, 29, 25, 29, 15, 29, 24, 0, 2, 26, 17, 19, 9, 20, -6, -13, -13, -11, -4, -4, 11, 23];

Can_MinT = [28, 28, 26, 19, 16, 24, 26, 24, 24, 29, 29, 27, 31, 26, 38, 23, 13, 14, 28, 19, 19, 17, 22, 2, 4, 5, 7, 8, 14, 14, 23];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Calculate moments of the data sets %%%

% Observed
Ith_mean = mean(Ith_MinT); % T mean
Can_mean= mean(Can_MinT); % Td mean
CV = cov(Ith_MinT,Can_MinT); % covariance of T and Td
[Evec, Eval]=eig(CV); % Eigen values and vectors of covariance matrix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Plot observed multivariate contours %%

% Observed data
xCenter = Ith_mean; % ellipses centered at sample averages
yCenter = Can_mean;
theta = 0 : 0.01 : 2*pi; % angles used for plotting ellipses

% compute angle for rotation of ellipse
% rotation angle will be angle between x axis and first eigenvector
x_vec= [1 ; 0]; % vector along x-axis
cosrotation =dot(x_vec,Evec(:,1))/(norm(x_vec)*norm(Evec(:,1))); 
rotation =pi/2-acos(cosrotation); % rotation angle
R  = [sin(rotation) cos(rotation); ...
      -cos(rotation)  sin(rotation)]; % create a rotation matrix

% create chi squared vector
chisq = [1.368 4.605 3.2188  5.991]; % percentiles of chi^2 dist df=2

% size ellipses for each quantile
for i = 1:length(chisq)
    % calculate the radius of the ellipse
    xRadius(i)=(chisq(i)*Eval(1,1))^.5; % primary
    yRadius(i)=(chisq(i)*Eval(2,2))^.5; % secondary
    % lines for plotting ellipse
    x{i} = xRadius(i)* cos(theta);
    y{i} = yRadius(i) * sin(theta);
    % rotate ellipse
    rotated_Coords{i} = R*[x{i} ; y{i}];
    % center ellipse
    x_plot{i}=rotated_Coords{i}(1,:)'+xCenter;
    y_plot{i}=rotated_Coords{i}(2,:)'+yCenter;
end

% Set up plot
figure
xlabel('Ithaca Minimum Temperature (F)')
ylabel('Canandagua Minimum Temperature (F)(F)')
hold on

% Plot data points
plot(Ith_MinT,Can_MinT,'o');

% Plot contours
for j = 1:length(chisq)
    plot(x_plot{j},y_plot{j})
end
legend('Data points','50%', '80%', '90%', '99%')

 

A simple command for plotting autocorrelation functions in Matlab

Autocorrelation is a measure of persistence within a data set, which can be defined as the tendency for successive data points to be similar (Wilks, 2011).  In atmospheric science temporal autocorrelation can be a helpful tool for model evaluation. Temporal autocorrelation is also a fundamental concept for synthetic weather generation (for more detail see Julie’s fantastic series of blog posts on synthetic weather generation here).  Calculating autocorrelation within a sample data set can also be a helpful for assessing the applicability of classical statistical methods requiring independence of data points within a sample. Should a data set prove to be strongly persistent, such methods will likely yield inaccurate results.

Autocorrelation is commonly computed by making a copy of the original data set, shifting the copy k points forward (where k is the lag over which you would like to compute the autocorrelation)  and computing the Pearson correlation coefficient between the original data set and the copy.

autocorrelation-equation

Where:

autocorr_variables

The calculation of autocorrelation for a number of different lags at once is known as the autocorrelation function. Plotting the autocorrelation graphically can be a helpful tool for quickly assessing the presence of autocorrelation within a data set.

You can generate such plots in Matlab using the simple command shown below:

autocorr(T,k) 
% T is your data set and k is the number of lags you would like to compute

The command generates a plot of the autocorrelation function. Below are two examples, the first is the autocorrelation function of a set of observed temperature values in Des Moines Iowa, the second is autocorrelation function of the temperature values at the same location as modeled by the MM5I regional climate model:

DM_HRM3.jpg

Figure 1: Temporal autocorrelation function of temperature observations from Des Moines Iowa (temperatures reported at 3 hour intervals)

 

dm_mm5i

Figure 2: Temporal autocorrelation function of temperature produced by the MM5I regional climate model for Des Moines Iowa (temperatures reported at 3 hour intervals)

Note the cyclical nature of the autocorrelation functions, this is a reflection of the daily temperature cycle. The autocorrelations function of the maximum or minimum temperatures would show more constant persistence.

Sources:

Wilks, D. S. (2011). Statistical methods in the atmospheric sciences. Burlington, MA: Academic Press.

Root finding in MATLAB, R, Python and C++

In dynamical systems, we are often interested in finding stable points, or equilibria. Some systems have multiple equilibria. As an example, take the lake problem, which is modeled by the equation below where Xt is the lake P concentration, at are the anthropogenic P inputs, Yt~LN(μ,σ2)  are random natural P inputs, b is the P loss rate, and q is a shape parameter controlling the rate of P recycling from the sediment. The first three terms on the right hand side make up the “Inputs” in the figure, while the last term represents the “Outputs.” A lake is in equilibrium when the inputs are equal to the outputs and the lake P concentration therefore is not changing over time.

lakeModel

For irreversible lakes this occurs at three locations, even in the absence of anthropogenic and natural inputs: an oligotrophic equilibrium, an unstable equilibrium (called the critical P threshold) and a eutrophic equilibrium (see figure below).

PcritDiagram

The unstable equilibrium in this case is called the critical P threshold because once it is crossed, it is impossible to return to an oligotrophic equilibrium by reducing anthropogenic and natural P inputs alone. In irreversible lakes like this, we would therefore like to keep the lake P concentration below the critical P threshold. How do we find the critical P threshold? With a root finding algorithm!

As stated earlier, the system above will be in equilibrium when the inputs are equal to the outputs and the P concentration is not changing over time, i.e. when

X_{t+1} - X_t = \frac{X^q_t}{1+X^q_t} - bX_t = 0

Therefore we simply need to find the zero, or “root” of the above equation.  Most of the methods for this require either an initial estimate or upper and lower bounds on the location of the root. These are important, since an irreversible lake will have three roots. If we are only interested in the critical P threshold, we have to make sure that we provide an estimate which leads to the unstable equilibrium, not either of the stable equilibria. If possible, you should plot the function whose root you are finding to make sure you are giving a good initial estimate or bounds, and check afterward to ensure the root that was found is the one you want! Here are several examples of root-finding methods in different programming languages.

In MATLAB, roots can be found with the function fzero(fun,x0) where ‘fun’ is the function whose root you want to find, and x0 is an initial estimate. This function uses Brent’s method, which combines several root-finding methods: bisection, secant, and inverse quadratic interpolation. Below is an example using the lake problem.

myfun = @(x,b,q) x^q/(1+x^q)-b*x;
b = 0.42;
q = 2.0;
fun = @(x) myfun(x,b,q);
pcrit = fzero(fun,0.75);

This returns pcrit = 0.5445, which is correct. If we had provided an initial estimate of 0.25 instead of 0.75, we would get pcrit = 2.6617E-19, basically 0, which is the oligotrophic equilibrium in the absence of anthropogenic and natural P inputs. If we had used 1.5 as an initial estimate, we would get pcrit = 1.8364, the eutrophic equilibrium.

MatlabScreenShot

In R, roots can be found with the function uniroot, which also uses Brent’s method. Dave uses this on line 10 of the function lake.eval in his OpenMORDM example. Instead of taking in an initial estimate of the root, this function takes in a lower and upper bound. This is safer, as you at least know that the root estimate will lie within these bounds. Providing an initial estimate that is close to the true value should do well, but is less predictable; the root finding algorithm may head in the opposite direction from what is desired.

b <- 0.42
q <- 2.0
pcrit <- uniroot(function(x) x^q/(1+x^q) - b*x, c(0.01, 1.5))$root

This returns pcrit = 0.5445145. Good, we got the same answer as we did with MATLAB! If we had used bounds of c(0.75, 2.0) we would have gotten 1.836426, the eutrophic equilibrium.

What if we had given bounds that included both of these equilibria, say c(0.5, 2.0)? In that case, R returns an error: ‘f() values at end points not of opposite sign’. That is, if the value returned by f(x) is greater than 0 for the lower bound, it must be less than 0 for the upper bound and vice versa. In this case both f(0.5) and f(2.0) are greater than 0, so the algorithm fails. What if we gave bounds for which one is greater than 0 and another less, but within which there are multiple roots, say c(-0.5,2.0)? Then R just reports the first one it finds, in this case pcrit = 0.836437, the eutrophic equilibrium. So it’s important to make sure you pick narrow enough bounds that include the root you want, but not roots you don’t!

RscreenShot

In Python, you can use either scipy.optimize.root or scipy.optimize.brentq, which is what Jon uses on line 14 here. scipy.optimize.root can be used with several different algorithms, but the default is Powell’s hybrid method, also called Powell’s dogleg method. This function only requires an initial estimate of the root.

from scipy.optimize import root
b = 0.42
q = 2.0
pcrit = root(lambda x: x**(1+x**q) - b*x, 0.75)

scipy.optimize.root returns an object with several attributes. The attribute of interest to us is the root, represented by x, so we want pcrit.x. In this case, we get the correct value of 0.54454. You can play around with initial estimates to see how pcrit.x changes.

PythonScreenShot1

Not surprisingly, scipy.optimize.brentq uses Brent’s method and requires bounds as an input.

from scipy.optimize import brentq as root
b = 0.42
q = 2.0
pcrit = root(lambda x: x**(1+x**q) - b*x, 0.01, 1.5)

This just returns the root itself, pcrit = 0.5445. Again, you can play around with the bounds to see how this estimate changes.

PythonScreenShot2

In C++, Dave again shows how this can be done in the function ‘main-lake.cpp’ provided in the Supplementary Material to OpenMORDM linked from this page under the “Publications” section. On lines 165-168 he uses the bisect tool to find the root of the function given on lines 112-114. I’ve copied the relevant sections of his code into the function ‘find_Pcrit.cpp’ below.


#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <iostream>
#include <boost/math/tools/roots.hpp>

namespace tools = boost::math::tools;
using namespace std;

double b, q, pcrit;

double root_function(double x) {
  return pow(x,q)/(1+pow(x,q)) - b*x;
}

bool root_termination(double min, double max) {
  return abs(max - min) <= 0.000001;
}

int main(int argc, char* argv[])
{
  b = 0.42;
  q = 2.0;

  std::pair<double, double> result = tools::bisect(root_function, 0.01, 1.0, root_termination);
  pcrit = (result.first + result.second)/2;
  cout << pcrit << endl;
}

This yields the desired root of pcrit = 0.54454, but of course, changing the bounds may result in different estimates. In case you missed it, the take home message is to be careful about your initial estimate and bounds ;).

CPPscreenShot

 

Plotting Code added to Lake Problem Diagnostics Repository

The Reed group Matlab scripts I used to generate attainment and control maps after performing algorithm diagnostics are now on Github (https://github.com/VictoriaLynn/Lake-Problem-Diagnostics/tree/master/Plotting-Code). I tried to provide comments in the code wherever others may need to make changes to use it on their results. Otherwise, it should be possible to use them as is.

I used Illustrator to improve the Matlab figures. See Jon’s post for tips on using Illustrator.