BYOM, byom_debkiss_no_egg.m

BYOM is a General framework for simulating model systems in terms of ordinary differential equations (ODEs). The model itself needs to be specified in derivatives.m, and call_deri.m may need to be modified to the particular problem as well. The files in the engine directory are needed for fitting and plotting. Results are shown on screen but also saved to a log file (results.out).

The model: standard DEBkiss model (see http://www.debtox.info/book_debkiss.html) following Jager et al (2013), http://dx.doi.org/10.1016/j.jtbi.2013.03.011. Parameters on length basis, and ignoring the embryonic stage.

This script: Example is for the pond snail at maximum food. This is a fit to data for juvenile/adults. The data set was also used in Zimmer et al (2012), http://dx.doi.org/10.1007/s10646-012-0973-5.

Contents

Initial things

Make sure that this script is in a directory somewhere below the BYOM folder.

clear, clear global % clear the workspace and globals
global DATA W X0mat % make the data set and initial states global variables
global glo          % allow for global parameters in structure glo
global pri zvd      % global structures for optional priors and zero-variate data
diary off           % turn of the diary function (if it is accidentaly on)
% set(0,'DefaultFigureWindowStyle','docked'); % collect all figure into one window with tab controls
set(0,'DefaultFigureWindowStyle','normal'); % separate figure windows

pathdefine % set path to the BYOM/engine directory
glo.basenm  = mfilename; % remember the filename for THIS file for the plots
glo.saveplt = 0; % save all plots as (1) Matlab figures, (2) JPEG file or (3) PDF (see all_options.txt)

The data set

Data are entered in matrix form, time in rows, scenarios (exposure concentrations) in columns. First column are the exposure times, first row are the concentrations or scenario numbers. The number in the top left of the matrix indicates how to calculate the likelihood:

% Shell length in mm over time in ad libitum feeding regime
DATA{1} = [0.5        1
            0       12.918
           14        18.65
           28       21.549
           42       23.759
           56       25.669
           70       27.146
           84       27.953
           98       28.189
          112        28.27
          128       29.242
          140        29.53];

% weight factors (number of replicates per observation)
W{1} = [30
    30
    29
    28
    28
    26
    26
    26
    25
    23
    21  ];

% Cumulative reproduction over time in ad libitum feeding regime; reduced data
DATA{2}= [0.5            1
           35            0
           49           76
           63        219.8
           77        389.7
           91        534.6
          105        676.8
          119        858.9
          133       1023.1    ];

W{2} = [28
    28
    28
    26
    26
    26
    24
    23   ];

% if weight factors are not specified, ones are assumed in start_calc.m

Initial values for the state variables

Initial states, scenarios in columns, states in rows. First row are the 'names' of all scenarios.

X0mat = [1       % the scenarios
         12.8    % initial shell length in mm
         0];     % initial cumul. repro

Initial values for the model parameters

Model parameters are part of a 'structure' for easy reference. Global parameters as part of the structure glo

glo.delM  = 0.401; % shape corrector (used in call_deri.m)
glo.dV    = 0.1;   % dry weight density (used in call_deri.m)
glo.len   = 1;     % switch to fit physical length (0=off, 1=on, 2=on and no shrinking) (used in call_deri.m)
glo.mat   = 1;     % include maturity maint. (0=off, 1=include)
% Note: if glo.len > 0 than the initial state for size in X0mat is length too!

% syntax: par.name = [startvalue fit(0/1) minval maxval optional:log/normal scale (0/1)];
par.sJAm = [0.15    1 0 1e6]; % specific assimilation rate
par.sJM  = [0.010   1 0 1e6]; % specific maintenance costs
par.WB0  = [0.15    0 0 1e6]; % initial weight of egg
par.Lwp  = [22      1 0 1e6]; % shell length at puberty
par.yAV  = [0.8     0 0 1];   % yield of assimilates on volume (starvation)
par.yBA  = [0.95    0 0 1];   % yield of egg buffer on assimilates
par.yVA  = [0.8     0 0 1];   % yield of structure on assimilates (growth)
par.kap  = [0.79    1 0 1];   % allocation fraction to soma
par.f    = [1       0 0 2];   % scaled food leve
par.fB   = [0.5     0 0 2];   % scaled food level, embryo
par.Lwf  = [0       0 0 500]; % half-saturation shell length

Time vector and labels for plots

Specify what to plot. If time vector glo.t is not specified, a default is used, based on the data set

glo.t   = linspace(0,150,100); % time vector for the model curves in days

% specify the y-axis labels for each state variable
glo.ylab{1} = 'shell length (mm)';
glo.ylab{2} = 'cumulative reproduction (eggs)';
% specify the x-axis label (same for all states)
glo.xlab    = 'time (days)';
glo.leglab1 = 'scenario '; % legend label before the 'scenario' number
glo.leglab2 = ''; % legend label after the 'scenario' number

prelim_checks % script to perform some preliminary checks and set things up
% Note: prelim_checks also fills all the options (opt_...) with defauls, so
% modify options after this call, if needed.

Calculations and plotting

Here, the function is called that will do the calculation and the plotting. Options for the plotting can be set using opt_plot (see prelim_checks.m). Options for the optimsation routine can be set using opt_optim. Options for the ODE solver are part of the global glo.

For the demo, the iterations are turned off (opt_optim.it = 0).

glo.eventson     = 1; % events function on (1) or off (0)

opt_optim.fit    = 1; % fit the parameters (1), or don't (0)
opt_optim.it     = 0; % show iterations of the optimisation (1, default) or not (0)
opt_plot.bw      = 0; % if set to 1, plots in black and white with different plot symbols
opt_plot.annot   = 0; % annotations in multiplot for fits: 1) box with parameter estimates 2) single legend

% optimise and plot (fitted parameters in par_out)
par_out = calc_optim(par,opt_optim); % start the optimisation
calc_and_plot(par_out,opt_plot); % calculate model lines and plot them
Provisional goodness-of-fit measures for each data set
    0.9967    0.9960

=================================================================================
Results of the parameter estimation with BYOM version 4.0
=================================================================================
   Filename      : byom_debkiss_no_egg 
   Analysis date : 28-Apr-2017 (09:36) 
   Data entered  :
     data state 1: 11x1, continuous data, power 0.50 transf.
     data state 2: 8x1, continuous data, power 0.50 transf.
   Search method: Nelder-Mead simplex direct search, 2 rounds. 
     The optimisation routine has converged to a solution
     Total 293 simplex iterations used to optimise. 
     Minus log-likelihood has reached the value 51.9159 (AIC=111.832). 
=================================================================================
sJAm       0.1545 (fit: 1, initial: 0.15) 
sJM       0.01028 (fit: 1, initial: 0.01) 
WB0          0.15 (fit: 0, initial: 0.15) 
Lwp         23.07 (fit: 1, initial: 22) 
yAV           0.8 (fit: 0, initial: 0.8) 
yBA          0.95 (fit: 0, initial: 0.95) 
yVA           0.8 (fit: 0, initial: 0.8) 
kap        0.7859 (fit: 1, initial: 0.79) 
f               1 (fit: 0, initial:  1) 
fB            0.5 (fit: 0, initial: 0.5) 
Lwf             0 (fit: 0, initial:  0) 
=================================================================================
Time required: 4.6 secs
Plots result from the optimised parameter values. 

Profiling the likelihood

By profiling you make robust confidence intervals for one or more of your parameters. Use the name of the parameter as it occurs in your parameter structure par above. You do not need to run the entire script before you can make a profile.

Options for the profiling can be set using opt_prof (see prelim_checks). For this demo, the level of detail of the profiling is set to 'course' and no sub-optimisations are used. However, consider these options when working on your own data (e.g., set opt_prof.subopt=10).

opt_prof.detail   = 2; % detailed (1) or a course (2) calculation
opt_prof.subopt   = 0; % number of sub-optimisations to perform to increase robustness

% % UNCOMMENT FOLLOWING LINE(S) TO CALCULATE
% % run a profile for selected parameters ...
% calc_proflik(par_out,'sJAm',opt_prof);  % calculate a profile
% calc_proflik(par_out,'sJM',opt_prof);  % calculate a profile

% % UNCOMMENT FOLLOWING LINE(S) TO CALCULATE
% % or run profiles for all fitted parameters.
% all_profiles(par_out,opt_prof);

% Automatically calculate profiles for all parameters, and redo
% optimisation when a better value is found.
opt_prof.subopt   = -1; % number of sub-optimisations to perform to increase robustness
% Note: the -1 makes a comparison between 0 and 10 sub-optimisations.
par_out = auto_profiles(par_out,opt_prof,opt_optim); % Experimental!
 
Starting automatic calculations 
 
Starting automatic calculations without sub-optimisations.
No results will be printed on screen or plotted until the analysis is finished.
  Starting a profile for parameter sJAm (1 of 4)
  Starting a profile for parameter sJM (2 of 4)
  Starting a profile for parameter Lwp (3 of 4)
  Starting a profile for parameter kap (4 of 4)
Starting second run with sub-optimisations.
  Starting a profile for parameter sJAm (1 of 4)
  Starting a profile for parameter sJM (2 of 4)
  Starting a profile for parameter Lwp (3 of 4)
  Starting a profile for parameter kap (4 of 4)
 
Parameters and bounds are for the last runs with 10 sub-optimisations.
Parameter, best fit value, interval
==========================================================
sJAm       0.1545 interval:     0.1438 - 0.1679 
sJM       0.01028 interval:   0.009406 - 0.01136 
WB0          0.15 parameter not fitted 
Lwp         23.07 interval:      22.75 - 23.48 
yAV           0.8 parameter not fitted 
yBA          0.95 parameter not fitted 
yVA           0.8 parameter not fitted 
kap        0.7859 interval:     0.7702 - 0.7954 
f               1 parameter not fitted 
fB            0.5 parameter not fitted 
Lwf             0 parameter not fitted 
==========================================================
Time required: 11 mins, 1.2 secs
 
very similar profile for parameter sJAm (max. diff. 0.027) 
very similar profile for parameter sJM  (max. diff. 0.016) 
very similar profile for parameter Lwp  (max. diff. 0.1) 
very similar profile for parameter kap  (max. diff. 0.0015) 
 

Other files: derivatives

To archive analyses, publishing them with Matlab is convenient. To keep track of what was done, the file derivatives.m can be included in the published result.

%% BYOM function derivatives.m (the model in ODEs)
%
%  Syntax: dX = derivatives(t,X,par,c)
%
% This function calculates the derivatives for the model system. It is
% linked to the script files <byom_debkiss_no_egg.html
% byom_debkiss_no_egg.m>. As input, it gets:
%
% * _t_   is the time point, provided by the ODE solver
% * _X_   is a vector with the previous value of the states
% * _par_ is the parameter structure
% * _c_   is the external concentration (or scenario number)
%
% Time _t_ and scenario name _c_ are handed over as single numbers by
% <call_deri.html call_deri.m> (you do not have to use them in this
% function). Output _dX_ (as vector) provides the differentials for each
% state at _t_.
%
% * Author: Tjalling Jager
% * Date: April 2017
% * Web support: <http://www.debtox.info/byom.html>
% * Back to index <walkthrough_debkiss.html>

%% Start

function dX = derivatives(t,X,par,c)

global glo          % allow for global parameters in structure glo

%% Unpack states
% The state variables enter this function in the vector _X_. Here, we give
% them a more handy name.

WV = X(1); % state 1 is the structural body mass
cR = X(2); % state 2 is the cumulative reproduction (not used in this function)
WB = 0; % no egg phase, so egg buffer is empty

%% Unpack parameters
% The parameters enter this function in the structure _par_. The names in
% the structure are the same as those defined in the byom script file.
% The 1 between parentheses is needed as each parameter has 5 associated
% values.

dV    = glo.dV;       % dry weight density of structure
delM  = glo.delM;     % shape correction coefficient (if needed)
% delM is used to allow Lwp as a parameter instead of WVp

sJAm  = par.sJAm(1);  % specific assimilation rate 
sJM   = par.sJM(1);   % specific maintenance costs 
WB0   = par.WB0(1);   % initial weight of egg
Lwp   = par.Lwp(1);   % shell length at puberty
yAV   = par.yAV(1);   % yield of assimilates on volume (starvation)
yBA   = par.yBA(1);   % yield of egg buffer on assimilates
yVA   = par.yVA(1);   % yield of structure on assimilates (growth)
kap   = par.kap(1);   % allocation fraction to soma
f     = par.f(1);     % scaled food level
fB    = par.fB(1);    % scaled food level for the embryo
Lwf   = par.Lwf(1);   % half-saturation length for initial food limitation

if glo.len ~= 0 % only if the switch is set to 1 or 2!
    % Translate shell length at puberty to body weight
    WVp = glo.dV*(Lwp * glo.delM).^3; % initial length to initial dwt
    WVf = glo.dV*(Lwf * glo.delM).^3; % initial length to initial dwt
end

%% Calculate the derivatives
% This is the actual model, specified as a system of ODEs.
% Note: some unused fluxes are calculated because they may be needed for
% future modules (e.g., respiration and ageing).

L = (WV/dV)^(1/3); % volumetric length

% select what to do with maturity maintenance
if glo.mat == 1
    sJJ = sJM * (1-kap)/kap; % add specific maturity maintenance with the suggested value
else
    sJJ = 0; % or ignore it
end

WVb = WB0 *yVA * kap; % body mass at birth

if WB > 0 % if we have an embryo
    f = fB; % assimilation at different rate
else
    if WVf > 0
        f = f / (1+WVf/WV); % hyperbolic relationship for f with body weight
    end
end

JA = f * sJAm * L^2;          % assimilation
JM = sJM * L^3;               % somatic maintenance
JV = yVA * (kap*JA-JM);       % growth

if WV < WVp                   % below size at puberty
    JR = 0;                   % no reproduction
    JJ = sJJ * L^3;           % maturity maintenance flux
    % JH = (1-kap) * JA - JJ; % maturation flux (not used!)
else
    JJ = sJJ * (WVp/dV);      % maturity maintenance
    JR = (1-kap) * JA - JJ;   % reproduction flux
end

% starvation rules may override these fluxes
if kap * JA < JM      % allocated flux to soma cannot pay maintenance
    if JA >= JM + JJ  % but still enough total assimilates to pay both maintenances
        JV = 0;       % stop growth
        if WV >= WVp  % for adults ...
            JR = JA - JM - JJ; % repro buffer gets what's left
        else
            % JH = JA - JM - JJ; % maturation gets what's left (not used!)
        end
    elseif JA >= JM   % only enough to pay somatic maintenance
        JV = 0;       % stop growth
        JR = 0;       % stop reproduction
        % JH = 0;     % stop maturation for juveniles (not used)
        JJ = JA - JM; % maturity maintenance flux gets what's left
    else              % we need to shrink
        JR = 0;       % stop reproduction
        JJ = 0;       % stop paying maturity maintenance
        % JH = 0;     % stop maturation for juveniles (not used)
        JV = (JA - JM) / yAV; % shrink; pay somatic maintenance from structure
    end
end

% we do not work with a repro buffer here, so a negative JR does not make
% sense; therefore, minimise to zero
JR = max(0,JR);

% Calcululate the derivatives
dWV = JV;             % change in body mass
dcR = yBA * JR / WB0; % continuous reproduction flux
if WB > 0             % for embryos ...
    dWB = -JA;        % decrease in egg buffer
else
    dWB = 0;          % for juveniles/adults, no change
end
if WV < WVb/4 && dWB == 0 % dont shrink below quarter size at birth
    dWV = 0; % simply stop shrinking ...
end

dX = [dWV;dcR];       % collect both derivatives in one vector

Other files: call_deri

To archive analyses, publishing them with Matlab is convenient. To keep track of what was done, the file call_deri.m can be included in the published result.

%% BYOM function call_deri.m (calculates the model output)
%
%  Syntax: [Xout TE] = call_deri(t,par,X0v)
%
% This function calls the ODE solver to solve the system of differential
% equations specified in <derivatives.html derivatives.m>, or the explicit
% function(s) in <simplefun.html simplefun.m>. As input, it gets:
%
% * _t_   the time vector
% * _par_ the parameter structure
% * _X0v_   a vector with initial states and one concentration (scenario number)
%
% The output _Xout_ provides a matrix with time in rows, and states in
% columns. This function calls <derivatives.html derivatives.m>. The
% optional output _TE_ is the time at which an event takes place (specified
% using the events function). The events function is set up to catch
% discontinuities. It should be specified according to the problem you are
% simulating. If you want to use parameters that are (or influence) initial
% states, they have to be included in this function.
%
% * Author: Tjalling Jager
% * Date: April 2017
% * Web support: <http://www.debtox.info/byom.html>
% * Back to index <walkthrough_debkiss.html>

%% Start

function [Xout TE] = call_deri(t,par,X0v)

global glo          % allow for global parameters in structure glo
global zvd   % global structure for zero-variate data

% NOTE: this file is modified so that data and output can be as body length
% whereas the state variable remains body weight. Also, there is a
% calculation to prevent shrinking in physical length (as shells, for
% example, do not shrink).

%% Initial settings
% This part extracts optional settings for the ODE solver that can be set
% in the main script (defaults are set in prelim_checks). The useode option
% decides whether to calculate the model results using the ODEs in
% <derivatives.html derivatives.m>, or the analytical solution in
% <simplefun.html simplefun.m>. Using eventson=1 turns on the events
% handling. Also modify the sub-function at the bottom of this function!
% Further in this section, initial values can be determined by a parameter
% (overwrite parts of X0), and zero-variate data can be calculated. See the
% example BYOM files for more information.

useode   = glo.useode; % calculate model using ODE solver (1) or analytical solution (0)
eventson = glo.eventson; % events function on (1) or off (0)
stiff    = glo.stiff; % set to 1 to use a stiff solver instead of the standard one

% Unpack the vector X0v, which is X0mat for one scenario
X0 = X0v(2:end); % these are the intitial states for a scenario

% % if needed, extract parameters from par that influence initial states in X0
% % start from specified initial size
% Lw0 = par.Lw0(1);  % initial body length (mm)
% X0(1) = glo.dV*((Lw0*glo.delM)^3); % recalculate to dwt, and put this parameter in the correct location of the initial vector

% start from the value provided in X0mat
if glo.len ~= 0 % only if the switch is set to 1 or 2! 
    % Assume that X0mat includes initial length; if it includes mass, no correction is needed here
    % The model is set up in masses, so we need to translate the initial length
    % (in mm body length) to body mass using delM. Assume that size is first state.
    X0(1) = glo.dV*((X0(1) * glo.delM)^3); % initial length to initial dwt
end

% % start at hatching, so calculate initial length from egg weight
% WB0   = par.WB0(1);     % initial dry weight of egg
% yVA   = par.yVA(1);     % yield of structure on assimilates (growth)
% kap   = par.kap(1);     % allocation fraction to soma
% WVb   = WB0 *yVA * kap; % dry body mass at birth
% X0(1) = WVb;            % put this estimate in the correct location of the initial vector

% % if needed, calculate model values for zero-variate data from parameter set
% if ~isempty(zvd)
%     zvd.ku(3) = par.Piw(1) * par.ke(1); % add model prediction as third value
% end

%% Calculations
% This part calls the ODE solver (or the explicit model in <simplefun.html
% simplefun.m>) to calculate the output (the value of the state variables
% over time). There is generally no need to modify this part. The solver
% ode45 generally works well. For stiff problems, the solver might become
% very slow; you can try ode15s instead.

c  = X0v(1);     % the concentration (or scenario number)
t  = t(:);       % force t to be a row vector (needed when useode=0)

% specify options for the ODE solver
options = odeset; % start with default options
if eventson == 1
    options = odeset(options, 'Events',@eventsfun); % add an events function
end
options = odeset(options, 'RelTol',1e-4,'AbsTol',1e-7); % specify tightened tolerances
% options = odeset(options,'InitialStep',max(t)/1000,'MaxStep',max(t)/100); % specify smaller stepsize

TE = 0; % dummy for time of events

if useode == 1 % use the ODE solver to calculate the solution
    % call the ODE solver (use ode15s for stiff problems, escpecially for pulses)
    if isempty(options.Events) % if no events function is specified ...
        switch stiff
            case 0
                [~,Xout] = ode45(@derivatives,t,X0,options,par,c);
            case 1
                [~,Xout] = ode113(@derivatives,t,X0,options,par,c);
            case 2
                [~,Xout] = ode15s(@derivatives,t,X0,options,par,c);
        end
    else % with an events functions ... additional output arguments for events:
        % TE catches the time of an event, YE the states at the event, and IE the number of the event
        switch stiff
            case 0
                [~,Xout,TE,YE,IE] = ode45(@derivatives,t,X0,options,par,c);
            case 1
                [~,Xout,TE,YE,IE] = ode113(@derivatives,t,X0,options,par,c);
            case 2
                [~,Xout,TE,YE,IE] = ode15s(@derivatives,t,X0,options,par,c);
        end
    end
else % alternatively, use an explicit function provided in simplefun!
    Xout = simplefun(t,X0,par,c);
end

if isempty(TE) || all(TE == 0) % if there is no event caught
    TE = +inf; % return infinity
end

%% Output mapping
% _Xout_ contains a row for each state variable. It can be mapped to the
% data. If you need to transform the model values to match the data, do it
% here. 

% This is the new part to avoid shrinking on length
if glo.len ~= 0 % only if the switch is set to 1 or 2!
    % Here, I translate the model output in body mass to estimated shell length
    W = Xout(:,1); % Assume that body weight is the first state
    L = (W/glo.dV).^(1/3)/glo.delM; % estimated shell length    
    if glo.len == 2 % when animal cannot shrink in length (but does on volume!)
        maxL = 0;%zeros(size(L)); % remember max size achieved
        for i = 1:length(L) % run through length data
            maxL = max([maxL;L(i)]); % new max is max of old max and previous size
            L(i) = maxL; % replace value in output vector
        end 
    end
    Xout(:,1) = L; % replace body weight by length
end

% % To obtain the output of the derivatives at each time point. The values in
% % dXout might be used to replace values in Xout, if the data to be fitted
% % are the changes (rates) instead of the state variable itself.
% % dXout = zeros(size(Xout)); % initialise with zeros
% for i = 1:length(t) % run through all time points
%     dXout(i,:) = derivatives(t(i),Xout(i,:),par,c); 
%     % derivatives for each stage at each time
% end

%% Events function
% Modify this part of the code if _eventson_=1. This subfunction catches
% the 'events': in this case, it looks for birth and the point where size
% exceeds the threshold for puberty. This function should be adapted to the
% problem you are modelling.
%
% Note that the eventsfun has the same inputs, in the same sequence, as
% <derivatives.html derivatives.m>.

function [value,isterminal,direction] = eventsfun(t,X,par,c)

global glo

Lwp      = par.Lwp(1); % shell length at puberty
if glo.len ~= 0 % only if the switch is set to 1 or 2!
    % Translate shell length at puberty to body weight
    WVp = glo.dV*(Lwp * glo.delM).^3; % initial length to initial dwt
end

nevents  = 1;          % number of events that we try to catch

value       = zeros(nevents,1); % initialise with zeros
value(1)    = X(1) - WVp;       % thing to follow is body mass (state 1) minus threshold

isterminal  = zeros(nevents,1); % do NOT stop the solver at an event
direction   = zeros(nevents,1); % catch ALL zero crossing when function is increasing or decreasing