Minnesota ad model builder short course october 22 24 2007
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Minnesota AD Model Builder Short Course October 22-24, 2007. Thanks to Jim Bence, Brian Linton, and Brian Irwin for providing materials used in previous courses QFC Supporting Partners – MSU, GLFC, Michigan DNR, Minnesota DNR, Ohio DNR, New York DEC, Illinois DNR, Ontario MNR.

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Minnesota AD Model Builder Short Course October 22-24, 2007

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Minnesota ad model builder short course october 22 24 2007

Minnesota AD Model Builder Short CourseOctober 22-24, 2007

  • Thanks to Jim Bence, Brian Linton, and Brian Irwin for providing materials used in previous courses

  • QFC Supporting Partners – MSU, GLFC, Michigan DNR, Minnesota DNR, Ohio DNR, New York DEC, Illinois DNR, Ontario MNR


Quantitative fisheries center qfc

Quantitative Fisheries Center (QFC)

  • Created July 2005

  • Co-directors: Jim Bence and Mike Jones

  • Staffing:

    • Associate Director

    • Computer Programmer

    • Post-Docs (2)

    • Graduate students (3 - PhD; 3 - MS)


Quantitative fisheries center qfc1

Quantitative Fisheries Center (QFC)

  • Provide research, outreach, and educational services to supporting partners

  • Outreach examples

    • Computer programming support to Michigan DNR inland creel database

    • SCAA consultation for Lake Erie percid assessments

    • River classifications in MI, WI, NY, PA

    • Power analysis for OhDNR Lake Erie gill net surveys


Quantitative fisheries center qfc2

Quantitative Fisheries Center (QFC)

  • Education

    • AD Model Builder short courses taught in East Lansing (2006, 2007) and Cornell Biological Field Station (2007)

    • Online Maximum Likelihood Estimation course (launched October 16, 2007)

    • Introduction to R short course (currently being converted to an online format)

    • Online Resampling Approaches to Data Analysis course (planned for summer 2008)


How this course will differ from previous offerings

How this course will differ from previous offerings

  • More emphasis on straightforward applications

  • More hands on programming (coding the whole program rather than only bits and pieces)

  • Less emphasis on coding efficiency (comes with practice)


What is ad model builder and why should you use it

What is AD Model Builder and why should you use it?

  • Auto Differentiation Model Builder

  • Software for creating computer programs to estimate parameters of statistical models


What are the advantages of using it

What are the advantages of using it?

  • Fast and accurate

  • Flexible

  • Designed for general maximum likelihood problems

  • Libraries for Bayesian and robust estimation methods

  • Includes many advanced programming options (estimation in phases)

  • Multi-dimensional arrays


How fast is it

How fast is it?

  • Evaluation by Schnute and Olsen

  • 100 parameter catch-at-age model from Schnute and Richards (2005)


Why is it so fast

Why is it so fast?

  • Auto differentiation – a method for approximating derivatives to within numerical precision

  • Most other computer programs actually calculate derivatives with respect to every parameter (finite differences)

    • Newton-Raphson – requires first and second order derivatives

    • Levenberg-Marquardt – requires first order derivatives


What are some of the most noticeable differences with other software packages

What are some of the most noticeable differences with other software packages?

  • Users must specify the objective function to be minimized (Note: ADMB only does minimization)


Minnesota ad model builder short course october 22 24 2007

Objective function

Parameter value


Admb differences with sas

ADMB Differences with SAS

data lenweight;

input length weight;

datalines;

358212

360242

382402

388285

394325

.

.

.

  • 12542

  • 15909

    ;

    Run;


Admb differences with sas1

ADMB Differences with SAS

procnlin data=lenweight;

parameters a=0 b=3;

model weight=a*length**b;

run;

Proc NLIN estimates parameters by (weighted) least squares; minimize the sum of square errors


Admb differences with sas2

ADMB Differences with SAS

procnlmixed data=lenweight;

ypred=alpha*length**beta;

parms alpha=0.001, beta=3, sigma=1;

model weight~normal(ypred,sigma);

run;

Proc NLMIXED estimates parameters by maximum likelihood


Admb differences with sas3

ADMB Differences with SAS

procnlp data=lenweight tech=newrap inest=par1 outest=opar1 maxiter=1000;

parms a, b, sigma;

ypred=a*Length**b;

nlogl = log(sigma)+0.5*((weight-ypred)/sigma)**2;

min nlogl;

run;

Proc NLP (NonLinear Programming) in SAS/OR is an estimation method similar in vein to that of ADMB in that analysts must specify their objective function


What are the most striking differences with other packages

What are the most striking differences with other packages?

  • Users specify the objective function to be minimized

  • Steps to running

    • Create an ADMB template

    • Convert template to C++ code

    • Compile – convert from programming code to machine code (creates an executable file)

    • Link the executable file to C++ libraries

    • Run your executable file

  • Resulting executable can be run on similar datasets on any computer


  • What are the difficulties associated with using admb

    What are the difficulties associated with using ADMB?

    • Requires a more intimate knowledge of statistical theory (probability distributions, likelihoods, Hessians)

    • Some knowledge of C++ is required

    • Code can be a little quirky (as you will soon see)


    Admb files

    ADMB Files

    Input

    .tpl – make the model

    .dat – input data

    .pin – initial values (optional; need to specify for all parameters)

    Output

    .par – parameters estimates

    .cor – correlation of parameters

    .std – parameter estimates with std. deviations

    .rep – user-defined outputs (optional)


    Admb files1

    ADMB Files

    Input

    ADMB will expect .dat and .pin files to have same name as .tpl

    e.g., MilleLacs.tpl, MilleLacs.dat (this can be overridden)

    Output

    • By default, output files will have same file name

      e.g., MilleLacs.rep, MilleLacs.par (this can be overridden)

    • Note: In the project folder,

    • ignore the files with the extra ~ on the extension…

    • e.g., Oneida.tpl~

    • they are temporary files (so be sure you open the right file).


    Dat file

    .dat file

    • Simply contains the data you will use when fitting your model

    Simple.dat

    #Simple linear regression example

    #For ADMB Short Course 1, August 2007

    #Created by D. Fournier, modified by B. Linton

    #Any text after "#" is ignored

    # number of observations

    10

    # observed Y values

    1.4 4.7 5.1 8.3 9.0 14.5 14.0 13.4 19.2 18

    # observed x values

    -1 0 1 2 3 4 5 6 7 8


    Tpl sections

    Each must be written just like that

    .tpl Sections

    • DATA_SECTION

    • PARAMETER_SECTION

    • INITIALIZATION_SECTION

    • PROCEDURE_SECTION

    • REPORT_SECTION

    • Other commonly used section

    • PRELIMINARY_CALCS_SECTION

    • LOCAL_CALCS


    Keep in mind

    Keep in mind

    • Different sections use different programming languages

      • Data, Parameter, Initialization sections used ADMB code

      • Procedure, Report, Local Calcs, Preliminary Calcs sections use C++ code

        • Lines typically must end with ;

        • Not absolute as in SAS (loops, conditional statements)


    Keep in mind1

    Keep in mind

    • Comments in .dat file are specified with ‘#”

    • Comments in .tpl are specified with ‘//’


    Keep in mind2

    Keep in mind

    • Section heads (DATA_SECTION, PARAMETER_SECTION) must be left justified

      • Except LOCAL_CALCS section, requires one space before typing LOCAL_CALCS

    • All other lines should have two spaces before the text


    Tpl sections1

    .tpl Sections

    • DATA_SECTION

    Identify values that will be read-in from .dat file

    Need to consider the order of numbers in your .dat file

    Can read your data in as integers, real numbers, matrices, arrays,…

    DATA_SECTION

    init_int first_year

    init_int last_year

    init_int first_age

    init_int last_age

    init_number lambda

    init_matrix obs_length(first_year,last_year,first_age,last_age)


    Tpl sections2

    .tpl Sections

    • DATA_SECTION

    Also where you can declare your looping variable; valid throughout your entire code

    DATA_SECTION

    init_int first_year

    init_int last_year

    init_int first_age

    init_int last_age

    init_number lambda

    init_matrix obs_length(first_year,last_year,first_age,last_age)

    int i

    int j


    If dat doesn t have the same name as tpl

    .tpl Sections

    If .dat doesn’t have the same name as .tpl

    • DATA_SECTION

    • Assume program is MyModel.tpl

    • Then, default search is for MyModel.dat

    • Code below will read-in a file named ControlFile.dat:

    • !!ad_comm::change_datafile_name("ControlFile.dat");

    • Can also go back:

    • !!ad_comm::change_datafile_name(“MyModel.dat");

      !! – tells ADMB that what follows is C++ code


    Minnesota ad model builder short course october 22 24 2007

    .tpl Sections

    Always a good idea to verify that your data have been read in correctlyIn .dat file, have -8888 as your last entryIn Data_section, specify init_int test as the last read in variable and type!!cout << test << endl;!!exit(99);

    • DATA_SECTION


    Minnesota ad model builder short course october 22 24 2007

    .tpl Sections

    DATA_SECTION//Read data in from simple.dat init_int nobs //number of observations init_vector Y(1,nobs) //observed Y values init_vector x(1,nobs) //observed x values init_int test //test variable !!cout << test << endl; !!exit(99);

    • DATA_SECTION


    Tpl sections3

    .tpl Sections

    • DATA_SECTION

    • PARAMETER_SECTION

    • Define Parameters – the values to be estimated (must have at least 1)

    • use loge scale, if only interested in non-negative parameter space

    • Identified by the prefix init_

    • Intermediary Variables - quantities that will change as a result of parameter estimation

    • Can also declare index variables here.

    • Also, if “containers” are needed just for output and not for calculations, then put those here too.

    • Name your Objective Function – the quantity to be minimized


    Tpl sections4

    .tpl Sections

    • DATA_SECTION

    • PARAMETER_SECTION

    PARAMETER_SECTION

    //Parameters to be estimated

    init_number a //slope parameter

    init_number b //intercept parameter

    //Quantities calculated from parameters

    vector pred_Y(1,nobs) //predicted Y values

    //Value to be minimized by ADMB

    objective_function_value rss //residual sum of squares


    Keep in mind3

    Keep in mind

    • Init_ in DATA_SECTION indicates a value that will be read in from the .dat file

    • Init_ in PARAMETER_SECTION specifies a variable that will be estimated


    Tpl sections5

    .tpl Sections

    • DATA_SECTION

    • PARAMETER_SECTION

    • INITIALIZATION_SECTION

    Set Initial values for parameters

    - use in place of .pin file

    log_F -1.0

    log_M -1.6


    Tpl sections6

    .tpl Sections

    • DATA_SECTION

    • PARAMETER_SECTION

    • INITIALIZATION_SECTION

    • PROCEDURE_SECTION

    Back transform parameters for use in functions (if needed)

    e.g., F = exp(log_F)

    Construct Functions

    Specify the equation for your Objective function

    Must have a PROCEDURE_SECTION for model to compile


    Tpl sections7

    .tpl Sections

    • PROCEDURE_SECTION

    DATA_SECTION

    init_int nobs //number of observations

    init_vector Y(1,nobs) //observed Y values

    init_vector x(1,nobs) //observed x values

    PARAMETER_SECTION

    init_number a //slope parameter

    init_number b //intercept parameter

    vector pred_Y(1,nobs) //predicted Y values

    objective_function_value rss //residual sum of squares

    PROCEDURE_SECTION

    //Simple linear model gives predicted Y values

    pred_Y=a*x+b;

    //Parameter estimates obtained by minimizing

    //objective function value (residual sum of squares)

    rss=norm2(Y-pred_Y); //norm2(x)=x1^2+x2^2+...+xn^2


    Tpl sections8

    .tpl Sections

    • DATA_SECTION

    • PARAMETER_SECTION

    • INITIALIZATION_SECTION

    • PROCEDURE_SECTION

    • REPORT_SECTION

    Specify output to go to .rep file

    Be sure to end .tpl with an empty line (hard return)


    Tpl sections9

    .tpl Sections

    • Report section useful for reporting values not otherwise needed in the model

    • Can be organized in many ways

    • Can still do calculations in REPORT_SECTION

      • e.g., report<< “S: ” << exp(-Z) <<endl;

    • Results (.rep file) can be read into other programs


    Create an output dat file

    Create an Output .dat file

    Append to file

    • Use an output file stream

      ofstream ofs(“MyOutput.dat”,ios::app);

      {

      ofs << “Output variable x: “ << x << endl;

      ofs << “Output variable y: “ << y << endl;

      }

    • Also can delete a file

      system(“del MyOutput.dat);

    Note: different system command for Linux


    Other tpl sections

    Other .tpl Sections

    • PRELIMINARY_CALCS_SECTION

      Uses C++ code

      Can do some preliminary calculations and manipulations with the data before getting into the model proper

      e.g., pi = 3.14;

    • RUNTIME_SECTION

    • Change behavior of function minimizer

    • TOP_OF_MAIN_SECTION

    • Change AUTODIFF global variables


    Compare with sas code

    Compare with SAS code

    DATA GROWTH;

    INPUT AGE LENGTH GENDER;

    DATALINES;

    PROCNLIN DATA=GROWTH METHOD=MARQUARDT;

    PARAMETERS LINF1 = 1100 K1=0.4 T01=0.0;

    YPRED= LINF1*(1-EXP(-K1*(AGE-T01)));

    MODEL LENGTH = YPRED;

    OUTPUT OUT=DATA_OUT PRED=PP RES=RR;

    RUN;

    Data Section

    Runtime Section

    Initialization Section

    Prelim Calcs Section

    Procedure Section

    Report Section


    Tpl file

    .tpl File

    • General rule: make .tpl file as general as possible (try to avoid hard coding) – will allow you to analyze future datasets

    • Must be “compiled” into C++ code

      1) tpl2cpp (makes .cpp file)

      2) compile (makes .exe file)

      3) link (connects libraries)

    • We’ll use Emacs (more later)


    Compiling your tpl

    Compiling your .tpl

    • Need a C++ compiler to run your code

    • After it is compiled, model will be a .exe

      • (so can be run on machines without ADMB)

    • If you change the .tpl file, it must be recompiled…

    • If you change and save data (values, sometimes dimensions), the existing model will still be ready to go…

      • So, advantage to putting starting values, ect…, into .dat or .pin files.


    How should i build my tpl

    How should I build my tpl

    Suggestions

    • Keep projects in separate folder

    • Name, describe, and date each file at the top

    • Start with a simple working program

    • Be sure data get read in correctly

    • Use unique names for files and parameters (don’t use “catch” as a variable name)

    • Avoid “hard coding” … make it flexible

    • Build it one step at a time

    • COMMENT, COMMENT, COMMENT


    About emacs

    About Emacs

    • For this class, you will Emacs to construct your .tpl file

    • A highly customizable text editor

    • We have modified Emacs so that an ADMB .tpl file is automatically linked to a C++ compiler

    • MINGW32 is a freeware C++ compiler – don’t need to buy both ADMB and Visual Studio


    Using emacs

    Using Emacs

    • Refer to Emacs Basics handout

    • Hotkeys are different

      • e.g., “control-v” will not paste

    • Highlighting text will automatically copy it

    • Remember to save files and recompile .tpl


    Let s try an example

    Let’s Try an Example

    Simple linear regression model

    Estimation by least squares


    Let s try an example1

    Let’s Try an Example

    • Start Emacs by double clicking the Emacs icon on the desktop


    Let s try an example2

    Let’s Try an Example

    • Open the simple.tpl and simple.dat files in the MNADMB folder located on your desktop


    Simple diagnostics

    Simple Diagnostics

    • Types of error messages:

      • Compile

      • Run-time

    • Modes of operation:

      • Safe mode

      • Optimization mode


    Compile errors

    Compile Errors

    Error in line 48 while reading

    r

    • Line number refers to tpl file

    • Need a space at start of each line of code

      • Except for comments and section headings

    • Need a “return” after last line of tpl


    Compile errors1

    Compile Errors

    c:/…/simple.cpp:36: error: expected `;' before "rss“

    c:/…/simple.cpp:35: error: `pred_Y' undeclared (first use this function)

    • Check designated line in cpp file

    • Make corrections to tpl file, not cpp file


    Run time errors

    Run-Time Errors

    Error reading in dat file – no error message

    • In DATA_SECTION, values made up for init_objects that are not assigned values from dat file

    • Use “cout” command to make sure dat file reads in properly


    Run time errors1

    Run-Time Errors

    VarValueGradient

    110.00000-1.#IND0e+000

    VarValueGradient

    20.000001.#QNANe+000

    • IND0: infinity or division by zero

    • QNAN: not a real number

    • Use “cout” command to check calculations


    Run time errors2

    Run-Time Errors

    Error in matrix inverse -- matrix singular in inv(dmatrix)

    • Hessian cannot be inverted to obtain asymptotic standard errors

    • Use different parameter starting values

    • Reparameterize model


    Run time errors3

    Run-Time Errors

    array bound exceeded -- index too high in prevariable::operator[]

    • Tried to assign value outside the defined range indices for vector or matrix

      • Define a vector to be 10 elements long

      • Write values to the vector using a loop with 11 steps

    • Use “cout” command to locate error in tpl

    • Error message only appears when in safe mode


    Modes of operation

    Modes of Operation

    • Safe mode: provides bounds checking on all array objects

      • ADModel > tpl2cpp > compile > link

      • ADModel > makeadms

    • Optimization mode: provides faster execution

      • ADModel > makeadm


    Essential theory

    Essential theory


    Goal model building

    Goal – Model Building

    • Models are tools for evaluating hypotheses

    • Uses of models

      • Improved understanding of a system

      • Prediction

      • Help in making decisions

    • Likely to have several competing models that you will want to fit and compare

    • Many different types of estimation procedures to consider – we will consider maximum likelihood


    Additional information

    Additional Information

    Online MLE Course Launched 16 Oct. 2007

    Registration at www.shop.msu.edu

    Normal Cost: $370 to QFC Supporting Partners

    $300 if you contact me ([email protected]) by November 5


    Background on maximum likelihood

    Background on maximum likelihood

    • Likelihood – a measure of how likely a set of parameters are to have produced your data

    • Can be confusing. Often written as: But not always –

    • Think of as function of parameters.

    • Depends upon data.

    • Likelihoods are mathematically the same as probability distributions; thus, you must consider what probability distribution gave rise to your data


    Probability density functions

    Probability density functions

    • Functions describing the probability of obtaining a particular outcome for a random variable

    • For discrete distributions, sometimes referred to as probability mass functions

    • Typically denoted as (x) or as (x|θ)


    Properties of probability distributions

    Properties of probability distributions

    • Probability density functions (pdf) for continuous distributions

    • Probability mass functions (pmf) for discrete distributions

    • Joint pdf/pmf for multiple independent observations


    A familiar example

    A familiar example

    Probability density function

    • A single observation from a normal distribution


    Minnesota ad model builder short course october 22 24 2007

    c1

    c2


    The likelihood function

    The likelihood function

    • Mathematically equal to the joint probability density function

    • But NOT a probability density for parameters

    Maximum likelihood estimates are values of parameters that maximize the likelihood


    Properties of maximum likelihood estimates

    Properties of maximum likelihood estimates

    • Invariant to transformations

    • Asymptotically efficient (lowest possible variance)

    • Asymptotically normally distributed

    • Asymptotically unbiased (expected value of the estimated parameter equals the true value)

    • If we assume independent, normally distributed errors, ML methods provide the same estimates of structural parameters as least squares.

    E.g., can be biased for small n

    Summary – versatile and widely used with a number of desirable properties, but not perfect!

    For normal distribution


    Back to really really really simple example normal one observation variance known

    Back to really, really, really simple example: Normal, one observation, variance known

    • Likelihood equal to the probability density function

    x = 12

    Maximum likelihood estimates are values of parameters that maximize the likelihood


    Slightly more complicated example normal sample multiple observations

    Slightly more complicated example– normal sample (multiple observations)


    To ease calculations typically take the negative log of the likelihood

    To ease calculations, typically take the negative log of the likelihood

    Parameter estimates that maximum the likelihood are the same values that will minimize the negative log likelihood


    Minnesota ad model builder short course october 22 24 2007

    Slightly more complicated example – regression


    Ignoring constants

    Ignoring constants

    • If you minimize negative log likelihood you can ignore constants because:

    Forsame q will minimize left and right hand side

    • Example of the reduced (ignored constants dropped) negative log likelihood (for normal). This depends on what you estimate.


    Concentrated likelihood

    Concentrated Likelihood

    Reduce the number of parameters by writing some parameters as a function of other parameters


    Concentrated likelihood1

    Concentrated Likelihood


    Combining normal data with different variances

    Combining normal data with different variances

    • Data are: {y11, y12, …y1k1, y21, y22, …y2k2} Plus known predictors X

    • First and second set of y have different distributions (variances)

    • -logL= L1+L2+IC:

    • Just special case of rule for getting joint pdf for independent data


    Concentrated negative log likelihood when there is more than one normal component

    Concentrated negative log likelihood when there is more than one normal component

    Lambda’s are a weighting factor – how strong a role should a particular data set play in influencing the overall fit of the model


    Likelihood for lognormal distribution

    Likelihood for Lognormal Distribution


    Von bertalanffy growth model

    Von Bertalanffy Growth Model

    Welcome to nonlinearity


    Example modeling size versus age

    Example – Modeling size versus age


    Von bertalanffy model

    Von Bertalanffy Model


    Let s write some code

    Let’s write some code…..

    • Check out your .dat file (number of observations, data for individual fish, dummy variable)

    • Start out by reading in your data

    • Use the simple.tpl as an example

    • Use the concentrated likelihood for the normal distribution as your obj. function

    • Initial values – Linf=1200 mm, t0 = 0, Growth coefficient = 0.4


    Hints

    Hints

    • Read age and length data in as a matrix

      e.g., init_matrix fish(1,nobs,1,2)

    • Create an age and length vector

      e.g., vector ages(1,nobs)

    • Extract age and length data from the matrix using the extract column command (column)

    • e.g. ages = column(fish,1)


    Let s write some code1

    Let’s write some code…..

    • Advanced things to try

      • Linf and Kappa must be positive so try estimating on a log scale

      • Use the full negative log likelihood as your objective function (sigma will be another parameter that will need to be estimated; also will need to be estimated on a log scale)


    Mortality estimation

    Mortality estimation

    • Tag-recovery studies widely used to estimate fishing and natural mortality rates

    • Expected number of tag recaptures generally considered to follow a multinomial distribution


    Mortality model

    Mortality model

    • Expected number of tag recaptures generally considered to follow a multinomial distribution


    Mortality model1

    Mortality model

    • Expected numbers of recaptures

    Recovery Periods

    S = probability a fish survives the year

    f = probability a fish is harvested by angler and its tag is retrieved and reported


    Minnesota ad model builder short course october 22 24 2007

    Instantaneous mortality formulation

    Prob. surviving previous time periods

    Prob. harvest during time period


    Let s write some code2

    Let’s write some code…..

    • Estimate Fs and Ms on the log scale

    • Assume =0.18 for all years

    • Use report section to calculate instantaneous total mortality for each year

    • Use report section to calculate exploitation rate


    Advanced things to try

    Advanced things to try…..

    • Create an output vector of predicted tag recoveries to see how well model agrees with observed data

    • Try different objective functions and see how results match with one another


    Minnesota ad model builder short course october 22 24 20071

    Minnesota AD Model Builder Short CourseOctober 22-24, 2007

    Day 2

    Questions before proceeding???


    Strategy for building your tpl from scratch

    Strategy for building your tpl from scratch

    • Start with a working file from another problem, as memory aid on coding syntax etc (section names, define variables, loops…).

    • First create a minimal program which consists of the required data, parameter, and procedure sections, has an objective function variable and one estimable parameter.

    • Check that the program reads the data in correctly (use cout and exit)


    Strategy for building your tpl from scratch1

    Strategy for building your tpl from scratch

    • Sequentially add calculations to the procedure section and check they work using cout and exit. Much easier to check things as you build them up rather than trying to find where the errors are after writing lots of code.

    • Use small steps and do not worry about efficiency too much at this stage.

    • Make sure the estimation procedure is working before investing time in defining derived variables in report section.


    Quick refresher

    Quick Refresher

    • In LN_Density folder is a dataset that was generated by random draw from a lognormal density (μ = 10, σ2 = 1.5)

    • Use ADMB to find MLEs of μ and σ2


    Alewife stock recruitment in lake huron

    Alewife stock recruitment in Lake Huron

    • What effect would reduction in salmonid stocking by OMNR and MiDNR have on fish communities?

    • Stock-recruitment relationships of prey species (alewife, rainbow smelt) recognized as major source of uncertainty


    Alewife spawners vs recruits

    Alewife spawners vs. recruits


    Ricker stock recruitment curve

    Ricker stock-recruitment curve

    Additive Error

    Multiplicative Error

    If X~N(μ,σ2) and Y = exp(X)

    then Y~LN(μ,σ2)


    Ricker stock recruitment curve1

    Ricker stock-recruitment curve

    Linearized form of multiplicative error

    To calculate R/S when R and S are vectors, use the command elem_div(R,S). Don’t forget to declare a vector where your predicted values will go


    Ricker stock recruitment curve2

    Ricker stock-recruitment curve

    Linearized form


    Things to try

    Things to try…..

    • Estimate the linearized version of the multiplicative error form of the recruitment model

    • Estimate the additive error form of the recruitment model (try using concentrated likelihood)


    Inference using ad model builder

    Inference using AD Model Builder


    An overview on inference

    An overview on inference

    • By inference, I mean going beyond point estimates and saying something about the quality of the estimates. How likely is it that the estimate is close to the true value?

    • Topics related to inference

      • Estimates of standard errors

      • Confidence intervals

      • Bayesian probability intervals


    Minnesota ad model builder short course october 22 24 2007

    • Inferences depend upon the variance-covariance matrix:Diagonal elements are variances of parameter estimates, off-diagonals are covariances among parameter estimates.


    What is a variance and covariance

    What is a variance and covariance?

    • Recall definition of expected value


    Covariances and parameter estimates

    Covariances and parameter estimates

    • The variances describe uncertainty in the parameter estimates.

    • The square-root of the variances gives the standard errors

    • The covariances describe how the estimation errors for two parameters are related. When parameter “a” is over-estimated does parameter “b” also tend to be over-estimated (+ cov), tend to be under-estimated (- cov) or is there no relationship (0 cov)?


    Correlation matrix

    Correlation matrix

    • Diagonals are 1.0

    • Off diagonals are correlations among parameter estimates:


    Asymptotic results for parameters

    Asymptotic results for parameters

    • Done automatically in ADMB (i.e., you don’t have to code anything)

    • Results are in *.std and *.cor

    • These are based on the Hessian matrix:


    Minnesota ad model builder short course october 22 24 2007

    measures how likelihood falls off away from best estimate


    Minnesota ad model builder short course october 22 24 2007

    Cross derivatives “twist” the likelihood surface. Not counting for them would cause underestimation of uncertainty!


    Minnesota ad model builder short course october 22 24 2007

    Example *.std output

    index name value std dev

    1 log_q -1.6219e+000 2.7145e+000

    2 log_popscale 7.4954e+000 1.6715e-001

    3 log_sel_par -6.0105e+000 2.7178e+000

    4 log_sel_par -3.1105e+000 2.7089e+000

    5 log_sel_par -1.3544e+000 2.7038e+000

    6 log_sel_par -1.4792e-001 2.6779e+000

    7 log_sel_par -4.7468e-002 2.5159e+000

    8 log_sel_par -7.7288e-001 2.0588e+000

    9 log_relpop 8.1995e-001 1.7816e-001

    10 log_relpop 1.5404e+000 1.7094e-001

    11 log_relpop 1.2639e+000 1.7262e-001

    ……. … …


    Minnesota ad model builder short course october 22 24 2007

    Example of *.cor file

    index name value std dev 1 2 3 4 5

    1 log_q -1.6219e+0002.7145e+000 1.0000

    2 log_popscale 7.4954e+0001.6715e-001 -0.6779 1.0000

    3 log_sel_par -6.0105e+0002.7178e+000 -0.9971 0.6695 1.0000

    4 log_sel_par -3.1105e+0002.7089e+000 -0.9997 0.6763 0.9970 1.0000

    5 log_sel_par -1.3544e+0002.7038e+000 -0.9999 0.6771 0.9971 0.9997 1.0000

    6 log_sel_pa …..

    ………………………

    52 ………………………


    What happens if we use rss instead of neg logl or neg logconc

    What happens if we use RSS instead of neg logL or neg logConc?

    Made this change to growth.tpl

    // conc=(nobs/2.0)*log(rss); //concentrated likelihood

    //changed obj function to just be RSS for illustrative purposes

    //DO NOT DO THIS

    conc=rss;


    Standard errors for derived quantities

    Standard errors for derived quantities

    • Often we are interested in assessing the uncertainty of derived quantities (quantities that are functions of one or model parameters)

      • biomass in last year of assessment, MSY for a logistic surplus production model, ratio of abundance in 2002 to abundance in 1995, SSBR based on recent mortality schedule,…

    • Calculated using the Delta method


    Delta method

    Delta method

    • Method for approximating statistical properties of nonlinear functions of random variables based on a Taylor series approximation of a function


    Delta method1

    Delta method

    • Method for approximating statistical properties of nonlinear functions of random variables based on a Taylor series approximation of a function


    Delta method2

    Delta method

    • For functions of several random variables, method requires partial derivatives, covariances, …

    • Approximation of variances less accurate then approximations of expectations (first versus second order Taylor series expansion)

    • Also used to estimate standard errors of derived quantities in SAS (Proc NLMIXED, PROC GLIMMIX)


    Standard error estimates for derived quantities continued

    Standard error estimates for derived quantities (continued)

    • Can be done for any type of variable (number, vector, matrix)

    • Specified in PARAMETER_SECTION

      • sdreport_number Z

      • sdreport_vector predicted_N (2,nages)

    • Results are included in *.std and *.cor files


    Refer to simple tpl

    Refer to simple.tpl


    Asymptotic standard errors can produce misleading inferences

    Asymptotic standard errors can produce misleading inferences

    • When sample sizes are small

    • The curvature of the likelihood surface changes substantially within the range of plausible estimates – i.e., “near to the maximum likelihood estimates


    Profile likelihood method

    Profile Likelihood Method

    • Typically, a method for constructing confidence intervals where analysts vary one or more parameters systematically and computes the values of the other parameters that maximize the likelihood

    • Surface of the likelihood used to construct confidence intervals based on a chi-square distribution


    Profile likelihood method cont

    Profile Likelihood Method (cont.)

    • Construct profile likelihood for growth coefficient of von Bertalanffy growth model (MLE = 0.281)


    Profile likelihood method cont1

    Profile Likelihood Method (cont.)


    Profile likelihood method1

    Profile Likelihood Method

    • This is NOT inverting a likelihood ratio test in ADMB land!

    • This is Bayesian in philosophy (in the same way that MCMC is). Can also be motivated by likelihood theory (support intervals)

    • Idea is to use the profile for g() to approximate the probability density function for g.


    How to use the profile method

    How to use the profile method

    • Declare a variable you would like to profile as type likeprof_number in the parameter section, and assign it the correct value in the procedure section.

    • When you run your program use the lprof switch: run -lprof

    • Results are saved in xxxxx.plt where xxxxx is the name of your likeprof_number variable

    • Your variable is varied over a “profile” of values and the best fit constrained to match each value of your variable is found


    Minnesota ad model builder short course october 22 24 2007

    PLT File contains list of point (x,y)

    x is value (say biomass)

    y is associated prob density

    Plot of Y vs X gives picture of prob distribution

    ADMB manual says estimate probability x in (xr,xs) by


    Profile likelihood options

    Profile likelihood options

    Switch

    -prsaveThis saves the parameter values associated with each step of profile in myvar.pv

    Options set in tpl (preliminary calcs section): e.g., for lprof var myvar:

    PRELIMINARY_CALCS_SECTION

    myvar.set_stepnumber(10); // default is 8

    myvar.set_stepsize(0.2); //default is 0.5

    Note manuals says stepsize is in estimated standard deviations but this appears to be altered adaptively during the profile

    WARNING -- LOTS OF STEPS CAN TAKE LOTS OF TIME!


    Exercises

    Exercises

    • Calculate asymptotic standard errors and likelihood profiles for

      • Growth model – parameters of growth model; predictions of length at age (asymptotic only)

      • Mortality model – Z=M+F and u=FA/Z

      • Stock recruitment model (linear version of the multiplicative error)

    Remember, with linear version and simple linear regression, β0 estimates log and β1 estimates β


    Essential programming skills

    Essential Programming Skills

    • New ADMB concepts and techniques

      • Loops

      • Conditional statements

      • Bounded objects

      • User-defined functions

      • Random number generation


    Loops

    Loops

    • Repeats code a specified number of times

      for (i=m;i<=n;i++)

      {

      . . . . . . ;

      }

      for (i=10, i>=0, i-=2)

    looping variable

    ‘i’ goes from ‘m’ to ‘n’ in increments of 1

    Code that is repeated for each increment of ‘i’

    ‘i’ goes from 10 to 0 in increments of -2


    Refer to loop tpl

    Refer to loop.tpl


    Conditional statements

    Conditional Statements

    • Runs code if conditions met

      if (condition)

      {

      . . . . . . ;

      }

    if condition is true

    then run this code


    Common conditional statements

    Common Conditional Statements

    • (X==Y)X equal to Y

    • (X!=Y)X not equal to Y

    • (X<Y)X less than Y

    • (X<=Y)X less than or equal to Y

    • (X>Y)X greater than Y

    • (X>=Y)X greater than or equal to Y

    • Use && for compound statements

      • e.g., if (iyear=1998 && iarea=north)

    • Use || for or statement

      • e.g., if (iyear=1998 || iyear=2000)


    Conditional statements1

    Conditional Statements

    if (condition)

    {

    . . . . . . ;

    }

    else

    {

    . . . . . . ;

    }

    if condition is true

    then run this code

    if condition is false

    then run this code


    Advanced conditional statements

    Advanced Conditional Statements

    • active(parameter)

      • Returns true if parameter is being estimated

    • last_phase()

      • Returns true if in last phase of estimation

    • mceval_phase()

      • Returns true if –mceval switch is used

    • sd_phase()

      • Returns true if in SD report phase


    Refer to conditional tpl

    Refer to conditional.tpl


    Bounded objects

    Bounded Objects

    • Bounds constrain what values a parameter can take

      init_bounded_number x(-10,10)

      init_bounded_vector y(1,nobs,-10,10)

    Lower bound

    Upper bound


    User defined functions

    User-Defined Functions

    • Organize code in PROCEDURE_SECTION

      function_name();

      FUNCTION function_name

      . . . . . . ;

    Call function for use

    Define function

    Code for function


    Functions that take arguments

    Functions that take arguments

    • Functions that do not take arguments can be used to organize code

      get_catch_at_age();

    • Functions that take arguments can simplify calculations

      rss=norm2(residuals);

    • Beware of functions that take parameters as arguments


    Refer to function tpl

    Refer to function.tpl


    Random number generator

    Random number seed

    Random Number Generator

    • Initialize random number generator (x)

      random_number_generator x(seed);

    • Fill object (y) with random numbers

      y.fill_randn(x); // yi~Normal(0,1)

      y.fill_randu(x); //yi~Uniform(0,1)


    Random number generator1

    Random Number Generator

    • Random number generator produces pseudo-random numbers

    • Pseudo-random numbers are generated from an algorithm which is a function of the random number seed

    • The same random number seed will always produce the same string of numbers


    Exercises1

    Exercises

    • Modify the growth.tpl so that you use a loop rather than the norm2 command to calculate the residual sum of squares


    Now to code it

    Now to code it…


    Piecewise regression catch curve

    Piecewise regression catch curve

    • Maceina (2007) recently proposed using piecewise regression to estimate size related mortality rates by catch curves


    Piecewise regression catch curve1

    Piecewise regression catch curve

    Knot or joinpoint


    Piecewise regression catch curve2

    Piecewise regression catch curve

    • 4 parameters (β0, β1, β2, knot) [actually 5 with 1 linear constraint]

    • Knot should be initialized as a bounded variable (minimum and maximum age)

    • Use a loop and conditional statement to estimate predicted ln catch at age for different fish ages

    • Use concentrated log likelihood (assume normal)


    Caveat

    Caveat

    • This estimation approach is similar to the one taken in Maceina (2007)

    • However, this is not the generally recommended approach for piecewise models

    • Grid search recommended

      • Search for regression parameters across a grid of knots

      • Fix the knot, estimate the regression parameters


    Now to code it1

    Now to code it…


    Convergence issues

    Convergence Issues

    • Convergence criteria

    • Diagnosing convergence problems

      • Convergence messages

      • Self diagnostics

    • Fixing convergence problems

      • Convergence criteria problems

      • Code problems


    Convergence criteria

    Convergence Criteria

    • Gradients close to zero

      • Maximum |gradient| < 1x10-4

    • Obj. function value fails to decrease

      • Change < 1x10-6 for 10 iterations in a row

    • Obj. function evaluated too many times

      • Maximum evaluations = 1,000

    • Line search fails to find parameters with lower objective function value

      • Step size adjusted 30 times


    Convergence messages

    Convergence Messages

    ic > imax in fminim is answer attained ?

    Function minimizer not making progress ... is minimum attained?

    Minimprove criterion = 0.0000e+000

    • Run-time messages indicating convergence problems


    Self diagnostics

    Self Diagnostics

    • Compare smallest and largest eigenvalues of Hessian in eva file

    • Is logarithm of determinant of Hessian small in cor file?

    • Are correlations large in cor file?

    • Are standard errors large compared to parameter value in std file?

    • Examine trajectory of iterations including objective function and key parameters


    Convergence criteria problems

    Convergence Criteria Problems

    • Is convergence criteria too strict or too loose?

      • Does objective function value change substantially as gradients approach convergence criterion?

      • Are results sensitive to changes in convergence criterion?

      • Try different parameter starting values


    Changing convergence criteria

    Changing Convergence Criteria

    • In tpl file

      RUNTIME_SECTION

      convergence_criteria 0.001

      maximum_function_evaluations 500

    • With runtime switches

      -crit 0.001 –maxfn 500

    • Switch to restart (after rescaling) if function not improving but gradients not near zero

      -rs


    Code problems

    Code Problems

    • Do predictions respond to parameter values?

      • If not possible can estimate parameters in different phases

      • Parameterize the current function differently

      • Or use a different function


    Improving efficiency

    Improving Efficiency

    • You do not need to worry about model efficiency in most cases

    • In general, it is only important when:

      • Your model is very complex

      • You are running your model many times (e.g., mcmc, simulation study)


    Efficiency rule number 1 calculate something only once if you can

    Efficiency rule number 1 – calculate something only once if you can!

    • Quantities that do not change but are needed during estimation should be calculated in PRELIMINARY_CALCS_SECTION

    • Quantities that are not needed for estimation but only for reporting should be calculated in REPORT_SECTION or if uncertainty estimates are needed conditional on phase too:

    • If (sd_phase())

      {…

      }


    Efficiency rule number 2 avoid unneeded loops

    Efficiency rule number 2 – avoid unneeded loops

    • Use admb built in functions (e.g., sum, rowsum, element by element multiplication and division, etc)

    • Combine loops over the same index


    Bayesian inference in ad model builder

    Bayesian Inference in AD Model Builder

    • Bayesian inference – a different philosophical approach to statistics than traditional inference

    • Essential element is the use of observed data to transform numerical estimates of our degree of belief in a hypothesis into posterior distributions that take into account the evidence in the observed data


    Minnesota ad model builder short course october 22 24 2007

    Bayesian Inference

    ADMB presumes we are going to start by finding the parameters that maximize the posterior density (called highest posterior or modal estimates), so just minimize the log posterior. Just like a negative log-likelihood but with new terms for priors


    Markov chain monte carlo

    Markov Chain Monte Carlo

    • Calculation of posterior distributions can be mathematically intractable accept for trivial scenarios

    • Markov Chain Monte Carlo method is an algorithmic way to generate samples from a complex multivariate pdf (in practice, usually the posterior distribution).

    • This is useful in looking at marginal distributions of derived quantities.

    • These marginal distributions are the same thing the profile likelihood method was approximating.


    Specifying priors

    Specifying Priors

    • If prior on M were log-normal with median of 0.2 and with sd for ln(M)=0.1, then just add to your likelihood:

    • For special case of diffuse prior ln(p()) is constant inside the bounds, so a bounded diffuse prior can be specified just by setting bounds on parameters.


    Doing an mcmc run

    Doing an MCMC run

    • Use -mcmc N switch to generate a chain of length N. Default N is 100,000.

    • Summarized output for parameters, sdreport variables and likeprof_numbers is in *.hst

    • This automatic summary is for the entire chain with no provision for discarding a burn-in and no built in diagnostics.

    • Serious evaluation of the validity of the MCMC results requires you gain access to the chain values.


    Gaining access to chain values

    Gaining access to chain values

    • When you do the MCMC run, add the switch -mcsave N, which saves in a binary file every Nth values from the chain.

    • You can rerun your program to read in the saved results and make one run through your model for each saved set of parameters. Use the switch -mceval

    • You can add code to your program to write out results (and do special calculations) during the mceval phase.

    • You can modify your program to do this even after you generate your chain, provided your change does influence the posterior density.


    Minnesota ad model builder short course october 22 24 2007

    Example of code to write results out

    when using mceval switch

    if (mceval_phase()) cout << Blast << endl;

    Important caution: this writes to standard output. Better redirect this to file or millions or numbers will go scrolling by!


    Some basic mcmc diagnostics

    Some basic MCMC diagnostics

    • Look at trace plot

    • Look at autocorrelation function for chain

    • Calculate “effective sample size”

    • Compare subchain CDFs (if the first and second half differ substantially then chain may be too short

    • Lots of other diagnostics and procedures

      • E.g., parallel chains and formal comparisons


    Minnesota ad model builder short course october 22 24 2007

    Trace plot example

    100,000 steps, sampled every 100


    Minnesota ad model builder short course october 22 24 2007

    First half

    Second half

    Entire chain

    burn-in excluded


    Minnesota ad model builder short course october 22 24 2007

    Autocorrelation example

    AR(1) shown for comparison (curve)


    Mcmc chain options to make changes to transition rule

    MCMC Chain Optionsto make changes to transition rule

    • -mcgrope p p is the proportion of “fat” tail

    • -mcrb N 1 to 9, smaller = weaker correlation

    • -mcdiag Hessian replaced with Identity

    • -mcmult N Scaler for Hessian

    • -mcnoscale No automatic adj to scaler


    Starting and restarting a chain

    Starting and Restarting a Chain

    • -mcr Restart from where it left off

    • -mcpin fn Start chain at params in fn

      • The output obtained by running with the switches -lprof -prsave (in *.prv) can be useful for this.


    Getting mcmc results into r

    Getting MCMC Results into R

    • To check chains, easiest to simply read the MCMC results into R and to use CODA functions


    Getting mcmc results into r1

    Getting MCMC Results into R

    Steps:

    1)Declare your parameters as sdreport_ objects (e.g., sdreport_number parameter1)

    2) In procedure section, include the following code

    if (mceval_phase())

      {

        cout << parameter1 << “ “ << parameter2 << “ “ << endl;

      }

    3) Use "Run ..." command from the ADModel menu.  Then in the mini-buffer, you enter the following switches "-mcmc XXX -mcsave YYY" where XXX is the number of MCMC cycles and YYY is how often the cycles are saved.

    4) Run the model a second time using the "Run ..."  In the minibuffer, you enter the switch "-mceval >> filename" where filename is the name of the file to which you want to save the parameters you are interested in


    Getting mcmc results into r2

    Getting MCMC Results into R

    • Open the chain output file in Excel and copy the chain results to the clipboard

    • Use the read.table command in R to copy the data into R

    • Convert to an .mcmc object and use CODA functions


    Huge literature on mcmc and diagnostics

    Huge literature on MCMC and Diagnostics

    • Gelman et al. Bayesian Data Analysis good general source on all things Bayesian

    • Cowles and Carlin, Markov Chain Monte Carlo convergence diagnostics: a comparative review. JASA 91:833-904


    Exercises2

    Exercises

    • Using models that you have estimated previously (i.e., growth, mortality, recruitment, piecewise) practice using Bayesian methods with both informative and non-informative priors and getting resulting MCMC chains into R and plotting with CODA functions (trace plots, density plots)


    Minnesota ad model builder short course october 22 24 20072

    Minnesota AD Model Builder Short CourseOctober 22-24, 2007

    Day 3

    Questions before proceeding???


    Sensitivity to parameter starting values

    Sensitivity to Parameter Starting Values

    • Why do we care about sensitivity to parameter starting values?

    • Methods for specifying starting values

      • Default values

      • In tpl file

      • In dat file

      • In pin file

    • Precedence between the methods


    Why do we care about sensitivity to starting values

    Why do we care about sensitivity to starting values?

    • Avoiding local minimums in the likelihood surface

      • If different starting values lead to solution with lower obj. function value, then you were at a local minimum

    • Identifying sensitive parameters

      • If small change to parameter starting value causes large change in results, then you may want to reparameterize model


    Default starting values

    Default Starting Values

    • Parameter with unspecified starting value has default starting value of zero

    • Bounded parameter has default starting value which is midway between lower and upper bounds


    Specify starting values in tpl file

    Specify starting values in tpl file

    INITIALIZATION_SECTION

    log_q -1.0

    • Must recompile tpl file everytime starting values are changed


    Specify starting values in dat file

    Specify starting values in dat file

    DATA_SECTION

    init_number start_log_q

    PRELIMINARY_CALCS_SECTION

    log_q = start_log_q;

    • Can change starting values without recompiling tpl file


    Specify starting values in pin file

    Specify starting values in pin file

    #Example pin file for model with 15 parameters

    0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0

    • Can change starting values without recompiling tpl file

    • Must specify a starting value for each parameter


    Precedence between methods

    Precedence Between Methods

    • Specifying starting values in dat file takes precedence over pin file and INITIALIZATION_SECTION

    • Specifying starting values in pin file takes precedence over INITIALIZATION_SECTION


    Matrix algebra in admb

    Matrix Algebra in ADMB

    • Usual rules of matrix algebra

    • Faster than loops

    • Also can do element-by-element tasks

      • element-by-element product

      • element-by-element division


    Matrix operations in admb

    Matrix Operations in ADMB

    Matrix Addition and Subtraction

    • Element-by-element calculations

    • Commutative: A+B = B+A

    • Associative: A±(B±C)=(A±B)±C

    Matrix Multiplication and Division

    • Not commutative: A*N usually ≠ N*A

    • ADMB can do both traditional matrix multiplication

    • and element-by-element operations

    Transpose

    See Gotelli and Ellison (2004) for a good primer on matrix operations


    Matrix multiplication

    Matrix Multiplication

    Column dimension of A must =

    Row dimension of N


    Array and matrix functions

    Array and Matrix Functions

    • Functions elem_prod and elem_div provide elementwise multiplication and division

    • For vector objects x, y and z

      z=elem_prod(x,y);//returns zi=xiyi

      z=elem_div(x,y);//returns zi=xi/yi

    • For matrix objects x, y and z

      z=elem_prod(x,y);//returns zi,j=xi,jyi,j

      z=elem_div(x,y);//returns zi,j=xi,j/yi,j


    Let s try an example3

    Let’s Try an Example

    (very) Simple simulation model

    • Leslie matrix calculator (following Gotelli 2001)

    • No estimation (but dummy parameter still needed in ADMB)

    Initial

    abundance-at-age

    matrix

    Transition matrix

    Fecundity

    Survival


    Let s try an example4

    Age 1

    Age 2

    Age 3

    Age 4

    Let’s Try an Example

    ADMB Tasks:

    • Perform matrix projection

    • Use a for loop (8 years)

    • Output

      • annual total abundance

      • age-specific abundance


    Advanced programming

    Advanced Programming

    • Phases

    • Reparameterization to improve estimation


    Phases

    Phases

    • Minimization of objective function can be carried out in phases

    • Parameter remains fixed at starting value until its phase is reached, then it become active

    • Allows difficult parameters to be estimated when other parameters are “almost” estimated


    Phases1

    Phases

    • Specified in PARAMETER_SECTION

      init_number x //estimate in phase 1

      init_number x(1) //estimate in phase 1

      init_number x(-1)//remains fixed

      init_vector x(1,n,2)//estimate in phase 2

      init_matrix x(1,n,1,m,3)//estimate in phase 3


    Parameterization issues

    Parameterization Issues

    • How do you estimate highly correlated parameters?

      • Catchabilities for multiple fisheries

      • Annual recruitments

    • Deviation method

    • Difference method

    • Random walk


    Deviation method

    Deviation Method

    • Estimate one free parameter

      X

    • Estimate m parameters as bounded_dev_vector

      w1, . . . , wm

    • Then

      logX1 = logX + w1

      . . . .

      logXm = logX + wm


    Bounded dev vector

    bounded_dev_vector

    • Specified in PARAMETER_SECTION

      init_bounded_dev_vector x(1,m,-10,10)

    • Each element must take value between lower and upper bounds

      -10 < xi < 10

    • All elements must sum to zero


    Difference method

    Difference Method

    • Estimate n free parameters:

      X1, y1, . . . , yn-1

    • Then

      logX1

      logX2 = logX1 + y1

      . . . . .

      logXn = logXn-1 + yn-1


    Random walk

    Random Walk

    • Estimate n parameters

      X1, w1, . . . , wn-1

    • Then


    Time varying von bertalanffy growth model

    Time-Varying Von Bertalanffy Growth Model

    • Asymptotic length varies over time

    • Mean length at age-1 (L1) and Brody growth coefficient (K) are constant over time


    Random walk1

    Random Walk

    • Model time-varying asymptotic length


    Model parameters

    Model Parameters

    Lyr=1…10,age=1 – mean length at age 1 for all years

    L,1– initial asymptotic length

    K – Brody growth coefficient

    w1, . . . , w9– annual deviation of L

    Lyr=1,age=2…9 – mean length at ages 2…9 for yr. 1


    Negative log likelihood

    Negative Log Likelihood


    Concentrated negative log likelihood

    Concentrated Negative Log Likelihood


    Now to look at the code

    Now to look at the code…..


    Overview of catch at age

    CAA estimates of population dynamics

    CAA predictions of observed data

    Overview of Catch-At-Age

    Observed data

    Negative log likelihood


    Basic relationships in catch at age models

    Basic Relationships in Catch-at-Age Models

    • Catch in relation to current abundance at fishing and natural mortalities

    • Number at any age in relation to initial cohort strength and cumulative fishing and natural mortality rates

    • Relationship between fishing mortality and fishing effort (catchability)


    Observed data

    Observed Data

    • Total annual fishery catch

    • Proportion of catch-at-age

    • Auxiliary data

      • Fishing effort

      • Survey index of relative abundance

      • Tagging data (to estimate M)


    Population submodel

    Initial numbers at age

    Recruitment

    Population Submodel

    . . . .

    logR+w1

    logN1,1+y1

    logN1,n-1+yn-1

    logR+w2

    . . . .

    logR+wm


    Population submodel1

    Numbers of fish

    Survival

    Total mortality

    Fishing mortality

    Natural mortality

    Population Submodel


    Population submodel2

    Selectivity

    Effort

    Catchability

    Effort error

    Population Submodel


    Observation submodel

    Observation Submodel

    • Baranov’s catch equation


    Observation submodel1

    Total catch

    Observed total catch

    Observation error

    Observation Submodel


    Observation submodel2

    Proportion of catch-at-age

    Numbers sampled at age

    Proportions

    Effective sample size

    Obs. proportion of catch-at-age

    Observation Submodel


    Negative log likelihood for multinomial

    Negative Log Likelihoodfor Multinomial


    Model parameters1

    Model Parameters

    R

    w1, . . . , wm

    y1, . . . , yn-1

    q

    s1, . . . , sn-1

    z1, . . . , zm

    se

    Ratio of relative variances (assumed known)


    Negative log likelihood ignoring constants

    Negative Log Likelihood (ignoring constants)


    Now to look at the code1

    Now to look at the code…..


    Simulating data

    Simulating Data


    Simulating data1

    Simulating Data

    • Data simulations are useful for testing models

    • How well does model perform when processes underlying “reality” are known?

      • The “true” values of parameters and variables can be compared to model estimates

    • Make sure model works before using real world data


    Simulating data2

    Simulating Data

    • “Parameter” values are read in from dat file

    • “Parameter” values used in estimation model equations to calculate true data

    • Random number generator creates random errors

    • Adding random error to true data gives observed data


    Data generating model

    Input “parameters”

    Data Generating Model

    • Recruitments generated from white noise process


    Data generating model1

    Data Generating Model

    • Numbers at age in first year came from applying mortality to randomly generated recruitments


    Input parameters

    Input “Parameters”

    sw

    M

    E1, . . . , Em

    s1, . . . , sn

    q

    sz

    se

    NE

    Must also provide seed for random number generator


    Tpl file must still contain

    Tpl file must still contain:

    • Data section

    • Parameter section

    • Procedure section

    • objective_function_value

    • One active parameter


    Data generating model2

    Data Generating Model

    • Most of work is done in preliminary calcs section or using local_calcs command

      • Operations involved only need to be run once

    • Use “exit(0);” command at end of local_calcs since no parameters or asymptotic standard errors need to be estimate


    Time for some code

    Time for some code…..


    Simulation study

    Simulation Study

    • Simulation study combines a data generating model with a control program to repeatedly fit an estimating model to many simulated data sets

    • This provides replicate model runs to better evaluate an estimating model’s performance

      • Only one replicate normally is available in the real world


    Simulation study1

    Simulation Study

    • Can evaluate how a model performs vs. different underlying “reality”

      • E.g., with different levels of observation error

    • Can evaluate how well different models can fit the same data sets

      • E.g., fit Ricker and Beverton-Holt stock-recruitment models to same data sets

    • Or can use a combination of the two approaches


    Simulation study practicum

    Simulation Study Practicum

    • You have seen:

      • Catch-at-age model

      • Data generating model for catch-at-age model

      • Control program for data generating and catch-at-age models

    • Now you need to modify the three programs to run a Monte Carlo simulation study


    Simulation study practicum1

    Simulation Study Practicum

    • Your simulation study will look at the effects of process error on performance of a catch-at-age model

    • Data sets will be generated using two levels (low and high) of process error for catchability


    Let s code a simulation study

    Let’s code a simulation study…..


    Miscellaneous topics and tricks

    Miscellaneous Topics and Tricks

    • Missing data

    • Advanced ADMB functions

    • Using dat file for flexibility


    Missing data

    Missing Data

    • It is not uncommon to have missing years of data in a time series of observed data

    • One solution is to interpolate the missing years of data outside the model fitting process by some ad hoc method

      • E.g., averaging data from the adjacent years

    • A better solution is to allow the model to predict values for the missing data

      • This takes advantage of all the available data


    Missing data implementation

    Missing DataImplementation

    • Use special value to denote missing data in dat file

      • E.g., a value you wouldn’t normally see in real data like -1

    • Use loops and conditional statements to exclude missing data values from objective function value

    • Otherwise, model will try to match predicted values to the missing data values


    Missing data multinomial case

    Missing DataMultinomial Case

    • Replace missing data with 0 and it will not contribute to negative log likelihood value


    Advanced functions

    Advanced Functions

    • Filling objects

    • Obtaining shape information

    • Extracting subobjects

    • Sorting vectors and matrices

    • Cumulative density functions


    Filling objects

    Filling Objects

    v.fill(“{1,2,3,6}”); // v=[1,2,3,6]

    v.fill_seqadd(1,0.5); // v=[1,1.5,2,2,5]

    m.rowfill_seqadd(3,1,0.5); // fill row 3 with sequence

    m.colfill_seqadd(2,1,0.5); // fill column 2 with sequence

    m.rowfill(3,v); // fill row 3 with vector v

    m.colfill(2,v); // fill column 2 with vector v


    Obtaining shape information

    Obtaining Shape Information

    i=v.indexmax(); // returns maximum index

    i=v.indexmin(); // returns minimum index

    i=m.rowmax(); // returns maximum row index

    i=m.rowmin(); // returns minimum row index

    i=m.colmax(); // returns maximum column index

    i=m.colmin(); // returns minimum column index


    Extracting subobjects

    Extracting Subobjects

    v=column(m,2); // extract column 2 of m

    v=extract_row(m,3); // extract row 3 of m

    v=extract_diagonal(m); // extract diagonal elements of m

    vector u(1,20)

    vector v(1,19)

    u(1,19)=v; // assign values of v to elements 1-19 of u

    --u(2,20)=v; // assign values of v to elements 2-20 of u

    u(2,20)=++v; // assign values of v to elements 2-20 of u

    u.shift(5); // new min is 5 new max is 24


    Sorting objects

    Sorting Objects

    • Sorting vectors

      w=sort(v); // sort elements of v in ascending order

    • Sorting matrices

      x=sort(m,3); // sort columns of m, with column 3 in ascending order


    Cumulative density functions

    Cumulative Density Functions

    • For standard normal distribution

      x=cumd_norm(z); // x=p(Z<=z), Z~N(0,1)

    • Also have CDF for Cauchy distribution

      cumd_cauchy()


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