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

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### Von Bertalanffy Growth Model

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)

- 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 (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 (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

- 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?

- Auto Differentiation Model Builder
- Software for creating computer programs to estimate parameters of statistical models

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?

- Evaluation by Schnute and Olsen
- 100 parameter catch-at-age model from Schnute and Richards (2005)

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?

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

Parameter value

ADMB Differences with SAS

data lenweight;

input length weight;

datalines;

358 212

360 242

382 402

388 285

394 325

.

.

.

- 12542
- 15909

;

Run;

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 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 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?

- 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?

- 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

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 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

- 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

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

- 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 mind

- Comments in .dat file are specified with ‘#”
- Comments in .tpl are specified with ‘//’

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 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 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

- 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

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

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 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 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 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 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 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 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 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 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

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

- 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

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

- 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

- 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

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

- 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

- 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

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

Let’s Try an Example

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

Simple Diagnostics

- Types of error messages:
- Compile
- Run-time
- Modes of operation:
- Safe mode
- Optimization mode

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 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

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 Errors

Var Value Gradient

1 10.00000 -1.#IND0e+000

Var Value Gradient

2 0.00000 1.#QNANe+000

- IND0: infinity or division by zero
- QNAN: not a real number
- Use “cout” command to check calculations

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 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

- Safe mode: provides bounds checking on all array objects
- ADModel > tpl2cpp > compile > link
- ADModel > makeadms
- Optimization mode: provides faster execution
- ADModel > makeadm

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

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 (brenden@msu.edu) by November 5

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

- 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

- Probability density functions (pdf) for continuous distributions
- Probability mass functions (pmf) for discrete distributions
- Joint pdf/pmf for multiple independent observations

c2

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

- 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

- Likelihood equal to the probability density function

x = 12

Maximum likelihood estimates are values of parameters that maximize 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

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

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

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

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

Welcome to nonlinearity

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

- 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 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

- 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

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

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

Instantaneous mortality formulation

Prob. surviving previous time periods

Prob. harvest during time period

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…..

- 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

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 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

- 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

- 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

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 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 curve

Linearized form

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)

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

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?

- Recall definition of expected value

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

- Diagonals are 1.0
- Off diagonals are correlations among parameter estimates:

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:

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

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

… …. … …

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?

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

- 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

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

Delta method

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

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)

- 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

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

- 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.)

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

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

- 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

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

Switch

-prsave This 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

- 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

- New ADMB concepts and techniques
- Loops
- Conditional statements
- Bounded objects
- User-defined functions
- Random number generation

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

Conditional Statements

- Runs code if conditions met

if (condition)

{

. . . . . . ;

}

if condition is true

then run this code

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 Statements

if (condition)

{

. . . . . . ;

}

else

{

. . . . . . ;

}

if condition is true

then run this code

if condition is false

then run this code

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

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

- 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 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

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 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

Exercises

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

Piecewise regression catch curve

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

Piecewise regression catch curve

Knot or joinpoint

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

- 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

Convergence Issues

- Convergence criteria
- Diagnosing convergence problems
- Convergence messages
- Self diagnostics
- Fixing convergence problems
- Convergence criteria problems
- Code problems

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

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

- 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

- 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

- 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

- 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

- 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!

- 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

- 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 – 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

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

- 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

- 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

- 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

- 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.

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

- 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

100,000 steps, sampled every 100

AR(1) shown for comparison (curve)

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

- -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

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

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 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

- 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

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)

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?

- 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

- 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

INITIALIZATION_SECTION

log_q -1.0

- Must recompile tpl file everytime starting values are changed

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

#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

- 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

- 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 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

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 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

Age 2

Age 3

Age 4

Let’s Try an ExampleADMB Tasks:

- Perform matrix projection
- Use a for loop (8 years)
- Output
- annual total abundance
- age-specific abundance

Advanced Programming

- Phases
- Reparameterization to improve estimation

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

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

- How do you estimate highly correlated parameters?
- Catchabilities for multiple fisheries
- Annual recruitments
- Deviation method
- Difference method
- Random walk

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

- 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

- Estimate n free parameters:

X1, y1, . . . , yn-1

- Then

logX1

logX2 = logX1 + y1

. . . . .

logXn = logXn-1 + yn-1

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 Walk

- Model time-varying asymptotic length

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

CAA estimates of population dynamics

CAA predictions of observed data

Overview of Catch-At-AgeObserved data

Negative log likelihood

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

- Total annual fishery catch
- Proportion of catch-at-age
- Auxiliary data
- Fishing effort
- Survey index of relative abundance
- Tagging data (to estimate M)

Observation Submodel

- Baranov’s catch equation

Numbers sampled at age

Proportions

Effective sample size

Obs. proportion of catch-at-age

Observation SubmodelModel Parameters

R

w1, . . . , wm

y1, . . . , yn-1

q

s1, . . . , sn-1

z1, . . . , zm

se

Ratio of relative variances (assumed known)

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 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

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

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:

- Data section
- Parameter section
- Procedure section
- objective_function_value
- One active parameter

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

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 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

- 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 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

Miscellaneous Topics and Tricks

- Missing data
- Advanced ADMB functions
- Using dat file for flexibility

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 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 DataMultinomial Case

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

Advanced Functions

- Filling objects
- Obtaining shape information
- Extracting subobjects
- Sorting vectors and matrices
- Cumulative density functions

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

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

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 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

- 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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