The FMM Procedure (Experimental)

specifies an absolute function convergence criterion. For minimization, termination requires , where is the vector of parameters in the optimization and is the objective function. The default value of is the negative square root of the largest double-precision value, which serves only as a protection against overflows.

specifies an absolute function difference convergence criterion. For all techniques except NMSIMP, termination requires a small change of the function value in successive iterations:

Here, denotes the vector of parameters that participate in the optimization, and is the objective function. The same formula is used for the NMSIMP technique, but is defined as the vertex with the lowest function value, and is defined as the vertex with the highest function value in the simplex. The default value is . The optional integer value specifies the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

specifies an absolute gradient convergence criterion. Termination requires the maximum absolute gradient element to be small:

Here, denotes the vector of parameters that participate in the optimization, and is the gradient of the objective function with respect to the th parameter. This criterion is not used by the NMSIMP technique. The default value is 1E5. The optional integer value specifies the number of successive iterations for which the criterion must be satisfied before the process can be terminated.

produces a table with additional details about the fitted model components.

produces the covariance matrix of the parameter estimates. For maximum likelihood estimation, this matrix is based on the inverse (projected) Hessian matrix. For Bayesian estimation, it is the empirical covariance matrix of the posterior estimates. The covariance matrix is shown for all parameters, even if they did not participate in the optimization or sampling.

produces the inverse of the covariance matrix of the parameter estimates. For maximum likelihood estimation, the covariance matrix is based on the inverse (projected) Hessian matrix. For Bayesian estimation, it is the empirical covariance matrix of the posterior estimates. This matrix is then inverted by sweeping, and rows and columns that correspond to linear dependencies or singularities are zeroed.

produces the correlation matrix of the parameter estimates. For maximum likelihood estimation this matrix is based on the inverse (projected) Hessian matrix. For Bayesian estimation, it is based on the empirical covariance matrix of the posterior estimates.

specifies the criterion by which the FMM procedure ranks models when multiple models are evaluated during maximum likelihood estimation. You can choose from the following keywords to rank models:

LOGL|LL

based on the mixture log likelihood

AIC

based on Akaike’s information criterion

AICC

based on the bias-corrected AIC criterion

BIC

based on the Bayesian information criterion

PEARSON

based on the Pearson statistic

GRADIENT

based on the largest element of the gradient (in absolute value)

The default is CRITERION=LOGL.

names the SAS data set to be used by PROC FMM. The default is the most recently created data set.

specifies how the FMM procedure handles support violations of observations. For example, in a mixture of two Poisson variables, negative response values are not possible. However, in a mixture of a Poisson and a normal variable, negative values are possible, and their likelihood contribution to the Poisson component is zero. An observation that violates the support of one component distribution of the model might be a valid response with respect to one or more other component distributions. This requires some nuanced handling of support violations in mixture models.

The default exclusion technique, EXCLUSION=ALL, removes an observation from the analysis only if it violates the support of all component distributions. The other extreme, EXCLUSION=NONE, permits an observation into the analysis regardless of support violations. EXCLUSION=ANY removes observations from the analysis if the response violates the support of any component distributions. In the single-component case, EXCLUSION=ALL and EXCLUSION=ANY are identical.

specifies a relative function convergence criterion. For all techniques except NMSIMP, termination requires a small relative change of the function value in successive iterations,

Here, denotes the vector of parameters that participate in the optimization, and is the objective function. The same formula is used for the NMSIMP technique, but is defined as the vertex with the lowest function value, and is defined as the vertex with the highest function value in the simplex. The

The default is , where FDIGITS is by default , and is the machine precision. The optional integer value specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.

requests that the "Optimization Information," "Iteration History," and "Fit Statistics" tables be produced for all optimizations when models with different number of components are evaluated. For example, the following statements fit a binomial regression model with up to three components and produces fit and optimization information for all three:

proc fmm fitdetails;
   model y/n  = x / kmax=3;
run;

Without the FITDETAILS option, only the "Fit Statistics" table for the selected model is displayed.

specifies a relative gradient convergence criterion. For all techniques except CONGRA and NMSIMP, termination requires that the normalized predicted function reduction be small,

Here, denotes the vector of parameters that participate in the optimization, is the objective function, and is the gradient. For the CONGRA technique (where a reliable Hessian estimate is not available), the following criterion is used:

This criterion is not used by the NMSIMP technique. The default value is 1E8. The optional integer value specifies the number of successive iterations for which the criterion must be satisfied before the process can terminate.

displays the Hessian matrix of the model. This option is not available for Bayesian estimation.

specifies the value assumed by the FMM procedure if a log likelihood cannot be computed (for example, because the value of the response variable falls outside of the response distribution’s support). The default value is E.

adds parameter estimates and gradients to the "Iteration History" table. If the FMM procedure centers or scales the model variables (or both), the parameter estimates and gradients reported during the iteration refer to that scale. You can suppress centering and scaling with the NOCENTER option.

specifies the maximum number of function calls in the optimization process. The default values are as follows, depending on the optimization technique:

• TRUREG, NRRIDG, and NEWRAP: 125

• QUANEW and DBLDOG: 500

• CONGRA: 1000

• NMSIMP: 3000

The optimization can terminate only after completing a full iteration. Therefore, the number of function calls that are actually performed can exceed the number that is specified by the MAXFUNC= option. You can choose the optimization technique with the TECHNIQUE= option.

specifies the maximum number of iterations in the optimization process. The default values are as follows, depending on the optimization technique:

• TRUREG, NRRIDG, and NEWRAP: 50

• QUANEW and DBLDOG: 200

• CONGRA: 400

• NMSIMP: 1000

These default values also apply when n is specified as a missing value. You can choose the optimization technique with the TECHNIQUE= option.

specifies an upper limit of r seconds of CPU time for the optimization process. The default value is the largest floating-point double representation of your computer. The time specified by the MAXTIME= option is checked only once at the end of each iteration. Therefore, the actual running time can be longer than that specified by the MAXTIME= option.

specifies the minimum number of iterations. The default value is 0. If you request more iterations than are actually needed for convergence to a stationary point, the optimization algorithms can behave strangely. For example, the effect of rounding errors can prevent the algorithm from continuing for the required number of iterations.

specifies the length to which long effect names are shortened. The default and minimum value is 20.

requests that regressor variables not be centered or scaled. By default the FMM procedure centers and scales columns of the matrix if the models contain intercepts. If NOINT options in MODEL statements are in effect, the columns of are scaled but not centered. Centering and scaling can help with the stability of estimation and sampling algorithms. The FMM procedure does not produce a table of the centered and scaled coefficients and provides no user control over the type of centering and scaling that is applied. The NOCENTER option turns any centering and scaling off and processes the raw values of the continuous variables.

suppresses the display of the "Class Level Information" table if you do not specify number. If you specify number, the values of the classification variables are displayed for only those variables whose number of levels is less than number. Specifying a number helps to reduce the size of the "Class Level Information" table if some classification variables have a large number of levels.

suppresses the display of the "Iteration History Information" table.

suppresses the normal display of tabular and graphical results. The NOPRINT option is useful when you want to create only one or more output data sets with the procedure. This option temporarily disables the Output Delivery System (ODS); see Chapter 20, Using the Output Delivery System, for more information.

specifies the sorting order for the levels of CLASS variables. This ordering determines which parameters in the model correspond to each level in the data.

You can specify the following values for order-type:

When the default ORDER=FORMATTED is in effect for numeric variables for which you have supplied no explicit format, the levels are ordered by their internal values. To order numeric class levels with no explicit format by their BEST12. formatted values, you can specify this format explicitly for the CLASS variables.

When FORMATTED and INTERNAL values are involved, the sort order is machine-dependent.

When the response variable appears in a CLASS statement, the ORDER= option in the PROC FMM statement applies to its sort order. For example, in the following statements the sort order of the wheeze variable is determined by the order of appearance in the input data set because the response variable appears in the CLASS statement:

proc fmm order=data;
   class city wheeze;
   model wheeze = city age / dist=binary s;
run;

However, in the following statements the sort order of the wheeze variable is determined by the formatted value (the default response-option in the MODEL statement):

proc fmm order=data;
   class city;
   model wheeze = city age / dist=binary s;
run;

The ORDER= option in the PROC FMM statement has no effect on the sort order of the wheeze variable because it does not appear in the CLASS statement.

When you specify a response-option in the MODEL statement, it overrides the ORDER= option in the PROC FMM statement.

For more information about sorting order, see the chapter on the SORT procedure in the Base SAS Procedures Guide and the discussion of BY-group processing in SAS Language Reference: Concepts.

specifies the display style for parameters and effects. The FMM procedure can display parameters in two styles:

• The EFFECT style (which is used by the MIXED and GLIMMIX procedure, for example) identifies a parameter with an "Effect" column and adds separate columns for the CLASS variables in the model.

• The LABEL style creates one column, named Parameter, that combines the relevant information about a parameter into a single column. If your model contains multiple CLASS variables, the LABEL style might use space more economically.

The EFFECT style is the default for models that contain effects; otherwise the LABEL style is used (for example, in homogeneous mixtures). You can change the display style with the PARMSTYLE= option. Regardless of the display style, ODS output data sets that contain information about parameter estimates contain columns for both styles.

specifies a variable in the input data set that identifies component membership. You can specify missing values for observations whose component membership is undetermined; this is known as a partial classification (McLachlan and Peel 2000, p. 75). For observations with known membership, the likelihood contribution is no longer a mixture. If observation is known to be a member of component , then its log likelihood contribution is

Otherwise, if membership is undertermined, it is

The variable specified in the PARTIAL= option can be numeric or character. In case of a character variable, the variable must appear in the CLASS statement. If the PARTIAL= variable appears in the CLASS statement, the membership assignment is made based on the levelized values of the variable, as shown in the "Class Level Information" table. Invalid values of the PARTIAL= variable are ignored.

In a model in which label switching is a problem, the switching can sometimes be avoided by assigning just a few observations to categories. For example, in a three-component model, switches might be prevented by assigning the observation with the smallest response value to the first component and the observation with the largest response value to the last component.

controls the plots produced through ODS Graphics.

ODS Graphics must be enabled before requesting plots. For example:

ods graphics on;
proc fmm data=yeast seed=12345;
   model count/n = / k=2;
   freq f;
   performance cpucount=2;
   bayes;
run;
ods graphics off;

For more information about enabling and disabling ODS Graphics, see the section Enabling and Disabling ODS Graphics in Chapter 21, Statistical Graphics Using ODS.

Global Plot Options

Specific Plot Options

determines the random number seed for analyses that depend on a random number stream. If you do not specify a seed or if you specify a value less than or equal to zero, the seed is generated from reading the time of day from the computer clock. The largest possible value for the seed is . The seed value is reported in the "Model Information" table.

You can use the SYSRANDOM and SYSRANEND macro variables after a PROC FMM run to query the initial and final seed values. However, using the final seed value as the starting seed for a subsequent analysis does not continue the random number stream where the previous analysis left off. The SYSRANEND macro variable provides a mechanism to pass on seed values to ensure that the sequence of random numbers is the same every time you run an entire program.

Analyses that use the same (nonzero) seed are not completely reproducible if they are executed with a different number of threads since the random number streams in separate threads are independent. You can control the number of threads used by the FMM procedure with system options or through the PERFORMANCE statement in the FMM procedure.

tunes the singularity criterion in Cholesky decompositions. The default is times the machine epsilon; this product is approximately 1E12 on most computers.

sets the tolerance for which the residual variance or scale parameter is considered to be zero. The default is 1E4 times the machine epsilon; this product is approximately 1E12 on most computers.

tunes the general singularity criterion applied by the FMM procedure in sweeps and inversions. The default is times the machine epsilon; this product is approximately 1E12 on most computers.

specifies the optimization technique to obtain maximum likelihood estimates. You can choose from the following techniques by specifying the appropriate keyword:

CONGRA

performs a conjugate-gradient optimization.

DBLDOG

performs a version of double-dogleg optimization.

NEWRAP

performs a Newton-Raphson optimization combining a line-search algorithm with ridging.

NMSIMP

performs a Nelder-Mead simplex optimization.

NONE

performs no optimization.

NRRIDG

performs a Newton-Raphson optimization with ridging.

QUANEW

performs a dual quasi-Newton optimization.

TRUREG

performs a trust-region optimization.

The default is TECH=QUANEW.

For more details about these optimization methods, see the section Choosing an Optimization Algorithm of Chapter 19, Shared Concepts and Topics.