Abstract

Mixture of autoregressions (MoAR) models provide a model-based approach to the clustering of time series data. The maximum likelihood (ML) estimation of MoAR models requires evaluating products of large numbers of densities of normal random variables. In practical scenarios, these products converge to zero as the length of the time series increases, and thus the ML estimation of MoAR models becomes infeasible without the use of numerical tricks. We propose a maximum pseudolikelihood (MPL) estimation approach as an alternative to the use of numerical tricks. The MPL estimator is proved to be consistent and can be computed with an EM (expectation-maximization) algorithm. Simulations are used to assess the performance of the MPL estimator against that of the ML estimator in cases where the latter was able to be calculated. An application to the clustering of time series data arising from a resting state fMRI experiment is presented as a demonstration of the methodology.

1  Introduction

The simultaneous acquisition of large numbers of time series arises in many areas of modern science. This is especially true in the areas of biological and medical image analyses, where multiple time series are commonly acquired in electrocardiogram (ECG), electroencephalography (EEG), and functional magnetic resonance imaging (fMRI) experiments. In such experiments, hundreds to hundreds of thousands of time series can be acquired simultaneously, each often thousands of periods long. Upon acquisition, a common approach in such experiments is to organize the time series into similarity groups (clusters) based on their properties.

The clustering of time series data has received much attention in years. For example, the literature reports in Liao (2005) and Esling and Agon (2012) illustrate the breadth of research in the area. It is clear from Esling and Agon (2012) that there are many potential directions for approaching the problem. Given the context of this letter, we concentrate only on mixture-model-based methods for clustering time series data. A brief review of recent developments in this direction is given below.

Cadez, Gaffney, and Smyth (2000) suggested a mixture of Markov chains model for the clustering of data based on web browsing behavior, time course gene expression, and red blood cell cytograms. Xiong and Yeung (2004) suggested mixture of autoregressive moving-average regressions (MoARMA) models for the clustering of ECG, EEG, population, and temperature data. Luan and Li (2003), Celeux, Martin, and Lavergne (2005), Ng, McLachlan, Ben-Tovim, and Ng (2006), and Scharl, Grun, and Leisch (2010) suggested various specifications of mixture of mixed effects models for the clustering of time course gene expression data. Wang, Ng, and McLachlan (2012) extended the methodology of Ng et al. (2006) by considering moving-average errors. Samé, Chamroukhi, Govaert, and Aknin (2011) suggested the use of mixture of linear experts for the clustering of electrical power consumption data.

Recently, Nguyen et al. (2016) reconsidered the work of Xiong and Yeung (2004) and proposed a mixture of autoregressions (MoAR) model for the clustering of spatially dependent time series data that arise from imaging-based experiments. In their work, a minorization–maximization (MM) algorithm (see Lange, 2013) was proposed, which both monotonically increased the marginal likelihood objective function and led to convergence to a stationary point of the log-marginal likelihood function. Furthermore, it was established that the maximum marginal likelihood estimator for the MoAR model was consistent under some regularity assumptions on the dependency structure of data. We note that marginal likelihood can be replaced by likelihood when the data are assumed to be independent.

The method that Nguyen, McLachlan, Ullmann, and Janke (2016) presented requires the evaluation of products of the form
formula
1.1
where is a vector containing realizations of , where arises from a finite normal mixture model with components (see McLachlan & Peel, 2000, regarding normal mixture models), and is a normal density function with mean and variance . Here the superscript indicates matrix transposition.

In standard application conditions, such products can decrease rapidly to values that are below usual machine precision for relatively small , where is the length of the time series under analysis. Numerical tricks can be applied (e.g., Press, Teukolsky, Vetterling, & Flannery, 2007) to avoid numerical underflows. We present an alternative to these tricks using pseudolikelihood (PL) functions.

In this letter, we formulate the maximum pseudolikelihood (MPL) estimator of the MoAR model for long time series, under the MPL estimation framework of Arnold and Strauss (1991; see also Molenberghs & Verbeke, 2005). We prove that the MPL estimator is consistent under mild regularity conditions. Also, we construct an expectation–maximization (EM) algorithm (Dempster, Laird, & Rubin, 1977) for the MPL estimation of the MoAR model. We show that the algorithm monotonically increases the PL value at each iteration and consequently leads to convergence to a stationary point of the log PL function.

Besides our algorithm and theoretical results, we also demonstrate the performance of our methodology using a simulation study. In this study, we demonstrate that the MPL estimator exhibits convergence toward the population parameter in finite samples. Also, we demonstrate that the MPL estimator can exhibit superefficiency for estimating of the mixing proportions when compared to the maximum likelihood (ML) estimator. We further demonstrate our methodology with an application to clustering data arising from an fMRI experiment.

The remainder of the letter proceeds as follows. In section 2, we present the MoAR model, review the work of Nguyen et al. (2016), and examine the problems associated with the calculation of equation 1.1. In section 3, we present the MPL estimator and construct an EM algorithm for its computation. In section 4, we examine aspects of statistical inference that arise from the use of the MPL estimator. In section 5, we present the results of our numerical simulations. In section 6, we present an example analysis of an fMRI data set. Conclusions are drawn in section 7.

2  ML Estimation of MoAR Models

2.1  Mixture of Autoregressive Models

Let be a random vector of length , indexed by . Suppose that is a latent random variable, such that , for , where and . We say that arises from a -component MoAR model of order if it can be characterized by the conditional density function,
formula
2.1
where , , and . Here is the model parameter vector.
Under the characterization of equation 2.1, if we suppose that the first elements of are nonstochastic (, for , almost everywhere), then we can further characterize via the joint conditional density function
formula
and hence the marginal density function,
formula
2.2
Using the characterization of equation 2.2, we can write the likelihood and log likelihood of an independent and identically distributed (i.i.d.) sample as
formula
and
formula
2.3
respectively.

Let the ML estimator be defined as an appropriate local-maximizer of equation 2.3. Due to the log summation form of equation 2.3, it is not possible to deduce a closed-form expression for . As such, an iterative algorithm is required for the computation of .

2.2  EM Algorithm for ML Estimation

Let be the initial value of for the application of the algorithm, and let be the th iterate. Nguyen et al. (2016) considered the following EM algorithm for computation of . We note that instead of using an EM algorithm, we could use for this problem an MM algorithm as, for example, in Nguyen et al. (2016) for their problem.

At the th iteration, the updates are given by
formula
2.4
formula
2.5
and
formula
2.6
for each , where for each .

As the updates of equations 2.4 to 2.6 are specified by an EM algorithm, the likelihood value increases monotonically at each iteration. Unfortunately, each iteration of the algorithm requires computating for every , which requires evaluating multiple products of equation 1.1. This can cause numerical underflow problems for large without the application of numerical tricks as mentioned earlier.

2.3  The Product Problem

We now consider the problem of computing equation 1.1 in a general context. Let be a vector of i.i.d. random variables with density function , where , , for each and . By independence and integration, we have the following result regarding the expectation of equation 1.1.

Proposition 1.
The expectation of equation 1.1 can be written as
formula
2.7

Note that the summation in equation 2.7 is less than one if for each For fixed , attains a global maximum at ; thus, the condition is fulfilled if we set for each (or simply since ). Under such a condition, it is easy to see that equation 2.7 goes to zero as increases. This degeneration can be very rapid for models with high variances in each component. For example, consider the following result:

Proposition 2.
For any and , if and , then
formula
2.8

Thus, numerical underflow can occur without the use of numerical tricks in a direct implementation of ML estimation using the EM algorithm. We now consider an alternative to ML estimation that addresses the product problem without the use of numerical tricks.

3  Maximum Pseudolikelihood Estimation

3.1  Pseudolikelihood Function

Using the equation 2.1 characterization and the PL definition of Arnold and Strauss (1991), we can write a PL function for a single time series as
formula
3.1
for each . We say “a” above since equation 3.1 is one of many possible PL functions that can be deduced from the equation 2.1 characterization. The chosen form of the PL function implicitly assumes that each of the random elements of can independently belong to each of the mixture components, conditioned on the previous elements. The specification allows the construction of a log PL function:
formula
3.2

Let the MPL estimator be defined as an appropriate local maximizer of equation 3.2. Like equation 2.3, equation 3.2 also contains terms of the log-summation form, and thus it is not possible to deduce a closed-form expression of . We now present an EM algorithm for the iterative computation of the MPL estimate.

3.2  EM Algorithm for MPL Estimation

We can specify a so-called complete-data version of the PL function in that it can be viewed as a joint density of the observed time series data and their unobservable component-indicator variables that imply the PL function. The logarithm of this joint density (the complete data log PL function) is given by
formula
3.3
where gathers up constants that do not depend on and is the component membership of time point of series , given the previous terms. Here, is the indicator variable that takes value 1 if proposition is true and 0 otherwise.
Starting from some initial value , the expectation of equation 3.3, computed using for , can be written as
formula
3.4
where
formula
3.5
The posterior probability is the conditional probability that belongs to the th component given and for , , and .
To perform the M-step, we maximize equation 3.4 under the restriction by constructing the Lagrangian and solving the equation corresponding to the first-order condition , where is the gradient operator. This yields the updates
formula
3.6
formula
3.7
and
formula
3.8
for each . Closely following the proof of Nguyen and McLachlan (2015), we obtain the following analog to Nguyen et al. (2016).
Proposition 3.
Given , if is obtained via the updates of equations 3.6 to 3.8 and
formula
3.9
then
formula

Proposition 3 implies that the log PL function monotonically increases at each iteration when the update steps 3.6 to 3.8 are used.

3.3  Convergence via the EM Algorithm

Given some initial value , the EM algorithm defined by updates 3.6 to 3.8 is run for some fixed number of iterations or until some convergence criterion is met, whereupon the final iterate of the algorithm is declared the MPL estimate (see Lange, 2013 for a description of various stopping criteria and their relative merits).

Let be a limit point of the EM algorithm, starting from some initial value . It is known that the EM algorithm is a special case of the MM algorithm (cf. Razaviyayn, Hong, & Luo, 2013). As such, the following theorem regarding the limit points of the EM algorithm can be adapted from the MM algorithm theory of Razaviyayn et al. (2013).

Theorem 1.

Starting from some initial value , if is a finite limit point of the sequence , obtained via updates 3.6 to 3.8, then is a saddle point or local maximum of equation 3.2.

As with the log-likelihood function from Nguyen et al. (2016), the log PL function is also unbounded. Because of this, the choice of initial value can be crucial to the success of the algorithm in finding an appropriate maximizer of equation 3.2. An example of a procedure that can be used to find good initial values is described in McLachlan and Peel (2000).

4  Statistical Inference

4.1  Consistency of MPL Estimator

Under usual regularity conditions, the MPL estimator is known to be consistent (see, e.g., Arnold & Strauss, 1991). Unfortunately, the log PL function is not identifiable, and thus the usual asymptotic formulations cannot be used. As such, we apply Amemiya (1985) to derive a result analogous to Nguyen et al. (2016).

Theorem 2.
Let be an i.i.d. random sample, such that for each , arises from a population with density function , where is a strict-local maximizer of . If (where we take , for some , if has no solution), then for any ,
formula

We omit the proof of theorem 5, as it follows closely the proof of Nguyen et al. (2016, theorem 5). We make the following remarks regarding theorem 5.

First, note that the theorem implies that the consistent roots of the log PL function are not necessarily the consistent roots of the log-likelihood equation. In many situations, the two sets of roots will correspond. Kenne Pagui, Salvan, and Sartori (2015) present a result regarding conditions under which such correspondence occurs. Unfortunately, it is difficult to verify the score and information conditions of Kenne Pagui et al. (2015), due to the nature of mixture-model densities. Second, the theorem suggests only that there may exist multiple roots of the log PL equation, of which one is consistent; as noted earlier, it is advisable to search for good initial values that lead to the correct root. Finally, the theorem can be extended to dependent identically distributed samples via conditioning on the dependence structure of . For example, as in Nguyen et al. (2016), one can assume ergodicity or -mixing conditions.

4.2  Cluster Analysis

When performing model-based clustering, one would generally use the plug-in Bayes’ rule for risk-minimal allocation. Let be the plug-in Bayes’ allocation of observation , and note that is the estimated posterior probability of observation belonging to cluster . In the current context, observation can be allocated via the plug-in Bayes’ rule,
formula
4.1
We cannot guarantee the convergence of equation 4.1 to the same allocation of as that obtained using the ML estimator, since we cannot establish the equivalence between and . Furthermore, the computation of equation 4.1 requires products of equation 1.1, which we are trying to avoid.
Unfortunately, we cannot overcome the first of these two caveats in a simple manner. Fortunately, the second can be addressed with an approximation. Using equation 3.5, we say that is the pseudoallocation of and define it as
formula
4.2
where and is the cluster pseudoallocation of .
Define the -mixing rate of observation over time as
formula
where is the -algebra generated by , for . The following result establishes the -mixing of , for each , and thus the convergence of to nontrivial limits under reasonable assumptions on the conditional characterizations of equation 2.1:
Proposition 4.

If the characteristic polynomials have roots inside the unit circle (with respect to ), for each , then for each ,

  1. The time series is -mixing.

  2. The cluster pseudoallocation converges to , for each .

Proof.
The hypothesis of the proposition guarantees that each of the conditional characterizations of equation 2.1 are -mixing (see Athreya & Pantula, 1986). Thus, if we denote the -mixing rate of conditioned on as , then as for each . Since can exhibit only one of the different behaviors of the conditional characterizations, we have
formula
as ; this implies part a.

Since are continuous functions of finitely many terms of , are also -mixing (see White, 2001, theorem 3.49) for each and . Because is bounded, it also has all of its moments, and thus White (2001, corollary 3.48) establishes the convergence of to , as . This proves part b.

In general, we do not expect the limit to equate to . We compare the performances of rules 4.1 and 4.2 in the next section.

5  Numerical Simulations

5.1  Simulation Setup

We report on three numerical simulation studies designated S1, S2, and S3. All studies refer to the classes of generative models () that are reported in Table 1. Examples of series of length 100 from each class are plotted in Figure 1.

Table 1:
Parameter Vectors of C1–C4, as Used in S1 and S2.
Class
 0.25 
 0.25 
 0.25 
 0.25 
Class
 0.25 
 0.25 
 0.25 
 0.25 
Figure 1:

Three realizations of time series of length from each of the classes C1–C4, as described in Table 1.

Figure 1:

Three realizations of time series of length from each of the classes C1–C4, as described in Table 1.

In S1, we generate time series of length from classes and with probabilities , where . This is repeated times for each combination of and . In S2, we generate time series of length for the same range of and as in S1, from classes to with probabilities , for each . This is also repeated times for each and . Simulation S3 is the same as S2 except that we generate time series from classes , , , and , with probabilities , , , and , respectively. This simulation scenario is offered as an unbalanced counterpoint to the balanced proportions design of S2.

For each combination and each study, we compute the MPL estimate and calculate the mean squared error (MSE) for each parameter element, where and denote the th element of the MPL estimate and the true parameter vector (as given in Table 1), respectively. Here in S1 and in S2 and S3. The MSE results for S1 and S2 are presented in Tables 2, 3, and 4, respectively. The MSE results for S3 are presented in Tables 5 and 6.

Table 2:
S1 MSE Results for the MPL Estimate Vector versus the True Parameter Vector, Averaged over Repetitions: .
100 100 1.4E-05 1.4E-05 3.5E-04 3.1E-04 2.9E-04 3.5E-04 3.1E-04 2.8E-04 6.9E-04 7.3E-04 
100 200 6.5E-06 6.5E-06 1.6E-04 1.4E-04 1.3E-04 1.6E-04 1.4E-04 1.3E-04 3.2E-04 2.9E-04 
100 500 4.0E-06 4.0E-06 4.6E-05 5.6E-05 5.5E-05 4.5E-05 5.6E-05 5.1E-05 1.4E-04 1.1E-04 
100 1000 1.3E-06 1.3E-06 2.9E-05 3.1E-05 1.8E-05 2.8E-05 3.1E-05 2.0E-05 5.8E-05 6.5E-05 
200 100 1.7E-05 1.7E-05 1.6E-04 1.5E-04 1.0E-04 1.6E-04 1.5E-04 1.0E-04 4.7E-04 3.5E-04 
200 200 6.2E-06 6.2E-06 9.5E-05 7.4E-05 5.5E-05 9.5E-05 7.2E-05 5.6E-05 1.9E-04 1.9E-04 
200 500 2.8E-06 2.8E-06 3.2E-05 3.5E-05 2.1E-05 3.1E-05 3.5E-05 2.2E-05 8.4E-05 1.0E-04 
200 1000 1.5E-06 1.5E-06 1.6E-05 1.3E-05 1.2E-05 1.6E-05 1.3E-05 1.2E-05 4.2E-05 4.2E-05 
500 100 1.1E-05 1.1E-05 5.5E-05 5.9E-05 4.7E-05 5.5E-05 5.9E-05 4.3E-05 1.6E-04 1.7E-04 
500 200 6.8E-06 6.8E-06 3.5E-05 2.6E-05 2.2E-05 3.5E-05 2.5E-05 2.5E-05 8.5E-05 9.2E-05 
500 500 3.0E-06 3.0E-06 1.2E-05 1.2E-05 9.7E-06 1.2E-05 1.2E-05 1.1E-05 3.5E-05 3.9E-05 
500 1000 1.2E-06 1.2E-06 6.0E-06 4.5E-06 4.5E-06 6.0E-06 4.6E-06 4.7E-06 1.8E-05 1.8E-05 
1000 100 1.3E-05 1.3E-05 3.0E-05 2.8E-05 3.0E-05 3.0E-05 2.8E-05 3.3E-05 7.5E-05 9.7E-05 
1000 200 6.6E-06 6.6E-06 1.8E-05 1.4E-05 1.5E-05 1.7E-05 1.4E-05 1.4E-05 4.6E-05 4.8E-05 
1000 500 2.8E-06 2.8E-06 6.9E-06 6.2E-06 5.6E-06 6.8E-06 6.0E-06 5.9E-06 2.6E-05 2.6E-05 
1000 1000 1.0E-06 1.0E-06 2.9E-06 3.0E-06 2.2E-06 2.9E-06 2.9E-06 2.5E-06 1.1E-05 1.1E-05 
100 100 1.4E-05 1.4E-05 3.5E-04 3.1E-04 2.9E-04 3.5E-04 3.1E-04 2.8E-04 6.9E-04 7.3E-04 
100 200 6.5E-06 6.5E-06 1.6E-04 1.4E-04 1.3E-04 1.6E-04 1.4E-04 1.3E-04 3.2E-04 2.9E-04 
100 500 4.0E-06 4.0E-06 4.6E-05 5.6E-05 5.5E-05 4.5E-05 5.6E-05 5.1E-05 1.4E-04 1.1E-04 
100 1000 1.3E-06 1.3E-06 2.9E-05 3.1E-05 1.8E-05 2.8E-05 3.1E-05 2.0E-05 5.8E-05 6.5E-05 
200 100 1.7E-05 1.7E-05 1.6E-04 1.5E-04 1.0E-04 1.6E-04 1.5E-04 1.0E-04 4.7E-04 3.5E-04 
200 200 6.2E-06 6.2E-06 9.5E-05 7.4E-05 5.5E-05 9.5E-05 7.2E-05 5.6E-05 1.9E-04 1.9E-04 
200 500 2.8E-06 2.8E-06 3.2E-05 3.5E-05 2.1E-05 3.1E-05 3.5E-05 2.2E-05 8.4E-05 1.0E-04 
200 1000 1.5E-06 1.5E-06 1.6E-05 1.3E-05 1.2E-05 1.6E-05 1.3E-05 1.2E-05 4.2E-05 4.2E-05 
500 100 1.1E-05 1.1E-05 5.5E-05 5.9E-05 4.7E-05 5.5E-05 5.9E-05 4.3E-05 1.6E-04 1.7E-04 
500 200 6.8E-06 6.8E-06 3.5E-05 2.6E-05 2.2E-05 3.5E-05 2.5E-05 2.5E-05 8.5E-05 9.2E-05 
500 500 3.0E-06 3.0E-06 1.2E-05 1.2E-05 9.7E-06 1.2E-05 1.2E-05 1.1E-05 3.5E-05 3.9E-05 
500 1000 1.2E-06 1.2E-06 6.0E-06 4.5E-06 4.5E-06 6.0E-06 4.6E-06 4.7E-06 1.8E-05 1.8E-05 
1000 100 1.3E-05 1.3E-05 3.0E-05 2.8E-05 3.0E-05 3.0E-05 2.8E-05 3.3E-05 7.5E-05 9.7E-05 
1000 200 6.6E-06 6.6E-06 1.8E-05 1.4E-05 1.5E-05 1.7E-05 1.4E-05 1.4E-05 4.6E-05 4.8E-05 
1000 500 2.8E-06 2.8E-06 6.9E-06 6.2E-06 5.6E-06 6.8E-06 6.0E-06 5.9E-06 2.6E-05 2.6E-05 
1000 1000 1.0E-06 1.0E-06 2.9E-06 3.0E-06 2.2E-06 2.9E-06 2.9E-06 2.5E-06 1.1E-05 1.1E-05 
Table 3:
S2 MSE Results for the First 10 MPL Estimate Elements versus the Respective True Parameter Elements, Averaged over Repetitions: .
100 100 8.4E-06 7.8E-06 2.3E-06 2.5E-06 4.8E-04 3.6E-04 4.3E-04 4.6E-04 4.5E-04 5.9E-04 
100 200 6.7E-06 6.1E-06 1.6E-06 1.2E-06 2.3E-04 1.4E-04 2.7E-04 2.0E-04 2.0E-04 3.2E-04 
100 500 6.1E-06 4.9E-06 5.2E-07 5.3E-07 1.1E-04 6.2E-05 1.1E-04 1.1E-04 5.3E-05 1.5E-04 
100 1000 6.0E-06 4.8E-06 2.7E-07 2.4E-07 6.1E-05 3.9E-05 6.3E-05 6.3E-05 4.0E-05 6.5E-05 
200 100 8.5E-06 7.6E-06 2.3E-06 2.4E-06 2.3E-04 1.3E-04 3.2E-04 2.6E-04 2.0E-04 4.0E-04 
200 200 6.0E-06 5.4E-06 9.0E-07 7.3E-07 1.4E-04 6.5E-05 2.0E-04 1.5E-04 1.2E-04 1.9E-04 
200 500 5.5E-06 4.4E-06 5.2E-07 4.4E-07 5.1E-05 2.6E-05 7.0E-05 5.7E-05 3.8E-05 7.0E-05 
200 1000 5.6E-06 4.4E-06 2.4E-07 2.4E-07 2.5E-05 1.9E-05 4.8E-05 3.0E-05 2.4E-05 4.2E-05 
500 100 7.4E-06 6.3E-06 2.1E-06 1.9E-06 1.0E-04 6.2E-05 1.3E-04 9.2E-05 9.6E-05 1.9E-04 
500 200 6.9E-06 5.0E-06 1.1E-06 1.0E-06 5.6E-05 3.2E-05 8.6E-05 6.9E-05 4.5E-05 1.0E-04 
500 500 5.2E-06 4.4E-06 3.5E-07 3.0E-07 2.3E-05 1.0E-05 3.0E-05 2.3E-05 1.9E-05 3.8E-05 
500 1000 5.6E-06 4.7E-06 1.9E-07 1.9E-07 1.0E-05 6.9E-06 2.0E-05 1.0E-05 1.0E-05 2.7E-05 
1000 100 6.1E-06 5.6E-06 1.3E-06 1.4E-06 5.7E-05 4.1E-05 1.1E-04 5.4E-05 4.8E-05 1.3E-04 
1000 200 5.8E-06 5.0E-06 7.6E-07 7.7E-07 3.0E-05 1.7E-05 4.9E-05 2.6E-05 2.9E-05 6.7E-05 
1000 500 5.3E-06 4.4E-06 2.8E-07 2.4E-07 9.3E-06 7.6E-06 2.8E-05 1.1E-05 1.2E-05 3.5E-05 
1000 1000 5.3E-06 4.4E-06 1.8E-07 1.9E-07 5.0E-06 2.7E-06 1.4E-05 5.4E-06 5.7E-06 1.3E-05 
100 100 8.4E-06 7.8E-06 2.3E-06 2.5E-06 4.8E-04 3.6E-04 4.3E-04 4.6E-04 4.5E-04 5.9E-04 
100 200 6.7E-06 6.1E-06 1.6E-06 1.2E-06 2.3E-04 1.4E-04 2.7E-04 2.0E-04 2.0E-04 3.2E-04 
100 500 6.1E-06 4.9E-06 5.2E-07 5.3E-07 1.1E-04 6.2E-05 1.1E-04 1.1E-04 5.3E-05 1.5E-04 
100 1000 6.0E-06 4.8E-06 2.7E-07 2.4E-07 6.1E-05 3.9E-05 6.3E-05 6.3E-05 4.0E-05 6.5E-05 
200 100 8.5E-06 7.6E-06 2.3E-06 2.4E-06 2.3E-04 1.3E-04 3.2E-04 2.6E-04 2.0E-04 4.0E-04 
200 200 6.0E-06 5.4E-06 9.0E-07 7.3E-07 1.4E-04 6.5E-05 2.0E-04 1.5E-04 1.2E-04 1.9E-04 
200 500 5.5E-06 4.4E-06 5.2E-07 4.4E-07 5.1E-05 2.6E-05 7.0E-05 5.7E-05 3.8E-05 7.0E-05 
200 1000 5.6E-06 4.4E-06 2.4E-07 2.4E-07 2.5E-05 1.9E-05 4.8E-05 3.0E-05 2.4E-05 4.2E-05 
500 100 7.4E-06 6.3E-06 2.1E-06 1.9E-06 1.0E-04 6.2E-05 1.3E-04 9.2E-05 9.6E-05 1.9E-04 
500 200 6.9E-06 5.0E-06 1.1E-06 1.0E-06 5.6E-05 3.2E-05 8.6E-05 6.9E-05 4.5E-05 1.0E-04 
500 500 5.2E-06 4.4E-06 3.5E-07 3.0E-07 2.3E-05 1.0E-05 3.0E-05 2.3E-05 1.9E-05 3.8E-05 
500 1000 5.6E-06 4.7E-06 1.9E-07 1.9E-07 1.0E-05 6.9E-06 2.0E-05 1.0E-05 1.0E-05 2.7E-05 
1000 100 6.1E-06 5.6E-06 1.3E-06 1.4E-06 5.7E-05 4.1E-05 1.1E-04 5.4E-05 4.8E-05 1.3E-04 
1000 200 5.8E-06 5.0E-06 7.6E-07 7.7E-07 3.0E-05 1.7E-05 4.9E-05 2.6E-05 2.9E-05 6.7E-05 
1000 500 5.3E-06 4.4E-06 2.8E-07 2.4E-07 9.3E-06 7.6E-06 2.8E-05 1.1E-05 1.2E-05 3.5E-05 
1000 1000 5.3E-06 4.4E-06 1.8E-07 1.9E-07 5.0E-06 2.7E-06 1.4E-05 5.4E-06 5.7E-06 1.3E-05 
Table 4:
S2 MSE Results for the Last 10 MPL Estimate Elements versus the Respective True Parameter Elements, Averaged over Repetitions: .
100 100 5.5E-04 6.7E-04 4.6E-04 4.8E-04 8.1E-04 4.1E-04 1.7E-03 1.6E-03 1.6E-03 1.6E-03 
100 200 3.2E-04 3.4E-04 1.6E-04 3.0E-04 3.6E-04 2.2E-04 7.1E-04 8.2E-04 7.8E-04 6.7E-04 
100 500 9.4E-05 1.6E-04 7.6E-05 1.0E-04 1.5E-04 6.5E-05 3.7E-04 4.1E-04 2.8E-04 2.5E-04 
100 1000 3.6E-05 8.7E-05 5.9E-05 4.7E-05 9.3E-05 4.3E-05 1.8E-04 2.5E-04 1.2E-04 1.1E-04 
200 100 3.1E-04 3.2E-04 1.6E-04 3.1E-04 3.4E-04 1.8E-04 8.4E-04 8.4E-04 7.1E-04 5.7E-04 
200 200 1.4E-04 1.9E-04 6.9E-05 1.6E-04 1.8E-04 1.3E-04 3.2E-04 5.2E-04 3.2E-04 3.7E-04 
200 500 5.5E-05 1.0E-04 4.1E-05 5.8E-05 1.1E-04 3.7E-05 2.1E-04 2.5E-04 1.3E-04 1.2E-04 
200 1000 2.7E-05 6.7E-05 3.0E-05 2.0E-05 6.1E-05 3.2E-05 1.7E-04 2.5E-04 6.2E-05 7.6E-05 
500 100 8.6E-05 1.8E-04 9.2E-05 8.8E-05 2.0E-04 9.0E-05 3.4E-04 4.6E-04 3.5E-04 3.2E-04 
500 200 5.7E-05 1.5E-04 5.3E-05 5.5E-05 1.6E-04 4.9E-05 2.2E-04 2.9E-04 1.2E-04 1.2E-04 
500 500 2.0E-05 6.7E-05 2.4E-05 1.8E-05 6.8E-05 2.4E-05 1.4E-04 2.0E-04 6.0E-05 5.8E-05 
500 1000 1.1E-05 4.2E-05 2.2E-05 1.1E-05 4.3E-05 1.8E-05 1.2E-04 1.7E-04 2.7E-05 3.0E-05 
1000 100 5.1E-05 1.1E-04 4.9E-05 4.7E-05 1.1E-04 6.0E-05 2.0E-04 2.7E-04 9.9E-05 9.4E-05 
1000 200 2.1E-05 8.6E-05 3.1E-05 2.4E-05 9.0E-05 3.6E-05 1.3E-04 2.0E-04 6.5E-05 6.9E-05 
1000 500 9.4E-06 5.1E-05 1.9E-05 8.7E-06 5.1E-05 1.7E-05 1.2E-04 1.8E-04 2.7E-05 2.7E-05 
1000 1000 4.3E-06 3.8E-05 1.4E-05 4.5E-06 3.8E-05 1.5E-05 1.0E-04 1.6E-04 1.6E-05 1.5E-05 
100 100 5.5E-04 6.7E-04 4.6E-04 4.8E-04 8.1E-04 4.1E-04 1.7E-03 1.6E-03 1.6E-03 1.6E-03 
100 200 3.2E-04 3.4E-04 1.6E-04 3.0E-04 3.6E-04 2.2E-04 7.1E-04 8.2E-04 7.8E-04 6.7E-04 
100 500 9.4E-05 1.6E-04 7.6E-05 1.0E-04 1.5E-04 6.5E-05 3.7E-04 4.1E-04 2.8E-04 2.5E-04 
100 1000 3.6E-05 8.7E-05 5.9E-05 4.7E-05 9.3E-05 4.3E-05 1.8E-04 2.5E-04 1.2E-04 1.1E-04 
200 100 3.1E-04 3.2E-04 1.6E-04 3.1E-04 3.4E-04 1.8E-04 8.4E-04 8.4E-04 7.1E-04 5.7E-04 
200 200 1.4E-04 1.9E-04 6.9E-05 1.6E-04 1.8E-04 1.3E-04 3.2E-04 5.2E-04 3.2E-04 3.7E-04 
200 500 5.5E-05 1.0E-04 4.1E-05 5.8E-05 1.1E-04 3.7E-05 2.1E-04 2.5E-04 1.3E-04 1.2E-04 
200 1000 2.7E-05 6.7E-05 3.0E-05 2.0E-05 6.1E-05 3.2E-05 1.7E-04 2.5E-04 6.2E-05 7.6E-05 
500 100 8.6E-05 1.8E-04 9.2E-05 8.8E-05 2.0E-04 9.0E-05 3.4E-04 4.6E-04 3.5E-04 3.2E-04 
500 200 5.7E-05 1.5E-04 5.3E-05 5.5E-05 1.6E-04 4.9E-05 2.2E-04 2.9E-04 1.2E-04 1.2E-04 
500 500 2.0E-05 6.7E-05 2.4E-05 1.8E-05 6.8E-05 2.4E-05 1.4E-04 2.0E-04 6.0E-05 5.8E-05 
500 1000 1.1E-05 4.2E-05 2.2E-05 1.1E-05 4.3E-05 1.8E-05 1.2E-04 1.7E-04 2.7E-05 3.0E-05 
1000 100 5.1E-05 1.1E-04 4.9E-05 4.7E-05 1.1E-04 6.0E-05 2.0E-04 2.7E-04 9.9E-05 9.4E-05 
1000 200 2.1E-05 8.6E-05 3.1E-05 2.4E-05 9.0E-05 3.6E-05 1.3E-04 2.0E-04 6.5E-05 6.9E-05 
1000 500 9.4E-06 5.1E-05 1.9E-05 8.7E-06 5.1E-05 1.7E-05 1.2E-04 1.8E-04 2.7E-05 2.7E-05 
1000 1000 4.3E-06 3.8E-05 1.4E-05 4.5E-06 3.8E-05 1.5E-05 1.0E-04 1.6E-04 1.6E-05 1.5E-05 
Table 5:
S3 MSE Results for the First 10 MPL Estimate Elements versus the Respective True Parameter Elements, Averaged over Repetitions: .
100 100 2.1E-06 1.1E-05 2.6E-06 1.2E-05 4.5E-04 2.4E-04 8.2E-04 5.5E-04 6.4E-04 1.1E-03 
100 200 6.4E-07 4.0E-06 2.0E-06 6.9E-06 2.5E-04 2.3E-04 1.1E-03 2.1E-04 2.4E-04 1.1E-03 
100 500 8.9E-07 5.3E-06 7.8E-07 6.1E-06 1.1E-04 1.2E-04 5.8E-04 1.4E-04 1.2E-04 7.8E-04 
100 1000 3.4E-07 5.1E-06 9.6E-07 7.7E-06 6.4E-05 4.7E-05 6.4E-04 7.6E-05 1.3E-04 8.6E-04 
200 100 1.3E-06 6.2E-06 4.4E-06 9.6E-06 2.1E-04 1.3E-04 6.9E-04 1.5E-04 3.5E-04 7.0E-04 
200 200 8.4E-07 6.1E-06 2.9E-06 1.0E-05 1.6E-04 7.7E-05 4.9E-04 1.5E-04 2.1E-04 5.9E-04 
200 500 4.6E-07 4.4E-06 1.5E-06 7.8E-06 5.5E-05 3.6E-05 5.3E-04 4.9E-05 1.0E-04 7.3E-04 
200 1000 2.4E-07 5.1E-06 7.7E-07 6.9E-06 1.6E-05 1.1E-05 5.8E-04 2.0E-05 7.7E-05 7.5E-04 
500 100 1.8E-06 6.5E-06 2.9E-06 8.3E-06 6.9E-05 5.9E-05 7.2E-04 7.7E-05 7.5E-05 8.9E-04 
500 200 5.9E-07 5.6E-06 1.7E-06 6.7E-06 3.9E-05 1.7E-05 6.8E-04 5.2E-05 7.4E-05 8.7E-04 
500 500 3.6E-07 5.2E-06 9.5E-07 7.1E-06 2.9E-05 3.6E-06 5.0E-04 3.2E-05 7.8E-05 6.5E-04 
500 1000 2.6E-07 5.5E-06 8.2E-07 7.0E-06 9.4E-06 9.6E-06 5.7E-04 1.1E-05 8.2E-05 7.4E-04 
1000 100 8.5E-07 5.4E-06 2.2E-06 7.9E-06 4.8E-05 5.1E-05 6.1E-04 4.6E-05 1.7E-04 8.5E-04 
1000 200 8.6E-07 5.3E-06 1.4E-06 7.6E-06 3.3E-05 1.8E-05 6.0E-04 3.8E-05 1.2E-04 7.6E-04 
1000 500 3.8E-07 5.2E-06 1.1E-06 7.6E-06 8.4E-06 1.6E-05 5.0E-04 1.1E-05 7.8E-05 6.8E-04 
1000 1000 2.5E-07 4.8E-06 8.3E-07 7.1E-06 4.1E-06 5.5E-06 5.2E-04 2.8E-06 6.3E-05 6.8E-04 
100 100 2.1E-06 1.1E-05 2.6E-06 1.2E-05 4.5E-04 2.4E-04 8.2E-04 5.5E-04 6.4E-04 1.1E-03 
100 200 6.4E-07 4.0E-06 2.0E-06 6.9E-06 2.5E-04 2.3E-04 1.1E-03 2.1E-04 2.4E-04 1.1E-03 
100 500 8.9E-07 5.3E-06 7.8E-07 6.1E-06 1.1E-04 1.2E-04 5.8E-04 1.4E-04 1.2E-04 7.8E-04 
100 1000 3.4E-07 5.1E-06 9.6E-07 7.7E-06 6.4E-05 4.7E-05 6.4E-04 7.6E-05 1.3E-04 8.6E-04 
200 100 1.3E-06 6.2E-06 4.4E-06 9.6E-06 2.1E-04 1.3E-04 6.9E-04 1.5E-04 3.5E-04 7.0E-04 
200 200 8.4E-07 6.1E-06 2.9E-06 1.0E-05 1.6E-04 7.7E-05 4.9E-04 1.5E-04 2.1E-04 5.9E-04 
200 500 4.6E-07 4.4E-06 1.5E-06 7.8E-06 5.5E-05 3.6E-05 5.3E-04 4.9E-05 1.0E-04 7.3E-04 
200 1000 2.4E-07 5.1E-06 7.7E-07 6.9E-06 1.6E-05 1.1E-05 5.8E-04 2.0E-05 7.7E-05 7.5E-04 
500 100 1.8E-06 6.5E-06 2.9E-06 8.3E-06 6.9E-05 5.9E-05 7.2E-04 7.7E-05 7.5E-05 8.9E-04 
500 200 5.9E-07 5.6E-06 1.7E-06 6.7E-06 3.9E-05 1.7E-05 6.8E-04 5.2E-05 7.4E-05 8.7E-04 
500 500 3.6E-07 5.2E-06 9.5E-07 7.1E-06 2.9E-05 3.6E-06 5.0E-04 3.2E-05 7.8E-05 6.5E-04 
500 1000 2.6E-07 5.5E-06 8.2E-07 7.0E-06 9.4E-06 9.6E-06 5.7E-04 1.1E-05 8.2E-05 7.4E-04 
1000 100 8.5E-07 5.4E-06 2.2E-06 7.9E-06 4.8E-05 5.1E-05 6.1E-04 4.6E-05 1.7E-04 8.5E-04 
1000 200 8.6E-07 5.3E-06 1.4E-06 7.6E-06 3.3E-05 1.8E-05 6.0E-04 3.8E-05 1.2E-04 7.6E-04 
1000 500 3.8E-07 5.2E-06 1.1E-06 7.6E-06 8.4E-06 1.6E-05 5.0E-04 1.1E-05 7.8E-05 6.8E-04 
1000 1000 2.5E-07 4.8E-06 8.3E-07 7.1E-06 4.1E-06 5.5E-06 5.2E-04 2.8E-06 6.3E-05 6.8E-04 
Table 6:
S3 MSE Results for the Last 10 MPL Estimate Elements versus the Respective True Parameter Elements, Averaged over Repetitions: .
100 100 5.3E-04 1.2E-03 3.6E-04 7.7E-04 1.8E-03 4.1E-04 1.4E-03 1.1E-03 1.0E-03 1.0E-03 
100 200 2.1E-04 5.2E-04 2.5E-04 2.4E-04 6.4E-04 2.4E-04 5.3E-04 9.1E-04 6.3E-04 5.4E-04 
100 500 1.3E-04 4.4E-04 1.1E-04 1.1E-04 5.5E-04 1.3E-04 2.3E-04 1.8E-04 1.2E-04 1.9E-04 
100 1000 6.0E-05 4.9E-04 7.8E-05 4.9E-05 6.1E-04 1.0E-04 1.6E-04 2.1E-04 1.1E-04 1.3E-04 
200 100 3.5E-04 1.3E-03 2.2E-04 4.8E-04 1.5E-03 1.8E-04 4.8E-04 5.6E-04 9.6E-04 9.4E-04 
200 200 1.6E-04 4.4E-04 1.5E-04 1.1E-04 5.3E-04 1.6E-04 5.9E-04 7.4E-04 4.1E-04 5.6E-04 
200 500 3.3E-05 4.1E-04 6.8E-05 3.4E-05 5.8E-04 6.7E-05 1.7E-04 1.8E-04 1.3E-04 1.5E-04 
200 1000 4.8E-05 4.6E-04 3.4E-05 2.8E-05 5.0E-04 4.7E-05 8.1E-05 1.8E-04 1.1E-04 1.4E-04 
500 100 8.9E-05 4.6E-04 4.3E-05 9.8E-05 6.0E-04 7.0E-05 3.2E-04 5.5E-04 2.1E-04 2.7E-04 
500 200 1.0E-04 5.1E-04 4.6E-05 7.5E-05 6.2E-04 3.9E-05 8.4E-05 9.9E-05 2.7E-04 1.5E-04 
500 500 3.9E-05 5.0E-04 3.8E-05 2.9E-05 5.0E-04 3.4E-05 1.4E-04 1.5E-04 7.2E-05 1.2E-04 
500 1000 9.7E-06 4.3E-04 4.8E-05 1.1E-05 5.2E-04 4.1E-05 8.1E-05 1.7E-04 3.3E-05 7.9E-05 
1000 100 4.5E-05 5.8E-04 9.0E-05 5.5E-05 5.8E-04 9.7E-05 2.1E-04 3.3E-04 1.6E-04 1.9E-04 
1000 200 3.3E-05 5.5E-04 5.6E-05 3.3E-05 5.7E-04 6.2E-05 1.3E-04 2.8E-04 9.2E-05 1.6E-04 
1000 500 1.3E-05 4.0E-04 4.6E-05 1.5E-05 5.0E-04 4.5E-05 7.2E-05 1.4E-04 5.0E-05 7.2E-05 
1000 1000 3.8E-06 4.2E-04 3.0E-05 3.2E-06 4.9E-04 3.3E-05 6.8E-05 1.3E-04 2.9E-05 5.9E-05 
100 100 5.3E-04 1.2E-03 3.6E-04 7.7E-04 1.8E-03 4.1E-04 1.4E-03 1.1E-03 1.0E-03 1.0E-03 
100 200 2.1E-04 5.2E-04 2.5E-04 2.4E-04 6.4E-04 2.4E-04 5.3E-04 9.1E-04 6.3E-04 5.4E-04 
100 500 1.3E-04 4.4E-04 1.1E-04 1.1E-04 5.5E-04 1.3E-04 2.3E-04 1.8E-04 1.2E-04 1.9E-04 
100 1000 6.0E-05 4.9E-04 7.8E-05 4.9E-05 6.1E-04 1.0E-04 1.6E-04 2.1E-04 1.1E-04 1.3E-04 
200 100 3.5E-04 1.3E-03 2.2E-04 4.8E-04 1.5E-03 1.8E-04 4.8E-04 5.6E-04 9.6E-04 9.4E-04 
200 200 1.6E-04 4.4E-04 1.5E-04 1.1E-04 5.3E-04 1.6E-04 5.9E-04 7.4E-04 4.1E-04 5.6E-04 
200 500 3.3E-05 4.1E-04 6.8E-05 3.4E-05 5.8E-04 6.7E-05 1.7E-04 1.8E-04 1.3E-04 1.5E-04 
200 1000 4.8E-05 4.6E-04 3.4E-05 2.8E-05 5.0E-04 4.7E-05 8.1E-05 1.8E-04 1.1E-04 1.4E-04 
500 100 8.9E-05 4.6E-04 4.3E-05 9.8E-05 6.0E-04 7.0E-05 3.2E-04 5.5E-04 2.1E-04 2.7E-04 
500 200 1.0E-04 5.1E-04 4.6E-05 7.5E-05 6.2E-04 3.9E-05 8.4E-05 9.9E-05 2.7E-04 1.5E-04 
500 500 3.9E-05 5.0E-04 3.8E-05 2.9E-05 5.0E-04 3.4E-05 1.4E-04 1.5E-04 7.2E-05 1.2E-04 
500 1000 9.7E-06 4.3E-04 4.8E-05 1.1E-05 5.2E-04 4.1E-05 8.1E-05 1.7E-04 3.3E-05 7.9E-05 
1000 100 4.5E-05 5.8E-04 9.0E-05 5.5E-05 5.8E-04 9.7E-05 2.1E-04 3.3E-04 1.6E-04 1.9E-04 
1000 200 3.3E-05 5.5E-04 5.6E-05 3.3E-05 5.7E-04 6.2E-05 1.3E-04 2.8E-04 9.2E-05 1.6E-04 
1000 500 1.3E-05 4.0E-04 4.6E-05 1.5E-05 5.0E-04 4.5E-05 7.2E-05 1.4E-04 5.0E-05 7.2E-05 
1000 1000 3.8E-06 4.2E-04 3.0E-05 3.2E-06 4.9E-04 3.3E-05 6.8E-05 1.3E-04 2.9E-05 5.9E-05 

Further, we also measure the similarity of the pseudoallocation 4.2 in comparison to the cluster allocation, equation 4.1. We make comparisons via the average similarity measurement , where is an indicator function that takes value 1 if proposition is true and 0 otherwise. The results for all studies are presented in Table 7.

Table 7:
Similarity Measurements of the Pseudoallocation, Equation 4.2 versus the Cluster Allocation, Equation 4.1, in S1 and S2.
1002005001000
S1 100 0.9929 0.9994 1.0000 1.0000 
 200 0.9947 0.9996 1.0000 1.0000 
 500 0.9963 0.9998 1.0000 1.0000 
 1000 0.9973 0.9999 1.0000 1.0000 
      
S2 100 0.8958 0.9667 0.9983 1.0000 
 200 0.9238 0.9758 0.9983 1.0000 
 500 0.9336 0.9788 0.9995 1.0000 
 1000 0.9382 0.9788 0.9995 1.0000 
      
S3 100 0.8625 0.9550 0.9960 1.0000 
 200 0.9028 0.9658 0.9988 1.0000 
 500 0.9051 0.9668 0.9985 1.0000 
 1000 0.9007 0.9635 0.9986 1.0000 
1002005001000
S1 100 0.9929 0.9994 1.0000 1.0000 
 200 0.9947 0.9996 1.0000 1.0000 
 500 0.9963 0.9998 1.0000 1.0000 
 1000 0.9973 0.9999 1.0000 1.0000 
      
S2 100 0.8958 0.9667 0.9983 1.0000 
 200 0.9238 0.9758 0.9983 1.0000 
 500 0.9336 0.9788 0.9995 1.0000 
 1000 0.9382 0.9788 0.9995 1.0000 
      
S3 100 0.8625 0.9550 0.9960 1.0000 
 200 0.9028 0.9658 0.9988 1.0000 
 500 0.9051 0.9668 0.9985 1.0000 
 1000 0.9007 0.9635 0.9986 1.0000 

Finally, we assess the efficiency of the MPL estimator relative to the ML estimator. We do this by computing the ML estimate and calculating the MSE for each parameter element, where is the th element of the ML estimate . We then compute the ratio of the ML MSE to the MPL MSE. The results for S1 and S2 are reported in Tables 8 to 10, respectively. The efficiency results for S3 are presented in Tables 11 and 12.

Table 8:
Relative Efficiency in S1, as Measured by the Ratio of the ML MSE to the MPL MSE for Each Parameter Element, Computed over Repetitions: .
100 100 1.9E+02 1.9E+02 7.0E-01 5.8E-01 8.7E-01 5.7E-01 5.4E-01 7.1E-01 6.5E-01 5.1E-01 
100 200 2.4E+02 2.4E+02 6.3E-01 4.7E-01 8.0E-01 7.1E-01 6.8E-01 7.3E-01 7.1E-01 7.5E-01 
100 500 1.3E+02 1.3E+02 7.6E-01 6.6E-01 6.0E-01 1.1E+00 7.6E-01 9.2E-01 6.7E-01 6.6E-01 
100 1000 2.0E+02 2.0E+02 6.6E-01 6.8E-01 1.3E+00 6.8E-01 6.7E-01 1.2E+00 5.7E-01 6.0E-01 
200 100 1.3E+02 1.3E+02 6.0E-01 5.9E-01 9.2E-01 6.9E-01 6.9E-01 9.3E-01 4.9E-01 5.3E-01 
200 200 2.3E+02 2.3E+02 4.9E-01 7.6E-01 8.0E-01 4.6E-01 5.3E-01 7.8E-01 5.9E-01 5.1E-01 
200 500 2.3E+02 2.3E+02 5.9E-01 4.9E-01 8.3E-01 6.1E-01 5.2E-01 8.5E-01 5.5E-01 4.5E-01 
200 1000 1.4E+02 1.4E+02 7.1E-01 7.5E-01 8.7E-01 6.4E-01 6.7E-01 8.2E-01 5.5E-01 5.2E-01 
500 100 2.3E+02 2.3E+02 6.6E-01 6.1E-01 7.5E-01 8.0E-01 6.1E-01 6.6E-01 5.1E-01 3.6E-01 
500 200 2.2E+02 2.2E+02 6.1E-01 7.8E-01 7.7E-01 4.1E-01 7.6E-01 6.1E-01 4.0E-01 3.4E-01 
500 500 1.8E+02 1.8E+02 7.8E-01 5.3E-01 7.0E-01 6.6E-01 4.6E-01 1.1E+00 4.4E-01 4.3E-01 
500 1000 2.2E+02 2.2E+02 6.2E-01 8.5E-01 9.1E-01 6.6E-01 8.9E-01 7.8E-01 4.2E-01 4.2E-01 
1000 100 1.7E+02 1.7E+02 5.2E-01 6.3E-01 6.1E-01 7.1E-01 7.8E-01 7.4E-01 6.0E-01 4.2E-01 
1000 200 1.3E+02 1.3E+02 7.2E-01 5.9E-01 7.1E-01 6.4E-01 6.2E-01 6.5E-01 4.8E-01 4.7E-01 
1000 500 1.5E+02 1.5E+02 7.5E-01 5.1E-01 5.9E-01 5.1E-01 4.9E-01 5.8E-01 3.1E-01 2.9E-01 
1000 1000 2.3E+02 2.3E+02 7.2E-01 6.5E-01 7.2E-01 6.9E-01 5.5E-01 7.5E-01 3.6E-01 3.7E-01 
100 100 1.9E+02 1.9E+02 7.0E-01 5.8E-01 8.7E-01 5.7E-01 5.4E-01 7.1E-01 6.5E-01 5.1E-01 
100 200 2.4E+02 2.4E+02 6.3E-01 4.7E-01 8.0E-01 7.1E-01 6.8E-01 7.3E-01 7.1E-01 7.5E-01 
100 500 1.3E+02 1.3E+02 7.6E-01 6.6E-01 6.0E-01 1.1E+00 7.6E-01 9.2E-01 6.7E-01 6.6E-01 
100 1000 2.0E+02 2.0E+02 6.6E-01 6.8E-01 1.3E+00 6.8E-01 6.7E-01 1.2E+00 5.7E-01 6.0E-01 
200 100 1.3E+02 1.3E+02 6.0E-01 5.9E-01 9.2E-01 6.9E-01 6.9E-01 9.3E-01 4.9E-01 5.3E-01 
200 200 2.3E+02 2.3E+02 4.9E-01 7.6E-01 8.0E-01 4.6E-01 5.3E-01 7.8E-01 5.9E-01 5.1E-01 
200 500 2.3E+02 2.3E+02 5.9E-01 4.9E-01 8.3E-01 6.1E-01 5.2E-01 8.5E-01 5.5E-01 4.5E-01 
200 1000 1.4E+02 1.4E+02 7.1E-01 7.5E-01 8.7E-01 6.4E-01 6.7E-01 8.2E-01 5.5E-01 5.2E-01 
500 100 2.3E+02 2.3E+02 6.6E-01 6.1E-01 7.5E-01 8.0E-01 6.1E-01 6.6E-01 5.1E-01 3.6E-01 
500 200 2.2E+02 2.2E+02 6.1E-01 7.8E-01 7.7E-01 4.1E-01 7.6E-01 6.1E-01 4.0E-01 3.4E-01 
500 500 1.8E+02 1.8E+02 7.8E-01 5.3E-01 7.0E-01 6.6E-01 4.6E-01 1.1E+00 4.4E-01 4.3E-01 
500 1000 2.2E+02 2.2E+02 6.2E-01 8.5E-01 9.1E-01 6.6E-01 8.9E-01 7.8E-01 4.2E-01 4.2E-01 
1000 100 1.7E+02 1.7E+02 5.2E-01 6.3E-01 6.1E-01 7.1E-01 7.8E-01 7.4E-01 6.0E-01 4.2E-01 
1000 200 1.3E+02 1.3E+02 7.2E-01 5.9E-01 7.1E-01 6.4E-01 6.2E-01 6.5E-01 4.8E-01 4.7E-01 
1000 500 1.5E+02 1.5E+02 7.5E-01 5.1E-01 5.9E-01 5.1E-01 4.9E-01 5.8E-01 3.1E-01 2.9E-01 
1000 1000 2.3E+02 2.3E+02 7.2E-01 6.5E-01 7.2E-01 6.9E-01 5.5E-01 7.5E-01 3.6E-01 3.7E-01 
Table 9:
Relative Efficiency in S2 (First Half of the Parameter Elements), as Measured by the Ratio of the ML MSE to the MPL MSE for Each Parameter Element, Computed over Repetitions: .
100 100 3.1E+02 3.5E+02 1.0E+03 6.9E+02 8.1E-01 2.0E+00 1.6E+00 1.1E+00 1.1E+00 9.1E-01 
100 200 1.6E+02 2.0E+02 7.4E+02 8.9E+02 1.2E+00 1.7E+00 1.2E+00 1.2E+00 1.1E+00 6.7E-01 
100 500 8.7E+01 8.8E+01 6.8E+02 7.2E+02 9.0E-01 1.8E+00 9.6E-01 1.0E+00 1.9E+00 7.6E-01 
100 1000 3.6E+01 5.2E+01 8.2E+02 9.9E+02 8.2E-01 2.0E+00 7.5E-01 6.9E-01 1.2E+00 8.5E-01 
200 100 2.9E+02 2.3E+02 9.4E+02 7.8E+02 7.8E-01 1.5E+00 6.7E-01 7.4E-01 1.0E+00 5.3E-01 
200 200 1.5E+02 1.9E+02 1.1E+03 1.1E+03 6.6E-01 1.8E+00 5.1E-01 6.4E-01 1.1E+00 4.5E-01 
200 500 6.2E+01 8.2E+01 6.2E+02 7.4E+02 7.8E-01 1.5E+00 6.7E-01 6.4E-01 1.3E+00 5.5E-01 
200 1000 3.2E+01 4.2E+01 7.6E+02 8.4E+02 6.8E-01 1.1E+00 5.1E-01 5.9E-01 7.0E-01 5.2E-01 
500 100 2.8E+02 3.5E+02 8.5E+02 1.2E+03 7.4E-01 1.4E+00 6.6E-01 8.9E-01 8.3E-01 4.3E-01 
500 200 1.3E+02 1.7E+02 7.8E+02 7.9E+02 5.7E-01 1.1E+00 4.1E-01 5.0E-01 7.7E-01 4.2E-01 
500 500 7.8E+01 7.2E+01 1.2E+03 1.6E+03 9.0E-01 1.1E+00 5.2E-01 7.8E-01 9.6E-01 4.0E-01 
500 1000 3.4E+01 4.3E+01 1.2E+03 1.0E+03 9.8E-01 1.0E+00 4.1E-01 9.9E-01 6.9E-01 3.8E-01 
1000 100 3.8E+02 2.8E+02 1.4E+03 1.4E+03 6.0E-01 7.7E-01 2.9E-01 6.0E-01 7.5E-01 3.3E-01 
1000 200 1.5E+02 1.5E+02 1.4E+03 1.0E+03 6.8E-01 1.2E+00 3.8E-01 8.7E-01 6.3E-01 2.8E-01 
1000 500 7.1E+01 1.1E+02 1.2E+03 1.5E+03 8.1E-01 9.3E-01 2.3E-01 7.2E-01 6.4E-01 2.0E-01 
1000 1000 3.6E+01 3.6E+01 8.7E+02 1.1E+03 8.5E-01 1.2E+00 2.6E-01 8.5E-01 6.8E-01 2.6E-01 
100 100 3.1E+02 3.5E+02 1.0E+03 6.9E+02 8.1E-01 2.0E+00 1.6E+00 1.1E+00 1.1E+00 9.1E-01 
100 200 1.6E+02 2.0E+02 7.4E+02 8.9E+02 1.2E+00 1.7E+00 1.2E+00 1.2E+00 1.1E+00 6.7E-01 
100 500 8.7E+01 8.8E+01 6.8E+02 7.2E+02 9.0E-01 1.8E+00 9.6E-01 1.0E+00 1.9E+00 7.6E-01 
100 1000 3.6E+01 5.2E+01 8.2E+02 9.9E+02 8.2E-01 2.0E+00 7.5E-01 6.9E-01 1.2E+00 8.5E-01 
200 100 2.9E+02 2.3E+02 9.4E+02 7.8E+02 7.8E-01 1.5E+00 6.7E-01 7.4E-01 1.0E+00 5.3E-01 
200 200 1.5E+02 1.9E+02 1.1E+03 1.1E+03 6.6E-01 1.8E+00 5.1E-01 6.4E-01 1.1E+00 4.5E-01 
200 500 6.2E+01 8.2E+01 6.2E+02 7.4E+02 7.8E-01 1.5E+00 6.7E-01 6.4E-01 1.3E+00 5.5E-01 
200 1000 3.2E+01 4.2E+01 7.6E+02 8.4E+02 6.8E-01 1.1E+00 5.1E-01 5.9E-01 7.0E-01 5.2E-01 
500 100 2.8E+02 3.5E+02 8.5E+02 1.2E+03 7.4E-01 1.4E+00 6.6E-01 8.9E-01 8.3E-01 4.3E-01 
500 200 1.3E+02 1.7E+02 7.8E+02 7.9E+02 5.7E-01 1.1E+00 4.1E-01 5.0E-01 7.7E-01 4.2E-01 
500 500 7.8E+01 7.2E+01 1.2E+03 1.6E+03 9.0E-01 1.1E+00 5.2E-01 7.8E-01 9.6E-01 4.0E-01 
500 1000 3.4E+01 4.3E+01 1.2E+03 1.0E+03 9.8E-01 1.0E+00 4.1E-01 9.9E-01 6.9E-01 3.8E-01 
1000 100 3.8E+02 2.8E+02 1.4E+03 1.4E+03 6.0E-01 7.7E-01 2.9E-01 6.0E-01 7.5E-01 3.3E-01 
1000 200 1.5E+02 1.5E+02 1.4E+03 1.0E+03 6.8E-01 1.2E+00 3.8E-01 8.7E-01 6.3E-01 2.8E-01 
1000 500 7.1E+01 1.1E+02 1.2E+03 1.5E+03 8.1E-01 9.3E-01 2.3E-01 7.2E-01 6.4E-01 2.0E-01 
1000 1000 3.6E+01 3.6E+01 8.7E+02 1.1E+03 8.5E-01 1.2E+00 2.6E-01 8.5E-01 6.8E-01 2.6E-01 
Table 10:
Relative Efficiency in S2 (Second Half of the Parameter Elements), as Measured by the Ratio of the ML MSE to the MPL MSE for Each Parameter Element, Computed over Repetitions: .
100 100 7.9E-01 6.9E-01 1.3E+00 1.2E+00 9.0E-01 2.1E+00 6.5E-01 6.0E-01 6.0E-01 6.7E-01 
100 200 8.4E-01 8.9E-01 2.0E+00 7.9E-01 5.0E-01 1.1E+00 7.3E-01 7.2E-01 6.2E-01 6.9E-01 
100 500 9.7E-01 6.3E-01 1.4E+00 9.7E-01 7.9E-01 1.7E+00 4.8E-01 4.3E-01 6.1E-01 8.2E-01 
100 1000 1.5E+00 5.8E-01 1.2E+00 1.1E+00 7.9E-01 1.3E+00 4.8E-01 3.4E-01 8.1E-01 7.0E-01 
200 100 9.0E-01 7.5E-01 1.5E+00 6.6E-01 7.9E-01 1.5E+00 6.7E-01 4.7E-01 5.6E-01 6.9E-01 
200 200 6.9E-01 5.7E-01 1.8E+00 5.1E-01 6.8E-01 9.2E-01 5.1E-01 5.1E-01 5.8E-01 5.3E-01 
200 500 5.4E-01 4.1E-01 1.2E+00 7.9E-01 3.6E-01 1.1E+00 3.8E-01 3.4E-01 6.6E-01 7.2E-01 
200 1000 8.4E-01 3.0E-01 8.4E-01 7.7E-01 3.0E-01 6.9E-01 2.7E-01 1.9E-01 5.2E-01 6.1E-01 
500 100 9.8E-01 4.9E-01 8.5E-01 8.6E-01 3.2E-01 6.9E-01 4.8E-01 4.4E-01 4.7E-01 4.4E-01 
500 200 5.8E-01 2.9E-01 6.7E-01 9.0E-01 2.6E-01 7.6E-01 3.5E-01 2.7E-01 7.4E-01 6.4E-01 
500 500 8.4E-01 1.8E-01 7.3E-01 9.1E-01 2.2E-01 7.5E-01 2.4E-01 1.7E-01 5.8E-01 3.8E-01 
500 1000 5.4E-01 2.2E-01 2.9E-01 7.9E-01 2.3E-01 5.6E-01 1.5E-01 1.1E-01 5.5E-01 4.5E-01 
1000 100 9.4E-01 4.4E-01 8.0E-01 1.0E+00 3.6E-01 7.6E-01 3.6E-01 3.1E-01 1.0E+00 1.1E+00 
1000 200 8.2E-01 1.8E-01 5.5E-01 7.4E-01 2.7E-01 4.0E-01 3.9E-01 1.8E-01 6.5E-01 5.2E-01 
1000 500 8.4E-01 1.5E-01 3.7E-01 8.4E-01 1.8E-01 4.0E-01 1.7E-01 9.2E-02 5.9E-01 5.7E-01 
1000 1000 9.8E-01 8.6E-02 2.9E-01 9.9E-01 1.2E-01 3.1E-01 6.0E-02 7.4E-02 4.6E-01 4.8E-01 
100 100 7.9E-01 6.9E-01 1.3E+00 1.2E+00 9.0E-01 2.1E+00 6.5E-01 6.0E-01 6.0E-01 6.7E-01 
100 200 8.4E-01 8.9E-01 2.0E+00 7.9E-01 5.0E-01 1.1E+00 7.3E-01 7.2E-01 6.2E-01 6.9E-01 
100 500 9.7E-01 6.3E-01 1.4E+00 9.7E-01 7.9E-01 1.7E+00 4.8E-01 4.3E-01 6.1E-01 8.2E-01 
100 1000 1.5E+00 5.8E-01 1.2E+00 1.1E+00 7.9E-01 1.3E+00 4.8E-01 3.4E-01 8.1E-01 7.0E-01 
200 100 9.0E-01 7.5E-01 1.5E+00 6.6E-01 7.9E-01 1.5E+00 6.7E-01 4.7E-01 5.6E-01 6.9E-01 
200 200 6.9E-01 5.7E-01 1.8E+00 5.1E-01 6.8E-01 9.2E-01 5.1E-01 5.1E-01 5.8E-01 5.3E-01 
200 500 5.4E-01 4.1E-01 1.2E+00 7.9E-01 3.6E-01 1.1E+00 3.8E-01 3.4E-01 6.6E-01 7.2E-01 
200 1000 8.4E-01 3.0E-01 8.4E-01 7.7E-01 3.0E-01 6.9E-01 2.7E-01 1.9E-01 5.2E-01 6.1E-01 
500 100 9.8E-01 4.9E-01 8.5E-01 8.6E-01 3.2E-01 6.9E-01 4.8E-01 4.4E-01 4.7E-01 4.4E-01 
500 200 5.8E-01 2.9E-01 6.7E-01 9.0E-01 2.6E-01 7.6E-01 3.5E-01 2.7E-01 7.4E-01 6.4E-01 
500 500 8.4E-01 1.8E-01 7.3E-01 9.1E-01 2.2E-01 7.5E-01 2.4E-01 1.7E-01 5.8E-01 3.8E-01 
500 1000 5.4E-01 2.2E-01 2.9E-01 7.9E-01 2.3E-01 5.6E-01 1.5E-01 1.1E-01 5.5E-01 4.5E-01 
1000 100 9.4E-01 4.4E-01 8.0E-01 1.0E+00 3.6E-01 7.6E-01 3.6E-01 3.1E-01 1.0E+00 1.1E+00 
1000 200 8.2E-01 1.8E-01 5.5E-01 7.4E-01 2.7E-01 4.0E-01 3.9E-01 1.8E-01 6.5E-01 5.2E-01 
1000 500 8.4E-01 1.5E-01 3.7E-01 8.4E-01 1.8E-01 4.0E-01 1.7E-01 9.2E-02 5.9E-01 5.7E-01 
1000 1000 9.8E-01 8.6E-02 2.9E-01 9.9E-01 1.2E-01 3.1E-01 6.0E-02 7.4E-02 4.6E-01 4.8E-01 
Table 11:
Relative Efficiency in S3 (First Half of the Parameter Elements), as Measured by the Ratio of the ML MSE to the MPL MSE for Each Parameter Element, Computed over Repetitions: .
100 100 1.0E+04 4.9E+02 3.1E+03 1.5E+03 2.1E+00 6.0E+00 2.3E+00 8.8E-01 1.8E+00 6.0E-01 
100 200 3.1E+04 1.3E+03 2.2E+03 3.3E+03 2.1E+00 3.3E+00 8.3E-01 2.7E+00 2.5E+00 4.6E-01 
100 500 2.2E+04 5.5E+02 3.2E+03 3.8E+03 1.8E+00 5.8E+00 6.2E-01 1.2E+00 1.4E+00 1.2E-01 
100 1000 5.8E+04 5.1E+02 2.4E+03 2.8E+03 2.2E+00 3.3E+00 2.7E-01 8.1E-01 2.3E-01 1.3E-01 
200 100 1.6E+04 6.7E+02 8.8E+02 2.9E+03 2.8E+00 2.2E+00 8.5E-01 1.5E+00 1.3E+00 4.1E-01 
200 200 2.6E+04 5.4E+02 9.5E+02 2.6E+03 1.5E+00 2.7E+00 7.6E-01 1.3E+00 6.5E-01 1.4E-01 
200 500 4.7E+04 5.5E+02 1.6E+03 3.0E+03 2.8E+00 2.7E+00 1.2E-01 7.5E-01 7.3E-01 1.0E-01 
200 1000 9.2E+04 4.8E+02 3.6E+03 3.0E+03 3.2E+00 5.3E+00 7.2E-02 2.0E+00 4.5E-01 2.2E-02 
500 100 1.3E+04 5.1E+02 1.4E+03 2.9E+03 3.0E+00 3.8E+00 3.7E-01 4.4E-01 1.1E+00 1.0E-01 
500 200 3.9E+04 5.1E+02 1.7E+03 3.2E+03 3.4E+00 7.1E+00 1.3E-01 6.9E-01 1.1E+00 6.2E-02 
500 500 6.8E+04 5.6E+02 3.1E+03 3.4E+03 1.1E+00 9.0E+00 5.6E-02 2.8E-01 2.1E-01 2.8E-02 
500 1000 8.8E+04 4.7E+02 2.8E+03 3.3E+03 1.4E+00 1.8E+00 1.8E-02 1.2E+00 1.0E-01 1.0E-02 
1000 100 2.5E+04 6.3E+02 1.3E+03 3.5E+03 2.7E+00 2.4E+00 1.5E-01 8.1E-01 2.9E-01 3.7E-02 
1000 200 2.7E+04 5.4E+02 2.7E+03 3.1E+03 1.5E+00 3.5E+00 4.4E-02 7.3E-01 1.8E-01 3.5E-02 
1000 500 5.7E+04 5.6E+02 3.1E+03 2.8E+03 2.2E+00 9.0E-01 2.0E-02 4.2E-01 9.6E-02 8.0E-03 
1000 1000 8.7E+04 5.8E+02 3.4E+03 3.2E+03 1.8E+00 1.4E+00 2.4E-02 2.8E+00 1.3E-01 2.8E-03 
100 100 1.0E+04 4.9E+02 3.1E+03 1.5E+03 2.1E+00 6.0E+00 2.3E+00 8.8E-01 1.8E+00 6.0E-01 
100 200 3.1E+04 1.3E+03 2.2E+03 3.3E+03 2.1E+00 3.3E+00 8.3E-01 2.7E+00 2.5E+00 4.6E-01 
100 500 2.2E+04 5.5E+02 3.2E+03 3.8E+03 1.8E+00 5.8E+00 6.2E-01 1.2E+00 1.4E+00 1.2E-01 
100 1000 5.8E+04 5.1E+02 2.4E+03 2.8E+03 2.2E+00 3.3E+00 2.7E-01 8.1E-01 2.3E-01 1.3E-01 
200 100 1.6E+04 6.7E+02 8.8E+02 2.9E+03 2.8E+00 2.2E+00 8.5E-01 1.5E+00 1.3E+00 4.1E-01 
200 200 2.6E+04 5.4E+02 9.5E+02 2.6E+03 1.5E+00 2.7E+00 7.6E-01 1.3E+00 6.5E-01 1.4E-01 
200 500 4.7E+04 5.5E+02 1.6E+03 3.0E+03 2.8E+00 2.7E+00 1.2E-01 7.5E-01 7.3E-01 1.0E-01 
200 1000 9.2E+04 4.8E+02 3.6E+03 3.0E+03 3.2E+00 5.3E+00 7.2E-02 2.0E+00 4.5E-01 2.2E-02 
500 100 1.3E+04 5.1E+02 1.4E+03 2.9E+03 3.0E+00 3.8E+00 3.7E-01 4.4E-01 1.1E+00 1.0E-01 
500 200 3.9E+04 5.1E+02 1.7E+03 3.2E+03 3.4E+00 7.1E+00 1.3E-01 6.9E-01 1.1E+00 6.2E-02 
500 500 6.8E+04 5.6E+02 3.1E+03 3.4E+03 1.1E+00 9.0E+00 5.6E-02 2.8E-01 2.1E-01 2.8E-02 
500 1000 8.8E+04 4.7E+02 2.8E+03 3.3E+03 1.4E+00 1.8E+00 1.8E-02 1.2E+00 1.0E-01 1.0E-02 
1000 100 2.5E+04 6.3E+02 1.3E+03 3.5E+03 2.7E+00 2.4E+00 1.5E-01 8.1E-01 2.9E-01 3.7E-02 
1000 200 2.7E+04 5.4E+02 2.7E+03 3.1E+03 1.5E+00 3.5E+00 4.4E-02 7.3E-01 1.8E-01 3.5E-02 
1000 500 5.7E+04 5.6E+02 3.1E+03 2.8E+03 2.2E+00 9.0E-01 2.0E-02 4.2E-01 9.6E-02 8.0E-03 
1000 1000 8.7E+04 5.8E+02 3.4E+03 3.2E+03 1.8E+00 1.4E+00 2.4E-02 2.8E+00 1.3E-01 2.8E-03 
Table 12:
Relative Efficiency in S3 (Second Half of the Parameter Elements), as Measured by the Ratio of the ML MSE to the MPL MSE for Each Parameter Element, Computed over Repetitions: .
100 100 5.4E-01 3.4E-01 1.0E+00 5.2E-01 2.7E-01 7.9E-01 1.6E+00 1.6E+00 8.9E-01 2.6E-01 
100 200 6.7E-01 3.6E-01 6.1E-01 7.5E-01 2.0E-01 6.6E-01 2.5E+00 6.6E-01 5.2E-01 4.1E-01 
100 500 7.9E-01 3.0E-01 8.8E-01 8.1E-01 1.2E-01 7.7E-01 1.8E+00 1.2E+00 1.5E+00 6.3E-01 
100 1000 7.2E-01 1.4E-01 5.6E-01 6.0E-01 2.8E-02 3.5E-01 1.2E+00 4.5E-01 4.3E-01 7.5E-01 
200 100 6.5E-01 1.8E-01 9.5E-01 3.0E-01 6.1E-02 4.5E-01 2.4E+00 1.2E+00 6.8E-01 2.0E-01 
200 200 5.9E-01 1.4E-01 5.7E-01 7.9E-01 1.0E-01 5.6E-01 8.6E-01 2.7E-01 2.4E-01 3.2E-01 
200 500 1.1E+00 9.8E-02 4.5E-01 8.5E-01 3.4E-02 3.7E-01 1.3E+00 4.3E-01 4.6E-01 4.5E-01 
200 1000 5.5E-01 3.2E-02 4.8E-01 3.5E-01 1.5E-02 3.1E-01 2.1E+00 2.1E-01 4.2E-01 2.1E-01 
500 100 4.7E-01 1.9E-01 2.0E+00 6.4E-01 5.4E-02 9.1E-01 1.4E+00 2.9E-01 4.1E-01 3.0E-01 
500 200 2.0E-01 1.0E-01 7.9E-01 2.3E-01 2.5E-02 5.7E-01 2.3E+00 1.6E+00 1.7E-01 2.8E-01 
500 500 3.4E-01 2.4E-02 4.4E-01 1.2E-01 1.1E-02 1.5E-01 6.4E-01 1.9E-01 6.3E-01 1.1E-01 
500 1000 4.2E-01 1.7E-02 8.8E-02 7.1E-01 6.6E-03 1.5E-01 3.8E-01 1.3E-01 6.1E-01 1.6E-01 
1000 100 7.1E-01 3.9E-02 3.8E-01 3.8E-01 2.9E-02 3.5E-01 1.2E+00 2.7E-01 4.1E-01 2.0E-01 
1000 200 8.8E-01 3.7E-02 6.1E-01 3.4E-01 3.9E-02 2.1E-01 9.9E-01 1.2E-01 2.4E-01 1.7E-01 
1000 500 3.4E-01 2.9E-02 2.3E-01 2.2E-01 7.1E-03 7.8E-02 7.2E-01 1.7E-01 1.4E-01 5.4E-02 
1000 1000 1.1E+00 6.7E-03 1.1E-01 7.0E-01 6.6E-03 7.8E-02 2.9E-01 8.0E-02 2.4E-01 1.1E-01 
100 100 5.4E-01 3.4E-01 1.0E+00 5.2E-01 2.7E-01 7.9E-01 1.6E+00 1.6E+00 8.9E-01 2.6E-01 
100 200 6.7E-01 3.6E-01 6.1E-01 7.5E-01 2.0E-01 6.6E-01 2.5E+00 6.6E-01 5.2E-01 4.1E-01 
100 500 7.9E-01 3.0E-01 8.8E-01 8.1E-01 1.2E-01 7.7E-01 1.8E+00 1.2E+00 1.5E+00 6.3E-01 
100 1000 7.2E-01 1.4E-01 5.6E-01 6.0E-01 2.8E-02 3.5E-01 1.2E+00 4.5E-01 4.3E-01 7.5E-01 
200 100 6.5E-01 1.8E-01 9.5E-01 3.0E-01 6.1E-02 4.5E-01 2.4E+00 1.2E+00 6.8E-01 2.0E-01 
200 200 5.9E-01 1.4E-01 5.7E-01 7.9E-01 1.0E-01 5.6E-01 8.6E-01 2.7E-01 2.4E-01 3.2E-01 
200 500 1.1E+00 9.8E-02 4.5E-01 8.5E-01 3.4E-02 3.7E-01 1.3E+00 4.3E-01 4.6E-01 4.5E-01 
200 1000 5.5E-01 3.2E-02 4.8E-01 3.5E-01 1.5E-02 3.1E-01 2.1E+00 2.1E-01 4.2E-01 2.1E-01 
500 100 4.7E-01 1.9E-01 2.0E+00 6.4E-01 5.4E-02 9.1E-01 1.4E+00 2.9E-01 4.1E-01 3.0E-01 
500 200 2.0E-01 1.0E-01 7.9E-01 2.3E-01 2.5E-02 5.7E-01 2.3E+00 1.6E+00 1.7E-01 2.8E-01 
500 500 3.4E-01 2.4E-02 4.4E-01 1.2E-01 1.1E-02 1.5E-01 6.4E-01 1.9E-01 6.3E-01 1.1E-01 
500 1000 4.2E-01 1.7E-02 8.8E-02 7.1E-01 6.6E-03 1.5E-01 3.8E-01 1.3E-01 6.1E-01 1.6E-01 
1000 100 7.1E-01 3.9E-02 3.8E-01 3.8E-01 2.9E-02 3.5E-01 1.2E+00 2.7E-01 4.1E-01 2.0E-01 
1000 200 8.8E-01 3.7E-02 6.1E-01 3.4E-01 3.9E-02 2.1E-01 9.9E-01 1.2E-01 2.4E-01 1.7E-01 
1000 500 3.4E-01 2.9E-02 2.3E-01 2.2E-01 7.1E-03 7.8E-02 7.2E-01 1.7E-01 1.4E-01 5.4E-02 
1000 1000 1.1E+00 6.7E-03 1.1E-01 7.0E-01 6.6E-03 7.8E-02 2.9E-01 8.0E-02 2.4E-01 1.1E-01 

All simulations are conducted in the R statistical programming environment (R Core Team, 2013). The autoregressive time series are generated using the arima.sim function in R. The EM algorithms are programmed in R, with the log PL value evaluations and EM algorithm updates coded in C via the Rcpp and RcppArmadillo packages (Eddelbuettel, 2013).

5.2  Results

Upon inspection of Tables 2 to 6, we observe a general decreasing trend in terms of increases in both and in all parameter elements. We see that the decreasing trend in is more gradual than in . Furthermore, the MSEs of the mixing proportions (i.e., ) appear not to be affected by the changes in . Also, we see that the decrease of the MSE with respect to is predicted by theorem 5. Comparing the results from S2 and S3, we see little difference between having balanced proportions or not.

The results from Table 7 indicate that the similarity of pseudoallocations and the cluster allocations increases with . Here, we observe that in S1, the concordance is perfect for , 1000, and in S2, the concordance is perfect for , across all values of . This is a good result since the pseudoallocation was considered for use in large scenarios. Furthermore, we note that there are only small differences between the concordance in S2 and S3. This implies that the MPL estimation process and pseudoallocations are robust to unbalanced proportions.

Finally, it follows from the general theory of PL estimation that there is an efficiency loss due to using MPL estimation, as compared to ML estimation (see Cox & Reid, 2004, and Kenne Pagui et al., 2015). The results from Tables 8 to 12 are in accordance with the general theory, as the large majority of MSE ratios are less than 1. However, we note that the MSE ratios of the mixing proportions are all greater than 1. The apparent super efficiency of the MPL estimates of the mixing proportions may be due to the fact that one could interpret each individual PL function as an approximate joint density of short time series that arise from -component mixture models with common mixing proportions. It is also interesting to note that we observe greater levels of superefficiency for the proportions from using the MPL. When comparing the results for S3 to S2, we notice that the difference in the efficiency between the estimates for the proportions can be up to three orders of magnitude.

6  Example Application

To demonstrate the application of our methodology, we consider an analysis of a time series data set arising from the fMRI of an individual in the resting state. The data set was obtained as part of the event-related task-based study in Orban et al. (2015).

6.1  Data Description

In this analysis, we use the resting-state fMRI time series of a single subject (26-year-old male), taken from an fMRI study (Orban et al., 2015). The data were acquired with consent from the individual after approval by the ethics committee at the Research Center of the Geriatric Institute, University of Montreal, Canada. The subject was right-handed and had no history of neurological or psychological disorders.

The brain imaging data were acquired on a 3-T MRI scanner (Magnetom Tim Trio, Siemens) with a 12-channel head coil. The image used in this experiment has spatial resolution voxels ( 56,470 voxels after inclusive masking gray-matter brain voxels; individual voxels have volume millimeters cubed), and a temporal resolution of volumes (repetition time of 2000 milliseconds). Data were preprocessed with the NIAK software (http://simexp.github.io/niak/; see also Bellec et al. (2012). The time series at each voxel, for , is mean normalized and detrended (i.e., each time series consists of the residuals of an ordinary least-square regression).

6.2  MoAR Estimation

Following the analysis in Orban et al. (2015), we fit an model to the data. Here, we note that corresponds to the number of clusters reported in Orban et al. (2015), and we found that was sufficiently rich for modeling the fMRI time series. The estimated parameter vectors are provided in Table 13. We have ordered the class labels with respect to the size of the component probability estimates .

Table 13:
Parameter Estimates of an Model of the Time Series Arising from a Resting-State fMRI.
1234
 0.044 0.023 0.019 0.024 
 0.097 1.077 0.651 0.329 
 0.141 0.028 0.014 0.010 
 0.008 0.390 0.118 0.136 
 0.133 0.254 0.010 0.183 
 0.003 0.120 0.022 0.135 
 0.033 0.047 0.008 0.056 
 0.066 0.180 0.097 0.083 
 0.015 0.175 0.0220 0.033 
 0.021 0.165 0.041 0.076 
 0.021 0.015 0.031 0.048 
 
 56.190 9.845 6.600 6.0657 
 0.136 0.262 0.264 0.338 
1234
 0.044 0.023 0.019 0.024 
 0.097 1.077 0.651 0.329 
 0.141 0.028 0.014 0.010 
 0.008 0.390 0.118 0.136 
 0.133 0.254 0.010 0.183 
 0.003 0.120 0.022 0.135 
 0.033 0.047 0.008 0.056 
 0.066 0.180 0.097 0.083 
 0.015 0.175 0.0220 0.033 
 0.021 0.165 0.041 0.076 
 0.021 0.015 0.031 0.048 
 
 56.190 9.845 6.600 6.0657 
 0.136 0.262 0.264 0.338 

6.3  Clustering of Voxels

Using the parameter estimates from Table 13, we cluster the voxels into the classes. We visualize the clustering at the midcoronal, midhorizontal, and midsagittal slices, as well as provide the variance over time of the voxel intensities (i.e., variance of the time series at each voxel) for the respective slices, for reference, in Figure 2. A point-wise mean and 95% confidence interval of the time series from each of the clusters and 200 voxels that are allocated to each cluster are graphed in Figure 3.

Figure 2:

(A1–A3) Visualizations of the clustering at the midcoronal, midhorizontal, and midsagittal slices, respectively. (B1–B3) Visualizations of the variance image at the respective slices to panels A1 to A3.

Figure 2:

(A1–A3) Visualizations of the clustering at the midcoronal, midhorizontal, and midsagittal slices, respectively. (B1–B3) Visualizations of the variance image at the respective slices to panels A1 to A3.

Figure 3:

Mean and 95% confidence intervals for each of the four clusters from section 6.2. The solid line indicates the point-wise mean, and the dashed line indicates the 95% point-wise confidence interval, in each plot. The 200 time series belonging to each cluster are plotted as colored lines.

Figure 3:

Mean and 95% confidence intervals for each of the four clusters from section 6.2. The solid line indicates the point-wise mean, and the dashed line indicates the 95% point-wise confidence interval, in each plot. The 200 time series belonging to each cluster are plotted as colored lines.

6.4  Discussion

We find it encouraging to observe that the clustering is overall symmetric with respect to the left and right brain hemispheres, as can be observed from an inspection of panels A1 and A2 in Figure 2. Furthermore, even without smoothing, the clusters across panels A1 to A3 appear to be contiguous, which indicates that adjacent regions of the brain behave similarly at rest, as would be anticipated given the higher strength of homotopic functional brain connections. Furthermore, we see that the majority of the highest-variance regions (as observable in panels B1 to B3) appear to be allocated to cluster 4. Thus, the MoAR clustering agrees with the sample variance image.

In Figure 3, the behaviors of the four clusters appear distinct. For example, cluster 3 has a lower variance around the mean than the other clusters. It will take further scientific investigation to explain the biological relevance of our observations.

We note that although there may be dependence between the image voxels, the conclusion of theorem 5 still holds under an assumption that the data are -mixing instead of i.i.d. A condition that implies -mixing is -dependence, whereupon each voxel depends on only a finite number of other voxels within the image (see Bradley, 2005).

If one wishes to explicitly account for the dependence between voxels, then the Markov random field (MRF) approach of Nguyen et al. (2016) can be applied to obtain a smoother image. Figure 4 displays slices of an MRF spatially smoothed version of the clustering from Figure 2. The two clusterings differ at approximately of the voxels, and it is debatable as to whether spatial smoothing is necessary.

Figure 4:

(A1–A3) Visualizations of the spatially smoothed clustering at the midcoronal, midhorizontal, and midsagittal slices, respectively. (B1–B3) Visualizations of the locations where the smoothed and original clustering differ from the respective slices to panels A1 to A3. Here, black indicates a difference.

Figure 4:

(A1–A3) Visualizations of the spatially smoothed clustering at the midcoronal, midhorizontal, and midsagittal slices, respectively. (B1–B3) Visualizations of the locations where the smoothed and original clustering differ from the respective slices to panels A1 to A3. Here, black indicates a difference.

7  Conclusion

In this article, we discussed the numerical problem inherent in the evaluation of expressions of equation 1.1 that arise in the ML estimation of MoAR models. In order to circumvent this problem, we considered instead the MPL estimator.

An EM algorithm was constructed for the computation of the MPL estimate. It was established that this algorithm increases the PL function after each iteration, and the sequence of iterates so produced converges to a stationary point of the log PL function. Furthermore, the MPL estimator was shown to be consistent.

Model-based clustering via the MoAR model requires evaluating estimated a posteriori probability terms that require computating expressions of equation 1.1. To circumvent the evaluation of such expressions, we propose a pseudoallocation rule as an approximation to the usual plug-in version of Bayes’ rule.

To assess the performance of the MPL estimator, we performed a number of simulation studies. We found that the MPL estimates converged in MSE to the true parameter, as increases, as established by the consistency result. However, like other PL estimators, the MPL suffers in efficiency when compared to the ML estimator for the same problem. Surprisingly, we found that the MPL estimates of the mixing proportions