We develop a random partition procedure based on a Dirichlet process prior with Laplace distribution. Gibbs sampling of a Laplace mixture of linear mixed regressions with a Dirichlet process is implemented as a random partition model when the number of clusters is unknown. Our approach provides simultaneous partitioning and parameter estimation with the computation of classification probabilities, unlike its counterparts. A full Gibbssampling algorithm is developed for an efficient Markov chain Monte Carlo posterior computation. The proposed method is illustrated with simulated data and one real data of the energy efficiency of Tsanas and Xifara (
Clustering algorithms attempt to understand a partition of a finite set of objects into a potentially predetermined number of nonempty subsets; in addition, the number of partitions is often unknown beforehand. We focus on probability models for partitions and avoid purely algorithmic methods. As a special case, product partition models (PPMs), introduced by Hartigan (1990) and Barry and Hartigan (1992), are based on modeling random partitions of the sample space. These assume that observations in different elements of a random partition of the data are actually independent. So if the probability distribution for the random partitions is in a product form prior to obtaining observations, it is also then in product form after obtaining the observations (Jordan
where
where
Cohesion is the measure of the strength of the functional relationship of the elements in each subsets that then controls the partition of subsets that can be roughly thought of as a probability. A popular choice is
where
A similarly popular prior on random partitions is model-based clustering and its extended models, which fit a finite mixture of multivariate Gaussian distributions with various variance structures to the data (Banfield and Raftery, 1993; Fraley and Raftery, 2002, 2007; McLachlan and Peel, 2000; Wolfe, 1970). It implements an Expectation-Maximization (EM) algorithm (Dempster
In the Bayesian literature, the nonparametric Bayesian clustering approach is usually based on a mixture of the DP (Antoniak, 1974; Ferguson, 1973) and an unknown number of clusters. Especially, a Dirichlet process mixture (DPM) of regression models has been widely used as a flexible semiparametric approach for clustering and density estimation (Escobar and West, 1995). The implementation of the DP mixture models has been made feasible by the modern method of Bayesian computation and efficient algorithms (MacEachern and Müller, 1998; Neal, 2000). Product partition type priors on a normal mixture of regression model also have been widely used for the tractable, probability-based, objective function to identify good partitions (Booth
Recently, a natural extension of the random partition model has been considered with incorporating covariate values in its definition. MacEachern (1999, 2000) proposed a collection of dependent random probability measures with marginal distributions given by the DP. This idea has been extended and applied to the construction of various types of random probability measures such as the density regression (Dunson
Park and Dunson (2010) argued that a semiparametric Bayesian approach with an infinite number of clusters can be considered by letting
Most of the mixture of the regression model are considered with a normality assumption for the distribution of subcluster
Song
In this research, we develop a full Bayesian estimation procedure for the linear regression mixture model of the full conditional posterior distribution with Laplace distribution. For the prior on the clustering structure, we consider a random partition model of the DP process based on a truncated approximation of stick-breaking priors (Ishwaran and Zarepour, 2000) because the proposed model leads to a tractable, probability-based, objective function to identify good partitions. For the full posterior distribution of Laplace distribution, we consider that the Laplace distribution is a scale mixture of a normal distribution with an exponential mixing density (Andrews and Mallows, 1974). Details are discussed in the following section.
We also apply a post process to posterior samples for parameters of the proposed model to choose a single clustering estimate to compromise the “label-switching” problem (Richardson and Green, 1997; Stephens, 2000). We follow Fritsch and Ickstadt (2009), which finds a single clustering estimate by maximizing the posterior expected adjusted Rand index with the posterior probabilities of two observations being clustered together.
We use hierarchical models and Gibbs sampling to obtain estimators for Laplace distribution mixture models. In Section 2, we consider the hierarchical structure of models and the basic identity of a scale mixture of a normal distribution for Laplace distribution. Section 3 provides details on Markov chain Monte Carlo (MCMC) procedures based on the full conditional distribution of parameters and post process based on the posterior similarity matrix to choose a single cluster and important oscillating functions in each curve based on the posterior expected adjusted Rand index. We compare the proposed Laplace regression mixture and the normal mixture in Section 4, using simulations and data sets. There is a discussion in Section 5.
We begin with construction of the random partition model with DP prior based on a Laplace linear regression. We discuss the mixture structure and the basic identity of the Laplace distribution which is a scale mixture of a normal distribution with an exponential mixing density.
We discussed in Section 1 that a PPM with a cohesion function
Blackwell and MacQueen (1973) proved that for
where
where
The algorithms of Bush and MacEachern (1996) are some of the most widely-used approaches for the posterior computation of Pólya urn DP. They argued that their approach first updates the configuration of subjects to clusters based on the Pólya urn scheme in (
Here, for the DP process prior, we consider the stick-breaking representation of the DP for the infinite number of clusters. According to Sethuraman (1994), if G is assigned a DP prior with precision
where
Ishwaran and James (2001) presented two Gibbs sampling methods for fitting Bayesian nonparametric hierarchical models based on stick-breaking priors. The first type of Gibbs sampler, referred to as a Pólya urn Gibbs sampler, applies to stick-breaking priors with a known Pólya urn characterization, that are priors with an explicit and simple prediction rule. The second method, the blocked Gibbs sampler, works by directly sampling values from the posterior of the random measure. They argue that the blocked Gibbs sampler avoids marginalizing over the prior and allows the prior to be directly involved in the Gibbs sampling scheme. This allows direct sampling of the nonparametric posterior and leads to several computational and inferential advantages. Thus, in this paper, we consider the blocked Gibbs sampler of Ishwaran and James (2001) based on the stick-breaking representation of the DP as a prior on the clustering structure.
For the index of cluster, we consider an indicator variable of mixture
where
For the density function of
where
The Laplace (double-exponential) distribution is a scale mixture of a normal distribution with an exponential mixing density (Andrews and Mallows, 1974), that is
Details of the equation and the proof have been discussed by Kyung
Ishwaran and Zarepour (2000) proposed a truncated approximation with
For the regression parameters in cluster
We begin with construction of a cluster structure and discuss how to estimate parameters in each cluster.
Step 1. Cluster structure
With an appropriate approximation level of
Upon sampling
for
Step 2. Model parameters
Given the cluster indices of each observation
where
where
and
Step 3. Post process
Mixture models suffer from a well-known “label-switching” problem, which arises due to the identical likelihood for any permutation of component-specific parameters. Inheriting the properties of the adjusted Rand index, Fritsch and Ickstadt (2009) proposed the posterior expected adjusted Rand index and showed the outperformance of the posterior expected adjusted Rand index over competing methods such as maximum a posteriori (MAP) estimate. In addition, implementing the method is easy with R package
We conduct simulation studies to evaluate our proposed random partition model with LAD regression. We implement the full conditional Gibbs sampler using DP prior on the cluster structure to analyze the energy efficiency data set. As a competing model, we consider the normal mixture model (NMM) of DP prior. Also, to compare the proposed method, we consider the model-based clustering of Fraley and Raftery (2002) based on the NMM, for which the R package
For the simulation studies, we considered the maximum truncation level is 30 (
We first evaluated the performance of our method with simulated data, where the classes are known. We simulated data according to the following regression models with
where
For regression, we generated two exploratory variables,
For various situation of data structure, we considered three different sets of errors for each clusters:
Set 1.
Set 2.
Set 3.
For normally distributed error data (Set 1), means of cluster
For the model-based normal mixture, to identify the optimal number of clusters and covariance structure for a given data set, the BIC is considered and the BIC plots of each data sets for the number of clusters are in Figure C.1 in
Based on the posterior mean and 95% credible interval, the proposed model correctly estimates the mean functions, but it fails to capture the linear trends correctly. However, the estimated
Table 1 shows the estimated scale parameters and 95% credible intervals of both Laplace partition models and NMMs. Estimated scale parameters of the proposed Laplace regression partition model are
For Laplace random partition data (Set 2), means of cluster
The estimated number of cluster is 5 based on our proposed model and 4 based on the NMM. Thus, the computed adjusted Rand index between the estimated
The proposed model correctly estimates the mean functions based on the posterior mean and 95% credible interval; however, it fails to capture the linear trends correctly and similar results of the NMM. Table 1 shows that the estimated scale parameters and 95% credible intervals of the proposed Laplace regression partition model are
With EM algorithm of fixed 3 clusters, the estimated scale parameters are
For
The estimated number of cluster is 3 based on our proposed model and 2 based on the NMM. Therefore, the computed adjusted Rand index between the estimated
The proposed model and the NMM correctly estimates the mean functions based on the posterior mean and 95% credible interval; however, it fails to capture the linear trends correctly. Figure 3 includes the estimated curves with true curves. With heavy tailed mixture, we observe that any method might be unable to capture the true clustering structure in data. Estimated scale parameters and 95% credible intervals in Table 1 of the proposed Laplace regression partition model are
For more complicated structure of the data generation process with clusters, we generated two more data sets to evaluate our proposed random partition model with LAD regression. We simulated data according to the following regression models with
where
For regression, we generated two exploratory variables,
We mimic the 5
Set 4.
Set 5.
The error in Set 4 is a mixture of two normal distributions and this complexity causes the generated data to appear to have at least four clusters and not easy to partition. This would produce 5% data likely to be low leverage outliers and unsmooth curved data. Based on the posterior mean and 95% credible interval of parameters in each clusters, we observe that the estimate fails to capture the linear trends correctly.
The true number of cluster is 2, and the NMM of Gibbs choose 2 clusters and the Laplace partition model of Gibbs have chosen 5 clusters based on the posterior expected adjusted Rand index. The BIC of the EM model based-cluster consider 2 to 3 clusters with “spherical, varying volume (VII)” variance structure, but we choose to have 2 clusters. Based on the chosen number of clusters of models, the computed adjusted Rand index between clusterings/partitions and the true indices of the clusters are in Table 2. We observe that the computed value of adjusted Rand index with true indices of the Gibbs NMM is slightly larger than that of the Gibbs Laplace partition model numerically. However, the EM model-based model seems not to detect true clustering indices correctly compared to other Gibbs models.
Figure 4 shows the estimated curves of NMM and LPM with 95% credible intervals on selected data. Based on the 95% credible intervals, we observe that the 95% credible interval of LPM is wider than NMM as discussed in the previous section. The estimated curves do not seem to adequately estimate the true curve at each data point due to the complexity of data generation setting. However, the true number of clusters is 2, and the estimated curves of NMM and LPM seem to capture the true number of clusters around the data points that can be easily partitioned.
In the generating setting of the 5
The energy efficient dataset was created and processed by Tsanas and Xifara (2012) using 12 different building shapes simulated in Ecotect. The buildings differ with respect to glazing area, glazing area distribution, and orientation, amongst other parameters. They originally simulate various settings as functions of the afore-mentioned characteristics to obtain 768 building shapes and the dataset comprises 768 samples and 8 features, aiming to predict two real valued responses. Two responses are “Heating Load” (
Tsanas and Xifara (2012) investigated the association strength of each input variable with each of the output variables using a variety of classical and non-parametrical statistical analysis tools to identify the most strongly related input variables. They compared a linear regression approach and random forests to estimate heating load (HL) and cooling load (CL). Tsanas and Xifara did not considered standardization and intercept in the model for the linear regression and random forest models. Tsanas and Xifara concluded that based on the random forest,
We instead consider a normal regression mixture model and a Laplace regression random partition model for HL and CL because a simple linear regression is inadequate to explain the relationship of input variables to output variable. Figure 6 includes histograms of HL and CL with estimated density based on Gaussian kernels. We observe from histograms that the HL might be able to explained with mixture of few normal distributions, and the CL can be explained with one normal distribution with small variance and one normal distribution with large variance.
Estimated cluster-specific parameters of normal linear regression mixture model and Laplace linear regression random partition model are in Table B.2 in
The estimated number of clusters of Laplace random partition model is 2, and cluster 1 is specified with positive parameters of
For the Cooling Load, estimated cluster-specific parameters of the normal linear regression mixture model and Laplace linear regression random partition model are in Table B.1 in
Unlike the conclusion of Tsanas and Xifara (2012), we observe that
We have developed a random partition procedure based on a DP prior with Laplace distribution. A full Gibbs-sampling algorithm for the linear regression mixture model of the full conditional posterior distribution with Laplace distribution is developed for an efficient MCMC posterior computation. For the prior on the clustering structure, we consider a random partition model of the DP, because the proposed model leads to a tractable, probability-based, objective function to identify good partitions. For the full posterior distribution of the Laplace distribution, we consider the fact the Laplace distribution is a scale mixture of a normal distribution with an exponential mixing density (Andrews and Mallows, 1974). We also have applied a post process to posterior samples for parameters of the proposed model to choose a single clustering estimate to compromise the “label-switching” problem based on maximizing the posterior expected adjusted Rand index of Fritsch and Ickstadt (2009).
For the comparison of the proposed methods, we considered the model-based clustering, Gaussian mixture model, based on the EM methods in our simulation studies. To choose the optimal number of clusters, we considered the BIC values on each sets. However, in our simulation studies, strangely for all data sets, the BIC has chosen only one cluster with most of multivariate covariance structures except “spherical, equal volume (EII)” and “spherical, varying volume (VII)” structures. It might be the reason of large scale parameter values of each sets of data for each clusters or the limitation of the BIC computation based on the EM algorithm. In the simulation, we already know the number of clusters and we fixed the number of clusters as 3 that is the true number of clusters for the data generation.
For the first set of simulations, we considered three different sets of error distributions, normal, Laplace, and
The two data sets in the second simulation section were with the 5% low leverage outliers and with the 5% high leverage outliers, respectively. The NMM and even the EM model based clustering algorithm failed to capture the linear trends correctly in the proposed model; in addition, the estimated curves were not on the generated data points correctly. However, for the data with 5% high leverage outliers, the 95% credible interval of the proposed Gibbs Laplace partition model seems to adequately estimate for the hidden structure of data with high leverage outliers, even though the 95% credible interval is wider.
The EM NMM seems to underestimate the scale parameters of each clusters on each set of data compared to other Gibbs methods. Also, with fixed number of clusters as the true number of clusters, the estimated curve of the EM on the histogram seem not to consider the distribution of data with heavy tailed error, but the indices of clusters based on the EM seem close to the true indices of clusters. It is best use the mixture models with Gibbs sampling if our goal is density estimation; however, the EM will provide more hidden information if our goal is the detection of the cluster indices.
We observe that
Minjung Kyung was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science, ICT & Future Planning (Grant No. NRF-2015R1C1A1A01051837).
The joint posterior distribution of parameters for cluster
For the posterior distribution of
where
Therefore,
For the posterior distribution of
Let
then
Therefore,
and
For the posterior distribution of
Therefore,
Posterior median and 95% credible interval of cluster-specific model parameters of NMM and LPM for Heating load
Model | Parameter | Cluster | ||
---|---|---|---|---|
1 | 2 | 3 | ||
NMM | 0.29 (−1.89, 2.28) | 0.23 (−1.81, 2.33) | −2.20 (−4.01, −0.71) | |
−0.01 (−0.83, 0.77) | −0.02 (−0.83, 0.75) | 0.00 (−0.78, 0.78) | ||
0.04 (−0.76, 0.86) | 0.06 (−0.70, 0.87) | 0.09 (−0.69, 0.88) | ||
−0.01 (−1.58, 1.63) | −0.02 (−1.56, 1.59) | −0.10 (−1.65, 1.49) | ||
1.87 (1.46, 2.77) | 2.01 (−0.41, 2.72) | 1.21 (0.82, 2.47) | ||
−0.01 (−0.44, 0.24) | −0.11 (−0.60, 0.41) | −0.02 (−0.11, 0.06) | ||
0.22 (−1.45, 2.63) | 0.67 (−1.41, 2.79) | 10.33 (2.27, 11.27) | ||
2.01 (1.05, 2.98) | 2.07 (−0.38, 3.44) | −0.09 (−0.15, 2.16) | ||
0.07 (0.04, 2.14) | 1.30 (0.35, 2.20) | 0.38 (0.32, 1.05) | ||
20 | 22 | 116 | ||
Model | Parameter | Cluster | ||
4 | 5 | 6 | ||
NMM | −1.64 (−2.72, −0.33) | −1.40 (−2.77, 0.07) | −0.15 (−2.20, 1.71) | |
0.01 (−0.84, 0.79) | 0.02 (−0.79, 0.80) | 0.07 (−0.71, 0.85) | ||
0.01 (−0.77, 0.86) | 0.03 (−0.76, 0.83) | −0.03 (−0.81, 0.76) | ||
−0.05 (−1.63, 1.64) | −0.07 (−1.63, 1.53) | 0.04 (−1.51, 1.60) | ||
4.11 (2.97, 4.25) | 3.10 (2.89, 3.27) | −0.62(−1.46, 0.32) | ||
−0.01 (−0.10, 0.05) | −0.03 (−0.16, 0.10) | 0.00 (−0.23, 0.24) | ||
11.85 (2.88, 12.32) | 3.47 (2.39, 4.40) | 7.82 (5.09, 10.68) | ||
−0.05 (−0.09, 0.06) | 0.00 (−0.10, 0.11) | −0.11 (−0.29, 0.08) | ||
0.39 (0.35, 0.71) | 0.65 (0.54, 0.79) | 1.32 (1.05, 1.65) | ||
371 | 125 | 114 | ||
Model | Parameter | Cluster | ||
1 | 2 | |||
LPM | −1.77 (−6.69, 1.42) | −1.38 (−3.30, 1.29) | ||
0.00 (−0.78, 0.76) | −0.04 (−0.87, 0.75) | |||
0.04 (−0.74, 0.82) | 0.13 (−0.88, 1.41) | |||
−0.04 (−1.58, 1.52) | −0.10 (−1.74, 1.52) | |||
3.68 (2.98, 4.28) | 3.62 (−1.43, 3.93) | |||
−0.03 (−0.19, 0.13) | −0.03 (−0.84, 0.91) | |||
11.63 (10.51, 14.38) | 11.25 (0.88, 12.48) | |||
0.09 (−0.03, 0.27) | 0.09 (−0.73, 0.95) | |||
2.23 (1.76, 2.43) | 2.21 (1.00, 2.47) | |||
729 | 39 |
NMM = normal mixture model; LPM = Laplace partition model.
Posterior median and 95% credible interval of cluster-specific model parameters of NMM and LPM for cooling load
Model | Parameter | Cluster | |||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
NMM | 0.10 (−1.54, 2.12) | 1.13 (−0.58, 2.77) | −0.21 (−2.24, 1.80) | −1.36 (−3.42, 1.99) | |
−0.01 (−0.80, 0.80) | −0.01 (−0.83, 0.79) | −0.04 (−0.84, 0.77) | −0.01 (−0.79, 0.76) | ||
0.08 (−0.72, 0.88) | 0.08 (−0.73, 0.89) | 0.26 (−0.55, 1.06) | 0.40 (−0.76, 0.85) | ||
−0.03 (−1.63, 1.56) | −0.03 (−1.64, 1.60) | −0.19 (−1.79, 1.42) | −0.02 (−1.58, 1.54) | ||
0.76 (0.20, 1.15) | 1.90 (0.77, 2.17) | 0.61 (−1.40, 1.43) | 5.31 (0.35, 5.62) | ||
−0.04 (−0.15, 0.08) | 0.14 (−0.07, 0.32) | −0.06 (−0.57, 0.48) | 0.04 (−0.12, 0.25) | ||
0.69 (−1.55, 2.52) | 9.52 (1.52, 10.94) | 1.81 (−0.36, 4.15) | 9.95 (0.19, 10.86) | ||
1.10 (−0.00, 2.42) | 0.03 (−0.10, 2.35) | 0.35 (−0.16, 0.80) | 0.01 (−0.07, 1.42) | ||
0.15 (0.08, 0.38) | 1.65 (0.14, 1.86) | 2.06 (1.42, 2.80) | 0.76 (0.13, 1.98) | ||
10 | 309 | 49 | 400 | ||
Model | Parameter | Cluster | |||
1 | 2 | ||||
LPM | −1.96 (−8.86, 0.82) | −1.02 (−3.39, 1.60) | |||
−0.00 (−0.81, 0.80) | −0.03 (−0.91, 0.84) | ||||
0.03 (−0.77, 0.83) | 0.11 (−0.93, 1.41) | ||||
−0.02 (−1.62, 1.58) | −0.10 (−1.84, 1.59) | ||||
3.80 (3.15, 5.05) | 3.57 (−1.71, 4.16) | ||||
0.12 (−0.13, 0.29) | 0.13 (−0.83, 1.06) | ||||
8.70 (7.37, 10.68) | 7.74 (1.20, 9.51) | ||||
0.06 (−0.11, 0.20) | 0.05 (−0.86, 1.04) | ||||
2.37 (1.98, 2.60) | 2.35 (1.00, 3.70) | ||||
739 | 29 |
NMM = normal mixture model; LPM = Laplace partition model.
BIC plots of model based clustering based on the generated data from normal mixture model (Set 1), Laplace random partition model (Set 2), and
Posterior median and 95% credible interval of cluster-specific model parameters of NMM and LPM for Heating load
Model | Parameter | Cluster | ||
---|---|---|---|---|
1 | 2 | 3 | ||
NMM | 0.29 (−1.89, 2.28) | 0.23 (−1.81, 2.33) | −2.20 (−4.01, −0.71) | |
−0.01 (−0.83, 0.77) | −0.02 (−0.83, 0.75) | 0.00 (−0.78, 0.78) | ||
0.04 (−0.76, 0.86) | 0.06 (−0.70, 0.87) | 0.09 (−0.69, 0.88) | ||
−0.01 (−1.58, 1.63) | −0.02 (−1.56, 1.59) | −0.10 (−1.65, 1.49) | ||
1.87 (1.46, 2.77) | 2.01 (−0.41, 2.72) | 1.21 (0.82, 2.47) | ||
−0.01 (−0.44, 0.24) | −0.11 (−0.60, 0.41) | −0.02 (−0.11, 0.06) | ||
0.22 (−1.45, 2.63) | 0.67 (−1.41, 2.79) | 10.33 (2.27, 11.27) | ||
2.01 (1.05, 2.98) | 2.07 (−0.38, 3.44) | −0.09 (−0.15, 2.16) | ||
0.07 (0.04, 2.14) | 1.30 (0.35, 2.20) | 0.38 (0.32, 1.05) | ||
20 | 22 | 116 | ||
Model | Parameter | Cluster | ||
4 | 5 | 6 | ||
NMM | −1.64 (−2.72, −0.33) | −1.40 (−2.77, 0.07) | −0.15 (−2.20, 1.71) | |
0.01 (−0.84, 0.79) | 0.02 (−0.79, 0.80) | 0.07 (−0.71, 0.85) | ||
0.01 (−0.77, 0.86) | 0.03 (−0.76, 0.83) | −0.03 (−0.81, 0.76) | ||
−0.05 (−1.63, 1.64) | −0.07 (−1.63, 1.53) | 0.04 (−1.51, 1.60) | ||
4.11 (2.97, 4.25) | 3.10 (2.89, 3.27) | −0.62(−1.46, 0.32) | ||
−0.01 (−0.10, 0.05) | −0.03 (−0.16, 0.10) | 0.00 (−0.23, 0.24) | ||
11.85 (2.88, 12.32) | 3.47 (2.39, 4.40) | 7.82 (5.09, 10.68) | ||
−0.05 (−0.09, 0.06) | 0.00 (−0.10, 0.11) | −0.11 (−0.29, 0.08) | ||
0.39 (0.35, 0.71) | 0.65 (0.54, 0.79) | 1.32 (1.05, 1.65) | ||
371 | 125 | 114 | ||
Model | Parameter | Cluster | ||
1 | 2 | |||
LPM | −1.77 (−6.69, 1.42) | −1.38 (−3.30, 1.29) | ||
0.00 (−0.78, 0.76) | −0.04 (−0.87, 0.75) | |||
0.04 (−0.74, 0.82) | 0.13 (−0.88, 1.41) | |||
−0.04 (−1.58, 1.52) | −0.10 (−1.74, 1.52) | |||
3.68 (2.98, 4.28) | 3.62 (−1.43, 3.93) | |||
−0.03 (−0.19, 0.13) | −0.03 (−0.84, 0.91) | |||
11.63 (10.51, 14.38) | 11.25 (0.88, 12.48) | |||
0.09 (−0.03, 0.27) | 0.09 (−0.73, 0.95) | |||
2.23 (1.76, 2.43) | 2.21 (1.00, 2.47) | |||
729 | 39 |
NMM = normal mixture model; LPM = Laplace partition model.
Posterior median and 95% credible interval of cluster-specific model parameters of NMM and LPM for cooling load
Model | Parameter | Cluster | |||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
NMM | 0.10 (−1.54, 2.12) | 1.13 (−0.58, 2.77) | −0.21 (−2.24, 1.80) | −1.36 (−3.42, 1.99) | |
−0.01 (−0.80, 0.80) | −0.01 (−0.83, 0.79) | −0.04 (−0.84, 0.77) | −0.01 (−0.79, 0.76) | ||
0.08 (−0.72, 0.88) | 0.08 (−0.73, 0.89) | 0.26 (−0.55, 1.06) | 0.40 (−0.76, 0.85) | ||
−0.03 (−1.63, 1.56) | −0.03 (−1.64, 1.60) | −0.19 (−1.79, 1.42) | −0.02 (−1.58, 1.54) | ||
0.76 (0.20, 1.15) | 1.90 (0.77, 2.17) | 0.61 (−1.40, 1.43) | 5.31 (0.35, 5.62) | ||
−0.04 (−0.15, 0.08) | 0.14 (−0.07, 0.32) | −0.06 (−0.57, 0.48) | 0.04 (−0.12, 0.25) | ||
0.69 (−1.55, 2.52) | 9.52 (1.52, 10.94) | 1.81 (−0.36, 4.15) | 9.95 (0.19, 10.86) | ||
1.10 (−0.00, 2.42) | 0.03 (−0.10, 2.35) | 0.35 (−0.16, 0.80) | 0.01 (−0.07, 1.42) | ||
0.15 (0.08, 0.38) | 1.65 (0.14, 1.86) | 2.06 (1.42, 2.80) | 0.76 (0.13, 1.98) | ||
10 | 309 | 49 | 400 | ||
Model | Parameter | Cluster | |||
1 | 2 | ||||
LPM | −1.96 (−8.86, 0.82) | −1.02 (−3.39, 1.60) | |||
−0.00 (−0.81, 0.80) | −0.03 (−0.91, 0.84) | ||||
0.03 (−0.77, 0.83) | 0.11 (−0.93, 1.41) | ||||
−0.02 (−1.62, 1.58) | −0.10 (−1.84, 1.59) | ||||
3.80 (3.15, 5.05) | 3.57 (−1.71, 4.16) | ||||
0.12 (−0.13, 0.29) | 0.13 (−0.83, 1.06) | ||||
8.70 (7.37, 10.68) | 7.74 (1.20, 9.51) | ||||
0.06 (−0.11, 0.20) | 0.05 (−0.86, 1.04) | ||||
2.37 (1.98, 2.60) | 2.35 (1.00, 3.70) | ||||
739 | 29 |
NMM = normal mixture model; LPM = Laplace partition model.
Posterior median and 95% CI for cluster-specific residual scale parameter
Model | Cluster | Truth | NMM | LPM | MAP | ||
---|---|---|---|---|---|---|---|
Mean | 95% CI | Mean | 95% CI | ||||
Normal | 1 | 0.71 | 0.58 | (0.47, 0.84) | 0.49 | (0.34, 0.98) | 0.28 |
2 | 0.45 | 0.44 | (0.36, 0.58) | 0.37 | (0.27, 0.95) | 0.22 | |
3 | 0.32 | 0.30 | (0.25, 0.54) | 0.24 | (0.17, 0.93) | 0.15 | |
Laplace | 1 | 0.71 | 0.73 | (0.45, 3.27) | 0.49 | (0.26, 1.00) | 0.43 |
2 | 0.45 | 0.41 | (0.19, 2.46) | 0.47 | (0.18, 1.67) | 0.36 | |
3 | 0.32 | 0.60 | (0.20, 5.38) | 0.85 | (0.11, 2.71) | 0.17 | |
4 | - | 0.36 | (0.19, 8.08) | 0.63 | (0.20, 2.35) | ||
5 | - | - | - | 0.21 | (0.06, 1.41) | ||
1 | 1.67 | 0.84 | (0.57, 11.80) | 0.40 | (0.25, 3.12) | 0.73 | |
2 | 1.67 | 2.75 | (2.12, 25.80) | 1.03 | (0.61, 5.20) | 0.66 | |
3 | 1.67 | - | - | 1.35 | (0.57, 4.24) | 0.30 |
CI = credible interval; NMM = normal mixture model; LPM = Laplace partition model; MAP = maximum a posteriori.
Computed the adjusted Rand index between partitions and the true indices of clusters of the same objects for NMM and LPM, and MAP estimate from the model-based clustering model of EM
Set 4 | Set 5 | |||||
---|---|---|---|---|---|---|
NMM | LPM | MAP | NMM | LPM | MAP | |
Rand index | 0.467 | 0.434 | 0.00 | 0.427 | 0.401 | 0.102 |
NMM = normal mixture model; LPM = Laplace partition model; MAP = maximum a posteriori.