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Introduction

As in the previous study, we consider a mixture model with \(K\) components:

\[ \boldsymbol{X}_i \mid z_i = k \sim \mathcal{N}(\boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k), \quad i = 1, \ldots, n. \]

We assume that the sequence of component means \(\{\boldsymbol{\mu}_1, \ldots, \boldsymbol{\mu}_K\}\) corresponds to evaluations of an underlying smooth function:

\[ \boldsymbol{\mu}_k = u\Bigl(\frac{k}{K}\Bigr), \quad k = 1, \ldots, K, \]

where \(u : [0,1] \to \mathbb{R}^d\) is an unknown smooth function. This formulation does not impose a specific probabilistic prior on \(u\), but instead assumes that \(\{\boldsymbol{\mu}_k\}\) vary smoothly across \(k\).

Algorithm steps

Our Smooth Threshold EM approach modifies the standard EM procedure as follows:

1. E-step

\[ \gamma_{ik}^{(t)} = \frac{\pi_k^{(t)} \, \mathcal{N}\bigl(\boldsymbol{X}_i \mid \boldsymbol{\mu}_k^{(t)}, \boldsymbol{\Sigma}_k^{(t)}\bigr)}{\sum_{j=1}^K \pi_j^{(t)} \, \mathcal{N}\bigl(\boldsymbol{X}_i \mid \boldsymbol{\mu}_j^{(t)}, \boldsymbol{\Sigma}_j^{(t)}\bigr)}. \]

2. Thresholding step

Hard assignment (deterministic):

\[ z_i^{(t)} = \arg\max_k \gamma_{ik}^{(t)}. \]

Optional stochastic assignment:

\[ z_i^{(t)} \sim \text{Categorical}\bigl(\gamma_{i1}^{(t)}, \ldots, \gamma_{iK}^{(t)}\bigr). \]

3. M-step (Smoother)

Given the assignments \(\{(z_i^{(t)}, \boldsymbol{X}_i)\}\), we fit a smoother that directly estimates the sequence of cluster means:

\[ \bigl(\boldsymbol{\mu}_1^{(t+1)}, \ldots, \boldsymbol{\mu}_K^{(t+1)}\bigr) \leftarrow \text{Smooth}\bigl(\{(z_i^{(t)}, \boldsymbol{X}_i)\}\bigr). \]

The smoother can be any technique enforcing the desired level of smoothness across component indices.

The cluster covariances are updated as:

\[ \boldsymbol{\Sigma}_k^{(t+1)} = \frac{1}{n_k^{(t)}} \sum_{i: z_i^{(t)} = k} \bigl(\boldsymbol{X}_i - \boldsymbol{\mu}_k^{(t+1)}\bigr)\bigl(\boldsymbol{X}_i - \boldsymbol{\mu}_k^{(t+1)}\bigr)^\top. \]

Simulate mixture in 2D

Here, we will simulate a mixture of Gaussians in 2D, for \(n = 500\) observations and \(K = 20\) components.

set.seed(123)
source("./code/simulate.R")
source("./code/stochasticEM.R")
library(Matrix)
library(matrixStats)
Warning: package 'matrixStats' was built under R version 4.3.3
library(MASS)
library(mvtnorm)
Warning: package 'mvtnorm' was built under R version 4.3.3
palette_colors <- rainbow(20)
alpha_colors <- sapply(palette_colors, function(clr) adjustcolor(clr, alpha.f=0.3))
sim <- simulate_mixture(n=500, K = 20, d=2, seed=123, proj_mat = matrix(c(1,-0.6,-0.6,1), nrow = 2, byrow = T))
plot(sim$X, col = alpha_colors[sim$z],
     pch = 19, cex = 0.5, 
     xlab = "X1", ylab = "X2",
     main = "Simulated mixture of Gaussians in 2D")

Now, let’s assume we don’t know there are 20 components, and we will fit a mixture model with \(K = 50\) components to this data. For simplicity, let’s assume \(\mathbf{\Sigma}_k = \mathbf{\Sigma}\) for all \(k\), where \(\mathbf{\Sigma}\) is a diagonal matrix.

Penalized B-spline smoother

We first consider the penalized B-spline smoother implemented through the package mgcv. This smoother uses a penalty term to enforce smoothness across the estimated means.

Let’s write a function to generate such smoother:

smoother_mgcv_generator <- function(K, pi = NULL, bs = "bs", num_basis = 30, locations = NULL, m = 2, sp = NULL, method = "ML") {
  if (is.null(locations)) {
    locations <- seq(0, 1, length.out = K)
  }
  if (is.null(pi)) {
    pi <- rep(1/K, K)
  } else if (length(pi) != K) {
    stop("pi must be a vector of length K")
  }

  smoother <- function(Z, X, params = NULL) {
    z_idx <- apply(Z, 1, which.max)
    t <- locations[z_idx]

    d <- ncol(X)
    U <- matrix(0, nrow = K, ncol = d)
    sigma <- numeric(d)

    for (i in 1:d) {
      df_current <- data.frame(x = X[, i], t = t)
      mod <- mgcv::gam(x ~ s(t, bs = bs, k = num_basis, m = m, sp = sp),
                       method = method,
                       family = gaussian(),
                       data = df_current)
      # mgcv::predict.gam returns a vector
      U[, i] <- predict(mod, newdata = data.frame(t = locations))
      sigma[i] <- var(mod$residuals)
    }

    list(pi = pi, mu = split(U, row(U)), sigma = sigma)
  }

  return(smoother)
}

Applying the smoother (with a fixed level of smoothness penalty) in the M-step to our simulated data:

smoother_mgcv <- smoother_mgcv_generator(K = 50, bs = "bs",
                                         num_basis = 20, m = c(3,2),
                                         sp = 100)
set.seed(123)
init_params <- make_default_init(sim$X, K=50)
result_mgcv <- stochastic_EM(
  X = sim$X,
  init_params = init_params,
  threshold = "hard",
  smoother = smoother_mgcv,
  max_iter = 100,
  tol = 1e-5,
  verbose = TRUE
)
Warning in smooth.construct.bs.smooth.spec(object, dk$data, dk$knots): basis
dimension is larger than number of unique covariates
Warning in smooth.construct.bs.smooth.spec(object, dk$data, dk$knots): basis
dimension is larger than number of unique covariates
Iteration 1 (initialization) 
Iteration 2:
  relΔU     = 1.183220e+00
  relΔSigma = 7.358657e-01
  relΔGamma = 9.679182e-01
Iteration 3:
  relΔU     = 2.192901e-01
  relΔSigma = 5.289045e-01
  relΔGamma = 1.135869e+00
Iteration 4:
  relΔU     = 1.339744e-01
  relΔSigma = 2.197193e-01
  relΔGamma = 6.204105e-01
Iteration 5:
  relΔU     = 9.890663e-02
  relΔSigma = 1.915638e-01
  relΔGamma = 3.838961e-01
Iteration 6:
  relΔU     = 6.595388e-02
  relΔSigma = 1.562570e-01
  relΔGamma = 3.482616e-01
Iteration 7:
  relΔU     = 5.100825e-02
  relΔSigma = 9.419623e-02
  relΔGamma = 2.871593e-01
Iteration 8:
  relΔU     = 4.177309e-02
  relΔSigma = 7.120505e-02
  relΔGamma = 2.487183e-01
Iteration 9:
  relΔU     = 4.257584e-02
  relΔSigma = 7.574855e-02
  relΔGamma = 2.226644e-01
Iteration 10:
  relΔU     = 2.985455e-02
  relΔSigma = 3.759246e-02
  relΔGamma = 2.316885e-01
Iteration 11:
  relΔU     = 2.985916e-02
  relΔSigma = 3.147648e-02
  relΔGamma = 1.803975e-01
Iteration 12:
  relΔU     = 2.694174e-02
  relΔSigma = 3.075809e-02
  relΔGamma = 1.840151e-01
Iteration 13:
  relΔU     = 2.368060e-02
  relΔSigma = 3.862209e-02
  relΔGamma = 1.734926e-01
Iteration 14:
  relΔU     = 1.979606e-02
  relΔSigma = 3.954792e-02
  relΔGamma = 1.575292e-01
Iteration 15:
  relΔU     = 1.916345e-02
  relΔSigma = 2.113260e-02
  relΔGamma = 1.424851e-01
Iteration 16:
  relΔU     = 1.831534e-02
  relΔSigma = 4.344799e-02
  relΔGamma = 1.375305e-01
Iteration 17:
  relΔU     = 1.815695e-02
  relΔSigma = 4.936822e-02
  relΔGamma = 1.428554e-01
Iteration 18:
  relΔU     = 1.662043e-02
  relΔSigma = 7.236334e-02
  relΔGamma = 1.453502e-01
Iteration 19:
  relΔU     = 1.446701e-02
  relΔSigma = 8.602045e-02
  relΔGamma = 1.408367e-01
Iteration 20:
  relΔU     = 1.586729e-02
  relΔSigma = 1.022498e-01
  relΔGamma = 1.383092e-01
Iteration 21:
  relΔU     = 1.796540e-02
  relΔSigma = 1.050712e-01
  relΔGamma = 1.572451e-01
Iteration 22:
  relΔU     = 2.129529e-02
  relΔSigma = 1.517928e-01
  relΔGamma = 1.809439e-01
Iteration 23:
  relΔU     = 3.704875e-02
  relΔSigma = 9.025651e-02
  relΔGamma = 2.381033e-01
Iteration 24:
  relΔU     = 1.867614e-02
  relΔSigma = 3.766447e-02
  relΔGamma = 3.079331e-01
Iteration 25:
  relΔU     = 1.555028e-02
  relΔSigma = 1.885227e-02
  relΔGamma = 2.111723e-01
Iteration 26:
  relΔU     = 1.517953e-02
  relΔSigma = 3.279566e-02
  relΔGamma = 2.042569e-01
Iteration 27:
  relΔU     = 1.848291e-02
  relΔSigma = 3.600861e-02
  relΔGamma = 2.017631e-01
Iteration 28:
  relΔU     = 1.774300e-02
  relΔSigma = 4.099660e-02
  relΔGamma = 2.189802e-01
Iteration 29:
  relΔU     = 1.027746e-02
  relΔSigma = 1.193345e-02
  relΔGamma = 2.030787e-01
Iteration 30:
  relΔU     = 1.185255e-02
  relΔSigma = 4.338463e-03
  relΔGamma = 1.402380e-01
Iteration 31:
  relΔU     = 1.052690e-02
  relΔSigma = 6.251619e-03
  relΔGamma = 1.386801e-01
Iteration 32:
  relΔU     = 9.163139e-03
  relΔSigma = 1.142382e-02
  relΔGamma = 1.186152e-01
Iteration 33:
  relΔU     = 6.977837e-03
  relΔSigma = 3.410998e-04
  relΔGamma = 1.226224e-01
Iteration 34:
  relΔU     = 9.168412e-03
  relΔSigma = 6.764842e-03
  relΔGamma = 9.466590e-02
Iteration 35:
  relΔU     = 7.959482e-03
  relΔSigma = 6.030971e-03
  relΔGamma = 1.090649e-01
Iteration 36:
  relΔU     = 4.745328e-03
  relΔSigma = 1.389693e-03
  relΔGamma = 7.962275e-02
Iteration 37:
  relΔU     = 5.445506e-03
  relΔSigma = 4.887827e-03
  relΔGamma = 3.969775e-02
Iteration 38:
  relΔU     = 8.701325e-03
  relΔSigma = 5.465738e-03
  relΔGamma = 4.402158e-02
Iteration 39:
  relΔU     = 3.620258e-03
  relΔSigma = 3.805160e-03
  relΔGamma = 6.051306e-02
Iteration 40:
  relΔU     = 4.416102e-03
  relΔSigma = 2.511656e-03
  relΔGamma = 5.792962e-02
Iteration 41:
  relΔU     = 6.176041e-03
  relΔSigma = 1.404402e-04
  relΔGamma = 6.644673e-02
Iteration 42:
  relΔU     = 5.423766e-03
  relΔSigma = 6.004494e-03
  relΔGamma = 7.646657e-02
Iteration 43:
  relΔU     = 4.951290e-03
  relΔSigma = 8.819709e-03
  relΔGamma = 8.487834e-02
Iteration 44:
  relΔU     = 4.777191e-03
  relΔSigma = 1.100042e-02
  relΔGamma = 6.139040e-02
Iteration 45:
  relΔU     = 4.760495e-03
  relΔSigma = 5.070052e-04
  relΔGamma = 6.652292e-02
Iteration 46:
  relΔU     = 3.164905e-03
  relΔSigma = 8.022960e-04
  relΔGamma = 5.834619e-02
Iteration 47:
  relΔU     = 1.363249e-03
  relΔSigma = 2.602167e-03
  relΔGamma = 4.658138e-02
Iteration 48:
  relΔU     = 1.223237e-03
  relΔSigma = 2.485901e-03
  relΔGamma = 2.816954e-02
Iteration 49:
  relΔU     = 5.813147e-04
  relΔSigma = 2.142353e-04
  relΔGamma = 2.920946e-02
Iteration 50:
  relΔU     = 6.828939e-04
  relΔSigma = 2.159854e-05
  relΔGamma = 1.242802e-02
Iteration 51:
  relΔU     = 9.459284e-04
  relΔSigma = 3.955822e-04
  relΔGamma = 1.814146e-02
Iteration 52:
  relΔU     = 0.000000e+00
  relΔSigma = 0.000000e+00
  relΔGamma = 1.848741e-02
Iteration 53:
  relΔU     = 0.000000e+00
  relΔSigma = 0.000000e+00
  relΔGamma = 0.000000e+00
Converged.
plot(sim$X, col=alpha_colors[sim$z], cex=1,
     xlab="X1", ylab="X2",
     pch=19, main="EM Fitted Means with MGCV Smoothing (Fixed)")
mu_matrix <- do.call(rbind, result_mgcv$params$mu)
for (k in 1:(nrow(mu_matrix)-1)) {
  if (sqrt(sum((mu_matrix[k+1,] - mu_matrix[k,])^2)) > 1e-6) {
    arrows(mu_matrix[k,1], mu_matrix[k,2],
           mu_matrix[k+1,1], mu_matrix[k+1,2],
           col="orange", lwd=2, length=0.1)
  }
}
points(mu_matrix, pch=8, cex=1, lwd=1, col="orange")

The estimated mean in each dimension:

plot(mu_matrix[,1] ~ seq(0,1, length.out = nrow(mu_matrix)),
     type = "b", col = "blue", pch = 19,
     xlab = "Index", ylab = "Mean",
     main = "Means from MGCV Smoother (Fixed)",
     ylim = range(mu_matrix))

plot(mu_matrix[,2] ~ seq(0,1, length.out = nrow(mu_matrix)),
     type = "b", col = "blue", pch = 19,
     xlab = "Index", ylab = "Mean",
     main = "Means from MGCV Smoother (Fixed)",
     ylim = range(mu_matrix))

Applying the smoother (with EB adapted smoothness penalty) in the M-step to our simulated data:

smoother_mgcv <- smoother_mgcv_generator(K = 50, bs = "bs",
                                         num_basis = 20, m = c(3,2),
                                         sp = NULL, method = "ML")
set.seed(123)
init_params <- make_default_init(sim$X, K=50)
result_mgcv <- stochastic_EM(
  X = sim$X,
  init_params = init_params,
  threshold = "hard",
  smoother = smoother_mgcv,
  max_iter = 100,
  tol = 1e-5,
  verbose = TRUE
)
Warning in smooth.construct.bs.smooth.spec(object, dk$data, dk$knots): basis
dimension is larger than number of unique covariates
Warning in smooth.construct.bs.smooth.spec(object, dk$data, dk$knots): basis
dimension is larger than number of unique covariates
Iteration 1 (initialization) 
Iteration 2:
  relΔU     = 6.684361e-02
  relΔSigma = 4.873505e-01
  relΔGamma = 7.277495e-01
Iteration 3:
  relΔU     = 2.862220e-02
  relΔSigma = 2.861013e-01
  relΔGamma = 3.494936e-01
Iteration 4:
  relΔU     = 9.985312e-03
  relΔSigma = 5.339118e-02
  relΔGamma = 1.816420e-01
Iteration 5:
  relΔU     = 7.500035e-03
  relΔSigma = 2.833092e-02
  relΔGamma = 9.892933e-02
Iteration 6:
  relΔU     = 6.302947e-03
  relΔSigma = 1.399816e-02
  relΔGamma = 8.060649e-02
Iteration 7:
  relΔU     = 9.475010e-03
  relΔSigma = 1.253006e-02
  relΔGamma = 4.534402e-02
Iteration 8:
  relΔU     = 8.641219e-03
  relΔSigma = 2.415722e-02
  relΔGamma = 6.335558e-02
Iteration 9:
  relΔU     = 7.284369e-03
  relΔSigma = 1.537775e-02
  relΔGamma = 4.084748e-02
Iteration 10:
  relΔU     = 2.650234e-03
  relΔSigma = 7.450274e-03
  relΔGamma = 2.934321e-02
Iteration 11:
  relΔU     = 3.492960e-03
  relΔSigma = 1.490281e-02
  relΔGamma = 2.301410e-02
Iteration 12:
  relΔU     = 3.896702e-03
  relΔSigma = 7.976918e-03
  relΔGamma = 2.921189e-02
Iteration 13:
  relΔU     = 3.675120e-03
  relΔSigma = 1.179966e-03
  relΔGamma = 1.828996e-02
Iteration 14:
  relΔU     = 7.085302e-03
  relΔSigma = 7.019274e-03
  relΔGamma = 2.115482e-02
Iteration 15:
  relΔU     = 4.649088e-03
  relΔSigma = 2.211740e-03
  relΔGamma = 2.920791e-02
Iteration 16:
  relΔU     = 9.028368e-03
  relΔSigma = 2.000472e-03
  relΔGamma = 2.078615e-02
Iteration 17:
  relΔU     = 6.397165e-03
  relΔSigma = 2.324601e-03
  relΔGamma = 3.548781e-02
Iteration 18:
  relΔU     = 2.358001e-03
  relΔSigma = 1.794774e-03
  relΔGamma = 3.189209e-02
Iteration 19:
  relΔU     = 4.747310e-03
  relΔSigma = 8.373788e-03
  relΔGamma = 1.853817e-02
Iteration 20:
  relΔU     = 4.803193e-03
  relΔSigma = 7.118830e-03
  relΔGamma = 3.669495e-02
Iteration 21:
  relΔU     = 4.669608e-03
  relΔSigma = 1.443668e-02
  relΔGamma = 5.085044e-02
Iteration 22:
  relΔU     = 2.057923e-03
  relΔSigma = 1.660992e-02
  relΔGamma = 5.093011e-02
Iteration 23:
  relΔU     = 4.794940e-03
  relΔSigma = 8.086061e-03
  relΔGamma = 2.891246e-02
Iteration 24:
  relΔU     = 4.416780e-03
  relΔSigma = 6.614696e-03
  relΔGamma = 3.950574e-02
Iteration 25:
  relΔU     = 4.713925e-03
  relΔSigma = 1.410874e-02
  relΔGamma = 3.542620e-02
Iteration 26:
  relΔU     = 6.561008e-03
  relΔSigma = 1.829404e-02
  relΔGamma = 2.826149e-02
Iteration 27:
  relΔU     = 3.667926e-03
  relΔSigma = 1.220594e-02
  relΔGamma = 4.072974e-02
Iteration 28:
  relΔU     = 1.465747e-03
  relΔSigma = 3.753192e-03
  relΔGamma = 2.973192e-02
Iteration 29:
  relΔU     = 0.000000e+00
  relΔSigma = 0.000000e+00
  relΔGamma = 1.328417e-02
Iteration 30:
  relΔU     = 0.000000e+00
  relΔSigma = 0.000000e+00
  relΔGamma = 0.000000e+00
Converged.
plot(sim$X, col=alpha_colors[sim$z], cex=1,
     xlab="X1", ylab="X2",
     pch=19, main="EM Fitted Means with MGCV Smoothing (EB)")
mu_matrix <- do.call(rbind, result_mgcv$params$mu)
for (k in 1:(nrow(mu_matrix)-1)) {
  if (sqrt(sum((mu_matrix[k+1,] - mu_matrix[k,])^2)) > 1e-6) {
    arrows(mu_matrix[k,1], mu_matrix[k,2],
           mu_matrix[k+1,1], mu_matrix[k+1,2],
           col="orange", lwd=2, length=0.1)
  }
}
points(mu_matrix, pch=8, cex=1, lwd=1, col="orange")

The estimated mean in each dimension:

plot(mu_matrix[,1] ~ seq(0,1, length.out = nrow(mu_matrix)),
     type = "b", col = "blue", pch = 19,
     xlab = "Index", ylab = "Mean",
     main = "Means from MGCV Smoother (EB)",
     ylim = range(mu_matrix))

plot(mu_matrix[,2] ~ seq(0,1, length.out = nrow(mu_matrix)),
     type = "b", col = "blue", pch = 19,
     xlab = "Index", ylab = "Mean",
     main = "Means from MGCV Smoother (EB)",
     ylim = range(mu_matrix))

Gaussian process smoother

Next, we consider the Gaussian process smoother based on the Integrated Wiener process (IWP). We use the package BayesGP to implement this smoother. The implementation is fully Bayesian, meaning we will assign prior to the smoothing parameter in each dimension as well as the noise variance. We will take the posterior mean/median as the final estimate.

smoother_iwp_generator <- function(K, pi = NULL, order = 2, locations = NULL, num_basis = 30, smooth.prior = NULL){
  if (is.null(locations)) {
    locations <- seq(0, 1, length.out = K)
  }
  if (is.null(pi)) {
    pi <- rep(1/K, K)
  } else if (length(pi) != K) {
    stop("pi must be a vector of length K")
  }

  smoother <- function(Z, X, params = NULL) {
    z_idx <- apply(Z, 1, which.max)
    t <- locations[z_idx]

    d <- ncol(X)
    U <- matrix(0, nrow = K, ncol = d)
    sigma <- numeric(d)

    for (i in 1:d) {
      df_current <- data.frame(x = X[, i], t = t)
      mod <- BayesGP::model_fit(x ~ f(t, model = "IWP",
                                      sd.prior = smooth.prior,
                                      order = order, k = num_basis),
                       family = "Gaussian",
                       data = df_current)
      U[, i] <- predict(mod, variable = "t", newdata = data.frame(t = locations))$mean
      sigma[i] <- BayesGP::post_table(mod)$median[3]^2
    }

    list(pi = pi, mu = split(U, row(U)), sigma = sigma)
  }

  return(smoother)
}
smoother_iwp <- smoother_iwp_generator(K = 50, order = 2, smooth.prior = 0.5,
                                         num_basis = 20)
set.seed(1234)
init_params <- make_default_init(sim$X, K=50)
result_iwp <- stochastic_EM(
  X = sim$X,
  init_params = init_params,
  threshold = "hard",
  smoother = smoother_iwp,
  max_iter = 100,
  tol = 1e-2,
  verbose = TRUE
)
Iteration 1 (initialization) 
Iteration 2:
  relΔU     = 1.460574e+00
  relΔSigma = 7.330318e-01
  relΔGamma = 9.611504e-01
Iteration 3:
  relΔU     = 4.346333e-01
  relΔSigma = 6.990058e-01
  relΔGamma = 1.463907e+00
Iteration 4:
  relΔU     = 1.325829e-01
  relΔSigma = 5.504142e-01
  relΔGamma = 1.163550e+00
Iteration 5:
  relΔU     = 8.018969e-02
  relΔSigma = 4.218150e-01
  relΔGamma = 5.245279e-01
Iteration 6:
  relΔU     = 2.824925e-02
  relΔSigma = 1.137275e-01
  relΔGamma = 3.418116e-01
Iteration 7:
  relΔU     = 1.958441e-02
  relΔSigma = 5.035668e-02
  relΔGamma = 1.538578e-01
Iteration 8:
  relΔU     = 1.303516e-02
  relΔSigma = 4.107821e-02
  relΔGamma = 9.094313e-02
Iteration 9:
  relΔU     = 1.412391e-02
  relΔSigma = 6.804256e-02
  relΔGamma = 7.603361e-02
Iteration 10:
  relΔU     = 1.879457e-02
  relΔSigma = 6.430619e-02
  relΔGamma = 1.006903e-01
Iteration 11:
  relΔU     = 1.451761e-02
  relΔSigma = 9.003162e-02
  relΔGamma = 7.791152e-02
Iteration 12:
  relΔU     = 9.808148e-03
  relΔSigma = 5.610747e-02
  relΔGamma = 1.088806e-01
Iteration 13:
  relΔU     = 1.353506e-02
  relΔSigma = 6.695791e-02
  relΔGamma = 7.988591e-02
Iteration 14:
  relΔU     = 1.307107e-02
  relΔSigma = 3.876638e-02
  relΔGamma = 7.979120e-02
Iteration 15:
  relΔU     = 8.210096e-03
  relΔSigma = 3.174851e-02
  relΔGamma = 6.288925e-02
Iteration 16:
  relΔU     = 6.149075e-03
  relΔSigma = 1.151352e-03
  relΔGamma = 6.495302e-02
Iteration 17:
  relΔU     = 4.989675e-03
  relΔSigma = 1.839257e-02
  relΔGamma = 4.850374e-02
Iteration 18:
  relΔU     = 7.842382e-03
  relΔSigma = 2.456145e-02
  relΔGamma = 4.638856e-02
Iteration 19:
  relΔU     = 7.393653e-03
  relΔSigma = 2.094358e-02
  relΔGamma = 5.890827e-02
Iteration 20:
  relΔU     = 2.946989e-03
  relΔSigma = 1.782861e-02
  relΔGamma = 3.834447e-02
Iteration 21:
  relΔU     = 4.415997e-03
  relΔSigma = 3.504158e-02
  relΔGamma = 3.781950e-02
Iteration 22:
  relΔU     = 1.974806e-03
  relΔSigma = 3.002725e-03
  relΔGamma = 2.782536e-02
Iteration 23:
  relΔU     = 1.372296e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 2.104529e-02
Iteration 24:
  relΔU     = 1.501882e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.049189e-02
Iteration 25:
  relΔU     = 1.900828e-03
  relΔSigma = 2.836835e-03
  relΔGamma = 1.202885e-02
Iteration 26:
  relΔU     = 1.440786e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.964937e-02
Iteration 27:
  relΔU     = 1.967807e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.101789e-02
Iteration 28:
  relΔU     = 1.595411e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.239784e-02
Iteration 29:
  relΔU     = 1.546608e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.056878e-02
Iteration 30:
  relΔU     = 3.289177e-03
  relΔSigma = 9.577742e-04
  relΔGamma = 1.298842e-02
Iteration 31:
  relΔU     = 2.275586e-03
  relΔSigma = 5.773719e-03
  relΔGamma = 4.108827e-02
Iteration 32:
  relΔU     = 3.262257e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 2.268453e-02
Iteration 33:
  relΔU     = 6.099177e-03
  relΔSigma = 2.990604e-03
  relΔGamma = 3.718201e-02
Iteration 34:
  relΔU     = 4.788426e-03
  relΔSigma = 9.379059e-04
  relΔGamma = 3.728146e-02
Iteration 35:
  relΔU     = 5.343617e-03
  relΔSigma = 9.339674e-04
  relΔGamma = 4.011198e-02
Iteration 36:
  relΔU     = 2.215458e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 4.144540e-02
Iteration 37:
  relΔU     = 4.006462e-03
  relΔSigma = 9.300269e-04
  relΔGamma = 2.559614e-02
Iteration 38:
  relΔU     = 3.858039e-03
  relΔSigma = 3.383990e-03
  relΔGamma = 4.328759e-02
Iteration 39:
  relΔU     = 3.853080e-03
  relΔSigma = 9.156118e-04
  relΔGamma = 4.813256e-02
Iteration 40:
  relΔU     = 4.773849e-03
  relΔSigma = 9.116741e-04
  relΔGamma = 4.320719e-02
Iteration 41:
  relΔU     = 2.166964e-03
  relΔSigma = 2.973816e-03
  relΔGamma = 5.438512e-02
Iteration 42:
  relΔU     = 2.615399e-03
  relΔSigma = 9.012538e-04
  relΔGamma = 2.755419e-02
Iteration 43:
  relΔU     = 4.246129e-03
  relΔSigma = 1.147849e-02
  relΔGamma = 3.654475e-02
Iteration 44:
  relΔU     = 2.671815e-03
  relΔSigma = 8.497419e-03
  relΔGamma = 4.968569e-02
Iteration 45:
  relΔU     = 1.857617e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 2.896816e-02
Iteration 46:
  relΔU     = 1.791708e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.818032e-02
Iteration 47:
  relΔU     = 1.767503e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.755026e-02
Iteration 48:
  relΔU     = 1.210919e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.649480e-02
Iteration 49:
  relΔU     = 1.480854e-03
  relΔSigma = 8.683240e-04
  relΔGamma = 1.043264e-02
Iteration 50:
  relΔU     = 2.147760e-03
  relΔSigma = 2.929820e-03
  relΔGamma = 1.256862e-02
Iteration 51:
  relΔU     = 2.369189e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.421194e-02
Iteration 52:
  relΔU     = 3.420338e-03
  relΔSigma = 8.707490e-04
  relΔGamma = 2.474637e-02
Iteration 53:
  relΔU     = 2.043789e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 4.266265e-02
Iteration 54:
  relΔU     = 1.089013e-03
  relΔSigma = 0.000000e+00
  relΔGamma = 1.913618e-02
Iteration 55:
  relΔU     = 9.060527e-04
  relΔSigma = 0.000000e+00
  relΔGamma = 2.428926e-03
Converged.
plot(sim$X, col=alpha_colors[sim$z], cex=1,
     xlab="X1", ylab="X2",
     pch=19, main="EM Fitted Means with IWP prior (fully Bayes)")
mu_matrix <- do.call(rbind, result_iwp$params$mu)
for (k in 1:(nrow(mu_matrix)-1)) {
  if (sqrt(sum((mu_matrix[k+1,] - mu_matrix[k,])^2)) > 1e-6) {
    arrows(mu_matrix[k,1], mu_matrix[k,2],
           mu_matrix[k+1,1], mu_matrix[k+1,2],
           col="orange", lwd=2, length=0.1)
  }
}
points(mu_matrix, pch=8, cex=1, lwd=1, col="orange")

Take a look at the estimated mean in each dimension:

plot(mu_matrix[,1] ~ seq(0,1, length.out = nrow(mu_matrix)),
     type = "b", col = "blue", pch = 19,
     xlab = "Index", ylab = "Mean",
     main = "Means from IWP Smoother",
     ylim = range(mu_matrix))

plot(mu_matrix[,2] ~ seq(0,1, length.out = nrow(mu_matrix)),
     type = "b", col = "blue", pch = 19,
     xlab = "Index", ylab = "Mean",
     main = "Means from IWP Smoother",
     ylim = range(mu_matrix))


sessionInfo()
R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.7.4

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Chicago
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] mvtnorm_1.3-1     MASS_7.3-60       matrixStats_1.4.1 Matrix_1.6-4     
[5] workflowr_1.7.1  

loaded via a namespace (and not attached):
 [1] mvQuad_1.0-8        jsonlite_2.0.0      compiler_4.3.1     
 [4] promises_1.3.3      Rcpp_1.0.14         stringr_1.5.1      
 [7] git2r_0.33.0        callr_3.7.6         later_1.4.2        
[10] jquerylib_0.1.4     splines_4.3.1       statmod_1.5.0      
[13] yaml_2.3.10         fastmap_1.2.0       lattice_0.22-6     
[16] R6_2.6.1            BayesGP_0.1.3       knitr_1.50         
[19] tibble_3.2.1        rprojroot_2.0.4     TMB_1.9.15         
[22] bslib_0.9.0         pillar_1.10.2       rlang_1.1.6        
[25] cachem_1.1.0        stringi_1.8.7       httpuv_1.6.16      
[28] xfun_0.52           getPass_0.2-4       fs_1.6.6           
[31] sass_0.4.10         cli_3.6.5           mgcv_1.9-1         
[34] magrittr_2.0.3      ps_1.9.1            digest_0.6.37      
[37] grid_4.3.1          processx_3.8.6      rstudioapi_0.16.0  
[40] nlme_3.1-166        lifecycle_1.0.4     vctrs_0.6.5        
[43] data.table_1.16.2   aghq_0.4.1          evaluate_1.0.3     
[46] glue_1.8.0          numDeriv_2016.8-1.1 whisker_0.4.1      
[49] rmarkdown_2.28      httr_1.4.7          tools_4.3.1        
[52] pkgconfig_2.0.3     htmltools_0.5.8.1