Latent projection predictive feature selection

General idea

The response families used in GLMs (McCullagh and Nelder 1989, chap. 2) (and in GLMMs, GAMs, and GAMMs) may be termed exponential dispersion (ED) families (Jørgensen 1987)¹. For a response family that is not an ED family, the Kullback-Leibler (KL) divergence minimization problem (see Piironen, Paasiniemi, and Vehtari 2020) is often not easy to solve analytically². In order to bypass this issue, the latent projection (Catalina, Bürkner, and Vehtari 2021) solves the KL minimization problem in the predictive space of the latent predictors³ instead of in the predictive space of the original response values.

To this end, the latent predictor is assumed to have a Gaussian distribution, since it (i) constitutes a combination of predictor data and regression parameters which is often linear (in the parameters, but—less—often also in the predictor data) or at least additive (across the predictor terms) and (ii) has infinite support. Furthermore, the Gaussian distribution has the highest differential entropy over all distributions with two finite moments and infinite support (see, e.g., Cover and Thomas 1991). In some cases, e.g., for the probit link, the Gaussian distribution is even part of the original statistical model. In case of the logit link, the Gaussian distribution with a standard deviation of 1.6 approximates the logistic distribution (with a scale parameter of 1).

The assumption of a Gaussian distribution for the latent predictors makes things a lot easier because it allows us to make use of projpred’s traditional projection.

As illustrated by the Poisson example below, the latent projection can not only be used for families not supported by projpred’s traditional projection, but it can also be beneficial for families supported by it.

Implementation

To use the latent projection in projpred, the new argument latent of extend_family() needs to be set to TRUE. Since extend_family() is called by init_refmodel() which in turn is called by get_refmodel() (more precisely, by the get_refmodel() methods) which in turn is called at the beginning of the top-level functions project(), varsel(), and cv_varsel(), it is possible to pass latent = TRUE from such a top-level function down to extend_family() via the ellipsis (...). However, for the latent projection, we recommend to define the reference model object of class refmodel explicitly (as illustrated in the examples below) to avoid repetitions⁴.

After performing the projection (either as a stand-alone feature via project() or embedded in a variable selection via varsel() or cv_varsel()), the post-processing (e.g., the calculation of the performance statistics in summary.vsel()) can be performed on the original response scale. For this, extend_family() has gained several new arguments accepting R functions responsible for the inverse-link transformation from latent scale to response scale (latent_ilink), for the calculation of log-likelihood values on response scale (latent_ll_oscale), and for drawing from the predictive distribution on response scale (latent_ppd_oscale). For some families, these arguments have internal defaults implemented natively in projpred. These families are listed in the main vignette (section “Supported types of models”). For all other families, projpred either tries to infer a reasonable function internally (in case of latent_ilink) or uses a dummy function returning only NAs (in case of latent_ll_oscale and latent_ppd_oscale), unless the user supplies custom functions to these arguments. When creating a reference model object for a family of the latter category (i.e., lacking full response-scale support by default), projpred will throw messages stating whether (and which) features will be unavailable unless at least some of these arguments are provided by the user. Again, the ellipsis (...) can be used to pass these arguments from a top-level function such as cv_varsel() down to extend_family(). In the post-processing functions, response-scale analyses can usually be deactivated by setting the new argument resp_oscale to FALSE, with the exception of predict.refmodel() and proj_linpred() where the existing arguments type and transform serve this purpose (see the documentation).

Apart from the arguments mentioned above, extend_family() has also gained a new argument latent_y_unqs whose purpose is described in the documentation.

While the latent projection is an approximate solution to the KL divergence minimization problem in the original response space⁵, the augmented-data projection (Weber and Vehtari 2023) gives the exact⁶ solution for some non-ED families, namely those where the response distribution has finite support. However, the augmented-data projection comes with a higher runtime than the latent projection. The families supported by projpred’s augmented-data projection are also listed in the main vignette (again section “Supported types of models”).

Data

First, we generate a training and a test dataset with a Poisson-distributed response:

# Number of observations in the training dataset (= number of observations in
# the test dataset):
N <- 71
sim_poiss <- function(nobs = 2 * N, ncon = 10, ngrpPL = 4, nnoise = 39) {
  # Regression coefficients for continuous predictors:
  coefs_con <- rnorm(ncon)
  # Continuous predictors:
  dat_sim <- matrix(rnorm(nobs * ncon), ncol = ncon)
  # Start linear predictor:
  linpred <- 2.1 + dat_sim %*% coefs_con
  
  # Population-level (PL) categorical predictor:
  dat_sim <- data.frame(
    x = dat_sim,
    grpPL = gl(n = ngrpPL, k = nobs %/% ngrpPL, length = nobs,
               labels = paste0("grpPL", seq_len(ngrpPL)))
  )
  # Regression coefficients for the PL categorical predictor:
  coefs_catPL <- rnorm(ngrpPL)
  # Continue linear predictor:
  linpred <- linpred + coefs_catPL[dat_sim$grpPL]
  
  # Noise predictors:
  dat_sim <- data.frame(
    dat_sim,
    xn = matrix(rnorm(nobs * nnoise), ncol = nnoise)
  )
  
  # Poisson response, using the log link (i.e., exp() as inverse link):
  dat_sim$y <- rpois(nobs, lambda = exp(linpred))
  # Shuffle order of observations:
  dat_sim <- dat_sim[sample.int(nobs), , drop = FALSE]
  # Drop the shuffled original row names:
  rownames(dat_sim) <- NULL
  return(dat_sim)
}
set.seed(300417)
dat_poiss <- sim_poiss()
dat_poiss_train <- dat_poiss[1:N, , drop = FALSE]
dat_poiss_test <- dat_poiss[(N + 1):nrow(dat_poiss), , drop = FALSE]

Reference model

Next, we fit the reference model that we consider as the best model in terms of predictive performance that we can construct (here, we assume that we don’t know about the true data-generating process even though the dataset was simulated):

library(rstanarm)
# Number of regression coefficients:
( D <- sum(grepl("^x|^grpPL", names(dat_poiss_train))) )

[1] 50

# Prior guess for the number of relevant (i.e., non-zero) regression
# coefficients:
p0 <- 10
# Prior guess for the overall magnitude of the response values, see Table 1 of
# Piironen and Vehtari (2017, DOI: 10.1214/17-EJS1337SI):
mu_prior <- 100
# Hyperprior scale for tau, the global shrinkage parameter:
tau0 <- p0 / (D - p0) / sqrt(mu_prior) / sqrt(N)
# Set this manually if desired:
ncores <- parallel::detectCores(logical = FALSE)
### Only for technical reasons in this vignette (you can omit this when running
### the code yourself):
ncores <- min(ncores, 2L)
###
options(mc.cores = ncores)
refm_fml <- as.formula(paste("y", "~", paste(
  grep("^x|^grpPL", names(dat_poiss_train), value = TRUE),
  collapse = " + "
)))
refm_fit_poiss <- stan_glm(
  formula = refm_fml,
  family = poisson(),
  data = dat_poiss_train,
  prior = hs(global_scale = tau0, slab_df = 100, slab_scale = 1),
  ### Only for the sake of speed (not recommended in general):
  chains = 2, iter = 500,
  ###
  seed = 7286013, QR = TRUE, refresh = 0
)

Variable selection using the latent projection

We now run a variable selection with latent = TRUE in the get_refmodel() call (see section “Implementation” for why get_refmodel() is called explicitly here) to request the latent projection. Since we have a hold-out test dataset available, we can use varsel() with argument d_test instead of cv_varsel(). Furthermore, we measure the runtime to compare it to the traditional projection’s later:

library(projpred)

d_test_lat_poiss <- list(
  data = dat_poiss_test,
  offset = rep(0, nrow(dat_poiss_test)),
  weights = rep(1, nrow(dat_poiss_test)),
  ### Here, we are not interested in latent-scale post-processing, so we can set
  ### element `y` to a vector of `NA`s:
  y = rep(NA, nrow(dat_poiss_test)),
  ###
  y_oscale = dat_poiss_test$y
)
refm_poiss <- get_refmodel(refm_fit_poiss, latent = TRUE)

Since `<refmodel>$dis` will consist of only `NA`s, downstream analyses based on this reference model won't be able to use log predictive density (LPD) values on latent scale. Furthermore, proj_predict() won't be able to draw from the latent Gaussian distribution.

time_lat <- system.time(vs_lat <- varsel(
  refm_poiss,
  d_test = d_test_lat_poiss,
  ### Only for the sake of speed (not recommended in general):
  nclusters_pred = 20,
  ###
  nterms_max = 14,
  seed = 95930
))

print(time_lat)

   user  system elapsed 
  1.787   0.004   1.791 

The message telling that <refmodel>$dis consists of only NAs will not concern us here because we will only be focusing on response-scale post-processing.

In order to decide for a submodel size, we first inspect the plot() results:

( gg_lat <- plot(vs_lat, stats = "mlpd", deltas = TRUE) )

Although the submodels’ MLPDs seem to be very close to the reference model’s MLPD from a submodel size of 6 on, a zoomed plot reveals that there is still some discrepancy at sizes 6 to 10 and that size 11 would be a better choice (further down below in the summary() output, we will also see that on absolute scale, the discrepancy at sizes 6 to 10 is not negligible):

gg_lat + ggplot2::coord_cartesian(ylim = c(-10, 0.05))

Thus, we decide for a submodel size of 11:

modsize_decided_lat <- 11

This is also the size that suggest_size() would suggest:

suggest_size(vs_lat, stat = "mlpd")

[1] 11

Only now, after we have made a decision for the submodel size, we inspect further results from the variable selection and, in particular, the solution path:

smmry_lat <- summary(
  vs_lat,
  stats = "mlpd",
  type = c("mean", "se", "lower", "upper", "diff", "diff.se")
)
print(smmry_lat, digits = 3)

------
Response-scale family:

Family: poisson 
Link function: log 

------
Latent-scale family:

Family: gaussian 
Link function: identity 

------
Formula: .y ~ x.1 + x.2 + x.3 + x.4 + x.5 + x.6 + x.7 + x.8 + x.9 + x.10 + 
    grpPL + xn.1 + xn.2 + xn.3 + xn.4 + xn.5 + xn.6 + xn.7 + 
    xn.8 + xn.9 + xn.10 + xn.11 + xn.12 + xn.13 + xn.14 + xn.15 + 
    xn.16 + xn.17 + xn.18 + xn.19 + xn.20 + xn.21 + xn.22 + xn.23 + 
    xn.24 + xn.25 + xn.26 + xn.27 + xn.28 + xn.29 + xn.30 + xn.31 + 
    xn.32 + xn.33 + xn.34 + xn.35 + xn.36 + xn.37 + xn.38 + xn.39
Observations (training set): 71
Observations (test set): 71
Projection method: latent
Search method: L1
Maximum submodel size for the search: 14
Number of projected draws in the search: 1 (from clustered projection)
Number of projected draws in the performance evaluation: 20 (from clustered projection)

Performance evaluation summary (response scale) with `deltas = FALSE`:
 size solution_terms     mlpd       se    lower    upper      diff  diff.se
    0           <NA> -6281.54 2793.417 -9074.95 -3488.12 -6278.324 2.79e+03
    1            x.4 -4190.86 1866.360 -6057.22 -2324.50 -4187.650 1.87e+03
    2            x.6 -1431.70  553.630 -1985.33  -878.07 -1428.491 5.53e+02
    3           x.10  -471.56  203.944  -675.50  -267.61  -468.345 2.04e+02
    4            x.3  -447.33  250.031  -697.36  -197.30  -444.118 2.50e+02
    5            x.1  -166.19   84.658  -250.85   -81.53  -162.976 8.45e+01
    6            x.2   -65.27   31.185   -96.45   -34.08   -62.058 3.10e+01
    7            x.8  -112.69   58.348  -171.04   -54.35  -109.483 5.82e+01
    8            x.7  -114.85   62.044  -176.89   -52.81  -111.639 6.19e+01
    9            x.5  -125.54   75.647  -201.19   -49.89  -122.330 7.55e+01
   10          grpPL   -13.04    6.509   -19.55    -6.54    -9.834 6.37e+00
   11            x.9    -2.89    0.278    -3.17    -2.61     0.318 6.41e-02
   12           xn.6    -2.89    0.279    -3.17    -2.61     0.319 6.66e-02
   13          xn.29    -2.90    0.283    -3.18    -2.62     0.310 6.70e-02
   14          xn.14    -2.98    0.304    -3.28    -2.67     0.236 7.80e-02

On absolute scale (column mlpd), we see that submodel size 10 leads to an MLPD of -13.045, i.e., a geometric mean predictive density (GMPD; due to the discrete response family, the “density” values are probabilities here, so we will report the GMPD as percentage) of exp(-13.045) which is ca. 2e-04 % whereas size 11 leads to a GMPD of ca. 5.54 %. This is a considerable improvement from size 10 to size 11, so another justification for size 11. (Size 12 would have resulted in a GMPD of ca. 5.55 % which is only a small improvement compared to size 11.)

In the solution path up to the selected size of 11, we can see that in this case, projpred has correctly selected the truly relevant predictors first and only then the noise predictors. We can see this more clearly using the following code:

soltrms_lat <- solution_terms(vs_lat)
( soltrms_lat_final <- head(soltrms_lat, modsize_decided_lat) )

 [1] "x.4"   "x.6"   "x.10"  "x.3"   "x.1"   "x.2"   "x.8"   "x.7"   "x.5"  
[10] "grpPL" "x.9"  

We will skip post-selection inference here (see the main vignette for a demonstration of post-selection inference), but note that proj_predict() has gained a new argument resp_oscale and that analogous response-scale functionality is available in proj_linpred() (argument transform) and predict.refmodel() (argument type).

Variable selection using the traditional projection

We will now look at what projpred’s traditional projection would have given:

d_test_trad_poiss <- d_test_lat_poiss
d_test_trad_poiss$y <- d_test_trad_poiss$y_oscale
d_test_trad_poiss$y_oscale <- NULL
time_trad <- system.time(vs_trad <- varsel(
  refm_fit_poiss,
  d_test = d_test_trad_poiss,
  ### Only for the sake of speed (not recommended in general):
  nclusters_pred = 20,
  ###
  nterms_max = 14,
  seed = 95930
))

print(time_trad)

   user  system elapsed 
  2.701   0.004   2.705 

( gg_trad <- plot(vs_trad, stats = "mlpd", deltas = TRUE) )

smmry_trad <- summary(
  vs_trad,
  stats = "mlpd",
  type = c("mean", "se", "lower", "upper", "diff", "diff.se")
)
print(smmry_trad, digits = 3)

Family: poisson 
Link function: log 

Formula: y ~ x.1 + x.2 + x.3 + x.4 + x.5 + x.6 + x.7 + x.8 + x.9 + x.10 + 
    grpPL + xn.1 + xn.2 + xn.3 + xn.4 + xn.5 + xn.6 + xn.7 + 
    xn.8 + xn.9 + xn.10 + xn.11 + xn.12 + xn.13 + xn.14 + xn.15 + 
    xn.16 + xn.17 + xn.18 + xn.19 + xn.20 + xn.21 + xn.22 + xn.23 + 
    xn.24 + xn.25 + xn.26 + xn.27 + xn.28 + xn.29 + xn.30 + xn.31 + 
    xn.32 + xn.33 + xn.34 + xn.35 + xn.36 + xn.37 + xn.38 + xn.39
Observations (training set): 71
Observations (test set): 71
Projection method: traditional
Search method: L1
Maximum submodel size for the search: 14
Number of projected draws in the search: 1 (from clustered projection)
Number of projected draws in the performance evaluation: 20 (from clustered projection)

Performance evaluation summary with `deltas = FALSE`:
 size solution_terms    mlpd     se lower   upper    diff diff.se
    0           <NA> -3469.3 1593.1 -5062 -1876.2 -3466.1  1592.9
    1            x.6 -2545.4 1068.0 -3613 -1477.4 -2542.2  1067.8
    2          xn.12 -3560.3 1666.6 -5227 -1893.7 -3557.1  1666.4
    3          grpPL -3774.1 1778.0 -5552 -1996.1 -3770.8  1777.8
    4            x.4 -2438.1 1131.2 -3569 -1306.8 -2434.9  1131.1
    5           x.10  -959.8  488.4 -1448  -471.4  -956.6   488.3
    6            x.3  -821.7  518.8 -1340  -302.8  -818.4   518.7
    7          xn.11  -686.5  439.2 -1126  -247.3  -683.3   439.0
    8           xn.2 -1330.4  779.8 -2110  -550.6 -1327.2   779.7
    9          xn.22 -1220.0  722.4 -1942  -497.6 -1216.8   722.3
   10            x.2  -744.2  395.4 -1140  -348.8  -741.0   395.3
   11            x.1  -107.8   37.0  -145   -70.8  -104.6    36.8
   12          xn.23   -83.1   33.0  -116   -50.1   -79.9    32.8
   13          xn.16  -120.7   67.8  -189   -52.9  -117.5    67.7
   14            x.8   -93.0   45.2  -138   -47.8   -89.7    45.1

As these results show, the traditional projection takes longer than the latent projection, although the difference is rather small on absolute scale (which is due to the fact that the L1 search is already quite fast). More importantly however, the submodel size selection plot is much more unstable and the solution path reveals that several noise terms enter the solution path before truly relevant ones.

Conclusion

In conclusion, this example showed that the latent projection can be advantageous also for families supported by projpred’s traditional projection by improving the runtime and the stability of the variable selection, eventually also leading to better variable selection results.

One important point is that we have used the L1 search here. In case of the latent projection, a forward search would have given only slightly different results (in particular, a slightly smoother submodel size selection plot). However, in case of the traditional projection, a forward search would have given markedly better results (in particular, the submodel size selection plot would have been much smoother and all of the noise terms would have been selected after the truly relevant ones). Thus, the conclusions made for the L1 search here cannot be transmitted easily to the forward search.

Data

We will re-use the data generated above in the Poisson example.

Reference model

We now fit a reference model with the negative binomial distribution as response family⁸:

refm_fit_nebin <- stan_glm(
  formula = refm_fml,
  family = neg_binomial_2(),
  data = dat_poiss_train,
  prior = hs(global_scale = tau0, slab_df = 100, slab_scale = 1),
  ### Only for the sake of speed (not recommended in general):
  chains = 2, iter = 500,
  ###
  seed = 7304, QR = TRUE, refresh = 0
)

Variable selection using the latent projection

To run a variable selection with latent = TRUE, we now also need to specify more arguments which will be passed to extend_family()⁹:

refm_prec <- as.matrix(refm_fit_nebin)[, "reciprocal_dispersion", drop = FALSE]
latent_ll_oscale_nebin <- function(ilpreds, y_oscale,
                                   wobs = rep(1, length(y_oscale)), cl_ref,
                                   wdraws_ref = rep(1, length(cl_ref))) {
  y_oscale_mat <- matrix(y_oscale, nrow = nrow(ilpreds), ncol = ncol(ilpreds),
                         byrow = TRUE)
  wobs_mat <- matrix(wobs, nrow = nrow(ilpreds), ncol = ncol(ilpreds),
                     byrow = TRUE)
  refm_prec_agg <- cl_agg(refm_prec, cl = cl_ref, wdraws = wdraws_ref)
  ll_unw <- dnbinom(y_oscale_mat, size = refm_prec_agg, mu = ilpreds, log = TRUE)
  return(wobs_mat * ll_unw)
}
latent_ppd_oscale_nebin <- function(ilpreds_resamp, wobs, cl_ref,
                                    wdraws_ref = rep(1, length(cl_ref)),
                                    idxs_prjdraws) {
  refm_prec_agg <- cl_agg(refm_prec, cl = cl_ref, wdraws = wdraws_ref)
  refm_prec_agg_resamp <- refm_prec_agg[idxs_prjdraws, , drop = FALSE]
  ppd <- rnbinom(prod(dim(ilpreds_resamp)), size = refm_prec_agg_resamp,
                 mu = ilpreds_resamp)
  ppd <- matrix(ppd, nrow = nrow(ilpreds_resamp), ncol = ncol(ilpreds_resamp))
  return(ppd)
}
refm_nebin <- get_refmodel(refm_fit_nebin, latent = TRUE,
                           latent_ll_oscale = latent_ll_oscale_nebin,
                           latent_ppd_oscale = latent_ppd_oscale_nebin)

Defining `latent_ilink` as a function which calls `family$linkinv`, but there is no guarantee that this will work for all families. If relying on `family$linkinv` is not appropriate or if this raises an error in downstream functions, supply a custom `latent_ilink` function (which is also allowed to return only `NA`s if response-scale post-processing is not needed).

Since `<refmodel>$dis` will consist of only `NA`s, downstream analyses based on this reference model won't be able to use log predictive density (LPD) values on latent scale. Furthermore, proj_predict() won't be able to draw from the latent Gaussian distribution.

vs_lat_nebin <- varsel(
  refm_nebin,
  d_test = d_test_lat_poiss,
  ### Only for the sake of speed (not recommended in general):
  nclusters_pred = 20,
  ###
  nterms_max = 14,
  seed = 95930
)

Again, the message telling that <refmodel>$dis consists of only NAs will not concern us here because we will only be focusing on response-scale post-processing. The message concerning latent_ilink can be safely ignored here (the internal default based on family$linkinv works correctly in this case).

Again, we first inspect the plot() results to decide for a submodel size:

( gg_lat_nebin <- plot(vs_lat_nebin, stats = "mlpd", deltas = TRUE) )

Again, a zoomed plot is more helpful:

gg_lat_nebin + ggplot2::coord_cartesian(ylim = c(-2.5, 0.25))

Although the submodels’ MLPDs approach the reference model’s already from a submodel size of 8 on (taking into account the uncertainty indicated by the error bars), the curve levels off only from a submodel size of 11 on. For a more informed decision, the GMPD could be taken into account again, but for the sake of brevity, we will simply decide for a submodel size of 11 here:

modsize_decided_lat_nebin <- 11

This is not the size that suggest_size() would suggest, but as mentioned in the main vignette and in the documentation, suggest_size() provides only a quite heuristic decision (so we stick with our manual decision here):

suggest_size(vs_lat_nebin, stat = "mlpd")

[1] 8

Only now, after we have made a decision for the submodel size, we inspect further results from the variable selection and, in particular, the solution path:

smmry_lat_nebin <- summary(
  vs_lat_nebin,
  stats = "mlpd",
  type = c("mean", "se", "lower", "upper", "diff", "diff.se")
)
print(smmry_lat_nebin, digits = 3)

------
Response-scale family:

Family: neg_binomial_2 
Link function: log 

------
Latent-scale family:

Family: gaussian 
Link function: identity 

------
Formula: .y ~ x.1 + x.2 + x.3 + x.4 + x.5 + x.6 + x.7 + x.8 + x.9 + x.10 + 
    grpPL + xn.1 + xn.2 + xn.3 + xn.4 + xn.5 + xn.6 + xn.7 + 
    xn.8 + xn.9 + xn.10 + xn.11 + xn.12 + xn.13 + xn.14 + xn.15 + 
    xn.16 + xn.17 + xn.18 + xn.19 + xn.20 + xn.21 + xn.22 + xn.23 + 
    xn.24 + xn.25 + xn.26 + xn.27 + xn.28 + xn.29 + xn.30 + xn.31 + 
    xn.32 + xn.33 + xn.34 + xn.35 + xn.36 + xn.37 + xn.38 + xn.39
Observations (training set): 71
Observations (test set): 71
Projection method: latent
Search method: L1
Maximum submodel size for the search: 14
Number of projected draws in the search: 1 (from clustered projection)
Number of projected draws in the performance evaluation: 20 (from clustered projection)

Performance evaluation summary (response scale) with `deltas = FALSE`:
 size solution_terms    mlpd      se   lower   upper     diff  diff.se
    0           <NA> -386.10 157.030 -543.13 -229.07 -382.421 156.8118
    1            x.4  -98.66  30.140 -128.80  -68.52  -94.985  29.8979
    2            x.6  -20.54   6.219  -26.76  -14.33  -16.867   6.0514
    3           x.10   -8.52   1.215   -9.73   -7.30   -4.839   1.0220
    4            x.3   -5.68   0.720   -6.40   -4.96   -2.003   0.4969
    5            x.1   -4.59   0.568   -5.15   -4.02   -0.908   0.3726
    6            x.2   -4.07   0.470   -4.54   -3.61   -0.397   0.2686
    7            x.8   -3.87   0.410   -4.28   -3.46   -0.194   0.1492
    8            x.5   -3.79   0.403   -4.19   -3.38   -0.108   0.1325
    9          xn.29   -3.78   0.403   -4.18   -3.38   -0.101   0.1332
   10            x.7   -3.66   0.384   -4.04   -3.28    0.018   0.0967
   11          grpPL   -3.43   0.343   -3.77   -3.09    0.247   0.0435
   12          xn.31   -3.43   0.342   -3.77   -3.08    0.252   0.0409
   13            x.9   -3.42   0.340   -3.76   -3.08    0.263   0.0374
   14           xn.2   -3.42   0.341   -3.77   -3.08    0.254   0.0388

As we can see from the solution path, our selected 11 predictor terms lack one truly relevant predictor (x.9) and include one noise term (xn.29). More explicitly, our selected predictor terms are:

soltrms_lat_nebin <- solution_terms(vs_lat_nebin)
( soltrms_lat_nebin_final <- head(soltrms_lat_nebin,
                                  modsize_decided_lat_nebin) )

 [1] "x.4"   "x.6"   "x.10"  "x.3"   "x.1"   "x.2"   "x.8"   "x.5"   "xn.29"
[10] "x.7"   "grpPL"

Again, we will skip post-selection inference here (see the main vignette for a demonstration of post-selection inference).

Conclusion

This example demonstrated how the latent projection can be used for those families which are neither supported by projpred’s traditional nor by projpred’s augmented-data projection. (In the future, this vignette will be extended to demonstrate how both, latent and augmented-data projection, can be applied to those non-ED families which are supported by both.)