  We present the development of a semi-supervised regression method using
variational autoencoders (VAE), which is customized for use in soft sensing
applications. We motivate the use of semi-supervised learning considering the
fact that process quality variables are not collected at the same frequency as
other process variables leading to many unlabelled records in operational
datasets. These unlabelled records are not possible to use for training quality
variable predictions based on supervised learning methods. Use of VAEs for
unsupervised learning is well established and recently they were used for
regression applications based on variational inference procedures. We extend
this approach of supervised VAEs for regression (SVAER) to make it learn from
unlabelled data leading to semi-supervised VAEs for regression (SSVAER), then
we make further modifications to their architecture using additional
regularization components to make SSVAER well suited for learning from both
labelled and unlabelled process data. The probabilistic regressor resulting
from the variational approach makes it possible to estimate the variance of the
predictions simultaneously, which provides an uncertainty quantification along
with the generated predictions. We provide an extensive comparative study of
SSVAER with other publicly available semi-supervised and supervised learning
methods on two benchmark problems using fixed-size datasets, where we vary the
percentage of labelled data available for training. In these experiments,
SSVAER achieves the lowest test errors in 11 of the 20 studied cases, compared
to other methods where the second best gets 4 lowest test errors out of the 20.
