Is there any regression allowing correlated values other than linear mixed model? In genomics, nearby SNPs are in LD (correlated) with each other. It violates the independence assumption in linear models and are being treated as random effect in linear mixed model in a method estimating the degree that phenotype is influenced by genotype (i.e., estimating heritability in GCTA). Random effect is a grouping variable, hence can only be integer. This is possible in genotypic estimation because there are only three genotypes considered (e.g., AA, AT, TT).
But what if they are correlated continuous variables? They are correlated so linear regression can't be used. They are not integer so can't be treated as random effect in linear mixed model.
 A: With correlated data, the idea is to assume or identify from the data the form of the correlation and take that into account in the analysis. That's what a linear mixed model does: it typically assumes a Gaussian distribution among cases of intercepts and possibly slopes, and fits the data accordingly.
Linear mixed models aren't the only way to deal with correlated observations within individuals or groups. Classic multivariate analysis of variance is one example, although there are more flexible tools available. Frank Harrell has a nice table comparing different approaches in his chapter on generalized least squares.
What you have in mind seems to be a single set of correlated continuous values rather than the correlations that occur within individuals or groups. That's what time series analysis accomplishes. Rob Hyndman and George Athanasopoulos have a superb online reference explaining the principles. Similar methods are extended into multiple dimensions for studies in geography and ecology.
