What has wikipedia to say about ?

In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability. It is also called the percent-point function or inverse cumulative distribution function.

For example, the cumulative distribution function of exponential ($\lambda$) (i.e. intensity \(\lambda\) and expected value (mean) \(\frac{1}{\lambda}\)) is:

A normal function isn’t so normal

The normal density function is:

$$ \large f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp^{-\frac{(x - \mu)^2}{(2 \sigma^2)}} $$

It doesn’t make sense to calculate the probability for a single value in a continuous probability function, it is by definition zero, but you can calculate relative likelihoods (heights). dnorm simply gives the value of the function for a given x, not the area under the curve for that x (which is basically nothing for a single value). To find the density (height) for a single x value on the normal distribution, use dnorm() in the following way (here each x value is treated as separate and vectorized over),

Linear mixed models are widely used in Agriculture and Plant Breeding, as of recent. With access to genotype data high resolution phenotype data, it has become more of a requirement to use this family of model.

Mixed models allow for experimental (design or outcome) variables’ parameter estimates to have probabilistic distributions – most commonly normal – with opportunity to specify different variance-covariance components among the levels of those variables. In this post, I wish to discuss on some of the popular mixed modeling tools and techniques in the R community with links and discussion of the concepts surrounding variations of modeling techniques.

Many find genetics, as a field of science on its own, charming. Many more are excited to learn about the science that fits seamlessly into complexity driven life of organisms, providing explanation for natural phenomena at both micro-evolutionary and macro-evolutionary scales. But only few find fascination with its deep running concepts, going down to more fundamental physical theories. This article tries to at least expose, if not laid satisfying argument, to some of fundamental questions in genetics concerning nature of code (mostly chemical behaviour). In particular, following are some of the questions I plan to touch upon (Credit goes to a student of mine who posed these questions one evening and left me pondering on details):

Take a grid and serpentine it row-wise or column-wise

This fn joins two matrices alternately columnwise, which is why this is the source of inspiration for generating serpentine design.

alternate.cols <- function(m1, m2) {
  cbind(m1, m2)[, order(c(seq(ncol(m1)), seq(ncol(m2))))]
}

A custom function to create a serpentine design in whatever fashion specified:

serpentine <- function(x, columnwise=TRUE){
  if (columnwise) {
    odd <- x[, seq(1, by=2, length.out = ncol(x)/2)] # odd x
    rev_even <- x[, seq(from = 2, 
                        by=2, 
                        length.out = (ifelse((ncol(x)%%2 != 0), 
                                             ((ncol(x)/2)-1), 
                                             (ncol(x)/2))))][seq(dim(x)[1],1),] # or, even[rev(1:nrow(x)),] # reversed even x
    alternate_cbind <-  cbind(odd, rev_even)[, order(c(seq(ncol(odd)), 
                                                       seq(ncol(rev_even))))]
    return(alternate_cbind)}
  else {
    odd <- x[seq(1, by=2, length.out = nrow(x)/2),] # odd x
    rev_even <- x[seq(from = 2, by=2, length.out = (ifelse((nrow(x)%%2 != 0), 
                                                           ((nrow(x)/2)-1), 
                                                           (nrow(x)/2)))), ][, seq(dim(x)[2],1)] # or, even[, rev(1:ncol(x))] # reversed even x
    alternate_rbind <-  rbind(odd, rev_even)[order(c(seq(nrow(odd)), 
                                                     seq(nrow(rev_even)))), ]
    return(alternate_rbind)
  }
}

Let’s see the function in action