# Concepts

## What is a weitrix?

A “weitrix” is a SummarizedExperiment object (or subclass thereof) with two assays, one containing the actual measurements and the other the associated weights. A “weitrix” metadata entry stores the names of these assays. There are several ways to construct a weitrix:

• as_weitrix(x, weights) constructs a weitrix, where x is a matrix of measurements and weights is a corresponding matrix of weights.

• A SummarizedExperiment can be marked as a weitrix using bless_weitrix. This requires specifying the names of the two assays to be used.

• Anything the limma package knows how to work with can be converted to a weitrix using as_weitrix. (Most functions in the weitrix package will attempt this conversion automatically.)

The usual SummarizedExperiment accessor functions can be used: assay rowData colData metadata

Additionally, the blessed assays be accessed using: weitrix_x weitrix_weights

## Rows and columns

weitrix follows the Bioconductor convention of placing features in rows and units of observation (samples, cells) in columns. This is the transpose of the normal R convention!

## Weights

A weight determines the importance of an observation. One way of thinking about a weight is that it is as if this one observation is actually an average over some number of real observations. For example if an observation is an average over some number of reads, the number of reads might be used as the weight.

The choice of weight is somewhat arbitrary. You can use it simply to tell model fitting (such as that in weitrix_components) what to pay most attention to. It’s better to pay more attention to more accurately measured observations.

A weight of 0 indicates completely missing data.

The concept of weights used in this package is the same as for weights specified to the lm function or the limma lmFit function.

Weights can be calibrated per row so they are one over the variance of a measurement. When testing using limma, a calibrated weitrix will produce better results than an uncalibrated one. A trend line or curve can be fitted to dispersions for each row, based on known predictors. This is similar to the trend option in limma’s eBayes function, but allows other predictors beyond the row average.

Some examples of possible measurements and weights:

• poly(A) tail length is measured for a collection of reads. The measurement is the average log tail length (number of non-templated “A”s), and the weight is the number of reads. See poly(A) tail length example.

• Several different polyadenylation sites are observed per gene. A “shift” score is assigned to each site: the proportion of all reads upstrand of the site minus the proportion of all reads downstrand of it. The measurement is an average over the site score for each read, and the weight is the number of reads. See alternative polyadenylation example.

• A read aligning to a gene is assigned to a particular exon. For a particular sample, gene, and exon, the measurement is the proportion of reads aligning to that exon out of all reads aligning to that gene, and the weight is the total reads aligning to the gene. counts_proportions can be used to construct an approporiate weitrix. Some further calibration is possible based on the average proportion in each exon over all samples (a somewhat rough-and-ready strategy compared to using a GLM).

## Linear models and components of variation

An important feature of the weitrix package is the ability to extract components of variation, similar to PCA. The novel feature compared to PCA is that this is possible with unevenly weighted matrices or matrices with many missing values. Also, by default components are varimax rotated for improved interpretability.

This is implemented as an extension of the idea of fitting a linear model to each row. It is implemented in weitrix_components, a major workhorse function in this package. A pre-specified design matrix can be given (by default this contains only an intercept term), and then zero or more additional components requested. The result is:

• a “col” matrix containing the specified design matrix and additionally the “scores” of novel components of variation for each sample.

• a “row” matrix containing for each row estimated coefficients and additionally the “loadings” of novel components of variation.

These two matrices can be multiplied together to approximate the original data. This will impute any missing values, as well as smoothing existing data.

The example vignettes contain examples of how this function is used.

## Dispersion

After constructing a model of systematic variation in a weitrix using weitrix_components, possibly with several components discoved from the data, each row’s residual “dispersion” can be estimated with weitrix_dispersion.

The term “dispersion” as used in this package is similar to variance but taking weights into account. For example if weights represent numbers of reads, it is the read-level variance. After calibration of a weitrix, it is also relative to the calibrated trend.

For a particular row with measurements $$y$$, weights $$w$$, design matrix $$X$$ (including discovered component scores), fitted coefficients $$\hat\beta$$, and residual degrees of freedom $$\nu$$ (number of non-zero weights minus number of columns in $$X$$), the dispersion $$\sigma^2$$ is estimated with:

$\hat\varepsilon = y-X\hat\beta$

$\hat\sigma^2 = {1 \over \nu} \sum_i w_i \hat\varepsilon_i^2$

Similarly where $$R^2$$ values are reported, these are proportions of weighted variation that have been explained.

## Use with limma and topconfects

A weitrix can be converted to an EList object for use with limma: weitrix_elist

The \$col matrix of a Components may be used as a design matrix for differential analysis with limma. Warning: This may produce liberal results, because the design matrix is itself uncertain and this isn’t taken into account. Use this with caution.

When there are a small number of columns (such as from a bulk RNA-Seq experiemtn), weights may be “calibrated” to a trend line before testing using limma, in order to make limma’s Empirical Bayes squeezing of dispersions more effective. This is done using weitrix_calibrate_trend, which scales the weights of each row to eliminate any trends with known predictors of the dispersion. This is very similar to the “trend” option in limma::eBayes, but more flexible. limma::eBayes can account for a trend curve relative to the average expression level. weitrix_calibrate_trend can use a different predictor of the dispersion, or combine several predictors at once.

When there are many columns (such as from a single cell experiment), the dispersion can be estimated accurately and such considerations are irrelevant. Rows can be tested individually by any means that can use weights, such as lm. limma is convenient and there’s no harm in using it, but also no advantage. If calibration is still desired, dispersions can be estimated with weitrix_dispersions and calibrated directly with weitrix_calibrate.

Calibrating weights does not change estimates of coefficients for each row (the “row” matrix). It may have some effect on the components of variation discovered.

Having tested each row, there is then the question of which results are most interesting. In happy data which has been collected from well behaved organisms (eg all the same strain or breed, grown under controlled conditions) and which has all been measured to the same accuracy, p-values can be abused as a proxy for effect size without much harm. This breaks down when different measurements have very different weights, and different rows have different dispersions. The common default of ordering results by p-value will tend to highly rank results that have been measured to high accuracy or have low variability, and may down-rank other results with much larger effect sizes. For this reason, we recommend following limma::lmFit with the use of topconfects::limma_confects from our topconfects package, to find large confident effect sizes.

# Practical details

## Big datasets

weitrix can use DelayedArray assays. Functions that produce weitrices will used DelayedArray output assays if given DelayedArray input assays.

weitrix will attempt to perform calculations blockwise in parallel. weitrix tries to use DelayedArray defaults. Adjust with DelayedArray::setRealizationBackend, DelayedArray::setAutoBlockSize, DelayedArray::setAutoBPPARAM.

It is always possible to convert an assay back to a normal R matrix with as.matrix.

Set the DelayedArray realization backend to HDF5Array if weitrices will be too big to fit in memory uncompressed. The HDF5Array backend stores data on disk, in temporary files.

If using DelayedArray::setRealizationBackend("HDF5Array") you may also want to set HDF5Array::setHDF5DumpDir.

A weitrix can be permanently stored to disk using HDF5Array::saveHDF5SummarizedExperiment.

Example setup:

library(DelayedArray)
library(HDF5Array)

# Store intermediate results in a directory called __dump__
# You may need to clean up this directory manually
setRealizationBackend("HDF5Array")
setHDF5DumpDir("__dump__")

## Parallelism fine tuning

### BiocParallel problems

Parallel processing in R and Bioconductor remains finicky but is necessary for large datasets. weitrix uses DelayedArray’s default parallel processing as its own default, see DelayedArray::getAutoBPPARAM() and DelayedArray::setAutoBPPARAM().

If weitrix hangs, try running it with serial processing:

DelayedArray::setAutoBPPARAM( BiocParallel::SerialParam() )

### OpenBLAS

If using parallel processing, multi-threaded linear algebra libraries will just slow things down. If you have installed OpenBLAS you may need to disable multi-threading. You can see the BLAS R is using in sessionInfo(). Disable multi-threading using the RhpcBLASctl package: