rho | A self documenting DSL built on http4s

It turns out the complicated code of laplacian_2D_array can be simplified as the following implementation:

This appears to be the result of some aggressive optimization by the compiler. With some algebra, you can determine that as of this line:

You can use the "filled" version of each shape (use for example ggpubr::show_point_shapes() to see a list), so here 22, 21, 24 and 23.

This is a hard problem to solve in general, and even a fairly hard problem to solve in specific cases.

I think the difference arose between version 1.1.27.1 and 1.1.28, probably from this NEWS item:

construction of interacting factors (e.g. when f1:f2 or f1/f2 occur in random effects terms) is now more efficient for partially crossed designs (doesn't try to create all combinations of f1 and f2) (GH #635 and #636)

My guess is that this changes the ordering of the components in the Z matrix, which in turn means that results of various linear algebra operations are not identical (e.g. floating point arithmetic is not associative, so while binary addition is commutative (a + b == b + a), left-to-right evaluation of a sum may not be the same as right-to-left evaluation ((a+b) + c != a + (b+c)) ...)

My attempt at reproducing the problem uses the same version of R ("under development 2022-02-25 r81818") and compares only lme4 package versions 1.18.1 with 1.1.28.9000 (development); any upstream packages such as Rcpp, RcppEigen, Matrix use the same versions. (I had to backport a few changes from the development version of lme4 to 1.1.18.1 to get it to install under the most recent version of R, but I don't think any of those modifications would affect numerical results.)

I did the comparison by installing different versions of the lme4 package before running the code in a fresh R session. My results differed between versions 1.1.18.1 and 1.1.28 less than yours did (both fits were singular, and the relative differences in the theta estimates were of the order of 2e-7 — still greater than your desired 1e-8 tolerance but much smaller than 1e-4 ...)

The results from 1.1.18.1 and 1.1.27.1 were identical.

• Q1: Why are your results more different between versions than mine?
  • in general/anecdotally, numerical results on Windows are slightly more unstable/differ more from other platforms
  • there are more differences between your two test platforms than among mine: R version, upstream packages (Matrix/Rcpp/RcppEigen/minqa), possibly the compiler versions and settings used to build everything [all of which could make a difference]
• Q2: how should one deal with this kind of problem?
  • as a minor frame challenge, why (other than not understanding what's going on, which is a perfectly legitimate reason to be concerned) does this worry you? The differences in the results are way smaller than the magnitude of statistical uncertainty, and differences this large are also likely to occur across different platforms (OS/compiler version/etc.) even for otherwise identical environments (versions of R, lme4, and other packages).
  • you could revert to version 1.1.27.1 for now ...
  • I do take the differences between 1.1.27.1 as a bug, of sorts — at the very least it's an undocumented change in the package. If it were sufficiently high-priority I could investigate the code changes described above and see if there is a way to fix the problems they addressed without breaking backward compatibility (in theory this should be possible, but it could be annoyingly difficult ...)

For improving the speed, you can see Numba, which is useable if you use NumPy a lot but not every code can be used with Numba. Apart from that, the formulation of the equation system is confusing. You are solving 3 equations and adding the result to a single dydt list by 3 elements each. You can simply create three lists, solve each equation and add them to their respective list. For this, you need to re-write my_system as:

Here's how far I've gotten:

• the exponential rates are supposed to be specified as logs of the rates (to make sure that the rates themselves stay positive, i.e. that we have exponential decay curves rather than growth curves)
• I simplified the model significantly, taking out the random effects in T1 and T2.

You need to convert DF2 into long format too. Then you can use the p values as the labels by passing DF2 as an argument to a different geom_text call. You can also map the fontface aesthetic to the the p values to get the bold font for "significant*" p values. Finally, you need to nudge the p value labels down a bit.

Find the center of the tile:

You can use a 2D FFT to find the general direction in which the crop is aligned (as proposed by mozway in the comments). The idea is that the general direction can be easily extracted from centred beaming rays appearing in the magnitude spectrum when the input contains many lines in the same direction. You can find more information about how it works in this previous post. It works directly with the input image, but it is better to apply the Gaussian + Canny filters.

Here is the interesting part of the magnitude spectrum of the filtered gray image:

The main beaming ray can be easily seen. You can extract its angle by iterating over many lines with an increasing angle and sum the magnitude values on each line as in the following figure:

Here is the magnitude sum of each line plotted against the angle (in radian) of the line:

Based on that, you just need to find the angle that maximize the computed sum.

Here is the resulting code: