It is often very useful to practise mathematics by automatically generated exercises. One approach is multiple choice quizzes (MCQ), but it turns out to be fairly difficult to generate authentic wrong answers. Instead, we want the user to input the answer and be able to parse the answer and check whether this is the correct answer. There are many fun challenges in this, e.g. to verify that 2 is equal to 1 + 1 (as text strings the two are different, but mathematically they are equal, at least to a convenient approximation in this case).

It is with great pleasure that I can announce that Ryacas version 1.1.0 has now been accepted into Journal of Open Source Software and same version released to CRAN. (The source code is available at https://github.com/mikldk/ryacas/.)
I already wrote about Ryacas many times before. I will refer you to the “Getting started” and “The high-level (symbol) interface” vignettes or one of the others available at the CRAN page or the package’s website.

It is with great pleasure that I can announce that Ryacas version 1.0.0 is now released to CRAN (https://cran.r-project.org/package=Ryacas). I wish to thank all co-authors: Rob Goedman, Gabor Grothendieck, Søren Højsgaard, Grzegorz Mazur, Ayal Pinkus.
It means that you can install the package by (possible after binaries have been built):
install.packages("Ryacas") Followed by:
library(Ryacas) (The source code is available at https://github.com/mikldk/ryacas/.)
Now you have the yacas computer algebra system fully available!

In a recent blog post I tried to get yacas to solve a system of polynomial equations. Unfortunately it could not do that, so I solved it numerically instead. Now it is possible – together with many other systems of polynomial equations thanks to fixing a small error in yacas. It has now been fixed, also in Ryacas (development version), so hurry up and update Ryacas to the latest version 0.

Finding prediction intervals (for future observations) is something different than finding confidence intervals (for unknown population parameters).
Here, I demonstrate one approach to doing so.
First we load the library and simulate some data:
library(mgcv) set.seed(1) dat <- gamSim(eg = 1, n = 400, dist = "normal", scale = 2) ## Gu & Wahba 4 term additive model The simulated in dat contains the “truth” in the f variables:

Let \(\rho_{XY}\) be the correlation between the stochastic variables \(X\) and \(Y\) and similarly for \(\rho_{XZ}\) and \(\rho_{YZ}\). If we know two of these, can we say anything about the third?
In a recent blog post I dealt with the problem mathematically and I used the concept of a partial correlation coefficient. Here I will take a simulation approach.
First z is simulated. Then x and y is simulated based on z in a regression context with a slope between \(-1\) and \(1\).

Update Aug 10, 2019: I wrote a new blog post about the same as below but using a simulation approach.
Update Aug 27, 2019: Minor change in how equations are solved (from version 0.9.0.9122).
Let \(\rho_{XY}\) be the correlation between the stochastic variables \(X\) and \(Y\) and similarly for \(\rho_{XZ}\) and \(\rho_{YZ}\). If we know two of these, can we say anything about the third?
Yes, sometimes, but not always.

Update Aug 27, 2019: I wrote a new blog post showing that Ryacas can now solve the system of equations directly without using optim().
As you may know, I am maintaining the Ryacas package (with online documentation) for doing symbolic mathematics (and other stuff) in R using the yacas software (with online documentation).
Søren Højsgaard and I have been preparing a new major release of Ryacas (see blog post on it).

Update Aug 27, 2019: Minor change in how equations are solved (from version 0.9.0.9122).
We (Søren Højsgaard and I) are preparing a new major release of Ryacas (GitHub). It will have a new interface to yacas that is thinner, cleaner and more robust. It relies on yacas’s RForm() function rather than an OpenMath XML interface.
It also means that the API has changed in Ryacas: new functions are introduced and old ones deprecated.

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