poly | Milner type system with extensible records

You can set the breaks of geom_contour_filled to start at your 75th centile, and make the NA value of scale_fill_manual transparent. You also need to draw in the default contour lines:

When you fit a stat_smooth() (or geom_smooth()) curve you are essentially creating data points i.e. you are generating a list of coordinates that the line will follow. When you changed the y axis limits, some of these coordinates ended up outside the limits and were removed. So, it isn't your original 16 points that are outside your limits, it is the 'calculated' coordinates for the geom_smooth() line.

Here is an example showing the new 'internal' data created by stat_smooth() in the ggplot object ("p2"):

As a follow-up to your comment, I have prepared a reprex so that you can test the code. It should work...

If it doesn't work, here are some suggestions:

1. Make sure all your libraries are up to date, especially tidyverse, mapview and sf. On my side, I run the code with the following versions: tidyverse 1.3.1, mapview 2.10.0 and sf 1.0.6
2. Close all open documents in your R session and close R. Reopen a new R session and open only the file that contains the code to test.
3. Load only the libraries needed to run the code.

Hope this helps. I'm crossing my fingers that these suggestions will unstuck you.

Reprex

• Your data

I have 6 sets of code for 16, 32, 64 bit crc, non-reflected and reflected here. The code is setup for Visual Studio. Comments have been added to the constants which were missing from Intel's github site.

https://github.com/jeffareid/crc

32 bit non-relfected is here:

https://github.com/jeffareid/crc/tree/master/crc32f

You'll need to change the polynomial in crc32fg.cpp, which generates the constants. The polynomial you want is actually:

note: although your question is about the lmer() function, this answer also applies to lm() and other R functions that fit linear models.

The way that coefficient estimates from linear models in R are presented can be confusing. To understand what's going on, you need to understand how R fits linear models when the predictor is a factor variable.

Before we look at factor variables, let's look at the more straightforward situation where the predictor is continuous. In your example dataset, one of the predictors is wind speed (continuous variable). The estimated coefficient is about -0.35. It's easy to interpret this: averaged across the other predictors, for every increase of 1 km/h in wind speed, your response value is predicted to decrease by 0.35.

But what about if the predictor is a factor? A categorical variable cannot increase or decrease by 1. Instead it can take several discrete values. So what the lmer() or lm() function does by default is automatically code your factor variable as a set of so-called "dummy variables." Dummy variables are binary (they can take values of 0 or 1). If the factor variable has n levels, you need n-1 dummy variables to encode it. The reference level or control group acts like an intercept.

In the case of your habitat variable, there are only 2 levels so you have only 1 dummy variable which will be 0 if habitat is not Forest and 1 if it is Forest. Now we can interpret the coefficient estimate of -68.8: the average value of your response is expected to be 68.8 less in forest habitat relative to the reference level of grassland habitat. You don't need a second dummy variable for grassland because you only need to estimate the one coefficient to compare the two habitats.

If you had a third habitat, let's say wetland, there would be a second dummy variable that would be 0 if not wetland and 1 if wetland. The coefficient estimate there would be the expected difference between the value of the response variable in wetland habitat compared to grassland habitat. Grassland will be the reference level for all the coefficients.

Now to directly address your question of why habitatForest is the coefficient name.

Because by default no reference level or control group is specified, the first one in the factor level ordering becomes the reference level to which all other levels are compared. Then the coefficients are named by appending the variable's name to the name of the level being compared to the reference level. Your factor is ordered with grassland first and forest second. So the coefficient is the effect of the habitat being forest habitat, compared to the reference level, which is grassland in this case. If you switched the habitat factor level ordering, Forest would be the reference level and you would get habitatGrassland as the coefficient instead. (Note that default factor level ordering is alphabetical, so without specifically ordering the factor levels as you seem to have done, Forest would be the reference level by default).

Incidentally, the two links you give in your question (guides to mixed models from Phillip Alday and Tufts) do in fact have the same kind of output as you are getting. For example in Alday's tutorial, the factor recipe has 3 levels: A, B, and C. There are two coefficients in the fixed effects summary, recipeB and recipeC, just as you would expect from dummy coding using A as reference level. You may be confusing the fixed effects summary with the ANOVA table presented elsewhere in his post. The ANOVA table does only have a single line for recipe which gives you the ratio of variance due to recipe (across all its levels) and the total variance. So that would only be one ratio regardless of how many levels recipe has.

This is not the place for a full discussion of contrast coding in linear models in R. The dummy coding (which you may also see called one-hot encoding) I described here is just one way to do it. These resources may be helpful:

• Documentation for the base R function contrasts()
• description of different types of categorical variable coding in R from UCLA IDRE Stats
• Marissa Barlaz' tutorial on R contrast coding

Computing with a square root of -2 amounts to working modulo the polynomial t^2 + 2.

The function power_mod can be used for that.

Instead of first powering and then reducing modulo t^2 + 2, which would be wasteful, it performs the whole powering process modulo t^2 + 2, which is a lot more efficient.

Here are two ways to write the (same) computation.

Before considering the problem of the translation, there is a more important aspect that has to be considered: if you want to create a transformation based on the center of a polygon, you must find that center. That point is called centroid, the geometric center of any polygon.

While there are simple formulas for all basic geometric shapes, finding the centroid of a (possibly irregular) polygon with an arbitrary number of vertices is a bit more complex.

Using the arithmetic mean of vertices is not a viable option, as even in a simple square you might have multiple points on a single side, which would move the computed "center" towards those points.

The formula can be found in the Wikipedia article linked above, while a valid python implementation is available in this answer.

I modified the formula of that answer in order to accept a sequence of QPoints, while improving readability and performance, but the concept remains the same:

Thanks to @BenBolker and @russ-lenth for confirming that the issue is related to the missing terms attribute "predvars" in the GLS object, which provides the fitted coefficients for poly. Notice how this works in an LM framework (original post) and the attribute is there (see also ?makepredictcall). Note that this can have potential implications for prediction.

I think you're mistaken about the pivot part because there's no pivot happening here. This can be achieved with IF() or CASE expression functions added to a very basic MySQL syntax. So, this: