BUG: fix scatter plots and regression lines in Moran plots

[martinfleis]

Fixes #178

The regressions were all just wrong. I have fixed that and manually verified that all of them are in line with what GeoDa is producing.

There was also wrong description of axes in the notebook which I believe is a result of a confusion of using x and y as variable names and passing those to bivariate Moran as y=x, x=y. So I changed those to explicitly mention the variable name.

Fixing here because it may take a bit before we'll be able to steer people towards plotting within esda and we should not produce erroneous outputs.

I suggest doing a patch release with this and #185 as soon as this is merged.

[knaaptime]

nice; good find. It still feels mildly strange that the plots occasionally miss the origin

but the implementation looks right now 🤷‍♂️

[martinfleis]

That's the same in GeoDa. And given this is a regression model on real world data, it is not even surprising.

[knaaptime]

well, since most moran plots do hit the origin, it is kinda surprising :)

[knaaptime]

plus, like, conceptually, I think of that line as originating at 0,0 in a moran plot... We intentionally center the scatter on z=0,0 because the quadrants are meaningful. It's the slope of the line that's conceptually meaningful, not the intercept in this plot.

it almost feels reasonable to me that we'd shift the intercept to be 0 in the event that it's not there already

[martinfleis]

I'll let this to decide to someone more knowledgable :). Now I'm satisfied that we match GeoDa. And I think that spdep is following the same model, but I'm too lazy to test that so guessing from the code.

[knaaptime]

agree. still curious about this issue though. I wonder if @sjsrey or @ljwolf have thoughts? Is the 'natural' intercept of a moran plot ==0? It feels to me like it is

[knaaptime]

tldr: the reason the intercept is not zero is because the y-axis is lag(z(y)), not z(lag(y)), so zero on the y-axis is not the mean of the distribution on the y-axis--it's the mean of the distribution on the x-axis. So when we classify an observation into a hotspot quadrant (even along the vertical axis for Wy), we're using mean(y) to define high/low, (not mean(lag(y))). I.e. a "large lag value" means above the mean of y, not above the mean of lag(y).Geoda and the original papers are clear about this, though I don't think its always obvious from the plots

so the intercept will move around based on whether lag(z(y)) is above or below z(y). I'm square on what's happening, though I still think you could argue for z(lag(y)) on the y-axis :)