Now, I’ve decided to break down the Barton and Schruben (1993) method into two parts as you need to know what an empirical cumulative distribution function (eCDF) is first. That’s empirical, don’t you know! So what is it? Well, it’s basically computes all the number of events that can happen at a certain point over the total number of points. So lets say Randy has an equal chance of catching Anton in one of the six justified realms, i.e., a 1/6th chance. If he looks into one realm, his chances of finding Anton is 1/6th. But if he thoroughly searches that realm and starts looking into a second realm, his chances jump up to 2/6. And then when he moves up to a third realm, his chances go up to 3/6, or 1/2. Yeah, I know I’m getting lazy with this post. I mean … how could Randy exhaust every nook and cranny of every dimension and how could I assume that each dimension carries an equally probability? Well, I don’t and that’s the thing. That’s why we will continue talking about mapping the eCDF onto other scales. Next time — until then:
Yeah, another play on Barton and Schruben (1993). Admit I’m kinda getting lazy with the pictures too — will work on that, promise!
So why would we want to do this? Well, sometimes we just wanna transform the data and make it easier to work with and then transform them back. That and it was sort of the basis of my dissertation and I kinda want to brag about it again. So anyway, lets say Randy’s scientists pinpoint their target to somewhere between Houston and Atlanta. So their target could be in New Orleans, one popular place for the APS March meetings and so would also be a good place for a movie prem … anyway. Or if they narrow their target to be between Baltimore and Boston, then they can pinpoint him to say, Philadelphia. Hmmm, Boston and Baltimore are also popular APS March meeting places and would be good for book sign … but anyway. One way, we could numerically pinpoint the location between two points, say using the empirical cumulative distribution function (eCDF) values using the Barton and Schruben method! And what is the Barton and Schruben method? And what do eCDFs look like? Good question — so we’ll cover that next time! But Barton and Schruben always had me thinking of Bart Simpson which made me think of this.
Oh, Bart! I’d be lucky just to make one movie! But perhaps my chalkboard should look like this then. “I will not yet obsess over book signings/movie premieres … I will not yet obsess over book signings/movie premieres … I will not yet obsess over book signings/movie premieres … I will not yet obsess over book signings/movie premieres … ”
Anyway Cowbunga until next time!
Mwahahaha … doesn’t that sound like something Anton would say? Well, yeah, probably, so of course, I’m gonna give you an example from my trilogy! So say Randy narrowed down to three justified dimensions and twelve unjustified dimensions where the Anton he’s looking for could be. Again, let’s have 1’s indicate he is really in that dimension and 0’s indicate he’s not.
Now say, Randy has his team of scientists do their team of scientists stuff so that they collapse the possibilities of finding Anton in one good dimension or two bad dimensions using the vector of probabilities [0.2,0.1,0.1,0.1,0.2]. So how do they do that? Well, they can multiply the matrix by the vector. And how do they do that? Well they can multiple the elements of each row of the matrix to each element of the vector and then sum up those products. Like multiplying the first row would give you (0.2)(1) + (0.1)(1) + (0.1)(1) + (0.1)(0) + (0.2)(0) = 0.4. And same with the second and thirds rows, giving us a vector of probabilities [0.4, 0.4, 0.1]. So Randy would have a 40% chance of getting the Anton he wanted in the good dimension, a 40% chance of getting the Anton he wanted in one bad dimension, and 20% chance of getting his target in the other bad dimensions. So again, that’s pretty cool, eh? Makes you wanna go to Amazon and check out … okay, stopping. Till, next time when we cover … I’ll figure it out still … but anyway, here’s what Randy found when he looked into one of the dimensions his scientists showed him.
So thought I’d do a quick post today about the transpose – which is when you flip a matrix so the rows become columns and the columns, rows. Simple enough concept, right? So again, lets say that Jane is a painter in, say, 3 dimensions and a physics student in 6 dimensions (you knew this example was coming! Don’t lie!). We could then present the possibility of some of her physics and some of her painter dimensions as crossing (indicated by a 1) using a matrix where the columns are painter dimensions and the rows are physics dimensions.
But its transpose would have columns as physics dimensions and the rows as painter dimensions, like this:
So do any of these dimensions cross? Well, now, if you wanna know, you know what to do. But until next time, strike a transpose! Yes, I am working on new material too.
So today I’m going to talk a little about how we can collapse many dimensions into fewer dimensions for easier handling. And we can do that through means of a matrix. Not as cool as the movie, Matrix, but pretty cool. You know what else would be cool if made into the movie(s)? Okay, I’ll stop. Because first, we’ll need a book release event! Like at an APS March meeting and … okay, wait … wait don’t go! Hear me out! So let’s say we have Tina in the final six justified dimension and six possibilities for cities hosting the APS March 2019 meeting. So there could be 6 x 6 = 36 possibilities where I could be for a book hosting event and where Tina could be. But let’s say not all possibilities exist and so we instead place a 1 in a 2×2 matrix where they do and a 0 where they don’t. And we come up with a matrix like this.
So if, for example, Tina is in Dimension C, my book release could be in Cities 1, 2, 4, 5, or 6. Or if she’s in Dimension E, my release could be in Cities 1, 3, 4, or 5 and so worth. Pretty cool, eh? And kind of like the way Anton’s scientists can keep track of all the possible dimensions that exist. So that’s kinda cool and another way to look at The Matrix.
Now, come to think of it Keanu Reeves would also make a good Randy and … yes, I’m stopping. Anyhoo, until next time — what if I told you that in another dimension … yes, stopping again!