It’s been a while since I had to do math. What does the dot symbol mean at the end of this? I’m not sure if it’s a superscript for the last xi, or inline with the “a”. On the page in my book, it is inline with the dot over the “i”.
----------L
= P(x[SIZE=“1”]1)Πa[/SIZE]xi-1xi•
-------- i=2
You’re either missing a bit and it denotes multiplication or it’s a full-stop.
a dot is sometimes used to denote a generic subscript, but there’d typically be one on each side of the equation.
x[sub].[/sub] = sum(j= 1 to n)y[sub]j.[/sub]
would mean it’s true for all x[sub]i[/sub]
It has to do with using Bayesian statistics and Markov chains in DNA sequence prediction. This has to do with the transitional probability of a certain residue following another.
That’s how it appears on the page, but they say it’s what the previous equation becomes; previous equations:
Ast = P(xi = t | xi-1 =s)
So ACG or AAT (or whatever dinucleotide) is the “transitional probability” of an G folowing a C, or a T following an A.
P(x) = P(xL, xL-1,…, x1)
= (PxL|xL-1,…,x1)P(xL-1|xL-2,…,x1)•••P(x1)
So, there’s a P(x) on the left hand side of the equation.
L is length of sequence. Hope that helps.
In which case I still think it is a full-stop
That notation isn’t familiar to me, either.
What book is this from? I’ve got two books on Bayesian sequence analysis, and if I have the same as you I’d be happy to look at it and attempt to figure it out from context.
Both my books (“Bayesian Data Analysis” and “Biological Sequence Analysis”) use periods or commas after most equations in the body of the text, so I’m guessing it’s just punctuation. Is that the normal style? I can’t really say since those are nearly all of my Real Grownup Math books…
I found this page. (PDF!) It appears to be a chapter from some (unnamed) textbook, giving a detailed discussion of the algebra of summations.
It has many examples showing dot-notation for subscripts. Strangely, the notation is used before it is explained (unless maybe the earlier usage means something else, or there’s an earlier explanation I didn’t notice).
The text is searchable. Search for these sections and look at the next several pages following each of these headings. There are lots of dot-subscript examples there.
2.4.4
Proving a Result With Summation Algebra
2.7.2
Dot Notation
P.S.: When I took introductory statistics Lo! these many years ago, I learned this notation. But I don’t recall if I got it from our class textbook or from some other source. It was used in connection with correlation coefficient computations, which involve summing rows or columns of matrices, or sums of products of corresponding rows and columns of various matrices. (ETA: Even as I write this, some vague details are coming back to me. I learned this in connection with doing ANOVA computations.)
Generally, a dot for a subscript is an abbreviation for “the sum of all the elements”. It is most commonly used (IIRC) in connection with a matrix, and may appear in place of either the row subscript or the column subscript, or both.
If X is a matrix of elements x[sub]ij[/sub] then x[sub]•j[/sub] means the sum of all elements in column j (that is, the column total), and x[sub]i•[/sub] means the sum of all elements in row i (the row total), and x[sub]••[/sub] means the sum of all of the elements in the matrix (the grand total).
It is a convenient abbreviation for sigma notation. (Since the OP’s example involves pi notation, maybe it has an analogous usage there too? I would be inclined to interpret this as some kind of product of sums, with the pi notation indicating the product and the dot-subscript notation indicating the sums. Perhaps?)
Not only is it is convenient abbreviation, but I found it very convenient to work with too. I tried, successfully, to prove some formulas using this notation. I expected that I would have to expand each such notation to the full sigma notation, then use all kinds of summation algebra (of exactly the sort discussed in the chapter I linked) to prove things. It turned out MUCH simpler than that. I found that I was able to do algebra directly with the dot notation without having to expand it. It all worked out very nicely. This is therefore a good notation.
It’s “Biological Sequence Analysis” by Durbin et al, equation 3.2 on page 49. It probably is just punctuation, but the punctuation seems inconsistent–some equations have periods at the end and others don’t; they put a comma after the next numbered equation, some equations don’t have either.
I don’t know if that’s the normal style. It’s been years since I’ve had to work through a textbook and I had to look up some symbols, and I was like “OK, what’s the dot?”
Ha! I found that pdf after posting here. Thanks, though. I was wondering if it did mean something like 2.7.2 where they use it to stand in for a column or a row in an array.
Yeah, in this equation, it looks like it’s X subscript i “period” as in end of sentence, but it kinda looks like xi•. (<–that last one is a period:))
Here is another usage example I found.
Computational Procedures for 1-Way ANOVA
This brief 3-page shows dot notation several times. I didn’t read the doc closely. At a quick glance, I’m not certain if the usage is the same as the row-total or column-total I described above. Take a look and see what you can make of it.