Monday, April 11, 2016

Linear Weight Ratio

Note: The series of four posts I will be posting over the next month were written a long time ago, apparently in 2009. Since I have not been prolific in producing new material lately, I figured I might as well post some older stuff I’ve written that at the time I didn’t deem good or interesting enough to post. I did not vet all of the material in them, so any inaccuracies are my fault but do not necessarily reflect my current thinking.

Linear Weights Ratio (LWR) is an offensive metric developed by Tango Tiger, based on Linear Weights. Since it was developed and explained by Tango, there is really no need for me to step in and write a post that may just serve to confuse you. And I have not defined everything in exactly the same way he did, which will only add to the confusion.

I have always liked to write descriptions of other people’s research, for a couple of reasons. One is as a sort of critique/peer review, which does not have to be critical--it can also point out the positives about an approach. A second is so that if I use something later (and I have an upcoming post that uses LWR in the vein of Bill James' "Willie Davis method"), my readers can have some degree of confidence that I understand the topic at hand. All too often you will see people use metrics that they don’t really understand. By writing about the ones I’m using, I will be presenting you with sufficient evidence to draw your own conclusions as to whether or not I understand the tools I am using.

Let’s begin by focusing only on the basic, mutually exclusive offensive events: singles, doubles, triples, home runs, walks, and batting outs (AB - H). For now, we will assume that those categories encompass every possible outcome of a plate appearance. Let us also assume that we have some set of linear weights which give the value of each of those events: s is the value of a single, d of a double, t of a triple, h of a home run, w of a walk, and x of an out. Additionally, I am approaching this problem with absolute (total runs scored) weights, so x is something like -.1, not -.3. Tango’s LWR used the -.3 type value.

Given those assumptions, we can of course write:

RC = sS + dD + tT + hHR + wW + xO

Let’s consider “S”, “D”, etc. to be per PA frequencies (again, these events are assumed to encompass all possible PA outcomes, so PA = S + D + T + HR + W + O). If that is the case, we can rewrite O as 1 - S - D - T - HR - W, and write an expression for RC/Out:

RC/O = (sS + dD + tT + hHR + wW + x(1 - S - D - T - HR - W))/(1 - S - D - T - HR - W)

The out term can be canceled out, leaving us with:

RC/O = (sS + dD + tT + hHR + wW)/(1 - S - D - T - HR - W) + x

You can see that there is no need for the out term to be included at all; we are still implicitly including outs, but we don’t need to include them in the equation. The numerator of the expression is the run contribution of each event, excluding outs, while the denominator is outs. This is what I will call rLWR, for run LWR:

rLWR = (sS + dD + tT + hHR + wW)/(1 - S - D - T - HR - W)

In figuring his Linear Weight Ratio, Tango adds an additional wrinkle, and sets the weight of a single equal to 1, with the other weights changing proportionally. We can define s' as 1/s, and use that to define d' = d*s', t' = t*s', etc., and write LWR as:

LWR = (S + d'*D + t'*T + h'*HR + w'*W)/(1 - S - D - T - HR - W)

At this point I’ll plug in some actual numbers from the basic ERP equation I use ((TB + .8H + W - .3AB)*.324). This is not an optimal equation, and that’s okay because my point here is not to present a formula that you should use, just to demonstrate how you can derive your own formula for LWR based on whatever set of linear weights you are using. When that ERP equation is expanded, it becomes:

ERP = .486S + .810D + 1.134T + 1.458HR + .324W - .097(AB - H)

Which yields s’ = 2.058 and the following LWR equation:

LWR = (S + 1.67D + 2.33T + 3HR + .67W)/(1 - S - D - T - HR - W)

If you are using the actual counts of each event rather than the per PA frequencies, this could be written the same except PA would replace 1 in the denominator.

It is easy to convert between LWR and R/O, and it is a linear process. The equations are:

R/O = rLWR + x
rLWR = R/O - x
R/O = LWR/s' + x
LWR = (R/O - x)*s'

What alterations do we have to make to include non-batting outs in our ratio? This can be tricky since we can no longer assume a uniform value for outs across types. But we just need to ensure that the above relationships still hold, and weight the event in the numerator accordingly. (LWR*s' + x)*Outs must equal RC. We can expand that out:

(LWR numerator/Outs*s' + x)*Outs = RC

which simplifies to:

LWR numerator*s' + x*Outs = RC

For any specific event, x is known (the -.097 value), Outs is known (each out is worth one out), s' is known (2.06 in this case), and the RC weight of the event in question in known (let’s say we have CS at an overall value of -.3), so all we need to do is solve for the needed coefficient in the LWR numerator:

(RC weight - x)/s' = LWR numerator

For the CS example:

(-.3 - (-.097))/2.06 ~ = -.1

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