How is my candidate doing in the polls?

Given the profusion of polls, it is difficult to accurately gauge how well a given candidate is doing. A simple average of poll numbers won't adequately capture momentum (if such a concept exists), and a few averages (say, one of polls done in the past week, another of polls done in the past month) are difficult to parse. We want one, single, simple number.

We fix the candidate we're interested in, and we have polls \(P_{n+1}\) and \(P_{n}\) released at times \(t_{n}\lt t_{n+1}\). Ideally, we should be able to truncate the N polls to the last k without "much loss".

We could take a moving average, something like \[M_{n+1} = \alpha(t_{n}, t_{n+1}) P_{n+1} + (1 - \alpha(t_{n}, t_{n+1}))M_{n}\tag{1}\] where \(P_{n}\) refers to the n^th most recent poll released on the date \(t_{n}\), with the initial condition \(M_{1} = P_{1}\) and the function \[\alpha(t_{n}, t_{n+1}) = 1 - \exp\left(-\frac{|t_{n+1} - t_{n}|}{30 W}\right)\tag{2}\] where W is the average of the intervals between polls, and the difference in dates is measured in days. The 30 in the denominator of the exponent reflects 30 days in a month

Exercise 1. Show (1) α will take values between 0 and 1, (2) the larger the α, the quicker it "forgets" older data, (3) "older data" will be forgotten faster [what happens for regularly released polling data? Say, weekly, a new poll is released, what does α look like?].

Weighing Pollsters

If we knew about poll quality, we could add this in as another factor. Suppose we had a function1The codomain is a little ambiguous, we have it here as \(0\lt Q(P)\lt 1\), but either inequality could be weakened to "less than or equal than" conditions. So it could be extended to include \(0\leq Q(P)\lt 1\) or \(0\lt Q(P)\leq 1\) or even \(0\leq Q(P)\leq 1\). \[Q\colon \mathrm{Polls}\to (0,1)\tag{3}\] which gives each poll its quality (higher quality polls are nearer to 1). Then we could modify our function in Eq (2) to be something like \[ \begin{split} \tilde{\alpha}(t_{n}, t_{n+1}, P_{n+1}) &= Q(P_{n})\cdot\alpha(t_{n},t_{n+1}) \\ &= Q(P_{n})\cdot\left(1 - \exp\left(-\frac{|t_{n+1} - t_{n}|}{30 W}\right)\right)\end{split}\tag{4}\] which penalizes "worse polls" from influencing the moving average. (Since worse polls have smaller Q values, which leads to higher \(1 - Q\) values.)

One lazy way to go about this is to use pollster ratings from FiveThirtyEight, discard "F" rated polls, then take the moving average with \(Q(-)\) the familiar grading scheme used in the United States. (Or, more precisely, the midpoint of the interval for the grade.)

Letter grade	Percentage	Q-value
A+	97–100%	0.985
A	93–96%	0.945
A−	90–92%	0.91
B+	87–89%	0.88
B	83–86%	0.845
B−	80–82%	0.81
C+	77–79%	0.78
C	73–76%	0.745
C-	70–72%	0.71
D+	67–69%	0.68
D	63–66%	0.645
D-	60–62%	0.61

The other "natural" choices include (a) equidistant spacing in the interval (0, 1] so D- is given the value \(1/13\) all the way to A+ given \(13/13\), or (b) the roots of an orthogonal family of polynomials defined on the interval [0, 1].

Exercise 2. How do the different possible choices of Q-values affect the running average? [Hint: using the table above, is \(\widetilde{\alpha}\leq 0.61\) is an upper bound? Consider different scenarios, good poll numbers from bad polls, bad numbers from good polls.]

Exercise 3. If we assign \(Q(\mathrm{F}) = 0\) as opposed to discarding F-scored polls, how does that affect the weighted running average?

Some computed examples are available on github, but they're what you'd expect.