More data

BAMM Documentation Assimilating more data points.

I've conducted another NB test and collected a new data point for my AeT: 158 bpm. With this new data point, I want to update my estimate for my unknown average AeT. But first, a note: while in my first NB test I used the average heart rate of a 30-minute run in which I could breathe comfortably through my nose, in my second test I measured it a bit differently: it was the maximum heart rate I could sustain at which I could comfortably breathe through my nose for a few minutes at a time. On multiple occasions, I purposefully increased my pace to see the heart rate at which I needed to open my mouth for air. Though I conducted this NB test slightly differently, I am already accounting for uncertainty in AeT measurements by using Bayesian modeling; specifically, in the previous page I assumed a standard deviation of 5 bpm for these measurements. I can then use a conservative estimate for my AeT (e.g., the lower 10% quantile), and define my Zone 2 using this number. I am therefore not concerned about my results being influenced by changing my measurement protocol, because I am already properly accounting for uncertainty in these measurements and choosing conservative estimates for my AeT.

With multiple data points, the math for the posterior distribution changes a bit. Recall the prior distribution of my average AeT:

$\text{AeT}\sim N(130,\sigma_{AeT}^2)$

We describe the likelihood of observing a number of heart rates using the nose-breathing test (denoted y⁽ⁱ⁾_NB for i=1,...,n) conditional on a true but unknown average AeT value similarly as before, but for convenience we will assume the standard deviation is the same as that of the AeT:

$y^{(i)}_{NB}|\text{AeT}\sim N(\text{AeT},\sigma_{AeT}^2),\quad i=1,...,n$

What we want is the distribution of AeT|y⁽ⁱ⁾_NB; that is, the distribution of my average AeT given the data points I've collected using the nose-breathing test. The math is skipped, but the posterior distribution of AeT given the data points is

$\text{AeT}|y^{(i)}_{NB}\sim N\left(\frac{\sum_{i=1}^n y_i+130}{n+1},\frac{\sigma^2_{AeT}}{n+1}\right)$

Plugging in our numbers, we get

$\text{AeT}|y^{(i)}_{NB}\sim N\left(145.33,2.89\right)$

This page is here to illustrate that as I keep collecting measurements, the posterior distribution will keep narrowing. However, I now have a better idea of my AeT based on the nose-breathing test, and I won't continue to measure it this way. I will perform endurance training using 140 at my AeT. I have ordered a lactate testing kit, which will allow me to collect more data, do different modeling, and get a more precise estimate of my AeT.

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