Motivation

This is probably the most unattractive topic to any mortal and probably anyone that reads this piece of text will stop doing so after this line. For those that still hang in there: measurement accuracy & precision, a proper idea of measurement accuracy & precision is fundamental to any statement following from these measurements! To differentiate between BS and non-BS. The concepts of accuracy & precision are somewhat hard to comprehend for my peanut-size brain, but to me the figure in this link explains it nicely.

In a previous post I claimed that a simple mathematical model based on the assumption that the air in my living room is well-mixed can describe the CO₂ profile. This model includes one fit parameter: the ventilation rate, more precisely: the number of air changes per unit of time. Before making any rigours claims that the ventilation rate depends on whatever parameter you could think of, I needed to have an idea of the accuracy with which I can determine this parameter .

Model deviation

I personally do not like an approach heavily based on statistical methods for a couple of reasons: no one understands it, it is boring and often abused. I prefer a “visual” approach: let us just look at images, people like visuals. So I ended up with a “banana”-plot, see Figure 1. This is the same figure as in the previous post for which I compared measurement data and the model prediction. But, I know added scenario’s for which the ventilation rate (1/\(\tau\) = Air Changes per Hour) deviated with ±10%, ±20% and ±30%. I learned that the deviation of model and measurement data was always within ±30% but often within ±10% (not bad he) and that a banana is now anchored in my brain as “a method to fit an exponential decaying curve” not as a piece of fruit.

Figure 1: CO₂ measurement and model comparison. Marked area’s resemble a deviation in the ventilation rate of ±x%

The relative deviation between model (see equation 1) and measurement was always within ±5%, see Figure 2.

\[Relative \, deviation (\%) = \frac{C_{Model}(t)-C_{Measurement}(t) }{C_{Measurement(t)}} \times 100\]

Figure 2: Deviation between measurement and model

An alternative way of visualizing the curve fitting data is by ‘linearizing’ the exponential equation that describes the model, see equation 2.

\[C = C_{in} – (C_{in} – C_{0}) e ^{(-t / \tau) }\]

We define the relative driving force as:

\[Y = \frac{Driving \, force \, at \, time \, = \, t}{Maximum \, driving \, force} = \frac{C_{in} – C(t)}{C_{in} – C_{0}}\]

In which C_in is the CO₂ concentration in the air, C₀ is the CO₂ concentration at t=0 and C is the concentration at t=t.

We then rewrite equation 2 to equation 3:

\[-ln(Y) = t \times 1/ \tau\]

Thus by plotting t vs -ln(Y) would result in a straight line, which slope is equal to the ventilation rate (1/ \(\tau\)), see Figure 3. Indeed, the measurement data (and obviously the model) follow a rather straight line. As this is just a translation of Figure 1, the message is the same; the model typically deviates less than ±10% from the measurement. This reverse-funnel suggests that the starting point is critical and that the endpoint does not matter so much. But hey, images can be deceptive as well, the logarithmic y-axis messes with your brain.

Figure 3: “Linearized” CO₂ measurement and model comparison. Marked area’s resemble a deviation in the ventilation rate of ±x%

To wrap up, I am pretty happy with the agreement of model and measurement data, at least for the example given. A ventilation rate ±10% captures the vast majority of measurements. Obviously, the sensor that collects these data is a source of inaccuracy as well, something that I will discuss next time…