# How to handle rational power of units when fitting experimental data

I collect urban aerosol data and I would like to post-process it for analysis purposes.

Raw data are diameter cut counts $N_i$ ($\mathrm{dm^{-3}}$) that stands for particulate concentrations: number of particulates with a diameter $\phi$ above $\phi_i$ ($\mathrm{\mu m}$) by units of volume.

I have found in Hinds, Aerosol Technology, Wiley, 1973 that urban aerosol diameter distribution where diameters are ranging in $\phi \in [0.01, 10[\mathrm{\mu m}$ is almost a straight line in a log-log plot and then can be fitted by a power law. There is no theoretical reason for that, it is an empirical observation.

When I fit distributions to a power law as stated in Hinds:

$$\frac{\Delta N_j}{\Delta \log_{10}(\phi_j)} = \beta_0 \phi_{g,j}^{-\beta_1}$$

Where:

• $\Delta N_j$ bin concentrations in $\mathrm{dm^{-3}}$;
• $\Delta \log_{10}(\phi_j) = \log_{10}\frac{\phi_{j+1}}{\phi_{j}}$ is dimensionless;
• $\phi_{g,j}$ is the central parameter of the bin, we choose the geometric mean in ($\mathrm{\mu m}$);
• $\beta_0, \beta_1$ are fit parameters.

I have estimates of parameters that are plausible and that capture most of the dynamic. What is tickling me is units of $\beta$ parameters.

I think $\beta_1$ should be dimensionless because it is an exponent. The problem is units of $\beta_0$ that in my fit must be $\mathrm{dm^{-3}\mu m^{\beta_1}}$. I feel odd about that, because $\beta_1$ is probably not a rational number and it makes my comparison of $\beta_0$ complexer because they all have different units (depending of $\beta_1$).

I know that I may rationalize $\phi_{g,j}$ by a arbitrary diameter, lets say $\phi_0 = 1 \mathrm{\mu m}$ and the problem is gone, because I will fit a reduced diameter instead of an absolute diameter. $\phi_0 = 1 \mathrm{\mu m}$ is not a bad choice because aerosol equivalent diameters are close to this value, so it might be a characteristic length of my problem. But it left me uncomfortable. Why should I not pick another value for $\phi_0$? Is there another techniques to solve this issues?

How am I supposed to properly handle this power law fit in order to have units consistency and comparable fit parameters among time?

• You might want to read on Buckingham's pi theorem: en.wikipedia.org/wiki/Buckingham_%CF%80_theorem – ZeroTheHero Feb 7 '17 at 14:48
• @ZeroTheHero, I read twice the article and I did the exercises. It was a good refresh but I cannot see how it can help me. My problem is that the power is a real variable and therefore it affects units. Could you elaborate how do you think it can help me! Thank you – jlandercy Feb 9 '17 at 12:31
• I'll have another look but clearly you cannot have exponents with units, we agree on that. – ZeroTheHero Feb 9 '17 at 12:35
• @ZeroTheHero, yes if we let it happens, we are screwed – jlandercy Feb 9 '17 at 12:38
• are you sure $\Delta N_j$ should have units? After all, it's a number of particulates... That would make $\beta_0$ also dimensionless. Still doesn't quite solve your $\beta_1$ problem... at least for now. – ZeroTheHero Feb 9 '17 at 14:34