4.2.5 Maximum-likelihood estimation in  4.2 Applications  4.2.3 Mean of the

4.2.4 An example: Fitting to CCN measurements

Example 4.1: A cloud condensation nucleus (CCN) counter usually operates by exposing an air sample to a supersaturated environment and counting the number of droplets n that form in a specific sample volume V. As a function of the supersaturation S, the CCN spectrum is often reported in the form

$$N(S) = C \bigl({{S}\over{S_{ref}}}\bigr)^k$$ (4.25)

 

where N(S) is the concentration of CCN active at supersaturations smaller than or equal to S, C is the concentration active at or below the reference supersaturation S_ref, and k characterizes the rate of increase of concentration with supersaturation. The estimated concentration at the supersaturation S_i is then N₍S_i)=n_i/V.

Often small numbers of CCN are detected at a given supersaturation, so statistical fluctuations have an important influence on the final estimates of the parameters C and k. Furthermore, when the number of droplets detected is small, the non-Gaussian nature of expected deviations also should be considered when fitted parameters are determined. The fit is complicated also by the correlations that are usually present between uncertainties in C and k.

The method of maximum likelihood can be used to estimate values of C and k using a set of observations $\{n_i\}$ corresponding to supersaturations $\{S_i\}$. As is always the case with maximum-likelihood approaches, the calculation of the likelihood starts with an initial assumption about the expected form of the probability function. In this case, we assume that the spectrum has the form (4.25) and that the departures from that spectrum are caused by statistical fluctuations characterized by the Poisson distribution function. Either of these assumptions may be incorrect, for example because the real shape of the supersaturation spectrum may be more complicated than (4.25) or the changes measured in repeated samples may represent changing aerosol characteristics with time instead of the assumed supersaturation dependence. The maximum-likelihood result will always be dependent on the validity of these initial assumptions, and they should be tested when possible.

If the spectrum obeys (4.25) and the measurements follow Poisson statistics, the likelihood can be calculated as follows:

• For given values of C and k, the mean number of droplets expected at a given supersaturation S_i is $\mu_i=C~(S_i/S_{ref})^k~V$ where V is the sample volume.
• The (normalized) probability of observing n_i droplets at supersaturation S_i is given by the Poisson probability function (3.7) for mean value $\mu_i$.
• The likelihood function is then

$${\cal L}(C,k) = \prod_i \Phi_P(n_i,\mu_i)$$ (4.26)

 

where $\Phi_P$ is the Poisson probability function and the sum extends over all observations. This function is dependent on the assume values of C and k, so best values of these parameters correspond to the pair of values that maximize ${\cal L}$.

The resulting contour plot of the likelihood function might look like Fig. 4.2:
 

Figure 4.2: Contours of an example likelihood function, in units relative to the maximum, for a fit to (4.18). Contours at -0.5 and -2.0 represent respectively the one- and two-standard-deviation limits for the fit. The maximum likelihood occurs for C=196 cm^-3 and k=0.87. The data used to construct this plot were: for supersaturations of 0.2, 0.4, 0.6, 0.8, 1.0, and 1.5%, respective droplet counts of 2, 14, 11, 12, 26, and 25 representing a sample volume of 0.1 cm^-3. (These values were generated for the case where C=200 cm^-3 and k=0.8, with random errors added to each assumed measurement.)

The uncertainties in C and k are correlated, and that correlation poses special difficulties that will be discussed in the next section. The best value of the parameters can be found by calculating enough values to plot the contours as in Fig. 4.2, or a numerical maximization procedure can be used to find the maximum value of the likelihood. A plot such as that shown is particularly informative because contours are chosen to correspond to deviation limits representing 1 and 2 standard deviations.

   
 4.2.5 Maximum-likelihood estimation ... 4.2 Applications  4.2.3 Mean of the binomial ...