4.3 Application to experimental  4.2 Applications  4.2.5 Maximum-likelihood estimation in

4.2.6 Estimating exponential-decay time constants

Exponentially decaying functions arise in many experimental situations, either because of instrumental response characteristics or characteristics of the physical condition being measured. Some examples are these:

• Many thermometers respond to temperature change such that the difference between the measured and true temperature decays exponentially.
• Some observations of liquid water content in clouds appear to decrease exponentially with time.
• The time intervals between randomly occurring events obey an exponential probability distribution.

Because these situations are so common, there is often a need to estimate the time constant characterizing the exponential decay from measurements; i.e., to estimate $\tau$ in the formula

$$F(t) = F_0 e^{-t/\tau} \ .$$ (4.59)

 

The maximum-likelihood method provides a straightforward approach to this problem.^4.1 The approach will be illustrated by an example.
 

Example 4.2: The following are measurements of liquid water content averaged over 1 km in the center of a decaying cumulus cloud, at three minute intervals: 2.50, 2.15, 1.59, 1.10, 1.12, 0.93, 0.55, 0.65, 0.45 g m^-3. A plot of these shows them to be approximately consistent with an exponential decay. If the decay follows (4.59), what is the best estimate of the time constant $\tau$, if the measurement uncertainty in each measurement of the liquid water content is the same and characterized by a Gaussian distribution about the true value?

To calculate the likelihood function for this case, assume that (4.59) describes the mean value and measurements are distributed about this mean according to (3.2), so that the probability of observing liquid water content w_i at time t_i is

$$P(w_i, t_i) = {{1}\over{\sqrt{2\pi} \sigma_i}} \exp{{{-(w_i-A^{-t_i/\tau})^2}\over{2\sigma_i^2}}} \$$ (4.60)

 

where A and $\tau$ are parameters to be determined from the fit. Then

$$W = -\sum_i{{(w_i-Ce^{-t_i/\tau})^2}\over{2\sigma_i^2}} + C^\prime$$ (4.61)

 

where $C^\prime$ is constant with respect to variable parameters in the problem.

The two equations obtained from (4.6) by differentiating (4.61) with respect to the two parameters A and $\tau$ both give equations for A, so eliminating A by equating these expressions leads to the requirement that

$$f(\tau) = {{\sum_i{{w_it_ie^{-t_i/\tau}}\over{\sigma_i^2}}}\......^2}}}\over{\sum_i{{e^{-2t_i/\tau}}\over{sigma_i^2}}}} = 0 \ .$$ (4.62)

 

While not amenable to analytical solution, (4.62) is readily solved numerically or graphically, using methods that will be reviewed in Chapter 9. The solution can be obtained graphically by plotting $f(\tau)$ as a function of $\tau$ and finding the point at which $f(\tau$)=0. This occurs for $\tau$=13.6 min for the data of this problem, and the result is the fit shown in Fig. 4.3:

 

Figure 4.3: Measurements of liquid water content as a function of time, with assumed constant uncertainty estimates, and the best-fit exponential obtained by maximum-likelihood analysis.

Exercise 4.2: For the data of the preceding example, show that if the assumed accuracy in liquid water content is 0.2 g m^-3 then the one-standard-deviation limit for $\tau$ is 1.6 minutes. Check also that the scatter of measurements about the exponential best-fit line is consistent with this estimate, and adjust it if necessary.

   
 4.3 Application to experimental design  4.2 Applications  4.2.5 Maximum-likelihood estimation ...