Project 7 tries to determine the parameters associated with data that exhibits exponential growth or decay.
You may have run across simple models of biological growth, in which the population of a bacteria colony, for example, can be estimated by a formula like
P(t) = P0 * exp ( k * t )
Here, P0 is the population at time 0, and k is a parameter
that determines the rate of growth.
We might be trying to construct such a model for a given colony. In that case, we might measure the population P(t) at several times, and from that data, determine the value k.
If the population followed the formula exactly, and if we had exact values for P0 and the population at one other time, it turns out that we can figure out K exactly.
In the real world, we might expect that even if the actual population closely follows the model, there might be slight deviations, and that our measurement of the populuation will also include small errors. We might try to counteract this error tendency by taking several measurements. This gives us several estimates for K. How do we choose a particular value of K then? Since we may assume that even our best guess for K is in error, how seriously does this error affect the difference between our model and the actual behavior of the system under study?
Exponential growth rates occur in financial mathematics (in interest rates, in particular), but also in chemical processes and biological systems.
The problem of estimating a hidden parameter for the model of a system based on measurements of the system's behavior can be much harder than in this case. This general problem is known as estimation or the parameter identification problem.
The data for this problem gives 601 observations of t and y(t):
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