The relationship between two variables is often not linear, but almost any function that can be written in closed form can be incorporated in a nonlinear regression model. Nonlinear least squares regression extends linear least squares regression for nonlinear relationships. The way in which the unknown parameters in the function are estimated is conceptually the same as it is in linear least squares regression. Typical nonlinear relationships are decay processes, bacterial growth or enzymatic reaction dynamics. MaxStat has its own equation library (see figure 1) to help users to select the most appropriate function. MaxStat makes also professional graphs as shown further below.
Figure 1: MaxStat’s user interface to select nonlinear regression functions.
It is always possible to find a polynomial curve to describe data points. A polynomial function as the following general form:
Y = a + b*x + c*x² + d*x³…
Polynomial equations are useful to interpolate unknown values between two data points if the true relationship between the points are unknown. For example, a technical device may respond to the measuring variable in a nonlinear way and polynomial functions can be used for calibration purposes of the device. However, very few natural systems follow polynomial functions, and, therefore such functions hardly contribute to a better understanding of natural processes. We have to be careful in the use of polynomial functions and their interpretation.
For example, let us consider an experiment in which several populations of a fish species are exposed to different concentrations of a pesticide. We observe the development of tumor rates among the different populations (see Figure 2). We realize that the relationship between the concentration of the pesticide and tumor rates is not linear, but a third-order polynomial function (y=a+b*x+c*x²+d*x³) describes the observed data quite well (r²=0.945). However, scientifically the model is not meaningful. It basically says that we should add 0.25 ug/L pesticides in every pond, lake or river for the lowest tumor rate. The tumor rate decrease even at the highest concentration.
Figure 2: Respond of tumor rates to the concentrations of a pesticide.
In an ideal scenario, we have an idea which equation will provide us with a meaningful model. For example, decay processes (i.e. atomic decay, decomposition of organic matter, photo degradation) follows an exponential equation of the following form:
m(t) = m(0) * e^(-k*t)
with m(0) and m(t) as the mass at the time zero and t, k as the decay constant , and t the time of the process (in minutes, hours, days, weeks…).
For example, let us consider the decomposition of organic matter by earthworms in a triplicate experiment. We know that the earthworms will decompose the organic matter, and we can choose exponential decay as a model. The obtained equation is 100.34 * e^(-0.035*t). The decay constant is -0.035 day‾¹.
Figure 3: Decomposition of organic matter with the time.
Cultures of bacteria or yeast grow exponentially in a sterile medium. The equation is identical to the exponential decay, except the growth constant is positive.
n(t) = n(0) * e^(r*t)
with n(0) and n(t) is the number of generations at the time zero and t, r as the growth constant , and t the time of the process (in minutes, hours, days, weeks…).
For example, let us consider the growth of a bacterium in a triplicate experiment. We know that the bacterium will growth exponentially, and we can choose exponential growth as a model. The obtained equation is 33.66 * e^(0.09*t). The growth constant is 0.09 hour‾¹.
Figure 4: Exponential growth of a bacterium
To describe the dynamics of enzymatic reactions, saturation equations are used. The most common is the Michaelis-Menten equation of the following form:
V = (VM * S) / (VM + S)
with V as reaction speed, VM the maximum possible reaction speed, and S as substrate concentration. An example of a fit for a Michaelis/menten equation is shown in Figure 4.
Figure 5: Speed of an enzymatic reaction as a function of the substrate concentration