Why Diagnose the Model?
A constant-relative-rate model describes a quantity whose instantaneous rate of change is proportional to its current amount. For a positive quantity \(P(t)\), its exponential form is \(P(t)=Ae^{kt}\), where \(A>0\) is the amount at the chosen time origin and \(k\) is the constant relative rate. This model is useful only when it is a reasonable description of the measurements in question.
Measurements will rarely follow a formula perfectly. They may contain measurement error, and the process itself may not maintain a constant relative rate. Before using an exponential model to interpret or predict, inspect the data for evidence that the relative rate is approximately constant. Three useful diagnostics are successive ratios, changes in logarithms, and residual patterns.
What Constant Relative Rate Predicts
Suppose measurements \(P_0,P_1,\ldots,P_n\) are taken at times \(t_0,t_1,\ldots,t_n\), and each \(P_i\) is positive. For the model \(P(t)=Ae^{kt}\), the ratio of the model values at two times is
This equation gives the first diagnostic. If all time intervals have the same length, an exact exponential model predicts the same ratio at every step. If intervals have different lengths, the ratios need not match; instead, the logarithm of each ratio divided by the interval length should be constant.
The log-change formula follows by taking natural logarithms of the model ratio:
Since \(\ln(P_{i+1}/P_i)=\ln P_{i+1}-\ln P_i\), dividing by the positive interval length gives \(r_i=k\). In particular, the natural logarithm of an exact exponential model is linear in time: \(\ln P(t)=\ln A+kt\). This is the key idea behind logarithmic linearization.
Diagnostic 1: Compare Successive Ratios
When measurements are equally spaced, calculate each successive ratio \(P_{i+1}/P_i\). Ratios that are approximately equal are consistent with a constant relative rate. A sequence of ratios that steadily increases or decreases may indicate that the relative rate is changing. Ratios are a quick check, but they must be interpreted in light of the time intervals.
Worked Example: Equal Intervals and Nearly Constant Ratios
Measurements are \(8,12,18,27\) at times \(0,1,2,3\), respectively. Each interval has length \(1\). Calculate the successive ratios:
The ratios are all \(3/2\). Thus the data agree exactly with the discrete pattern of multiplying by \(3/2\) at each one-unit time step. The corresponding continuous exponential model through the first measurement is \(P(t)=8e^{kt}\), where \(e^k=3/2\), so \(k=\ln(3/2)\).
Check the predicted values at the measured times:
Each model value matches the corresponding measurement, so the values are exactly consistent with this exponential model. This agreement does not establish that the underlying process must continue exponentially beyond the observed times.
Diagnostic 2: Adjust for Unequal Time Intervals
If the measurements are not equally spaced, raw ratios alone can mislead. A longer interval allows more time for change, so its ratio may be larger even when the relative rate is constant. Instead, compare the interval-adjusted log changes \(r_i\). If they are close to one another, the data are consistent with an approximately constant value of \(k\); substantial systematic differences count against that interpretation.
Worked Example: Unequal Intervals with the Same Rate
Suppose measurements are \(10,20,80\) at times \(0,1,3\). The first interval has length \(1\), and the second has length \(2\). The successive ratios are
The raw ratios differ, but the intervals differ too. Calculate the adjusted log changes:
The adjusted log changes agree, so these measurements are exactly consistent with \(k=\ln 2\). Using the value \(P(0)=10\), the model is \(P(t)=10e^{(\ln 2)t}\). Check the data:
The model reproduces all three measurements. The different raw ratios do not contradict it: the second ratio covers twice as much time.
Diagnostic 3: Inspect Residual Patterns
A residual is the difference between an observed value and the value predicted by a specified model. For a positive measurement \(P_i\) and an exponential prediction \(\widehat P_i\), the raw residual is \(P_i-\widehat P_i\). It is also useful to examine the log residual \(\ln P_i-\ln\widehat P_i\), which measures the difference from the model on a logarithmic scale. A positive log residual means the measurement is above the prediction; a negative one means it is below.
Equivalently, plot \(\ln P_i\) against time and compare the points with a straight line. Residuals that vary irregularly around zero may be compatible with measurement variation. A visible, sustained pattern—such as several positive residuals in the middle and residuals near zero at both ends—suggests that a single straight line, and therefore a single constant relative rate, may not describe the data well.
Worked Example: A Curved Log Pattern Counts Against One Rate
Consider measurements \(10,20,30,40\) at times \(0,1,2,3\). Their successive ratios are
Because the intervals are equal, an exact exponential model would require equal ratios. These ratios decrease, so the data do not follow one exact exponential progression. The successive log changes also decrease:
To inspect residuals, take the exponential curve passing through the first and last measurements. Its rate is determined by \(10e^{3k}=40\), which gives \(e^{3k}=4\) and \(k=\frac{\ln 4}{3}\). Thus \(\widehat P(t)=10e^{(\ln 4)t/3}\). The predictions at times \(0\) and \(3\) are \(10\) and \(40\), so their residuals are zero.
At time \(1\), the prediction and log residual are
because \(2^3=8>4\), and hence \(2>4^{1/3}\). At time \(2\),
because \(3^3=27>16=4^2\), so \(3>4^{2/3}\). Both intermediate measurements lie above the endpoint exponential curve, while the endpoint residuals are zero. This systematic pattern is evidence against describing all four values with one constant relative rate. The check is consistent with the observations: the model matches the endpoints but predicts less than \(20\) and \(30\) at the two intermediate times.
How to Use the Diagnostics
Confirm that the measurements are positive before using logarithms, and note each interval length.
Use successive ratios for equal intervals; for unequal intervals, compare \((\ln P_{i+1}-\ln P_i)/(t_{i+1}-t_i)\).
Look for systematic structure rather than relying only on whether individual residuals are small.
Describe the data as consistent or inconsistent with a constant relative rate at the observed times; do not treat a finite data set as proof of future behavior.
These checks are related: the adjusted log change is the slope between successive points on a plot of \(\ln P_i\) against \(t_i\). If those slopes are nearly constant, the points lie approximately along a straight line. Residuals then provide a way to see whether deviations from a chosen line show a pattern. The diagnostics complement one another, but none removes the need to consider how measurements were collected and how much variation is expected.
Common Pitfall: Treating Unequal-Interval Ratios as Comparable
A common mistake is to reject an exponential model because ratios from intervals of different lengths are unequal. For the measurements \(10,20,80\) at times \(0,1,3\), one might incorrectly argue that \(2\ne4\), therefore the relative rate is not constant. That calculation compares a one-unit change with a two-unit change. The correction is to adjust for elapsed time: \(\ln 2/1=\ln 4/2=\ln 2\). The data are consistent with a constant rate after this adjustment.
Another limitation is that logarithms require positive data. If a measurement is zero or negative, its natural logarithm is not defined as a real number, so the log-difference diagnostic cannot be applied directly. Nor should positivity be silently assumed: it is a condition of the calculation, not a conclusion from it.
What the Evidence Can and Cannot Say
Nearly constant adjusted log changes and unstructured residuals support the use of an exponential model as an approximation over the observed range. Strongly changing adjusted log changes or a clear residual pattern count against a constant relative rate. However, finite measurements cannot prove that a process has a constant rate, and small discrepancies do not automatically disprove a useful model. Measurement precision, sample size, and the purpose of the model all affect how much deviation matters.
In particular, a model can fit the observed interval well and still fail outside it. The diagnostics assess whether the recorded data support the model over the times measured; they do not guarantee accurate extrapolation or identify why a pattern departs from exponential behavior.
Check Your Understanding
Use the definitions and diagnostics above to answer these questions.
- For equally spaced positive measurements, what successive-ratio pattern does an exact exponential model predict?
- Why can raw ratios differ when intervals have different lengths even if the relative rate is constant?
- For measurements \(P_i\) at times \(t_i\), write the interval-adjusted log change and state its value under an exact exponential model.
- What can a run of positive log residuals between two zero endpoint residuals suggest?
- Why can the logarithmic diagnostics not be applied directly to a zero measurement?
