# Autocorrelation Test for Experimental Kinetic Data?

Serial correlation in temporal concentration data obtained from experiment (not conventional time-series). How to determine the presence of serial autocorrelation?

I have a monotonically decreasing kinetic trace (reaction profile) with data points plotted as concentration of reactant (response variable-y(t) affected with measurement error or noise ε) versus time (input variable-t assumed without or with negligible measurement error when compared to y(t)), and can be expressed as:

Observation = model function + noise

How can one confirm that the kinetic data is serially correlated in the whole reaction profile or in a particular segment of it?

(Serial Autocorrelation means that what happens at one instant of time effects what happens at the next, since observations taken close together in time tend to be correlated. Correlated errors are very common occurrences in digital traces obtained from spectrophotometers and other analytical instruments. It is well known that data collected in a time sequence are likely to exhibit serial correlation – i.e., the errors measured in the observations of the response variable are not truly independent statistically. In such a circumstance the deviation at any point taken along the curve is often quite similar in both sign and magnitude to that of its neighbours.)

Is there a particular statistical test or methods that one can apply do determine this using data from a single temporal concentration profile affected with error? I could only find information regarding conventional time-series data, like the AR(1) model, or linear regressions using ordinary least-squares, incl. the Durbin-Watson statistic, but in my case the experimental observations decrease non-linearly with time obeying a proposed mechanistic kinetic law (see link below), and therefore should not be considered as time-series in the conventional sense.

The kinetic expression representing the model function is given in implicit form as F(t, y(t); θ) = 0, (see the link below) where θ is a parameter set containing two elementary rate constants, k1 and k2, which need to be determined numerically via non-linear least-squares fitting. My aim is to assess if the level of autocorrelation of the errors manifested in the observations is too high or not to decide whether least-squares should be employed to the whole data range of the raw kinetic profile (or a particular segment of it) or make a selection of that data to extract values of the two kinetic parameters using least-squares.

In order to estimate values of θ the errors in the measurements must be independent (i.e., uncorrelated), since least-squares should be applied only when errors in the measurements are truly independent statistically from one another. Therefore, the need to assess if there is any significant level of serial (auto)correlation in the original observations taken over time. Knowing this, it would help me select a minimal interval or range $$\Delta$$t between data points from the raw data where the measurements contain errors that are independent from those of the other points selected; this in order to maximize the number of data points to do the fitting. Then I could employ non-linear least-squares regression using F(t, y(t); θ) = 0 and numerically extract meaningful values of the kinetic constants from this subset of selected data.

The non-linear kinetic model, F(t, y(t); θ) = 0, is given as a transcendental implicit equation according to Profile Expression. Rate constants are denoted by k1 and k2 and their values are not known beforehand.

Reference material for consultation, such as scientific research papers and books, would also be highly appreciated directly related or within the context of my query. Thank you.

## 1 Answer

You can apply a transformation to the data to make it look like a conventional time series. For example, the model follows a pattern $$e^{-ct}$$, then apply a transformation $$y'(t)=y(t)e^{ct}$$, and you can apply conventional ARMA analysis on the transformed series.

• @ Paul: Your transformation works well when one has an explicit function by removing the underlying curve and then making the time-series analysis on y'(t). But how is that actually translated to the implicit form? Your non-linear model y(t) is expressed explicitly, whereas in my case the model function is given implicitly. Is there any way to make a time-series analysis directly on the original kinetic profile instead of making a transformation to look for serial autocorrelation - anybody? – CCMJ Mar 3 at 9:23
• You said in the question that the pattern is "obeying a proposed mechanistic kinetic law." If you have that proposed law, fine, use it to transform the data. If not, edit the question to get to what you are really asking. – Paul Mar 3 at 16:35