9.2 What are the Experimental Unit and the Unit of Prediction?

Nearly 2600 wavelengths are measured each day for two weeks for each of 15 small-scale bioreactors. This type of data forms a hierarchical structure in which wavelengths are measured within each day and within each bioreactor. Another way to say this is that the wavelength measurements are nested within day which is further nested within bioreactor. A characteristic of nested data structures is that the measurements within each nesting are more related to each other than the measurements between nesting. Here, the wavelengths within a day are more related or correlated to each other than the wavelengths between days. And the wavelengths between days are more correlated to each other than the wavelengths-between-bioreactors.

These intricate relationships among wavelengths within a day and bioreactor can be observed by through a plot of the autocorrelations. An autocorrelation is the correlation between the original series and each sequential lagged version of the series. In general, the autocorrelation decreases as the lag value increases, and can increase at later lags with trends in seasonality. The autocorrelation plot for the wavelengths of the first bioreactor and several days are displayed in Figure 9.4 (a) (which is representative of the relationships among wavelengths among other bioreactor and day combinations). This figure indicates that the correlation between wavelengths are different across days. Later days tend to have higher between-wavelength correlations but, in all cases, it takes many hundreds of lags to get the correlations below zero.

(a) Autocorrelations for selected lags of wavelengths for small-scale bioreactor 1 on the first day. (b) Autocorrelations for lags of days for average wavelength intensity for small-scale bioreactor 1.

Figure 9.4: (a) Autocorrelations for selected lags of wavelengths for small-scale bioreactor 1 on the first day. (b) Autocorrelations for lags of days for average wavelength intensity for small-scale bioreactor 1.

The next level up in the hierarchy is within bioreactor across days. Autocorrelations can also be used here to understand the correlation structure at this level of the hierarchy. To create the autocorrelations, the intensities across wavelengths will be averaged within a bioreactor and day. The average intensities will then be lagged across day and the correlations between lags will be created. Here again, we will use small-scale bioreactor 1 as a representative example. Figure 9.4 (b) shows the autocorrelations for the first 13 lagged days. Here correlations for the first lag is greater than 0.95, with correlations tailing off fairly quickly.

The top-most hierarchical structure is the bioreactor. Since the reactions occurring within one bioreactor do not affect what is occurring within a different reactor, the data at these levels are independent of one another. As discussed above, data within bioreactor are likely to be correlated with one another. What is the unit of prediction? Since the spectra are all measured simultaneously, we can think of this level of the hierarchy to be below the unit of prediction; we would not make a prediction at a specific wavelength. The use case for the model is to make a prediction for a bioreactor for a specific number of days that the cells have been growing. For this reason, the unit of prediction is day within bioreactor.

Understanding the units will guide the selection of cross validation method (Section 3.4) and is crucial for getting an honest assessment of a model’s predictive ability on new days. Consider, for example, if each day (within each bioreactor) was taken to be independent experimental unit and V-fold cross-validation was used as the resampling technique. In this scenario, days within the same bioreactor will likely be in both the analysis and assessment sets (Figure 3.5). Given the amount of correlated data within day, this is a bad idea since it will lead to artificially optimistic characterizations of the model.

A more appropriate resampling technique is to leave out all of the data for one or more bioreactors out en mass. A day effect will be used in the model, so the collection of data corresponding to different days should move with each bioreactor as they are allocated with the analysis or assessment sets.

In the last section, a comparison is made between these two methods of resampling.

The next few sections describe sequences of preprocessing methods that are applied to this type of data. While these methods are most useful for spectroscopy, they illustrate how the preprocessing and feature engineering can be applied to different layers of data.