System identification procedures

Previous lymphocyte recirculation and migration studies have been reviewed in Chapter 1. None of these studies appear to have considered the possibility of system identification modelling. The system identification adopted in this thesis has been strongly influenced by the work of Ljung (Ljung, 1987; Ljung, 1991), Modarreszadeh (Modarreszadeh et al., 1995) and Lai (Lai and Bruce, 1997). In those studies, these researchers were able to identify the dynamics of the respiratory system in several conditions using PEM models (Modarreszadeh et al., 1995; Lai and Bruce, 1997).

5.3.1 Lymph node system

The lymph node under investigation is assumed to be represented as a linear time invariant (LTI) system with single-input-single-output (SISO) as shown in the diagram in Figure 5.2(b). The system input u(t) and output y(t) represent the percentages of labelled lymphocytes in the venous blood and the efferent lymph of the studied lymph node respectively. Such an ideal lymph node model is in accord with the fact that, under normal physiological conditions, most

lymphocytes (ie more than 95%) found in the efferent lymph are derived from blood rather than from other sources (Hall and Morris, 1962; Hall and Morris,

1965b). In this model, the inputs from other sources, such as from the afferent lymph, are ignored as they contribute less than 5% to the model output.

5.3.2 Experimental data

As mentioned in section 3.1 of Chapter 3, the present experiments can be categorised into two groups namely the total lymphocyte population and several subset (ie CD4, CD5, CD8 and CD45R) studies. To identify the lymph node system, it is essential that adequate experimental data are collected. In the present study, after some preliminary trials, any experiments that provided less than 100 hours of data (ie the experimental time was less than 100 hours) were discarded. The remainder of the experiments (15 out of 21 for the total

population studies and 7 out of 11 for the subset studies) were subsequently incorporated into the whole system identification procedures.

5.3.3 Matlab as a tool

The system identification procedure and all of the graphic presentations in this chapter were performed by commands and algorithms implemented in Matlab

and the Matlab System Identification Toolbox (Mathsoft Company-MA, USA). Matlab is a powerful interactive commercial program for numeric computation and data presentation. It has been widely used in solving complex problems in several fields including applied mathematics, physics, chemistry and engineering. The system identification toolbox of Matlab using an iterative Gauss-Newton algorithm has been satisfactorily implemented (Ljung, 1991). Details of this algorithm and the commands used in the present study are included in Appendix 2 to this thesis.

5.3.4 Determination of the system delay

The first step in the system identification procedure is to determine the system delay, nk, of selected candidate models. Selection of the appropriate value of the system delay will ensure a good performance from the final model. The following several methods can be used to determine the system delay.

• Generally, observation of a graph of the actual input-output data in the time domain, and of the time difference between the first appearance of the input and of the output may indicate an approximate value for the system delay. However, this procedure can not ensure reliability. Indeed, in the case of many data sets, this procedure would fail to provide clear indications for the identification of the system delay.

• Alternatively, the system delay can be detected by an impulse response estimation obtained from a correlation analysis of the input and the output. This can be undertaken in the Matlab environment (using the command “era”). The determination of the system delay can be identified from visual observation of the identified impulse response.

• In this thesis, a more “ad hoc” searching procedure has been used for system delay determination. In general practice, a low order ARX model

(na=0, nb— 1, nk =d) has often been used as a prototype model to identify the

system delay (d). Modarreszadeh et al used a low order Box-Jenkins model to obtain an optimal delay which gave a minimal FPE value (Modarreszadeh

et al., 1995). After several trials, a low order output error (OE) model (/?&= 1,

nj= 1, nk =d) was selected as a good prototype model for the data in the present study. As the first step in using this, the prediction cost function (VN)

is calculated for this OE model with a delay of 1 (^= 1 , n/= 1, nk =1). Then the value of the system delay is increased by 1 and the prediction cost function is re-calculated. The value of the system delay is defined over the range 0..10. Because a smaller value of the prediction cost function implies a better model, this function was used as an indication for determination of the

system delay. The prediction cost is depicted as a function of the system delay and the value of the system delay is selected at the region where an abrupt change of the prediction cost occurs (this can be observed by a knee

in the graph that shows the prediction cost as a function of the system delay). Subsequently, the system delay is definitively selected and maintained as a constant value throughout the remaining steps of the identification procedure (in Section 5.3.5). The system delay could be also obtained by identifying a given range of ARX models (na, nb, nk)and the prediction cost versus nkfor a series values of naand nbcould be plotted using the command “arxstruc”.

However, in the present study, OE models were chosen to determine the system delay on the basis of their established good simulation results.

5.3.5 Selection of model structures from candidate models

The four candidate models suggest a variety of parameters that have to be selected for the final model. In this step, a vector of model parameters,#, is identified for each candidate model structure (ie ARX, ARMAX , OE and BJ model structures with a different number of parameters). The maximal order of each model parameter vector is, however, limited to the value of 3. For example, the vectors of OE and ARMAX model parameters are (nb, n/, nk)and (na, nb, nc, nk)respectively with na, nb, nc, nf <3. The system delay identified from Section 5.3.4 is used for the four candidate models. The individual value of the final prediction error (FPE) is calculated by means of (5.27) for the corresponding model structures. In a process similar to that used for the selection of the system delay, a graph representing the relationship between the FPE and the model dimension (ie the total number of identified parameters in each model) provides an indication of the model to be selected in each model structure.

In Matlab, the parameter estimation algorithm, which requires an iterative search for the minimum of (5.25) with a special start-up procedure, can be well

implemented. The Gauss-Newton minimisation procedure is carried out until the norm of the Gauss-Newton direction is less than a certain level (ie tolerance) (Ljung, 1991). The four candidate models have been identified according to the prediction criterion (5.25) whereas a good fit to the actual data, ie a good

simulation performance, has been the main goal in the present study. Therefore, the selection of the final model will be dependent upon the basis of the simulation criterion (5.28). Consequently, it would be expected that the final model will be either an OE or a BJ model as previously described in section 5.2.4.

5.3.6 Model validation

Because of the nature of the present study, most of the experimental data sets (after interpolation with a sampling period =1 hour) contain only about 100-240 data points. Such data sets are relatively small. The technique of “cross

validation”, a validation method that is performed on a data set that is different

from the data set used to identify the model, can not be performed in the present study as it requires a considerable number of data points. Thus in the present study, model validation had to be undertaken on the same set of data that had been used to identify the four candidate models.

At this stage, the final four models have been selected according to the steps described in Sections 5.3.4-5.3.5. A validation test is required to ensure the adequacy of these models. In the present study, the residual analysis method

has been selected as a validation test and will be performed on the final four models. Briefly, the residual (or error) is the difference between the actual output and the model output. The correlation of the residual with the input signal (cross correlation of the residual against the input, R&) can reveal information (in the residual) that may help to explain the process, ie information that has not been used, R ^ s used for validation of the input-output dynamics. The correlation of the residual with itself (autocorrelation of the residual, R££) is used for the validation of the noise model. However, R££ is not of significance in this study since it is concerned only with simulation results, ie validation of the input-output dynamics. More details on residual analysis can be found in Ljung (Ljung, 1987; Ljung, 1991).