## 2 Materials and Methods
### Kinetic Analyses  
The dynamic response of physiological variables towards an increase in physical load can be described by a number of features (Figure 1). These features were extracted using both exponential models and BJ-TF models. The following first order model was used as a basis for exponential modelling:  
\[0\leq t\leq TD:\qquad\qquad\Delta VO_{2}(t)=0, \tag{1}\] \[TD<t\leq 180:\quad\Delta VO_{2}(t)=Amp\big{(}1-e^{-(\frac{t-TD}{TC}) }\big{)},\]  
where \(\Delta VO_{2}(t)\) is the difference between the pre-processed VO\({}_{2}\) time series and the baseline VO\({}_{2}\) level (\(VO_{2,baseline}\)), i.e., \(\Delta VO_{2}(t)=VO_{2}(t)-VO_{2,baseline}\). \(VO_{2,baseline}\) is calculated as the mean VO\({}_{2}\) value of the last 30 s of unloaded cycling. The model parameters (\(Amp\), \(TD\) and \(TC\)) were estimated using non-linear least squares (Matlab 2015b, MathWorks Inc., Natick, Massachusetts, USA). Three mono-exponential modelling methods have been described in COPD literature. They will be referred to as method A [6; 8], B [10; 11] and C [9; 12]. Method A assumes there is no \(TD\) in the VO\({}_{2}\) response (\(TD=0\) s). Method B estimates the three model parameters using data from t = 0 s (load onset) until t = 180 s, whereas method C only uses data between t = 20 s and t = 180 s to exclude the contribution of the cardiodynamic phase (phase I). Time evolutions and autocorrelation tests of the model residuals were checked to assess whether the model residuals retained any meaningful structure. Remaining structure in the model residuals indicates that the residuals contain information that is not captured by the model. The relative amplitude of these residuals was assessed by the normalised root-mean-squared error (\(NRMSE\)), defined as the root-mean-squared error divided by the amplitude of the VO\({}_{2}\) response:  
\[NRMSE=\sqrt{\frac{\sum_{t=1}^{n}\text{residuals}(t)^{2}}{n}}.Amp^{-1}. \tag{2}\]  
A discrete-time, single-input, single-output BJ-TF model with time-invariant parameters can generally be written as follows:  
\[y(t)=\frac{B\big{(}z^{-1}\big{)}}{A(z^{-1})}u(t-TD)+\frac{C\big{(}z^{-1}\big{)} }{D(z^{-1})}e(t), \tag{3}\]  
where \(y(t)\) is the model output at time t, \(u(t)\) is the model input at time t, \(z^{-1}\) is the backwards shift operator (i.e. z\({}^{-\text{s}}\) y(t) = y(t-s)), \(e(t)\) is zero mean white noise and \(A(z^{-1})\), \(B(z^{-1})\), \(C(z^{-1})\) and \(D(z^{-1})\) are polynomials containing the model parameters:  
\[ \begin{split}& A\big{(}z^{-1}\big{)}=1+a_{1}z^{-1}+\cdots+a_{m}z^{- m},\\ & B\big{(}z^{-1}\big{)}=b_{0}+b_{1}z^{-1}+\cdots+b_{n}z^{-n},\\ & C\big{(}z^{-1}\big{)}=1+c_{1}z^{-1}+\cdots+c_{p}z^{-p},\\ & D\big{(}z^{-1}\big{)}=1+d_{1}z^{-1}+\cdots+d_{q}z^{-q}.\end{split}  \tag{4}\]  
Thus, the full BJ-TF model (3) is a combination of a 'noise-free' system model (first term) and a noise model (second term). As phase II is known to follow a first-order exponential response,the system model was defined as a first order model with one gain parameter. This transformed the full model (3) as follows:  
\[\Delta VO_{2}(t)=\frac{b_{0}}{1+a_{1}z^{-1}}W(t-TD)+\frac{C\big{(}z^{-1}\big{)}}{ D(z^{-1})}e(t), \tag{5}\]  
where the input variable \(W(t)\) was defined as:  
\[ \begin{array}{ll}t\leq 0:&W=0\;Watt,\\ t>0:&W=75\%\;of\;the\;maximal\;load\;reached\;during\;CPET.\end{array} \]  
The order of the noise model, given by the order of polynomial \(D(z^{-1})\), was unknown and was therefore identified by comparing the performances of models with noise models up to order three in \(C(z^{-1})\) and/or \(D(z^{-1})\). Full model fits were inspected visually and the residuals were checked using autocorrelation tests and NRMSE values. In addition, the full models with different noise structures were compared by their Young Identification Criterion (YIC) value [16]. YIC combines goodness of fit and reliability of the parameter estimations, while penalising for over-parameterisation. A more negative YIC value indicates a better model performance.