## 3 Results
### Introduction of Online Personalisation and Adaptive Modelling Algorithm
#### 3.4.3 Online Parameter Estimation Algorithm  
As explained earlier (Section 2.1.3), the general setting of regression analysis is to identify a relationship between a response variable (\(Y\)) and one or several explanatory variables (predictors) (\(X\)) by using a learning sample [43]. In a prediction framework, the main assumption for predicting \(Y\) on a new sample of \(X\) observations is that the regression model (with the general form \(Y\ =\ f(X)\ +\ \epsilon\), where \(\epsilon\) represents the model residuals) is still valid. Unfortunately, this assumption is not valid in the present case, where the thermal comfort of the individual cyclist is strongly dependent on many personal- and time-dependent factors [11]. Therefore, in this study, we adapted the original regression model (3) to a new sample (observations) by estimating a transformation [41; 43] between the original regression function (\(f(X)\)) and the new one (\(\tilde{f}(\tilde{X})\)) while still using the same model variables and structure. Ordinary least squares (LS) is one of the most popular regression techniques, which was used here for parameter estimation of the developed offline model. However, for the online parameter estimation and in the presence of unknown parameter changes, its adaptive versions--the sliding or moving LS, recursive least squares (RLS) and recursive partial least squares (PLS)--are widely used [41; 44; 45].  
In the present paper, the RLS algorithm is suggested for online personalisation and adaptive modelling of under-helmet thermal comfort. The suggested RLS algorithm has the advantage of being simple and computationally efficient for wearable and adaptive sensing, which was the case in the present work. In the RLS algorithm, the new regression vector \(\tilde{\beta}\), as in Equation (4), can be estimated recursively as follows [25; 46; 47]:  
\[\mathbf{P}_{n+1}=\mathbf{P}_{n}-\frac{\mathbf{P}_{n}\big{(}\mathbf{X}_{n+1} \big{)}^{T}\mathbf{X}_{n+1}\mathbf{P}_{n}}{1+\mathbf{X}_{n+1}\mathbf{P}_{n} (\mathbf{X}_{n+1})^{T}}, \tag{5}\]  
\[\tilde{\beta}_{n+1}=\tilde{\beta}_{n}+\mathbf{P}_{n+1}(\mathbf{X}_{n+1})^{T} \Big{(}\tilde{\mathbf{T}}_{C_{n+1}}-\mathbf{X}_{n+1}\tilde{\beta}_{n}\Big{)} \tag{6}\]where \(\mathbf{P}_{n}=\left(\mathbf{X}_{n}{}^{T}\mathbf{X}_{n}\right)^{-1}\). This recursive algorithm is efficient for cases where the regression vector \(\tilde{\mathbf{\beta}}\) is a function of time (time varying). However, in the case of adaptive modelling with streaming data, due to the arrival of new samples, the influence of new observations decreases gradually and the ability to track the changes in \(\tilde{\mathbf{\beta}}\) will be lost. Hence, to mitigate this, the widely used and popular forgetting factor approach [48] is proposed in this paper. The approach of forgetting here is based on gradually discarding older data in favour of more recent information. In the least-squares method, forgetting can be viewed as giving less weight to older data and more weight to recent data [48; 49]. Hence, the forgetting factor, \(\lambda\), was introduced to (5) as follows [41]:  
\[\mathbf{P}_{n+1}=\frac{1}{\lambda}\bigg{(}\mathbf{P}_{n}-\frac{ \mathbf{P}_{n}(\mathbf{X}_{n+1})^{T}\mathbf{X}_{n+1}\mathbf{P}_{n}}{\lambda+ \mathbf{X}_{n+1}\mathbf{P}_{n}(\mathbf{X}_{n+1})^{T}}\bigg{)} \tag{7}\]  
where \(\lambda\in(0,\,1]\). The forgettiing factor \(\lambda\) operates as a weight, which diminishes for more remote data and expands for more recent data [48; 49]. The main difference here between (5) and (7) is that in conventional RLS (5), the covariance vanishes to zero with time, losing its capability to keep track of changes in the regression vector \(\tilde{\mathbf{\beta}}\). In (7), however, the covariance matrix is divided by \(0\leq\lambda<1\) at each update. This slows down the fading out of the covariance matrix [49].