(Gibson, W.). T comes from the equation which links positions to speed and acceleration. Because it represents the time spent between to position (i.e. choices in the movies database), we put it equal to 1 (simulations have shown that its value does not influence the computations).

In the state space formulation given in equations (2.a) and (2.b), we have:

wkis assumed to be Gaussian random vector which allows us to consider random behaviours of the observed users and wk N 0,Q
vkdenotes perturbations on the measurements (in our case, this perturbations are minimized) 
vk *N(0,R*)
Measurement matrix H has the following structure:
Knowing that we can derive the 

![0_image_0.png](0_image_0.png)

equations of the Kalman predictor. This predictor will be able to predict the future state on the trajectory. These equations are the following (for further details, concerning how to obtain these equations, see (Gevers, M. and Vandendorpe)):
The computed prediction is given by:

$$\left\{\begin{array}{l l}{{\hat{X}_{k+1/k}=\hat{X}_{k/k-1}+K_{k}\big(Z_{k}-H\hat{X}_{k/k-1}\big)}}&{{(4)}}\\ {{}}&{{=\big(A-K_{k}C\big)\hat{X}_{k/k-1}+K_{k}Z_{k}}}\end{array}\right.$$
$${\mathrm{The~gain~of~the~filter~is:}}$$
$$K_{k}=A P_{k/k-1}H^{T}\left(H P_{k/k-1}H^{T}+R\right)^{-1}\ \ \ \ \ (5)$$

The evolution of the uncertainty on the state estimation is given by:

$$P_{k+1/k}=AP_{k/k-k}A^{T}\tag{6}$$ $$-AP_{k/k-1}H^{T}\Big{(}HP_{k/k-1}H^{T}+R\Big{)}^{-1}HP_{k/k-1}A^{T}$$

Where: 

$$\begin{array}{c}{{\hat{X}_{0/-1}=X_{0}}}\\ {{P_{0/-1}=P_{0}}}\\ {{\hat{X}_{k+1/k}}}\\ {{\ }}\\ {{\ }}\\ {{\left(Z_{k}-H\hat{X}_{k/k-1}\right)}}\\ {{\ }}\\ {{\ }}\\ {{\hat{X}_{k+1/k}}}\end{array}$$
are the initial conditions
ˆ X k 1/ kis the prediction of the state; it is the optimal estimation of the state of the model ˆ X k / k 1is called the Innovation sequence ˆ X k 1/ kis the state prediction at time k+1 knowing states from time 0 to time k
Using this algorithm, we are going to 

![0_image_1.png](0_image_1.png)

consider our target as described in figure 2: