= Kalman filtering of a moving scalar state
{c}
{title2=$P_t^- =P_{t-1}+Q,\quad K_t=P_t^-/(P_t^-+R)$}
Consider a <local-level state-space model> $x_t=x_{t-1}+w_t$, $y_t=x_t+v_t$, with mutually <independent> centered <Gaussian> noises of <variances> $Q=1/4$ and $R=1$. Start with $x_0\sim N(10,4)$, independently of these noises. Before the first observation, the <Kalman prediction step> gives $m_1^-=10$, $P_1^-=17/4$. If $y_1=14$, the <Kalman gain> is $17/21$, and the <posterior mean> and <posterior variance> become
$$
m_1=10+\frac{17}{21}(14-10)=\frac{278}{21},\qquad P_1=\frac{17}{21}.
$$
The next <Kalman prediction step> keeps the mean $m_2^-=278/21$ but adds fresh <process noise> <variance>, giving $P_2^-=89/84$. If $y_2=12$, then
$$
K_2=\frac{89}{173},\qquad
m_2=\frac{278}{21}+\frac{89}{173}\left(12-\frac{278}{21}\right)=\frac{2180}{173},\qquad
P_2=\frac{89}{173}.
$$
Thus new uncertainty is introduced at each transition and reduced by each measurement. It does not shrink to zero merely because many measurements have arrived: the state continues to change.
\Image[/kalman-local-level-tracking.png]
{title=A <Kalman filter> follows a changing scalar state and retains uncertainty about each new transition}
{description=Top: one seeded synthetic <random walk>, its noisy observations and the filtered <posterior mean>, starting from $N(10,4)$ with $Q=1/4$, $R=1$. The shaded region is the mean plus or minus two posterior <standard deviations>, a pointwise conditional band under the stated model. Bottom: the <Kalman gain> approaches its positive steady value. Setting $Q=0$ instead asserts a constant state and eventually stops adapting appreciably. This single generated record illustrates the mechanism; it is not an empirical accuracy study.}
The <steady-state local-level Kalman gain> here is
$$
K_\infty=\frac{\sqrt{17}-1}{8}\approx0.390388.
$$
With $R=1$, the limiting <posterior variance> has the same numerical value. By contrast, the static-state model $Q=0$ with $P_0=4$ gives $P_t=1/(t+1/4)$ and $K_t=1/(t+1/4)$. That decreasing gain is appropriate for a state that truly remains constant. Applied to the changing <random walk>, it builds confidence in the wrong model. The choice of <process noise> is therefore a statement about state evolution, rather than just an instruction to make a plot look smoother.
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