Keep two running trackers: - tracker A = “how jumpy prices have been lately” (volatility) - tracker B = “how prices have moved lately” (returns), but scaled by tracker A

Update both after every new price. The ratio tells you how many standard deviations today’s price move is, smoothed over time. Large positive = trend up, large negative = trend down.

Step-by-step

1. Compute a scale-free return
   $$r_{t}= \frac{P_t-P_{t-1}}{P_{t-1}}\quad\text{or}\quad\ln(P_t/P_{t-1}) $$ Using percent/log makes the metric independent of the absolute price level.

2. Update volatility (σ²) first
   Exponential moving average of squared returns with decay η:
   $$\sigma^{2}{t}= (1-\eta)\,\sigma^{2}{t-1}+\eta\,r_{t}^{2} $$ This is a one-line “memory” of recent squared moves; smaller η = longer memory.

3. Normalise today’s return $$z_t = \frac{r_t}{\sigma_t+\epsilon} $$ ε is a tiny constant to prevent division by zero.
4. Update the signal
   Another exponential average, but of the z-scores: $$ \phi_t = (1-\eta)\,\phi_{t-1} + \sqrt{\eta}\,z_t $$ The \(\sqrt{\eta}\) factor keeps the signal’s variance near 1 when returns are white noise.

So φ = +2.0 means: the exponentially weighted average of 112 day returns is two standard deviations above zero.