Expand the cascaded components before comparing the lines
Let x be the current source price, N a positive integer period, alpha = 2 / (N + 1), and q = 1 − alpha. Each stage uses the current output of the preceding stage and its own previous-bar state. The subtraction in the final combinations changes the weighting; the number of EMA stages alone does not describe the smoothness of the output.
- E2 by itself is a twice-smoothed EMA; E3 by itself is a three-times-smoothed EMA. Neither is the final DEMA or TEMA.
- At N = 5, alpha = 1/3. A one-unit change in the current source, with previous states held fixed, changes DEMA by 2alpha − alpha² = 5/9 and TEMA by 3alpha − 3alpha² + alpha³ = 19/27. These sensitivities are approximately 0.556 and 0.704, despite identical periods.
- For N = 1, alpha = 1 and all stages equal the current source. Both outputs equal price, so a strict price-versus-average crossing rule produces no signal. Missing observations and different seeding conventions require separate handling.
| Component | Current-bar calculation | Role |
|---|---|---|
| E1 | alpha × x + q × previous E1 | EMA of the source. |
| E2 | alpha × current E1 + q × previous E2 | EMA of E1, with the same N. |
| E3 | alpha × current E2 + q × previous E3 | EMA of E2, with the same N. |
| DEMA | 2 × E1 − E2 | E1 plus the correction E1 − E2. |
| TEMA | 3 × E1 − 3 × E2 + E3 | DEMA plus the extra correction E1 − 2 × E2 + E3. |