TEMA1 from nested exponential averages, then a two-horizon MACD

Assemble TEMA1 from three nested averages

TEMA1 is assembled as three times a first exponential moving average, minus three times that average applied to its own output, plus the same average applied a third time. Exponential smoothing is a recursively weighted average of recent observations; applying the same smoother once, twice, and three times supplies those three terms.

A moving average in this construction is a lookback smoother of ordered closing prices. The same TEMA1 series can stand alone or serve as each leg of a later two-horizon difference.

A selectable lookback for the worked series

The worked TEMA1 example uses a 26-observation lookback, and that lookback is selectable by the user.

A related double-smoothed average is not a finite nest

A related double-smoothed average cannot be written as a finite nest of moving-average calls because the definition refers to its own prior values. TEMA1 stays inside a finite nest: the second term is the first exponential moving average applied to its own output, and the third term applies that same average once more.

MACD-TEMA1 as a short TEMA1 minus a long TEMA1

MACD is a two-horizon difference of smoothers, written here as a short TEMA1 minus a long TEMA1 rather than a short exponential average minus a long one. MACD-TEMA1 is the difference between a 12-period TEMA1 and a 26-period TEMA1.

The two TEMA1 legs may be written as one difference only if each nested formula is fully parenthesized before subtraction.

Smoothing constants and replaceable lengths

The two moving-average lengths in the TEMA1 difference are presented as replaceable parameters rather than fixed constants. The smoothing constant is the exponential weight; the conventional MACD pair 0.15 and 0.075 maps to the non-integer periods 12.3333 and 25.6667 and may replace the rounded 12 and 26 lookbacks.