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Miha Kralj
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- LTMA (Linear Trend Moving Average): Introduces a predictive moving average using dual cascaded EMAs for trend estimation. - MCNMA (McNicholl EMA): Implements a zero-lag TEMA using a cascaded EMA structure for enhanced responsiveness. - NLMA (Non-Lag Moving Average): Utilizes a damped cosine kernel to achieve reduced lag in moving averages. - NMA (Natural Moving Average): Adapts smoothing based on volatility profiles using a square-root kernel. - NYQMA (Nyquist Moving Average): Applies the Nyquist-Shannon theorem to prevent aliasing in cascaded moving averages. - RAIN (Rainbow Moving Average): Combines multiple SMA layers with weighted averages for multi-scale smoothing. - TRAMA (Trend Regularity Adaptive Moving Average): Adapts smoothing based on the frequency of new highs and lows in price data.
114 lines
4.2 KiB
Markdown
114 lines
4.2 KiB
Markdown
# SOLAR: Solar Cycle Indicator
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SOLAR models Earth's seasonal position relative to the Sun using astronomical ephemeris calculations. Output oscillates continuously from $-1.0$ (Winter Solstice) through $0.0$ (Equinoxes) to $+1.0$ (Summer Solstice), providing a smooth, mathematically precise seasonal phase for econometric modeling. Like LUNAR, the indicator is purely time-based, requires no price data, and has zero warmup since the calculation is deterministic from any timestamp.
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## Historical Context
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Seasonal adjustments are fundamental to econometric analysis. Agricultural commodities, retail sales, energy consumption, and tourism all exhibit strong annual patterns. Traditional approaches use monthly dummy variables or calendar-based lookup tables, creating discontinuities at month boundaries. Astronomical seasonality offers a continuous, smooth alternative: the Sun's ecliptic longitude provides an exact phase position within the annual cycle at any time resolution. The implementation derives from Jean Meeus' *Astronomical Algorithms* (1998), computing the Sun's geometric mean longitude, mean anomaly, and equation of center with sufficient precision ($\pm 0.01°$) for financial applications. Unlike lunar cycles, the tropical year's length varies by only seconds over centuries, making solar seasonality highly predictable.
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## Architecture & Physics
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### 1. Julian Date Conversion
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$$JD = \frac{UnixMs}{86400000} + 2440587.5$$
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$$T = \frac{JD - 2451545.0}{36525.0}$$
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where $T$ is Julian centuries from the J2000.0 epoch.
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### 2. Geometric Mean Longitude
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The Sun's mean position in its apparent orbit:
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$$L_0 = 280.46646 + 36000.76983T + 0.0003032T^2$$
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### 3. Mean Anomaly
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Angular distance from perihelion:
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$$M = 357.52911 + 35999.05029T - 0.0001537T^2$$
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### 4. Equation of Center
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Correction for orbital eccentricity ($e \approx 0.0167$):
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$$C = (1.914602 - 0.004817T - 0.000014T^2) \sin M + (0.019993 - 0.000101T) \sin 2M + 0.000289 \sin 3M$$
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### 5. True Ecliptic Longitude
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$$\lambda_{Sun} = L_0 + C$$
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### 6. Seasonal Index
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$$Solar = \sin(\lambda_{Sun})$$
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This maps: Vernal Equinox ($\lambda = 0°$) $\to 0$, Summer Solstice ($\lambda = 90°$) $\to +1$, Autumnal Equinox ($\lambda = 180°$) $\to 0$, Winter Solstice ($\lambda = 270°$) $\to -1$.
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### 7. Complexity
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$O(1)$ per timestamp. No state required. Zero warmup. The tropical year is approximately 365.242 days.
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## Mathematical Foundation
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### Parameters
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| Parameter | Description | Default | Constraint |
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|-----------|-------------|---------|------------|
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| (none) | No user-configurable parameters | | |
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The calculation is entirely determined by the input timestamp.
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### Pseudo-code
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```
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function SOLAR(timestamp):
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// Julian date
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JD ← timestamp_to_unix_ms / 86400000 + 2440587.5
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T ← (JD - 2451545.0) / 36525.0
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// Geometric mean longitude
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L0 ← FMA(T, FMA(T, 0.0003032, 36000.76983), 280.46646)
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L0 ← mod(L0, 360)
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// Mean anomaly
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M ← FMA(T, FMA(T, -0.0001537, 35999.05029), 357.52911)
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M ← mod(M, 360)
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// Equation of center
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C ← FMA(T, FMA(T, -0.000014, -0.004817), 1.914602) · sin(M)
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+ FMA(T, -0.000101, 0.019993) · sin(2M)
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+ 0.000289 · sin(3M)
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// True ecliptic longitude
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λ ← L0 + C
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// Seasonal index
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solar ← sin(λ · π / 180)
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emit solar
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```
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### Seasonal Correspondence (Northern Hemisphere)
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| Date (approx.) | $\lambda_{Sun}$ | Solar Value | Season |
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|-----------------|-----------------|-------------|--------|
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| March 20 | $0°$ | $0.0$ | Vernal Equinox |
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| June 21 | $90°$ | $+1.0$ | Summer Solstice |
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| September 22 | $180°$ | $0.0$ | Autumnal Equinox |
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| December 21 | $270°$ | $-1.0$ | Winter Solstice |
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### Output Interpretation
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| Condition | Meaning |
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|-----------|---------|
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| $Solar \approx +1$ | Peak summer (Northern Hemisphere) |
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| $Solar \approx -1$ | Peak winter (Northern Hemisphere) |
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| $Solar = 0$ (rising) | Spring equinox crossing |
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| $Solar = 0$ (falling) | Autumn equinox crossing |
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| Southern Hemisphere | Negate the output |
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## Resources
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- **Meeus, J.** *Astronomical Algorithms*. 2nd ed., Willmann-Bell, 1998.
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- **USNO** *Astronomical Almanac*. U.S. Government Publishing Office (annual reference for solstice/equinox verification).
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