# T3: Tillson T3 Moving Average

## What It Does

The T3 Moving Average is a hyper-smooth, low-lag indicator developed by Tim Tillson. It uses a unique "volume factor" to control how aggressively the moving average tracks the price. Unlike standard moving averages that simply smooth data, T3 applies multiple layers of smoothing (specifically, a generalized DEMA) to create a curve that is exceptionally smooth yet responsive to significant price moves.

## Historical Context

Tim Tillson introduced the T3 in his article "Smoothing Techniques for More Accurate Signals" in *Technical Analysis of Stocks & Commodities* (January 1998). His goal was to improve upon the lag characteristics of traditional moving averages and the overshoot problems of DEMA (Double Exponential Moving Average).

## How It Works

### The Core Idea

T3 is essentially a "moving average of a moving average of a moving average..." but using a generalized DEMA (GD) instead of a simple EMA.

- **GD (Generalized DEMA):** A mix of EMA and DEMA controlled by a volume factor $v$.
- **T3:** Applying the GD filter six times in sequence ($GD(GD(GD(GD(GD(GD(Price))))))$).

The "Volume Factor" ($v$) determines how much "DEMA" (fast, overshooting) vs "EMA" (slow, lagging) is mixed in.

- $v=0$: T3 behaves like a triple EMA (very smooth, some lag).
- $v=1$: T3 behaves like a DEMA (very fast, prone to overshoot).
- $v=0.7$: The standard default, offering a balance.

### Mathematical Foundation

1. **Generalized DEMA (GD):**
   $$ GD(x, v) = EMA(x) \times (1 + v) - EMA(EMA(x)) \times v $$

2. **T3 Sequence:**
   $$ e1 = GD(Price) $$
   $$ e2 = GD(e1) $$
   $$ e3 = GD(e2) $$
   $$ ... $$
   $$ T3 = e6 $$

### Implementation Details

Our implementation uses the recursive GD formula for O(1) updates.

- **Complexity:** O(1) per update (6 GD calculations).
- **Stability:** Requires a warmup period to stabilize all 6 internal layers.

## Configuration

| Parameter | Default | Purpose | Adjustment Guidelines |
|-----------|---------|---------|----------------------|
| Period | 14 | Smoothing period | Standard lookback. |
| Volume Factor (v) | 0.7 | Responsiveness | 0.7 is standard. Lower (0.1-0.5) = smoother/slower. Higher (0.8-1.0) = faster/responsive. |

## Performance Profile

| Operation | Complexity | Description |
|-----------|------------|-------------------|
| Streaming update | O(1) | 6 layers of GD calculation |
| Bar correction | O(1) | Efficient state rollback |
| Batch processing | O(N) | Single pass through data |
| Memory footprint | O(1) | Stores state for 6 internal layers |

## Interpretation

### Trading Signals

#### Trend Identification

- **Smoothness:** T3 is famous for filtering out "noise" better than almost any other MA. If T3 is rising, the trend is likely real, not just a blip.
- **Crossovers:** Price crossing T3 is a significant event due to the indicator's smoothness.

### When It Works Best

- **Noisy Markets:** T3 shines in markets with lots of wicks and erratic movement, where standard EMAs would get chopped up.

### When It Struggles

- **Lag:** Despite its clever math, applying a filter 6 times introduces lag. It will turn after the market turns, not with it.

## Architecture Notes

This implementation makes specific trade-offs:

### Choice: 6 Layers

- **Implementation:** We implement the standard "T3" which implies 6 layers of smoothing.
- **Rationale:** While "T2" or "T4" are possible, "T3" (6 layers) is the industry standard definition.

## References

- Tillson, Tim. "Smoothing Techniques for More Accurate Signals." *Technical Analysis of Stocks & Commodities*, V. 16:1 (33-37), 1998.

## C# Usage

### Streaming Updates (Single Instance)

```csharp
using QuanTAlib;

var t3 = new T3(period: 14, vFactor: 0.7);

// Process each new bar
TValue result = t3.Update(new TValue(timestamp, closePrice));
Console.WriteLine($"T3: {result.Value:F2}");

// Check if buffer is full
if (t3.IsHot)
{
    // Indicator is fully initialized
}
```

### Batch Processing (Historical Data)

```csharp
// TSeries API
TSeries prices = ...;
TSeries t3Values = T3.Batch(prices, period: 14, vFactor: 0.7);

// Span API (High Performance)
double[] prices = new double[1000];
double[] output = new double[1000];
T3.Calculate(prices.AsSpan(), output.AsSpan(), period: 14, vFactor: 0.7);
```

### Bar Correction (isNew Parameter)

```csharp
var t3 = new T3(14);

// New bar
t3.Update(new TValue(time, 100), isNew: true);

// Intra-bar update
t3.Update(new TValue(time, 101), isNew: false); // Replaces 100 with 101