# Exponential Moving Average (EMA)

EMA is a weighted moving average that gives more importance to recent prices. It responds faster to price changes than SMA, making it useful for capturing emerging trends earlier.

**Formula**

```
Initial EMA = SMA for the first calculation

For subsequent periods:
EMA(t) = α × ClosePrice(t) + (1 - α) × EMA(t-1)
```

Where:

* `α` (alpha) = smoothing factor = 2 / (n + 1)
* `n` = number of periods
* `t` = current time period

**How It Works**\
EMA applies a multiplier (alpha) that gives more weight to recent prices. The smoothing factor decreases as the period length increases, meaning a 200-period EMA responds much slower than a 9-period EMA.<br>

**Example Calculation**

Using the same example price data from SMA:

**Step 1: Calculate Initial EMA (Period 5)**\
The first EMA equals the SMA:

```
EMA₅ = SMA₅ = 22.0
```

*Available after Period 5 closes (on Period 6)*

**Step 2: Calculate Smoothing Factor**

```
α = 2 / (n + 1) = 2 / (5 + 1) = 0.333
```

**Step 3: Calculate EMA for Period 6**

```
EMA₅(Period 6) = α × ClosePrice(Period 6) + (1 - α) × EMA₅(Period 5)
                = 0.333 × 26 + (1 - 0.333) × 22.0
                = 8.658 + 14.674
                = 23.33
```

*Available after Period 6 closes (on Period 7)*

**Step 4: Calculate EMA for Period 7 (ClosePrice = 27)**

```
EMA₅(Period 7) = α × ClosePrice(Period 7) + (1 - α) × EMA₅(Period 6)
                = 0.333 × 27 + (1 - 0.333) × 23.33
                = 8.991 + 15.56
                = 24.55
```

**Key Characteristics**

* **Responsiveness**: Reacts faster to price changes than SMA
* **Trend Sensitivity**: Better at identifying trend changes early
* **Complexity**: Carries forward historical data through the exponential calculation
* **Memory**: Each new EMA depends on the previous EMA value\ <br>

### Reading Exponential Moving Average (EMA) on Supra L1

```move
#[view]
public fun compute_ema(
    pair_id: u32,                                  // Unique identifier of the trading pair
    period: u64,                                   // EMA period [9, 20, 50, 200]
    candle_duration: u64,                          // Duration in milliseconds
    missing_candles_tolerance_percentage: u64      // Max missing candles (two-decimal fixed-point)
): (Option<u128>, Option<u64>, Option<u64>)
```

**Returns**: A tuple with:

* `Option<u128>`: Latest EMA value (scaled), or `none` if unavailable
* `Option<u64>`: Number of missing candles since last EMA update
* `Option<u64>`: Total candles formed from first candle to latest update

**Example**:

```move
// Get 9-period and 50-period EMA for ETH on 15-minute timeframe
let (fast_ema, missing1, total1) = supra_oracle_ti::compute_ema(
    3,         // pair_id (ETH)
    9,         // period
    900_000,   // candle_duration (15 min)
    1000       // 10% tolerance
);

let (slow_ema, missing2, total2) = supra_oracle_ti::compute_ema(
    3,         // pair_id (ETH)
    50,        // period
    900_000,   // candle_duration (15 min)
    1000       // 10% tolerance
);

// Check for golden cross (fast EMA crosses above slow EMA)
if (option::is_some(&fast_ema) && option::is_some(&slow_ema)) {
    let fast = option::extract(&mut fast_ema);
    let slow = option::extract(&mut slow_ema);
    
    if (fast > slow) {
        // Bullish signal
    }
}
```


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