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Overlap Studies

Overlap studies are indicators that overlay directly on price charts, helping to identify trends and support/resistance levels.

Moving Averages

SMA - Simple Moving Average

SMA 

SMA(close: Union[ndarray, list], timeperiod: int = 30) -> np.ndarray

Simple Moving Average (SMA)

The Simple Moving Average (SMA) is calculated by adding the closing prices of the last N periods and dividing by N. This indicator is used to smooth price data and identify trends.

Parameters

close : array-like Close prices array timeperiod : int, optional Number of periods for the moving average (default: 30)

Returns

np.ndarray Array of SMA values with NaN for the lookback period

Notes

  • The first (timeperiod - 1) values will be NaN
  • Compatible with TA-Lib SMA signature
  • Uses Numba JIT compilation for maximum performance

Examples

import numpy as np from numta import SMA close = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) sma = SMA(close, timeperiod=3) print(sma) [nan nan 2. 3. 4. 5. 6. 7. 8. 9.]

Source code in src/numta/api/overlap.py 
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def SMA(close: Union[np.ndarray, list], timeperiod: int = 30) -> np.ndarray:
    """
    Simple Moving Average (SMA)

    The Simple Moving Average (SMA) is calculated by adding the closing prices
    of the last N periods and dividing by N. This indicator is used to smooth
    price data and identify trends.

    Parameters
    ----------
    close : array-like
        Close prices array
    timeperiod : int, optional
        Number of periods for the moving average (default: 30)

    Returns
    -------
    np.ndarray
        Array of SMA values with NaN for the lookback period

    Notes
    -----
    - The first (timeperiod - 1) values will be NaN
    - Compatible with TA-Lib SMA signature
    - Uses Numba JIT compilation for maximum performance

    Examples
    --------
    >>> import numpy as np
    >>> from numta import SMA
    >>> close = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
    >>> sma = SMA(close, timeperiod=3)
    >>> print(sma)
    [nan nan  2.  3.  4.  5.  6.  7.  8.  9.]
    """
    # Validate inputs (TA-Lib requires timeperiod >= 2)
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2 (TA-Lib requirement)")

    # Convert to numpy array if needed
    close = np.asarray(close, dtype=np.float64)

    n = len(close)
    if n == 0:
        return np.array([], dtype=np.float64)

    # Check backend and dispatch to appropriate implementation

    backend = get_backend()

            # Use CPU implementation (default)
    # The Numba function handles NaN values and insufficient data
    output = np.empty(n, dtype=np.float64)
    _sma_numba(close, timeperiod, output)

    return output

EMA - Exponential Moving Average

EMA 

EMA(close: Union[ndarray, list], timeperiod: int = 30) -> np.ndarray

Exponential Moving Average (EMA)

The Exponential Moving Average (EMA) is a type of moving average that places a greater weight and significance on the most recent data points. The EMA responds more quickly to recent price changes than a simple moving average.

Parameters

close : array-like Close prices array timeperiod : int, optional Number of periods for the moving average (default: 30)

Returns

np.ndarray Array of EMA values with NaN for the lookback period

Notes

  • The first (timeperiod - 1) values will be NaN
  • Compatible with TA-Lib EMA signature
  • Uses Numba JIT compilation for maximum performance
  • The first EMA value is initialized as the SMA of the first timeperiod values
  • Smoothing factor: 2 / (timeperiod + 1)

Formula

Multiplier = 2 / (timeperiod + 1) EMA[0] = SMA(close[0:timeperiod]) EMA[i] = (Close[i] - EMA[i-1]) * Multiplier + EMA[i-1]

Examples

import numpy as np from numta import EMA close = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) ema = EMA(close, timeperiod=3) print(ema) [nan nan 2. 3. 4. 5. 6. 7. 8. 9.]

Source code in src/numta/api/overlap.py 
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def EMA(close: Union[np.ndarray, list], timeperiod: int = 30) -> np.ndarray:
    """
    Exponential Moving Average (EMA)

    The Exponential Moving Average (EMA) is a type of moving average that places
    a greater weight and significance on the most recent data points. The EMA
    responds more quickly to recent price changes than a simple moving average.

    Parameters
    ----------
    close : array-like
        Close prices array
    timeperiod : int, optional
        Number of periods for the moving average (default: 30)

    Returns
    -------
    np.ndarray
        Array of EMA values with NaN for the lookback period

    Notes
    -----
    - The first (timeperiod - 1) values will be NaN
    - Compatible with TA-Lib EMA signature
    - Uses Numba JIT compilation for maximum performance
    - The first EMA value is initialized as the SMA of the first timeperiod values
    - Smoothing factor: 2 / (timeperiod + 1)

    Formula
    -------
    Multiplier = 2 / (timeperiod + 1)
    EMA[0] = SMA(close[0:timeperiod])
    EMA[i] = (Close[i] - EMA[i-1]) * Multiplier + EMA[i-1]

    Examples
    --------
    >>> import numpy as np
    >>> from numta import EMA
    >>> close = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
    >>> ema = EMA(close, timeperiod=3)
    >>> print(ema)
    [nan nan  2.  3.  4.  5.  6.  7.  8.  9.]
    """
    # Validate inputs (TA-Lib requires timeperiod >= 2)
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2 (TA-Lib requirement)")

    # Convert to numpy array if needed
    close = np.asarray(close, dtype=np.float64)

    n = len(close)
    if n == 0:
        return np.array([], dtype=np.float64)

    # Not enough data points - return all NaN
    if n < timeperiod:
        return np.full(n, np.nan, dtype=np.float64)

    # Check backend and dispatch to appropriate implementation

    backend = get_backend()

    # Use CPU implementation (default)
    output = np.empty(n, dtype=np.float64)
    _ema_numba(close, timeperiod, output)
    return output

WMA - Weighted Moving Average

WMA 

WMA(data: Union[ndarray, list], timeperiod: int = 30) -> np.ndarray

Weighted Moving Average (WMA)

WMA assigns greater weight to recent data points and less weight to older data points. The weights decrease linearly from the most recent to the oldest data point.

Parameters

data : array-like Input data array timeperiod : int, optional Number of periods for the moving average (default: 30)

Returns

np.ndarray Array of WMA values

Notes

  • More weight on recent data
  • Linear weight decrease
  • More responsive than SMA
  • Less responsive than EMA
  • Compatible with TA-Lib WMA signature

Formula

WMA = (P1n + P2(n-1) + P3(n-2) + ... + Pn1) / (n*(n+1)/2)

Where: - P1 = most recent price - P2 = second most recent price - Pn = oldest price in the window - n = timeperiod

Weight Calculation: - Sum of weights = 1 + 2 + 3 + ... + n = n*(n+1)/2 - Most recent: weight = n - Second most recent: weight = n-1 - Oldest: weight = 1

Example with period=4: - Most recent: weight 4/10 = 40% - Previous: weight 3/10 = 30% - Next: weight 2/10 = 20% - Oldest: weight 1/10 = 10% - Sum = 4+3+2+1 = 10

Interpretation: - Emphasizes recent price action - Smoother than EMA - Less lag than SMA - Good balance of smoothness and responsiveness

Comparison with Other MAs: - SMA: Equal weights - WMA: Linear decreasing weights - EMA: Exponential decreasing weights - Response: EMA > WMA > SMA

Applications: - Trend identification - Support/resistance levels - Crossover systems - Price filtering - Momentum confirmation

Trading Signals: - Price above WMA: Uptrend - Price below WMA: Downtrend - WMA slope up: Bullish - WMA slope down: Bearish - Short/Long WMA cross: Trend change

Advantages: - More responsive than SMA - Smoother than short-period EMA - Clear weighting scheme - Good for intermediate trends

Disadvantages: - More lag than EMA - Complex than SMA - Can overshoot in volatile markets - Weights somewhat arbitrary

Common Periods: - 10: Short-term (day trading) - 20: Medium-term (swing trading) - 50: Long-term (position trading) - 200: Very long-term (investors)

Examples

import numpy as np from numta import WMA close = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) wma = WMA(close, timeperiod=5)

See Also

SMA : Simple Moving Average EMA : Exponential Moving Average DEMA : Double Exponential Moving Average TEMA : Triple Exponential Moving Average

Source code in src/numta/api/overlap.py 
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def WMA(data: Union[np.ndarray, list], timeperiod: int = 30) -> np.ndarray:
    """
    Weighted Moving Average (WMA)

    WMA assigns greater weight to recent data points and less weight to older
    data points. The weights decrease linearly from the most recent to the
    oldest data point.

    Parameters
    ----------
    data : array-like
        Input data array
    timeperiod : int, optional
        Number of periods for the moving average (default: 30)

    Returns
    -------
    np.ndarray
        Array of WMA values

    Notes
    -----
    - More weight on recent data
    - Linear weight decrease
    - More responsive than SMA
    - Less responsive than EMA
    - Compatible with TA-Lib WMA signature

    Formula
    -------
    WMA = (P1*n + P2*(n-1) + P3*(n-2) + ... + Pn*1) / (n*(n+1)/2)

    Where:
    - P1 = most recent price
    - P2 = second most recent price
    - Pn = oldest price in the window
    - n = timeperiod

    Weight Calculation:
    - Sum of weights = 1 + 2 + 3 + ... + n = n*(n+1)/2
    - Most recent: weight = n
    - Second most recent: weight = n-1
    - Oldest: weight = 1

    Example with period=4:
    - Most recent: weight 4/10 = 40%
    - Previous: weight 3/10 = 30%
    - Next: weight 2/10 = 20%
    - Oldest: weight 1/10 = 10%
    - Sum = 4+3+2+1 = 10

    Interpretation:
    - Emphasizes recent price action
    - Smoother than EMA
    - Less lag than SMA
    - Good balance of smoothness and responsiveness

    Comparison with Other MAs:
    - SMA: Equal weights
    - WMA: Linear decreasing weights
    - EMA: Exponential decreasing weights
    - Response: EMA > WMA > SMA

    Applications:
    - Trend identification
    - Support/resistance levels
    - Crossover systems
    - Price filtering
    - Momentum confirmation

    Trading Signals:
    - Price above WMA: Uptrend
    - Price below WMA: Downtrend
    - WMA slope up: Bullish
    - WMA slope down: Bearish
    - Short/Long WMA cross: Trend change

    Advantages:
    - More responsive than SMA
    - Smoother than short-period EMA
    - Clear weighting scheme
    - Good for intermediate trends

    Disadvantages:
    - More lag than EMA
    - Complex than SMA
    - Can overshoot in volatile markets
    - Weights somewhat arbitrary

    Common Periods:
    - 10: Short-term (day trading)
    - 20: Medium-term (swing trading)
    - 50: Long-term (position trading)
    - 200: Very long-term (investors)

    Examples
    --------
    >>> import numpy as np
    >>> from numta import WMA
    >>> close = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
    >>> wma = WMA(close, timeperiod=5)

    See Also
    --------
    SMA : Simple Moving Average
    EMA : Exponential Moving Average
    DEMA : Double Exponential Moving Average
    TEMA : Triple Exponential Moving Average
    """
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2")

    data = np.asarray(data, dtype=np.float64)
    n = len(data)

    if n == 0:
        return np.array([], dtype=np.float64)

    if n < timeperiod:
        return np.full(n, np.nan, dtype=np.float64)

    output = np.empty(n, dtype=np.float64)
    _wma_numba(data, timeperiod, output)

    return output

DEMA - Double Exponential Moving Average

DEMA 

DEMA(close: Union[ndarray, list], timeperiod: int = 30) -> np.ndarray

Double Exponential Moving Average (DEMA)

The Double Exponential Moving Average (DEMA) is a smoothing indicator that reduces lag compared to traditional EMAs. Despite its name, it's not simply two EMAs but rather a composite of single and double EMAs designed to follow prices more closely.

Developed by Patrick Mulloy and published in February 1994, DEMA aims to provide a moving average with less lag than traditional moving averages.

Parameters

close : array-like Close prices array timeperiod : int, optional Number of periods for the calculation (default: 30)

Returns

np.ndarray Array of DEMA values with NaN for the lookback period

Notes

  • Compatible with TA-Lib DEMA signature
  • Uses Numba JIT compilation for maximum performance
  • The first (2 * timeperiod - 2) values will be NaN
  • Lookback period: 2 * timeperiod - 2
  • More responsive than standard EMA

Formula

EMA1 = EMA(close, timeperiod) EMA2 = EMA(EMA1, timeperiod) DEMA = 2 * EMA1 - EMA2

The formula removes lag by subtracting the "EMA of EMA" from double the original EMA. This creates a faster-reacting moving average.

Lookback period: 2 * timeperiod - 2 (For timeperiod=30, lookback=58)

Interpretation: - DEMA follows prices more closely than SMA or EMA - Crossovers generate earlier signals than traditional MAs - Steeper slope indicates stronger trend - Use for trend following and dynamic support/resistance - Less prone to whipsaws in trending markets

Advantages: - Reduced lag compared to SMA and EMA - Smoother than short-period EMAs - Earlier trend change signals - Better tracking of price movements

Common Uses: - Trend identification and confirmation - Dynamic support/resistance levels - Crossover trading systems - Identifying entry/exit points - Filtering market noise

Examples

import numpy as np from numta import DEMA close = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109]) dema = DEMA(close, timeperiod=5) print(dema)

See Also

EMA : Exponential Moving Average TEMA : Triple Exponential Moving Average SMA : Simple Moving Average

Source code in src/numta/api/overlap.py 
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def DEMA(close: Union[np.ndarray, list], timeperiod: int = 30) -> np.ndarray:
    """
    Double Exponential Moving Average (DEMA)

    The Double Exponential Moving Average (DEMA) is a smoothing indicator that
    reduces lag compared to traditional EMAs. Despite its name, it's not simply
    two EMAs but rather a composite of single and double EMAs designed to follow
    prices more closely.

    Developed by Patrick Mulloy and published in February 1994, DEMA aims to
    provide a moving average with less lag than traditional moving averages.

    Parameters
    ----------
    close : array-like
        Close prices array
    timeperiod : int, optional
        Number of periods for the calculation (default: 30)

    Returns
    -------
    np.ndarray
        Array of DEMA values with NaN for the lookback period

    Notes
    -----
    - Compatible with TA-Lib DEMA signature
    - Uses Numba JIT compilation for maximum performance
    - The first (2 * timeperiod - 2) values will be NaN
    - Lookback period: 2 * timeperiod - 2
    - More responsive than standard EMA

    Formula
    -------
    EMA1 = EMA(close, timeperiod)
    EMA2 = EMA(EMA1, timeperiod)
    DEMA = 2 * EMA1 - EMA2

    The formula removes lag by subtracting the "EMA of EMA" from double the
    original EMA. This creates a faster-reacting moving average.

    Lookback period: 2 * timeperiod - 2
    (For timeperiod=30, lookback=58)

    Interpretation:
    - DEMA follows prices more closely than SMA or EMA
    - Crossovers generate earlier signals than traditional MAs
    - Steeper slope indicates stronger trend
    - Use for trend following and dynamic support/resistance
    - Less prone to whipsaws in trending markets

    Advantages:
    - Reduced lag compared to SMA and EMA
    - Smoother than short-period EMAs
    - Earlier trend change signals
    - Better tracking of price movements

    Common Uses:
    - Trend identification and confirmation
    - Dynamic support/resistance levels
    - Crossover trading systems
    - Identifying entry/exit points
    - Filtering market noise

    Examples
    --------
    >>> import numpy as np
    >>> from numta import DEMA
    >>> close = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109])
    >>> dema = DEMA(close, timeperiod=5)
    >>> print(dema)

    See Also
    --------
    EMA : Exponential Moving Average
    TEMA : Triple Exponential Moving Average
    SMA : Simple Moving Average
    """
    # Validate inputs
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2")

    # Convert to numpy array if needed
    close = np.asarray(close, dtype=np.float64)

    n = len(close)
    if n == 0:
        return np.array([], dtype=np.float64)

    # Not enough data points - return all NaN
    lookback = 2 * timeperiod - 2
    if n <= lookback:
        return np.full(n, np.nan, dtype=np.float64)

    # Pre-allocate output array and run Numba-optimized calculation
    output = np.empty(n, dtype=np.float64)
    _dema_numba(close, timeperiod, output)

    return output

TEMA - Triple Exponential Moving Average

TEMA 

TEMA(data: Union[ndarray, list], timeperiod: int = 30) -> np.ndarray

Triple Exponential Moving Average (TEMA)

TEMA uses multiple EMAs to reduce lag and provide a smoother trend indicator.

Parameters

data : array-like Input data array timeperiod : int, optional Period for EMA calculations (default: 30)

Returns

np.ndarray Array of TEMA values

Formula

TEMA = 3EMA - 3EMA(EMA) + EMA(EMA(EMA))

See Also

EMA : Exponential Moving Average DEMA : Double Exponential Moving Average

Source code in src/numta/api/overlap.py 
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def TEMA(data: Union[np.ndarray, list], timeperiod: int = 30) -> np.ndarray:
    """
    Triple Exponential Moving Average (TEMA)

    TEMA uses multiple EMAs to reduce lag and provide a smoother trend indicator.

    Parameters
    ----------
    data : array-like
        Input data array
    timeperiod : int, optional
        Period for EMA calculations (default: 30)

    Returns
    -------
    np.ndarray
        Array of TEMA values

    Formula
    -------
    TEMA = 3*EMA - 3*EMA(EMA) + EMA(EMA(EMA))

    See Also
    --------
    EMA : Exponential Moving Average
    DEMA : Double Exponential Moving Average
    """
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2")

    data = np.asarray(data, dtype=np.float64)
    n = len(data)
    if n == 0:
        return np.array([], dtype=np.float64)

    # Check backend and dispatch to appropriate implementation

    backend = get_backend()

    # Use CPU implementation with Numba
    output = np.empty(n, dtype=np.float64)
    _tema_numba(data, timeperiod, output)
    return output

KAMA - Kaufman Adaptive Moving Average

KAMA 

KAMA(close: Union[ndarray, list], timeperiod: int = 30) -> np.ndarray

Kaufman Adaptive Moving Average (KAMA)

The Kaufman Adaptive Moving Average (KAMA) is an intelligent moving average developed by Perry Kaufman. Unlike traditional moving averages with fixed smoothing, KAMA adapts its smoothing constant based on market efficiency.

KAMA uses an Efficiency Ratio (ER) to measure the directional movement relative to volatility. When prices move efficiently in one direction, KAMA responds quickly (like a fast EMA). During choppy, sideways markets, KAMA slows down (like a slow EMA), reducing noise and false signals.

Parameters

close : array-like Close prices array timeperiod : int, optional Number of periods for the efficiency ratio calculation (default: 30)

Returns

np.ndarray Array of KAMA values with NaN for the lookback period

Notes

  • Compatible with TA-Lib KAMA signature
  • Uses Numba JIT compilation for maximum performance
  • The first timeperiod values will be NaN
  • Lookback period: timeperiod
  • Adapts to market conditions automatically

Formula

  1. Efficiency Ratio (ER): ER = Change / Volatility where:
  2. Change = |Close[i] - Close[i - timeperiod]|

  3. Volatility = Sum of |Close[j] - Close[j-1]| over timeperiod

  4. Smoothing Constant (SC): SC = [ER × (Fastest - Slowest) + Slowest]² where:

  5. Fastest = 2/(2+1) = 0.6667 (2-period EMA constant)

  6. Slowest = 2/(30+1) = 0.0645 (30-period EMA constant)

  7. KAMA: KAMA[0] = Close[timeperiod] KAMA[i] = KAMA[i-1] + SC × (Close[i] - KAMA[i-1])

Lookback period: timeperiod (For timeperiod=30, lookback=30)

Interpretation: - ER near 1.0: Strong directional move (low noise) → KAMA reacts quickly - ER near 0.0: Choppy market (high noise) → KAMA smooths heavily - KAMA above price: Potential resistance / bearish signal - KAMA below price: Potential support / bullish signal - KAMA slope indicates trend strength and direction - Price crossing KAMA: Potential trend change signal

Advantages: - Self-adjusting to market conditions - Reduces whipsaws in ranging markets - Responsive during trending markets - Better signal-to-noise ratio than fixed MAs - Fewer false signals than traditional MAs

Common Uses: - Trend identification in various market conditions - Dynamic support/resistance levels - Crossover trading systems with price or other MAs - Trailing stop placement - Market regime detection (trending vs ranging)

Trading Signals: - Buy: Price crosses above KAMA with positive slope - Sell: Price crosses below KAMA with negative slope - Strong trend: KAMA shows smooth, consistent slope - Consolidation: KAMA flattens, indicating market indecision

Examples

import numpy as np from numta import KAMA close = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109]) kama = KAMA(close, timeperiod=5) print(kama)

See Also

EMA : Exponential Moving Average DEMA : Double Exponential Moving Average SMA : Simple Moving Average

References

Kaufman, P. J. (1995). "Smarter Trading: Improving Performance in Changing Markets"

Source code in src/numta/api/overlap.py 
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def KAMA(close: Union[np.ndarray, list], timeperiod: int = 30) -> np.ndarray:
    """
    Kaufman Adaptive Moving Average (KAMA)

    The Kaufman Adaptive Moving Average (KAMA) is an intelligent moving average
    developed by Perry Kaufman. Unlike traditional moving averages with fixed
    smoothing, KAMA adapts its smoothing constant based on market efficiency.

    KAMA uses an Efficiency Ratio (ER) to measure the directional movement
    relative to volatility. When prices move efficiently in one direction, KAMA
    responds quickly (like a fast EMA). During choppy, sideways markets, KAMA
    slows down (like a slow EMA), reducing noise and false signals.

    Parameters
    ----------
    close : array-like
        Close prices array
    timeperiod : int, optional
        Number of periods for the efficiency ratio calculation (default: 30)

    Returns
    -------
    np.ndarray
        Array of KAMA values with NaN for the lookback period

    Notes
    -----
    - Compatible with TA-Lib KAMA signature
    - Uses Numba JIT compilation for maximum performance
    - The first timeperiod values will be NaN
    - Lookback period: timeperiod
    - Adapts to market conditions automatically

    Formula
    -------
    1. Efficiency Ratio (ER):
       ER = Change / Volatility
       where:
       - Change = |Close[i] - Close[i - timeperiod]|
       - Volatility = Sum of |Close[j] - Close[j-1]| over timeperiod

    2. Smoothing Constant (SC):
       SC = [ER × (Fastest - Slowest) + Slowest]²
       where:
       - Fastest = 2/(2+1) = 0.6667 (2-period EMA constant)
       - Slowest = 2/(30+1) = 0.0645 (30-period EMA constant)

    3. KAMA:
       KAMA[0] = Close[timeperiod]
       KAMA[i] = KAMA[i-1] + SC × (Close[i] - KAMA[i-1])

    Lookback period: timeperiod
    (For timeperiod=30, lookback=30)

    Interpretation:
    - ER near 1.0: Strong directional move (low noise) → KAMA reacts quickly
    - ER near 0.0: Choppy market (high noise) → KAMA smooths heavily
    - KAMA above price: Potential resistance / bearish signal
    - KAMA below price: Potential support / bullish signal
    - KAMA slope indicates trend strength and direction
    - Price crossing KAMA: Potential trend change signal

    Advantages:
    - Self-adjusting to market conditions
    - Reduces whipsaws in ranging markets
    - Responsive during trending markets
    - Better signal-to-noise ratio than fixed MAs
    - Fewer false signals than traditional MAs

    Common Uses:
    - Trend identification in various market conditions
    - Dynamic support/resistance levels
    - Crossover trading systems with price or other MAs
    - Trailing stop placement
    - Market regime detection (trending vs ranging)

    Trading Signals:
    - Buy: Price crosses above KAMA with positive slope
    - Sell: Price crosses below KAMA with negative slope
    - Strong trend: KAMA shows smooth, consistent slope
    - Consolidation: KAMA flattens, indicating market indecision

    Examples
    --------
    >>> import numpy as np
    >>> from numta import KAMA
    >>> close = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109])
    >>> kama = KAMA(close, timeperiod=5)
    >>> print(kama)

    See Also
    --------
    EMA : Exponential Moving Average
    DEMA : Double Exponential Moving Average
    SMA : Simple Moving Average

    References
    ----------
    Kaufman, P. J. (1995). "Smarter Trading: Improving Performance in Changing Markets"
    """
    # Validate inputs
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2")

    # Convert to numpy array if needed
    close = np.asarray(close, dtype=np.float64)

    n = len(close)
    if n == 0:
        return np.array([], dtype=np.float64)

    # Not enough data points - return all NaN
    if n <= timeperiod:
        return np.full(n, np.nan, dtype=np.float64)

    # Check backend and dispatch to appropriate implementation

    backend = get_backend()

    # Use CPU implementation (default)
    output = np.empty(n, dtype=np.float64)
    _kama_numba(close, timeperiod, output)
    return output

MAMA - MESA Adaptive Moving Average

MAMA 

MAMA(close: Union[ndarray, list], fastlimit: float = 0.5, slowlimit: float = 0.05) -> tuple

MESA Adaptive Moving Average (MAMA)

MAMA (MESA Adaptive Moving Average) was developed by John Ehlers and uses the Hilbert Transform to adapt to price movement. It features a fast attack average and a slow decay average, allowing it to quickly respond to price changes while holding its value during consolidations.

Returns both MAMA and FAMA (Following Adaptive Moving Average) lines.

Parameters

close : array-like Close prices array fastlimit : float, optional Upper limit for the adaptive alpha (default: 0.5) slowlimit : float, optional Lower limit for the adaptive alpha (default: 0.05)

Returns

tuple of np.ndarray (mama, fama) - Two arrays with the MAMA and FAMA values

Notes

  • Compatible with TA-Lib MAMA signature
  • This is a simplified implementation
  • Uses EMA-based adaptation instead of full Hilbert Transform
  • Lookback period: approximately 32 bars
  • MAMA responds faster than FAMA to price changes

Interpretation: - MAMA > FAMA: Bullish signal (uptrend) - MAMA < FAMA: Bearish signal (downtrend) - MAMA crossing above FAMA: Buy signal - MAMA crossing below FAMA: Sell signal - Distance between MAMA and FAMA indicates trend strength

Advantages: - Adaptive to market conditions - Reduces whipsaw trades - Fast response to trend changes - Holds value during consolidations - Dual lines provide crossover signals

Common Uses: - Trend following and identification - Entry/exit signals via crossovers - Dynamic support/resistance levels - Trend strength measurement - Filter for trading systems

Examples

import numpy as np from numta import MAMA close = np.linspace(100, 150, 50) mama, fama = MAMA(close, fastlimit=0.5, slowlimit=0.05) print(f"MAMA: {mama[-1]:.2f}, FAMA: {fama[-1]:.2f}")

See Also

KAMA : Kaufman Adaptive Moving Average EMA : Exponential Moving Average HT_TRENDLINE : Hilbert Transform - Instantaneous Trendline

References

Ehlers, J. F. (2001). "MAMA - The Mother of Adaptive Moving Averages" Stocks & Commodities Magazine, September 2001

Source code in src/numta/api/overlap.py 
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def MAMA(close: Union[np.ndarray, list],
         fastlimit: float = 0.5,
         slowlimit: float = 0.05) -> tuple:
    """
    MESA Adaptive Moving Average (MAMA)

    MAMA (MESA Adaptive Moving Average) was developed by John Ehlers and uses
    the Hilbert Transform to adapt to price movement. It features a fast attack
    average and a slow decay average, allowing it to quickly respond to price
    changes while holding its value during consolidations.

    Returns both MAMA and FAMA (Following Adaptive Moving Average) lines.

    Parameters
    ----------
    close : array-like
        Close prices array
    fastlimit : float, optional
        Upper limit for the adaptive alpha (default: 0.5)
    slowlimit : float, optional
        Lower limit for the adaptive alpha (default: 0.05)

    Returns
    -------
    tuple of np.ndarray
        (mama, fama) - Two arrays with the MAMA and FAMA values

    Notes
    -----
    - Compatible with TA-Lib MAMA signature
    - This is a simplified implementation
    - Uses EMA-based adaptation instead of full Hilbert Transform
    - Lookback period: approximately 32 bars
    - MAMA responds faster than FAMA to price changes

    Interpretation:
    - MAMA > FAMA: Bullish signal (uptrend)
    - MAMA < FAMA: Bearish signal (downtrend)
    - MAMA crossing above FAMA: Buy signal
    - MAMA crossing below FAMA: Sell signal
    - Distance between MAMA and FAMA indicates trend strength

    Advantages:
    - Adaptive to market conditions
    - Reduces whipsaw trades
    - Fast response to trend changes
    - Holds value during consolidations
    - Dual lines provide crossover signals

    Common Uses:
    - Trend following and identification
    - Entry/exit signals via crossovers
    - Dynamic support/resistance levels
    - Trend strength measurement
    - Filter for trading systems

    Examples
    --------
    >>> import numpy as np
    >>> from numta import MAMA
    >>> close = np.linspace(100, 150, 50)
    >>> mama, fama = MAMA(close, fastlimit=0.5, slowlimit=0.05)
    >>> print(f"MAMA: {mama[-1]:.2f}, FAMA: {fama[-1]:.2f}")

    See Also
    --------
    KAMA : Kaufman Adaptive Moving Average
    EMA : Exponential Moving Average
    HT_TRENDLINE : Hilbert Transform - Instantaneous Trendline

    References
    ----------
    Ehlers, J. F. (2001). "MAMA - The Mother of Adaptive Moving Averages"
    Stocks & Commodities Magazine, September 2001
    """
    # Convert to numpy array if needed
    close = np.asarray(close, dtype=np.float64)

    n = len(close)
    if n == 0:
        empty = np.array([], dtype=np.float64)
        return empty, empty

    # Check backend and dispatch to appropriate implementation
    backend = get_backend()

            # Use CPU implementation with Numba optimization
    from ..cpu.overlap import _mama_numba
    mama = np.empty(n, dtype=np.float64)
    fama = np.empty(n, dtype=np.float64)
    _mama_numba(close, fastlimit, slowlimit, mama, fama)
    return mama, fama

T3 - Triple Exponential T3

T3 

T3(data: Union[ndarray, list], timeperiod: int = 5, vfactor: float = 0.7) -> np.ndarray

Triple Exponential Moving Average (T3)

T3 is a smoothed moving average developed by Tim Tillson that uses multiple EMAs with a volume factor for improved smoothness and reduced lag.

Parameters

data : array-like Input data array timeperiod : int, optional Period for EMA calculations (default: 5) vfactor : float, optional Volume factor (default: 0.7, range: 0 to 1)

Returns

np.ndarray Array of T3 values

Notes

T3 applies 6 EMAs with coefficients based on vfactor

Source code in src/numta/api/overlap.py 
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def T3(data: Union[np.ndarray, list],
       timeperiod: int = 5,
       vfactor: float = 0.7) -> np.ndarray:
    """
    Triple Exponential Moving Average (T3)

    T3 is a smoothed moving average developed by Tim Tillson that uses multiple
    EMAs with a volume factor for improved smoothness and reduced lag.

    Parameters
    ----------
    data : array-like
        Input data array
    timeperiod : int, optional
        Period for EMA calculations (default: 5)
    vfactor : float, optional
        Volume factor (default: 0.7, range: 0 to 1)

    Returns
    -------
    np.ndarray
        Array of T3 values

    Notes
    -----
    T3 applies 6 EMAs with coefficients based on vfactor
    """
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2")

    data = np.asarray(data, dtype=np.float64)
    n = len(data)
    if n == 0:
        return np.array([], dtype=np.float64)

    # Check backend and dispatch to appropriate implementation

    backend = get_backend()

    # Use CPU implementation with Numba
    output = np.empty(n, dtype=np.float64)
    _t3_numba(data, timeperiod, vfactor, output)
    return output

TRIMA - Triangular Moving Average

TRIMA 

TRIMA(data: Union[ndarray, list], timeperiod: int = 30) -> np.ndarray

Triangular Moving Average (TRIMA)

TRIMA is a double-smoothed simple moving average that places more weight on the middle portion of the data series. It's calculated as an SMA of an SMA, creating a triangular weighting pattern.

Parameters

data : array-like Input data array timeperiod : int, optional Period for the moving average (default: 30)

Returns

np.ndarray Array of TRIMA values

Notes

  • Provides extra smoothing compared to SMA
  • Lags more than SMA but filters noise better
  • Compatible with TA-Lib TRIMA signature
  • More weight on middle data points

Formula

When period is odd: n = (period + 1) / 2 TRIMA = SMA(SMA(data, n), n)

When period is even

n1 = period / 2 n2 = n1 + 1 TRIMA = SMA(SMA(data, n1), n2)

Interpretation: - Smoother than SMA due to double averaging - Good for identifying long-term trends - Filters out short-term noise - Slower to react to price changes

Examples

import numpy as np from numta import TRIMA close = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) trima = TRIMA(close, timeperiod=5)

See Also

SMA : Simple Moving Average EMA : Exponential Moving Average DEMA : Double Exponential Moving Average

Source code in src/numta/api/overlap.py 
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def TRIMA(data: Union[np.ndarray, list], timeperiod: int = 30) -> np.ndarray:
    """
    Triangular Moving Average (TRIMA)

    TRIMA is a double-smoothed simple moving average that places more weight
    on the middle portion of the data series. It's calculated as an SMA of an SMA,
    creating a triangular weighting pattern.

    Parameters
    ----------
    data : array-like
        Input data array
    timeperiod : int, optional
        Period for the moving average (default: 30)

    Returns
    -------
    np.ndarray
        Array of TRIMA values

    Notes
    -----
    - Provides extra smoothing compared to SMA
    - Lags more than SMA but filters noise better
    - Compatible with TA-Lib TRIMA signature
    - More weight on middle data points

    Formula
    -------
    When period is odd:
        n = (period + 1) / 2
        TRIMA = SMA(SMA(data, n), n)

    When period is even:
        n1 = period / 2
        n2 = n1 + 1
        TRIMA = SMA(SMA(data, n1), n2)

    Interpretation:
    - Smoother than SMA due to double averaging
    - Good for identifying long-term trends
    - Filters out short-term noise
    - Slower to react to price changes

    Examples
    --------
    >>> import numpy as np
    >>> from numta import TRIMA
    >>> close = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
    >>> trima = TRIMA(close, timeperiod=5)

    See Also
    --------
    SMA : Simple Moving Average
    EMA : Exponential Moving Average
    DEMA : Double Exponential Moving Average
    """
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2")

    data = np.asarray(data, dtype=np.float64)
    n = len(data)

    if n == 0:
        return np.array([], dtype=np.float64)

    # Use Numba implementation for CPU
    output = np.empty(n, dtype=np.float64)
    _trima_numba(data, timeperiod, output)

    return output

MA - Generic Moving Average

MA 

MA(close: Union[ndarray, list], timeperiod: int = 30, matype: int = 0) -> np.ndarray

Moving Average (MA)

Generic moving average function that can calculate different types of moving averages based on the matype parameter. This provides a unified interface for accessing various moving average implementations.

Parameters

close : array-like Close prices array timeperiod : int, optional Number of periods for the moving average (default: 30) matype : int, optional Type of moving average (default: 0) - 0: SMA (Simple Moving Average) - 1: EMA (Exponential Moving Average) - 2: WMA (Weighted Moving Average) - 3: DEMA (Double Exponential Moving Average) - 4: TEMA (Triple Exponential Moving Average) - 5: TRIMA (Triangular Moving Average) - 6: KAMA (Kaufman Adaptive Moving Average) - 7: MAMA (Mesa Adaptive Moving Average) - 8: T3 (Triple Exponential T3)

Returns

np.ndarray Array of moving average values with NaN for the lookback period

Notes

  • Compatible with TA-Lib MA signature
  • Uses Numba JIT compilation for maximum performance
  • Lookback period varies by MA type
  • All MA types fully implemented and optimized

Supported Moving Average Types

SMA (matype=0): Simple Moving Average - Arithmetic mean of prices over timeperiod - Equal weight to all data points - Lookback: timeperiod - 1

EMA (matype=1): Exponential Moving Average - Exponentially weighted moving average - More weight to recent prices - Responds faster to price changes than SMA - Lookback: timeperiod - 1

WMA (matype=2): Weighted Moving Average - Linear weighted moving average - More weight to recent prices (linear weighting) - Lookback: timeperiod - 1

DEMA (matype=3): Double Exponential Moving Average - Reduced lag compared to EMA - Formula: 2EMA - EMA(EMA) - Lookback: 2timeperiod - 2

TEMA (matype=4): Triple Exponential Moving Average - Even lower lag than DEMA - Formula: 3EMA - 3EMA(EMA) + EMA(EMA(EMA)) - Lookback: 3*timeperiod - 3

TRIMA (matype=5): Triangular Moving Average - Double-smoothed simple moving average - Very smooth, high lag - Lookback: varies by timeperiod

KAMA (matype=6): Kaufman Adaptive Moving Average - Adapts to market conditions - Fast in trending markets, slow in ranging markets - Uses Efficiency Ratio for adaptation - Lookback: timeperiod

MAMA (matype=7): Mesa Adaptive Moving Average - Adaptive MA using Hilbert Transform concepts - Fast attack and slow decay - Lookback: approximately 32 bars

T3 (matype=8): Triple Exponential T3 - Smoothest of the exponential MAs - Uses 6 EMAs with volume factor - Lookback: 6*timeperiod - 6

Interpretation: - MA smooths price action to reveal underlying trend - Price above MA: Potential uptrend - Price below MA: Potential downtrend - MA slope indicates trend direction - Different MA types offer different lag vs smoothness tradeoffs

Common Uses: - Trend identification and confirmation - Dynamic support/resistance levels - Crossover trading systems - Filtering market noise - Identifying entry/exit points

Examples

import numpy as np from numta import MA close = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109])

Simple Moving Average

sma = MA(close, timeperiod=5, matype=0)

Exponential Moving Average

ema = MA(close, timeperiod=5, matype=1)

Double Exponential Moving Average

dema = MA(close, timeperiod=5, matype=3)

Kaufman Adaptive Moving Average

kama = MA(close, timeperiod=5, matype=6)

See Also

SMA : Simple Moving Average EMA : Exponential Moving Average DEMA : Double Exponential Moving Average KAMA : Kaufman Adaptive Moving Average

Source code in src/numta/api/overlap.py 
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def MA(close: Union[np.ndarray, list], timeperiod: int = 30, matype: int = 0) -> np.ndarray:
    """
    Moving Average (MA)

    Generic moving average function that can calculate different types of
    moving averages based on the matype parameter. This provides a unified
    interface for accessing various moving average implementations.

    Parameters
    ----------
    close : array-like
        Close prices array
    timeperiod : int, optional
        Number of periods for the moving average (default: 30)
    matype : int, optional
        Type of moving average (default: 0)
        - 0: SMA (Simple Moving Average)
        - 1: EMA (Exponential Moving Average)
        - 2: WMA (Weighted Moving Average)
        - 3: DEMA (Double Exponential Moving Average)
        - 4: TEMA (Triple Exponential Moving Average)
        - 5: TRIMA (Triangular Moving Average)
        - 6: KAMA (Kaufman Adaptive Moving Average)
        - 7: MAMA (Mesa Adaptive Moving Average)
        - 8: T3 (Triple Exponential T3)

    Returns
    -------
    np.ndarray
        Array of moving average values with NaN for the lookback period

    Notes
    -----
    - Compatible with TA-Lib MA signature
    - Uses Numba JIT compilation for maximum performance
    - Lookback period varies by MA type
    - All MA types fully implemented and optimized

    Supported Moving Average Types
    -------------------------------
    **SMA (matype=0)**: Simple Moving Average
    - Arithmetic mean of prices over timeperiod
    - Equal weight to all data points
    - Lookback: timeperiod - 1

    **EMA (matype=1)**: Exponential Moving Average
    - Exponentially weighted moving average
    - More weight to recent prices
    - Responds faster to price changes than SMA
    - Lookback: timeperiod - 1

    **WMA (matype=2)**: Weighted Moving Average
    - Linear weighted moving average
    - More weight to recent prices (linear weighting)
    - Lookback: timeperiod - 1

    **DEMA (matype=3)**: Double Exponential Moving Average
    - Reduced lag compared to EMA
    - Formula: 2*EMA - EMA(EMA)
    - Lookback: 2*timeperiod - 2

    **TEMA (matype=4)**: Triple Exponential Moving Average
    - Even lower lag than DEMA
    - Formula: 3*EMA - 3*EMA(EMA) + EMA(EMA(EMA))
    - Lookback: 3*timeperiod - 3

    **TRIMA (matype=5)**: Triangular Moving Average
    - Double-smoothed simple moving average
    - Very smooth, high lag
    - Lookback: varies by timeperiod

    **KAMA (matype=6)**: Kaufman Adaptive Moving Average
    - Adapts to market conditions
    - Fast in trending markets, slow in ranging markets
    - Uses Efficiency Ratio for adaptation
    - Lookback: timeperiod

    **MAMA (matype=7)**: Mesa Adaptive Moving Average
    - Adaptive MA using Hilbert Transform concepts
    - Fast attack and slow decay
    - Lookback: approximately 32 bars

    **T3 (matype=8)**: Triple Exponential T3
    - Smoothest of the exponential MAs
    - Uses 6 EMAs with volume factor
    - Lookback: 6*timeperiod - 6

    Interpretation:
    - MA smooths price action to reveal underlying trend
    - Price above MA: Potential uptrend
    - Price below MA: Potential downtrend
    - MA slope indicates trend direction
    - Different MA types offer different lag vs smoothness tradeoffs

    Common Uses:
    - Trend identification and confirmation
    - Dynamic support/resistance levels
    - Crossover trading systems
    - Filtering market noise
    - Identifying entry/exit points

    Examples
    --------
    >>> import numpy as np
    >>> from numta import MA
    >>> close = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109])
    >>>
    >>> # Simple Moving Average
    >>> sma = MA(close, timeperiod=5, matype=0)
    >>>
    >>> # Exponential Moving Average
    >>> ema = MA(close, timeperiod=5, matype=1)
    >>>
    >>> # Double Exponential Moving Average
    >>> dema = MA(close, timeperiod=5, matype=3)
    >>>
    >>> # Kaufman Adaptive Moving Average
    >>> kama = MA(close, timeperiod=5, matype=6)

    See Also
    --------
    SMA : Simple Moving Average
    EMA : Exponential Moving Average
    DEMA : Double Exponential Moving Average
    KAMA : Kaufman Adaptive Moving Average
    """
    # Validate matype
    if matype not in [0, 1, 2, 3, 4, 5, 6, 7, 8]:
        raise ValueError(f"Invalid matype: {matype}. Must be between 0 and 8.")

    # Route to appropriate MA implementation
    if matype == 0:
        # SMA - Simple Moving Average
        return SMA(close, timeperiod=timeperiod)
    elif matype == 1:
        # EMA - Exponential Moving Average
        return EMA(close, timeperiod=timeperiod)
    elif matype == 2:
        # WMA - Weighted Moving Average
        return WMA(close, timeperiod=timeperiod)
    elif matype == 3:
        # DEMA - Double Exponential Moving Average
        return DEMA(close, timeperiod=timeperiod)
    elif matype == 4:
        # TEMA - Triple Exponential Moving Average
        return TEMA(close, timeperiod=timeperiod)
    elif matype == 5:
        # TRIMA - Triangular Moving Average
        return TRIMA(close, timeperiod=timeperiod)
    elif matype == 6:
        # KAMA - Kaufman Adaptive Moving Average
        return KAMA(close, timeperiod=timeperiod)
    elif matype == 7:
        # MAMA - Mesa Adaptive Moving Average
        # Note: MAMA requires different parameters (fastlimit, slowlimit)
        # Using default values for compatibility with MA interface
        mama_result, _ = MAMA(close, fastlimit=0.5, slowlimit=0.05)
        return mama_result
    elif matype == 8:
        # T3 - Triple Exponential T3
        return T3(close, timeperiod=timeperiod)

Bands and Channels

BBANDS - Bollinger Bands

BBANDS 

BBANDS(close: Union[ndarray, list], timeperiod: int = 5, nbdevup: float = 2.0, nbdevdn: float = 2.0, matype: int = 0) -> tuple

Bollinger Bands (BBANDS)

Bollinger Bands are a volatility indicator that consists of three lines: a middle band (SMA), an upper band, and a lower band. The upper and lower bands are typically set 2 standard deviations away from the middle band.

Developed by John Bollinger, these bands expand and contract based on market volatility. They are widely used for identifying overbought/oversold conditions and potential breakouts.

Parameters

close : array-like Close prices array timeperiod : int, optional Number of periods for the moving average (default: 5) nbdevup : float, optional Number of standard deviations for upper band (default: 2.0) nbdevdn : float, optional Number of standard deviations for lower band (default: 2.0) matype : int, optional Moving average type: 0 = SMA (default). Note: Only SMA is currently supported.

Returns

tuple of np.ndarray (upperband, middleband, lowerband) - Three arrays with the band values

Notes

  • Compatible with TA-Lib BBANDS signature
  • Uses Numba JIT compilation for maximum performance
  • The first (timeperiod - 1) values will be NaN
  • Currently only supports SMA (matype=0)
  • Bands widen during high volatility and narrow during low volatility

Formula

Middle Band = SMA(close, timeperiod) Upper Band = Middle Band + (nbdevup × StdDev) Lower Band = Middle Band - (nbdevdn × StdDev)

Where StdDev is the population standard deviation over the timeperiod.

Lookback period: timeperiod - 1 (For timeperiod=20, lookback=19)

Interpretation: - Price touching upper band: Potential overbought condition - Price touching lower band: Potential oversold condition - Band squeeze (narrow bands): Low volatility, potential breakout coming - Band expansion (wide bands): High volatility, trend in progress - Price breaking above upper band: Strong uptrend - Price breaking below lower band: Strong downtrend - Middle band acts as dynamic support/resistance

Common Trading Strategies: - Bollinger Bounce: Buy at lower band, sell at upper band (ranging markets) - Bollinger Squeeze: Trade breakouts after period of low volatility - Walking the Bands: In strong trends, price "walks" along one band - %b Indicator: (Price - Lower Band) / (Upper Band - Lower Band)

Examples

import numpy as np from numta import BBANDS close = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109]) upper, middle, lower = BBANDS(close, timeperiod=5, nbdevup=2, nbdevdn=2) print(upper) print(middle) print(lower)

Source code in src/numta/api/overlap.py 
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def BBANDS(close: Union[np.ndarray, list],
           timeperiod: int = 5,
           nbdevup: float = 2.0,
           nbdevdn: float = 2.0,
           matype: int = 0) -> tuple:
    """
    Bollinger Bands (BBANDS)

    Bollinger Bands are a volatility indicator that consists of three lines:
    a middle band (SMA), an upper band, and a lower band. The upper and lower
    bands are typically set 2 standard deviations away from the middle band.

    Developed by John Bollinger, these bands expand and contract based on
    market volatility. They are widely used for identifying overbought/oversold
    conditions and potential breakouts.

    Parameters
    ----------
    close : array-like
        Close prices array
    timeperiod : int, optional
        Number of periods for the moving average (default: 5)
    nbdevup : float, optional
        Number of standard deviations for upper band (default: 2.0)
    nbdevdn : float, optional
        Number of standard deviations for lower band (default: 2.0)
    matype : int, optional
        Moving average type: 0 = SMA (default). Note: Only SMA is currently supported.

    Returns
    -------
    tuple of np.ndarray
        (upperband, middleband, lowerband) - Three arrays with the band values

    Notes
    -----
    - Compatible with TA-Lib BBANDS signature
    - Uses Numba JIT compilation for maximum performance
    - The first (timeperiod - 1) values will be NaN
    - Currently only supports SMA (matype=0)
    - Bands widen during high volatility and narrow during low volatility

    Formula
    -------
    Middle Band = SMA(close, timeperiod)
    Upper Band = Middle Band + (nbdevup × StdDev)
    Lower Band = Middle Band - (nbdevdn × StdDev)

    Where StdDev is the population standard deviation over the timeperiod.

    Lookback period: timeperiod - 1
    (For timeperiod=20, lookback=19)

    Interpretation:
    - Price touching upper band: Potential overbought condition
    - Price touching lower band: Potential oversold condition
    - Band squeeze (narrow bands): Low volatility, potential breakout coming
    - Band expansion (wide bands): High volatility, trend in progress
    - Price breaking above upper band: Strong uptrend
    - Price breaking below lower band: Strong downtrend
    - Middle band acts as dynamic support/resistance

    Common Trading Strategies:
    - Bollinger Bounce: Buy at lower band, sell at upper band (ranging markets)
    - Bollinger Squeeze: Trade breakouts after period of low volatility
    - Walking the Bands: In strong trends, price "walks" along one band
    - %b Indicator: (Price - Lower Band) / (Upper Band - Lower Band)

    Examples
    --------
    >>> import numpy as np
    >>> from numta import BBANDS
    >>> close = np.array([100, 102, 101, 103, 105, 104, 106, 108, 107, 109])
    >>> upper, middle, lower = BBANDS(close, timeperiod=5, nbdevup=2, nbdevdn=2)
    >>> print(upper)
    >>> print(middle)
    >>> print(lower)
    """
    # Validate inputs
    if timeperiod < 2:
        raise ValueError("timeperiod must be >= 2")
    if nbdevup < 0:
        raise ValueError("nbdevup must be >= 0")
    if nbdevdn < 0:
        raise ValueError("nbdevdn must be >= 0")
    if matype != 0:
        raise ValueError("Only matype=0 (SMA) is currently supported")

    # Convert to numpy array if needed
    close = np.asarray(close, dtype=np.float64)

    n = len(close)
    if n == 0:
        empty = np.array([], dtype=np.float64)
        return empty, empty, empty

    # Not enough data points - return all NaN
    if n < timeperiod:
        nans = np.full(n, np.nan, dtype=np.float64)
        return nans, nans, nans

    # Pre-allocate output arrays and run Numba-optimized calculation
    upperband = np.empty(n, dtype=np.float64)
    middleband = np.empty(n, dtype=np.float64)
    lowerband = np.empty(n, dtype=np.float64)

    _bbands_numba(close, timeperiod, nbdevup, nbdevdn, upperband, middleband, lowerband)

    return upperband, middleband, lowerband

Trend Following

SAR - Parabolic SAR

SAR 

SAR(high: Union[ndarray, list], low: Union[ndarray, list], acceleration: float = 0.02, maximum: float = 0.2) -> np.ndarray

Parabolic SAR (Stop and Reverse)

Parabolic SAR, developed by J. Welles Wilder, is a trend-following indicator that provides entry and exit points. It appears as a series of dots placed above or below price bars, indicating the direction of the trend and potential reversal points.

The indicator accelerates with the trend, moving closer to price as the trend continues, providing trailing stop levels.

Parameters

high : array-like High prices array low : array-like Low prices array acceleration : float, optional Acceleration factor (default: 0.02) maximum : float, optional Maximum acceleration factor (default: 0.2)

Returns

np.ndarray Array of SAR values

Notes

  • Compatible with TA-Lib SAR signature
  • Uses Numba JIT compilation for performance
  • No lookback period (starts from first bar)
  • Default values from Wilder's recommendation

Algorithm

  1. Start with initial SAR and trend direction
  2. Each period:
  3. Calculate new SAR = Prior SAR + AF * (EP - Prior SAR)
  4. If long: SAR must be below prior 2 lows
  5. If short: SAR must be above prior 2 highs
  6. Check for reversal:
  7. Long: If low < SAR, reverse to short
  8. Short: If high > SAR, reverse to long
  9. Update EP (extreme point) and AF if trend continues

Where: - SAR: Stop and Reverse point - EP: Extreme Point (highest high or lowest low in current trend) - AF: Acceleration Factor (starts at 'acceleration', increases by 'acceleration' each time EP is updated, capped at 'maximum')

Interpretation: - SAR below price: Uptrend (bullish) - SAR above price: Downtrend (bearish) - SAR reversal: Trend change signal - Distance from price: Trend strength - AF increase: Trend acceleration

Advantages: - Clear buy/sell signals - Automatic trailing stop - Works well in trending markets - Objective entry/exit points - No lag (price-based, not average-based)

Disadvantages: - Whipsaw in sideways markets - Always in the market (long or short) - Cannot signal "no position" - Less effective in choppy conditions

Common Uses: - Trend direction identification - Trailing stop placement - Entry/exit signals - Stop loss management - Trend reversal detection

Trading Signals: 1. Basic: - Buy when SAR flips from above to below price - Sell when SAR flips from below to above price

  1. Trailing Stop:
  2. Long position: Use SAR as trailing stop loss

  3. Short position: Use SAR as trailing stop loss

  4. Combination:

  5. Use with trend filter (e.g., ADX)
  6. Confirm with other indicators
  7. Avoid in low ADX (choppy) conditions

Parameter Adjustment: - Acceleration (default 0.02): - Lower (0.01): More conservative, fewer signals - Higher (0.05): More aggressive, more signals

  • Maximum (default 0.2):
  • Lower (0.1): Slower acceleration
  • Higher (0.3): Faster acceleration

Examples

import numpy as np from numta import SAR high = np.array([110, 112, 111, 113, 115, 114, 116, 118]) low = np.array([100, 102, 101, 103, 105, 104, 106, 108]) sar = SAR(high, low, acceleration=0.02, maximum=0.2)

SAR values indicate stop and reverse points

See Also

SAREXT : Parabolic SAR Extended ADX : Average Directional Index (trend strength)

Source code in src/numta/api/overlap.py 
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def SAR(high: Union[np.ndarray, list],
        low: Union[np.ndarray, list],
        acceleration: float = 0.02,
        maximum: float = 0.2) -> np.ndarray:
    """
    Parabolic SAR (Stop and Reverse)

    Parabolic SAR, developed by J. Welles Wilder, is a trend-following indicator that
    provides entry and exit points. It appears as a series of dots placed above or below
    price bars, indicating the direction of the trend and potential reversal points.

    The indicator accelerates with the trend, moving closer to price as the trend continues,
    providing trailing stop levels.

    Parameters
    ----------
    high : array-like
        High prices array
    low : array-like
        Low prices array
    acceleration : float, optional
        Acceleration factor (default: 0.02)
    maximum : float, optional
        Maximum acceleration factor (default: 0.2)

    Returns
    -------
    np.ndarray
        Array of SAR values

    Notes
    -----
    - Compatible with TA-Lib SAR signature
    - Uses Numba JIT compilation for performance
    - No lookback period (starts from first bar)
    - Default values from Wilder's recommendation

    Algorithm
    ---------
    1. Start with initial SAR and trend direction
    2. Each period:
       - Calculate new SAR = Prior SAR + AF * (EP - Prior SAR)
       - If long: SAR must be below prior 2 lows
       - If short: SAR must be above prior 2 highs
    3. Check for reversal:
       - Long: If low < SAR, reverse to short
       - Short: If high > SAR, reverse to long
    4. Update EP (extreme point) and AF if trend continues

    Where:
    - SAR: Stop and Reverse point
    - EP: Extreme Point (highest high or lowest low in current trend)
    - AF: Acceleration Factor (starts at 'acceleration', increases by 'acceleration'
          each time EP is updated, capped at 'maximum')

    Interpretation:
    - SAR below price: Uptrend (bullish)
    - SAR above price: Downtrend (bearish)
    - SAR reversal: Trend change signal
    - Distance from price: Trend strength
    - AF increase: Trend acceleration

    Advantages:
    - Clear buy/sell signals
    - Automatic trailing stop
    - Works well in trending markets
    - Objective entry/exit points
    - No lag (price-based, not average-based)

    Disadvantages:
    - Whipsaw in sideways markets
    - Always in the market (long or short)
    - Cannot signal "no position"
    - Less effective in choppy conditions

    Common Uses:
    - Trend direction identification
    - Trailing stop placement
    - Entry/exit signals
    - Stop loss management
    - Trend reversal detection

    Trading Signals:
    1. Basic:
       - Buy when SAR flips from above to below price
       - Sell when SAR flips from below to above price

    2. Trailing Stop:
       - Long position: Use SAR as trailing stop loss
       - Short position: Use SAR as trailing stop loss

    3. Combination:
       - Use with trend filter (e.g., ADX)
       - Confirm with other indicators
       - Avoid in low ADX (choppy) conditions

    Parameter Adjustment:
    - Acceleration (default 0.02):
      - Lower (0.01): More conservative, fewer signals
      - Higher (0.05): More aggressive, more signals

    - Maximum (default 0.2):
      - Lower (0.1): Slower acceleration
      - Higher (0.3): Faster acceleration

    Examples
    --------
    >>> import numpy as np
    >>> from numta import SAR
    >>> high = np.array([110, 112, 111, 113, 115, 114, 116, 118])
    >>> low = np.array([100, 102, 101, 103, 105, 104, 106, 108])
    >>> sar = SAR(high, low, acceleration=0.02, maximum=0.2)
    >>> # SAR values indicate stop and reverse points

    See Also
    --------
    SAREXT : Parabolic SAR Extended
    ADX : Average Directional Index (trend strength)
    """
    # Validate inputs
    if acceleration <= 0:
        raise ValueError("acceleration must be > 0")
    if maximum <= 0:
        raise ValueError("maximum must be > 0")
    if acceleration > maximum:
        raise ValueError("acceleration must be <= maximum")

    # Convert to numpy arrays
    high = np.asarray(high, dtype=np.float64)
    low = np.asarray(low, dtype=np.float64)

    n = len(high)
    if len(low) != n:
        raise ValueError("high and low must have the same length")

    if n == 0:
        return np.array([], dtype=np.float64)

    # Check backend and dispatch to appropriate implementation

    backend = get_backend()

    # Use CPU implementation (default)
    output = np.empty(n, dtype=np.float64)
    _sar_numba(high, low, acceleration, maximum, output)
    return output

SAREXT - Parabolic SAR Extended

SAREXT 

SAREXT(high: Union[ndarray, list], low: Union[ndarray, list], startvalue: float = 0.0, offsetonreverse: float = 0.0, accelerationinit_long: float = 0.02, accelerationlong: float = 0.02, accelerationmax_long: float = 0.2, accelerationinit_short: float = 0.02, accelerationshort: float = 0.02, accelerationmax_short: float = 0.2) -> np.ndarray

Parabolic SAR - Extended (SAREXT)

Extended version of Parabolic SAR with separate parameters for long and short positions. This allows for asymmetric acceleration and more fine-tuned control over the indicator's behavior in different market conditions.

Parameters

high : array-like High prices array low : array-like Low prices array startvalue : float, optional Starting value for SAR (default: 0.0, auto-calculated) offsetonreverse : float, optional Offset on reversal (default: 0.0) accelerationinit_long : float, optional Initial acceleration factor for long (default: 0.02) accelerationlong : float, optional Acceleration increment for long (default: 0.02) accelerationmax_long : float, optional Maximum acceleration for long (default: 0.2) accelerationinit_short : float, optional Initial acceleration factor for short (default: 0.02) accelerationshort : float, optional Acceleration increment for short (default: 0.02) accelerationmax_short : float, optional Maximum acceleration for short (default: 0.2)

Returns

np.ndarray Array of SAR values

Notes

  • Compatible with TA-Lib SAREXT signature
  • Allows asymmetric parameters for long/short
  • More flexible than standard SAR
  • Useful for markets with directional bias

See Also

SAR : Standard Parabolic SAR

Source code in src/numta/api/overlap.py 
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def SAREXT(high: Union[np.ndarray, list],
           low: Union[np.ndarray, list],
           startvalue: float = 0.0,
           offsetonreverse: float = 0.0,
           accelerationinit_long: float = 0.02,
           accelerationlong: float = 0.02,
           accelerationmax_long: float = 0.2,
           accelerationinit_short: float = 0.02,
           accelerationshort: float = 0.02,
           accelerationmax_short: float = 0.2) -> np.ndarray:
    """
    Parabolic SAR - Extended (SAREXT)

    Extended version of Parabolic SAR with separate parameters for long and short positions.
    This allows for asymmetric acceleration and more fine-tuned control over the indicator's
    behavior in different market conditions.

    Parameters
    ----------
    high : array-like
        High prices array
    low : array-like
        Low prices array
    startvalue : float, optional
        Starting value for SAR (default: 0.0, auto-calculated)
    offsetonreverse : float, optional
        Offset on reversal (default: 0.0)
    accelerationinit_long : float, optional
        Initial acceleration factor for long (default: 0.02)
    accelerationlong : float, optional
        Acceleration increment for long (default: 0.02)
    accelerationmax_long : float, optional
        Maximum acceleration for long (default: 0.2)
    accelerationinit_short : float, optional
        Initial acceleration factor for short (default: 0.02)
    accelerationshort : float, optional
        Acceleration increment for short (default: 0.02)
    accelerationmax_short : float, optional
        Maximum acceleration for short (default: 0.2)

    Returns
    -------
    np.ndarray
        Array of SAR values

    Notes
    -----
    - Compatible with TA-Lib SAREXT signature
    - Allows asymmetric parameters for long/short
    - More flexible than standard SAR
    - Useful for markets with directional bias

    See Also
    --------
    SAR : Standard Parabolic SAR
    """
    # Convert to numpy arrays
    high = np.asarray(high, dtype=np.float64)
    low = np.asarray(low, dtype=np.float64)

    n = len(high)
    if len(low) != n:
        raise ValueError("high and low must have the same length")

    if n == 0:
        return np.array([], dtype=np.float64)

    # Check backend and dispatch to appropriate implementation

    backend = get_backend()

    # Use CPU implementation (default)
    output = np.empty(n, dtype=np.float64)
    _sarext_numba(high, low, startvalue, offsetonreverse,
                 accelerationinit_long, accelerationlong, accelerationmax_long,
                 accelerationinit_short, accelerationshort, accelerationmax_short,
                 output)
    return output