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"""
Written By: Rajeev Jain, 2025-07-23
Trader Tips python code for TAS&C Magazine article "The Continuation Index"

"""

# import required python libraries
%matplotlib inline

import pandas as pd
import numpy as np
import math
import datetime as dt
import yfinance as yf
print(yf.__version__)

0.2.58

# Use Yahoo Finance python package to obtain OHLCV data for the SP500 index
symbol = '^GSPC'
ohlcv = yf.download(
    symbol, 
    start="2019-01-01", 
    end="2025-07-24", 
    #group_by="Ticker", 
    auto_adjust=True,
    multi_level_index=False,
    progress=False,
)

# Python functions to implement UltimateSmoother, Laguerre Filter and 
# Continuation Index as defined in John Ehlers’ article. 
def ultimate_smoother(price_series, period):
    """
    Ultimate Smoother function converted from EasyLanguage to Python.
    
    Parameters:
        price_series (list or np.array): Time series of prices (e.g., closing
        prices)
        period (float): Smoothing period

    Returns:
        np.array: Smoothed price series
    """
    price_series = np.asarray(price_series)
    n = len(price_series)
    US = np.zeros(n)

    # Filter coefficients
    Q = math.exp(-1.414 * math.pi / period)
    c1 = 2 * Q * math.cos(1.414 * math.pi / period)
    c2 = Q * Q
    a0 = (1 + c1 + c2) / 4

    for i in range(n):
        if i < 3:
            US[i] = price_series[i]
        else:
            US[i] = ((1 - a0) * price_series[i] +
                     (2 * a0 - c1) * price_series[i - 1] +
                     (c2 - a0) * price_series[i - 2] +
                     c1 * US[i - 1] -
                     c2 * US[i - 2])
    return US

def laguerre_filter(price_series, gama, order, length):
    """
    Laguerre Filter Function converted from EasyLanguage to Python.
    Parameters:
        price_series (list or np.array): Input price data
        gama (float): Laguerre parameter, 0 <= gama < 1
        order (int): Filter order (integer <= 10)
        length (float): Length for Ultimate Smoother
    Returns:
        np.array: Laguerre filtered series
    """
    assert 0 <= gama < 1, "gama must be in [0, 1)"
    assert order <= 10 and order >= 1, "order must be integer between 1 and 10"
    price_series = np.asarray(price_series)
    n = len(price_series)
    output = np.zeros(n)

    # Initialize Laguerre matrix: shape (order+1, 2)
    LG = np.zeros((order + 1, 2))

    # Precompute ultimate smoother once
    smoothed_price = ultimate_smoother(price_series, length)
    for t in range(n):
        # Shift previous values: LG[:, 2] = LG[:, 1]
        LG[:, 1] = LG[:, 0]
        # Update LG[1] using the smoothed price at current time
        LG[1, 0] = smoothed_price[t]
        # Compute rest of the Laguerre components recursively
        for count in range(2, order + 1):
            LG[count, 0] = (
                -gama * LG[count - 1, 1] +
                LG[count - 1, 1] +
                gama * LG[count, 1]
            )
        # Sum current values of LG[1] to LG[order]
        FIR = np.sum(LG[1:order + 1, 0])
        output[t] = FIR / order
    return output
def continuation_index(close_prices, gama=0.8, order=8, length=40):
    """
    Continuation Index by John F. Ehlers (converted from EasyLanguage).
    
    Parameters:
        close_prices (list or np.array): Series of closing prices
        gama (float): Laguerre gamma (0 <= gama < 1)
        order (int): Order of Laguerre filter (<= 10)
        length (int): Base smoothing period
    Returns:
        np.array: Continuation Index values
    """
    close_prices = np.asarray(close_prices)
    n = len(close_prices)

    # Step 1: Ultimate Smoother with Length / 2
    US = ultimate_smoother(close_prices, length / 2)
    # Step 2: Laguerre Filter
    LG = laguerre_filter(close_prices, gama, order, length)
    # Step 3: Variance = avg(abs(US - LG)) over Length
    diff = np.abs(US - LG)
    variance = np.convolve(diff, np.ones(length)/length, mode='same')
    # Step 4: Normalized difference, scaled by 2
    with np.errstate(divide='ignore', invalid='ignore'):
        ref = np.where(variance != 0, 2 * (US - LG) / variance, 0)
    # Step 5: Inverse Fisher Transform compression
    CI = (np.exp(2 * ref) - 1) / (np.exp(2 * ref) + 1)
    return CI

# Compare Laguerre filter response for gamma setting of 0.4 vs gamma setting of 0.8
gama1 = 0.4
gama2 = 0.8
order = 8
length = 40
df = ohlcv.copy()
df['Laguerre_0.4'] = laguerre_filter(df['Close'], gama1, order, length)
df['Laguerre_0.8'] = laguerre_filter(df['Close'], gama2, order, length)
cols = ['Close', 'Laguerre_0.4', 'Laguerre_0.8']
ax = df[-255:][cols].plot(marker='.', grid=True, figsize=(9,6), title=f'Ticker={symbol}, Laguerre Filder Response, Gamma Parameter Comparison')
ax.set_xlabel('')
plt.show()

# Compare Laguerre filter response for order of 4 vs order of 8
order1 = 4
order2 = 8
length = 40
gama = 0.8
df = ohlcv.copy()
df['Laguerre_4th'] = laguerre_filter(df['Close'], gama, order1, length)
df['Laguerre_8th'] = laguerre_filter(df['Close'], gama, order2, length)
df
cols = ['Close', 'Laguerre_4th', 'Laguerre_8th']
ax = df[-255:][cols].plot(marker='.', grid=True, figsize=(9,6), title=f'Ticker={symbol}, Laguerre Filder Response, Order Parameter Comparison')
ax.set_xlabel('')
plt.show()

# Example of continuation index using default setting for 
# daily close of S&P 500 index
def plot_ci(df):
    ax = df[['Close', 'Laguerre']].plot(
        marker='.', 
        grid=True, 
        figsize=(9,4), 
        title=f'Ticker={symbol}'
    )
    ax.set_xlabel('')
    ax = df[['CI']].plot(
        marker='.', 
        grid=True, 
        figsize=(9,2), 
        title=f'Continuation Index'
    )
    ax.set_xlabel('')

gama=0.8
order=8
length=40
df = ohlcv.copy()
df['Laguerre'] = laguerre_filter(df['Close'], gama, order, length)
df['CI'] = continuation_index(df['Close'], gama, order, length)
plot_ci(df[-252:])

Code: Select all

"""
Written By: Rajeev Jain, jainraje@yahoo.com

Python code to implement concepts in Technical Analysis of S&C Magazine 
January 2026 article "The Reversion Index" by John F Ehlers. This python 
code is provided for TraderTips section of the magazine.

2025-11-12  Initial implementation
2025-11-14  Uploaded to github

https://github.com/jainraje/TraderTipArticles/

"""

# Import required python libraries

%matplotlib inline

import pandas as pd
import numpy as np
import yfinance as yf
import math
import datetime as dt
import matplotlib.pyplot as plt



# Use Yahoo Finance python package to obtain OHLCV data for desired instrument.

symbol = '^GSPC'
start = "2023-11-13"
end = dt.datetime.now().strftime('%Y-%m-%d') 
end = '2025-11-15'
ohlcv = yf.download(
    symbol, 
    start, 
    end,
    group_by="Ticker",  
    auto_adjust=True
)
ohlcv = ohlcv[symbol]
ohlcv



# three functions to implement concepts: 1) super_smoother 2) reversion_index 
# and 3) plot_reversion_index to plot relevant signals

def super_smoother(price: pd.Series, period: float) -> pd.Series:
    """
    Vectorized Ehlers SuperSmoother filter (© John F. Ehlers)
    
    Parameters
    ----------
    price : pd.Series
        Input price series (e.g., closing prices)
    period : float
        Smoothing period

    Returns
    -------
    pd.Series
        Smoothed price series
    """
    q = np.exp(-1.414 * np.pi / period)
    c1 = 2 * q * np.cos(np.radians(1.414 * 180 / period))
    c2 = q * q
    a0 = (1 - c1 + c2) / 2

    price_vals = price.to_numpy()
    out_values = np.zeros_like(price_vals)
    
    # Initialize first four values to the original price
    out_values[:4] = price_vals[:4]

    # Apply recursive filter from index 4 onward
    for i in range(4, len(price_vals)):
        out_values[i] = a0 * (price_vals[i] + price_vals[i-1]) + c1 * out_values[i-1] - c2 * out_values[i-2]

    return pd.Series(out_values, index=price.index)


def reversion_index(close: pd.Series, length: int = 20) -> pd.DataFrame:
    """
    Ehlers Reversion Index (© 2025 John F. Ehlers)
    Correct and fully vectorized Python version.
    """
    close = close.astype(float)
    
    # Delta: change from previous bar
    delta = close.diff().fillna(0)
    
    # Rolling sums of delta and absolute delta
    delta_sum = delta.rolling(window=length, min_periods=1).sum()
    abs_delta_sum = delta.abs().rolling(window=length, min_periods=1).sum()
    
    # Ratio: safely avoid division by zero
    ratio = delta_sum / abs_delta_sum.replace(0, np.nan)
    ratio = ratio.fillna(0)
    
    # Smooth and trigger lines
    smooth = super_smoother(ratio, period=8)
    trigger = super_smoother(ratio, period=4)
    
    return pd.DataFrame({
        'ReversionSmooth': smooth,
        'ReversionTrigger': trigger
    }, index=close.index)


def plot_reversion_index(df):
    
    import matplotlib.pyplot as plt
    import matplotlib.dates as mdates
    
    fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 9), sharex=True, gridspec_kw={'height_ratios':[2,2]})
    
    # ----- Top: Price & SuperSmoother -----
    ax1.plot(df.index, df['Close'], label='Close', linewidth=1)
    ax1.plot(df.index, df['SuperSmoother'], label='SuperSmoother', linewidth=2, alpha=0.6, color='orange')
    
    # Shading for bullish/bearish signals
    ax1.fill_between(df.index, df['Close'].min(), df['Close'].max(), 
                     where=df['Signal'] == 1, color='green', alpha=0.1)
    ax1.fill_between(df.index, df['Close'].min(), df['Close'].max(), 
                     where=df['Signal'] == -1, color='red', alpha=0.1)
    
    ax1.set_title('Price & SuperSmoother with Signal')
    ax1.grid(True)
    ax1.legend(loc='center left', bbox_to_anchor=(1, 0.5))
    
    # ----- Bottom: ReversionSmooth & ReversionTrigger -----
    ax2.plot(df.index, df['ReversionSmooth'], label='ReversionSmooth', linewidth=1, color='red')
    ax2.plot(df.index, df['ReversionTrigger'], label='ReversionTrigger', linewidth=1, color='darkblue')
    ax2.axhline(0, color='gray', linestyle='--', linewidth=1)
    
    # Overlay Signal as step plot
    #ax2.step(df.index, df['Signal'], where='mid', label='Signal', color='purple', linewidth=1.5, alpha=0.5)
    
    ax2.set_title('Reversion Index with Signal')
    ax2.grid(True)
    ax2.legend(loc='center left', bbox_to_anchor=(1, 0.5))
    
    # ----- Improve date formatting on the bottom subplot -----
    ax2.xaxis.set_major_locator(mdates.AutoDateLocator())
    ax2.xaxis.set_major_formatter(mdates.ConciseDateFormatter(mdates.AutoDateLocator()))
    plt.setp(ax2.get_xticklabels(), rotation=0, ha='center')  # horizontal, centered
    
    #plt.xlabel('Date')
    plt.tight_layout()
    plt.show()
	

 

# Call functions to run required calculations. Set buy when ReversionTrigger 
# swings above ReversionSmooth and sell for when ReversionTrigger swings below. 
# Different period and lengths can be tested to determine optimum settings. 
# Top plot close vs the SuperSmoother and red and white background shading to 
# show highlight when buy and sell signals are active. Using slicing to plot 
# last 1 year (aka 252 trading days).

df = ohlcv.copy()
df['SuperSmoother'] = super_smoother(df['Close'], period=10)
df = df.join(reversion_index(df['Close'], length=20))
df['Signal'] = np.where(df['ReversionTrigger'] > df['ReversionSmooth'], 1, -1)
plot_reversion_index(df[-252:])

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"""
# import required python libraries

import pandas as pd
import numpy as np
import yfinance as yf
import math
import datetime as dt
import matplotlib.pyplot as plt
import mplfinance as mpf

print(yf.__version__)

# three helper functions to 1) run one euro filter calculations 2) run 
# oscillator calculations and to 3) plot the close vs one euro filter output 
# on one subplot and the oscillator on a second subplot using built-in 
# matplotlib capabilities.

def calc_one_euro_filter(close, period_min=10, beta=0.2):
    """
    One Euro Filter Indicator
    Adapted from Georges Casiez et al. (CHI 2012)
    Converted from EasyLanguage to Python 
    
    Args:
        close (array-like): List or NumPy array of closing prices
        period_min (float): Minimum cutoff period (default=10)
        beta (float): Responsiveness factor (default=0.2)
        
    Returns:
        np.ndarray: Smoothed signal
    """
    close = np.asarray(close)
    n = len(close)
    
    # Initialize arrays
    smoothed = np.zeros(n)
    smoothed_dx = np.zeros(n)
    
    period_dx = 10.0
    alpha_dx = 2 * np.pi / (4 * np.pi + period_dx)
    
    # Initialization
    smoothed[0] = close[0]
    smoothed_dx[0] = 0.0
    
    for i in range(1, n):
        # Delta smoothing (EMA of price change)
        smoothed_dx[i] = alpha_dx * (close[i] - close[i - 1]) + (1 - alpha_dx) * smoothed_dx[i - 1]
        
        # Adjust cutoff based on rate of change
        cutoff = period_min + beta * abs(smoothed_dx[i])
        
        # Compute adaptive alpha
        alpha3 = 2 * np.pi / (4 * np.pi + cutoff)
        
        # Adaptive smoothing
        smoothed[i] = alpha3 * close[i] + (1 - alpha3) * smoothed[i - 1]
    
    return smoothed


def calc_highpass(price, period):
    """
    Two-pole high-pass filter (John Ehlers style)
    
    Args:
        price (array-like): Sequence of prices
        period (float): Filter period
    
    Returns:
        np.ndarray: High-pass filtered values
    """
    price = np.asarray(price, dtype=float)
    n = len(price)
    out = np.zeros(n)

    # Filter coefficients
    a1 = np.exp(-1.414 * np.pi / period)
    b1 = 2 * a1 * np.cos(math.radians(1.414 * 180 / period))
    c2 = b1
    c3 = -a1 ** 2
    c1 = (1 + c2 - c3)/4

    # Main filter loop
    for i in range(2, n):
        out[i] = (
            c1 * (price[i] - 2 * price[i - 1] + price[i - 2])
            + c2 * out[i - 1]
            + c3 * out[i - 2]
        )

    # Initialize first values (pass-through)
    out[0:2] = price[0:2]

    return out


def plot_one_euro_filter(df):
    
    ax = df[['Close', 'OEF']].plot(grid=True, title=f"Ticker={symbol}, Close vs One Euro Filter", figsize=(9,4), marker='.')
    ax.set_xlabel('')

    ax = df[['Osc']].plot(grid=True, title=f"One Euro Filter Oscilator", figsize=(9,2))
    ax.set_xlabel('')


 

# Use Yahoo Finance python package to obtain OHLCV data for desired symbol

symbol = '^GSPC'
start = "1990-01-01"
end = dt.datetime.now().strftime('%Y-%m-%d') 

ohlcv = yf.download(
    symbol, 
    start, 
    end,
    group_by="Ticker",  
    auto_adjust=False
)
ohlcv = ohlcv[symbol]



# call helper functions to calculate the One Euro Filter output (named OEF) and the 
# Oscillaor output (named Osc) and then plot close, OEF and Ocs values.

df['OEF'] = calc_one_euro_filter(df['Close'], period_min=10, beta=0.2)
df['Osc'] = calc_highpass(df['Close'], period=54)
plot_one_euro_filter(df['2025':])

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#
# import required python libraries
#
%matplotlib inline
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import yfinance as yf
import math
import datetime as dt
print(yf.__version__)

#
# Use Yahoo Finance python package to obtain OHLCV data 
#
symbol = '^GSPC'
symbol = 'SPY'
ohlcv = yf.download(symbol, start="1995-01-01", end="2025-04-18", group_by="Ticker", auto_adjust=True)
ohlcv = ohlcv[symbol]

#
# Use pandas built in plot function to see simple price chart 
#
ax = ohlcv['Close'].plot(
    figsize=(9,6), 
    grid=True, 
    title=f'{symbol}', 
    #marker='.'
)


#
# Building block / routines used to implement cybernetic oscillaor indicator 
#
def calc_highpass(price, period):
    
    a1 = np.exp(-1.414 * np.pi / period)
    b1 = 2 * a1 * np.cos(math.radians(1.414 * 180 / period))
    c2 = b1
    c3 = -a1 * a1
    c1 = (1 + c2 - c3)/4

    out_values = []
    for i in range(len(price)):
        if i >= 4:
            out_values.append(
                c1*(price[i] - 2*price[i-1] + price[i-2]) + c2*out_values[i-1] + c3*out_values[i-2]
            )
        else:
            out_values.append(price[i])
    
    return out_values

def calc_super_smoother(price, period):
        
        a1 = np.exp(-1.414 * np.pi / period)
        b1 = 2 * a1 * np.cos(math.radians(1.414 * 180 / period))
        c2 = b1
        c3 = -a1 * a1
        c1 = (1 - c2 - c3)
    
        out_values = []
        for i in range(len(price)):
            if i >= 4:
                out_values.append(c1*(price[i]+price[i-1])/2 + c2*out_values[i-1] + c3*out_values[i-2])
            else:
                out_values.append(price[i])
        
        return out_values

def calc_rms(price):
    
    length = len(price)
    sum_sq = 0
    for count in range(length):
        sum_sq += price[count] * price[count]
    return np.sqrt(sum_sq / length)

def calc_cybernetic_oscillator(close, params=(30, 20)):

    hp_length = params[0]
    lp_length = params[1]
    
    df = pd.DataFrame(index=close.index, data=close)
    df['HP'] = calc_highpass(close, hp_length)
    df['LP'] = calc_super_smoother(df['HP'], lp_length)
    df['RMS'] = df['LP'].rolling(100).apply(calc_rms)
    df['CO'] = df['LP']/df['RMS']

    return df['CO']

#
# Exampe python code to create two versions of the cybernetic oscillator 
# as presented in June 2005 Trader Tip article
#

lp_length = 20
hp1_length = 30
hp2_length = 250

co1_params=(hp1_length, lp_length)
co2_params=(hp2_length, lp_length)

df = ohlcv.copy()
df['CO1'] = calc_cybernetic_oscillator(ohlcv['Close'], params=co1_params)
df['CO2'] = calc_cybernetic_oscillator(ohlcv['Close'], params=co2_params)
df


#
# MatPlotLib routine to plot Close, CO1 and CO2 values
#
def plot_indicators1(df):
    # Create a figure with three subplots stacked vertically
    fig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(9, 6), sharex=True)

    # Plotting the first subplot (e.g., Price Data)
    ax1.set_title(f"S&P 500")
    ax1.plot(df.index, df['Close'], label='Close', color='blue', marker='.')
    ax1.grid(True, linestyle='--', alpha=0.5)
    ax1.set_ylabel('Price Plot')
    ax1.legend(loc='upper left', bbox_to_anchor=(1, 1))

    # Plotting the second subplot
    ax2.set_title(f'Cybernetic Oscillator {co1_params}')
    ax2.plot(df.index, df['CO1'], label='CO1', color='red')
    ax2.axhline(y=0, color='black', linestyle='-', label='zero')
    #ax2.set_ylabel('Linear Slope')
    ax2.grid(True, linestyle='-', alpha=0.5)
    ax2.legend(loc='upper left', bbox_to_anchor=(1, 1))

    # Plotting the third subplot
    ax3.set_title(f'Cybernetic Oscillator {co2_params}')
    ax3.plot(df.index, df['CO2'], label='CO2', color='blue')
    ax3.axhline(y=0, color='black', linestyle='-', label='zero')
    ax3.grid(True, linestyle='-', alpha=0.5)
    ax3.legend(loc='upper left', bbox_to_anchor=(1, 1))
    
    # Rotate x-axis labels for better readability
    plt.xticks(rotation=45)

    # Improve overall layout
    plt.tight_layout()

    # Show the plot
    plt.show()



plot_indicators1(df['2024-03':'2025'])


#
# Example python code to implement the simple dual ROC strategy showcased 
# in trader tip article the python code is bundled in a *strategy* routine 
# so it can easily be called, paramater as passed into the routine as a variable
#
def strategy(ohlcv, params):
    
    df = ohlcv.copy()
    df['LP'] = calc_super_smoother(df['Close'], params[0])
    df['BP1'] = calc_highpass(df['LP'],params[1] )
    df['ROC1'] = df['BP1'] - df['BP1'].shift(2)
    
    df['BP2'] = calc_highpass(df['LP'],params[2])
    df['ROC2'] = df['BP2'] - df['BP2'].shift(2)
    
    df['Signal'] = np.where((df['ROC1'] > 0 ) & (df['ROC2'] >0 ), 1, np.nan)
    df['Signal'] = np.where((df['ROC1'] < 0 ) & (df['ROC2'] <0 ), 0, df['Signal'])
    df['Signal'] = df['Signal'].fillna(method='ffill')
    return df


params=(20, 55, 156)
data = strategy(ohlcv, params)
data


#
# The Simple Dual ROC strategy indicators are visualized in the MatplotLib 
# routine below. Here LP, BP1 and BP2 can be see observed along with price
#
def plot_indicators2(df):
    
    # Create a figure with three subplots stacked vertically
    fig, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=(9, 6), sharex=True)

    # Plotting the first subplot (e.g., Price Data)
    ax1.set_title(f"Ticker={symbol}")
    ax1.plot(df.index, df['Close'], label='Close', color='blue',)
    ax1.plot(df.index, df['LP'], label='LP', color='orange', )
    ax1.grid(True, linestyle='--', alpha=0.5)
    ax1.set_ylabel('Price Plot')
    ax1.legend(loc='upper left', bbox_to_anchor=(1, 1))

    # Plotting the second subplot
    ax2.set_title(f'BP1')
    ax2.plot(df.index, df['BP1'], label='BP1', color='red')
    ax2.axhline(y=0, color='black', linestyle='--', label='zero')
    #ax2.set_ylabel('Linear Slope')
    ax2.grid(True, linestyle='-', alpha=0.5)
    ax2.legend(loc='upper left', bbox_to_anchor=(1, 1))

    # Plotting the third subplot
    ax3.set_title(f'BP2')
    ax3.plot(df.index, df['BP2'], label='BP2', color='blue')
    ax3.axhline(y=0, color='black', linestyle='--', label='zero')
    ax3.grid(True, linestyle='-', alpha=0.5)
    ax3.legend(loc='upper left', bbox_to_anchor=(1, 1))
    
    # Rotate x-axis labels for better readability
    plt.xticks(rotation=45)

    # Improve overall layout
    plt.tight_layout()

    # Show the plot
    plt.show()

plot_indicators2(data['2024-03':'2025'])


#
# A simple backtest framework/routines are provided which allow changing 
# parameters and visualize performance results. Performance can be compared 
# against Buy&Hold of same ticker by setting. All buy and sell occur on the 
# day of the buy/sell signal using market close values
#
def backtest(data):

    df = data.copy()
    
    df['Return Ratio'] = np.where(df['Signal'].shift()==1, 1+df['Close'].pct_change(),1 )
    df['Strategy'] = df['Return Ratio'].cumprod()
    df['BH'] = (1+df['Close'].pct_change()).cumprod()

    df['Peak'] = df['Strategy'].cummax()
    df['% DD'] = 100*(df['Strategy']/df['Peak']-1)

    df['Peak'] = df['BH'].cummax()
    df['BH % DD'] = 100*(df['BH']/df['Peak']-1)

    df.at[df.index[0], 'Strategy'] = 1
    df.at[df.index[0], 'BH'] = 1

    return df


def plot_backtest_results(df, strategy_name="Simple Dual ROC Strategy", compare_bh_ena=False):

    fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(9, 6), sharex=True)

    ax1.set_title(f"{strategy_name}, Ticker={symbol},")
    ax1.plot(df.index, df['Strategy'], label='Strategy Equity Curve', color='blue',)

    if compare_bh_ena:
        ax1.plot(df.index, df['BH'], label='Buy & Hold Equity Curve', color='orange',)
    ax1.grid(True, linestyle='--', alpha=0.5)
    ax1.set_ylabel('Cumulative Return')
    ax1.legend(loc='upper left', bbox_to_anchor=(1, 1))

    ax2.set_title(f"% Drawdown")
    ax2.plot(df.index, df['% DD'], label='Strategy Drawdown', color='blue',)

    if compare_bh_ena:
        ax2.plot(df.index, df['BH % DD'], label='Buy & Hold Drawdown', color='orange',)

    ax2.grid(True, linestyle='--', alpha=0.5)
    ax2.set_ylabel('% drawdown')
    ax2.legend(loc='upper left', bbox_to_anchor=(1, 1))

#
# Putting the building blocks toegther, parameters can be changed and results viewed visually
# Here using default values presenyed in article
#
params=(20, 55, 156)
data = strategy(ohlcv, params)
df = backtest(data['2009':'2024'])
plot_backtest_results(df, strategy_name="Simple Dual ROC Strategy", compare_bh_ena=False)


#
# Here change parameter and see results compared to buy & hold of same ticker
#
params=(50, 55, 156)
data = strategy(ohlcv, params)
df = backtest(data['2009':'2024'])
plot_backtest_results(df, compare_bh_ena=True)