Dynamic Gradient Filter

Sigmoid Functions:

History and Mathematical Basis:

1. Sigmoid functions have a rich history in mathematics and are widely used in various fields, including statistics, machine learning, and signal processing.
   
2. The term "sigmoid" originates from the Greek words "sigma" (meaning "S-shaped") and "eidos" (meaning "form" or "type").
   
3. The sigmoid curve is characterized by its smooth S-shaped appearance, which allows it to map any real-valued input to a bounded output range, typically between 0 and 1.
   
4. The most common form of the sigmoid function is the logistic function:
   

Logistic Function (σ):

• Defined as σ(x) = 1 / (1 + e^(-x)), where:
  

'x' is the input value,
'e' is Euler's number (approximately 2.71828).

• This function was first introduced by Belgian mathematician Pierre François Verhulst in the 1830s to model population growth with limiting factors.
  
• It gained popularity in the early 20th century when statisticians like Ronald Fisher began using it in regression analysis.
  

Specific Sigmoid Functions Used in the Indicator:

• sig(val):

The 'sig' function in this indicator is a modified version of the logistic function, clamping a value between 0 and 1 on the sigmoid curve.

• siga(val):
  

The 'siga' function adjusts values between -1 and 1 on the sigmoid curve, offering a centered variation of the sigmoid effect.

• sigmoid(val):
  

The 'sigmoid' function provides a standard implementation of the logistic function, calculating the sigmoid value of the input data.

Adaptive Smoothing Factor:
The 'adaptiveSmoothingFactor(gradient, k)' function computes a dynamic smoothing factor for the filter based on the gradient of the price data and the user-defined sensitivity parameter 'k'.

• Gradient:
  

The gradient represents the rate of change in price, calculated as the absolute difference between the current and previous close prices.

• Sensitivity (k):
  

The 'k' parameter adjusts how quickly the filter reacts to changes in the gradient. Higher values of 'k' lead to a more responsive filter, while lower values result in smoother outputs.

Usage in the Indicator:
The "close" value refers to the closing price of each period in the chart's time frame

• The indicator calculates the gradient by measuring the absolute difference between the current "close" price and the previous "close" price.
  
• This gradient represents the strength or magnitude of the price movement within the chosen time frame.
  
• The "close" value plays a pivotal role in determining the dynamic behavior of the "Dynamic Gradient Filter," as it directly influences the smoothing factor.
  

What Makes This Special:
The "Dynamic Gradient Filter" indicator stands out due to its adaptive nature and responsiveness to changing market conditions.

Dynamic Smoothing Factor:

• The indicator's dynamic smoothing factor adjusts in real-time based on the rate of change in price (gradient) and the user-defined sensitivity '(k)' parameter.
  
• This adaptability allows the filter to respond promptly to both minor fluctuations and significant price movements.
  

Smoothed Price Action:

• The final output of the filter is a smoothed representation of the price action, aiding traders in identifying trends and potential reversals.
  

Customizable Sensitivity:

• Traders can adjust the 'Sensitivity' parameter '(k)' to suit their preferred trading style, making the indicator versatile for various strategies.
  

Visual Clarity:

• The plotted "Dynamic Gradient Filter" line on the chart provides a clear visual guide, enhancing the understanding of market dynamics.
  

Usage:
Traders and analysts can utilize the "Dynamic Gradient Filter" to:

• Identify trends and reversals in price movements.
  
• Filter out noise and highlight significant price changes.
  
• Fine-tune trading strategies by adjusting the sensitivity parameter.
  
• Enhance visual analysis with a dynamically adjusting filter line on the chart.
  

Literature:

1. https://en.wikipedia.org/wiki/Pierre_François_Verhulst
   
2. https://medium.com/in-maths-garden-with-julia/behind-the-logistic-function-47a8430a8f46
   
3. https://en.wikipedia.org/wiki/Sigmoid_function
   

Release Notes:

minor adjustment to

m = input.float(1, title="Multiplier",minval=0.01, maxval=10000, step=0.01)