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docs/indicators/averages/afirma/afirma.md
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## AFIRMA: Adaptive Filtering Integrated Recursive Moving Average
# AFIRMA: Autoregressive Finite Impulse Response Moving Average
### Concept
## Concept
AFIRMA combines elements of simple moving averages (`SMA`) with adaptive filtering techniques from signal processing. It aims to provide a more responsive indicator that can quickly adjust to market changes while maintaining stability.
AFIRMA indicator is a hybrid moving average that combines the benefits of digital filtering and cubic spline fitting to provide a smooth and accurate representation of price movement without significant time lag.
### Origin
## Origin
AFIRMA (Adaptive Filtering Integrated Recursive Moving Average) was developed by Clive Bowsher and Roland Meeks in their 2008 paper titled "*The Dynamics of Economic Functions: Modeling and Forecasting the Yield Curve.*" It was created as an improvement over traditional moving averages, designed to adapt more quickly to changes in financial time series data.
The AFIRMA indicator is based on the principles of digital signal processing and curve fitting. It was developed to address the limitations of traditional moving averages, which often suffer from time lag or fail to accurately track price movements.
### Key Features
## Key Features
1. **Adaptive Nature**: AFIRMA adjusts its behavior based on recent price movements, allowing it to respond more quickly to significant changes.
2. **Error-based Adaptation**: It uses the error between its last output and the current input to determine how much to adapt.
3. **Customizable Sensitivity**: The alpha parameter allows fine-tuning the indicator's responsiveness.
- **Digital Filter**: The AFIRMA indicator uses a digital filter to smooth out price movements.
- **Cubic Spline Fitting**: The latest candlesticks are smoothed using cubic spline fitting with the least square method to ensure a seamless transition.
- **Combined Moving Average**: The indicator combines the digital filter and cubic spline fitting to create a smooth moving average that accurately tracks prices without time lag.
- **Customizable Parameters**: The AFIRMA indicator allows users to adjust the Periods, Taps, and Window parameters to fine-tune the indicator's performance.
### Usage
## Advantages
1. **Trend Identification**: AFIRMA can help identify the start or end of trends more quickly than traditional moving averages.
2. **Signal Generation**: Traders may use crossovers of AFIRMA with price or other indicators to generate buy/sell signals.
3. **Dynamic Support/Resistance**: The AFIRMA line can act as a dynamic support or resistance level.
4. **Volatility Analysis**: The adaptive nature of AFIRMA can provide insights into market volatility.
- **Accurate Price Tracking**: The AFIRMA indicator provides a smooth and accurate representation of price movement without time lag.
- **Hybrid Approach**: The combination of digital filtering and cubic spline fitting provides a unique and effective approach to moving average calculation.
### Advantages
## Considerations
- More responsive to market changes compared to traditional moving averages.
- Reduces lag typically associated with moving averages.
- Customizable through its `alpha` parameter to suit different market conditions or trading styles.
### Considerations
- **Adaptive Factor**: Alpha serves as the base adaptive factor. It has a range between 0.0 and 1.0 that determines how quickly the AFIRMA responds to changes in the input data.
- **Error Sensitivity**: Alpha is used in calculating the adaptive factor, which is based on the error between the current input and the last AFIRMA value.
- **Balancing Stability and Responsiveness**: A smaller alpha (closer to 0.0) makes the AFIRMA more stable but less responsive to recent changes.
A larger alpha (closer to 1) makes the AFIRMA more responsive to recent changes but potentially more volatile.
- **Fine-tuning the Indicator**:
- In trending markets, a higher alpha might be preferred to capture price movements more quickly.
- In ranging markets, a lower alpha might be better to reduce false signals from price noise.
- **Adaptive Nature**: The use of alpha allows AFIRMA to adapt its behavior based on recent price movements. When there are significant changes (large errors), the adaptive factor increases, allowing AFIRMA to adjust more quickly.
**Complexity**: The AFIRMA indicator is a complex filter that requires some understanding of digital signal processing and curve fitting to use it right.
- **Parameter Optimization**: Finding the optimal parameters for the AFIRMA indicator may require some experimentation and testing.
- **Computational Resources**: The AFIRMA indicator is computationally more intensive than traditional moving averages due to the use of cubic spline fitting.
docs/indicators/averages/afirma/analysis.md
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# AFIRMA: Benchmark Analysis
This analysis evaluates the Autoregressive Finite Impulse Response Moving Average (AFIRMA) across four core benchmarks: accuracy, timeliness, overshooting, and smoothness. These benchmarks provide a comprehensive view of AFIRMA's performance characteristics and serve as a basis for comparison with other moving averages.
## Accuracy (closeness to the original data)
AFIRMA generally exhibits high accuracy in representing the original price data due to its sophisticated approach combining digital filtering and cubic spline fitting.
- **Strengths**:
- The digital filter component helps to reduce noise while preserving important price trends.
- Cubic spline fitting for recent candlesticks ensures that the most current price movements are accurately represented.
- **Considerations**:
- Accuracy can vary based on parameter settings. Incorrect parameter selection might lead to over-smoothing or under-smoothing, potentially reducing accuracy.
- In highly volatile markets, AFIRMA may sacrifice some accuracy for smoothness, especially if the parameters are set to prioritize noise reduction.
## Timeliness (amount of lag)
AFIRMA is designed to minimize lag, which is one of its key advantages over traditional moving averages.
- **Strengths**:
- The combination of ARMA modeling and FIR filtering allows AFIRMA to respond quickly to price changes.
- Cubic spline fitting of recent data points further reduces lag for the most current price movements.
- **Considerations**:
- While AFIRMA generally has less lag than traditional MAs, it's not entirely lag-free. Some minimal lag may still be present, especially with longer period settings.
- The amount of lag can be influenced by parameter settings. Optimizing for minimal lag might come at the cost of increased noise sensitivity.
## Overshooting (overcompensation during reversals)
AFIRMA's design helps to mitigate overshooting during price reversals, but the extent can vary based on settings and market conditions.
- **Strengths**:
- The digital filtering component helps to dampen extreme price movements, reducing the likelihood of significant overshooting.
- Cubic spline fitting allows for smoother transitions during reversals, potentially minimizing overshoot.
- **Considerations**:
- Overshooting can still occur, especially in markets with sudden, sharp reversals.
- The degree of overshooting can be influenced by parameter settings. More aggressive settings might increase responsiveness but also the risk of overshooting.
## Smoothness (continuous 2nd derivative, less jagged flow)
AFIRMA generally produces a smoother line than many traditional moving averages, which is one of its defining characteristics.
- **Strengths**:
- The digital filtering component effectively smooths out minor price fluctuations and noise.
- Cubic spline fitting ensures a smooth transition between historical and current data points.
- The combination of these techniques results in a visually smooth line that can make trend identification easier.
- **Considerations**:
- The degree of smoothness can be adjusted through parameter settings. Excessive smoothing might lead to a loss of responsiveness to genuine price changes.
- In some cases, the smooth line might mask short-term volatility that could be relevant for certain trading strategies.
## Conclusion
AFIRMA demonstrates strong performance across all four benchmarks, particularly excelling in accuracy, timeliness, and smoothness. Its complex approach allows it to balance these often competing characteristics more effectively than many traditional moving averages.
However, it's important to note that AFIRMA's performance can be significantly influenced by its parameter settings. Optimal use of AFIRMA requires careful tuning of these parameters to balance accuracy, timeliness, overshooting resistance, and smoothness for the specific asset and timeframe being analyzed.
When compared to other moving averages, AFIRMA generally offers superior or comparable performance across these benchmarks. However, this comes at the cost of increased complexity and computational requirements. Traders and analysts should weigh these factors when deciding whether to incorporate AFIRMA into their technical analysis toolkit.
docs/indicators/averages/afirma/calc.md
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# The Math Behind AFIRMA
## Components of AFIRMA
AFIRMA is a hybrid beast, combining two main components:
- Autoregressive Moving Average (ARMA)
- Finite Impulse Response (FIR) filter
### ARMA Component
$ X_t = c + \epsilon_t + \sum_{i=1}^p \phi_i X_{t-i} + \sum_{j=1}^q \theta_j \epsilon_{t-j} $
Where:
- $X_t$ is the time series value at time $t$<br>
- $c$ is a constant<br>
- $\phi_i$ are the parameters of the autoregressive term<br>
- $\theta_j$ are the parameters of the moving average term<br>
- $\epsilon_t$ is white noise<br>
### FIR Component
$ y[n] = \sum_{i=0}^{N-1} b_i \cdot x[n-i] $
Where:
- $y[n]$ is the output signal
- $x[n]$ is the input signal
- $b_i$ are the filter coefficients
- $N$ is the filter order
### AFIRMA: Putting It All Together
AFIRMA combines these components and adds cubic spline fitting to the mix. The general form can be expressed as:
$ AFIRMA_t = ARMA_t + FIR_t + CS_t $
Where:
- $ARMA_t$ is the ARMA component at time $t$
- $FIR_t$ is the FIR component at time $t$
- $CS_t$ is the cubic spline fitting component at time $t$
### Digital Filtering Process
- The price data is passed through the digital filter to smooth out fluctuations.
- The filter coefficients are optimized based on the specified parameters (Periods, Taps, Window).
### Cubic Spline Fitting
For the most recent bars:
- A cubic spline is fitted to the data points using the least squares method.
- This ensures a smooth transition between the filtered data and the most recent price movements.
### Parameter Definitions
The AFIRMA indicator allows for the adjustment of three main parameters:
- **Periods**: Affects the overall smoothness of the indicator.
- *Taps*: Influences the complexity of the digital filter.
- *Window*: Determines the number of recent bars to which the cubic spline fitting is applied.
### Computational Process
- Apply the ARMA model to the price data.
- Pass the result through the FIR filter.
- Apply cubic spline fitting to the most recent data points.
- Combine the results to produce the final AFIRMA value.
docs/indicators/averages/afirma/charts.dib
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#!csharp
String Name = "AFIRMA";
int p = 10;
double alpha = 0.5;
Func<int, AbstractBase> Indicator = period => new Afirma(period,alpha);
int taps = 6;
int periods = 6;
Afirma.WindowType window = Afirma.WindowType.BlackmanHarris;
Func<int, int, Afirma.WindowType, AbstractBase> Indicator = (taps, periods, windows) => new Afirma(taps: taps, periods: periods, window: window);
foreach (var item in Data) {
string Signal = item.Key;
double[] Input = item.Value;
TSeries Output = new();
var ma = Indicator(p);
var ma = Indicator(taps, periods, window);
foreach (var value in Input) { Output.Add(ma.Calc(value)); }
Plot plt = new();
var p1a = plt.Add.Signal(Input[24..]); p1a.Color = ScottPlot.Colors.Red; p1a.LineWidth = 2;
var p1b = plt.Add.Signal(Output.v.ToArray()[24..]); p1b.Color = ScottPlot.Colors.Blue; p1b.LineWidth = 4;
plt.Title($"{Signal} - {Name}({p}, {alpha})");
plt.Title($"{Signal} - {Name}({taps}, {periods}, {window.ToString()})");
plt.Display();
plt.SaveSvg($"img/{Name}{p}_{Signal}.svg", 450, 300);
plt.SaveSvg($"img/{Name}{taps}_{Signal}.svg", 450, 300);
}
docs/indicators/averages/afirma/charts.md
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# AFIRMA Charts
# AFIRMA: Charts
![](img/AFIRMA10_Spike.svg) ![](img/AFIRMA10_Impulse.svg) ![](img/AFIRMA10_Triangle.svg) ![](img/AFIRMA10_Sawtooth.svg) ![](img/AFIRMA10_Sine.svg) ![](img/AFIRMA10_Chirp.svg) ![](img/AFIRMA10_White.svg) ![](img/AFIRMA10_Gauss.svg) ![](img/AFIRMA10_B.svg) ![](img/AFIRMA10_HF.svg) ![](img/AFIRMA10_ImpulseHF.svg) ![](img/AFIRMA10_SawtoothHF.svg) ![](img/AFIRMA10_SineG.svg) ![](img/AFIRMA10_ChirpG.svg) ![](img/AFIRMA10_Complex.svg) ![](img/AFIRMA10_Market.svg)
![](img/AFIRMA6_Spike.svg) ![](img/AFIRMA6_Impulse.svg) ![](img/AFIRMA6_Triangle.svg) ![](img/AFIRMA6_Sawtooth.svg) ![](img/AFIRMA6_Sine.svg) ![](img/AFIRMA6_Chirp.svg) ![](img/AFIRMA6_White.svg) ![](img/AFIRMA6_Gauss.svg) ![](img/AFIRMA6_B.svg) ![](img/AFIRMA6_HF.svg) ![](img/AFIRMA6_ImpulseHF.svg) ![](img/AFIRMA6_SawtoothHF.svg) ![](img/AFIRMA6_SineG.svg) ![](img/AFIRMA6_ChirpG.svg) ![](img/AFIRMA6_Complex.svg) ![](img/AFIRMA6_Market.svg)
docs/indicators/averages/afirma/img/AFIRMA10_B.svg → docs/indicators/averages/afirma/img/AFIRMA6_B.svg
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Side by Side Swipe Overlay

Before

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After

Width:  |  Height:  |  Size: 30 KiB

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Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 30 KiB

After

Width:  |  Height:  |  Size: 30 KiB

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+79 -79
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ChirpG - AFIRMA(6, 6, BlackmanHarris)
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Side by Side Swipe Overlay

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After

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Complex - AFIRMA(6, 6, BlackmanHarris)
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Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 31 KiB

After

Width:  |  Height:  |  Size: 31 KiB

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Side by Side Swipe Overlay

Before

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After

Width:  |  Height:  |  Size: 30 KiB

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HF - AFIRMA(6, 6, BlackmanHarris)
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Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 30 KiB

After

Width:  |  Height:  |  Size: 30 KiB

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Impulse - AFIRMA(10, 0.5)
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Impulse - AFIRMA(6, 6, BlackmanHarris)
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Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 31 KiB

After

Width:  |  Height:  |  Size: 31 KiB

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+79 -79
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Side by Side Swipe Overlay

Before

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After

Width:  |  Height:  |  Size: 30 KiB

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Side by Side Swipe Overlay

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After

Width:  |  Height:  |  Size: 33 KiB

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Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 33 KiB

After

Width:  |  Height:  |  Size: 33 KiB

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Side by Side Swipe Overlay

Before

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After

Width:  |  Height:  |  Size: 31 KiB

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Side by Side Swipe Overlay

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After

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Side by Side Swipe Overlay

Before

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After

Width:  |  Height:  |  Size: 32 KiB

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Side by Side Swipe Overlay

Before

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After

Width:  |  Height:  |  Size: 31 KiB

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Side by Side Swipe Overlay

Before

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After

Width:  |  Height:  |  Size: 33 KiB

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