OPEN-SOURCE SCRIPT
SSA-like Trend (Online PCA + Cycle Filter)
# SSA-like Trend (Online PCA + Cycle Filter)
## Overview
This indicator is an **online approximation to Singular Spectrum Analysis (SSA)** using **Oja's Principal Component Analysis (PCA) learning rule**. It attempts to separate the dominant low-frequency structure (trend) from shorter-term fluctuations by continuously learning the strongest component in a lagged embedding of price.
Unlike classical SSA, which requires constructing a trajectory matrix and performing eigendecomposition, this implementation updates the dominant component incrementally on each bar and is therefore suitable for Pine Script's real-time execution model.
The indicator produces:
* A dominant trend estimate
* ±1 standard deviation envelope around the trend
* A detrended residual series
* An estimate of the strongest cycle length
---
# Theory
## 1. Delay Embedding
For each bar, a lagged vector is constructed:
[
X_t =
[x_t,;x_{t-1},;x_{t-2},;\ldots,;x_{t-L+1}]
]
where:
* (x_t) = normalized price
* (L) = embedding length
This is equivalent to the trajectory matrix concept used in SSA.
The embedding transforms a one-dimensional time series into a higher-dimensional state space.
---
## 2. Online Principal Component Analysis
The indicator learns the first principal component using Oja's learning rule.
The projection of the embedded vector onto the dominant component is:
[
y_t = w^T X_t
]
where:
* (w) = learned eigenvector
* (X_t) = embedded state vector
The weight update is:
[
w_{new}
=======
w + \eta y (X - yw)
]
where:
* (\eta) = learning rate
* (y) = projection score
Weights are normalized after each update:
[
w \leftarrow \frac{w}{|w|}
]
This converges toward the dominant eigenvector of the covariance matrix.
---
## 3. Trend Reconstruction
The current sample's trend estimate is reconstructed as:
[
Trend_Z = y \cdot w_0
]
where:
* (w_0) is the weight associated with the current bar
The result is transformed back into price space:
[
Trend
=====
Mean + Trend_Z \times StdDev
]
This yields a smoothed estimate of the dominant low-frequency component.
---
## 4. Residual (Detrended Signal)
The residual is:
[
Residual = Price - Trend
]
The residual contains:
* Cyclic activity
* Noise
* Short-term oscillations
* Mean-reverting behavior
Many traders use this series similarly to a detrended oscillator.
---
## 5. Envelope
A volatility envelope is built around the trend:
[
Upper = Trend + \sigma
]
[
Lower = Trend - \sigma
]
where:
[
\sigma = StdDev(Residual)
]
computed over the user-selected envelope length.
This creates adaptive trend bands.
---
## 6. Dominant Cycle Detection
The script estimates the strongest cycle by searching for the lag with maximum autocorrelation.
For each lag:
[
Corr(lag)
=========
Correlation(x_t,x_{t-lag})
]
The lag producing the highest correlation is selected:
[
Cycle = \arg\max Corr(lag)
]
This provides an estimate of the dominant repeating structure in the market.
---
# Inputs
### Embedding Length
**Default:** 20
Controls the dimension of the lagged state vector.
Smaller values:
* Faster adaptation
* More noise
Larger values:
* Smoother trend
* Slower response
Typical range:
| Market Style | Suggested L |
| ------------ | ----------- |
| Intraday | 10–20 |
| Swing | 20–40 |
| Position | 30–50 |
---
### Learning Rate
**Default:** 0.001
Controls adaptation speed of the PCA component.
Smaller values:
* More stable
* Slower convergence
Larger values:
* Faster adaptation
* More instability
Recommended:
[
0.0001 \le \eta \le 0.005
]
---
### Envelope Length
**Default:** 50
Controls volatility estimation for trend bands.
Smaller:
* More reactive
Larger:
* More stable
---
# Outputs
## SSA Trend
Orange line.
Represents the dominant learned component.
Can be interpreted as:
* Adaptive trend
* Low-frequency structure
* Market baseline
---
## Upper Envelope
[
Trend + 1\sigma
]
Potential overextension zone.
---
## Lower Envelope
[
Trend - 1\sigma
]
Potential underextension zone.
---
## Detrended Signal
[
Price - Trend
]
Useful for:
* Mean reversion
* Oscillator analysis
* Cycle analysis
---
## Dominant Cycle Length
Displayed as a separate series.
Represents the lag with the highest autocorrelation.
Can be interpreted as the market's currently strongest repeating cycle.
---
# Relationship to Classical SSA
This indicator is **not a full SSA implementation**.
Classical SSA:
1. Builds trajectory matrix
2. Computes covariance matrix
3. Performs eigendecomposition
4. Reconstructs selected components via diagonal averaging
This indicator:
1. Builds lagged vectors
2. Learns dominant eigenvector online
3. Reconstructs dominant component approximately
Advantages:
* Real-time
* Computationally lightweight
* Pine compatible
Disadvantages:
* Only learns one dominant component
* No full eigenspectrum
* Approximate reconstruction
---
# Practical Interpretation
### Rising Trend
When the SSA Trend slopes upward:
* Dominant market structure is bullish.
### Falling Trend
When the SSA Trend slopes downward:
* Dominant market structure is bearish.
### Large Positive Residual
Price significantly above trend:
[
Price \gg Trend
]
Possible:
* Momentum burst
* Overextension
### Large Negative Residual
Price significantly below trend:
[
Price \ll Trend
]
Possible:
* Panic move
* Undershoot
### Cycle Compression
If estimated cycle length contracts:
* Faster market rhythm
* Higher activity
### Cycle Expansion
If estimated cycle length expands:
* Slower market rhythm
* Trend-dominated regime
---
# Best Use Cases
This indicator is most useful for:
* Trend extraction
* Regime detection
* Adaptive smoothing
* Cycle-aware analysis
* Mean reversion around trend
It is less suitable as a standalone entry signal and is generally strongest when combined with volatility, cycle, or momentum analysis.
## Overview
This indicator is an **online approximation to Singular Spectrum Analysis (SSA)** using **Oja's Principal Component Analysis (PCA) learning rule**. It attempts to separate the dominant low-frequency structure (trend) from shorter-term fluctuations by continuously learning the strongest component in a lagged embedding of price.
Unlike classical SSA, which requires constructing a trajectory matrix and performing eigendecomposition, this implementation updates the dominant component incrementally on each bar and is therefore suitable for Pine Script's real-time execution model.
The indicator produces:
* A dominant trend estimate
* ±1 standard deviation envelope around the trend
* A detrended residual series
* An estimate of the strongest cycle length
---
# Theory
## 1. Delay Embedding
For each bar, a lagged vector is constructed:
[
X_t =
[x_t,;x_{t-1},;x_{t-2},;\ldots,;x_{t-L+1}]
]
where:
* (x_t) = normalized price
* (L) = embedding length
This is equivalent to the trajectory matrix concept used in SSA.
The embedding transforms a one-dimensional time series into a higher-dimensional state space.
---
## 2. Online Principal Component Analysis
The indicator learns the first principal component using Oja's learning rule.
The projection of the embedded vector onto the dominant component is:
[
y_t = w^T X_t
]
where:
* (w) = learned eigenvector
* (X_t) = embedded state vector
The weight update is:
[
w_{new}
=======
w + \eta y (X - yw)
]
where:
* (\eta) = learning rate
* (y) = projection score
Weights are normalized after each update:
[
w \leftarrow \frac{w}{|w|}
]
This converges toward the dominant eigenvector of the covariance matrix.
---
## 3. Trend Reconstruction
The current sample's trend estimate is reconstructed as:
[
Trend_Z = y \cdot w_0
]
where:
* (w_0) is the weight associated with the current bar
The result is transformed back into price space:
[
Trend
=====
Mean + Trend_Z \times StdDev
]
This yields a smoothed estimate of the dominant low-frequency component.
---
## 4. Residual (Detrended Signal)
The residual is:
[
Residual = Price - Trend
]
The residual contains:
* Cyclic activity
* Noise
* Short-term oscillations
* Mean-reverting behavior
Many traders use this series similarly to a detrended oscillator.
---
## 5. Envelope
A volatility envelope is built around the trend:
[
Upper = Trend + \sigma
]
[
Lower = Trend - \sigma
]
where:
[
\sigma = StdDev(Residual)
]
computed over the user-selected envelope length.
This creates adaptive trend bands.
---
## 6. Dominant Cycle Detection
The script estimates the strongest cycle by searching for the lag with maximum autocorrelation.
For each lag:
[
Corr(lag)
=========
Correlation(x_t,x_{t-lag})
]
The lag producing the highest correlation is selected:
[
Cycle = \arg\max Corr(lag)
]
This provides an estimate of the dominant repeating structure in the market.
---
# Inputs
### Embedding Length
**Default:** 20
Controls the dimension of the lagged state vector.
Smaller values:
* Faster adaptation
* More noise
Larger values:
* Smoother trend
* Slower response
Typical range:
| Market Style | Suggested L |
| ------------ | ----------- |
| Intraday | 10–20 |
| Swing | 20–40 |
| Position | 30–50 |
---
### Learning Rate
**Default:** 0.001
Controls adaptation speed of the PCA component.
Smaller values:
* More stable
* Slower convergence
Larger values:
* Faster adaptation
* More instability
Recommended:
[
0.0001 \le \eta \le 0.005
]
---
### Envelope Length
**Default:** 50
Controls volatility estimation for trend bands.
Smaller:
* More reactive
Larger:
* More stable
---
# Outputs
## SSA Trend
Orange line.
Represents the dominant learned component.
Can be interpreted as:
* Adaptive trend
* Low-frequency structure
* Market baseline
---
## Upper Envelope
[
Trend + 1\sigma
]
Potential overextension zone.
---
## Lower Envelope
[
Trend - 1\sigma
]
Potential underextension zone.
---
## Detrended Signal
[
Price - Trend
]
Useful for:
* Mean reversion
* Oscillator analysis
* Cycle analysis
---
## Dominant Cycle Length
Displayed as a separate series.
Represents the lag with the highest autocorrelation.
Can be interpreted as the market's currently strongest repeating cycle.
---
# Relationship to Classical SSA
This indicator is **not a full SSA implementation**.
Classical SSA:
1. Builds trajectory matrix
2. Computes covariance matrix
3. Performs eigendecomposition
4. Reconstructs selected components via diagonal averaging
This indicator:
1. Builds lagged vectors
2. Learns dominant eigenvector online
3. Reconstructs dominant component approximately
Advantages:
* Real-time
* Computationally lightweight
* Pine compatible
Disadvantages:
* Only learns one dominant component
* No full eigenspectrum
* Approximate reconstruction
---
# Practical Interpretation
### Rising Trend
When the SSA Trend slopes upward:
* Dominant market structure is bullish.
### Falling Trend
When the SSA Trend slopes downward:
* Dominant market structure is bearish.
### Large Positive Residual
Price significantly above trend:
[
Price \gg Trend
]
Possible:
* Momentum burst
* Overextension
### Large Negative Residual
Price significantly below trend:
[
Price \ll Trend
]
Possible:
* Panic move
* Undershoot
### Cycle Compression
If estimated cycle length contracts:
* Faster market rhythm
* Higher activity
### Cycle Expansion
If estimated cycle length expands:
* Slower market rhythm
* Trend-dominated regime
---
# Best Use Cases
This indicator is most useful for:
* Trend extraction
* Regime detection
* Adaptive smoothing
* Cycle-aware analysis
* Mean reversion around trend
It is less suitable as a standalone entry signal and is generally strongest when combined with volatility, cycle, or momentum analysis.
Script open-source
Nello spirito di TradingView, l'autore di questo script lo ha reso open source, in modo che i trader possano esaminarne e verificarne la funzionalità. Complimenti all'autore! Sebbene sia possibile utilizzarlo gratuitamente, ricordiamo che la ripubblicazione del codice è soggetta al nostro Regolamento.
Declinazione di responsabilità
Le informazioni e le pubblicazioni non sono intese come, e non costituiscono, consulenza o raccomandazioni finanziarie, di investimento, di trading o di altro tipo fornite o approvate da TradingView. Per ulteriori informazioni, consultare i Termini di utilizzo.
Script open-source
Nello spirito di TradingView, l'autore di questo script lo ha reso open source, in modo che i trader possano esaminarne e verificarne la funzionalità. Complimenti all'autore! Sebbene sia possibile utilizzarlo gratuitamente, ricordiamo che la ripubblicazione del codice è soggetta al nostro Regolamento.
Declinazione di responsabilità
Le informazioni e le pubblicazioni non sono intese come, e non costituiscono, consulenza o raccomandazioni finanziarie, di investimento, di trading o di altro tipo fornite o approvate da TradingView. Per ulteriori informazioni, consultare i Termini di utilizzo.