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.
Mã nguồn mở
Theo đúng tinh thần TradingView, tác giả của tập lệnh này đã công bố nó dưới dạng mã nguồn mở, để các nhà giao dịch có thể xem xét và xác minh chức năng. Chúc mừng tác giả! Mặc dù bạn có thể sử dụng miễn phí, hãy nhớ rằng việc công bố lại mã phải tuân theo Nội quy.
Thông báo miễn trừ trách nhiệm
Thông tin và các ấn phẩm này không nhằm mục đích, và không cấu thành, lời khuyên hoặc khuyến nghị về tài chính, đầu tư, giao dịch hay các loại khác do TradingView cung cấp hoặc xác nhận. Đọc thêm tại Điều khoản Sử dụng.
Mã nguồn mở
Theo đúng tinh thần TradingView, tác giả của tập lệnh này đã công bố nó dưới dạng mã nguồn mở, để các nhà giao dịch có thể xem xét và xác minh chức năng. Chúc mừng tác giả! Mặc dù bạn có thể sử dụng miễn phí, hãy nhớ rằng việc công bố lại mã phải tuân theo Nội quy.
Thông báo miễn trừ trách nhiệm
Thông tin và các ấn phẩm này không nhằm mục đích, và không cấu thành, lời khuyên hoặc khuyến nghị về tài chính, đầu tư, giao dịch hay các loại khác do TradingView cung cấp hoặc xác nhận. Đọc thêm tại Điều khoản Sử dụng.