### Technical Guide for Forex Time Series Analysis Using AI/ML Models

#### Objective
This guide provides a comprehensive overview of the methodologies and machine learning models used in analyzing forex time series data, focusing on EUR/USD and other major and minor pairs. The goal is to understand the underlying technical principles, implement feature engineering, perform correlation analysis, identify trends, train AI/ML models, and evaluate their performance using RMSE.

### Key Components

1. **Data Preparation**
2. **Feature Engineering**
3. **Correlation Analysis**
4. **Trend Identification**
5. **Model Training**
6. **Model Evaluation**

### 1. Data Preparation

#### Context
Forex data is high-frequency time series data that requires careful preprocessing to handle missing values, outliers, and ensure consistency. TimescaleDB is used for efficient storage and retrieval due to its scalability and time-series optimizations.

**Technical Details:**
- **Data Sourcing**: Forex data is typically retrieved from APIs such as OANDA, which provide real-time and historical data.
- **Preprocessing**: This includes filling missing values using forward fill or interpolation methods, handling outliers through techniques like z-score normalization, and converting timestamps to a uniform format.

### 2. Feature Engineering

#### Context
Feature engineering transforms raw data into meaningful features that enhance the model's predictive capabilities. This process is critical for time series analysis as it captures temporal dependencies and seasonality.

**Technical Details:**
- **Lag Features**: Introducing past values (lags) as predictors helps capture temporal dependencies.
    - **Mathematical Formulation**: \( \text{Lag}(k) = X_{t-k} \)
- **Rolling Statistics**: Calculating rolling mean, variance, and standard deviation captures local trends and volatility.
    - **Mathematical Formulation**: \( \text{Rolling Mean}(w) = \frac{1}{w} \sum_{i=t-w+1}^{t} X_i \)
- **Scaling**: Normalization or standardization ensures that features are on a similar scale, which is essential for models like LSTM and Transformers.

### 3. Correlation Analysis

#### Context
Correlation analysis identifies relationships between different forex pairs, which can inform trading strategies and portfolio management. 

**Technical Details:**
- **Pearson Correlation**: Measures linear correlation between pairs.
    - **Formula**: \( \rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \)
    - **Properties**: Symmetric, bounded between -1 and 1.
- **Visualization**: Heatmaps are used to visualize the correlation matrix, highlighting highly correlated pairs.

### 4. Trend Identification

#### Context
Identifying trends helps in understanding the market direction and making informed trading decisions. Techniques like moving averages smooth out short-term fluctuations and highlight longer-term trends.

**Technical Details:**
- **Moving Averages**: Simple and exponential moving averages (SMA, EMA) are used.
    - **SMA Formula**: \( \text{SMA}(n) = \frac{1}{n} \sum_{i=0}^{n-1} X_{t-i} \)
    - **EMA Formula**: \( \text{EMA}(t) = \alpha \cdot X_t + (1-\alpha) \cdot \text{EMA}(t-1) \)
- **Trend Lines**: Connecting significant highs or lows in price data to form resistance and support lines.

### 5. Model Training

#### Context
Different machine learning models have different strengths in time series forecasting. This project uses ARIMA, LSTM, and Transformer models.

**Technical Details:**

**ARIMA (AutoRegressive Integrated Moving Average):**
- **Components**: AR (p) - AutoRegression, I (d) - Integration, MA (q) - Moving Average.
    - **AR**: \( X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \dots + \phi_p X_{t-p} + \epsilon_t \)
    - **I**: \( Y_t = X_t - X_{t-1} \) (d times differencing)
    - **MA**: \( X_t = \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2} + \dots + \theta_q \epsilon_{t-q} \)
- **Use Case**: Effective for univariate time series with trends and seasonality.

**LSTM (Long Short-Term Memory):**
- **Architecture**: Special type of RNN capable of learning long-term dependencies.
    - **Gates**: Input, forget, and output gates control the cell state.
    - **Equations**:
        - Forget Gate: \( f_t = \sigma(W_f \cdot [h_{t-1}, X_t] + b_f) \)
        - Input Gate: \( i_t = \sigma(W_i \cdot [h_{t-1}, X_t] + b_i) \)
        - Output Gate: \( o_t = \sigma(W_o \cdot [h_{t-1}, X_t] + b_o) \)
        - Cell State: \( C_t = f_t * C_{t-1} + i_t * \tilde{C_t} \)
- **Use Case**: Suitable for capturing long-term dependencies in time series data.

**Transformers:**
- **Architecture**: Self-attention mechanism allows the model to weigh the importance of different parts of the input sequence.
    - **Attention Mechanism**: \( \text{Attention}(Q, K, V) = \text{softmax}\left( \frac{QK^T}{\sqrt{d_k}} \right) V \)
    - **Components**: Multi-head attention, feed-forward networks, and positional encodings.
- **Use Case**: Powerful for sequence modeling tasks, especially when capturing global dependencies.

### 6. Model Evaluation

#### Context
Model evaluation is crucial to assess the accuracy and reliability of predictions. RMSE (Root Mean Squared Error) is a standard metric for this purpose.

**Technical Details:**
- **RMSE**: Measures the average magnitude of the error.
    - **Formula**: \( \text{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=1}^n (Y_i - \hat{Y_i})^2 } \)
    - **Interpretation**: Lower RMSE indicates better model performance.

### Workflow Summary (Pseudocode)

#### Data Preparation
1. Ingest data from OANDA.
2. Preprocess data: handle missing values, outliers.
3. Store preprocessed data in TimescaleDB.

#### Feature Engineering
1. Create lag features and rolling statistics.
2. Store engineered features in TimescaleDB.

#### Correlation Analysis and Storage
1. Calculate correlation matrix.
2. Store correlation results in TimescaleDB.

#### Trend Identification and Storage
1. Calculate moving averages and trend indicators.
2. Store trend data in TimescaleDB.

#### Model Training (ARIMA, LSTM, Transformers)
1. Retrieve feature-engineered data from TimescaleDB.
2. Train ARIMA, LSTM, and Transformer models.
3. Store trained models and scalers.

#### Model Evaluation and Storage
1. Evaluate models using RMSE.
2. Store evaluation results in TimescaleDB.

### Conclusion
This guide provides a detailed, technical overview of the methodologies used in forex time series analysis, leveraging advanced AI/ML models like ARIMA, LSTM, and Transformers. Each step is designed to ensure robustness, scalability, and accuracy in forecasting and trend identification, making it suitable for high-frequency trading environments and financial analytics.