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📊 Advanced Time Series Topics Tutorial

1. 🔍 Anomaly Detection in Time Series

✅ Definition:

Detect points or periods where the behavior of the series deviates significantly from normal patterns.

📐 Methods:

  • Statistical Thresholding:

  • \(z = \frac{x - \mu}{\sigma}\)

  • Rolling Statistics:

  • Use rolling mean and standard deviation.

  • Model-based:

  • Train a forecast model → compare actual vs predicted residuals.

  • Isolation Forest / One-Class SVM / Autoencoders

✅ Code Example (Statistical):

import numpy as np
threshold = 3  # Z-score
z_scores = (df['value'] - df['value'].mean()) / df['value'].std()
anomalies = df[np.abs(z_scores) > threshold]

✅ Code Example (Autoencoder in PyTorch):

Let me know if you want the complete deep learning version.

2. 👯 Time Series Clustering

✅ Definition:

Grouping similar time series based on their shape, pattern, or frequency domain characteristics.

📐 Distance Metrics:

  • Euclidean
  • Dynamic Time Warping (DTW): aligns sequences by warping time.
\[ DTW(x, y) = \min \sum \text{dist}(x_i, y_j) \text{ with time alignment} \]

✅ Code Example (with tslearn):

from tslearn.clustering import TimeSeriesKMeans
model = TimeSeriesKMeans(n_clusters=3, metric="dtw")
labels = model.fit_predict(series_array)

3. 🔗 Multivariate Time Series

✅ Definition:

Multiple time-dependent variables recorded over the same time period.

📘 Vector Autoregression (VAR):

\[ Y_t = A_1 Y_{t-1} + A_2 Y_{t-2} + \dots + A_p Y_{t-p} + e_t \]

✅ Code Example:

from statsmodels.tsa.api import VAR

model = VAR(df[['temp', 'humidity']])
results = model.fit(maxlags=5)
forecast = results.forecast(df.values[-5:], steps=10)

4. 📎 Exogenous Variables

✅ Definition:

External variables that influence the target time series.

🔢 SARIMAX (ARIMA with exogenous):

\[ Y_t = ARIMA(p,d,q) + \beta X_t + \epsilon_t \]

✅ Code Example:

from statsmodels.tsa.statespace.sarimax import SARIMAX

model = SARIMAX(df['sales'], exog=df[['promo']], order=(1,1,1), seasonal_order=(1,1,1,12))
results = model.fit()

🔢 VAR with Exogenous Variables:

model = VAR(df[['temp', 'humidity']])
results = model.fit()

5. ❓ Forecasting with Missing Data

✅ Strategies:

  • Imputation (mean, median, linear, interpolation)
  • Model-based Imputation (Kalman, EM)
  • Multivariate Filling

✅ Code Example:

df['value'] = df['value'].interpolate(method='linear')

For more sophisticated:

from fancyimpute import KNN
filled = KNN(k=3).fit_transform(df.values)

6. 📊 Probabilistic Forecasting

✅ Definition:

Forecasts that give uncertainty estimates (intervals or distributions), not just point estimates.

📘 Techniques:

  • Quantile Regression: predict multiple quantiles
  • Bayesian models: e.g., Pyro, TensorFlow Probability
  • Prophet: gives prediction intervals
  • DeepAR (Amazon): RNN-based probabilistic model

✅ Quantile Loss:

\[ L_q(y, \hat{y}) = \max(q(y - \hat{y}), (q-1)(y - \hat{y})) \]

✅ Example using Prophet:

from prophet import Prophet
model = Prophet()
model.fit(df.rename(columns={"value": "y", "date": "ds"}))
forecast = model.predict(model.make_future_dataframe(periods=30))