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Time Series Forecasting Project — JJ EPS & Amazon Stock

This project explores classical and deep learning approaches to time series forecasting using two real-world datasets.

• Johnson & Johnson (JJ) quarterly earnings per share (EPS)
• Amazon (AMZN) daily closing stock prices

Both case studies follow a similar pipeline

• Exploratory time series visualization
• Seasonal decomposition
• Stationarity testing (ADF) and differencing
• ACF/PACF analysis
• ARIMA model selection (grid search + auto-ARIMA)
• Final ARIMA model fit and forecasting
• LSTM-based forecasting as a deep learning alternative
• FFT-based frequency-domain analysis

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📂 Project Structure

You might have the code organized roughly as:

• part1_jj.ipynb — JJ EPS analysis & forecasting
• part2_amzn.ipynb — Amazon stock analysis & forecasting
• jj.csv — Johnson & Johnson EPS data
• AMZN.csv — Amazon daily stock data
• decomposejj.png — JJ seasonal decomposition plot
• sales_pred.png — JJ ARIMA forecast with CI
• lstmjj.png — JJ LSTM 24-step forecast
• lstmamz.png — AMZN LSTM forecast (≈2 years)

(Names can be adjusted to match your actual files.)

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🔧 Dependencies

Install required Python packages:

pip install numpy pandas matplotlib statsmodels scikit-learn pmdarima tqdm tensorflow

Main libraries used:

• Core: numpy, pandas
• Plots: matplotlib
• Time Series: statsmodels (ARIMA, SARIMAX, ADF, seasonal_decompose)
• Model selection: pmdarima (auto_arima)
• Metrics & scaling: scikit-learn (MinMaxScaler, mean_squared_error)
• Deep learning: tensorflow.keras (LSTM, Dense, Sequential)
• Progress bars: tqdm
• Frequency analysis: numpy.fft (fft, ifft)

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📊 Dataset 1 — Johnson & Johnson EPS (Quarterly)

Data

• File: jj.csv

• Columns:

  • date — Quarterly dates
  • data — EPS values (Johnson & Johnson)

The series spans 84 quarterly observations.

Workflow

1️⃣ Visualization

• Load data, set date as DateTime index
• Plot EPS over time to observe trend and clear seasonality

2️⃣ Seasonal Decomposition

• Multiplicative decomposition with quarterly period:

  seasonal_decompose(data['data'], model='multiplicative', period=4)

• Plots:

  • Observed
  • Trend
  • Seasonal
  • Residuals

• Figure saved as decomposejj.png

3️⃣ Stationarity & Transformations

• ADF test on original series → non-stationary (p ≈ 1.0)

• Apply log transform & first difference:

  data['data_log'] = np.log(data['data'])
  data['data_tr_1'] = data['data_log'].diff()

• ADF on transformed series → stationary (p < 0.001), supporting d = 1 for ARIMA.

4️⃣ ACF & PACF

• Plots computed on the differenced log series to guide AR (p) and MA (q) orders.

5️⃣ ARIMA Model Selection (Grid Search)

• Search over p, q ∈ {0,…,7} with d = 1

  order_list = [(p,1,q) for p in range(8) for q in range(8)]

• For each order, fit ARIMA and record AIC.

• Best model (lowest AIC):

  ARIMA(6,1,3) with AIC ≈ 115.50
  

6️⃣ Auto-ARIMA Check

• pmdarima.auto_arima is run on a train subset and also supports (6,1,3) as optimal under constraints, reinforcing the grid search result.

7️⃣ Final ARIMA(6,1,3) Model

• Fitted on full data['data']:

  best_model = ARIMA(data['data'], order=(6,1,3))
  best_model_fit = best_model.fit()

• Residual diagnostics:

  • Ljung–Box test
  • Jarque–Bera test
  • Residual plots

These indicate no major autocorrelation left and decent residual behavior (though not perfectly normal).

8️⃣ In-Sample Fit & Metrics

• Predictions over the entire historical range.

• Metrics used:

  • MAPE
  • MAE
  • MPE
  • RMSE
  • Correlation
  • MinMax error

Representative results:

• RMSE ≈ 0.401
• MAPE ≈ 8.95%
• Corr ≈ 0.996

9️⃣ Forecasting Future Quarters

• get_forecast(steps=26) for the next 26 quarters:

  • Predicted mean
  • 95% confidence intervals

• Forecast index built via quarterly DateTime, and plots include:

  • Historical EPS
  • Future predictions
  • Confidence interval band

• Figure saved as sales_pred.png

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🤖 LSTM on JJ EPS

Preprocessing

• Raw values reshaped to (n,1)
• MinMaxScaler used for normalization
• Sliding window: window_size = 24, forecast_steps = 24 (24-step-ahead multi-output forecast).

Model

• Stacked LSTM with Dense layers:

  model = Sequential([
      LSTM(64, return_sequences=True, input_shape=(window_size, 1)),
      LSTM(32),
      Dense(64, activation='relu'),
      Dense(32, activation='relu'),
      Dense(forecast_steps)
  ])

• Trained for 50 epochs with Adam optimizer.

Performance & Plots

• Rolling 24-step forecast RMSE ≈ 1.636

• Plots:

  • LSTM forecast vs actual for evaluation
  • Extended 24-month forecast appended to the original series

• Figure saved as lstmjj.png

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🔉 FFT Analysis for JJ EPS

• FFT applied to the EPS series to inspect dominant frequencies:

  X_jj = fft(jj_array)

• Plots:

  • FFT amplitude spectrum (stem plot)
  • IFFT reconstruction to confirm original signal recovery

This helps confirm the presence of strong seasonal components.

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📊 Dataset 2 — Amazon (AMZN) Daily Close Prices

Data

• File: AMZN.csv

• Columns:

  • Date — Trading date
  • Close — Daily closing stock price

The series contains ~1,258 daily data points.

Workflow

1️⃣ Visualization

• Load AMZN data, keep Date and Close.
• Plot the closing prices to inspect trend, volatility, and structural breaks.

2️⃣ Seasonal Decomposition

• Multiplicative decomposition with period=12:

  result_amazon = seasonal_decompose(data['Close'], model='multiplicative', period=12)

• Same 4 components plotted: observed, trend, seasonal, residuals.

3️⃣ Stationarity & Differencing

• ADF test on Close:

  • Statistic ≈ -1.66
  • p-value ≈ 0.45 → non-stationary

• First difference:

  data['Close_diff'] = data['Close'].diff()

• ADF on Close_diff:

  • Statistic ≈ -36.25
  • p-value = 0.0 → stationary

Thus d = 1 is appropriate.

4️⃣ ACF & PACF

• ACF and PACF computed on Close_diff to explore AR and MA orders.

5️⃣ ARIMA Model Selection (Grid Search)

• Similar optimize_ARIMA function used as in the JJ example.

• Candidate orders: p, q ∈ {0,…,7} with d=1.

• Best model by AIC:

  ARIMA(2,1,2) with AIC ≈ 6118.42
  

6️⃣ Auto-ARIMA Baseline

• auto_arima on a train subset selects a simpler ARIMA(0,1,0) model, but the full grid search on complete data supports ARIMA(2,1,2) as superior.

7️⃣ Final ARIMA(2,1,2) Model

• Fit:

  best_model = ARIMA(data['Close'], order=(2,1,2))
  best_model_fit = best_model.fit()

• Summary includes AR/MA coefficients and diagnostics (Ljung–Box, JB, heteroskedasticity).

8️⃣ In-Sample Fit & Metrics

• Predictions over the full historical index.

• Metrics like MAPE, MAE, RMSE, correlation are computed:

  • RMSE ≈ 0.3123
  • MAPE ≈ 0.21%
  • Corr ≈ 0.99996

The in-sample fit is extremely tight.

9️⃣ Long-Horizon ARIMA Forecast (504 Steps)

• get_forecast(steps=504) (~2 years of trading days).

• Combine predicted mean with lower/upper confidence intervals.

• Plot:

  • Historical AMZN close prices
  • Forecasted path
  • Shaded region for 95% CI

(You can specify a filename in plt.savefig() to store this plot.)

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🤖 LSTM on AMZN

Preprocessing

• Use only Close values.
• Normalize with MinMaxScaler.
• Build sequences with time_steps = 60 (use 60 past days to predict next day).

X, y = create_sequences(scaled_data, time_steps)
train_size = int(len(X) * 0.9)
X_train, y_train = X[:train_size], y[:train_size]

Model

• Stacked LSTM:

  model = Sequential([
      LSTM(100, return_sequences=True, activation='relu', input_shape=(time_steps, 1)),
      LSTM(50, activation='relu'),
      Dense(1)
  ])

• Trained for 50 epochs with batch size 32.

• Training loss steadily decreases to ~1e-3 range.

Recursive Forecasting (≈2 Years)

• Start from the last 60 normalized observations.
• Predict 1-step ahead, append, slide window, repeat.
• future_steps = 504 (≈ 2 years of business days).
• Inverse transform predictions back to price scale.
• Build forecast index with business-day frequency and combine with historical data.

Plot:

• Original AMZN Close
• LSTM forecast (dashed, dark orange)

Saved as: lstmamz.png

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🔉 FFT Analysis for AMZN

• FFT applied to daily closing prices:

  X = fft(close_array)

• Plots:

  • Amplitude spectrum (stem plot)
  • IFFT reconstruction to confirm signal consistency

This gives a frequency-domain perspective on the stock’s price behavior.

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⚖️ ARIMA vs LSTM — High-Level Comparison

• ARIMA

  • Pros: Interpretable coefficients, confidence intervals, well-suited to linear structures and low-frequency series (like JJ EPS).
  • Cons: Limited in capturing complex nonlinear patterns; assumptions may break in highly volatile financial data.

• LSTM

  • Pros: Flexible for nonlinear dynamics and long-range dependencies; can handle complex patterns in stock prices.
  • Cons: Less interpretable, needs more data & tuning, no built-in statistical confidence intervals.

In both JJ and AMZN cases:

• ARIMA produces strong in-sample fits with low RMSE and high correlation.
• LSTM offers an alternative approach, especially for richer horizons and potentially capturing nonlinear behavior, at the cost of interpretability.

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🚀 How to Run

1. Ensure jj.csv and AMZN.csv are in your working directory.

2. Open the notebooks/scripts for each part (e.g. assignment_1.ipynb and assignment_1_part2.ipynb).

3. Run all cells in order:

   • Data loading & visualization
   • Decomposition & stationarity tests
   • ARIMA model selection & forecast
   • LSTM training & forecast
   • FFT analysis

4. Check the generated plots:

   • decomposejj.png, sales_pred.png, lstmjj.png
   • lstmamz.png (and any ARIMA forecast plot you save for AMZN)

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