Time Series Analysis

1️⃣ Decomposition (Trend, Seasonality, Noise)

📌 Why?

• To separate the time series into trend, seasonality, and random noise components.

• Helps in understanding patterns before applying advanced models.

  
  

2️⃣ ARIMA (AutoRegressive Integrated Moving Average)

📌 Why?

• ARIMA is one of the best models for forecasting univariate time series data.

• Removes trends and makes data stationary before forecasting.

• ARIMA assumes that past values influence future values and removes trends by differencing.

📌 When to Use?

• When you have only one time series (e.g., predicting AAPL stock price).

• When there is a clear trend in the data but no strong seasonality.

3️⃣ VAR (Vector Autoregression)

📌 Why?

• ARIMA works well for one time series, but if we have multiple interrelated time series (e.g., AAPL, MSFT, GOOGL), we use VAR.

• VAR captures the relationship between multiple variables that influence each other.

• Here, we are using AAPL, MSFT, and GOOGL together in a multivariate system.

📌 When to Use?

• When dealing with multiple interdependent time series.

• Example: Predicting stock market movements based on other stocks.

4️⃣ GARCH (Generalized Autoregressive Conditional Heteroskedasticity)

📌 Why?

• Financial time series (like stock prices) show volatility clustering (large price changes followed by more large changes).

• ARIMA assumes constant variance, but stock markets don’t behave like that.

• GARCH models volatility (risk) and improves forecasting accuracy.

• This helps estimate volatility and can be used to model risk in finance.

📌 When to Use?

• When data shows high volatility and clustering.

• Example: Modeling stock market crashes or currency exchange rate fluctuations.

5️⃣ Stationarity Tests (ADF, KPSS)

📌 Why?

• Most time series models require stationary data.

• If a series has a unit root (non-stationary), then standard forecasting models won’t work properly.

• The ADF test checks if a time series is stationary.

• If the p-value > 0.05, the data is non-stationary, and we need differencing.

📌 When to Use?

• Before applying ARIMA, VAR, or GARCH, we need to check stationarity.

🚀 Summary: Why Use These Models?

Model	Purpose	When to Use?
Decomposition	Breaks data into trend, seasonality, and noise	When you want to understand the pattern of the time series
ARIMA	Forecasts a single time series	When predicting one variable like AAPL stock price
VAR	Forecasts multiple dependent time series	When multiple variables influence each other (AAPL, MSFT, GOOGL)
GARCH	Models volatility	When data has volatility clustering (e.g., stock prices)
ADF, KPSS	Checks if the data is stationary	Before applying ARIMA, VAR, or GARCH

Would you like to add deep learning models like LSTM for better predictions? 🚀

4o

You said:

What is smoothing explain

ChatGPT said:

What is Smoothing in Time Series?

📌 Smoothing is a technique used in time series analysis to remove noise and highlight trends, cycles, or patterns in the data. It helps in making the data clearer for forecasting and analysis by reducing short-term fluctuations.

Why Use Smoothing?

✅ Removes Random Variations → Makes patterns clearer.

✅ Enhances Trend Detection → Helps in spotting long-term trends.

✅ Improves Forecasting Accuracy → Used in predictive models.

✅ Filters Out Noise → Helps to identify true patterns in financial, climate, and economic data.

Types of Smoothing Techniques

1️⃣ Moving Average (MA)

📌 Averages a specific number of past observations to smooth fluctuations.

🔹 Simple Moving Average (SMA) → Uses equal weights.

🔹 Weighted Moving Average (WMA) → Recent values get more weight.

🔹 Exponential Moving Average (EMA) → More emphasis on recent data using a smoothing factor.

2️⃣ Exponential Smoothing

📌 Assigns exponentially decreasing weights to past observations.It reacts faster to changes than SMA.

🔹 Single Exponential Smoothing → Best for stationary data (no trend/seasonality).

🔹 Double Exponential Smoothing (Holt’s Method) → Captures trend.

🔹 Triple Exponential Smoothing (Holt-Winters Method) → Captures trend and seasonality.

.

3️⃣ LOWESS (Locally Weighted Scatterplot Smoothing)

📌 A non-parametric method that fits local regressions to smooth data.🔹 Best for

When to Use Different Smoothing Methods?

Method	Best for	Handles Trends?	Handles Seasonality?
Moving Average	Short-term fluctuations	❌ No	❌ No
Exponential Smoothing	Stationary data	✔️ Yes (Holt’s)	✔️ Yes (Holt-Winters)
LOWESS	Nonlinear trends	✔️ Yes	❌ No

🚀 Summary

Smoothing is used to reduce noise, highlight trends, and improve forecasting. Different methods like Moving Averages, Exponential Smoothing, and LOWESS help depending on the nature of the time series.

Would you like to apply this to real stock market data? 📊🚀

4o

You said:

What is the meaning of autoregressive

ChatGPT said:

What is the Meaning of Autoregressive?

📌 Autoregressive (AR) models are a type of time series model where the current value of the series is based on its past values. This means the series has a dependence on its own past observations, hence the name "auto" (self) + "regressive" (predicting using past values).

🔹 Interpretation of ϕ\phiϕ (Phi)

• If ∣ϕ∣<1|\phi| < 1∣ϕ∣<1 → The series is stationary (values remain around a constant mean).

• If ϕ=1\phi = 1ϕ=1 → The series follows a random walk (unit root process, non-stationary).

• If ∣ϕ∣>1|\phi| > 1∣ϕ∣>1 → The series is explosive (values grow over time, unstable).

🔹 Why is it Called Autoregressive?

• "Auto" → The model uses its own past values.

• "Regressive" → The model performs regression on past values to predict future ones.

📊 Interpretation:

• The AR(1) process generates a time series where each value depends on the previous one.

• If ϕ\phiϕ is positive, the series moves smoothly.

• If ϕ\phiϕ is negative, the series oscillates.

🔹 Real-World Examples of Autoregressive Processes

1️⃣ Stock Prices 📈

• Stock prices often follow an AR(1) process where today's price is highly correlated with yesterday’s.

2️⃣ Inflation Rates 💰

• Inflation tends to follow an AR(p) model, meaning past inflation rates influence future inflation.

3️⃣ Interest Rates 📊

• Central banks’ interest rate policies often exhibit autoregressive behavior, where past rates affect future rates.

🔹 Summary

✔ Autoregressive (AR) models predict future values based on past values.✔ The key parameter ϕ\phiϕ determines if the series is stationary, non-stationary, or explosive.✔ AR models are widely used in finance, economics, and weather forecasting.

🔹 Steps We Will Follow

1️⃣ Load real stock price data (e.g., META, AAPL, GOOG).

2️⃣ Plot the time series to visualize trends.

3️⃣ Check stationarity using Augmented Dickey-Fuller (ADF) test.

4️⃣ Fit an AR model and interpret results.

🔹 Interpretation of Results

• If ADF test rejects the null hypothesis (p-value < 0.05) → The series is stationary, and we can apply an AR model directly.

• If ADF test does NOT reject (p-value > 0.05) → The series is non-stationary, and we take first differences.

• AR(1) Coefficient (Φ1) tells us:

  • Φ1 < 1 → The series is stationary (mean-reverting).

  • Φ1 = 1 → The series follows a random walk (unit root).

  • Φ1 > 1 → The series is explosive (unstable).