L2 Examples of Time Series Data

  • Why is Time Series important?
    • Understand underlying trend and patterns, which helps us to eventually guess how the data will move in the future
    • Predict the likelihood of future events (will stock price go up or down?)
    • Make inference about the probability models governing series
    • Confidence intervals and model parameters
  • TS Notation
    • \(X_{t},X_{t+h},\dots\) (\(h\) is the time between observations, \(t\) is the time point of the first observation)
    • \(\dfrac{1}{h}\) stands for the sampling frequency
    • Order is important \(\implies\) Observations are dependent
    • e.g.
      • \(X_{t}, X_{t+2},\dots\)
      • gap = h = 2
      • Sampling frequency = 1/2 (half per day)
  • Example 1: Carbon-dioxide levels in the atmosphere
    • Upward trend
    • Repetitions (Seasonality: due to some underlying seasonality)
  • Example 2: Oil spot price in dollar/barrel
    • L2_Examples of Time Series Data-1768798387533.webp
    • No overall trend
    • No apparent seasonality
  • Example 3: SGST (state GST) revenues of India
    • L2_Examples of Time Series Data-1768798436417.webp
    • There is a mild upward trend
  • Example 4: Delhi's air quality index between 2017-2020
    • L2_Examples of Time Series Data-1768798467804.webp
    • Mild trend in batches (no overall trend)
    • Seasonality is present
    • Time scale: Dataset is monthly

  • Example 5: Quarterly sugarcane prices in India

    • L2_Examples of Time Series Data-1768798508065.webp
    • Time scale: Dataset is quarterly (chosen based on purpose)

  • Choosing Time Scales

    • Stock data are available at ticker level
    • Consider two things
      • scale of required forecasts
      • level of noise in data
    • Forecast next-day sales:
      • Why spend time to collect minute-by-minute data? There will be a lot of noise
      • Daily can be used as that's what we need
    • Noise = Lot of random repetitions / fluctuations in the data
  • Long Vs Short Time Series
    • Long = Time series contains a lot of observations
      • Weekly interest rates over 5 years
      • Daily closing stock prices over 5 years
    • Short = Doesn't contain a lot of observations
      • Daily closing stock price over a month: 22-23
  • Example 6: Indian Population
  • Example 7: Intl. Airline Data (Monthly totals of international passengers)
    • L2_Examples of Time Series Data-1768799057531.webp
    • Non-stationary seasonal time series:
      • Upward trend
      • Seasonal (low in winter, high in summer)
      • Variability increases with time
  • Goals of TS analysis
    • Explanatory: Understanding the underlying stochastic mechanism that gives rise to new data points
    • Predictive: Use that understanding, and the history of the series to predict new values.