Time Series Analysis : Stationary, Trend, Seasonal, Decomposition, Differences, Windowing, Running Values, Exponential Smoothing, Fill Methods, Resampling, Jackknife Estimation, Bootstrapping, Fourier Transform , Filtering, Correlation, Auto-Correlation, Partial Auto-Correlation, Random Walks, Dickey-Fuller Test, ARIMA Models

1. notes.md

Time Series

• A set of values measured sequentially in time
• Values are typically (but not always) measured at equal interval x₁, x₂, x₃, etc..
• Values can be:
  • Continuous
  • discrete or symbolic (words)
• Associated with empirical observation of time varying phenomena:
  • Stock market price (day, hour, minute, tick etc...)
  • Temperature (day, minute, second, etc..)
  • Number of paitents (week, month, etc..)
  • GDP (quarter, year, etc..)
• Forecasting requires predicting future values based on past behaviour

Mathematical Conventions

• The value of the time series at time t is given by X_t
• The values at a given lag l are given by X_t-l
• The mean of the overall signal is $\mu $ and the corresponding running value is $\mu_{t}^{w}$
• The variance of the overall signal is $\sigma $ and the corresponding running value is $\sigma_{t}^{w}$
• Running values are calculated over window of width w

Time Series Analysis

• The Fundamental assumption is that previous values determine the current value:
  x_t+1 = f(x_t,x_t-1,...)

Three Fundamental Behaviors

2. stationarity.md

Stationarity

• A time series is said to be stationary if its basic statistical properties are independent of time
• In particular:
  • Mean Average value stays constant
  • Variance : The width of the curve is bounded
  • Covariance: Correlation between points is indepedent of time
• Stationary processes are easier to analyze
• Many time series analysis algorithms assume the time series to be stationary
• Several rigorous tests for stationarity have been developed such as the (Augmented) Dickey-Fuller and Hurst Exponent
• Typically, the first step of any analysis is to transform the series to make it stationary

3. Trend.md

Trends

• Many time series have a clear trend or tendency:
  • Stock market indices tend to go up over time
  • Number of cases of preventable diseases tends to go down over time
• Trends can be additive or multiplicative:

• Trends can be removed by subtraction or division of the correct values
• One simple way to determine the trend is to calculate a running average over the series

def running_average(x, order):
  current = x[:order].sum()
  running = []
  
  for i in range(order, x.shape[0]):
    current += x[i]
    current -= x[i-order]
    running.append(current/order)
  
  return np.array(running)

4. Seasonality.md

Seasonality

• Many of the phenomena we might be interested in varying in time in a cyclical or seasonal fashion

  • Ice-cream sales peak in the summer and drop in the winter
  • Number of cell phone calls made is larger during the day than at night
  • Many types of crime are more frequent at night than during the day
  • Visits to museums are more frequent in the weekend than in weekdays
  • The stock market grows during bull periods and shrinks during bear periods

• Understanding the seasonality of a time series provides important information about its long term behavior and is extremely useful in predicting future values

• If the period is fixed it’s called seasonality, while if the period is irregular it's called cyclical