Intro to Time Series Forecasting

ARIMA

Notation

• T-n ： a prior or lag time
• T: current time and point of reference
• T+n: future or forecast time

components

• Level: baseline value
• trend: often linear increasing or decreasing over time
• Seasonality: repeating patterns over time
• Noise: cannot be explained by the model

Some concerns

• Sample size

• Updated frequently over time or be made once and remain static

• Down-sampling or up-sampling

  • Frequency
  • outliers
  • Missing

• As a supervised Machine Learning

  • Sliding window with univariate time series / multivariate time series

Q&A

• (Python) Difference between autocorrelation_plot and plot_acf / plot_pacf ?
  • autocorrelation_plot and plot_acf are the same
• Definition
  • { Yt } 严平稳: 对一切 k 和时点 t1, t2, …, tn, 都有T_t1, T_t2, … T_tn 与T_{t1-k}, T_{t2-k}, …., T_{tn-k} 的联合分布相关
    • { Yt } 弱平稳条件
      • 均值函数在所有时间上恒为常数
      • Gamma_{t, t-k} = gamma_{0, k}, 对所有时间 t 和 滞后 k
  • methods
    • Line plot
    • Randomly split data into 2 or more parts then check the mean and covariance
  • Statistical test - ADF(augmented Dicky-Fuller test)
    • Explanation
    • H0: time series has a unit root, meaning is is non-stationary
• Transforms
  • Difference
  • Log
    • 当序列散度与序列值有正相关关系时，即序列的值越大，围绕该值的波动就越大
    • 对数的差分通常称为收益率
  • Box-Cox/幂变换
    • 估计lambda
    • 当lambda = 0 时，退化为log变换
• Add Seasonality
  • https://www.statsmodels.org/dev/examples/notebooks/generated/statespace_sarimax_stata.html
• How to interpret the key results for ARIMA :
  • For each coef, the null hypothesis is that the term is not significantly different from 0, which indicates that no association exists between the term and the response.
  • https://support.minitab.com/en-us/minitab/18/help-and-how-to/modeling-statistics/time-series/how-to/arima/interpret-the-results/key-results/?SID=117600
• Residuals test
  • Residuals time series -> exist trend or not
  • qq plot -> lies in a line
  • Residuals acf graph
  • Residual Ljung-box test
    • 将相关系数的值作为一个组来检验，定义统计量 Q