Autoregressive model

在下面build_model的代码中涉及到tfp中的sts.Autoregressive.很长时间不用，恍惚中忘记了何为Autoregressive？通过阅读Autoregressive Model: Definition & The AR Process学习到以下几点：

• auto不是英文单词，而是表示self的希腊语.这可能也说明这个模型有多么经典.
• autoregressive model旨在寻找当前时刻的值，与相近历史值的线性关系，因此又叫做markov model.与过去几个值，由order这个参数来指定.
• autoregressive model本身是一个包含随机信息的过程，可以很好的预测未来趋势，但无法得到准确的point estimation.(The AR process is an example of a stochastic process, which have degrees of uncertainty or randomness built in. The randomness means that you might be able to predict future trends pretty well with past data, but you’re never going to get 100 percent accuracy. Usually, the process gets “close enough” for it to be useful in most scenarios.)

数学公式还是最简洁的表达方式，AR model表示如下：

$$y_t={\phi }_1{y}_{t-1}+{\phi }_2{y}_{t-2}+...+{\phi }_p{y}_{t-p}+A_t+\delta$$

• $y_{t-1}$, $y_{t-2}$, …, $y_{t-p}$ are the past series values.
• $A_t$ is white noise (i.e. randomness)
• $\delta$ is defined by the following equation:

$$\delta =(1-\sum _{i=1}^p{\varphi }_i)\mu$$
where $\mu$ is the process mean.

为在markdown中写出上述公式，参照了LaTeX Math Symbols and Motivating Examples.

def build_model(observed_time_series):
  hour_of_day_effect = sts.Seasonal(
      num_seasons=24,
      observed_time_series=observed_time_series,
      name='hour_of_day_effect')
  day_of_week_effect = sts.Seasonal(
      num_seasons=7, num_steps_per_season=24,
      observed_time_series=observed_time_series,
      name='day_of_week_effect')
  temperature_effect = sts.LinearRegression(
      design_matrix=tf.reshape(temperature - np.mean(temperature),
                               (-1, 1)), name='temperature_effect')
  autoregressive = sts.Autoregressive(
      order=1, # scalar Python positive int specifying the number of past timesteps to regress on.
      observed_time_series=observed_time_series,
      name='autoregressive')
  model = sts.Sum([hour_of_day_effect,
                   day_of_week_effect,
                   temperature_effect,
                   autoregressive],
                   observed_time_series=observed_time_series)
  return model

代码来自Structural Time Series Modeling Case Studies: Atmospheric CO2 and Electricity Demand