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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

=

φ

1

y

t

−

1

+

φ

2

y

t

−

2

+

.

.

.

+

φ

p

y

t

−

p

+

A

t

+

δ

y_t = \varphi_1 y_{t-1} + \varphi_2 y_{t-2} + … + \varphi_p y_{t-p} + A_t+ \delta







y










t





​














=









φ










1





​













y











t


−


1






​














+









φ










2





​













y











t


−


2






​














+








.


.


.




+









φ










p





​













y











t


−


p






​














+









A










t





​














+








δ

• 
  
  
  y

  t

  −

  1

  y_{t-1}
  
  
  
  
  
  
  
  y
  
  
  
  
  
  
  
  
  
  
  
  t
  
  
  −
  
  
  1
  
  
  
  
  
  
  ​
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  ,
  
  
  
  y

  t

  −

  2

  y_{t-2}
  
  
  
  
  
  
  
  y
  
  
  
  
  
  
  
  
  
  
  
  t
  
  
  −
  
  
  2
  
  
  
  
  
  
  ​
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  , …,
  
  
  
  y

  t

  −

  p

  y_{t-p}
  
  
  
  
  
  
  
  y
  
  
  
  
  
  
  
  
  
  
  
  t
  
  
  −
  
  
  p
  
  
  
  
  
  
  ​
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  are the past series values.

• 
  
  
  A

  t

  A_t
  
  
  
  
  
  
  
  A
  
  
  
  
  
  
  
  
  
  
  t
  
  
  
  
  
  ​
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  is white noise (i.e. randomness)

• 
  
  
  δ

  \delta
  
  
  
  
  
  
  δ
  
  
  
  
  
  is defined by the following equation:

δ

=

(

1

−

∑

i

=

1

p

ϕ

i

)

μ

\delta = (1 – \sum_{i=1}^p \phi_i) \mu






δ




=








(


1




−

















i


=


1









∑








p





​














ϕ










i





​












)


μ







where



μ

\mu






μ





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