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مطالعات موردی مدل‌سازی سری‌های زمانی ساختاری: دی‌اکسید کربن اتمسفر و تقاضای الکتریسیته

این نوت بوک دو نمونه از برازش مدل های سری زمانی ساختاری به سری های زمانی و استفاده از آنها برای تولید پیش بینی ها و توضیحات را نشان می دهد.

مشاهده در TensorFlow.org

در Google Colab اجرا شود

مشاهده منبع در GitHub

دانلود دفترچه یادداشت

وابستگی ها و پیش نیازها

واردات و راه اندازی

%matplotlib inline
import matplotlib as mpl
from matplotlib import pylab as plt
import matplotlib.dates as mdates
import seaborn as sns

import collections

import numpy as np
import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp

from tensorflow_probability import distributions as tfd
from tensorflow_probability import sts

tf.enable_v2_behavior()

همه چیز را سریع کنید!

قبل از شیرجه رفتن، مطمئن شویم که از یک GPU برای این نسخه نمایشی استفاده می کنیم.

برای انجام این کار، "Runtime" -> "Change runtime type" -> "Hardware accelerator" -> "GPU" را انتخاب کنید.

قطعه زیر تأیید می کند که ما به یک GPU دسترسی داریم.

if tf.test.gpu_device_name() != '/device:GPU:0':
  print('WARNING: GPU device not found.')
else:
  print('SUCCESS: Found GPU: {}'.format(tf.test.gpu_device_name()))

SUCCESS: Found GPU: /device:GPU:0

راه اندازی توطئه

روش های کمکی برای ترسیم سری های زمانی و پیش بینی ها

from pandas.plotting import register_matplotlib_converters
register_matplotlib_converters()

sns.set_context("notebook", font_scale=1.)
sns.set_style("whitegrid")
%config InlineBackend.figure_format = 'retina'

def plot_forecast(x, y,
                  forecast_mean, forecast_scale, forecast_samples,
                  title, x_locator=None, x_formatter=None):
  """Plot a forecast distribution against the 'true' time series."""
  colors = sns.color_palette()
  c1, c2 = colors[0], colors[1]
  fig = plt.figure(figsize=(12, 6))
  ax = fig.add_subplot(1, 1, 1)

  num_steps = len(y)
  num_steps_forecast = forecast_mean.shape[-1]
  num_steps_train = num_steps - num_steps_forecast


  ax.plot(x, y, lw=2, color=c1, label='ground truth')

  forecast_steps = np.arange(
      x[num_steps_train],
      x[num_steps_train]+num_steps_forecast,
      dtype=x.dtype)

  ax.plot(forecast_steps, forecast_samples.T, lw=1, color=c2, alpha=0.1)

  ax.plot(forecast_steps, forecast_mean, lw=2, ls='--', color=c2,
           label='forecast')
  ax.fill_between(forecast_steps,
                   forecast_mean-2*forecast_scale,
                   forecast_mean+2*forecast_scale, color=c2, alpha=0.2)

  ymin, ymax = min(np.min(forecast_samples), np.min(y)), max(np.max(forecast_samples), np.max(y))
  yrange = ymax-ymin
  ax.set_ylim([ymin - yrange*0.1, ymax + yrange*0.1])
  ax.set_title("{}".format(title))
  ax.legend()

  if x_locator is not None:
    ax.xaxis.set_major_locator(x_locator)
    ax.xaxis.set_major_formatter(x_formatter)
    fig.autofmt_xdate()

  return fig, ax

def plot_components(dates,
                    component_means_dict,
                    component_stddevs_dict,
                    x_locator=None,
                    x_formatter=None):
  """Plot the contributions of posterior components in a single figure."""
  colors = sns.color_palette()
  c1, c2 = colors[0], colors[1]

  axes_dict = collections.OrderedDict()
  num_components = len(component_means_dict)
  fig = plt.figure(figsize=(12, 2.5 * num_components))
  for i, component_name in enumerate(component_means_dict.keys()):
    component_mean = component_means_dict[component_name]
    component_stddev = component_stddevs_dict[component_name]

    ax = fig.add_subplot(num_components,1,1+i)
    ax.plot(dates, component_mean, lw=2)
    ax.fill_between(dates,
                     component_mean-2*component_stddev,
                     component_mean+2*component_stddev,
                     color=c2, alpha=0.5)
    ax.set_title(component_name)
    if x_locator is not None:
      ax.xaxis.set_major_locator(x_locator)
      ax.xaxis.set_major_formatter(x_formatter)
    axes_dict[component_name] = ax
  fig.autofmt_xdate()
  fig.tight_layout()
  return fig, axes_dict

def plot_one_step_predictive(dates, observed_time_series,
                             one_step_mean, one_step_scale,
                             x_locator=None, x_formatter=None):
  """Plot a time series against a model's one-step predictions."""

  colors = sns.color_palette()
  c1, c2 = colors[0], colors[1]

  fig=plt.figure(figsize=(12, 6))
  ax = fig.add_subplot(1,1,1)
  num_timesteps = one_step_mean.shape[-1]
  ax.plot(dates, observed_time_series, label="observed time series", color=c1)
  ax.plot(dates, one_step_mean, label="one-step prediction", color=c2)
  ax.fill_between(dates,
                  one_step_mean - one_step_scale,
                  one_step_mean + one_step_scale,
                  alpha=0.1, color=c2)
  ax.legend()

  if x_locator is not None:
    ax.xaxis.set_major_locator(x_locator)
    ax.xaxis.set_major_formatter(x_formatter)
    fig.autofmt_xdate()
  fig.tight_layout()
  return fig, ax

رکورد CO2 Mauna Loa

ما تطبیق مدلی را برای خوانش CO2 اتمسفر از رصدخانه Mauna Loa نشان خواهیم داد.

داده ها

# CO2 readings from Mauna Loa observatory, monthly beginning January 1966
# Original source: http://scrippsco2.ucsd.edu/data/atmospheric_co2/primary_mlo_co2_record
co2_by_month = np.array('320.62,321.60,322.39,323.70,324.08,323.75,322.38,320.36,318.64,318.10,319.78,321.03,322.33,322.50,323.04,324.42,325.00,324.09,322.54,320.92,319.25,319.39,320.73,321.96,322.57,323.15,323.89,325.02,325.57,325.36,324.14,322.11,320.33,320.25,321.32,322.89,324.00,324.42,325.63,326.66,327.38,326.71,325.88,323.66,322.38,321.78,322.85,324.12,325.06,325.98,326.93,328.14,328.08,327.67,326.34,324.69,323.10,323.06,324.01,325.13,326.17,326.68,327.17,327.79,328.92,328.57,327.36,325.43,323.36,323.56,324.80,326.01,326.77,327.63,327.75,329.73,330.07,329.09,328.04,326.32,324.84,325.20,326.50,327.55,328.55,329.56,330.30,331.50,332.48,332.07,330.87,329.31,327.51,327.18,328.16,328.64,329.35,330.71,331.48,332.65,333.09,332.25,331.18,329.39,327.43,327.37,328.46,329.57,330.40,331.40,332.04,333.31,333.97,333.60,331.90,330.06,328.56,328.34,329.49,330.76,331.75,332.56,333.50,334.58,334.88,334.33,333.05,330.94,329.30,328.94,330.31,331.68,332.93,333.42,334.70,336.07,336.75,336.27,334.92,332.75,331.59,331.16,332.40,333.85,334.97,335.38,336.64,337.76,338.01,337.89,336.54,334.68,332.76,332.55,333.92,334.95,336.23,336.76,337.96,338.88,339.47,339.29,337.73,336.09,333.92,333.86,335.29,336.73,338.01,338.36,340.07,340.77,341.47,341.17,339.56,337.60,335.88,336.02,337.10,338.21,339.24,340.48,341.38,342.51,342.91,342.25,340.49,338.43,336.69,336.86,338.36,339.61,340.75,341.61,342.70,343.57,344.14,343.35,342.06,339.81,337.98,337.86,339.26,340.49,341.38,342.52,343.10,344.94,345.76,345.32,343.98,342.38,339.87,339.99,341.15,342.99,343.70,344.50,345.28,347.06,347.43,346.80,345.39,343.28,341.07,341.35,342.98,344.22,344.97,345.99,347.42,348.35,348.93,348.25,346.56,344.67,343.09,342.80,344.24,345.56,346.30,346.95,347.85,349.55,350.21,349.55,347.94,345.90,344.85,344.17,345.66,346.90,348.02,348.48,349.42,350.99,351.85,351.26,349.51,348.10,346.45,346.36,347.81,348.96,350.43,351.73,352.22,353.59,354.22,353.79,352.38,350.43,348.73,348.88,350.07,351.34,352.76,353.07,353.68,355.42,355.67,355.12,353.90,351.67,349.80,349.99,351.30,352.52,353.66,354.70,355.38,356.20,357.16,356.23,354.81,352.91,350.96,351.18,352.83,354.21,354.72,355.75,357.16,358.60,359.34,358.24,356.17,354.02,352.15,352.21,353.75,354.99,355.99,356.72,357.81,359.15,359.66,359.25,357.02,355.00,353.01,353.31,354.16,355.40,356.70,357.17,358.38,359.46,360.28,359.60,357.57,355.52,353.69,353.99,355.34,356.80,358.37,358.91,359.97,361.26,361.69,360.94,359.55,357.48,355.84,356.00,357.58,359.04,359.97,361.00,361.64,363.45,363.80,363.26,361.89,359.45,358.05,357.75,359.56,360.70,362.05,363.24,364.02,364.71,365.41,364.97,363.65,361.48,359.45,359.61,360.76,362.33,363.18,363.99,364.56,366.36,366.80,365.63,364.47,362.50,360.19,360.78,362.43,364.28,365.33,366.15,367.31,368.61,369.30,368.88,367.64,365.78,363.90,364.23,365.46,366.97,368.15,368.87,369.59,371.14,371.00,370.35,369.27,366.93,364.64,365.13,366.68,368.00,369.14,369.46,370.51,371.66,371.83,371.69,370.12,368.12,366.62,366.73,368.29,369.53,370.28,371.50,372.12,372.86,374.02,373.31,371.62,369.55,367.96,368.09,369.68,371.24,372.44,373.08,373.52,374.85,375.55,375.40,374.02,371.48,370.70,370.25,372.08,373.78,374.68,375.62,376.11,377.65,378.35,378.13,376.61,374.48,372.98,373.00,374.35,375.69,376.79,377.36,378.39,380.50,380.62,379.55,377.76,375.83,374.05,374.22,375.84,377.44,378.34,379.61,380.08,382.05,382.24,382.08,380.67,378.67,376.42,376.80,378.31,379.96,381.37,382.02,382.56,384.37,384.92,384.03,382.28,380.48,378.81,379.06,380.14,381.66,382.58,383.71,384.34,386.23,386.41,385.87,384.45,381.84,380.86,380.86,382.36,383.61,385.07,385.84,385.83,386.77,388.51,388.05,386.25,384.08,383.09,382.78,384.01,385.11,386.65,387.12,388.52,389.57,390.16,389.62,388.07,386.08,384.65,384.33,386.05,387.49,388.55,390.07,391.01,392.38,393.22,392.24,390.33,388.52,386.84,387.16,388.67,389.81,391.30,391.92,392.45,393.37,394.28,393.69,392.59,390.21,389.00,388.93,390.24,391.80,393.07,393.35,394.36,396.43,396.87,395.88,394.52,392.54,391.13,391.01,392.95,394.34,395.61,396.85,397.26,398.35,399.98,398.87,397.37,395.41,393.39,393.70,395.19,396.82,397.92,398.10,399.47,401.33,401.88,401.31,399.07,397.21,395.40,395.65,397.23,398.79,399.85,400.31,401.51,403.45,404.10,402.88,401.61,399.00,397.50,398.28,400.24,401.89,402.65,404.16,404.85,407.57,407.66,407.00,404.50,402.24,401.01,401.50,403.64,404.55,406.07,406.64,407.06,408.95,409.91,409.12,407.20,405.24,403.27,403.64,405.17,406.75,408.05,408.34,409.25,410.30,411.30,410.88,408.90,407.10,405.59,405.99,408.12,409.23,410.92'.split(',')).astype(np.float32)

co2_by_month = co2_by_month
num_forecast_steps = 12 * 10 # Forecast the final ten years, given previous data
co2_by_month_training_data = co2_by_month[:-num_forecast_steps]

co2_dates = np.arange("1966-01", "2019-02", dtype="datetime64[M]")
co2_loc = mdates.YearLocator(3)
co2_fmt = mdates.DateFormatter('%Y')

fig = plt.figure(figsize=(12, 6))
ax = fig.add_subplot(1, 1, 1)
ax.plot(co2_dates[:-num_forecast_steps], co2_by_month_training_data, lw=2, label="training data")
ax.xaxis.set_major_locator(co2_loc)
ax.xaxis.set_major_formatter(co2_fmt)
ax.set_ylabel("Atmospheric CO2 concentration (ppm)")
ax.set_xlabel("Year")
fig.suptitle("Monthly average CO2 concentration, Mauna Loa, Hawaii",
             fontsize=15)
ax.text(0.99, .02,
        "Source: Scripps Institute for Oceanography CO2 program\nhttp://scrippsco2.ucsd.edu/data/atmospheric_co2/primary_mlo_co2_record",
        transform=ax.transAxes,
        horizontalalignment="right",
        alpha=0.5)
fig.autofmt_xdate()

png

مدل و اتصال

ما این سری را با یک روند خطی محلی، به‌علاوه یک اثر فصلی ماه از سال مدل‌سازی می‌کنیم.

def build_model(observed_time_series):
  trend = sts.LocalLinearTrend(observed_time_series=observed_time_series)
  seasonal = tfp.sts.Seasonal(
      num_seasons=12, observed_time_series=observed_time_series)
  model = sts.Sum([trend, seasonal], observed_time_series=observed_time_series)
  return model

ما مدل را با استفاده از استنتاج تغییرات برازش خواهیم کرد. این شامل اجرای یک بهینه ساز برای به حداقل رساندن یک تابع تلفات متغیر، کران پایین شواهد منفی (ELBO) است. این با مجموعه ای از توزیع های پسین تقریبی برای پارامترها مطابقت دارد (در عمل ما فرض می کنیم که اینها نرمال های مستقل هستند که به فضای پشتیبانی هر پارامتر تبدیل شده اند).

روش‌های پیش‌بینی tfp.sts به نمونه‌های پسینی به‌عنوان ورودی نیاز دارند، بنابراین ما با رسم مجموعه‌ای از نمونه‌ها از خلف متغیر به پایان می‌رسیم.

co2_model = build_model(co2_by_month_training_data)

# Build the variational surrogate posteriors `qs`.
variational_posteriors = tfp.sts.build_factored_surrogate_posterior(
    model=co2_model)

از دست دادن تغییرات را به حداقل برسانید.

# Allow external control of optimization to reduce test runtimes.
num_variational_steps = 200 # @param { isTemplate: true}
num_variational_steps = int(num_variational_steps)

# Build and optimize the variational loss function.
elbo_loss_curve = tfp.vi.fit_surrogate_posterior(
    target_log_prob_fn=co2_model.joint_distribution(
        observed_time_series=co2_by_month_training_data).log_prob,
    surrogate_posterior=variational_posteriors,
    optimizer=tf.optimizers.Adam(learning_rate=0.1),
    num_steps=num_variational_steps,
    jit_compile=True)

plt.plot(elbo_loss_curve)
plt.show()

# Draw samples from the variational posterior.
q_samples_co2_ = variational_posteriors.sample(50)

WARNING:tensorflow:From /usr/local/lib/python3.6/dist-packages/tensorflow_core/python/ops/linalg/linear_operator_diag.py:166: calling LinearOperator.__init__ (from tensorflow.python.ops.linalg.linear_operator) with graph_parents is deprecated and will be removed in a future version.
Instructions for updating:
Do not pass `graph_parents`.  They will  no longer be used.

png

print("Inferred parameters:")
for param in co2_model.parameters:
  print("{}: {} +- {}".format(param.name,
                              np.mean(q_samples_co2_[param.name], axis=0),
                              np.std(q_samples_co2_[param.name], axis=0)))

Inferred parameters:
observation_noise_scale: 0.17199112474918365 +- 0.009443143382668495
LocalLinearTrend/_level_scale: 0.17671072483062744 +- 0.01510554924607277
LocalLinearTrend/_slope_scale: 0.004302256740629673 +- 0.0018349259626120329
Seasonal/_drift_scale: 0.041069451719522476 +- 0.007772190496325493

پیش بینی و انتقاد

حال بیایید از مدل برازش برای ساخت یک پیش بینی استفاده کنیم. ما فقط tfp.sts.forecast را فراخوانی می کنیم که یک نمونه توزیع TensorFlow را نشان می دهد که توزیع پیش بینی را در مراحل زمانی آینده نشان می دهد.

co2_forecast_dist = tfp.sts.forecast(
    co2_model,
    observed_time_series=co2_by_month_training_data,
    parameter_samples=q_samples_co2_,
    num_steps_forecast=num_forecast_steps)

به طور خاص، mean و stddev توزیع پیش‌بینی به ما یک پیش‌بینی با عدم قطعیت حاشیه‌ای در هر مرحله می‌دهد، و همچنین می‌توانیم نمونه‌هایی از آینده‌های احتمالی ترسیم کنیم.

num_samples=10

co2_forecast_mean, co2_forecast_scale, co2_forecast_samples = (
    co2_forecast_dist.mean().numpy()[..., 0],
    co2_forecast_dist.stddev().numpy()[..., 0],
    co2_forecast_dist.sample(num_samples).numpy()[..., 0])

fig, ax = plot_forecast(
    co2_dates, co2_by_month,
    co2_forecast_mean, co2_forecast_scale, co2_forecast_samples,
    x_locator=co2_loc,
    x_formatter=co2_fmt,
    title="Atmospheric CO2 forecast")
ax.axvline(co2_dates[-num_forecast_steps], linestyle="--")
ax.legend(loc="upper left")
ax.set_ylabel("Atmospheric CO2 concentration (ppm)")
ax.set_xlabel("Year")
fig.autofmt_xdate()

png

ما می‌توانیم تناسب مدل را با تجزیه آن به سهم سری‌های زمانی فردی بیشتر درک کنیم:

# Build a dict mapping components to distributions over
# their contribution to the observed signal.
component_dists = sts.decompose_by_component(
    co2_model,
    observed_time_series=co2_by_month,
    parameter_samples=q_samples_co2_)

co2_component_means_, co2_component_stddevs_ = (
    {k.name: c.mean() for k, c in component_dists.items()},
    {k.name: c.stddev() for k, c in component_dists.items()})

_ = plot_components(co2_dates, co2_component_means_, co2_component_stddevs_,
                    x_locator=co2_loc, x_formatter=co2_fmt)

png

پیش بینی تقاضای برق

حال بیایید مثال پیچیده‌تری را در نظر بگیریم: پیش‌بینی تقاضای برق در ویکتوریا استرالیا.

ابتدا مجموعه داده را می سازیم:

# Victoria electricity demand dataset, as presented at
# https://otexts.com/fpp2/scatterplots.html
# and downloaded from https://github.com/robjhyndman/fpp2-package/blob/master/data/elecdaily.rda
# This series contains the first eight weeks (starting Jan 1). The original
# dataset was half-hourly data; here we've downsampled to hourly data by taking
# every other timestep.
demand_dates = np.arange('2014-01-01', '2014-02-26', dtype='datetime64[h]')
demand_loc = mdates.WeekdayLocator(byweekday=mdates.WE)
demand_fmt = mdates.DateFormatter('%a %b %d')

demand = np.array("3.794,3.418,3.152,3.026,3.022,3.055,3.180,3.276,3.467,3.620,3.730,3.858,3.851,3.839,3.861,3.912,4.082,4.118,4.011,3.965,3.932,3.693,3.585,4.001,3.623,3.249,3.047,3.004,3.104,3.361,3.749,3.910,4.075,4.165,4.202,4.225,4.265,4.301,4.381,4.484,4.552,4.440,4.233,4.145,4.116,3.831,3.712,4.121,3.764,3.394,3.159,3.081,3.216,3.468,3.838,4.012,4.183,4.269,4.280,4.310,4.315,4.233,4.188,4.263,4.370,4.308,4.182,4.075,4.057,3.791,3.667,4.036,3.636,3.283,3.073,3.003,3.023,3.113,3.335,3.484,3.697,3.723,3.786,3.763,3.748,3.714,3.737,3.828,3.937,3.929,3.877,3.829,3.950,3.756,3.638,4.045,3.682,3.283,3.036,2.933,2.956,2.959,3.157,3.236,3.370,3.493,3.516,3.555,3.570,3.656,3.792,3.950,3.953,3.926,3.849,3.813,3.891,3.683,3.562,3.936,3.602,3.271,3.085,3.041,3.201,3.570,4.123,4.307,4.481,4.533,4.545,4.524,4.470,4.457,4.418,4.453,4.539,4.473,4.301,4.260,4.276,3.958,3.796,4.180,3.843,3.465,3.246,3.203,3.360,3.808,4.328,4.509,4.598,4.562,4.566,4.532,4.477,4.442,4.424,4.486,4.579,4.466,4.338,4.270,4.296,4.034,3.877,4.246,3.883,3.520,3.306,3.252,3.387,3.784,4.335,4.465,4.529,4.536,4.589,4.660,4.691,4.747,4.819,4.950,4.994,4.798,4.540,4.352,4.370,4.047,3.870,4.245,3.848,3.509,3.302,3.258,3.419,3.809,4.363,4.605,4.793,4.908,5.040,5.204,5.358,5.538,5.708,5.888,5.966,5.817,5.571,5.321,5.141,4.686,4.367,4.618,4.158,3.771,3.555,3.497,3.646,4.053,4.687,5.052,5.342,5.586,5.808,6.038,6.296,6.548,6.787,6.982,7.035,6.855,6.561,6.181,5.899,5.304,4.795,4.862,4.264,3.820,3.588,3.481,3.514,3.632,3.857,4.116,4.375,4.462,4.460,4.422,4.398,4.407,4.480,4.621,4.732,4.735,4.572,4.385,4.323,4.069,3.940,4.247,3.821,3.416,3.220,3.124,3.132,3.181,3.337,3.469,3.668,3.788,3.834,3.894,3.964,4.109,4.275,4.472,4.623,4.703,4.594,4.447,4.459,4.137,3.913,4.231,3.833,3.475,3.302,3.279,3.519,3.975,4.600,4.864,5.104,5.308,5.542,5.759,6.005,6.285,6.617,6.993,7.207,7.095,6.839,6.387,6.048,5.433,4.904,4.959,4.425,4.053,3.843,3.823,4.017,4.521,5.229,5.802,6.449,6.975,7.506,7.973,8.359,8.596,8.794,9.030,9.090,8.885,8.525,8.147,7.797,6.938,6.215,6.123,5.495,5.140,4.896,4.812,5.024,5.536,6.293,7.000,7.633,8.030,8.459,8.768,9.000,9.113,9.155,9.173,9.039,8.606,8.095,7.617,7.208,6.448,5.740,5.718,5.106,4.763,4.610,4.566,4.737,5.204,5.988,6.698,7.438,8.040,8.484,8.837,9.052,9.114,9.214,9.307,9.313,9.006,8.556,8.275,7.911,7.077,6.348,6.175,5.455,5.041,4.759,4.683,4.908,5.411,6.199,6.923,7.593,8.090,8.497,8.843,9.058,9.159,9.231,9.253,8.852,7.994,7.388,6.735,6.264,5.690,5.227,5.220,4.593,4.213,3.984,3.891,3.919,4.031,4.287,4.558,4.872,4.963,5.004,5.017,5.057,5.064,5.000,5.023,5.007,4.923,4.740,4.586,4.517,4.236,4.055,4.337,3.848,3.473,3.273,3.198,3.204,3.252,3.404,3.560,3.767,3.896,3.934,3.972,3.985,4.032,4.122,4.239,4.389,4.499,4.406,4.356,4.396,4.106,3.914,4.265,3.862,3.546,3.360,3.359,3.649,4.180,4.813,5.086,5.301,5.384,5.434,5.470,5.529,5.582,5.618,5.636,5.561,5.291,5.000,4.840,4.767,4.364,4.160,4.452,4.011,3.673,3.503,3.483,3.695,4.213,4.810,5.028,5.149,5.182,5.208,5.179,5.190,5.220,5.202,5.216,5.232,5.019,4.828,4.686,4.657,4.304,4.106,4.389,3.955,3.643,3.489,3.479,3.695,4.187,4.732,4.898,4.997,5.001,5.022,5.052,5.094,5.143,5.178,5.250,5.255,5.075,4.867,4.691,4.665,4.352,4.121,4.391,3.966,3.615,3.437,3.430,3.666,4.149,4.674,4.851,5.011,5.105,5.242,5.378,5.576,5.790,6.030,6.254,6.340,6.253,6.039,5.736,5.490,4.936,4.580,4.742,4.230,3.895,3.712,3.700,3.906,4.364,4.962,5.261,5.463,5.495,5.477,5.394,5.250,5.159,5.081,5.083,5.038,4.857,4.643,4.526,4.428,4.141,3.975,4.290,3.809,3.423,3.217,3.132,3.192,3.343,3.606,3.803,3.963,3.998,3.962,3.894,3.814,3.776,3.808,3.914,4.033,4.079,4.027,3.974,4.057,3.859,3.759,4.132,3.716,3.325,3.111,3.030,3.046,3.096,3.254,3.390,3.606,3.718,3.755,3.768,3.768,3.834,3.957,4.199,4.393,4.532,4.516,4.380,4.390,4.142,3.954,4.233,3.795,3.425,3.209,3.124,3.177,3.288,3.498,3.715,4.092,4.383,4.644,4.909,5.184,5.518,5.889,6.288,6.643,6.729,6.567,6.179,5.903,5.278,4.788,4.885,4.363,4.011,3.823,3.762,3.998,4.598,5.349,5.898,6.487,6.941,7.381,7.796,8.185,8.522,8.825,9.103,9.198,8.889,8.174,7.214,6.481,5.611,5.026,5.052,4.484,4.148,3.955,3.873,4.060,4.626,5.272,5.441,5.535,5.534,5.610,5.671,5.724,5.793,5.838,5.908,5.868,5.574,5.276,5.065,4.976,4.554,4.282,4.547,4.053,3.720,3.536,3.524,3.792,4.420,5.075,5.208,5.344,5.482,5.701,5.936,6.210,6.462,6.683,6.979,7.059,6.893,6.535,6.121,5.797,5.152,4.705,4.805,4.272,3.975,3.805,3.775,3.996,4.535,5.275,5.509,5.730,5.870,6.034,6.175,6.340,6.500,6.603,6.804,6.787,6.460,6.043,5.627,5.367,4.866,4.575,4.728,4.157,3.795,3.607,3.537,3.596,3.803,4.125,4.398,4.660,4.853,5.115,5.412,5.669,5.930,6.216,6.466,6.641,6.605,6.316,5.821,5.520,5.016,4.657,4.746,4.197,3.823,3.613,3.505,3.488,3.532,3.716,4.011,4.421,4.836,5.296,5.766,6.233,6.646,7.011,7.380,7.660,7.804,7.691,7.364,7.019,6.260,5.545,5.437,4.806,4.457,4.235,4.172,4.396,5.002,5.817,6.266,6.732,7.049,7.184,7.085,6.798,6.632,6.408,6.218,5.968,5.544,5.217,4.964,4.758,4.328,4.074,4.367,3.883,3.536,3.404,3.396,3.624,4.271,4.916,4.953,5.016,5.048,5.106,5.124,5.200,5.244,5.242,5.34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temperature = np.array("18.050,17.200,16.450,16.650,16.400,17.950,19.700,20.600,22.350,23.700,24.800,25.900,25.300,23.650,20.700,19.150,22.650,22.650,22.400,22.150,22.050,22.150,21.000,19.500,18.450,17.250,16.300,15.700,15.500,15.450,15.650,16.500,18.100,17.800,19.100,19.850,20.300,21.050,22.800,21.650,20.150,19.300,18.750,17.900,17.350,16.850,16.350,15.700,14.950,14.500,14.350,14.450,14.600,14.600,14.700,15.450,16.700,18.300,20.100,20.650,19.450,20.200,20.250,20.050,20.250,20.950,21.900,21.000,19.900,19.250,17.300,16.300,15.800,15.000,14.400,14.050,13.650,13.500,14.150,15.300,14.800,17.050,18.350,19.450,18.550,18.650,18.850,19.800,19.650,18.900,19.500,17.700,17.350,16.950,16.400,15.950,14.900,14.250,13.050,12.000,11.500,10.950,12.300,16.100,17.100,19.600,21.100,22.600,24.350,25.250,25.750,20.350,15.550,18.300,19.400,19.250,18.550,17.700,16.750,15.800,14.900,14.050,14.100,13.500,13.000,12.950,13.300,13.900,15.400,16.750,17.300,17.750,18.400,18.500,18.800,19.450,18.750,18.400,16.950,15.800,15.350,15.250,15.150,14.900,14.500,14.600,14.400,14.150,14.300,14.500,14.950,15.550,15.800,15.550,16.450,17.500,17.700,18.750,19.600,19.900,19.350,19.550,17.900,16.400,15.550,14.900,14.400,13.950,13.300,12.950,12.650,12.450,12.350,12.150,11.950,14.150,15.850,17.750,19.450,22.150,23.850,23.450,24.950,26.850,26.100,25.150,23.250,21.300,19.850,18.900,18.250,17.450,17.100,16.400,15.550,15.050,14.400,14.550,15.150,17.050,18.850,20.850,24.250,27.700,28.400,30.750,30.700,32.200,31.750,30.650,29.750,28.850,27.850,25.950,24.700,24.850,24.050,23.850,23.500,22.950,22.200,21.750,22.350,24.050,25.150,27.100,28.050,29.750,31.250,31.900,32.950,33.150,33.950,33.850,33.250,32.500,31.500,28.300,23.900,22.900,22.300,21.250,20.500,19.850,18.850,18.300,18.100,18.200,18.150,18.000,17.700,18.250,19.700,20.750,21.800,21.500,21.600,20.800,19.400,18.400,17.900,17.600,17.550,17.550,17.650,17.400,17.150,16.800,17.000,16.900,17.200,17.350,17.650,17.800,18.400,19.300,20.200,21.050,21.700,21.800,21.800,21.500,20.000,19.300,18.200,18.100,17.700,16.950,16.250,15.600,15.500,15.300,15.450,15.500,15.750,17.350,19.150,21.650,24.700,25.200,24.300,26.900,28.100,29.450,29.850,29.450,26.350,27.050,25.700,25.150,23.850,22.450,21.450,20.850,20.700,21.300,21.550,20.800,22.300,26.300,32.600,35.150,36.800,38.150,39.950,40.850,41.250,42.300,41.950,41.350,40.600,36.350,36.150,34.600,34.050,35.400,36.300,35.550,33.700,30.650,29.450,29.500,31.000,33.300,35.700,36.650,37.650,39.400,40.600,40.250,37.550,37.300,35.400,32.750,31.200,29.600,28.350,27.500,28.750,28.900,29.900,28.700,28.650,28.150,28.250,27.650,27.800,29.450,32.500,35.750,38.850,39.900,41.100,41.800,42.750,39.900,39.750,40.800,37.950,31.250,34.600,30.250,28.500,27.900,27.950,27.300,26.900,26.800,26.050,26.100,27.700,31.850,34.850,36.350,38.000,39.200,41.050,41.600,42.350,43.100,33.500,30.700,29.100,26.400,23.900,24.700,24.350,23.450,23.450,23.550,23.050,22.200,22.100,22.000,21.900,22.050,22.550,22.850,22.450,22.250,22.650,22.350,21.900,21.000,20.950,20.200,19.700,19.400,19.200,18.650,18.150,18.150,17.650,17.350,17.150,16.800,16.750,16.400,16.500,16.700,17.300,17.750,19.200,20.400,20.900,21.450,22.000,22.100,21.600,21.700,20.500,19.850,19.750,19.500,19.200,19.800,19.500,19.200,19.200,19.150,19.050,19.100,19.250,19.550,20.200,20.550,21.450,23.150,23.500,23.400,23.500,23.300,22.850,22.250,20.950,19.750,19.450,18.900,18.450,17.950,17.550,17.300,16.950,16.900,16.850,17.100,17.250,17.400,17.850,18.100,18.600,19.700,21.000,21.400,22.650,22.550,22.000,21.050,19.550,18.550,18.300,17.750,17.800,17.650,17.800,17.450,16.950,16.500,16.900,17.050,16.750,17.300,18.800,19.350,20.750,21.400,21.900,21.950,22.800,22.750,23.200,22.650,20.800,19.250,17.800,16.950,16.550,16.050,15.750,15.150,14.700,14.150,13.900,13.900,14.000,15.800,17.650,19.700,22.500,25.300,24.300,24.650,26.450,27.250,26.550,28.800,27.850,25.200,24.750,23.750,22.550,22.350,21.700,21.300,20.300,20.050,20.500,21.250,20.850,21.000,19.400,18.900,18.150,18.650,20.200,20.000,21.650,21.950,21.150,20.400,19.500,19.150,18.400,18.050,17.750,17.600,17.150,16.750,16.350,16.250,15.900,15.850,15.900,16.200,18.500,18.750,18.800,19.850,19.750,19.600,19.300,20.000,20.250,19.700,18.600,17.400,17.100,16.650,16.250,16.250,15.800,15.350,14.800,14.250,13.500,13.400,14.350,15.800,17.700,19.000,21.050,22.200,22.450,24.950,24.750,25.050,26.400,26.200,26.500,25.850,24.400,23.600,22.650,21.500,20.150,19.900,18.850,18.700,18.750,18.650,20.050,23.450,24.900,26.450,28.550,30.600,31.550,32.800,33.500,33.700,34.450,34.200,33.650,32.900,31.750,30.500,29.250,28.100,26.450,25.400,25.400,25.150,25.400,25.100,25.950,28.100,30.400,32.000,33.750,34.700,35.800,37.000,39.050,39.750,41.200,41.050,36.050,28.250,24.450,23.150,22.050,21.600,21.450,20.800,20.250,19.700,19.400,19.650,19.100,18.650,18.900,19.400,20.700,21.750,22.350,24.100,23.350,24.400,22.950,22.400,20.950,19.600,18.900,18.000,17.400,16.800,16.550,16.300,16.250,16.750,16.700,17.100,17.500,18.150,18.850,20.650,22.600,25.600,28.500,26.750,27.200,27.300,27.500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num_forecast_steps = 24 * 7 * 2 # Two weeks.
demand_training_data = demand[:-num_forecast_steps]

colors = sns.color_palette()
c1, c2 = colors[0], colors[1]

fig = plt.figure(figsize=(12, 6))
ax = fig.add_subplot(2, 1, 1)
ax.plot(demand_dates[:-num_forecast_steps],
        demand[:-num_forecast_steps], lw=2, label="training data")
ax.set_ylabel("Hourly demand (GW)")

ax = fig.add_subplot(2, 1, 2)

ax.plot(demand_dates[:-num_forecast_steps],
        temperature[:-num_forecast_steps], lw=2, label="training data", c=c2)
ax.set_ylabel("Temperature (deg C)")
ax.set_title("Temperature")
ax.xaxis.set_major_locator(demand_loc)
ax.xaxis.set_major_formatter(demand_fmt)
fig.suptitle("Electricity Demand in Victoria, Australia (2014)",
             fontsize=15)
fig.autofmt_xdate()

png

مدل و تناسب

مدل ما یک ساعت از روز و یک روز از هفته فصلی را با یک رگرسیون خطی مدل‌سازی اثر دما و یک فرآیند اتورگرسیو برای رسیدگی به باقیمانده‌های واریانس محدود ترکیب می‌کند.

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,
      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

همانطور که در بالا ذکر شد، مدل را با استنتاج تغییراتی مطابقت می دهیم و نمونه هایی را از قسمت عقبی ترسیم می کنیم.

demand_model = build_model(demand_training_data)

# Build the variational surrogate posteriors `qs`.
variational_posteriors = tfp.sts.build_factored_surrogate_posterior(
    model=demand_model)

از دست دادن تغییرات را به حداقل برسانید.

# Allow external control of optimization to reduce test runtimes.
num_variational_steps = 200 # @param { isTemplate: true}
num_variational_steps = int(num_variational_steps)

# Build and optimize the variational loss function.
elbo_loss_curve = tfp.vi.fit_surrogate_posterior(
    target_log_prob_fn=demand_model.joint_distribution(
        observed_time_series=demand_training_data).log_prob,
    surrogate_posterior=variational_posteriors,
    optimizer=tf.optimizers.Adam(learning_rate=0.1),
    num_steps=num_variational_steps,
    jit_compile=True)
plt.plot(elbo_loss_curve)
plt.show()

# Draw samples from the variational posterior.
q_samples_demand_ = variational_posteriors.sample(50)

png

print("Inferred parameters:")
for param in demand_model.parameters:
  print("{}: {} +- {}".format(param.name,
                              np.mean(q_samples_demand_[param.name], axis=0),
                              np.std(q_samples_demand_[param.name], axis=0)))

Inferred parameters:
observation_noise_scale: 0.010157477110624313 +- 0.0026443174574524164
hour_of_day_effect/_drift_scale: 0.0019522204529494047 +- 0.0011986979516223073
day_of_week_effect/_drift_scale: 0.013334915973246098 +- 0.01825258508324623
temperature_effect/_weights: [0.06648794] +- [0.00411669]
autoregressive/_coefficients: [0.9871232] +- [0.00413899]
autoregressive/_level_scale: 0.14199139177799225 +- 0.002658574376255274

پیش بینی و انتقاد

باز هم، ما یک پیش بینی را به سادگی با فراخوانی tfp.sts.forecast با مدل، سری زمانی و پارامترهای نمونه خود ایجاد می کنیم.

demand_forecast_dist = tfp.sts.forecast(
    model=demand_model,
    observed_time_series=demand_training_data,
    parameter_samples=q_samples_demand_,
    num_steps_forecast=num_forecast_steps)

num_samples=10

(
    demand_forecast_mean,
    demand_forecast_scale,
    demand_forecast_samples
) = (
    demand_forecast_dist.mean().numpy()[..., 0],
    demand_forecast_dist.stddev().numpy()[..., 0],
    demand_forecast_dist.sample(num_samples).numpy()[..., 0]
    )

fig, ax = plot_forecast(demand_dates, demand,
                        demand_forecast_mean,
                        demand_forecast_scale,
                        demand_forecast_samples,
                        title="Electricity demand forecast",
                        x_locator=demand_loc, x_formatter=demand_fmt)
ax.set_ylim([0, 10])
fig.tight_layout()

png

بیایید تجزیه سری مشاهده شده و پیش بینی شده را به اجزای مجزا تجسم کنیم:

# Get the distributions over component outputs from the posterior marginals on
# training data, and from the forecast model.
component_dists = sts.decompose_by_component(
    demand_model,
    observed_time_series=demand_training_data,
    parameter_samples=q_samples_demand_)

forecast_component_dists = sts.decompose_forecast_by_component(
    demand_model,
    forecast_dist=demand_forecast_dist,
    parameter_samples=q_samples_demand_)

demand_component_means_, demand_component_stddevs_ = (
    {k.name: c.mean() for k, c in component_dists.items()},
    {k.name: c.stddev() for k, c in component_dists.items()})

(
    demand_forecast_component_means_,
    demand_forecast_component_stddevs_
) = (
    {k.name: c.mean() for k, c in forecast_component_dists.items()},
    {k.name: c.stddev() for k, c in forecast_component_dists.items()}
    )

# Concatenate the training data with forecasts for plotting.
component_with_forecast_means_ = collections.OrderedDict()
component_with_forecast_stddevs_ = collections.OrderedDict()
for k in demand_component_means_.keys():
  component_with_forecast_means_[k] = np.concatenate([
      demand_component_means_[k],
      demand_forecast_component_means_[k]], axis=-1)
  component_with_forecast_stddevs_[k] = np.concatenate([
      demand_component_stddevs_[k],
      demand_forecast_component_stddevs_[k]], axis=-1)


fig, axes = plot_components(
  demand_dates,
  component_with_forecast_means_,
  component_with_forecast_stddevs_,
  x_locator=demand_loc, x_formatter=demand_fmt)
for ax in axes.values():
  ax.axvline(demand_dates[-num_forecast_steps], linestyle="--", color='red')

png

اگر بخواهیم ناهنجاری‌ها را در سری‌های مشاهده‌شده تشخیص دهیم، ممکن است به توزیع‌های پیش‌بینی یک مرحله‌ای نیز علاقه‌مند باشیم: پیش‌بینی برای هر مرحله، تنها با توجه به گام‌های زمانی تا آن نقطه. tfp.sts.one_step_predictive تمام توزیع های پیش بینی یک مرحله ای را در یک پاس محاسبه می کند:

demand_one_step_dist = sts.one_step_predictive(
    demand_model,
    observed_time_series=demand,
    parameter_samples=q_samples_demand_)

demand_one_step_mean, demand_one_step_scale = (
    demand_one_step_dist.mean().numpy(), demand_one_step_dist.stddev().numpy())

یک طرح ساده تشخیص ناهنجاری این است که تمام مراحل زمانی را که در آن مشاهدات بیش از سه stddev از مقدار پیش‌بینی‌شده فاصله دارند، علامت‌گذاری کنید - طبق مدل، این گام‌های زمانی شگفت‌انگیزترین هستند.

fig, ax = plot_one_step_predictive(
    demand_dates, demand,
    demand_one_step_mean, demand_one_step_scale,
    x_locator=demand_loc, x_formatter=demand_fmt)
ax.set_ylim(0, 10)

# Use the one-step-ahead forecasts to detect anomalous timesteps.
zscores = np.abs((demand - demand_one_step_mean) /
                 demand_one_step_scale)
anomalies = zscores > 3.0
ax.scatter(demand_dates[anomalies],
           demand[anomalies],
           c="red", marker="x", s=20, linewidth=2, label=r"Anomalies (>3$\sigma$)")
ax.plot(demand_dates, zscores, color="black", alpha=0.1, label='predictive z-score')
ax.legend()
plt.show()

png