此笔记本展示了将结构化时间序列模型拟合到时间序列并使用它们生成预测和解释的两个示例。
 在 TensorFlow.org 上查看
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 在 Google Colab 中运行
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 在 GitHub 上查看源代码
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 下载笔记本
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依赖项和先决条件
导入和设置
加速！
在深入研究之前，让我们确保我们正在使用 GPU 进行此演示。
为此，请选择“运行时”->“更改运行时类型”->“硬件加速器”->“GPU”。
以下代码段将验证我们是否可以访问 GPU。
if jax.default_backend() != 'gpu':
print('WARNING: GPU device not found.')
else:
print('SUCCESS: Found GPU.')
SUCCESS: Found GPU.
绘图设置
用于绘制时间序列和预测的辅助方法。
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(jnp.min(forecast_samples), jnp.min(y)), max(jnp.max(forecast_samples), jnp.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
莫纳罗亚 CO2 记录
我们将演示将模型拟合到来自莫纳罗亚天文台的大气 CO2 读数。
数据
# 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('320.62,321.6,322.39,323.7,324.08,323.75,322.37,320.36,318.64,318.1,319.78,321.02,322.33,322.5,323.03,324.41,325,324.09,322.54,320.92,319.25,319.39,320.72,321.95,322.57,323.15,323.89,325.02,325.57,325.36,324.14,322.11,320.33,320.25,321.32,322.89,324,324.41,325.63,326.66,327.38,326.71,325.88,323.66,322.38,321.78,322.85,324.11,325.06,325.99,326.93,328.13,328.08,327.67,326.34,324.68,323.1,323.07,324.01,325.13,326.17,326.68,327.18,327.78,328.93,328.57,327.36,325.43,323.36,323.56,324.8,326.01,326.77,327.63,327.75,329.72,330.07,329.09,328.04,326.32,324.84,325.2,326.5,327.55,328.55,329.56,330.3,331.5,332.48,332.07,330.87,329.31,327.52,327.19,328.16,328.65,329.36,330.71,331.49,332.65,333.1,332.26,331.18,329.4,327.44,327.38,328.46,329.58,330.41,331.41,332.05,333.32,333.98,333.62,331.91,330.06,328.57,328.35,329.5,330.77,331.76,332.58,333.5,334.59,334.89,334.34,333.06,330.95,329.31,328.95,330.32,331.69,332.94,333.43,334.71,336.08,336.76,336.28,334.93,332.76,331.6,331.17,332.41,333.86,334.98,335.4,336.65,337.76,338.02,337.91,336.55,334.69,332.77,332.56,333.93,334.96,336.24,336.77,337.97,338.89,339.48,339.3,337.74,336.1,333.93,333.87,335.3,336.74,338.03,338.37,340.09,340.78,341.48,341.19,339.57,337.61,335.9,336.03,337.12,338.23,339.25,340.5,341.4,342.52,342.93,342.27,340.5,338.45,336.71,336.88,338.38,339.63,340.77,341.63,342.72,343.59,344.16,343.37,342.07,339.83,338,337.88,339.28,340.51,341.4,342.54,343.12,344.96,345.78,345.34,344,342.4,339.89,340.01,341.16,342.98,343.82,344.62,345.38,347.15,347.52,346.88,345.47,343.34,341.13,341.4,343.02,344.25,344.99,346.01,347.43,348.34,348.92,348.24,346.54,344.64,343.06,342.78,344.21,345.53,346.28,346.93,347.83,349.53,350.19,349.54,347.92,345.88,344.83,344.15,345.64,346.88,348,348.47,349.41,350.97,351.84,351.25,349.5,348.08,346.44,346.1,347.54,348.69,350.16,351.47,351.96,353.33,353.97,353.55,352.14,350.19,348.5,348.66,349.85,351.12,352.55,352.86,353.48,355.21,355.47,354.92,353.7,351.47,349.61,349.79,351.09,352.32,353.46,354.5,355.19,356,356.96,356.04,354.62,352.71,350.77,350.99,352.64,354.02,354.53,355.55,356.96,358.4,359.14,358.04,355.98,353.81,351.95,352.02,353.55,354.79,355.79,356.52,357.61,358.95,359.46,359.05,356.82,354.8,352.81,353.11,353.96,355.2,356.5,356.97,358.18,359.26,360.08,359.4,357.38,355.33,353.5,353.8,355.15,356.62,358.19,358.73,359.79,361.09,361.51,360.78,359.38,357.31,355.68,355.83,357.42,358.87,359.81,360.84,361.48,363.3,363.64,363.11,361.75,359.31,357.91,357.62,359.42,360.56,361.91,363.11,363.89,364.58,365.29,364.84,363.52,361.35,359.32,359.48,360.64,362.21,363.06,363.87,364.44,366.23,366.68,365.52,364.36,362.39,360.08,360.67,362.32,364.17,365.22,366.04,367.2,368.5,369.19,368.77,367.53,365.67,363.8,364.13,365.36,366.87,368.05,368.77,369.49,371.04,370.9,370.25,369.17,366.83,364.54,365.04,366.58,367.92,369.05,369.37,370.42,371.57,371.74,371.6,370.02,368.03,366.53,366.64,368.2,369.44,370.2,371.42,372.04,372.78,373.94,373.23,371.54,369.47,367.88,368.02,369.6,371.16,372.36,373,373.44,374.77,375.48,375.33,373.95,371.41,370.63,370.18,372.01,373.71,374.61,375.55,376.04,377.58,378.28,378.07,376.54,374.42,372.92,372.94,374.29,375.63,376.73,377.31,378.33,380.44,380.56,379.49,377.71,375.77,373.99,374.17,375.79,377.39,378.29,379.56,380.07,382.01,382.21,382.05,380.63,378.64,376.38,376.77,378.27,379.92,381.33,381.98,382.53,384.33,384.89,384,382.25,380.44,378.77,379.03,380.11,381.63,382.55,383.68,384.31,386.2,386.38,385.85,384.42,381.81,380.83,380.83,382.32,383.58,385.04,385.81,385.8,386.74,388.48,388.02,386.22,384.05,383.05,382.75,383.98,385.08,386.63,387.1,388.5,389.54,390.15,389.6,388.05,386.06,384.64,384.32,386.05,387.48,388.55,390.08,391.02,392.39,393.24,392.26,390.35,388.53,386.85,387.18,388.69,389.83,391.33,391.96,392.49,393.4,394.33,393.75,392.64,390.25,389.05,388.98,390.3,391.86,393.13,393.42,394.43,396.51,396.96,395.97,394.6,392.61,391.2,391.09,393.03,394.42,395.69,396.94,397.35,398.44,400.06,398.96,397.45,395.49,393.47,393.77,395.27,396.9,398.01,398.18,399.56,401.44,401.98,401.41,399.17,397.3,395.49,395.74,397.32,398.88,399.94,400.4,401.6,403.52,404.03,402.81,401.54,398.93,397.43,398.22,400.17,401.82,402.58,404.09,404.79,407.5,407.59,406.94,404.43,402.17,400.95,401.43,403.57,404.48,406,406.57,406.99,408.88,409.84,409.05,407.13,405.17,403.2,403.57,405.1,406.68,407.98,408.36,409.21,410.24,411.23,410.81,408.83,407.02,405.53,405.93,408.04,409.17,410.85,411.59,411.91,413.46,414.76,413.89,411.78,410.01,408.48,408.4,410.16,411.81,413.3,414.05,414.45,416.11,417.15,416.29,414.42,412.52,411.18,411.12,412.88,413.89,415.15,416.47,417.16,418.24,418.95,418.7,416.65,414.34,412.91,413.55,414.82,416.43,418.01,418.99,418.45,420.02,420.77,420.68,418.68,416.76,415.41,415.31'.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", "2022-11", 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()
模型和拟合
我们将使用局部线性趋势加上一年中的月份季节性效应对该序列进行建模。
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
我们将使用变分推理来拟合模型。这涉及运行优化器以最小化变分损失函数，即负证据下界 (ELBO)。这将为参数拟合一组近似后验分布（在实践中，我们假设这些分布是独立的正态分布，并转换为每个参数的支持空间）。
tfp.sts 预测方法需要后验样本作为输入，因此我们将通过从变分后验中抽取一组样本来结束。
co2_model = build_model(co2_by_month_training_data)
# Build the variational surrogate posteriors `qs`.
init_fn, build_surrogate_fn = (
tfp.sts.build_factored_surrogate_posterior_stateless(model=co2_model))
最小化变分损失。
# Allow external control of optimization to reduce test runtimes.
num_variational_steps = 200 # @param { isTemplate: true}
num_variational_steps = int(num_variational_steps)
seed = jax.random.PRNGKey(42)
init_seed, fit_seed, sample_seed = jax.random.split(seed, 3)
initial_parameters = init_fn(init_seed)
jd = co2_model.joint_distribution(co2_by_month_training_data)
# Build and optimize the variational loss function.
optimized_parameters, elbo_loss_curve = tfp.vi.fit_surrogate_posterior_stateless(
target_log_prob_fn=jd.log_prob,
initial_parameters=initial_parameters,
build_surrogate_posterior_fn=build_surrogate_fn,
optimizer=optax.adam(0.1),
num_steps=num_variational_steps,
seed=fit_seed)
plt.plot(elbo_loss_curve)
plt.show()
# Draw samples from the variational posterior.
variational_posteriors = build_surrogate_fn(optimized_parameters)
q_samples_co2_ = variational_posteriors.sample(50, seed=sample_seed)
print("Inferred parameters:")
for param in co2_model.parameters:
print("{}: {} +- {}".format(param.name,
jnp.mean(q_samples_co2_[param.name], axis=0),
jnp.std(q_samples_co2_[param.name], axis=0)))
Inferred parameters: observation_noise_scale: 0.1685197800397873 +- 0.007219966035336256 LocalLinearTrend_level_scale: 0.18049846589565277 +- 0.011273686774075031 LocalLinearTrend_slope_scale: 0.009398984722793102 +- 0.0021420123521238565 Seasonal_drift_scale: 0.03475992754101753 +- 0.005793483462184668
预测和批评
现在让我们使用拟合的模型来构建预测。我们只需调用 tfp.sts.forecast，它将返回一个 TensorFlow Distribution 实例，表示未来时间步长的预测分布。
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)
特别是，预测分布的 mean 和 stddev 为我们提供了在每个时间步长具有边缘不确定性的预测，我们还可以绘制可能的未来的样本。
num_samples=10
co2_forecast_mean, co2_forecast_scale, co2_forecast_samples = (
co2_forecast_dist.mean()[..., 0],
co2_forecast_dist.stddev()[..., 0],
co2_forecast_dist.sample(num_samples, seed=sample_seed)[..., 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()
我们可以通过将模型分解为各个时间序列的贡献来进一步了解模型的拟合情况
# 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)
电力需求预测
现在让我们考虑一个更复杂的示例：预测澳大利亚维多利亚州的电力需求。
首先，我们将构建数据集
# 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()
模型和拟合
我们的模型结合了每日和每周的季节性，使用线性回归对温度的影响进行建模，并使用自回归过程来处理有界方差残差。
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=jnp.reshape(temperature - jnp.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
与上面一样，我们将使用变分推理来拟合模型，并从后验中抽取样本。
demand_model = build_model(demand_training_data)
# Build the variational surrogate posteriors `qs`.
# variational_posteriors = tfp.sts.build_factored_surrogate_posterior(
# model=demand_model)
init_fn, build_surrogate_fn = (
tfp.sts.build_factored_surrogate_posterior_stateless(model=demand_model))
最小化变分损失。
# Allow external control of optimization to reduce test runtimes.
num_variational_steps = 200 # @param { isTemplate: true}
num_variational_steps = int(num_variational_steps)
seed = tfp.random.sanitize_seed(jax.random.PRNGKey(42), salt='fit_stateless')
init_seed, fit_seed, sample_seed = tfp.random.split_seed(seed, n=3)
initial_parameters = init_fn(init_seed)
jd = demand_model.joint_distribution(demand_training_data)
# Build and optimize the variational loss function.
optimized_parameters, elbo_loss_curve = tfp.vi.fit_surrogate_posterior_stateless(
target_log_prob_fn=jd.log_prob,
initial_parameters=initial_parameters,
build_surrogate_posterior_fn=build_surrogate_fn,
optimizer=optax.adam(learning_rate=0.1),
num_steps=num_variational_steps,
seed=fit_seed)
plt.plot(elbo_loss_curve)
plt.show()
# Draw samples from the variational posterior.
surrogate_posterior = build_surrogate_fn(optimized_parameters)
q_samples_demand_ = surrogate_posterior.sample(50, seed=sample_seed)
print("Inferred parameters:")
for param in demand_model.parameters:
print("{}: {} +- {}".format(param.name,
jnp.mean(q_samples_demand_[param.name], axis=0),
jnp.std(q_samples_demand_[param.name], axis=0)))
Inferred parameters: observation_noise_scale: 0.007361860014498234 +- 0.001575619913637638 hour_of_day_effect_drift_scale: 0.002189201768487692 +- 0.0007748314528726041 day_of_week_effect_drift_scale: 0.01211678609251976 +- 0.018613168969750404 temperature_effect_weights: [0.06205687] +- [0.00406887] autoregressive_coefficients: [0.9839599] +- [0.00560341] autoregressive_level_scale: 0.14477692544460297 +- 0.003696543164551258
预测和批评
同样，我们通过调用 tfp.sts.forecast 并传入我们的模型、时间序列和采样参数来创建预测。
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_dist.mean()[..., 0]
demand_forecast_scale = demand_forecast_dist.stddev()[..., 0]
demand_forecast_samples =demand_forecast_dist.sample(
num_samples, seed=sample_seed)[..., 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()
让我们将观察到的序列和预测序列分解为各个组件进行可视化
# 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] = jnp.concatenate([
demand_component_means_[k],
demand_forecast_component_means_[k]], axis=-1)
component_with_forecast_stddevs_[k] = jnp.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')
如果我们想检测观察到的序列中的异常，我们可能还会对单步预测分布感兴趣：给定到该点为止的时间步长，对每个时间步长的预测。 tfp.sts.one_step_predictive 在一次传递中计算所有单步预测分布
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(), demand_one_step_dist.stddev())
一个简单的异常检测方案是标记所有观察值与预测值相差超过三个标准差的时间步长——这些是根据模型最“令人惊讶”的时间步长。
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 = jnp.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()
