이 노트북은 구조적 시계열 모델을 시계열에 맞추고 예측 및 설명을 생성하는 데 사용하는 두 가지 예를 보여줍니다.
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종속성 및 전제 조건
가져오기 및 설정
%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()
일을 빨리 만드십시오!
본격적으로 시작하기 전에 이 데모에 GPU를 사용하고 있는지 확인하겠습니다.
이렇게 하려면 "런타임" -> "런타임 유형 변경" -> "하드웨어 가속기" -> "GPU"를 선택합니다.
다음 스니펫은 GPU에 대한 액세스 권한이 있는지 확인합니다.
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
플로팅 설정
시계열 및 예측을 플로팅하기 위한 도우미 메서드입니다.
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
마우나로아 CO2 기록
우리는 마우나 로아 천문대의 대기 CO2 수치에 모델을 맞추는 것을 시연할 것입니다.
데이터
# 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()
모델 및 피팅
지역 선형 추세와 월별 계절 효과를 사용하여 이 시리즈를 모델링합니다.
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
우리는 변형 추론을 사용하여 모델을 맞출 것입니다. 여기에는 옵티마이저를 실행하여 변이 손실 함수인 ELBO(음의 증거 하한)를 최소화하는 작업이 포함됩니다. 이것은 매개변수에 대한 대략적인 사후 분포 세트에 맞습니다(실제로 우리는 이것이 각 매개변수의 지원 공간으로 변환된 독립적인 법선이라고 가정합니다).
tfp.sts
예측 방법에는 입력으로 사후 샘플이 필요하므로 가변 사후에서 샘플 세트를 그리는 것으로 마무리하겠습니다.
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)
변동 손실을 최소화합니다.
# 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.
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
예측과 비판
이제 적합 모델을 사용하여 예측을 구성해 보겠습니다. tfp.sts.forecast
를 호출하면 미래 시간 단계에 대한 예측 분포를 나타내는 TensorFlow Distribution 인스턴스를 반환합니다.
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)
특히 예측 분포의 mean
과 stddev
는 각 시간 단계에서 한계 불확실성이 있는 예측을 제공하고 가능한 미래의 샘플을 추출할 수도 있습니다.
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()
모델을 개별 시계열의 기여도로 분해하여 모델의 적합성을 더 이해할 수 있습니다.
# 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)
전력 수요 예측
이제 더 복잡한 예를 살펴보겠습니다. 빅토리아 오스트레일리아의 전력 수요 예측입니다.
먼저 데이터 세트를 빌드합니다.
# 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()
모델 및 피팅
우리 모델은 시간 및 요일 계절성과 온도의 영향을 모델링하는 선형 회귀 및 유계 분산 잔차를 처리하기 위한 자기회귀 프로세스를 결합합니다.
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
위와 같이 모델을 변형 추론으로 맞추고 사후 샘플을 추출합니다.
demand_model = build_model(demand_training_data)
# Build the variational surrogate posteriors `qs`.
variational_posteriors = tfp.sts.build_factored_surrogate_posterior(
model=demand_model)
변동 손실을 최소화합니다.
# 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)
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
예측과 비판
다시 말하지만, 모델, 시계열 및 샘플링된 매개변수를 사용하여 tfp.sts.forecast
를 호출하여 예측을 생성합니다.
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()
관측 및 예측 계열을 개별 구성 요소로 분해하는 것을 시각화해 보겠습니다.
# 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')
관찰된 시리즈에서 이상을 탐지하려면 1단계 예측 분포에 관심이 있을 수도 있습니다. 즉, 해당 시점까지의 시간 단계만 주어진 각 시간 단계에 대한 예측입니다. tfp.sts.one_step_predictive
는 단일 패스에서 모든 1단계 예측 분포를 계산합니다.
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())
간단한 이상 탐지 체계는 관측치가 예측된 값에서 3개 이상의 표준 편차인 모든 시간 단계에 플래그를 지정하는 것입니다. 이는 모델에 따라 가장 '놀라운' 시간 단계입니다.
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()