Bu defter, yapısal zaman serisi modellerinin zaman serilerine uydurulmasına ve bunların tahminler ve açıklamalar oluşturmak için kullanılmasına ilişkin iki örneği göstermektedir.
 TensorFlow.org'da görüntüleyin |  Google Colab'da çalıştırın |  Kaynağı GitHub'da görüntüleyin |  Not defterini indir |
Bağımlılıklar ve Ön Koşullar
İthalat ve kurulumlar
%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()
İşleri Hızlandırın!
Dalmadan önce, bu demo için bir GPU kullandığımızdan emin olalım.
Bunu yapmak için "Çalışma Zamanı" -> "Çalışma zamanı türünü değiştir" -> "Donanım hızlandırıcı" -> "GPU" öğesini seçin.
Aşağıdaki kod parçası, bir GPU'ya erişimimiz olduğunu doğrulayacaktır.
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
Çizim kurulumu
Zaman serileri ve tahminleri çizmek için yardımcı yöntemler.
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
Mauna Loa CO2 kaydı
Mauna Loa gözlemevinden alınan atmosferik CO2 okumalarına bir model uydurmayı göstereceğiz.
Veri
# 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()
Model ve Montaj
Bu seriyi yerel bir lineer trend ve ayrıca bir aylık mevsimsel etki ile modelleyeceğiz.
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
Varyasyon çıkarımını kullanarak modele uyacağız. Bu, negatif kanıt alt sınırı (ELBO) olan değişken kayıp fonksiyonunu en aza indirmek için bir optimize edicinin çalıştırılmasını içerir. Bu, parametreler için bir dizi yaklaşık sonsal dağılıma uyar (pratikte bunların her parametrenin destek alanına dönüştürülmüş bağımsız Normaller olduğunu varsayıyoruz).
tfp.sts tahmin yöntemleri girdi olarak sonsal örnekler gerektirir, bu nedenle varyasyon sonsalından bir dizi örnek çizerek bitireceğiz.
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)
Varyasyon kaybını en aza indirin.
# 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
Tahmin ve eleştiri
Şimdi bir tahmin oluşturmak için uygun modeli kullanalım. Biz sadece, gelecekteki zaman adımlarında tahmine dayalı dağılımı temsil eden bir TensorFlow Distribution örneği döndüren tfp.sts.forecast çağırırız.
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)
Özellikle, tahmin dağılımının mean ve stddev bize her zaman adımında marjinal belirsizliğe sahip bir tahmin verir ve ayrıca olası gelecek örnekleri de çizebiliriz.
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()
Modelin uyumunu, onu bireysel zaman serilerinin katkılarına ayırarak daha iyi anlayabiliriz:
# 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)
Elektrik talep tahmini
Şimdi daha karmaşık bir örneği ele alalım: Victoria Avustralya'daki elektrik talebini tahmin etmek.
İlk olarak, veri kümesini oluşturacağız:
# 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()
Model ve uydurma
Modelimiz, günün saati ve haftanın günü mevsimselliğini, sıcaklığın etkisini modelleyen doğrusal bir regresyon ve sınırlı varyans artıklarını işlemek için otoregresif bir süreçle birleştirir.
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
Yukarıdaki gibi, modele varyasyon çıkarımı uygulayacağız ve arkadan örnekler çizeceğiz.
demand_model = build_model(demand_training_data)
# Build the variational surrogate posteriors `qs`.
variational_posteriors = tfp.sts.build_factored_surrogate_posterior(
model=demand_model)
Varyasyon kaybını en aza indirin.
# 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
Tahmin ve eleştiri
Yine, modelimiz, zaman serimiz ve örneklenmiş parametrelerimizle tfp.sts.forecast çağırarak bir tahmin oluşturuyoruz.
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()
Gözlenen ve tahmin serilerinin ayrı ayrı bileşenlere ayrıştırılmasını görselleştirelim:
# 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')
Gözlenen serilerdeki anormallikleri tespit etmek istiyorsak, tek adımlı tahmin dağılımlarıyla da ilgilenebiliriz: yalnızca o noktaya kadar olan zaman adımları verilen her bir zaman adımı için tahmin. tfp.sts.one_step_predictive , tüm tek adımlı tahmine dayalı dağılımları tek geçişte hesaplar:
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())
Basit bir anormallik tespit şeması, gözlemlerin tahmin edilen değerden üç stddev'den fazla olduğu tüm zaman adımlarını işaretlemektir - bunlar modele göre en 'şaşırtıcı' zaman adımlarıdır.
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()
