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這個筆記本示範如何使用 TFP 近似推論工具，在擬合和預測結構性時間序列 (STS) 模型時，納入 (非高斯分佈) 觀測模型。在這個範例中，我們將使用 Poisson 觀測模型來處理離散計數資料。
import time
import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
import tf_keras
import tensorflow_probability as tfp
from tensorflow_probability import bijectors as tfb
from tensorflow_probability import distributions as tfd
合成資料
首先，我們將產生一些合成計數資料。
num_timesteps = 30
observed_counts = np.round(3 + np.random.lognormal(np.log(np.linspace(
num_timesteps, 5, num=num_timesteps)), 0.20, size=num_timesteps))
observed_counts = observed_counts.astype(np.float32)
plt.plot(observed_counts)
[<matplotlib.lines.Line2D at 0x7f940ae958d0>]
模型
我們將指定一個具有隨機漫步線性趨勢的簡單模型。
def build_model(approximate_unconstrained_rates):
trend = tfp.sts.LocalLinearTrend(
observed_time_series=approximate_unconstrained_rates)
return tfp.sts.Sum([trend],
observed_time_series=approximate_unconstrained_rates)
這個模型將作用於控制觀測的 Poisson 率參數序列，而不是作用於觀測到的時間序列。
由於 Poisson 率必須為正值，我們將使用雙射函數將實值 STS 模型轉換為正值的分佈。Softplus 轉換 \(y = \log(1 + \exp(x))\) 是一個自然的選擇，因為它對於正值幾乎是線性的，但其他選擇，例如 Exp (將常態隨機漫步轉換為對數常態隨機漫步) 也是可能的。
positive_bijector = tfb.Softplus() # Or tfb.Exp()
# Approximate the unconstrained Poisson rate just to set heuristic priors.
# We could avoid this by passing explicit priors on all model params.
approximate_unconstrained_rates = positive_bijector.inverse(
tf.convert_to_tensor(observed_counts) + 0.01)
sts_model = build_model(approximate_unconstrained_rates)
為了對非高斯分佈觀測模型使用近似推論，我們將把 STS 模型編碼為 TFP JointDistribution。此聯合分佈中的隨機變數是 STS 模型的參數、潛在 Poisson 率的時間序列和觀測到的計數。
def sts_with_poisson_likelihood_model():
# Encode the parameters of the STS model as random variables.
param_vals = []
for param in sts_model.parameters:
param_val = yield param.prior
param_vals.append(param_val)
# Use the STS model to encode the log- (or inverse-softplus)
# rate of a Poisson.
unconstrained_rate = yield sts_model.make_state_space_model(
num_timesteps, param_vals)
rate = positive_bijector.forward(unconstrained_rate[..., 0])
observed_counts = yield tfd.Poisson(rate, name='observed_counts')
model = tfd.JointDistributionCoroutineAutoBatched(sts_with_poisson_likelihood_model)
推論準備
我們想要根據觀測到的計數，推斷模型中未觀測到的量。首先，我們將聯合對數密度以觀測到的計數為條件。
pinned_model = model.experimental_pin(observed_counts=observed_counts)
我們還需要一個約束雙射函數，以確保推論尊重 STS 模型參數的約束 (例如，尺度必須為正值)。
constraining_bijector = pinned_model.experimental_default_event_space_bijector()
使用 HMC 進行推論
我們將使用 HMC (特別是 NUTS) 從模型參數和潛在率的聯合後驗中取樣。
這將比使用 HMC 擬合標準 STS 模型慢得多，因為除了模型的 (相對較少數量的) 參數外，我們還必須推斷整個 Poisson 率序列。因此，我們將運行相對較少的步驟；對於推論品質至關重要的應用，增加這些值或運行多個鏈可能是有意義的。
取樣器配置
首先，我們指定一個取樣器，然後使用 sample_chain 運行該取樣核心以產生樣本。
sampler = tfp.mcmc.TransformedTransitionKernel(
tfp.mcmc.NoUTurnSampler(
target_log_prob_fn=pinned_model.unnormalized_log_prob,
step_size=0.1),
bijector=constraining_bijector)
adaptive_sampler = tfp.mcmc.DualAveragingStepSizeAdaptation(
inner_kernel=sampler,
num_adaptation_steps=int(0.8 * num_burnin_steps),
target_accept_prob=0.75)
initial_state = constraining_bijector.forward(
type(pinned_model.event_shape)(
*(tf.random.normal(part_shape)
for part_shape in constraining_bijector.inverse_event_shape(
pinned_model.event_shape))))
# Speed up sampling by tracing with `tf.function`.
@tf.function(autograph=False, jit_compile=True)
def do_sampling():
return tfp.mcmc.sample_chain(
kernel=adaptive_sampler,
current_state=initial_state,
num_results=num_results,
num_burnin_steps=num_burnin_steps,
trace_fn=None)
t0 = time.time()
samples = do_sampling()
t1 = time.time()
print("Inference ran in {:.2f}s.".format(t1-t0))
Inference ran in 24.83s.
我們可以透過檢查參數軌跡來對推論進行健全性檢查。在這種情況下，它們似乎已經探索了資料的多種解釋，這很好，儘管更多的樣本將有助於判斷鏈的混合程度。
f = plt.figure(figsize=(12, 4))
for i, param in enumerate(sts_model.parameters):
ax = f.add_subplot(1, len(sts_model.parameters), i + 1)
ax.plot(samples[i])
ax.set_title("{} samples".format(param.name))
現在是回報：讓我們看看 Poisson 率的後驗分佈！我們還將繪製觀測計數的 80% 預測區間，並且可以檢查此區間是否包含我們實際觀測到的大約 80% 的計數。
param_samples = samples[:-1]
unconstrained_rate_samples = samples[-1][..., 0]
rate_samples = positive_bijector.forward(unconstrained_rate_samples)
plt.figure(figsize=(10, 4))
mean_lower, mean_upper = np.percentile(rate_samples, [10, 90], axis=0)
pred_lower, pred_upper = np.percentile(np.random.poisson(rate_samples),
[10, 90], axis=0)
_ = plt.plot(observed_counts, color="blue", ls='--', marker='o', label='observed', alpha=0.7)
_ = plt.plot(np.mean(rate_samples, axis=0), label='rate', color="green", ls='dashed', lw=2, alpha=0.7)
_ = plt.fill_between(np.arange(0, 30), mean_lower, mean_upper, color='green', alpha=0.2)
_ = plt.fill_between(np.arange(0, 30), pred_lower, pred_upper, color='grey', label='counts', alpha=0.2)
plt.xlabel("Day")
plt.ylabel("Daily Sample Size")
plt.title("Posterior Mean")
plt.legend()
<matplotlib.legend.Legend at 0x7f93ffd35550>
預測
為了預測觀測到的計數，我們將使用標準 STS 工具來建立潛在率的預測分佈 (在無約束空間中，再次因為 STS 旨在對實值資料建模)，然後將取樣的預測傳遞到 Poisson 觀測模型中。
def sample_forecasted_counts(sts_model, posterior_latent_rates,
posterior_params, num_steps_forecast,
num_sampled_forecasts):
# Forecast the future latent unconstrained rates, given the inferred latent
# unconstrained rates and parameters.
unconstrained_rates_forecast_dist = tfp.sts.forecast(sts_model,
observed_time_series=unconstrained_rate_samples,
parameter_samples=posterior_params,
num_steps_forecast=num_steps_forecast)
# Transform the forecast to positive-valued Poisson rates.
rates_forecast_dist = tfd.TransformedDistribution(
unconstrained_rates_forecast_dist,
positive_bijector)
# Sample from the forecast model following the chain rule:
# P(counts) = P(counts | latent_rates)P(latent_rates)
sampled_latent_rates = rates_forecast_dist.sample(num_sampled_forecasts)
sampled_forecast_counts = tfd.Poisson(rate=sampled_latent_rates).sample()
return sampled_forecast_counts, sampled_latent_rates
forecast_samples, rate_samples = sample_forecasted_counts(
sts_model,
posterior_latent_rates=unconstrained_rate_samples,
posterior_params=param_samples,
# Days to forecast:
num_steps_forecast=30,
num_sampled_forecasts=100)
forecast_samples = np.squeeze(forecast_samples)
def plot_forecast_helper(data, forecast_samples, CI=90):
"""Plot the observed time series alongside the forecast."""
plt.figure(figsize=(10, 4))
forecast_median = np.median(forecast_samples, axis=0)
num_steps = len(data)
num_steps_forecast = forecast_median.shape[-1]
plt.plot(np.arange(num_steps), data, lw=2, color='blue', linestyle='--', marker='o',
label='Observed Data', alpha=0.7)
forecast_steps = np.arange(num_steps, num_steps+num_steps_forecast)
CI_interval = [(100 - CI)/2, 100 - (100 - CI)/2]
lower, upper = np.percentile(forecast_samples, CI_interval, axis=0)
plt.plot(forecast_steps, forecast_median, lw=2, ls='--', marker='o', color='orange',
label=str(CI) + '% Forecast Interval', alpha=0.7)
plt.fill_between(forecast_steps,
lower,
upper, color='orange', alpha=0.2)
plt.xlim([0, num_steps+num_steps_forecast])
ymin, ymax = min(np.min(forecast_samples), np.min(data)), max(np.max(forecast_samples), np.max(data))
yrange = ymax-ymin
plt.title("{}".format('Observed time series with ' + str(num_steps_forecast) + ' Day Forecast'))
plt.xlabel('Day')
plt.ylabel('Daily Sample Size')
plt.legend()
plot_forecast_helper(observed_counts, forecast_samples, CI=80)
VI 推論
當推斷完整時間序列時，變分推論可能會出現問題，例如我們的近似計數 (而不是僅僅是時間序列的參數，如標準 STS 模型中)。變數具有獨立後驗的標準假設是完全錯誤的，因為每個時間步都與其鄰居相關，這可能導致低估不確定性。因此，對於完整時間序列的近似推論，HMC 可能是一個更好的選擇。但是，VI 可能會快很多，並且可能適用於模型原型設計或在其效能可以經驗證明「足夠好」的情況下。
為了使用 VI 擬合我們的模型，我們只需構建和最佳化替代後驗。
surrogate_posterior = tfp.experimental.vi.build_factored_surrogate_posterior(
event_shape=pinned_model.event_shape,
bijector=constraining_bijector)
# Allow external control of optimization to reduce test runtimes.
num_variational_steps = 1000 # @param { isTemplate: true}
num_variational_steps = int(num_variational_steps)
t0 = time.time()
losses = tfp.vi.fit_surrogate_posterior(pinned_model.unnormalized_log_prob,
surrogate_posterior,
optimizer=tf_keras.optimizers.Adam(0.1),
num_steps=num_variational_steps)
t1 = time.time()
print("Inference ran in {:.2f}s.".format(t1-t0))
Inference ran in 11.37s.
plt.plot(losses)
plt.title("Variational loss")
_ = plt.xlabel("Steps")
posterior_samples = surrogate_posterior.sample(50)
param_samples = posterior_samples[:-1]
unconstrained_rate_samples = posterior_samples[-1][..., 0]
rate_samples = positive_bijector.forward(unconstrained_rate_samples)
plt.figure(figsize=(10, 4))
mean_lower, mean_upper = np.percentile(rate_samples, [10, 90], axis=0)
pred_lower, pred_upper = np.percentile(
np.random.poisson(rate_samples), [10, 90], axis=0)
_ = plt.plot(observed_counts, color='blue', ls='--', marker='o',
label='observed', alpha=0.7)
_ = plt.plot(np.mean(rate_samples, axis=0), label='rate', color='green',
ls='dashed', lw=2, alpha=0.7)
_ = plt.fill_between(
np.arange(0, 30), mean_lower, mean_upper, color='green', alpha=0.2)
_ = plt.fill_between(np.arange(0, 30), pred_lower, pred_upper, color='grey',
label='counts', alpha=0.2)
plt.xlabel('Day')
plt.ylabel('Daily Sample Size')
plt.title('Posterior Mean')
plt.legend()
<matplotlib.legend.Legend at 0x7f93ff4735c0>
forecast_samples, rate_samples = sample_forecasted_counts(
sts_model,
posterior_latent_rates=unconstrained_rate_samples,
posterior_params=param_samples,
# Days to forecast:
num_steps_forecast=30,
num_sampled_forecasts=100)
forecast_samples = np.squeeze(forecast_samples)
plot_forecast_helper(observed_counts, forecast_samples, CI=80)
