建模欧洲的 COVID-19 传播及其干预措施的效果

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在 TensorFlow.org 上查看 在 Google Colab 中运行 在 GitHub 上查看源代码 下载笔记本

为了减缓 2020 年初 COVID-19 的传播，欧洲国家采取了诸如关闭非必要企业、隔离个别病例、旅行禁令和其他鼓励社会疏远的措施等非药物干预措施。Imperial College COVID-19 应对小组在其论文 "Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries" 中分析了这些措施的有效性，使用贝叶斯层次模型结合机械流行病学模型。

此 Colab 包含该分析的 TensorFlow Probability (TFP) 实现，组织如下

  • "模型设置" 定义疾病传播和由此导致的死亡的流行病学模型、模型参数上的贝叶斯先验分布以及条件于参数值死亡人数的分布。
  • "数据预处理" 加载每个国家/地区干预措施的时间和类型、随时间推移的死亡人数以及受感染者的估计死亡率数据。
  • "模型推理" 构建贝叶斯层次模型并运行哈密顿蒙特卡罗 (HMC) 以从参数的后验分布中进行采样。
  • "结果" 显示感兴趣量的后验预测分布，例如预测死亡人数以及在没有干预措施的情况下反事实死亡人数。

该论文发现有证据表明各国已设法减少每位受感染者传播的新感染人数 (\(R_t\))，但可信区间包含 \(R_t=1\)（高于该值时疫情将继续蔓延），并且现在就干预措施的有效性得出明确结论还为时过早。该论文的 Stan 代码位于作者的 Github 存储库中，此 Colab 再现了 版本 2。

pip3 install -q git+git://github.com/arviz-devs/arviz.git
pip3 install -q tf-nightly tfp-nightly

导入

1 模型设置

1.1 感染和死亡的机械模型

感染模型模拟每个国家/地区随时间推移的感染人数。输入数据是干预措施的时间和类型、人口规模和初始病例。参数控制干预措施的有效性和疾病传播率。死亡人数的预期值模型将死亡率应用于预测的感染人数。

感染模型对先前的每日感染人数与序列间隔分布（受感染者与感染他人之间天数的分布）进行卷积。在每个时间步长，时间 \(t\) 的新感染人数 \(n_t\) 计算为

\begin{equation} \sum_{i=0}^{t-1} n_i \mu_t \text{p} (\text{caught from someone infected at } i | \text{newly infected at } t) \end{equation} 其中 \(\mu_t=R_t\) 并且条件概率存储在下面定义的 conv_serial_interval 中。

预期死亡模型对每日感染人数和感染与死亡之间的时间分布进行卷积。也就是说，第 \(t\) 天的预期死亡人数计算如下

\begin{equation} \sum_{i=0}^{t-1} n_i\text{p(第 \(t\) 天死亡|第 \(i\) 天感染)} \end{equation} 其中条件概率存储在 conv_fatality_rate 中，定义如下。

from tensorflow_probability.python.internal import broadcast_util as bu

def predict_infections(
    intervention_indicators, population, initial_cases, mu, alpha_hier,
    conv_serial_interval, initial_days, total_days):
  """Predict the number of infections by forward-simulation.

  Args:
    intervention_indicators: Binary array of shape
      `[num_countries, total_days, num_interventions]`, in which `1` indicates
      the intervention is active in that country at that time and `0` indicates
      otherwise.
    population: Vector of length `num_countries`. Population of each country.
    initial_cases: Array of shape `[batch_size, num_countries]`. Number of cases
      in each country at the start of the simulation.
    mu: Array of shape `[batch_size, num_countries]`. Initial reproduction rate
      (R_0) by country.
    alpha_hier: Array of shape `[batch_size, num_interventions]` representing
      the effectiveness of interventions.
    conv_serial_interval: Array of shape
      `[total_days - initial_days, total_days]` output from
      `make_conv_serial_interval`. Convolution kernel for serial interval
      distribution.
    initial_days: Integer, number of sequential days to seed infections after
      the 10th death in a country. (N0 in the authors' Stan code.)
    total_days: Integer, number of days of observed data plus days to forecast.
      (N2 in the authors' Stan code.)
  Returns:
    predicted_infections: Array of shape
      `[total_days, batch_size, num_countries]`. (Batched) predicted number of
      infections over time and by country.
  """
  alpha = alpha_hier - tf.cast(np.log(1.05) / 6.0, DTYPE)

  # Multiply the effectiveness of each intervention in each country (alpha)
  # by the indicator variable for whether the intervention was active and sum
  # over interventions, yielding an array of shape
  # [total_days, batch_size, num_countries] that represents the total effectiveness of
  # all interventions in each country on each day (for a batch of data).
  linear_prediction = tf.einsum(
      'ijk,...k->j...i', intervention_indicators, alpha)

  # Adjust the reproduction rate per country downward, according to the
  # effectiveness of the interventions.
  rt = mu * tf.exp(-linear_prediction, name='reproduction_rate')

  # Initialize storage array for daily infections and seed it with initial
  # cases.
  daily_infections = tf.TensorArray(
      dtype=DTYPE, size=total_days, element_shape=initial_cases.shape)
  for i in range(initial_days):
    daily_infections = daily_infections.write(i, initial_cases)

  # Initialize cumulative cases.
  init_cumulative_infections = initial_cases * initial_days

  # Simulate forward for total_days days.
  cond = lambda i, *_: i < total_days
  def body(i, prev_daily_infections, prev_cumulative_infections):
    # The probability distribution over days j that someone infected on day i
    # caught the virus from someone infected on day j.
    p_infected_on_day = tf.gather(
        conv_serial_interval, i - initial_days, axis=0)

    # Multiply p_infected_on_day by the number previous infections each day and
    # by mu, and sum to obtain new infections on day i. Mu is adjusted by
    # the fraction of the population already infected, so that the population
    # size is the upper limit on the number of infections.
    prev_daily_infections_array = prev_daily_infections.stack()
    to_sum = prev_daily_infections_array * bu.left_justified_expand_dims_like(
        p_infected_on_day, prev_daily_infections_array)
    convolution = tf.reduce_sum(to_sum, axis=0)
    rt_adj = (
        (population - prev_cumulative_infections) / population
        ) * tf.gather(rt, i)
    new_infections = rt_adj * convolution

    # Update the prediction array and the cumulative number of infections.
    daily_infections = prev_daily_infections.write(i, new_infections)
    cumulative_infections = prev_cumulative_infections + new_infections
    return i + 1, daily_infections, cumulative_infections

  _, daily_infections_final, last_cumm_sum = tf.while_loop(
      cond, body,
      (initial_days, daily_infections, init_cumulative_infections),
      maximum_iterations=(total_days - initial_days))
  return daily_infections_final.stack()

def predict_deaths(predicted_infections, ifr_noise, conv_fatality_rate):
  """Expected number of reported deaths by country, by day.

  Args:
    predicted_infections: Array of shape
      `[total_days, batch_size, num_countries]` output from
      `predict_infections`.
    ifr_noise: Array of shape `[batch_size, num_countries]`. Noise in Infection
      Fatality Rate (IFR).
    conv_fatality_rate: Array of shape
      `[total_days - 1, total_days, num_countries]`. Convolutional kernel for
      calculating fatalities, output from `make_conv_fatality_rate`.
  Returns:
    predicted_deaths: Array of shape `[total_days, batch_size, num_countries]`.
      (Batched) predicted number of deaths over time and by country.
  """
  # Multiply the number of infections on day j by the probability of death
  # on day i given infection on day j, and sum over j. This yields the expected
  result_remainder = tf.einsum(
      'i...j,kij->k...j', predicted_infections, conv_fatality_rate) * ifr_noise

  # Concatenate the result with a vector of zeros so that the first day is
  # included.
  result_temp = 1e-15 * predicted_infections[:1]
  return tf.concat([result_temp, result_remainder], axis=0)

1.2 参数值先验

这里我们定义模型参数的联合先验分布。许多参数值被假定为独立的，因此先验可以表示为

\(\text p(\tau, y, \psi, \kappa, \mu, \alpha) = \text p(\tau)\text p(y|\tau)\text p(\psi)\text p(\kappa)\text p(\mu|\kappa)\text p(\alpha)\text p(\epsilon)\)

其中

  • \(\tau\) 是每个国家初始病例数的指数分布的共享速率参数，\(y = y_1, ... y_{\text{num_countries} }\)。
  • \(\psi\) 是死亡人数的负二项分布中的一个参数。
  • \(\kappa\) 是每个国家初始再生数的半正态分布的共享尺度参数，\(\mu = \mu_1, ..., \mu_{\text{num_countries} }\)（表示每个感染者传播的额外病例数）。
  • \(\alpha = \alpha_1, ..., \alpha_6\) 是六项干预措施中每项的有效性。
  • \(\epsilon\)（在代码中称为 ifr_noise，以作者的 Stan 代码命名）是感染致死率 (IFR) 中的噪声。

我们将此模型表示为 TFP 联合分布，这是一种 TFP 分布，能够表达概率图模型。

def make_jd_prior(num_countries, num_interventions):
  return tfd.JointDistributionSequentialAutoBatched([
      # Rate parameter for the distribution of initial cases (tau).
      tfd.Exponential(rate=tf.cast(0.03, DTYPE)),

      # Initial cases for each country.
      lambda tau: tfd.Sample(
          tfd.Exponential(rate=tf.cast(1, DTYPE) / tau),
          sample_shape=num_countries),

      # Parameter in Negative Binomial model for deaths (psi).
      tfd.HalfNormal(scale=tf.cast(5, DTYPE)),

      # Parameter in the distribution over the initial reproduction number, R_0
      # (kappa).
      tfd.HalfNormal(scale=tf.cast(0.5, DTYPE)),

      # Initial reproduction number, R_0, for each country (mu).
      lambda kappa: tfd.Sample(
          tfd.TruncatedNormal(loc=3.28, scale=kappa, low=1e-5, high=1e5),
          sample_shape=num_countries),

      # Impact of interventions (alpha; shared for all countries).
      tfd.Sample(
          tfd.Gamma(tf.cast(0.1667, DTYPE), 1), sample_shape=num_interventions),

      # Multiplicative noise in Infection Fatality Rate.
      tfd.Sample(
          tfd.TruncatedNormal(
              loc=tf.cast(1., DTYPE), scale=0.1, low=1e-5, high=1e5),
              sample_shape=num_countries)
  ])

1.3 在参数值条件下观察到的死亡率的可能性

似然模型表示 \(p(\text{deaths} | \tau, y, \psi, \kappa, \mu, \alpha, \epsilon)\)。它应用了在参数条件下感染人数和预期死亡的模型，并假设实际死亡遵循负二项分布。

def make_likelihood_fn(
    intervention_indicators, population, deaths,
    infection_fatality_rate, initial_days, total_days):

  # Create a mask for the initial days of simulated data, as they are not
  # counted in the likelihood.
  observed_deaths = tf.constant(deaths.T[np.newaxis, ...], dtype=DTYPE)
  mask_temp = deaths != -1
  mask_temp[:, :START_DAYS] = False
  observed_deaths_mask = tf.constant(mask_temp.T[np.newaxis, ...])

  conv_serial_interval = make_conv_serial_interval(initial_days, total_days)
  conv_fatality_rate = make_conv_fatality_rate(
      infection_fatality_rate, total_days)

  def likelihood_fn(tau, initial_cases, psi, kappa, mu, alpha_hier, ifr_noise):
    # Run models for infections and expected deaths
    predicted_infections = predict_infections(
        intervention_indicators, population, initial_cases, mu, alpha_hier,
        conv_serial_interval, initial_days, total_days)
    e_deaths_all_countries = predict_deaths(
        predicted_infections, ifr_noise, conv_fatality_rate)

    # Construct the Negative Binomial distribution for deaths by country.
    mu_m = tf.transpose(e_deaths_all_countries, [1, 0, 2])
    psi_m = psi[..., tf.newaxis, tf.newaxis]
    probs = tf.clip_by_value(mu_m / (mu_m + psi_m), 1e-9, 1.)
    likelihood_elementwise = tfd.NegativeBinomial(
        total_count=psi_m, probs=probs).log_prob(observed_deaths)
    return tf.reduce_sum(
        tf.where(observed_deaths_mask,
                likelihood_elementwise,
                tf.zeros_like(likelihood_elementwise)),
        axis=[-2, -1])

  return likelihood_fn

1.4 感染后死亡的概率

本节计算感染后死亡的天数分布。它假设从感染到死亡的时间是两个伽马变量量的总和，分别表示从感染到发病的时间和从发病到死亡的时间。死亡时间分布与 Verity 等人 (2020) 的感染致死率数据相结合，以计算感染后死亡的天数的概率。

def daily_fatality_probability(infection_fatality_rate, total_days):
  """Computes the probability of death `d` days after infection."""

  # Convert from alternative Gamma parametrization and construct distributions
  # for number of days from infection to onset and onset to death.
  concentration1 = tf.cast((1. / 0.86)**2, DTYPE)
  rate1 = concentration1 / 5.1
  concentration2 = tf.cast((1. / 0.45)**2, DTYPE)
  rate2 = concentration2 / 18.8
  infection_to_onset = tfd.Gamma(concentration=concentration1, rate=rate1)
  onset_to_death = tfd.Gamma(concentration=concentration2, rate=rate2)

  # Create empirical distribution for number of days from infection to death.
  inf_to_death_dist = tfd.Empirical(
      infection_to_onset.sample([5e6]) + onset_to_death.sample([5e6]))

  # Subtract the CDF value at day i from the value at day i + 1 to compute the
  # probability of death on day i given infection on day 0, and given that
  # death (not recovery) is the outcome.
  times = np.arange(total_days + 1., dtype=DTYPE) + 0.5
  cdf = inf_to_death_dist.cdf(times).numpy()
  f_before_ifr = cdf[1:] - cdf[:-1]
  # Explicitly set the zeroth value to the empirical cdf at time 1.5, to include
  # the mass between time 0 and time .5.
  f_before_ifr[0] = cdf[1]

  # Multiply the daily fatality rates conditional on infection and eventual
  # death (f_before_ifr) by the infection fatality rates (probability of death
  # given intection) to obtain the probability of death on day i conditional
  # on infection on day 0.
  return infection_fatality_rate[..., np.newaxis] * f_before_ifr

def make_conv_fatality_rate(infection_fatality_rate, total_days):
  """Computes the probability of death on day `i` given infection on day `j`."""
  p_fatal_all_countries = daily_fatality_probability(
      infection_fatality_rate, total_days)

  # Use the probability of death d days after infection in each country
  # to build an array of shape [total_days - 1, total_days, num_countries],
  # where the element [i, j, c] is the probability of death on day i+1 given
  # infection on day j in country c.
  conv_fatality_rate = np.zeros(
      [total_days - 1, total_days, p_fatal_all_countries.shape[0]])
  for n in range(1, total_days):
    conv_fatality_rate[n - 1, 0:n, :] = (
        p_fatal_all_countries[:, n - 1::-1]).T
  return tf.constant(conv_fatality_rate, dtype=DTYPE)

1.5 序列间隔

序列间隔是疾病传播链中连续病例之间的时间，并且假设服从 Gamma 分布。我们使用序列间隔分布计算在第 \(i\) 天感染的人在第 \(j\) 天从之前感染的人那里感染病毒的概率（predict_infections 的 conv_serial_interval 参数）。

def make_conv_serial_interval(initial_days, total_days):
  """Construct the convolutional kernel for infection timing."""

  g = tfd.Gamma(tf.cast(1. / (0.62**2), DTYPE), 1./(6.5*0.62**2))
  g_cdf = g.cdf(np.arange(total_days, dtype=DTYPE))

  # Approximate the probability mass function for the number of days between
  # successive infections.
  serial_interval = g_cdf[1:] - g_cdf[:-1]

  # `conv_serial_interval` is an array of shape
  # [total_days - initial_days, total_days] in which entry [i, j] contains the
  # probability that an individual infected on day i + initial_days caught the
  # virus from someone infected on day j.
  conv_serial_interval = np.zeros([total_days - initial_days, total_days])
  for n in range(initial_days, total_days):
    conv_serial_interval[n - initial_days, 0:n] = serial_interval[n - 1::-1]
  return tf.constant(conv_serial_interval, dtype=DTYPE)

2 数据预处理

COUNTRIES = [
    'Austria',
    'Belgium',
    'Denmark',
    'France',
    'Germany',
    'Italy',
    'Norway',
    'Spain',
    'Sweden',
    'Switzerland',
    'United_Kingdom'
]

2.1 获取并预处理干预数据

2.2 获取病例/死亡数据并加入到干预措施中

2.3 获取并处理感染致死率和人口数据

2.4 预处理特定国家/地区的数据

# Model up to 75 days of data for each country, starting 30 days before the
# tenth cumulative death.
START_DAYS = 30
MAX_DAYS = 102
COVARIATE_COLUMNS = any_intervention_list + ['any_intervention']

# Initialize an array for number of deaths.
deaths = -np.ones((num_countries, MAX_DAYS), dtype=DTYPE)

# Assuming every intervention is still inplace in the unobserved future
num_interventions = len(COVARIATE_COLUMNS)
intervention_indicators = np.ones((num_countries, MAX_DAYS, num_interventions))

first_days = {}
for i, c in enumerate(COUNTRIES):
  c_data = data.loc[c]

  # Include data only after 10th death in a country.
  mask = c_data['deaths'].cumsum() >= 10

  # Get the date that the epidemic starts in a country.
  first_day = c_data.index[mask][0] - pd.to_timedelta(START_DAYS, 'days')
  c_data = c_data.truncate(before=first_day)

  # Truncate the data after 28 March 2020 for comparison with Flaxman et al.
  c_data = c_data.truncate(after='2020-03-28')

  c_data = c_data.iloc[:MAX_DAYS]
  days_of_data = c_data.shape[0]
  deaths[i, :days_of_data] = c_data['deaths']
  intervention_indicators[i, :days_of_data] = c_data[
    COVARIATE_COLUMNS].to_numpy()
  first_days[c] = first_day

# Number of sequential days to seed infections after the 10th death in a
# country. (N0 in authors' Stan code.)
INITIAL_DAYS = 6

# Number of days of observed data plus days to forecast. (N2 in authors' Stan
# code.)
TOTAL_DAYS = deaths.shape[1]

3 模型推理

Flaxman 等人 (2020) 使用 Stan 通过哈密顿蒙特卡罗 (HMC) 和无 U 形采样器 (NUTS) 从参数后验中进行采样。

在此，我们应用 HMC，采用双重平均步长自适应。我们使用 HMC 的试运行进行预处理和初始化。

推理可在 GPU 上在几分钟内运行。

3.1 为模型构建先验和似然

jd_prior = make_jd_prior(num_countries, num_interventions)
likelihood_fn = make_likelihood_fn(
    intervention_indicators, population_value, deaths,
    infection_fatality_rate, INITIAL_DAYS, TOTAL_DAYS)

3.2 实用工具

def get_bijectors_from_samples(samples, unconstraining_bijectors, batch_axes):
  """Fit bijectors to the samples of a distribution.

  This fits a diagonal covariance multivariate Gaussian transformed by the
  `unconstraining_bijectors` to the provided samples. The resultant
  transformation can be used to precondition MCMC and other inference methods.
  """
  state_std = [    
      tf.math.reduce_std(bij.inverse(x), axis=batch_axes)
      for x, bij in zip(samples, unconstraining_bijectors)
  ]
  state_mu = [
      tf.math.reduce_mean(bij.inverse(x), axis=batch_axes)
      for x, bij in zip(samples, unconstraining_bijectors)
  ]
  return [tfb.Chain([cb, tfb.Shift(sh), tfb.Scale(sc)])
          for cb, sh, sc in zip(unconstraining_bijectors, state_mu, state_std)]

def generate_init_state_and_bijectors_from_prior(nchain, unconstraining_bijectors):
  """Creates an initial MCMC state, and bijectors from the prior."""
  prior_samples = jd_prior.sample(4096)

  bijectors = get_bijectors_from_samples(
      prior_samples, unconstraining_bijectors, batch_axes=0)

  init_state = [
    bij(tf.zeros([nchain] + list(s), DTYPE))
    for s, bij in zip(jd_prior.event_shape, bijectors)
  ]

  return init_state, bijectors

@tf.function(autograph=False, experimental_compile=True)
def sample_hmc(
    init_state,
    step_size,
    target_log_prob_fn,
    unconstraining_bijectors,
    num_steps=500,
    burnin=50,
    num_leapfrog_steps=10):

    def trace_fn(_, pkr):
        return {
            'target_log_prob': pkr.inner_results.inner_results.accepted_results.target_log_prob,
            'diverging': ~(pkr.inner_results.inner_results.log_accept_ratio > -1000.),
            'is_accepted': pkr.inner_results.inner_results.is_accepted,
            'step_size': [tf.exp(s) for s in pkr.log_averaging_step],
        }

    hmc = tfp.mcmc.HamiltonianMonteCarlo(
        target_log_prob_fn,
        step_size=step_size,
        num_leapfrog_steps=num_leapfrog_steps)

    hmc = tfp.mcmc.TransformedTransitionKernel(
        inner_kernel=hmc,
        bijector=unconstraining_bijectors)

    hmc = tfp.mcmc.DualAveragingStepSizeAdaptation(
        hmc,
        num_adaptation_steps=int(burnin * 0.8),
        target_accept_prob=0.8,
        decay_rate=0.5)

    # Sampling from the chain.
    return tfp.mcmc.sample_chain(
        num_results=burnin + num_steps,
        current_state=init_state,
        kernel=hmc,
        trace_fn=trace_fn)

3.3 定义事件空间双射

当从各向同性多元高斯分布中进行采样时，HMC 的效率最高（Mangoubi 和 Smith (2017)），因此第一步是预处理目标密度，使其尽可能地看起来像这样。

首先，我们将受限（例如，非负）变量变换到非受限空间，这是 HMC 所需的。此外，我们采用 SinhArcsinh 双射来控制变换目标密度的尾部的重量；我们希望这些尾部大致呈 \(e^{-x^2}\) 下降。

unconstraining_bijectors = [
    tfb.Chain([tfb.Scale(tf.constant(1 / 0.03, DTYPE)), tfb.Softplus(),
                tfb.SinhArcsinh(tailweight=tf.constant(1.85, DTYPE))]), # tau
    tfb.Chain([tfb.Scale(tf.constant(1 / 0.03, DTYPE)), tfb.Softplus(),
                tfb.SinhArcsinh(tailweight=tf.constant(1.85, DTYPE))]), # initial_cases
    tfb.Softplus(), # psi
    tfb.Softplus(), # kappa
    tfb.Softplus(), # mu
    tfb.Chain([tfb.Scale(tf.constant(0.4, DTYPE)), tfb.Softplus(),
                tfb.SinhArcsinh(skewness=tf.constant(-0.2, DTYPE), tailweight=tf.constant(2., DTYPE))]), # alpha
    tfb.Softplus(), # ifr_noise
]

3.4 HMC 试运行

我们首先运行由先验预处理的 HMC，从变换空间中的 0 初始化。我们不使用先验样本初始化链，因为在实践中，由于数值较差，这些样本通常会导致链卡住。

%%time

nchain = 32

target_log_prob_fn = lambda *x: jd_prior.log_prob(*x) + likelihood_fn(*x)
init_state, bijectors = generate_init_state_and_bijectors_from_prior(nchain, unconstraining_bijectors)

# Each chain gets its own step size.
step_size = [tf.fill([nchain] + [1] * (len(s.shape) - 1), tf.constant(0.01, DTYPE)) for s in init_state]

burnin = 200
num_steps = 100

pilot_samples, pilot_sampler_stat = sample_hmc(
    init_state,
    step_size,
    target_log_prob_fn,
    bijectors,
    num_steps=num_steps,
    burnin=burnin,
    num_leapfrog_steps=10)

CPU times: user 56.8 s, sys: 2.34 s, total: 59.1 s
Wall time: 1min 1s

3.5 可视化试运行样本

我们正在寻找卡住的链并观察收敛。我们可以在此进行正式诊断，但鉴于这只是试运行，因此不是特别必要。

import arviz as az
az.style.use('arviz-darkgrid')

var_name = ['tau', 'initial_cases', 'psi', 'kappa', 'mu', 'alpha', 'ifr_noise']

pilot_with_warmup = {k: np.swapaxes(v.numpy(), 1, 0)
                     for k, v in zip(var_name, pilot_samples)}

我们在预热期间观察到偏差，主要是因为双重平均步长适应使用非常激进的搜索来寻找最优步长。一旦适应关闭，偏差也会消失。

az_trace = az.from_dict(posterior=pilot_with_warmup,
                        sample_stats={'diverging': np.swapaxes(pilot_sampler_stat['diverging'].numpy(), 0, 1)})
az.plot_trace(az_trace, combined=True, compact=True, figsize=(12, 8));

png

plt.plot(pilot_sampler_stat['step_size'][0]);

png

3.6 运行 HMC

原则上，我们可以将试点样本用于最终分析（如果我们运行更长时间以获得收敛），但启动另一个 HMC 运行会更有效，这次由试点样本进行预处理和初始化。

%%time

burnin = 50
num_steps = 200

bijectors = get_bijectors_from_samples([s[burnin:] for s in pilot_samples],
                                       unconstraining_bijectors=unconstraining_bijectors,
                                       batch_axes=(0, 1))

samples, sampler_stat = sample_hmc(
    [s[-1] for s in pilot_samples],
    [s[-1] for s in pilot_sampler_stat['step_size']],
    target_log_prob_fn,
    bijectors,
    num_steps=num_steps,
    burnin=burnin,
    num_leapfrog_steps=20)

CPU times: user 1min 26s, sys: 3.88 s, total: 1min 30s
Wall time: 1min 32s

plt.plot(sampler_stat['step_size'][0]);

png

3.7 可视化样本

import arviz as az
az.style.use('arviz-darkgrid')

var_name = ['tau', 'initial_cases', 'psi', 'kappa', 'mu', 'alpha', 'ifr_noise']

posterior = {k: np.swapaxes(v.numpy()[burnin:], 1, 0)
             for k, v in zip(var_name, samples)}
posterior_with_warmup = {k: np.swapaxes(v.numpy(), 1, 0)
             for k, v in zip(var_name, samples)}

计算链的摘要。我们正在寻找高 ESS 和接近 1 的 r_hat。

az.summary(posterior)

az_trace = az.from_dict(posterior=posterior_with_warmup,
                        sample_stats={'diverging': np.swapaxes(sampler_stat['diverging'].numpy(), 0, 1)})
az.plot_trace(az_trace, combined=True, compact=True, figsize=(12, 8));

png

查看所有维度上的自相关函数很有启发性。我们正在寻找快速下降的函数，但不要下降到负值（这表明 HMC 达到共振，这对遍历性不利，并且可能引入偏差）。

with az.rc_context(rc={'plot.max_subplots': None}):
  az.plot_autocorr(posterior, combined=True, figsize=(12, 16), textsize=12);

png

4 结果

以下图表分析了 \(R_t\) 的后验预测分布、死亡人数和感染人数，类似于 Flaxman 等人 (2020) 中的分析。

total_num_samples = np.prod(posterior['mu'].shape[:2])

# Calculate R_t given parameter estimates.
def rt_samples_batched(mu, intervention_indicators, alpha):
  linear_prediction = tf.reduce_sum(
      intervention_indicators * alpha[..., np.newaxis, np.newaxis, :], axis=-1)
  rt_hat = mu[..., tf.newaxis] * tf.exp(-linear_prediction, name='rt')
  return rt_hat

alpha_hat = tf.convert_to_tensor(
    posterior['alpha'].reshape(total_num_samples, posterior['alpha'].shape[-1]))
mu_hat = tf.convert_to_tensor(
    posterior['mu'].reshape(total_num_samples, num_countries))
rt_hat = rt_samples_batched(mu_hat, intervention_indicators, alpha_hat)
sampled_initial_cases = posterior['initial_cases'].reshape(
    total_num_samples, num_countries)
sampled_ifr_noise = posterior['ifr_noise'].reshape(
    total_num_samples, num_countries)
psi_hat = posterior['psi'].reshape([total_num_samples])

conv_serial_interval = make_conv_serial_interval(INITIAL_DAYS, TOTAL_DAYS)
conv_fatality_rate = make_conv_fatality_rate(infection_fatality_rate, TOTAL_DAYS)
pred_hat = predict_infections(
    intervention_indicators, population_value, sampled_initial_cases, mu_hat,
    alpha_hat, conv_serial_interval, INITIAL_DAYS, TOTAL_DAYS)
expected_deaths = predict_deaths(pred_hat, sampled_ifr_noise, conv_fatality_rate)

psi_m = psi_hat[np.newaxis, ..., np.newaxis]
probs = tf.clip_by_value(expected_deaths / (expected_deaths + psi_m), 1e-9, 1.)
predicted_deaths = tfd.NegativeBinomial(
    total_count=psi_m, probs=probs).sample()

# Predict counterfactual infections/deaths in the absence of interventions
no_intervention_infections = predict_infections(
    intervention_indicators,
    population_value,
    sampled_initial_cases,
    mu_hat,
    tf.zeros_like(alpha_hat),
    conv_serial_interval,
    INITIAL_DAYS, TOTAL_DAYS)

no_intervention_expected_deaths = predict_deaths(
    no_intervention_infections, sampled_ifr_noise, conv_fatality_rate)
probs = tf.clip_by_value(
    no_intervention_expected_deaths / (no_intervention_expected_deaths + psi_m),
    1e-9, 1.)
no_intervention_predicted_deaths = tfd.NegativeBinomial(
    total_count=psi_m, probs=probs).sample()

4.1 干预措施的有效性

类似于 Flaxman 等人 (2020) 的图 4。

def intervention_effectiveness(alpha):

  alpha_adj = 1. - np.exp(-alpha + np.log(1.05) / 6.)
  alpha_adj_first = (
      1. - np.exp(-alpha - alpha[..., -1:] + np.log(1.05) / 6.))

  fig, ax = plt.subplots(1, 1, figsize=[12, 6])
  intervention_perm = [2, 1, 3, 4, 0]
  percentile_vals = [2.5, 97.5]
  jitter = .2

  for ind in range(5):
    first_low, first_high = tfp.stats.percentile(
        alpha_adj_first[..., ind], percentile_vals)
    low, high = tfp.stats.percentile(
        alpha_adj[..., ind], percentile_vals)

    p_ind = intervention_perm[ind]
    ax.hlines(p_ind, low, high, label='Later Intervention', colors='g')
    ax.scatter(alpha_adj[..., ind].mean(), p_ind, color='g')
    ax.hlines(p_ind + jitter, first_low, first_high,
              label='First Intervention', colors='r')
    ax.scatter(alpha_adj_first[..., ind].mean(), p_ind + jitter, color='r')

    if ind == 0:
      plt.legend(loc='lower right')
  ax.set_yticks(range(5))
  ax.set_yticklabels(
      [any_intervention_list[intervention_perm.index(p)] for p in range(5)])
  ax.set_xlim([-0.01, 1.])
  r = fig.patch
  r.set_facecolor('white') 

intervention_effectiveness(alpha_hat)

png

4.2 按国家划分的感染、死亡和 R_t

类似于 Flaxman 等人 (2020) 的图 2。

png

4.3 有无干预措施的预测/预报每日死亡人数

png