示例：有偏 Kalman 滤波器

import argparse
import itertools
import math
import os

import pyro.distributions as dist
import torch
import torch.nn as nn
from torch.optim import Adam

import funsor
import funsor.ops as ops
import funsor.torch.distributions as f_dist
from funsor.domains import Reals
from funsor.pyro.convert import dist_to_funsor, funsor_to_mvn
from funsor.tensor import Tensor, Variable

# We use a 2D continuous-time NCV dynamics model throughout.
# See http://webee.technion.ac.il/people/shimkin/Estimation09/ch8_target.pdf
TIME_STEP = 1.0
NCV_PROCESS_NOISE = torch.tensor(
    [
        [1 / 3, 0.0, 1 / 2, 0.0],
        [0.0, 1 / 3, 0.0, 1 / 2],
        [1 / 2, 0.0, 1.0, 0.0],
        [0.0, 1 / 2, 0.0, 1.0],
    ]
)
NCV_TRANSITION_MATRIX = torch.tensor(
    [
        [1.0, 0.0, 0.0, 0.0],
        [0.0, 1.0, 0.0, 0.0],
        [1.0, 0.0, 1.0, 0.0],
        [0.0, 1.0, 0.0, 1.0],
    ]
)


@torch.no_grad()
def generate_data(num_frames, num_sensors):
    """
    Generate data from a damped NCV dynamics model
    """
    dt = TIME_STEP
    bias_scale = 4.0
    obs_noise = 1.0
    trans_noise = 0.3

    # define dynamics
    z = torch.cat([10.0 * torch.randn(2), torch.rand(2)])  # position  # velocity
    damp = 0.1  # damp the velocities
    f = torch.tensor(
        [
            [1, 0, 0, 0],
            [0, 1, 0, 0],
            [dt * math.exp(-damp * dt), 0, math.exp(-damp * dt), 0],
            [0, dt * math.exp(-damp * dt), 0, math.exp(-damp * dt)],
        ]
    )
    trans_dist = dist.MultivariateNormal(
        torch.zeros(4),
        scale_tril=trans_noise * torch.linalg.cholesky(NCV_PROCESS_NOISE),
    )

    # define biased sensors
    sensor_bias = bias_scale * torch.randn(2 * num_sensors)
    h = torch.eye(4, 2).unsqueeze(-1).expand(-1, -1, num_sensors).reshape(4, -1)
    obs_dist = dist.MultivariateNormal(
        sensor_bias, scale_tril=obs_noise * torch.eye(2 * num_sensors)
    )

    states = []
    observations = []
    for t in range(num_frames):
        z = z @ f + trans_dist.sample()
        states.append(z)

        x = z @ h + obs_dist.sample()
        observations.append(x)

    states = torch.stack(states)
    observations = torch.stack(observations)
    assert observations.shape == (num_frames, num_sensors * 2)
    return observations, states, sensor_bias


class Model(nn.Module):
    def __init__(self, num_sensors):
        super(Model, self).__init__()
        self.num_sensors = num_sensors

        # learnable params
        self.log_bias_scale = nn.Parameter(torch.tensor(0.0))
        self.log_obs_noise = nn.Parameter(torch.tensor(0.0))
        self.log_trans_noise = nn.Parameter(torch.tensor(0.0))

    def forward(self, observations, add_bias=True):
        obs_dim = 2 * self.num_sensors
        bias_scale = self.log_bias_scale.exp()
        obs_noise = self.log_obs_noise.exp()
        trans_noise = self.log_trans_noise.exp()

        # bias distribution
        bias = Variable("bias", Reals[obs_dim])
        assert not torch.isnan(bias_scale), "bias scales was nan"
        bias_dist = dist_to_funsor(
            dist.MultivariateNormal(
                torch.zeros(obs_dim),
                scale_tril=bias_scale * torch.eye(2 * self.num_sensors),
            )
        )(value=bias)

        init_dist = dist.MultivariateNormal(
            torch.zeros(4), scale_tril=100.0 * torch.eye(4)
        )
        self.init = dist_to_funsor(init_dist)(value="state")

        # hidden states
        prev = Variable("prev", Reals[4])
        curr = Variable("curr", Reals[4])
        self.trans_dist = f_dist.MultivariateNormal(
            loc=prev @ NCV_TRANSITION_MATRIX,
            scale_tril=trans_noise * torch.linalg.cholesky(NCV_PROCESS_NOISE),
            value=curr,
        )

        state = Variable("state", Reals[4])
        obs = Variable("obs", Reals[obs_dim])
        observation_matrix = Tensor(
            torch.eye(4, 2)
            .unsqueeze(-1)
            .expand(-1, -1, self.num_sensors)
            .reshape(4, -1)
        )
        assert observation_matrix.output.shape == (
            4,
            obs_dim,
        ), observation_matrix.output.shape
        obs_loc = state @ observation_matrix
        if add_bias:
            obs_loc += bias
        self.observation_dist = f_dist.MultivariateNormal(
            loc=obs_loc, scale_tril=obs_noise * torch.eye(obs_dim), value=obs
        )

        logp = bias_dist
        curr = "state_init"
        logp += self.init(state=curr)
        for t, x in enumerate(observations):
            prev, curr = curr, "state_{}".format(t)
            logp += self.trans_dist(prev=prev, curr=curr)
            logp += self.observation_dist(state=curr, obs=x)
            # marginalize out previous state
            logp = logp.reduce(ops.logaddexp, prev)
        # marginalize out bias variable
        logp = logp.reduce(ops.logaddexp, "bias")

        # save posterior over the final state
        assert set(logp.inputs) == {"state_{}".format(len(observations) - 1)}
        posterior = funsor_to_mvn(logp, ndims=0)

        # marginalize out remaining variables
        logp = logp.reduce(ops.logaddexp)
        assert isinstance(logp, Tensor) and logp.shape == (), logp.pretty()
        return logp.data, posterior


def track(args):
    results = {}  # keyed on (seed, bias, num_frames)
    for seed in args.seed:
        torch.manual_seed(seed)
        observations, states, sensor_bias = generate_data(
            max(args.num_frames), args.num_sensors
        )
        for bias, num_frames in itertools.product(args.bias, args.num_frames):
            print(
                "tracking with seed={}, bias={}, num_frames={}".format(
                    seed, bias, num_frames
                )
            )
            model = Model(args.num_sensors)
            optim = Adam(model.parameters(), lr=args.lr, betas=(0.5, 0.8))
            losses = []
            for i in range(args.num_epochs):
                optim.zero_grad()
                log_prob, posterior = model(observations[:num_frames], add_bias=bias)
                loss = -log_prob
                loss.backward()
                losses.append(loss.item())
                if i % 10 == 0:
                    print(loss.item())
                optim.step()

            # Collect evaluation metrics.
            final_state_true = states[num_frames - 1]
            assert final_state_true.shape == (4,)
            final_pos_true = final_state_true[:2]
            final_vel_true = final_state_true[2:]

            final_state_est = posterior.loc
            assert final_state_est.shape == (4,)
            final_pos_est = final_state_est[:2]
            final_vel_est = final_state_est[2:]
            final_pos_error = float(torch.norm(final_pos_true - final_pos_est))
            final_vel_error = float(torch.norm(final_vel_true - final_vel_est))
            print("final_pos_error = {}".format(final_pos_error))

            results[seed, bias, num_frames] = {
                "args": args,
                "observations": observations[:num_frames],
                "states": states[:num_frames],
                "sensor_bias": sensor_bias,
                "losses": losses,
                "bias_scale": float(model.log_bias_scale.exp()),
                "obs_noise": float(model.log_obs_noise.exp()),
                "trans_noise": float(model.log_trans_noise.exp()),
                "final_state_estimate": posterior,
                "final_pos_error": final_pos_error,
                "final_vel_error": final_vel_error,
            }
        if args.metrics_filename:
            print("saving output to: {}".format(args.metrics_filename))
            torch.save(results, args.metrics_filename)
    return results


def main(args):
    funsor.set_backend("torch")
    if (
        args.force
        or not args.metrics_filename
        or not os.path.exists(args.metrics_filename)
    ):
        # Increase compression threshold for numerical stability.
        with funsor.gaussian.Gaussian.set_compression_threshold(3):
            results = track(args)
    else:
        results = torch.load(args.metrics_filename)

    if args.plot_filename:
        import matplotlib

        matplotlib.use("Agg")
        import numpy as np
        from matplotlib import pyplot

        seeds = set(seed for seed, _, _ in results)
        X = args.num_frames
        pyplot.figure(figsize=(5, 1.4), dpi=300)

        pos_error = np.array(
            [
                [results[s, 0, f]["final_pos_error"] for s in seeds]
                for f in args.num_frames
            ]
        )
        mse = (pos_error**2).mean(axis=1)
        std = (pos_error**2).std(axis=1) / len(seeds) ** 0.5
        pyplot.plot(X, mse**0.5, "k--")
        pyplot.fill_between(
            X, (mse - std) ** 0.5, (mse + std) ** 0.5, color="black", alpha=0.15, lw=0
        )

        pos_error = np.array(
            [
                [results[s, 1, f]["final_pos_error"] for s in seeds]
                for f in args.num_frames
            ]
        )
        mse = (pos_error**2).mean(axis=1)
        std = (pos_error**2).std(axis=1) / len(seeds) ** 0.5
        pyplot.plot(X, mse**0.5, "r-")
        pyplot.fill_between(
            X, (mse - std) ** 0.5, (mse + std) ** 0.5, color="red", alpha=0.15, lw=0
        )

        pyplot.ylabel("Position RMSE")
        pyplot.xlabel("Track Length")
        pyplot.xticks((5, 10, 15, 20, 25, 30))
        pyplot.xlim(5, 30)
        pyplot.tight_layout(0)
        pyplot.savefig(args.plot_filename)


def int_list(args):
    result = []
    for arg in args.split(","):
        if "-" in arg:
            beg, end = map(int, arg.split("-"))
            result.extend(range(beg, 1 + end))
        else:
            result.append(int(arg))
    return result


if __name__ == "__main__":
    parser = argparse.ArgumentParser(description="Biased Kalman filter")
    parser.add_argument(
        "--seed",
        default="0",
        type=int_list,
        help="random seed, comma delimited for multiple runs",
    )
    parser.add_argument(
        "--bias",
        default="0,1",
        type=int_list,
        help="whether to model bias, comma deliminted for multiple runs",
    )
    parser.add_argument(
        "-f",
        "--num-frames",
        default="5,10,15,20,25,30",
        type=int_list,
        help="number of sensor frames, comma delimited for multiple runs",
    )
    parser.add_argument("--num-sensors", default=5, type=int)
    parser.add_argument("-n", "--num-epochs", default=50, type=int)
    parser.add_argument("--lr", default=0.1, type=float)
    parser.add_argument("--metrics-filename", default="", type=str)
    parser.add_argument("--plot-filename", default="", type=str)
    parser.add_argument("--force", action="store_true")
    args = parser.parse_args()
    main(args)