动态因子图模型（Dynamic Factor Graphs）

最新推荐文章于 2024-05-06 10:38:45 发布

颹蕭蕭

最新推荐文章于 2024-05-06 10:38:45 发布

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分类专栏：时间序列 # 深度学习 # 机器学习文章标签： DFG 动态因子图隐变量 pytorch 时间序列

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本文链接：https://blog.youkuaiyun.com/itnerd/article/details/106120988

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文章目录

DFG 相关论文
DFG 原理
pytorch 实现

DFG 相关论文

1.《Dynamic Factor Graphs for Time Series Modeling》

在这里插入图片描述
论文地址：https://link.springer.com/chapter/10.1007%2F978-3-642-04174-7_9

这篇2009年文章提出了动态因子图模型，用于时间序列建模。作者是图灵奖得主 Yann LeCun 的学生。

2.《DYNAMIC FACTOR GRAPHS –A NEW WIND POWER FORECASTING APPROACH》

在这里插入图片描述
这篇2014年的文章将动态因子图模型用于风能预测。对比 DFG 模型与 ARIMA 模型在极短时间（未来六小时）风能预测的性能。

3. 其它拓展

STNN 时空神经网络就是将 DFG 拓展成时空序列建模。

DFG 原理

模型结构

实际上就是隐变量模型，但考虑的应用场景是：隐变量是连续的、（可能）高维的，内在动力学过程是确定性的、（可能）高度非线性的。 隐变量与观测变量之间的关系也可以是非线性的。

在这里插入图片描述

为什么需要隐变量模型？

对时间序列建模的常用两种方法：

无隐变量 $Y_t = f(Y_{t-1}, \ldots, Y_{t-p}) + \epsilon_t$ 当前时刻的值仅仅由最近的 $p$ 个历史值确定。

有隐变量 $Z_t = f(\mathbf{Z}_{t-p}^{t-1}) + \epsilon_t \\ Y_t = g(Z_t) + w_t$ 隐变量模型是对不完全观测数据建模的较合适的方法，即观测量 $Y$ 不一定能反映动态过程的完整性质，它可能只是更高维状态 $Z$ 的低维投影。

DFG 模型由两部分组成：观测模型、动态模型。

观测模型：
$Y_t^* \equiv g(W_o, Z_t) \\ Y_t = Y_t^* + \omega_t$
$g(\cdot)$ 可以选取简单的线性观测模型。 $\omega_t$ 为高斯噪声。
动态模型：
$Z_t^* \equiv f(W_d, \mathbf{Z}_{t-p}^{t-1}) \\ Z_t = Z_t^* + \epsilon_t$
$\mathbf{Z}_{t-p}^{t-1}$ 表示 $p$ 个历史状态， $f(\cdot)$ 可以选择多元自回归模型，也可以采用多层神经网络模型来拟合非线性动态。 $\epsilon_t$ 为高斯噪声。

模型推断

由于 Yann LeCun 主推基于能量的机器学习方法，这个模型的学习问题也是基于能量的思路，实际上就是抛开概率分布，一心做优化。

模型推断的目的是为了找到最优的隐状态序列 $Z_{t_a}^{t_b}$ 以最小化观测时间 $t_a, t_b]$ 内的能量：
$E(W_d, W_o, Y_{t_a}^{t_b}) = \sum_{t=t_a}^{t_b} E_o(t) + E_d(t)$
其中，
$E_d(t) \equiv \min_{Z_t} E_d(W_d, \mathbf{Z}_{t-p}^{t-1}, Z_t) = \min ||Z^*_t - Z_t||^2\\ E_o(t) \equiv \min_{Z_t} E_o(W_o, Z_{t}, Y_t) = \min ||Y^*_t - Y_t||^2$

模型训练

模型训练视为了找的最优的参数 $\mathbf{W} = [W_o, W_d]$ 以最小化损失函数：
$L(\mathbf{W}, Y, Z) = E(\mathbf{W}, Y) + R_z(Z) + R(\mathbf{W})$

其中 $R(\mathbf{W})$ 是对参数的正则化， $R_z(Z)$ 是对隐状态的正则化，例如平滑性约束： $R_z(Z_t^{t+1}) = \sum_i (z_{i,t+1} -z_{i,t})^2$ .

模型训练的过程由如下两步交替进行：

$\tilde{Z} = \argmin_{Z} L(\mathbf{\tilde{W}}, Y, Z)$
$\tilde{W} = \argmin_{W} L(\mathbf{W}, Y, \tilde{Z})$

这可以视为 EM 算法的两步，

E 步：推断最优的隐变量 $Z$ ；
M 步：优化参数 $W$ 使得能量最低（概率最大）。

pytorch 实现

import


import torch
import torch.nn as nn
import torch.nn.functional as F
import torch.optim as optim
import torch.backends.cudnn as cudnn
from torch.autograd import Variable

import os
import random
import json
from collections import defaultdict, OrderedDict
import shutil

utils

def rmse(x_pred, x_target, reduce=True):
    if reduce:
        return x_pred.sub(x_target).pow(2).sum(-1).sqrt().mean().item()
    return x_pred.sub(x_target).pow(2).sum(-1).sqrt().squeeze()


def identity(input):
    return input


class Logger(object):
    def __init__(self, log_dir, name, chkpt_interval):
        super(Logger, self).__init__()
        os.makedirs(os.path.join(log_dir, name))
        self.log_path = os.path.join(log_dir, name, 'logs.json')
        self.model_path = os.path.join(log_dir, name, 'model.pt')
        self.logs = defaultdict(list)
        self.logs['epoch'] = 0
        self.chkpt_interval = chkpt_interval

    def log(self, key, value):
        if isinstance(value, dict):
            for k, v in value.items():
                self.log('{}.{}'.format(key, k), v)
        else:
            self.logs[key].append(value)

    def checkpoint(self, model):
        if (self.logs['epoch'] + 1) % self.chkpt_interval == 0:
            self.save(model)
        self.logs['epoch'] += 1

    def save(self, model):
        with open(self.log_path, 'w') as f:
            json.dump(self.logs, f, sort_keys=True, indent=4)
        torch.save(model.state_dict(), self.model_path)

dataset

def dataset_factory(data_dir, name):
    # get dataset
    if name == 'lorentz':
        opt, data = lorentz(data_dir, '{}.csv'.format(name))
    else:
        raise ValueError('Non dataset named `{}`.'.format(name))
   
    # split train / test
    train_data = data[:opt.nt_train]
    test_data = data[opt.nt_train:]
    return opt, (train_data, test_data)


def lorentz(data_dir, file='lorentz.csv'):
    # dataset configuration
    opt = DotDict()
    opt.nt = 200
    opt.nt_train = 100
    opt.nx = 3
    opt.periode = opt.nt
    # loading data
    data = torch.Tensor(np.genfromtxt(os.path.join(data_dir, file))).view(opt.nt, opt.nx)

    return opt, data

modules

class MLP(nn.Module):
    def __init__(self, ninp, nhid, nout, nlayers, dropout):
        super(MLP, self).__init__()
        self.ninp = ninp
        # modules
        if nlayers == 1:
            self.module = nn.Linear(ninp, nout)
        else:
            modules = [nn.Linear(ninp, nhid), nn.ReLU(), nn.Dropout(dropout)]
            nlayers -= 1
            while nlayers > 1:
                modules += [nn.Linear(nhid, nhid), nn.ReLU(), nn.Dropout(dropout)]
                nlayers -= 1
            modules.append(nn.Linear(nhid, nout))
            self.module = nn.Sequential(*modules)

    def forward(self, input):
        return self.module(input)

DFG model

class DFG(nn.Module):
    def __init__(self, nx, nt, nz, nhid=0, nlayers=1,
                 activation='identity', device='cuda'):
        super(DFG, self).__init__()
        assert (nhid > 0 and nlayers > 1) or (nhid == 0 and nlayers == 1)
        # attributes
        self.nt = nt
        self.nx = nx
        self.nz = nz

        # kernel
        self.activation = F.tanh if activation == 'tanh' else identity if activation == 'identity' else None
        
        # modules
        self.factors = nn.Parameter(torch.Tensor(nt, nz))
        self.dynamic = MLP(nz, nhid, nz, nlayers, 0)
        self.decoder = nn.Linear(nz, nx, bias=False)

        # init
        self.factors.data.uniform_(-0.1, 0.1)
        

    def update_z(self, z):
        z = z.view(-1,self.nz)
        z_next = self.dynamic(z)
        return self.activation(z_next)

    def decode_z(self, z):
        x_rec = self.decoder(z)
        return x_rec

    def dec_closure(self, t_idx):
        z_inf = self.factors[t_idx]
        x_rec = self.decoder(z_inf)
        return x_rec

    def dyn_closure(self, t_idx):
        z_input = self.factors[t_idx]
        z_input = z_input.view(-1, self.nz)
        z_gen = self.dynamic(z_input)
        return self.activation(z_gen)

    def generate(self, nsteps):
        z = self.factors[-1]
        z_gen = []
        for t in range(nsteps):
            z = self.update_z(z)
            z_gen.append(z)
        z_gen = torch.stack(z_gen).view(-1,self.nz)
        x_gen = self.decode_z(z_gen)
        return x_gen, z_gen

    def factors_parameters(self):
        yield self.factors

load data


# parse
opt = DotDict()

opt.datadir = 'data'
opt.dataset = 'lorentz'
opt.outputdir = 'output_lorentz'
opt.xp = 'dfg'

opt.lr = 0.001
opt.beta1 = 0.9
opt.beta2 = 0.999
opt.eps = 1e-9
opt.wd = 1e-6
opt.wd_z = 1e-7
opt.xp = 'stnn_d'
opt.device = 0
opt.patience = 300
opt.l2_z = 0.01
opt.l1_rel = 0.01
opt.nz = 6
opt.nhid = 0
opt.nlayers = 1
opt.lambd = 1
opt.activation = 'identity'
opt.batch_size = 100
opt.nepoch = 10000
opt.manualSeed = random.randint(1, 10000)


# cudnn
if opt.device > -1:
    os.environ["CUDA_VISIBLE_DEVICES"] = str(opt.device)
    device = torch.device('cuda:0')
else:
    device = torch.device('cpu')

    
random.seed(opt.manualSeed)
torch.manual_seed(opt.manualSeed)
if opt.device > -1:
    torch.cuda.manual_seed_all(opt.manualSeed)


#######################################################################################################################
# Data
#######################################################################################################################
# -- load data
setup, (train_data, test_data) = dataset_factory(opt.datadir, opt.dataset)
train_data = train_data.to(device)
test_data = test_data.to(device)

for k, v in setup.items():
    opt[k] = v

# -- train inputs
t_idx = torch.arange(1,opt.nt_train, out=torch.LongTensor()).contiguous()

train


if os.path.exists(opt.outputdir):
    shutil.rmtree(opt.outputdir)
#######################################################################################################################
# Model
#######################################################################################################################
model = DFG(opt.nx, opt.nt_train, opt.nz, opt.nhid, opt.nlayers,
                        opt.activation).to(device)


#######################################################################################################################
# Optimizer
#######################################################################################################################
params = [{'params': model.factors_parameters(), 'weight_decay': opt.wd_z},
          {'params': model.dynamic.parameters()},
          {'params': model.decoder.parameters()}]

optimizer = optim.Adam(params, lr=opt.lr, betas=(opt.beta1, opt.beta2), eps=opt.eps, weight_decay=opt.wd)

if opt.patience > 0:
    lr_scheduler = optim.lr_scheduler.ReduceLROnPlateau(optimizer, patience=opt.patience)


#######################################################################################################################
# Logs
#######################################################################################################################
logger = Logger(opt.outputdir, opt.xp, opt.nt_train)
with open(os.path.join(opt.outputdir, opt.xp, 'config.json'), 'w') as f:
    json.dump(opt, f, sort_keys=True, indent=4)


#######################################################################################################################
# Training
#######################################################################################################################
lr = opt.lr

from tqdm.notebook import tqdm
pb = tqdm(range(opt.nepoch))
for e in pb:
    # ------------------------ Train ------------------------
    model.train()
    # --- decoder ---
    idx_perm = torch.randperm(opt.nt_train-1).to(device)
    batches = idx_perm.split(opt.batch_size)
    logs_train = defaultdict(float)
    for i, batch in enumerate(batches):

        optimizer.zero_grad()
        # data
        input_t = t_idx[batch]
        x_target = train_data[input_t]
        # closure
        x_rec = model.dec_closure(input_t)


        mse_dec = F.mse_loss(x_rec, x_target)
        # backward
        mse_dec.backward()
        # step
        optimizer.step()
        # log
        logger.log('train_iter.mse_dec', mse_dec.item())
        logs_train['mse_dec'] += mse_dec.item() * len(batch)

    # --- dynamic ---
    idx_perm = torch.randperm(opt.nt_train-1).to(device)
    batches = idx_perm.split(opt.batch_size)
    for i, batch in enumerate(batches):
        optimizer.zero_grad()
        # data
        input_t = t_idx[batch]

        # closure
        z_inf = model.factors[input_t]
        z_pred = model.dyn_closure(input_t - 1)

        # loss
        mse_dyn = z_pred.sub(z_inf).pow(2).mean()
        loss_dyn = mse_dyn * opt.lambd
        
        if opt.l2_z > 0:
            loss_dyn += opt.l2_z * model.factors[input_t - 1].sub(model.factors[input_t]).pow(2).mean()

        # backward
        loss_dyn.backward()
        # step
        optimizer.step()
        
        # log
        logger.log('train_iter.mse_dyn', mse_dyn.item())
        logs_train['mse_dyn'] += mse_dyn.item() * len(batch)
        logs_train['loss_dyn'] += loss_dyn.item() * len(batch)
    
    # --- logs ---
    logs_train['mse_dec'] /= opt.nt_train
    logs_train['mse_dyn'] /= opt.nt_train
    logs_train['loss_dyn'] /= opt.nt_train
    logs_train['loss'] = logs_train['mse_dec'] + logs_train['loss_dyn']
    logger.log('train_epoch', logs_train)
    
    # ------------------------ Test ------------------------
   
    model.eval()
    with torch.no_grad():
        x_pred, _ = model.generate(opt.nt - opt.nt_train)
        score_ts = rmse(x_pred, test_data, reduce=False)
        score = rmse(x_pred, test_data)

    logger.log('test_epoch.rmse', score)
    logger.log('test_epoch.ts', {t: {'rmse': scr.item()} for t, scr in enumerate(score_ts)})
    
    # checkpoint
    logger.log('train_epoch.lr', lr)
    pb.set_postfix(loss=logs_train['loss'], rmse_test=score, lr=lr)
    logger.checkpoint(model)
    
    # schedule lr
    if opt.patience > 0 and score < 10:
        lr_scheduler.step(score)
    lr = optimizer.param_groups[0]['lr']
    if lr <= 1e-5:
        break
logger.save(model)

result


"""
result
"""

model.eval()
with torch.no_grad():
    prediction, _ = model.generate(100)
    mse = rmse(prediction, test_data)


import matplotlib.pyplot as plt
plt.figure('Test plots', figsize=(17, 4), dpi=90)

with open(os.path.join(opt.outputdir, opt.xp, 'logs.json'), 'r') as f:
    logs = json.load(f)

plt.plot([logs['test_epoch.ts.{}.rmse'.format(ts)][-1] for ts in range(100)], label=opt.xp, alpha=0.8)
plt.grid()
plt.title('Prediction RMSE')
plt.xlabel('timestep')
plt.legend()

在这里插入图片描述

plt.figure('Results',figsize=(15,6))

plt.subplot(1, 3, 1)
plt.imshow(test_data.squeeze().cpu().numpy().T, aspect='auto', cmap='jet')
plt.colorbar()
plt.title('ground truth')


plt.subplot(1, 3, 2)
plt.imshow(prediction.squeeze().cpu().numpy().T, aspect='auto', cmap='jet')
plt.colorbar()
plt.title('{} prediction'.format('DFG'))

plt.subplot(1, 3, 3)
plt.imshow(test_data.sub(prediction).abs().cpu().numpy().T, aspect='auto')
plt.colorbar()
plt.title('absolute error')

在这里插入图片描述

plt.figure()
plt.plot(model.decode_z(model.factors).squeeze().detach().cpu().numpy())
plt.plot(range(100,200),prediction.squeeze().detach().cpu().numpy())
plt.plot(range(100,200),test_data.squeeze().cpu().numpy())
plt.show()