残差如果有某种结构，我们可以对残差建模，进一步捕捉信息
可以为剩余误差时间序列创建一个模型，并预测模型的预期误差。然后可以从模型预测中减去预测误差，从而提高性能。
一个简单有效的残差模型是自回归模型。这是使用一些滞后误差值来预测下一个时间步骤的误差的地方。这些滞后误差组合在线性回归模型中，很像直接时间序列观测的自回归模型。
error(t) = b0 + b1*error(t-1) + b2*error(t-2) ...+ bn*error(t-n)
残差时间序列的自回归称为移动平均模型--MA。这是令人困惑的，因为它与移动平均平滑过程无关。可以把它看作自回归(AR)过程的兄弟，除了滞后残差而不是滞后原始观测。
1.建立模型（这里选取了最naive的模型）
from pandas import Series
from pandas import DataFrame
from pandas import concat
from statsmodels.tsa.ar_model import AR
series = Series.from_csv('daily-total-female-births.csv', header=0)
# create lagged dataset
values = DataFrame(series.values)
dataframe = concat([values.shift(1), values], axis=1)
dataframe.columns = ['t-1', 't+1']
# split into train and test sets
X = dataframe.values
train_size = int(len(X) * 0.66)
train, test = X[1:train_size], X[train_size:]
train_X, train_y = train[:,0], train[:,1]
test_X, test_y = test[:,0], test[:,1]
# persistence model on training set
train_pred = [x for x in train_X]
# calculate residuals
train_resid = [train_y[i]-train_pred[i] for i in range(len(train_pred))]
# model the training set residuals
model = AR(train_resid)
model_fit = model.fit()
window = model_fit.k_ar
coef = model_fit.params
print('Lag=%d, Coef=%s' % (window, coef))MA的结果如下
Lag=15, Coef=[ 0.10120699 -0.84940615 -0.77783609 -0.73345006 -0.68902061 -0.59270551
-0.5376728 -0.42553356 -0.24861246 -0.19972102 -0.15954013 -0.11045476
-0.14045572 -0.13299964 -0.12515801 -0.03615774]
接下来看看MA就是对残差的预测效果如何把（还是滚动预测）
from pandas import Series
from pandas import DataFrame
from pandas import concat
from statsmodels.tsa.ar_model import AR
from matplotlib import pyplot
series = Series.from_csv('daily-total-female-births.csv', header=0)
# create lagged dataset
values = DataFrame(series.values)
dataframe = concat([values.shift(1), values], axis=1)
dataframe.columns = ['t-1', 't+1']
# split into train and test sets
X = dataframe.values
train_size = int(len(X) * 0.66)
train, test = X[1:train_size], X[train_size:]
train_X, train_y = train[:,0], train[:,1]
test_X, test_y = test[:,0], test[:,1]
# persistence model on training set
train_pred = [x for x in train_X]
# calculate residuals
train_resid = [train_y[i]-train_pred[i] for i in range(len(train_pred))]
# model the training set residuals
model = AR(train_resid)
model_fit = model.fit()
window = model_fit.k_ar
coef = model_fit.params
# walk forward over time steps in test
history = train_resid[len(train_resid)-window:]
history = [history[i] for i in range(len(history))]
predictions = list()
expected_error = list()
for t in range(len(test_y)):
# persistence
yhat = test_X[t]
error = test_y[t] - yhat
expected_error.append(error)
# predict error
length = len(history)
lag = [history[i] for i in range(length-window,length)]
pred_error = coef[0]
for d in range(window):
pred_error += coef[d+1] * lag[window-d-1]
predictions.append(pred_error)
history.append(error)
print('predicted error=%f, expected error=%f' % (pred_error, error))
# plot predicted error
pyplot.plot(expected_error)
pyplot.plot(predictions, color='red')
pyplot.show()
那我们怎么提高我们的预测效果呢？
improved forecast = forecast + estimated error！
from pandas import Series
from pandas import DataFrame
from pandas import concat
from statsmodels.tsa.ar_model import AR
from matplotlib import pyplot
from sklearn.metrics import mean_squared_error
series = Series.from_csv('daily-total-female-births.csv', header=0)
# create lagged dataset
values = DataFrame(series.values)
dataframe = concat([values.shift(1), values], axis=1)
dataframe.columns = ['t-1', 't+1']
# split into train and test sets
X = dataframe.values
train_size = int(len(X) * 0.66)
train, test = X[1:train_size], X[train_size:]
train_X, train_y = train[:,0], train[:,1]
test_X, test_y = test[:,0], test[:,1]
# persistence model on training set
train_pred = [x for x in train_X]
# calculate residuals
train_resid = [train_y[i]-train_pred[i] for i in range(len(train_pred))]
# model the training set residuals
model = AR(train_resid)
model_fit = model.fit()
window = model_fit.k_ar
coef = model_fit.params
# walk forward over time steps in test
history = train_resid[len(train_resid)-window:]
history = [history[i] for i in range(len(history))]
predictions = list()
for t in range(len(test_y)):
# persistence
yhat = test_X[t]
error = test_y[t] - yhat
# predict error
length = len(history)
lag = [history[i] for i in range(length-window,length)]
pred_error = coef[0]
for d in range(window):
pred_error += coef[d+1] * lag[window-d-1]
# correct the prediction
yhat = yhat + pred_error
predictions.append(yhat)
history.append(error)
print('predicted=%f, expected=%f' % (yhat, test_y[t]))
# error
mse = mean_squared_error(test_y, predictions)
print('Test MSE: %.3f' % mse)
# plot predicted error
pyplot.plot(test_y)
pyplot.plot(predictions, color='red')
pyplot.show()
提升之后的model
https://machinelearningmastery.com/model-residual-errors-correct-time-series-forecasts-python/