残差=y-yhat

一般我们就停止在这里了

但是如果残差表现的有某种形式，代表我们的模型需要进一步改进，如果残差表现的杂乱无章，代表确实没什么别的信息好提取了

现在用最naive的model--上一个时间的值=yhat看看残差表现吧

关于残差，可以看我的另一篇文章https://mp.csdn.net/postedit/82989567

from pandas import Series
from pandas import DataFrame
from pandas import concat
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
predictions = [x for x in test_X]
# calculate residuals
residuals = [test_y[i]-predictions[i] for i in range(len(predictions))]
residuals = DataFrame(residuals)
print(residuals.head())
residuals.plot()
pyplot.show()

残差表现如下：

现在看看基本信息

1.均值--越接近0越好

A value close to zero suggests no bias in the forecasts, whereas positive and negative values suggest a positive or negative bias in the forecasts made.

print(residuals.describe())

结果如下

count  125.000000
mean     0.064000
std      9.187776
min    -28.000000
25%     -6.000000
50%     -1.000000
75%      5.000000
max     30.000000

mean和0还是有点差距

2.直方图密度图about残差

我们希望残差分布越接近正太越好

If the plot showed a distribution that was distinctly non-Gaussian, it would suggest that assumptions made by the modeling process were perhaps incorrect and that a different modeling method may be required.

A large skew may suggest the opportunity for performing a transform to the data prior to modeling, such as taking the log or square root.

# histogram plot

residuals.hist()

pyplot.show()

# density plot

residuals.plot(kind='kde')

pyplot.show()

3.QQ图检验正太更快速的方式

from pandas import Series
from pandas import DataFrame
from pandas import concat
from matplotlib import pyplot
import numpy
from statsmodels.graphics.gofplots import qqplot
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
predictions = [(x-0.064000) for x in test_X]
# calculate residuals
residuals = [test_y[i]-predictions[i] for i in range(len(predictions))]
residuals = numpy.array(residuals)
qqplot(residuals)
pyplot.show()

 越接近对角线越好

4.自回归图

残差的自回归越小越好！

We do not see an obvious autocorrelation trend across the plot. There may be some positive autocorrelation worthy of further investigation at lag 7 that seems significant.

https://machinelearningmastery.com/visualize-time-series-residual-forecast-errors-with-python/