Linear Regression With Time Series

产生 Time 序列

df = tunnel.copy()

df['Time'] = np.arange(len(tunnel.index))

df.head()

使用LinearRegression模型

from sklearn.linear_model import LinearRegression

# Training data
X = df.loc[:, ['Time']]  # features
y = df.loc[:, 'NumVehicles']  # target

# Train the model
model = LinearRegression()
model.fit(X, y)

# Store the fitted values as a time series with the same time index as
# the training data
y_pred = pd.Series(model.predict(X), index=X.index)
ax = y.plot(**plot_params)
ax = y_pred.plot(ax=ax, linewidth=3)
ax.set_title('Time Plot of Tunnel Traffic');

产生 Lag feature

df['Lag_1'] = df['NumVehicles'].shift(1)
df.head()

使用LinearRegression模型

from sklearn.linear_model import LinearRegression

X = df.loc[:, ['Lag_1']]
X.dropna(inplace=True)  # drop missing values in the feature set
y = df.loc[:, 'NumVehicles']  # create the target
y, X = y.align(X, join='inner')  # drop corresponding values in target

model = LinearRegression()
model.fit(X, y)

y_pred = pd.Series(model.predict(X), index=X.index)

ax = y.plot(**plot_params)
ax = y_pred.plot()

Trend 趋势

一个计算均值的方法：

moving_average = tunnel.rolling(
    window=365,       # 365-day window
    center=True,      # puts the average at the center of the window
    min_periods=183,  # choose about half the window size
).mean()              # compute the mean (could also do median, std, min, max, ...)

ax = tunnel.plot(style=".", color="0.5")
moving_average.plot(
    ax=ax, linewidth=3, title="Tunnel Traffic - 365-Day Moving Average", legend=False,
);

一个获取时间序列的方法：

from statsmodels.tsa.deterministic import DeterministicProcess

dp = DeterministicProcess(
    index=tunnel.index,  # dates from the training data
    constant=True,       # dummy feature for the bias (y_intercept)
    order=1,             # the time dummy (trend)
    drop=True,           # drop terms if necessary to avoid collinearity
)
# `in_sample` creates features for the dates given in the `index` argument
X = dp.in_sample()

X.head()

通过线性回归拟合，往往就是在拟合趋势（个人理解）

from sklearn.linear_model import LinearRegression

y = tunnel["NumVehicles"]  # the target

# The intercept is the same as the `const` feature from
# DeterministicProcess. LinearRegression behaves badly with duplicated
# features, so we need to be sure to exclude it here.
model = LinearRegression(fit_intercept=False)
model.fit(X, y)

y_pred = pd.Series(model.predict(X), index=X.index)
ax = tunnel.plot(style=".", color="0.5", title="Tunnel Traffic - Linear Trend")
_ = y_pred.plot(ax=ax, linewidth=3, label="Trend")

X = dp.out_of_sample(steps=30)

y_fore = pd.Series(model.predict(X), index=X.index)

y_fore.head()

Seasonality 季节性序列

给 X 添加季节性faeture

from statsmodels.tsa.deterministic import CalendarFourier, DeterministicProcess

# A  表示 每年， order 表示 几个周期，order = 10 表示按照每年 1、2、3、、、10个周期添加数据，正弦余弦对应的函数值
fourier = CalendarFourier(freq="A", order=10)  # 10 sin/cos pairs for "A"nnual seasonality

# seasonal=True 表示添加季节性 feature，因为数据的 freq 是 Day，所以会按照7天为一个周期，如果数据的 freq 是 Month，就会按照12个数据（月）为一个周期。
dp = DeterministicProcess(
    index=tunnel.index,
    constant=True,               # dummy feature for bias (y-intercept)
    order=1,                     # trend (order 1 means linear)
    seasonal=True,               # weekly seasonality (indicators)
    additional_terms=[fourier],  # annual seasonality (fourier)
    drop=True,                   # drop terms to avoid collinearity
)

X = dp.in_sample()  # create features for dates in tunnel.index
X.head(10)

这里还是用线性模型拟合，跟通常一样，没有差，主要区别在于给数据添加周期和傅立叶标签

y = tunnel["NumVehicles"]

model = LinearRegression(fit_intercept=False)
_ = model.fit(X, y)

y_pred = pd.Series(model.predict(X), index=y.index)
X_fore = dp.out_of_sample(steps=90)
y_fore = pd.Series(model.predict(X_fore), index=X_fore.index)

ax = y.plot(color='0.25', style='.', title="Tunnel Traffic - Seasonal Forecast")
ax = y_pred.plot(ax=ax, label="Seasonal")
ax = y_fore.plot(ax=ax, label="Seasonal Forecast", color='C3')
_ = ax.legend()

Time Series as Features

就是把前面的 target 也作为 fetaure
核心代码

import pandas as pd

# Federal Reserve dataset: https://www.kaggle.com/federalreserve/interest-rates
reserve = pd.read_csv(
    "../input/ts-course-data/reserve.csv",
    parse_dates={'Date': ['Year', 'Month', 'Day']},
    index_col='Date',
)

y = reserve.loc[:, 'Unemployment Rate'].dropna().to_period('M')
df = pd.DataFrame({
    'y': y,
    'y_lag_1': y.shift(1),
    'y_lag_2': y.shift(2),    
})

df.head()