Time series analysis 41 - time series prediction tbats model

Timing analysis 41

Timing prediction TBATS Model

     We introduced (S)ARIMA(X) Series model 、Prophet And other methods to predict time series data , In this article, we try to use TBATS Model for time series prediction .TBATS The main goal of the model is to model complex seasonal factors in an exponential smoothing way . Please look at the chart below. ,

TBATS: Trigonometric seasonality, Box-Cox transformation, ARMA errors, Trend and Seasonal components.

brief introduction

     Generally speaking, the problem of time series prediction is to predict the future value of time series according to the historical observations of the time series data . In the process of modeling, one of the headache problems of menstruation is the uncertainty of time series , It changes according to the data scenario .
And this article will introduce TBATS The model mainly uses exponential smoothing method to solve complex periodic problems .

    TBATS Different options will be considered to model the period of time series , Include ：

• Conduct Box-Cox Convert and do not Box-Cox transformation
• Consider timing trends and ignore trends
• Use trend attenuation and inapplicable trend attenuation
• Use for model residuals ARIMA(p,q) Or not
• No seasonal model
• Balance seasonal factors to varying degrees
  Please look at the chart below. ：
  
  The final model selection uses Akaike information (AIC, Akaike information criterion).
  Be careful ： Automatically ARIMA It is used to determine whether the residuals need modeling or find appropriate parameters (p,q).

A simple example

Let's take a look at the simplest example

from tbats import TBATS
import numpy as np

# required on windows for multi-processing,
# see https://docs.python.org/2/library/multiprocessing.html#windows
if __name__ == '__main__':
    np.random.seed(2342)
    t = np.array(range(0, 160))
    y = 5 * np.sin(t * 2 * np.pi / 7) + 2 * np.cos(t * 2 * np.pi / 30.5) + \
        ((t / 20) ** 1.5 + np.random.normal(size=160) * t / 50) + 10
    
    # Create estimator
    estimator = TBATS(seasonal_periods=[14, 30.5])
    
    # Fit model
    fitted_model = estimator.fit(y)
    
    # Forecast 14 steps ahead
    y_forecasted = fitted_model.forecast(steps=14)
    
    # Summarize fitted model
    print(fitted_model.summary())

Use Box-Cox: True
Use trend: True
Use damped trend: True
Seasonal periods: [14. 30.5]
Seasonal harmonics [2 1]
ARMA errors (p, q): (0, 0)
Box-Cox Lambda 0.654044
Smoothing (Alpha): 0.132425
Trend (Beta): 0.023495
Damping Parameter (Phi): 0.947184
Seasonal Parameters (Gamma): [-9.18866163e-09 -4.37211750e-09 -1.04956292e-08 -1.81209981e-08]
AR coefficients []
MA coefficients []
Seed vector [ 5.69510575e+00 -1.36715628e-01 4.70120645e-04 4.91577774e-02
-3.14318406e-02 1.93662153e+00 6.22414387e-01 9.31711655e-02]

AIC 1028.837243

# Time series analysis
print(fitted_model.y_hat) # in sample prediction
print(fitted_model.resid) # in sample residuals
print(fitted_model.aic)

[12.00026315 15.37130609 15.92513946 12.92209113 8.86829808 6.55044357
6.98379669 9.80510678 12.96461526 13.46459656 10.73401852 6.96869927
4.88388936 5.2411647 7.89921391 11.16766831 12.01242589 9.84585836
6.61337902 4.99937038 5.90163047 9.22357233 13.21064516 14.63622867
12.54077865 9.32236216 7.62704064 8.86253335 12.72529442 16.732014
17.97565652 15.35674111 11.53204899 9.35938536 10.22652865 13.76843469
…
[-2.63150260e-04 4.92298548e-01 7.81058008e-01 1.04674185e+00
4.74485095e-01 -2.89650437e-01 -1.09380563e-02 7.11659349e-01
7.52445725e-01 1.52818057e+00 7.74776582e-01 2.21684967e-01
-1.03614988e+00 -3.83613709e-01 1.26200022e+00 1.11129016e+00
2.11411936e+00 9.74278300e-01 5.82900889e-01 -3.77720695e-01
-1.42384359e-01 1.28154205e+00 1.23632142e+00 9.87393000e-01
1.61135725e+00 5.23587924e-01 3.71403105e-01 4.83700871e-01
…
1028.8372428992702

# Reading model parameters
print(fitted_model.params.alpha)
print(fitted_model.params.beta)
print(fitted_model.params.x0)
print(fitted_model.params.components.use_box_cox)
print(fitted_model.params.components.seasonal_harmonics)

0.13242488070375835
0.023495267045578822
[ 5.69510575e+00 -1.36715628e-01 4.70120645e-04 4.91577774e-02
-3.14318406e-02 1.93662153e+00 6.22414387e-01 9.31711655e-02]
True
[2 1]

A more complicated example

We still use the front SARIMAX Data used in the series of blog posts , This data is a daily sales data , contain 5 year 10 In storage 50 Sales data of products . In this example, we use warehousing 1 Products in 1 related data .

import pandas as pd
df = pd.read_csv('walmart/train.csv')
df = df[(df['store'] == 1) & (df['item'] == 1)] # item 1 in store 1
df = df.set_index('date')
y = df['sales']
y_to_train = y.iloc[:(len(y)-365)]
y_to_test = y.iloc[(len(y)-365):] # last year for testing

We can see the weekly and annual periodicity in the figure , This indicates that there are multiple cycles in the sequence

TBATS Model

from tbats import TBATS, BATS # Fit the model
estimator = TBATS(seasonal_periods=(7, 365.25))
model = estimator.fit(y_to_train)# Forecast 365 days ahead
y_forecast = model.forecast(steps=365)

Note that the annual cycle is defined as 365,25, Not an integer , But it doesn't matter ,TBATS The model can be supported .

it seems ,TBATS The model fits well for two kinds of mixed periodicity .

Annual periodic fitting

Weekly periodic fitting

The model parameters are as follows ：
Use Box-Cox: True
Use trend: False
Use damped trend: False
Seasonal periods: [ 7. 365.25]
Seasonal harmonics [ 3 11]
ARMA errors (p, q): (0, 0)
Box-Cox Lambda 0.234955
Smoothing (Alpha): 0.015789

TBATS Three equilibrium strategies are used for weekly periodicity , For annual periodicity 11 An equilibrium strategy ; At the same time Box-Cox Method ,lambda by 0.234955; There is no trend modeling , Not used ARMA Modeling residuals .

SARIMA Model weekly periodicity

SARIMA Only one cycle can be modeled , And the cycle cannot be too long . Let's try to use SARIMA Model the time series data , It can be done with TBATS Compare the .

from pmdarima import auto_arima
arima_model = auto_arima(y_to_train, seasonal=True, m=7)
y_arima_forecast = arima_model.predict(n_periods=365)

Autoarima I chose SARIMA(0, 1, 1)x(1, 0, 1, 7) model, Annual periodicity is not modeled .

SARIMAX + Fourier term

We can use SARIMAX Model , Take Fourier term as external variable to fit the second periodic factor .

# prepare Fourier terms
exog = pd.DataFrame({
    'date': y.index})
exog = exog.set_index(pd.PeriodIndex(exog['date'], freq='D'))
exog['sin365'] = np.sin(2 * np.pi * exog.index.dayofyear / 365.25)
exog['cos365'] = np.cos(2 * np.pi * exog.index.dayofyear / 365.25)
exog['sin365_2'] = np.sin(4 * np.pi * exog.index.dayofyear / 365.25)
exog['cos365_2'] = np.cos(4 * np.pi * exog.index.dayofyear / 365.25)
exog = exog.drop(columns=['date'])
exog_to_train = exog.iloc[:(len(y)-365)]
exog_to_test = exog.iloc[(len(y)-365):]# Fit model
arima_exog_model = auto_arima(y=y_to_train, exogenous=exog_to_train, seasonal=True, m=7)# Forecast
y_arima_exog_forecast = arima_exog_model.predict(n_periods=365, exogenous=exog_to_test)

Here we use two Fourier terms as external variables . Now? SARIMAX The model completes the modeling of two cyclical factors .

Model comparison

We use Mean Absolute Error Compare the three models ：

TBATS: 3.8527
SARIMA：7.2249
SARIMAX + 2 Fourier term ：3.9045

Advantages and disadvantages

advantage

TBATS The model can model complex cyclical factors , Such as non integer period 、 Long period, etc .

shortcoming

because TBATS The model mixes and tries many methods at the bottom , So its calculation speed is relatively slow .
TBATS The model cannot be like SARIMAX Add external variables like that .