Time series analysis of data mining [easy to understand]

Hello everyone , I meet you again , I'm your friend, Quan Jun .

A set of random variables arranged in chronological order X1,X2,…,Xt Represents a time series of random events .

The purpose of time series analysis is to give an observed time series , Predict the future value of the sequence .

The model name	describe
Smoothing method	It is often used for trend analysis and prediction , Using smoothing technology , Weaken the impact of short-term random fluctuations on the series , Smoothes the sequence . Depending on the smoothing technique used , It can be divided into moving average method and exponential smoothing method .
Trend fitting method	Take time as an independent variable , The corresponding sequence observations are used as dependent variables , Build a regression model . According to the characteristics of the sequence , It can be divided into linear fitting and curve fitting .
Combination model	The change of time series is mainly affected by the long-term trend （T）、 Seasonal changes （S）、 Cyclical changes （C） And irregular changes （） The influence of these four factors . According to the characteristics of the sequence , We can build addition model and multiplication model . additive model ：x = T+S+C+ Multiplication model ：x = TSC
AR Model	before p The sequence value of the period is an independent variable , A random variable Xt Establish a linear regression model for dependent variables
MA Model	A random variable Xt The value of is independent of the sequence value of previous periods , establish Xt With the former q Random disturbance of period （） The linear regression model of
ARMA Model	A random variable Xt The value of , Not only with the former p The sequence value of the period , Also with the former q Random disturbance of period （） of
ARIMA Model	Many non-stationary sequences will show the properties of stationary sequences after difference , Call this nonstationary sequence a differential stationary sequence . For differential stationary sequences, we can use ARIMA Model fitting
ARCH Model	It can accurately simulate the volatility of time series variables , It is suitable for sequences with heteroscedasticity and short-term autocorrelation of heteroscedasticity function
GARCH Model and its derivative models	It is called generalized ARCH Model , yes ARCH The expansion of the model . It can better reflect the long-term memory in the actual sequence 、 The asymmetry of information

1、 Before time series analysis , You need to preprocess the sequence , Including pure randomness and stability test . According to the test results, the sequences can be divided into different types , Take different analytical methods .

Pure random sequence	Also called white noise sequence , There is no correlation between the items of the sequence , The sequence is undergoing completely disordered random fluctuations . White noise sequence is a stationary sequence without information to extract , Analysis can be terminated .
Stationary non white noise sequence	Mean and variance are constants . A linear model is usually established to fit the development of the sequence , So as to extract useful information . ARMA Stationary series fitting model is the most commonly used model .
Nonstationary sequence	Mean and variance are unstable . It is generally transformed into a stationary sequence , Using the analysis method of stationary time series , Such as ARMA Model . If the time series is processed by difference operation , It is stable , The sequence is called a difference stationary sequence , Use ARIMA Model analysis .

（1） Pure randomness test

If the sequence is a pure randomness test , There should be no relationship between the sequence values . In fact, the sample autocorrelation coefficient of pure random sequence is not absolutely zero , But it's close to zero , And random fluctuations around zero .

Pure randomness test , Also called white noise test , Generally, we construct test statistics to test . Commonly used test statistics are Q statistic 、LB statistic , From the autocorrelation coefficient of each delay period of the sample , You can calculate the test statistics , And then calculate the corresponding p value , If p The value is greater than the significance level , It means to accept the original hypothesis , It's a pure random sequence , Stop analyzing .

（2） Stability test

If the time series fluctuate near a constant and the range of fluctuation is limited , There are constant mean and constant variance , And delay k The autocorrelation and autocorrelation coefficients of the series variables are equal , Or delay k The influence degree between the series variables of the period is the same , The time series are called stationary series .

Two test methods ：

a. A graph test that makes a judgment according to the characteristics of a sequence diagram and an autocorrelation diagram , The method is simple to operate 、 Widely applied , The disadvantage is subjectivity ;

Sequence diagram test ： According to the property that the mean and variance of stationary time series are constant , The sequence diagram of the stationary sequence shows that the sequence value always fluctuates randomly around a constant , And the range of fluctuation is bounded .

If there is a clear trend or periodicity , Usually not a stationary sequence .

Autocorrelation graph test ： Stationary series have short-term correlation , So in a stationary sequence , Only the recent sequence value has a significant impact on the current value , The farther the interval between past values, the less the impact on current values .

With the number of delay periods k An increase in , The autocorrelation coefficient of stationary series will decay rapidly and tend to zero , And random fluctuations around zero , The autocorrelation coefficient of non-stationary series decays slowly .

b. Construct test statistics , At present, the most commonly used method is the unit root test .

Unit root test refers to checking whether there is a unit root in the sequence , Because the existence of unit root is nonstationary time series .

2、 Stationary time series analysis

ARMA The full name of the model is the autoregressive moving average model , Can be subdivided into AR Model 、MA Models and ARMA There are three types of models , Can be regarded as a multiple linear regression model .

Modeling steps ：

（1） Calculate the autocorrelation coefficient （ACF） And partial autocorrelation coefficient （PACF）

（2）ARMA Model recognition , It's also called model order determination , from AR（p） Model 、MA（q） Models and ARMA（p,q） The properties of autocorrelation coefficient and partial autocorrelation coefficient of , Choose the right model .

Model	Autocorrelation coefficient （ACF）	Partial autocorrelation coefficient （PACF）
AR（p）	trailing	p Order truncation
MA（q）	q Order truncation	trailing
ARMA（p,q）	trailing	trailing

（3） Estimate the value of the unknown parameter in the model , And carry out parameter test

（4） Model test

（5） Model optimization

（6） Model application ： Make short-term forecasts .

3、 Nonstationary time series analysis

actually , In nature, most sequences are nonstationary .

Analytical methods fall into two categories ：

（1） Time series analysis of deterministic factor decomposition

Put all the changes in the sequence down to four factors , Long term trends 、 Seasonal changes 、 The combined effect of cyclical and random changes .

The disadvantage is that the fluctuation caused by random factors is difficult to determine and analyze , The waste of random information is serious , It will lead to irrational model fitting accuracy .

（2） Random time series analysis

According to the different characteristics of time series , The models of stochastic time series analysis are as follows ARIMA Model 、 Residual autoregressive model 、 Seasonal models 、 Heteroscedasticity model, etc .

ARIMA Model modeling steps ：

a. Check the stability of the sequence

b. Difference the original sequence , And the stability and white noise test

c. choice ARIMA Model

Need to be for ARIMA（p、d、q） Model assignment parameters p、d、q. among d Is the difference number .

Or use forecast The inside of the bag auto.arima Function to achieve optimal ARIMA Automatic model selection .

Model	ACF	PACF
ARIMA(p,d,0)	Gradually decrease to zero	stay p The order decreases to zero
ARIMA(0,d,q)	q The order decreases to zero	Gradually decrease to zero
ARIMA(p,d,q)	Gradually decrease to zero	Gradually decrease to zero

d. Fitting model

e. forecast

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Illustrate with examples ARIMA Application of the model .

R Language implementation ：

1、 Read data Set

2、 Generate timing objects , Test for stationarity

sales = ts(data) # Generate timing objects 

plot.ts(sales,xlab=" Time ",ylab=" sales ") # Make a sequence diagram 

acf(sales) # Make an autocorrelation diagram 

library(fUnitRoots)
unitrootTest(sales) # Unit root test 

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The sequence diagram is as follows , With monotonic increasing trend .

The autocorrelation diagram is as follows , The autocorrelation coefficient is greater than zero for a long time , It shows that there is a strong long-term correlation between sequences .

Unit root test give the result as follows ,p The value is significantly greater than 0.05, It is judged as non-stationary sequence .（ The nonstationary sequence must not be a white noise sequence ）

3、 First order difference of the original sequence , And conduct stability test .

difsales = diff(sales) # First order difference 

plot.ts(difsales,xlab=" Time ",ylab=" Sales residual ") # Make a sequence diagram 

acf(difsales) # Make an autocorrelation diagram 

unitrootTest(difsales) # Unit root test 

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The sequence diagram is as follows , Relatively stable fluctuation near the mean value .

The autocorrelation diagram is as follows , There is a strong short-term correlation .

Unit root test give the result as follows ,p Less than 0.05, So the sequence after the first-order difference is a stationary sequence .

4、 White noise test

Box.test(difsales,type = "Ljung-Box") # White noise test 

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The result is

p The value is significantly less than 0.05, Therefore, the sequence after the first-order difference is a stationary non white noise sequence .

5、 fitting ARIMA Model

The first method ：

pacf(difsales) # Make partial autocorrelation diagram 

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The partial autocorrelation diagram is as follows ,

According to the table 4 How to choose , selected ARIMA（1,1,0） Model .

fit = arima(sales,order=c(1,1,0)) #ARIMA（1,1,0） Model 

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The second method ：

auto.arima(CWD) # Another example 

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The output is as follows ：

6、 Model test

After the model is determined , Check whether the residual is white noise , If it's not white noise , It indicates that there is still useful information in the residuals , Model parameters need to be modified , Further extraction .

fit = arima(CWD,order = c(0,1,1)) # Another example 

r3 = fit$residuals

Box.test(r3,type = "Ljung-Box")

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7、 forecast

library(forecast) 
forecast(fit,5) # Predict the next 5 The sequence value of the period 

plot(forecast(fit,5)) # Make a prediction chart , Dark areas are 80% and 95% The confidence interval of 

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The result is

8、 Model evaluation

Three statistical indicators are used to measure the prediction accuracy of the model ： Mean absolute error 、 Root mean square error 、 Mean absolute percent error . These three indicators reflect the prediction accuracy of the model from different aspects .

mae = mean(abs(pre-real)) # Mean absolute error 

rmse = mean((pre-real)^2) # Root mean square error 

mape = mean(abs(pre-real)/real) # Mean absolute percent error 

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Combined with actual business analysis , Set the error threshold to a value , Such as 1.5, Evaluate the prediction accuracy of the model .

Python Realization ：

#ARIMA Time series model 
import pandas as pd

forecastnum = 5
data = pd.read_excel("arima_data.xls",index_col=u' date ') #pandas Automatically put “ date ” The column is identified as datetime Format 

# Sequence diagram 
import matplotlib.pyplot as plt
plt.rcParams['font.sans-serif'] = ['SimHei'] 
plt.rcParams['axes.unicode_minus'] = False
data.plot()
plt.show()

# Autocorrelation diagram 
from statsmodels.graphics.tsaplots import plot_acf
plot_acf(data).show()

#ADF Unit root test 
from statsmodels.tsa.stattools import adfuller as ADF
print(u' Original sequence ADF The inspection result is ：',ADF(data[u' sales ']))
# The return values are adf、pvalue

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The output is ：

It can be concluded that the time series is unstable , Differential operation is required .

D_data = data.diff().dropna() # Difference 
D_data.columns = [u' Sales difference ']
D_data.plot() # Sequence diagram 
plt.show()

plot_acf(D_data).show() # Autocorrelation diagram 

from statsmodels.graphics.tsaplots import plot_pacf
plot_pacf(D_data).show() # Partial autocorrelation graph 

print(u' Of differential sequence ADF The inspection result is ：',ADF(D_data[u' Sales difference ']))

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The output is ：

At this time, the sequence after difference conforms to stationarity .

# White noise test 
from statsmodels.stats.diagnostic import acorr_ljungbox
print(u' The white noise test result of the difference sequence is ：',acorr_ljungbox(D_data,lags=1)) # Returns the sum of statistics p value 

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The output is ：

p The value is less than the significance level , So non white noise .

from statsmodels.tsa.arima_model import ARIMA
data[u' sales '] = data[u' sales '].astype(float)
# Order determination 
pmax = int(len(D_data)/10) # Generally, the order does not exceed length/10
qmax = int(len(D_data)/10) 
bic_matrix =[] #bic matrix 
for p in range(pmax+1):
    tmp = []
    for q in range(qmax+1):
        try: # There are some errors 
            tmp.append(ARIMA(data,(p,1,q)).fit().bic)
        except:
            tmp.append(None)
    bic_matrix.append(tmp)
    
bic_matrix = pd.DataFrame(bic_matrix)
p,q = bic_matrix.stack().idxmin() # First use stack Flattening , use idxmin Find the minimum position 
print (u'BIC The smallest p Values and q The value is ：%s,%s' % (p,q))

model = ARIMA(data,(p,1,q)).fit() # establish ARIMA（0,1,1） Model 
model.summary2() # Model report 

model.forecast(5) # forecast 5 Days of data , Return forecast results 、 Standard error 、 confidence interval 

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The output is ：

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