Mathematical modeling learning (38): time series model and grey prediction model

0. Preface

The most important part of the model is fitting, such as linear and exponential fitting. For the known formula, the fitting type can be seen directly, and the unknown type needs code fitting.

1, Time series

The time series is a little: ARMA model is generally used to fit the time series and predict the future value of the time series. Daniel tests stationarity. Automatic regression AR(Auto regressive) and moving average MA(Moving Average) prediction models. The disadvantage is that when there are great changes in the outside world, there are often large deviations. The effect of time series prediction method for medium and short-term prediction is better than that of long-term prediction.
Based on time series, there are three types:

Time series processing method

Suppose we guide the values from 1 to T and let us predict the value of T+1, then we take the average value as the predicted value of T+1D.

Of course, there will be errors:

This example is smooth. Take corporate income as an example:

This method is suitable for data with stable numerical value. y10 is a moving average, standard error.

2, Grey prediction

When dealing with less eigenvalue data, it does not need a large enough sample space, which can solve the problems of less historical data, low sequence integrity and reliability, and generate irregular original data to obtain a strong regular generation sequence. The disadvantage is that it is only suitable for medium and short-term prediction, and only suitable for prediction similar to exponential growth.

2.1 GM(1,1) model

Applicable to: grey model with first-order differential equation and only one variable. It is suitable for sequences with strong exponential law. The sequences involved are: cumulative sequence and mean sequence

If we process the data and add an explanation, we should subtract this constant from the final prediction result. Don't forget.

Process:
One time accumulation generation sequence:

Mean generation sequence

According to the above results, the differential equation will be obtained:

White differential equation

Find out a and B to get the relationship between x(1) and t.
How to solve a and b? least square method

The white differential equation is obtained:

x(0) is the original data.

2.2 summary steps

(1) Original sequence (reference data column):

(2) Primary accumulation sequence (1-AGO):

(3) One time progressive subtraction sequence (1-IAGO):

(that is, in the original sequence, the latter item successively subtracts the value of the previous item, for example, [x(2)-x(1),x(3-x(2),..., x(n)-x(n-1))].)

(4) Mean generation sequence: (this is the result of the cumulative sequence "(previous + latter) / 2")

Draw a diagram to clarify the idea:

2.3GM(1,1) actual combat

The prediction test of GM (1,1) is used to test "the average sound level data of road noise traffic in a city in the north from 1986 to 1992:

The code is:

x0 = [71.1 72.4 72.4 72.1 71.4 72 71.6]'; %Here is the column vector, which is equivalent to the dependent variable in the original data
n = length(x0);
lamda = x0(1:n-1)./x0(2:n) %Calculated stage ratio
range = minmax(lamda') %Range of calculated stage ratio
x1 = cumsum(x0)
B = [-0.5*(x1(1:n-1)+x1(2:n)),ones(n-1,1)]; %This is the constructed data matrix B
Y = x0(2:n); %data vector Y
u = B\Y  %Fitting parameters u(1)=a,u(2)=b
syms x(t)
x = dsolve(diff(x)+u(1)*x==u(2),x(0)==x0(1)); %Modeling and solving
xt = vpa(x,6) %Displays the solution of the differential equation in decimal format
prediction1 = subs(x,t,[0:n-1]); %Calculate the predicted value of known data
prediction1 = double(prediction1); %Convert signed numbers to numeric types for differential operations
prediction = [x0(1),diff(prediction1)] %Differential operation, restore data
epsilon = x0'-prediction %Calculate residual
delta = abs(epsilon./x0') %Calculate relative residual
rho = 1-(1-0.5*u(1))/(1+0.5*u(1))*lamda'%Calculate the stage ratio deviation value, u(1)=a

return:

2.4 Python implementation

# coding=gbk
"""
Author: Chuan Chuan
@time  : 2021/9/2 3:57
 Group: 970353786
"""
import numpy as np
import math
#Carry out level ratio test on the data. If it does not meet the requirements, carry out translation transformation, and return the transformed data and transformation times
def deal_data(data, c):
    times = 0
    while True:
        i = 1
        while i < len(data):
            proportion = data[i - 1] / data[i]
            if math.exp(-2 / (len(data) + 1)) < proportion < math.exp(2 / (len(data) + 1)):
                i = i + 1
            else:
                data = data + [c] * len(data)
                times += 1
                break
        if i == len(data):
            break
    return data, times

#forecast
def predict(data):
    x1 = data.cumsum()
    B1 = -(x1[:len(x1) - 1] + x1[1:]) / 2.0
    B = np.array([B1, np.ones_like(B1)]).T
    Y = data[1:]
    u = np.dot(np.dot(np.linalg.inv(np.dot(B.T, B)), B.T), Y)
    a, b = u[0], u[1]
    return np.array([(data[0] - b / a) * math.exp(-a * i) + b / a for i in range(len(data))])

#The above algorithms are fixed and can be copied
if __name__ == '__main__':
    x = np.array([71.1, 72.4, 72.4, 72.1, 71.4, 72.0, 71.6])#Existing data
    c = 0.1   # Translation constant, self setting
    data, times = deal_data(x, c)  # Data verification, and return the translated data and the number of horizontal shifts
    predict_data = predict(data)   # Forecast x1
    result = np.ediff1d(predict_data)  # Diminishing
    result = result - c * times   # Subtract the effect of translation
    print(result)#Prediction results
    print((np.array(result[:len(x)]) - np.array(x[1 :len(x)])) / np.array(x[1 :len(x)]))#standard deviation

return:

summary

It seems a little around. I also refer to a lot of materials here. The algorithms are consistent and the codes are basically the same. Let's talk about it first. Tomorrow I'll talk about a practical battle. At that time, I'll see how to modify this article. It's sent out first for the time being. If there's a problem, leave a message to correct it.

Tags: Machine Learning Deep Learning

Posted on Wed, 01 Sep 2021 22:06:02 -0400 by powlow