Wind speed prediction using ARIMA model
21 views (last 30 days)
Show older comments
Dear All,
I am trying to predicte the next 2 hours wind speed of 10-min wind speed reading (12-point ahead forecasting). for that i am trying to compare an ANN-NAR model with ARIMA model. for the last one i am getting problems in the predicted wind speed.
in my code i am using a very simple method which has the following steps:
1- Calculated the Autocorrelation & Partial Autocorrelation functions on the row data in order to:
A- see if there is a need for data differencing (Identifiy the d value of the ARIMA model)
B- try to identify the p,q order of the AR and MA filters respectivly.
2- re-calculate the Autocorrelation & Partial Autocorrelation function on the differenced data in order to see if it changes and to identifiy the correct d-value of the ARIMA model.
3- as this Autocorrelation calculation is time consuming it can be shutdown by the if condition.
4- after reading the results of the "correlation- test" an ARIMA model is created, in this mfile i have created random ARIMA models which need to be tested.
5- after building the ARIMA model the standerd steps are carried out:
A- estimating the model
B- calculating the residual (optional at the time being don't care of the results)
C- testing the residual (optional at the time being don't care of the results)
D- forecasting the wind speed for the next 2 hours (12- points ahead of 10-min reading)
E- running the simulation (Monte Carlo simulation) (optional)
F- comparing the forecasted results with the measured results.
MY QUESTION: Why i am getting a non-realistic forecasting results of wind speed when using the forecast function on the selected (estimated) ARIMA model?
my code is as the following:
if true
%%introduction for the usage of this mfile
% Creating an ARIMA model in matlab to intgrate with the ANN script to
% compare the different prediction results of those two differnt models in
% the report was called:
% 1- STSFMs-ARIMA
% 2- ANNTSFMs_NAR
clear
clc
tic
%%pre-processing of the wind speed data by doing the ac.f and pac.f
% this part will try to describe the available data which will help use
% understand the best fit model to be used in next steps.
% ------------- Loading the data --------------------
load('C:\Users\00037218\Desktop\Wind Speed prediction\E82_Wind_2_2013.mat')
%----------------------- End ------------------------
% for doing the AutoCorrelation Function and the Partial AutoCorrelation
% Function please use the following functions in matlab:
k_max = round(length(Final_test)/3);% maximum number of lags to be correlated
Test = 0;
if Test
tic
[ACF, lags, bounds] = autocorr(Final_test,k_max);
Feedback_delays = lags((ACF > bounds(1,1)) | (ACF < bounds(2,1)));
q = Feedback_delays;
[PACF, Plags, Pbounds] = parcorr(Final_test,k_max);
P = lags((PACF > Pbounds(1,1)) | (PACF < Pbounds(2,1)));
figure();
subplot(2,1,1)
lineHandles = stem(lags,ACF,'filled','r-o');
set(lineHandles(1),'MarkerSize',4)
grid('on')
xlabel('Lag')
ylabel('Sample Autocorrelation')
title('Sample Autocorrelation Function')
hold('on')
plot(lags,ACF);hold on; plot(lags,repmat(bounds(1,1),1,max(lags)+1));hold on;plot(lags,repmat(bounds(2,1),1,max(lags)+1))
subplot(2,1,2)
lineHandles = stem(Plags,PACF,'filled','r-o');
set(lineHandles(1),'MarkerSize',4)
grid('on')
xlabel('Lag')
ylabel('Sample Partial Autocorrelations')
title('Sample Partial Autocorrelation Function')
hold('on')
plot(Plags,PACF);hold on; plot(Plags,repmat(Pbounds(1,1),1,max(Plags)+1));hold on;plot(Plags,repmat(Pbounds(2,1),1,max(Plags)+1))
toc
end
% if we have a linearly decaying sample ACF indicates a nonstationary
% process (time-serie) for that a diffrencing should be done using the diff
% function in matlab as follows:
if Test
tic
Final_test_stationary = diff(Final_test);
[ACF_D, lags_D, bounds_D] = autocorr(Final_test_stationary,k_max);
Feedback_delays_D = lags_D((ACF_D > bounds_D(1,1)) | (ACF_D < bounds_D(2,1)));
q_D = Feedback_delays_D;
[PACF_D, Plags_D, Pbounds_D] = parcorr(Final_test_stationary,k_max);
P_D = lags((PACF_D > Pbounds_D(1,1)) | (PACF_D < Pbounds_D(2,1)));
figure();
subplot(2,1,1)
lineHandles = stem(lags_D,ACF_D,'filled','r-o');
set(lineHandles(1),'MarkerSize',4)
grid('on')
xlabel('Lag')
ylabel('Sample Autocorrelation')
title('Sample Autocorrelation Function')
hold('on')
plot(lags_D,ACF_D);hold on; plot(lags_D,repmat(bounds_D(1,1),1,max(lags_D)+1));hold on;plot(lags_D,repmat(bounds_D(2,1),1,max(lags_D)+1))
subplot(2,1,2)
lineHandles = stem(lags_D,PACF_D,'filled','r-o');
set(lineHandles(1),'MarkerSize',4)
grid('on')
xlabel('Lag')
ylabel('Sample Partial Autocorrelations')
title('Sample Partial Autocorrelation Function')
hold('on')
plot(Plags_D,PACF_D);hold on; plot(Plags_D,repmat(Pbounds_D(1,1),1,max(Plags_D)+1));hold on;plot(Plags_D,repmat(Pbounds_D(2,1),1,max(Plags_D)+1))
toc
end
%%creating the ARIMA-mode
% this part of this mfile will show how to creat an STSFMs-ARIMA model and
% how to adjust the different setting of such a model in matlab
if 0 == 1
P = 1;
D = 1;
Q = 1;
STSFMs_ARIMA = arima(P,D,Q); % the creation of an ARIMA model with the following factors: AR-Factor = 2 (first input), Integration_Factor = 2 (second input), MA_Factor = 2 (third input).
% the built ARIMA model will looks like this:
% ARIMA(2,2,2) Model:
% --------------------
% Distribution: Name = 'Gaussian'
% P: 4
% D: 2
% Q: 2
% Constant: NaN
% AR: {NaN NaN} at Lags [1 2]
% SAR: {}
% MA: {NaN NaN} at Lags [1 2]
% SMA: {}
% Variance: NaN
% now to adjust the settings of such a model (seasonallity, lags ,and so
% on) we can use the following
STSFMs_ARIMA.Seasonality = 12;
% the model now will looks like this:
% ARIMA(2,2,2) Model Seasonally Integrated:
% ------------------------------------------
% Distribution: Name = 'Gaussian'
% P: 16
% D: 2
% Q: 2
% Constant: NaN
% AR: {NaN NaN} at Lags [1 2]
% SAR: {}
% MA: {NaN NaN} at Lags [1 2]
% SMA: {}
% Seasonality: 12
% Variance: NaN
% as it can be seen from this detailed represntation of the ARIMA model the
% following paramerters can be adjusted:
% 1- D (defrencing Factor)
% 2- Constant (Constant term of the selected model depend on the predicted time series)
% 3- AR (Non-seasonal AutoRegression coefficents)
% 4- SAR (Seasonal AutoRegression Coefficents)
% 5- MA (Non-seasonal MovingAverage coefficents)
% 6- SMA (Seasonal MovingAverage coefficents)
% 7- Seasonality
% 8- Variance.
% all the previouslly stated parameters could be ajusted in the same way
% the seasonality of the model was changed
STSFMs_ARIMA.AR = [];
STSFMs_ARIMA.MA = [];
Lags = num2cell(nan(1,STSFMs_ARIMA.Seasonality));
STSFMs_ARIMA.SAR = Lags;
STSFMs_ARIMA.SMA = Lags;
else
STSFMs_ARIMA = arima('SMALags',12,'D',1,'SARLags',12,...
'Seasonality',12);
D1 = LagOp({1 -1},'Lags',[0,1]);
D12 = LagOp({1 -1},'Lags',[0,12]);
D = D1*D12;
dFinal_test = filter(D,Final_test);
STSFMs_ARIMA_1 = arima('SMALags', 12, 'D',1,'SARLags',12,'MALags',1,'ARLags',1,...
'Seasonality',12);
STSFMs_ARIMA_Minitab = arima('SMALags', 12,'SMA',0.9852, 'D',1,'SARLags',12,'SAR',0.1264,'MALags',1,'MA',0.0500,'ARLags',1,'AR',0.9688,...
'Seasonality',12,'Constant',-0.0002256,'Variance',2);
max_order = 3;
[aic_matrix,bic_matrix] = ARMA_model_order_tuning(Final_test,max_order);
end
%%tuning (training or estimating) stage of the built ARIMA model
% for tuning our ARIMA model we will need to use the estimate function in
% matlab with a training time series. as following
Est_STSFMs_ARIMA = estimate(STSFMs_ARIMA,Final_test,'Display','full');
Est_STSFMs_ARIMA_1 = estimate(STSFMs_ARIMA_1,Final_test,'Display','full');
Est_STSFMs_ARIMA_Minitab = estimate(STSFMs_ARIMA_Minitab,Final_test,'Display','full');
% ther reuslts of tuning such model will be shown in matlab as following:
% ARIMA(2,2,2) Model Seasonally Integrated:
% ------------------------------------------
% Conditional Probability Distribution: Gaussian
%
% Standard t
% Parameter Value Error Statistic
% ----------- ----------- ------------ -----------
% Constant 0.000339604 0.000342367 0.99193
% AR{1} -1.07354 0.012271 -87.4857
% AR{2} -0.163044 0.0125264 -13.0161
% MA{1} 0.00845896 0.00237307 3.56456
% MA{2} -0.980359 0.00239701 -408.992
% Variance 0.551794 0.00897504 61.481
% for changing the estimation settings use the optimset as following:
options = optimset('fmincon');
options = optimset(options,'Algorithm','sqp','TolCon',1e-7,'MaxFunEvals',...
1000,'Display','iter','Diagnostics','on');
%%check goodness of fit
% now we will try to check of the residuals are normally distributed and
% uncorrelated by using the infer function in matlab and finaly plot the
% resutls by the following code:
res = infer(Est_STSFMs_ARIMA,Final_test);
res_1 = infer(Est_STSFMs_ARIMA_1,Final_test);
res_Minitab = infer(Est_STSFMs_ARIMA_Minitab,Final_test);
% for plotting the results
figure();
subplot(2,2,1)
plot(res./sqrt(Est_STSFMs_ARIMA.Variance));grid on
title('Standardized Residuals')
subplot(2,2,2)
qqplot(res);grid on
subplot(2,2,3)
autocorr(res)
subplot(2,2,4)
parcorr(res)
figure();
subplot(2,2,1)
plot(res_1./sqrt(Est_STSFMs_ARIMA_1.Variance));grid on
title('Standardized Residuals')
subplot(2,2,2)
qqplot(res_1);grid on
subplot(2,2,3)
autocorr(res_1)
subplot(2,2,4)
parcorr(res_1)
figure();
subplot(2,2,1)
plot(res_1./sqrt(Est_STSFMs_ARIMA_Minitab.Variance));grid on
title('Standardized Residuals')
subplot(2,2,2)
qqplot(res_Minitab);grid on
subplot(2,2,3)
autocorr(res_Minitab,108)
subplot(2,2,4)
parcorr(res_Minitab,108)
%%additional tests on the residuals
% the Durbin-Watson Statistic test with the P-value
[P_value,DW]=dwtest(res,Final_test);
% the Ljung-Box-test
[hLBQ,P_valueLBQ] = lbqtest(res,'lags',[12,24,36,48]);
% Engle's ARCH test is used to identify residual heteroscedasticity
[HARCH,p_valueARCH] = archtest(res,'lags',[12,24,36,48]);
% if the results of the first vlaues of the used test method are 1 means
% that evidence is establisted that the test is positive and the tested
% relation exist yet 0 indicate the the test fail and no evidence of the
% tested relation exists.
%%Forecasting stage using the Est_STSFMs_ARIMA (where the coefficents are now known)
% for foreacasting use the forecast matlab function as following:
N = 12;% forecast horizon
[Yc,YcMSE,U] = forecast (Est_STSFMs_ARIMA,N);% the second input is the forecast horizon for more details refer to the created report by the author and matlab documentation.
[Yc_1,YcMSE_1,U_1] = forecast (Est_STSFMs_ARIMA_1,N);
[Yc_Minitab,YcMSE_Minitab,U_Minitab] = forecast (Est_STSFMs_ARIMA_Minitab,12);
% for showing the forecasted results use the plot function in matlab as
% following:
figure();
subplot(3,1,1);
plot(Target_wind);grid on;hold on;plot(YcMSE,'-.r');xlabel('Prediction Step Index');ylabel('Wind Speed in m/s');title('Est_STSFMs_ARIMA_12 model for wind speed forecasting');legend('Expected Outputs','ARIMA Predictions');hold off
subplot(3,1,2);
plot(Target_wind);grid on;hold on;plot(Yc_1,'-.r');xlabel('Prediction Step Index');ylabel('Wind Speed in m/s');title('Est_STSFMs_ARIMA_12 model for wind speed forecasting (with non-seasonal terms)');legend('Expected Outputs','ARIMA Predictions');hold off
subplot(3,1,3);
plot(Target_wind);grid on;hold on;plot(Yc_Minitab,'-.r');xlabel('Prediction Step Index');ylabel('Wind Speed in m/s');title('Est_STSFMs_ARIMA_12 model for wind speed forecasting (with non-seasonal terms as in Minitab)');legend('Expected Outputs','ARIMA Predictions');hold off
%%Monte Carlo simulation of ARIMA
% for simulating different paths of the built model the Monte Carlo
% simulation can be used in Matlab as following:
number_of_paths = 10;
number_of_observation = N;%length(Final_test);
[Y,E,C] = simulate(Est_STSFMs_ARIMA,number_of_observation,'NumPaths',number_of_paths);%length(Final_test));
% for showing the different results of such simulation use the following
% commands:
figure();
subplot(3,1,1);
plot(Y);grid on
title('Simulated Response')
subplot(3,1,2);
plot(E);grid on
title('Simulated Innovations')
subplot(3,1,3);
plot(Y(:,1),'-.r');grid on;hold on; plot(Target_wind);hold off
title('Compare between Simulated and measured wind speed reading')
%%compare model output and measured output
% this part will try to show the usage of the compare function in matlab
% for doing so please follow those steps:
nx = 1:10;% the model order
sys = ssest(iddata(Final_test),nx);%#ok<*NASGU> % this will give the optimum model order number
%nx = 1; % chnage this value to that given by the previous analysis
sys = ssest(iddata(Final_test),nx);
compare(Target_wind,sys,N);%length(Final_test));%N)
compare(Final_test,sys,length(Final_test));
toc
end
2 Comments
Peter Mills
on 24 Jul 2018
Edited: Peter Mills
on 24 Jul 2018
what is the function "ARMA_model_order_tuning"? Thank you in advance
Accepted Answer
Hang Qian
on 20 May 2014
To forecast an ARIMA model, we want to provide both a fitted model as well as data. The former only carries model coefficients, but not observations.
Replace
forecast (Est_STSFMs_ARIMA,N)
by
forecast (Est_STSFMs_ARIMA,N,’Y0’,Y)
It will resolve the anomaly of unrealistic values.
Hang Qian
3 Comments
JiashenTeh
on 2 Mar 2015
Hi Hang,
Why when I simulate the ARMA model which I have fitted to my wind speed data , the wind speed I get is negative? this doesn't make any sense to me.
Can you please explain this? Thank you.
using the example above, let's say i do this simulate(Est_STSFMs_ARIMA,N)
and I get negative wind speed..
More Answers (5)
Lourence Ferrera
on 30 Oct 2017
Edited: Lourence Ferrera
on 30 Oct 2017
Is your data has seasonality ? How can I code the data to show that it has seasonality and forecast this using SARIMA or ARIMA in matlab?
3 Comments
Lourence Ferrera
on 4 Nov 2017
if we use henderson trend -estimation filter then we don't need the unit root test for our data right ?
Alex
on 16 Apr 2019
Hi Aubui,
Could you explain this statement "k_max = round(length(Final_test)/3);% maximum number of lags to be correlated"?
I could not understand the "max no. of lags to be correlated".
Thanks
AD
0 Comments
See Also
Categories
Find more on Conditional Mean Models in Help Center and File Exchange
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!