 Hello everyone, welcome back to the session of ARIMA. In the previous few sessions, we have discussed the AR process, the MA process and the ARMA process also. Now, in this session, we will extend this AR, MA and ARMA process into ARIMA model. This is the last part of ARIMA process, entire ARIMA process where the data will be non-stationary. That means, in the previous couple of models of ARIMA that we have discussed whether it is AR, MA or ARMA, we assume that data are stationary. So, the over a period of time data are stable, but if the data are non-stationary, I have already discussed that earlier, but we will concentrate on that part now. When the data are non-stationary, you cannot directly use AR or MA process or ARMA process. You need to first convert the non-stationary data into stationary and then you can start your ARIMA model. So, ARIMA model is only for non-stationary data. So the process, if you look at the screen here, we have completed the basic understanding of SCF and PACF function because these are the two important ingredients or components which is required for the entire ARIMA process. Because you need to read the graph, the correlogram and understand the order of AR and MA process. So that for that reason, we need SCF and PACF. We understood that, then we have completed the AR process, the MA process and ARMA process. Now, we will be concentrating on ARIMA. Remember, ARIMA is nothing but when the data are stationary, then only you can use ARIMA. So effectively, if the data are non-stationary, you convert the data, say time series data. If the time series data are non-stationary, you convert them into stationary. This is the differencing part or I part. This comes as integrated. This term is noted as integrated. I will discuss more detail about it. And then once the data are become stationary, how to do that, how to convert non-stationary into stationary that we will discuss. And then you check once the data become stationary, either you which as it is AR process or MA process or ARMA process. So effectively, there is no ARIMA, ARIMA is all put together ARIMA only if data are non-stationary. If the data are stationary, your model completes here only. If data are stationary, you just either use AR process or MA process or ARMA process. ARIMA will not come to the picture. If the data are non-stationary, then only the entire steps you have to do. That means, first convert the non-stationary data into stationary, then AR process as per the correlogram graph AR or MA or ARMA and you complete the ARIMA process. So let us go to the last part, last leg of entire ARIMA process that is called ARIMA, auto regressive integrated moving average process for non-stationary data. Now if you talk about non-stationary data, you can recall the initial session, different type of data pattern that we have discussed or different type of data understanding of time series data. There we have discussed detail about stationary and non-stationary data. Let us recap that for a while. So if the data are stationary, what happens? You can see that this data, there is a variation, but it falls inside a band line to some extent. The outlayer or too much of you know uncertainty is not involved here. So if you take any block of data, you can see the mean, variance, standard deviation, etcetera, correlation all are to some extent constant. Here if you think the mean is here for this data block, you go to say next 3 months or say next couple of period for further. If you can calculate the mean, look at the mean is almost same, almost same. Little variation will be there if the data are straight line, then there is no point of prediction right. So little variation could be there, but overall data are steady, so these type of data are called stationary data for which you can use AR or MA or ARMA right. You can use any one of them, but if the data are non-stationary, in that case what happens? The basic definition says that data are non-stable right. So here if you think for this block, for say couple of in the middle of the data, if you take a 3 months pure data, you can see the mean is here, but if you come here, you can think the mean is here, you can calculate that. So therefore, mean has been shifted, mean has been shifted right. So you can see the changes of mean. Next time you do not know where the data will go, it might come back here also. So it is unpredictable. So therefore, this type of data are called as a non-stationary data. To some extent uptrend or downtrend data or you know seasonal data are also falls under to some extent non-stationary pattern. But here we are assuming that the pure non-stationary whether data are to some extent random in nature, stochastic in nature and it is not steady, the mean is shifting from time to time. So therefore, this data we will be considering as a part of ARIMA discussion, ARIMA not ARMA or ARMA that we have done only for stationary purpose. Now let us think about that. So suppose this is a non-stationary data, you have to convert that into stationary data and then you use whatever the model that we have discussed so far in the previous sessions. Now the point here is that if the data are non-stationary, we are discussing the non-stationary data, right? If the data are non-stationary, how to convert non-stationary data into stationary? One part I told about that you can take the logarithm, you can use some scaling of the data, like scale change, you can do that. Many way you can do the like you know conversion from the non-stationary to stationary. Remember that we have done the decentralization process in seniority class. How to do decentralization of the data by dividing by the index? That also we understood. There are many way to manage the non-stationary data to convert the non-stationary data into stationary. Here, we will understand one of the interesting process called differencing of non-stationary data and that differencing process we will use to convert the stationary and we will use that stationary data in the ARIMA process. Let us see how does it work. Suppose, you have a time series data say, time series data say y, ytc in general, y1, y2, y3, y4, y5, y6, y7 like this, right? You have the data. Now, what you do? If you take a differencing of that say you know delta y equals to yt minus yt minus 1 say, suppose minus 1, so if you take this it is nothing but y2 minus y1, y3 minus y2, y4 minus y3, y5 minus y4, y6 minus y5, so this way you know y7 minus y6 dot, you can calculate this. This is called delta, right? Delta y or delta yt whatever you can define or sometimes people define as a yt star or delta yt whatever you want, right? So, this is the difference you need to use. This will reduce the variation of the data. You can see one example and that data, this particular differencing data, these data sets, these data sets you need to use for your further calculation of SCAP, SCAP, ARMA process, MA process, AR process, etcetera. These data you cannot use because these data are non-stationary. In case this is non-stationary, then you convert this into stationary by taking a differencing. This process are called to some extent integration process or differencing process. Therefore, we call it as a autoregressive, integrated or differencing MA process. So, this is nothing but the differencing of ARMA or I-part of ARMA. Let us see one example. Here you can see the data, suppose sample data I have kept, how this data? If you plot this data, it is up-trend data. But if you take a differencing of that 4 minus 5, it is coming out to minus 1, 6 minus 4, it is coming out to be 2, 7 minus 6, it is coming out to be 1, 9 minus 7, it is coming out to be 2, say 12 minus 3, 9, it is coming out to be 3. So, if you draw this graph, it will be like 5, 4, 6, 7, 9, 12, it is like this. But if you draw this graph, it could be like say minus 1, minus 1, 2, 1, 2, 3, like the data are like this. Look at this. So, this is quite steady data. So, this data has been scaled down, it has been converted into this differencing data. So, this data is to some extent you can see steady. Over a period of time, you take this is steady, but this data is up-trend data, right, up-trend data. So, you cannot take that data because it is non-stationary to use for AR or MA process. So, you need this data. You can take another example, say suppose you take a data say 2, 4, 7, 9, say 13, like this say. It is up-trend data or down-trend data, but if you take a differencing, 4 minus 2 will be 4 minus 2, will be 2, 7 minus 4 will be 3, say 9 minus 7 will be 2, 13 minus 9 will be 4. So, this if you draw this 2, 3, 2, 4, you can see data are steady. So, these data are converting into a steady data. So, this is called the differencing process. You have to do that to scale down data to convert non-stationary data, non-stationary data into a stationary data. In seasonality we have done, here we are not talking about seasonality, up-trend or down-trend or different zigzag data that we are converting through differencing process. So, in this case if you think that first order differencing is not sufficient till the data suppose, suppose your data are like this, data are like this and if you do the first order differencing, first round of differencing your data might be like this. Still it is up-trend data, right, curvy data. So, if you do one more round of differencing, look at one more round of differencing here you can do delta say you know say delta star y t, delta star y t say or delta 2 whatever. You do one more round of differencing this minus this, this value minus this value. In excel you can do this, right. So, you will get one round of data, one round another in excel sheet, you will get one other column. So, this is called one further round of differencing, second order differencing. This data might be steady. Now, suppose here we have done this is a first on main data y t data. So, this is a delta y t say and you found that this is also non-stationary. How to check the non-stationary whether you have converted the data into stationary or not? That we will do in the next slide through the Kepler test. The testing part we will do when to stop. How many differencing you should do so that you are you can confirm that data becomes stationary that part we will discuss later. But suppose you are seeing that your data are non-stationary then you do one more round of differencing. In that case your data might be to some extent steady. You can think about this look at the third graph look at the third graph to some extent it is steady, right it is steady. So, mean is not shifting much. So, therefore, some epsilon error will be there, but data has becomes and now it is a this this this differencing we have done say for this particular data sets. Third round differencing second round differencing we have done. So, now data becomes stationary take this data the scale down data into your ARMA process ARMA process I part is over Arima's enter discussion is over now suppose. That means, I part is over differencing is done now you go to AR, MA or ARMA that is it this is what the enter process of differencing and Arima. Let us see the next part that you have a time series data and you found that data are non-stationary by drawing the graph or understanding the different calculation process then you need to convert it into stationary. But the point here is that how many round of differencing you need to do one round of differencing, two round of differencing, three round of differencing you have to stop somewhere right. So, that you can start your ARMA process this I part has to be completed now. So, for that there is a you know checking criteria whether the data has converted into non-stationary to stationary or not and when to stop that that been evaluated through Dickey-Fleur test. Dickey-Fleur has evolved on methodology, mathematical structure through which you can test the data in Python and different software it is readily available, but here I will give you the technical aspects of the testing part how does it work. So, Dickey-Fleur test says that this Dickey-Fleur test will be used for data testing whether the data becomes stationary or not. So, non-stationary data if you have let me use the pen again if you have a non-stationary data how to convert that into stationary and when to stop this we are going to discuss now through Dickey-Fleur test. So, it is a unit root test sometimes they call it is a unit root test which is a mathematical structure to check the stationarity of the data. So, let us consider a stochastic process where such that you know for your data say time series data Yt, Y1, Y2, Y3, Y4, Y5, Y6 like that you have and you can develop a you know formula like this say this is your error part say for the timing we are not focusing suppose you have found a correlation like you know linear formula of that of data with past data past lag and you found a correlation among them say. So, linear regression whatever some relationship you found with your stochastic data process. Now, if you subtract that Yt minus 1 from Yt you will find this formula right just from both side you subtract Yt minus 1 Yt minus Yt. So, suppose Y5 minus Y4 like this Yt minus Yt minus 1 you will get this formula now you have a constant phi minus 1 minus 1. So, this constant if it become 0 after some it out of iteration if this constant become 0 your data differencing will become constant look at that the error part are there to some extent it is become 0. So, data become steady. So, no movement will be there now data become steady. So, therefore, this difference will become 0 if this become 0 if this does not become 0 then your data are non stationary still you need to do one more round of differencing and carry forward the testing part or differencing process. So, for that reason they proposed one hypothesis simple hypothesis that the hypothesis is that assume that phi minus 1 is 0 as a null hypothesis and alternative hypothesis is that phi not equals to 0 alternative is phi not equals to 0 right. So, if phi minus 1 equals to 0 that means, phi equals to 1 say whatever we are assuming that phi phi minus 1 not equals to 0 if you consider phi minus 1 not equals to 0 effectively alternative hypothesis is active, but if phi minus 1 equals to 0 in that case your null hypothesis will be active it means that you stop there this become 0 means this term let me use the highlight point this term become 0 means this term become constant this term become constant. So, stop here you do not need to do further differencing further differencing you just stop here in first round or maybe after one round. So, this criteria checking throw non stationary criteria checking throw hypothesis testing are done by Dickie Fuller to see whether that data becomes stationary or not. To illustrate it further let us come to a fresh slide and I will explain in detail. So, that you know a layman person can also understand if you go to regression if you go to regression suppose you have a data set say x and say y right you have the data and you can fit a regression line say y equals to alpha plus beta x right epsilon part we are not discussing suppose alpha plus beta x suppose you have fit a regression say and in that case what do you do in in ANOVA analysis for overall testing of regression and the individual variables also we are considering single variables say. So, we propose a null hypothesis right what is the null hypothesis beta equals to 0 and alternative is what beta not equals to 0 what does it mean in regression it means that if now if p greater than say 0.05 0.05 say 0.05 we say that null hypothesis is active and if p less than 0.05 we say that alternative is active. So, that means, if alternative is active that means beta not equals to 0. So, there is a relationship between y and x and you can fit your regression fit the regression fit the line right, but if p is quite high greater than 0.05 then null hypothesis will active and in that case beta equals to 0 beta 0 means what there is no relationship between y and x. So, you cannot fit a regression line x is not explaining y or y is not explained by y. So, this is the basic understanding of regression analysis through ANOVA test and the hypothesis concept we all know this one right. Now, come back to our time series data suppose you have a time series data same concept we will use here now. Now, suppose you have a time series data y y 1 y 2 y 3 y 4 y 5 y 6 dot dot dot say. So, now, if you take the differencing as I mentioned say y t minus y t minus 1 right. So, it is nothing, but y 2 minus y 1 y 3 minus y 2 y 4 minus y 3 y 5 minus y 4 like this this right these are your first order differencing now. Now, think about the basic AR model think about the basic AR model what is the basic AR model generally we write y t equals to alpha plus beta y t minus 1 right this is what the basic and plus epsilon part will be there this is what the basic AR model of order 1 correct. Now, what you do we all know the basic AR model AR model of order 1 let us consider let us consider the basic AR model right and then this is the basic formula of AR model we all know subtract y t minus 1 from both side that if you will attest we are understanding now. So, now, what you do y t minus I know the hat part is there, but overall your basic understanding I am trying to pass to you subtract y t minus 1 this part you can think about this y t minus 1 u it will be what alpha plus say beta minus 1 into y t minus 1 t minus 1 plus epsilon that we are not discussing right. So, now, what happened this is nothing, but what this is nothing, but your delta y t delta y t right. So, the differencing is expressed in through this formula now this is what I have subtracted y t minus 1 from both side. So, it will be look like this. Now, this beta minus 1 is nothing, but this phi minus 1 of this phi minus 1 of the stochastic process of decay flux test look at phi minus 1 same logic. Now, what we will do we will propose a hypothesis say null hypothesis say beta minus 1 equals to 0 and alternative we are proposing beta minus 1 not equals to 0. Now, you do the p test if p value check say 0.05 you can check if the test is significant what happens alternative is active right. If p less than 0.05 this null h 1 h alternative is active now alternative hypothesis is active that means, beta minus 1 not equals to 0. If beta minus 1 become not equals to 0 let me put a highlight point you will get to know if beta minus if beta minus 1 not equals to 0 what does it mean. If not equals to 0 it means that this is not constant there is a relationship between the previous data and the current data. So, effectively what happens effectively what happens the data earlier you have taken and you have done the first order of differencing say still not this delta y say and this is y say this is still not stationary because beta minus 1 not equals to 0 not equals to 0. So, h 1 is active. So, you do one more round of differencing unless until the data become stationary, but in case your test is insignificant and you found beta minus 1 equals to 0 that means, null is active null is active in case null is active not the alternative is active. Suppose null is active then beta minus 1 equals to 0 null hypothesis is active if beta minus 1 equals to 0 what happens this part become 0 right. So, your data become constant data become data become constant. So, the differencing become constant if the differencing become constant what does it mean the data to some extent become stationary because this become constant right almost this look at this constant now little bit of variation will be there we are not focusing on that, but beta minus 1 equals to 0. So, it means that this data differencing is fixed now it is now steady now look at the data differencing first order differencing this is a delta y t right this is a delta y t this become steady now. So, delta y t steady means data become stationary you do not need to do further differencing stop there and go to AR or MA or ARMA process because data has become stationary now, but until unless your beta minus 1 or this differencing process does not hold true for null hypothesis you need to continuously do the differencing process. Here I have mentioned here you can see if the alternative hypothesis of the first degree of differencing cannot be accepted we apply AR process because first differencing are not required in the beginning itself your data become stationary, but in case the alternative is accepted alternative is accepted means what h 1 is active now active now in that case what happens if h 1 is active what happens you need to do differencing process differencing of differencing you continue that differencing of the degree D first order differencing D could be 1 D could be initially if the data are stationary D will be 0 if data are non stationary then do first order of differencing then you do the second order differencing third order differencing you continue this unless data become stationary this is what the Dickie Fuller test and once it is done you finalize your D you stop the criteria from this exercise of hypothesis testing and complete the Dickie Fuller test or non stationary to stationary conversion. This part is nothing, but the non stationary to stationary conversion of data and the stopping criteria when to stop through Dickie Fuller test there are many more method but Dickie Fuller test is very popular I thought of explaining that. So, I believe it is clear now now here you can see couple of examples. Suppose if D is 1 only one round of differencing you are doing right in that case you can select this model look at here D equals to 1 generally in ARMA process in ARMA process we use 3 component PDQ P stands for AR model we have already discussed that AR model of order P Q stands for order of MA model we have already discussed that D stands for the differencing part the differencing right. So, if D is 1 then one round of differencing if D is 2 two round of differencing that means, first initial data were not stationary you need one round of differencing and then take that data to your ARMA process. Look at the illustration here suppose this model here we can say that D equals to 0. So, data are stationary data are stationary it is nothing, but ARMA 2 1 because D equals to 0 data are stationary data are initial data were stationary right. So, you your D is 0 you do not have to do differencing look at here 2 actual data lack we have taken and one error this is the new noise white noise new white noise of your model you do not have to think on that just think the main error the model combination this is the AR part this is the MA part right this is the MA part MA order 1 AR order 2 if you add that how to add when to select the ARMA process we have discussed when are confused or you are not able to read the correlogram in that case you can take that or you know only 2 lack of like PSCF graph is positively standing line and SCF has a single standing in that case you can take that. Now, look at this this is what ARMA 3 1 3 order of lack old lack you are taking say it would be T minus 3 sorry and here only one error you are taking same logic here you can think it is we are simple taking AR 1 model, but differencing D equals to 1 one round of differencing you have to do then you select AR model look at one round of differencing you have done and then simply you have done the AR model look at alpha plus this some constant will be there could be in your model, but this is the white noise part this is what AR model look at this case here also you have 2 order of differencing AR 2 1 0 what means that you are not doing any MA process just first round of differencing and then you are using your AR process. In case your model are like this in case your model are like this say 0 1 1 in that case you have to select say you know say gamma 1 epsilon 1 plus say error part not here you have to take the first round of differencing say you know say delta this data you have to take and then there delta y t data you have to take and then there are error part formula the MA process that I have talked about is using this data you have to calculate your error part and the corresponding forecast then only you can develop this model of MA process with differencing data just take the differencing and follow the MA process you will be able to get a forecast, but you have to take the differencing data because that data is stationary the previous data original data was non-stationary. So, this way you can illustrate any type of AR MA or AREMA model right this is what the overall AREMA process auto regressive integrated moving average process I believe it is clear to everybody. Now, let us understand the overall steps of AREMA we have discussed the AR process we have discussed the MA process we have discussed the AR process MA process together now we discussed the AREMA also right PDQ D is the differencing the number of differencing of data you have done to convert the non-stationary into stationary. So, all this we have discussed now right 1, 2, 3, 4 now let us summarize this through the couple of steps first what you have to do the entire conclusion of AREMA process that 3, 4 sessions that I have taken that summary I am sharing with you now. Suppose take first take the data visualize the data understand the pattern draw the graph think whether it is a stationary or not if the data are stationary you directly come this step if the data are stationary you plot your SCF graph and PSA graph and select P and Q your D will not come into the picture select P Q depending on the your data pattern and the plot and the kind of correlogram structure you can select P Q and you can complete your model you can run the model. But if the data are non-stationary if the data are non-stationary suppose data are non-stationary if not look at if not in that case what you do you first do the differencing which we have discussed in this session today you first do the differencing. So, this is your actual data and do the differencing part take this data now and start your SCF and PSAF and check the ARMA etcetera because here in that case you might have a D equals to 1 or D equals to 2 depending on the number of differencing steps that you have done. So, if the data are non-stationary first you do the differencing and check the D key flow test which we have discussed now and stop there unless until data become stationary continue the process of differencing once it is done your D will be finalized D will be finalized here now D will be finalized and then you select P and Q based on that data and go for AR or MA or ARMA process there is no ARMA only ARMA or ARMA effectively only the differencing part if the data are non-stationary the differencing process has to be done and D key flow test has to be used to check whether the data become stationary or not. This conversion process comes under I part differencing part, integrated part because it is a to some extent a part of mathematical integration therefore, they call it is integrated but effectively it is a AR autoregressive differencing MA process that is it. Now one more part remember if the data are seasonal if the data are seasonal in that case you cannot do the differencing this way here what you are doing you are doing y 1 y 2 y 3 y 4 and you are taking a differencing y 2 minus y 1 right y 3 minus y 2. So, this way you are doing the differencing right, but if the data are seasonal in that case you have to take year on year basis differencing. So, quarter to quarter suppose you have a quarter on quarter to quarter basis data right you have like this say quarter 5 quarter 6. So, you write quarter 5 here quarter 6 here quarter 7 here quarter 8 here like this way is any quarter 9. So, this way suppose what I am trying to say share with you is that you when you take the quarterly data seasonal data in that case do not subtract from q 2 to q 1 to check your D key fuller or say differencing process you subtract q 5 minus q 1 because that is your say first quarter say fill quarter this is your a fill quarter of next year this is your say June quarter this is your say January quarter last quarter this is your December quarter. So, you quarter to quarter differencing you have to do look at here if the data follow seasonality then use 12 points differencing scale down the data by taking differencing if not immediate 12 point means it is a monthly if it is a monthly, but if it is a quarterly then you can take 4 period. So, it is depending on say 12 or 4 depending on the quarter like you know period whether it is a monthly data or it is a quarter leader if it is a monthly data then you have to subtract January to January February to February do not subtract from February minus January because it is not a sequential data it is a seasonal data. So, otherwise your variation will not be checked effectively and seasonality conversion to decennialize data or to some extra stationary data will not be done effectively because seasonality involved there, but if it is a quarter leader then you subtract from quarter to quarter April to April previous year April. So, the in this case 4 differencing will be done and then you can take the data and you can carry forward your forecasting process I believe it is clear to everybody. So, this is what the overall summary first you check the data stationary or not if stationary go to directly SCF and PSF graph select P or D and complete the model AR or MA or ARMA if not that means data are non-stationary you do the differencing if the data are seasonal then do the differencing in seasonal manner either month to month differencing or quarter to quarter basis, but in general if the data are non-stationary and steady data like you know in sequential data are they are like you know time temperature or etcetera or say crude oil price you are checking device or say stock price in that case what you do just take the older data immediate pass data and lack lacks and take the differencing. So, far what we have discussed and check the decoupler test once the differencing are done non-stationary conversion are done decoupler testing are over and then you go to that data and start developing your SCF PSCF and you know select your D select D selection are done PD and complete your ARMA model and make the future prediction this is the overall process of ARMA right. Here if you would like to see the final summary you can see for AR model overall summary your you know we know that correlogram reading your SCF will have exponential decay, but PSCF will have a cutoff right PSCF will have a cutoff SCF will have a exponential decay this is for AR model. In MA model in MA model for MA case this will become for MA case this will become your SCF this will become your PSCF detail we have discussed because SCF is sufficient to select for MA process because you are considering only the error term right only error term. So, they are already independent you do not need to check the multicollinearity or you do not need to remove the partial effect direct data you can take and you can because you have already done the check the data and only error for term you are adding in your MA process. Now, if you are confused and you are not able to select the model suppose both are having some you know standing point in that case you can select ARMA PQ or both are having exponential decay you are confused to select then you can you can select ARMA 1 or ARMA 2 to maximum. This is what the total summary of AR, MA and ARMA right and there is no ARMA ARMA comes only in the beginning effectively ARMA should come in the beginning, but I have discussed in my entire module of ARMA the ARMA part the final part at the end we have started with AR process we have spent significant time on that because that is the major part of model actual lakhs you are taking actual data you are taking and actual regression doing you are doing, but it is auto regression that is it and then you go to MA process if you want to include the error terms with the mean data that we have discussed then we understood the ARMA process and then we are going to ARMA because if the data are non-stationary then only you need to convert non-stationary to stationary and then this data you get and then you check either of them that is it. So, this is what the overall process of ARMA. So, now let us illustrate this ARMA model through one Excel data here we have come to the Excel and if you see this column, column number B here we have taken a new data not the TCS data. So, this data it is a recycling company US product handling company. So, cerebrine integrated technologies we have taken their data and this data to some extent follow the non-stationarity. The TCS data that we have illustrated earlier does not follow non-stationarity to some extent that was stationary. So, therefore, we have taken a new data a plus 3 months say like you know October to December same way and then this data we have plotted here you can see this data let me you know make it like this and then you know here you can see the graph look at this graph. So, this graph is nothing but a non-stationary data you can see here block wise it is changing. So, just we have taken 3 month data if you take more data you will get to know suppose this data is following non-stationarity here you can see that data is shifting from location to location. So, it is to some extent non-stationarity. So, you cannot directly run this data for your Arima model right. So, you need differencing. So, we have done the differencing here you can see the differencing this graph let us see first the process of differencing look at here we have done the differencing here look at y t minus y t minus 1. So, delta y t the way we have illustrated. So, if you make it increase the length of this you will get to know some yeah this all this calculation we have done here you can see suppose this one say you can see y t minus y t minus 1. So, this is what your delta y t. So, we have done this first order differencing and then this data this column number d will be your input data sets because this data is a stationary data here you can see the graph. So, suppose only first order differencing process we have illustrated here and the entire Arima calculation we have done like Arima process in the previous session we have done it, but you can use your Python and you can test all this DQ fuller process that I have told you. Now, suppose these data are stationary now first order differencing are stationary you can see the data now look at the data range it is 0.62 you know point minus 0.8 and it is in between the range it is the fluctuation has gone down too much right. So, it in between 1 gap 1.5 gap in the range. So, you can say that to some extent this data are stationary it is a it is a quite small range, but this one has a like 5.5 to almost 8.5. So, it is a big range where they are, but this data does not have much range. So, we are assuming that this data are say stationary. So, we have taken the first order differencing as our input data to the Arima. So, this is your stationary data suppose this column number d now what we will do let me put a color so that you can understand better you should not have any confusion. So, if you come here and if you put a different color say this color right. So, this is your differencing data first order differencing data we are assuming that this is stationary the first one are not stationary. Now after differencing you do simple Arima process that we have done you take the average of this data then calculate the center of them calculate the center of them right center of them and then next step you initialize your residual suppose it is 0 then you start your forecast. So, this is your beta and gamma you have to optimize it the way we have done in Arima process same we are repeating the Arima now just we have done the differencing right and then you get your forecast with your center data after differencing and then again you get the residual with your forecast and then again use this residual and the center data you get your next forecast look at here you get your next forecast and drag that you get the forecast. Now this is the forecast with your center data and the error term right. So, one term of 1 1 Arima 1 1 we have taken to some extent you can say finally Arima 1 1 1 you can see Arima 1 1 1 right because we are taking first order differencing d equals to 1 in this case d equals to 1 p equals to 1 q equals to 1. So, this we are taking right. So, now you get the forecast so this is your final forecast. Now you can optimize this here in this not in this sheet, but other in other sheet I have done the optimization also here you can see the optimization of beta 1 and gamma 1 you can go to data and you can go to solver and you can optimize the data. So, you have done the optimization you got the final beta 1 beta gamma 1 and the corresponding forecast come back to this sheet again. So, now here what you do you do the final forecast. So, final forecast here you I have written here you can see the final forecast look at the yellow cell let me increase the font size this final forecast are nothing, but in terms of delta yt because this is your center data this is your center data and this is your delta yt right. So, this forecast whatever you will get in terms of delta yt. So, here this is actually this forecast look at column number n this forecast it is nothing, but you are based on the center data like the Arma process. Now that if you go back that will be in your with in terms of delta yt. So, this delta yt minus mean data minus this mean data we have placed with small yt remember the small yt term of Arma model. So, that we have replaced now this is in terms of forecast of small yt. Now that you have to replace that you can read just this read just after calculation you will find this. Now this is your forecast in terms of delta yt not the final. Now what you have to do you have to replace delta yt with yt minus yt minus 1 that if you do here you will get any forecast of any period. Suppose here we have done the forecast for 25th period you can see here. So, now if you see forecast for 25th period are nothing, but the actual yt minus 1 of 24th period look at here yt minus 1. So, you replace this yt forecast equals to yt minus 1 will come here. So, yt minus 1 y24 b24 plus 0.002 the adjustment plus the coefficient beta 1 coefficient you can say into the corresponding delta yt minus 1 this is your delta yt minus 1 minus this error term here it comes as a minus, but in general it is another gamma right. So, gamma into the error term the previous error term you just do it. So, this is your forecast for 25th period with your Arma model in excel we have done the illustration. Similarly, I have shown one more illustration suppose forecast for the say 48th period. So, in order to do the forecast for 48th period you take the data of 47th period look at the data of 47th period the actual yt minus 1 and the delta yt and the error term because final formula we have done here. So, this in this final formula you put the data and you get the forecast this is your forecast you would like to see the accuracy level here you can see. Suppose for 47th I am talking about if you think the data here look at here. So, 48th period actually 6.8 and the error term if you see the error term is nothing, but just you know minus 0.21 and here if you see the forecast is coming up to be 7.1. So, it is almost closer to the actual forecast and the error term with your data sets also. So, therefore, this is the forecast for Arama model. So, what is the summary of this final illustration? First you do the differencing that means you convert the non-stationary data into stationary then select your order based on the differencing process. Suppose you have taken one order of differencing and we are assuming that this is the stationary data then you start your Arama process. So, this all this actually nothing, but the Arama calculation it is nothing, but the Arama calculation you can recall the Arama session. So, that we have done, but now you have to do the adjustment of your data. This adjustment because you have taken the difference we have done here and then you can make forecast of any period that you want you just follow the process and drag it you will get the forecast. This is what the Arama of final module or final process of autoregressive integrated moving average process. I hope it is clear to everybody. Thank you.