 Hello everyone, welcome to the class of business forecasting. In the previous session, we discussed exponential smoothing model. Today, we will discuss trend projection and hold model. Hold model is extension of exponential smoothing model. So, if you look at the you know overall syllabus here, we have already entered into different type of time series methods and last couple of sessions, we have covered the basic name method of time series. We have discussed the different type of moving average methods and also the exponential series methods that the first on the simple exponential smoothing model. Today, we will extend the exponential smoothing model and we will enter into the trend analysis and in order to address the trend analysis, which model is most suitable that is called hold model. We will study that hold model today with illustration as well. So, let us first discuss about the trend projections. We all know in the session of components of time series, we discussed there are major four components. One is the trend and the seasonality and irregularity or say randomness and say cyclical. So, in that session, we discussed detail about trend right. If the data, the historical data has a trend, then how to you know address this issue or first you have to understand the behavior of the data and in which whether there is a trend or there is a downtrend, that understanding of the historical data, the past behavior of the data, you need to understand. Suppose you realize the data after drawing the graph or analyzing the behavior of the data, that there is a trend in the underlying data. So, in that case, you have to make a forecast according to the trend of the data, pattern of the data. So, there are many methods called you know say regression analysis or say trend line projections using different type of extensive time series methods. So, here you can see a sample illustration. Suppose you all know right, if you have a two variable data say independent variable x and the dependent variable y and then you can draw a trend line or say basic regression y equals to mx plus c. Here, I have written y equals to a plus bx and you can see the trend line here. So, this is the trend line right and look at the data pattern. So, this is data or nothing but the scattered data and if you want to find a relationship between that data, you will realize that the data is a uptrend. So, you can draw a line y equals to a plus bx or y equals to mx plus c. Effectively, you have considered a linear trend here using least square line, but if you see the behavior of the data, it might follow it might follow a uptrend. So, therefore, you need to fit a trend line or say regression line say to capture the pattern of the data. This is what the simple trend analysis. You have done it in couple of other sessions also. Even we have the regression analysis session where different type of extension of regression analysis and there also we will discuss detail of trend analysis. So, what are the major components just for your quick recap? What are the major components of a trend line or trend projections? You know you have to calculate the intercept and the slope and using that you can calculate the y equals to mx plus c y equals to a plus bx. So, this is what the basic trend line projections and today itself I will show you through say excel illustration how this can be used for trend projection. Now, the question is that suppose data has a trend say uptrend or downtrend. So, in that case in which situation this trend analysis or trend line can be captured or can be integrated to a time series data model. There are many models which can be used. There is no specific utilization or you know aspects of trend analysis or say trend projections. It is an integral part of different type of models to capture the trend of the data. Look at the figure here. It is just uptrend right. It can be seasonality also with uptrend. So, this type of data you have to make projection also. So, there will be different type of you know pattern of the data which captures uptrend or downtrend movement. Here the most popular is the linear regression where you have a dependent variable and independent variable and you can find the like causal relationship or you can say how the independent variable is explaining the dependent variable and can create a regression line which I have shown you just in the previous slide. There is another method called decomposition method which actually you know takes care of the seasonally and trend. Look at the figure, the second figure that I have drawn. So, there is uptrend and there is a seasonality. In order to address that you can use the decomposition method. We will discuss detail of decomposition method in a separate session. There is another logic called Arima. So, in Arima also you use trend line analysis. It is effectively say auto regression analysis. So, there also we use the trend line or the trend projections through Arima model. So, everywhere you are using it and look at the main point that of today's discussion that is called in exponential smoothing also you know when you enter into the whole model which we are going to discuss today. You will realize that there is a trend analysis or you can see that if there is uptrend in the data in the basic model where exponential smoothing you have used and you failed to get a base prediction because exponential smoothing simple exponential smoothing cannot capture the trend part. So, therefore, you need to use the hold model or hold exponential smoothing for trend analysis. There is another method which is extension of exponential smoothing as well as the hold model which is called the hold winter method where you take the seasonality and trend together which is alternative to decomposition method. We will discuss all these one by one. So, let us today discuss about the trend analysis through hold model. Just now I mentioned that hold model is extension of exponential smoothing model. If you think about the moving average concept that we have discussed in the in one of the session, there we talked the data to some extent steady and we are taking the trend we are capturing the trend where to some extent data you can average the data and average data are been you know taken as a forecasting value or the future value and that we are dragging that we are carrying forward as a movement of the data through basic moving average concept. And when it comes to the simple exponential smoothing there similar data similar concept just we do not take the fix combination of the you know range of the data as a moving average. Here in every average process every iteration we consider all the old data we do not do not drop anything here we always consider all the previous data and you drag the smoothing values and that weightage smoothing constant need to be calculated need to be optimized which we have done in the last session just I am giving a recap of that and then using the basic data or stationary data or steady data you can make a use of simple exponential smoothing model and make a forecast. But remember that in both the cases whether it is a moving average model or whether it is a simple exponential smoothing model we did not consider the trend part because trend was not present over there in case trend present over there and if you consider the simple exponential smoothing model for that data you will not get a good forecast there will be huge error because trend part cannot be captured through exponential smoothing model to avoid that drawback of exponential smoothing model hold has come forward or hold has developed model which is called hold exponential smoothing model where he capture the trend part. So, here you know he has taken the two base component one is the label another is the trend if you remember the exponential smoothing model it was like say you know y t plus 1 say hat equals to y t actual alpha of this plus 1 minus alpha alpha is the smoothing constant the weighted average actually of y t hat say. So, this is the model in exponential smoothing we discussed right this y t you can expand hat you can expand and you can add all the time series data and you can get a exponential smoothing model. We have discussed that hold has played with this part he has extended this exponential smoothing model and he has played with this part and he has put this into two component one is the label another is the to some extent you know trend say trend. So, he has developed these two component put together which will be replaced you know in place of y t hat and he has dragged that process let us see how it works. So, in that case since he has taken two information two new component one is the label value and another is the you know trend value. So, one more smoothing component will come that is called beta alpha we discussed in the exponential smoothing model that is the smoothing component or the you know that we optimize initially specify and then through optimization process we optimize that I have shown you in excel illustration the best alpha you will get it for the data to the initial period and rest you can distribute to the all the older period. Now, here same alpha will remain there here like as a input data for the label value base value, but since you are capturing one more component you are adding one more component in hold model that is called trend part that is called the trend part. So, we will add one more parameter called beta which will be a smoothing component for trend. So, these two parameter now we will optimize once the structure or the algorithm or the flow is ready the model is ready we will use we will optimize alpha and beta together. One more thing here sometimes people do not call this hold model as a you know simple extension of exponential smoothing model people sometimes called is a double exponential smoothing model why it is a double exponential smoothing model because you know here in basic exponential smoothing model you have only one you know exponential series right one data sets you are taking and you are calculating the forecast by taking combination of this approach right and you are expanding this in terms of alpha or say you know you are smoothing out the data and you are taking a combination of that through the smoothing constant alpha. So, one exponential series are there, but now here you are doing one more exponential smoothing that is called the trend part also. So, level part also you are doing one exponential smoothing and trend part also you are doing and we are capturing both and we are taking some of them and we are making a forecast for the future. So, therefore, we call it as a sometimes as a alternative to hold method as a double exponential smoothing models. Let us see how hold model work. So, as I mentioned it required actually it depends on the selection of the smoothing parameter alpha and beta. So, that parameter will optimize no worry we can see that parameters throw excel solver like the way we have optimized the alpha for exponential smoothing simple exponential smoothing model similarly here will optimize alpha and beta for hold model also through excel I will show you that. Now remember the range of alpha like same similar like exponential simple exponential smoothing model the range of alpha beta will try between 0 to 1 and you have initially you can give any value and you can optimize it later. And once you optimize it and you found the best alpha beta you can use that using these two equations you can get the final forecast for the forthcoming period. So, these are the you know ingredients you require. So, level at time period t initially you need to assign some level say l 0 you can say initial value. Initial value how to calculate the initial value that also I will show you say you know maybe initial couple of period average you can consider as the level value or just immediate value the first period you can consider as a initial guess to start the iteration. Similarly here also you have to start with the initial level and then get the iteration or the effect of hold model and then the trend level also the initial value of trend. So, for that also you can take the gap between the data set and that you can consider as the trend or if you do not understand that or if you do not think that no I will not take the gap I will take the 0 as the initial trend no doubt you can take that also. So, all options are open with you I will show you through illustration and excel also. So, once you get the initial value of level and trend say l 0 and say you know t 0 initial value if you get it through your assumptions then your model iteration will start and then you can drag the iterations automatically the new level and trend will be calculated automatically and your total sum will be added one by one and you will get the forecast for the future period. So, this way you know you may carry forward your forecast and you may get the trend projections through hold model. Now this two parameter are required alpha and beta for calculating the level value and trend value in every iteration and actual forecast will be coming as a sum of them and these are the general notation for forecast calculations in the iteration. Let us see how the model work here. So, I will illustrate this model now here let us recall the basic exponential smoothing model which we discussed in the previous it will be easier for us to understand. So, it is like you know y t plus 1 forecast equals to remember alpha into y t actual plus 1 minus alpha into y t forecast. So, weighted combination of the actual and forecast of the previous period we discussed this way. For example, say suppose you are at say y 5. So, then y 5 equals to alpha into y 4 plus 1 minus alpha of y 4 forecast that immediately you got you drag it you will get y 6 y 7. So, this way you will get the forecast by changing the actual data of previous period and the forecast period and you will get the exponential smoothing model the simple exponential smoothing model. Now this concepts will be using here in a different manner. Now let us see this basic model let us keep the general formula of exponential smoothing basic simple exponential smoothing and let us see what changes we have done here. Actually if you see this part simple this part forecast part we are replacing with this level and trend with this level and trend look at this look at same the rest all are same alpha of y t plus 1 minus alpha is as it is only for level calculation then the trend part we will discuss. So, you can see actual value say weighted combination say 50% of actual plus 50% of say the forecast. So, that forecast we are replacing with the initial guess of level and trend. So, that sum is nothing but your forecast. Now once you get it like the way here I have told you. So, if you get it that will give you the support for initial level calculations. So, that level you calculate as L t and then once you get it you calculate your trend look at the trend now trend part is nothing but here also it is a weighted combination say 50% say beta say 50% or so, 40% 50% of the level gap the stairs gap and the immediate level that you have calculated that and the previous level that you have calculated that are the stairs to two stairs value that gap plus the previous trend you had in the previous period that 50% of that. So, this combination this weighted combination say 50% beta say 0.5 50% of the level gap plus 50% of the previous trend that weighted combination is your new trend. Once you get the level and trend you add that you will get the forecast. Let us see one example suppose here suppose you calculate say for example say Y5 suppose. So, your Y5 is nothing but it is nothing but say you know L4 sorry plus T4 right. So, this is what your forecast now what will be your L4? L4 will be L4 will be alpha into Y say 4 plus 1 minus alpha of rather than alpha 4 forecast we are writing you know L3 plus T3. So, this combination is nothing but your Y4 forecast that value into corresponding weightage plus the actual value of fourth period into the corresponding weightage. So, this way you calculate the L4. So, this L4 you will use to calculate your T4 using this formula. So, this way what will be the calculation for T4? So, T4 formula calculations will be say you know beta into level gap say L4 minus L3 plus 1 minus beta into you know previous period weight. So, this combination is your forecast for period as a trend forecast for that particular intermediate period we call it is intermediate why I will tell you through excel. So, this way you can calculate your forecast. So, now for example say one more example I can tell you suppose here you know you want to calculate say you want to calculate your say L2 because you will start from there right. So, suppose L2. So, L2 will be alpha of Y2 plus 1 minus alpha of you know L1 plus T1. Now, what was your L1? L1 as nothing but alpha of Y1 plus 1 minus alpha of you know L0 plus T0 this value you had from there you are calculating. So, this is what the forecast for you know hold model by adding two component level value and trend value. Let us see this through a illustrations you will get a better picture actually. So, here you see the sample data look at the data look at this data you have a uptrend right. So, it is clear suppose you have analyzed in excel and graph and after processing of the data you realize the data has a trend. Now, that trend you have to make a forecast for the future as per the trend line now. So, how we will calculate the trend line separately through hold model and you can say add that with the level value. Remember again in exponential smoothing model you do not have the trend part it is simple forecast weighted combination of actual plus forecasted value and you are dragging at that value that is the simple exponential smoothing model. But here you are segregating the data the forecasted into two component one is the level as a trend and then you are adding them together again then you are adding them and you are getting a forecast for the forthcoming period. So, here suppose you have the data an initial guess I told you L0 you have to calculate the L0 L0 and T0. So, how we have calculated say suppose here we have assumed that initial period was 11. So, that we have taken as the forecast right and say gap between initial period and previous period current current period suppose the gap you have written here is a trend value you might say here it should be the one why if you consider 11 then to suppose here I am as for your information suppose it could be 10 or 11 whatever whatever. So, that gap you can consider or say suppose you had another period say previous period suppose some other value say you know say 8 something. So, you have taken a gap between this 8 and 11 and then 11 and 12 and then suppose take that average of that gap suppose here you will get say 3 and here you will get 1. So, take the average of that as the trend whatever as a 2. So, whatever initial value you can assume not a matter couple of initial period you can consider and then the average you can consider as the initial level also or the same period as it is you can consider as the initial level to start with and the trend also you can consider 0 also if you are not interested take the average of couple of previous periods gap and that as the trend what you can consider the 0 as the initial trend also. After certain time period you will see that the system the iteration will capture the actual trend of the data not a matter. So, these are the initial as you know assumptions to start with the level and trend. So, once you get the level and trend what happens you know you add these 2 we will get the forecast for the next period. So, for L1 you got the forecast with level plus trend. So, initial forecast we found it now look at level plus trend right we got the error you can write down error could be say you know 12 minus 13 equals to 1 we are not discussing the error part here in excel we will discuss because that part we have already discussed in measure of accuracy session. So, now let us focus about only the calculation of hold model now. So, now you found the forecast for the second say first period say with initial value. So, L0 and T0 worked as a intermediate level and trend to start with your initial level and trend to start with your hold model. So, you got the for now actual hold model will start calculation will start now. How we got the forecast intermediate level value for you know for this particular first period. So, how to calculate the forecast for the next period? So, let us say how we have calculated here. Suppose here you have the actual value say actual value of first period how much it is 12 it is 12 look at here it is actual value 12 plus say 50 say 20 percent we have taken alpha for this particular data sense we have taken alpha 0.2 and beta 0.4 initially we will optimize this in excel later. Now, suppose alpha 0.2 20 percent of actual 12 plus 1 minus 0.2 say 80 percent of the level and trend initially you already calculated level and trend what is 11 plus 2. So, that you can take you will get the forecast for level intermediate level for that period which will be used for second period actually which will be used look at here T plus 1 equals to L2 a LT plus in a trend. So, that means, if you calculate Y2 hat forecast it is nothing but L1 plus T1 remember this this L1 and T1 are the intermediate level and trend. So, that we are calculating right. So, here you see. So, this 12.2 I have I think in the next slide I have written here look at this 12.2 is the you know intermediate level calculations hold it. Now, the trend also you have to calculate then only this T1 also you have to calculate then only you will get the forecast for you know by adding these two you will get the forecast for the second period. So, let us see how we calculate the trend path now. Now, what happens this new intermediate level you found which is nothing but your L1 and L0 you have assumed it. So, take the gap. So, this gap is nothing but the one component of your trend say 40 percent of that because beta you have assumed 0.4 and 60 percent of the previous trend. So, weighted combination you are taking every time. So, that is nothing but forecast for or intermediate value of trend or say forecast of trend. So, this you got as 1.92 add these two you will get the forecast. So, you look at I have kept this in the previous row. So, that you may get the actual forecast and you may get the error here. So, what will be the next forecast you add these two you will get the forecast value for next period as 14.17. So, it is nothing but the forecast for the second period what is that it is nothing but actually why 2 had actually clear. So, now, if you get the difference between this and this you will get the error now for this particular period right. And then if you drag the formula you will get the forecast for the next period. Let me show you one more calculations of the level here. Look at here this we have calculated the sum of level plus trend actual forecast we found now. Now, for one more iteration for your information I am calculating one more level to get a better insight or clarity because first time you are understanding this. So, here you see. So, how I have calculated next trend now this is actually L 0 this is L 1 this is L 2. So, this L 2 and T 2 if you calculate you will get actually you know y 3 you will get the forecast of next y 3 hat sorry y 3 hat. So, how we will get this 15.12 look at what is your you know look at 50 percent say 20 percent of the actual value of that period actual value of that period plus say 80 percent of the previous period level and trend what we found previous period level and trend you found as you know 12.8 and 1.92 which is nothing, but 14.72 you take the 50 80 percent of that you will get the new level intermediate level which is nothing, but 15.15 clear now. So, this is what the intermediate level. So, once you get this level you will get one more using these value and the previous trend previous level you will get the gap and the previous trend you already have take the weighted combination with beta you will get the forecast for trend. So, these two once you get you added you will get the forecast for the next period. So, this way you can drag actually look at here you will get the similar calculations I have shown you also trend calculations you are now you all intermediate level you found. So, take the gap between these two and the corresponding previous period trend you already had take the weighted combination and you get the forecast for the intermediate period of trend and that trend which is what is this trend this is actually T 0 this is T 1 and this is T 2 you found it now. Now, you got say you know here L 2 you found here it was L 1 it was L 0. So, L 2 and T 2 you got you add it you will find the forecast for next period what is the forecast forecast is 17.8 final forecast for that period you drag this process drag all you will get the forecast for the 10th period here using this calculation and this calculations level 9 and level 10 9 as intermediate. So, you will get the forecast for the 10th period here I have done it look at here. So, you find the forecast this is what you know intermediate level and intermediate trend through the iteration and here you will find the forecast for the next period which is nothing but the sum of them and you have to write in the next row. So, that you know actual data minus forecast you can make a difference and you get the you know error part or residual part. So, now you get the forecast using hold model by calculating two component or two series therefore, we call it is a double expansion smoothing model one is the level calculation level series another the trend series both series you are calculating every iteration and you are adding them you are adding them you are getting the forecast you are adding them you are getting the forecast right. So, this way you are getting the final forecast through hold model this actually trend capture the trend look at the data now forecast. So, 13 14.7 to 17 point it is increasing right 20, 22, 24, 26, 29, 31, 35. So, effectively it is increasing despite data might have little bit of here and there, but the hold model actually making a forecast with the trend line analysis. So, this is what the advantage of hold model who takes care the trend part and the trend analysis aspects also. Now, the corresponding error you can calculate actual minus forecast and the corresponding MAD by taking the absolute value of the error and the average of them or you can take the square of the value of a all error and then take the sum and then the average of it will get the mean square error and if you take RMS square root you will get the RMS. This we have already discussed I will show you in excel. So, this is what the hold model you want to see the who capture the trend the graph look at the graph. See how accurately it is actually making the forecast as per the pattern of the data, but if you would have used explicit smoothing model your forecast would have like this your forecast would have like. So, here we could have get more error over here. So, this is what the advantage of hold model. Now, let us go to the excel and understand this throw excel illustration. Here you see the excel model if you look at here. So, initial value we have assumed 11 and 2 as per the data I have shown in the property. So, same data we have taken and look at the first and this is the forecast for initial period right by adding L 0 and trend 0. Now, you calculate L 1 what how we will calculate L 1? L 1 is nothing, but say 20 percent alpha 20 we have kept here alpha 20, 20 percent of the previous value plus say 80 percent of the previous level plus trend or you can say the previous forecast value whatever. So, take the weighted combination you get the forecast here. Now, if you drag it if you drag this you will get the forecast of level. Now, trend, trend is nothing, but this level initially for say for this trend how you calculate 1.92 the gap between this level value look at my mouse the gap between these two level value say 40 percent weightage of that plus 60 percent of the previous trend. So, this is your forecast for the trend. Now, you drag it you will get the forecast you drag this you will get the forecast for trend also. Now, once you get the last period level and trend we can get the forecast for 10th period or the forthcoming period actual forecast what is that say 35.16 with corresponding NEC and RNEC. Now, the question here is that whether this alpha and beta are the best or not? We have discussed that this is the initial assumption right and we have to optimize the best alpha whoever whatever the initial value you start that is not a matter based on the assumption the data and the corresponding initial guess you can start with if you have a large amount of data initial level is not a much matter, but you have to put a reasonable value if you are confused. So, you can take initial level as a trend value as 0 and level value as it is suppose here you had 11 actual value you can take as it is not a matter after that also you will get similar forecast. So, initial level and trend is not a matter or couple of periods as I mentioned you can take the average of couple of periods as the initial level average of the gaps of couple of periods can be trend or the initial trend can be 0. So, then you can drag the model of hold model. So, it is you are adding two series therefore, we are getting the trend separately along with level base and you are adding them you are making a forecast right. Now, the question here is that two point one is the how to find the alpha and beta best alpha and beta for everybody all participants if you run it with initial guess of alpha beta at the end you will find the best combination of alpha beta I will show you. Second point once that is done this trend you will find so effectively once the optimum value of alpha beta are found you will get to know not only for you will get advantage of not only getting a forecast for the next period say 10th period even for you know say period 11 also you can make a forecast here by taking this trend value. For example, suppose you have a say level value is this much and trend is this much. So, for next period your forecast will be say 32.48 plus 2 into here you have added 1 right 1 into level 2 into 2.68 say. So, you will get the forecast for 11th period also. So, it is so accurate the trend has been captured so accurately that you may get forecast for initial couple of period also. This is for the advantage of you know hold model where once you capture the actual trend path for the last period by multiplying 2 times 3 times etcetera couple of initial forthcoming period also you can make a forecast and you can make your business plan. It is not like that you cannot take another 12 months etcetera or 12 periods or say so many periods that is not acceptable, but initial couple of 2, 3 periods 4 periods you can actually you know use this trend path and you can make a forecast. Therefore, sometimes people call it as this final forecast of hold model as y t plus 1 forecast is nothing, but level plus m into trend. So, this m is nothing, but the additional period. So, if m is 1 then only for 10th period you are making forecast. If m is 2 you are making 11th period forecast right. If m is 3 you are making 12th period forecast. So, couple of additional period forecast also you are getting or you are making prediction by calculating the final trend and the level. You know you do not have to wait for the 11th period actual data 12th period actual data then you make a forecast of hold model that up to whatever the data you have using that you can make a forecast for the forthcoming also. This is for the strength of the hold model. Now, let us go to finalizing of alpha and beta or getting the optimum value of alpha and beta. So, what we will do now? Let me erase this part now. We will finalize the alpha beta now. So, here if you see go to data and go to solver as in the previous session I have discussed which value you want to minimize the RMSC right the error you want to minimize. So, select the RMSC. So, suppose here I have kept it in F18 cell. So, I have F17 cell. So, I have kept here as a base value as a subjective value F17. So, if you keep in other cell calculate the RMSC value or error value or MAD value or MAP value you can select that also not a matter. So, it is a minimization problem it is not a maximization you have to minimize the error. So, do not select the max select the minimization. What is your guess cell which value you have to finalize alpha beta right you can write alpha, beta here this JA3 and JA4 cell. So, rather than writing 1 by 1 since I have kept both together in the same place go and select both together 1 by 1 you can select also by taking comma by giving comma or you can drag both together because both we have kept in the same column. Now, the variables we have been selected now in this particular changing variable cell now you have to put the conditions right restriction like optimization you have been. So, what are the restrictions the range of alpha and beta cannot be more than 1 it cannot be less than 0. So, this range you have to put and look at here I have kept here and variables are non negative that you have to mention also. So, I can you know here how will you add this conditions here I can delete this and I can show you again suppose I will delete all this and I will add it again from the scratch. So, add the conditions. So, these two variables initial guess should be less equals to 1 you add and these two variable should be greater equals to 0 right. So, you put them done you have put the conditions here and also make sure that it is a non-linear problem because RMSE is a square root. So, it is non-linear you do not select the simplex LP. So, non-linear solve it you got the best combination of alpha beta for these data sets what is alpha 10 percent beta 100 percent. So, this is your forecast for the data and your final forecast is coming up to be 35.16 and corresponding RMSE is 2.83 you put any other alpha say 0.5 0.6 initial guess and you draw run the date run the forecast you will come up the same alpha beta look at here again you are coming up with the same alpha beta 10 percent and 1 and RMSE is same. So, this is what the forecast throw hold model now if you see suppose for downtrain uptrend data we found suppose I have changed the data look at the downtrend pattern of the data have been captured now same data kept in a in a reverse manner now suppose downtrend is the data the data pattern is in downtrend now will hold model work here also let us see what could be the forecast suppose here initial guess I have assumed 37 and trained as a 2 initially and I have run the same model same calculations let us say initial alpha beta have kept 0.2.4 and I will run the model go to data go to solver and same as it is I have kept the formula just solve it look at change changes in the alpha and beta and the corresponding RMSE look at the optimum alpha beta is 60 percent and 61 percent best whatever initial value you skip not a matter they will take care the best optimization value of alpha beta or best combination of alpha and beta the optimization solver will provide you the best in detail I have discussed in the measure of accuracy session or you know alpha selection in moving average method sorry exponentials moving average method as well as the exponential smoothing model. So, here also we are exploding the same logic and you got the RMSE and the forecast look at the forecast for 10 period now 10.59 and using this trend you can make the forecast for the next period also suppose 11 period forecast you want to make what could be the 11 period forecast using hold model you do not have to wait for the actual data of 10 period. So, 13 plus 2 into now m is 2 now 2 into the trend this trend is very strong now. So, you can get the forecast for 11 period also couple of period not all period you cannot make forecast for the so many previous forthcoming period succeeding periods initial couple of period forecast also you can make using this trend analysis like hold model. So, this is the hold model and the advantage of hold model is that it takes care of both the you know uptrend and non-trend data with a better accuracy of forecast and this model is more reliable as far as you know trend data is concerned. Now, let us come back to today's session and one more part I have to discuss with you these examples particular remember the data this example we have discussed through exponential simple exponential smoothing model also and there we realize that this data has a downtrend data and exponential smoothing when you use exponential smoothing we found that that it ended up with 1210 as the forecast 1210 like name method as the forecast it did not capture that trend part right and we stopped there with exponential smoothing model and concluded that exponential smoothing could not capture that trend part effectively. Now, we have run the same data using hold model and optimum alpha beta we found and look at the forecast 11.62 look at 11.62 is the forecast smoothing model ended up with 1210 it could not go down further because alpha cannot be more than one remember the discussion and here you are finding the forecast of 1162 which is much better as far as prediction is concerned to some extent is capture the trend of the data effectively look at the last next graph look at here this graph remember we have discussed detail in the simple exponential smoothing session. So, look at with alpha 0.2 forecast was here how much 1320 around look at 1324 I think if I remember something the forecast was like this and when you increase the alpha value to have 20 something and then when you ask the excel to find the best alpha it ended up alpha equals to 1 and the forecast was exactly 1210 remember. So, it could not capture the trend effectively because there is a limitation of exponential smoothing model and exponential smoothing model is very suitable very much suitable for steady data stationary data, but when it does not stationary it comes into the picture or trend come into the picture you select the hold model and look at the forecast of hold model with the best alpha beta combination it is giving a very accurate forecast this is what hold model and if you go back to the excel and you see here look at the excel look at the same data initially alpha have kept 0.2 0.4 and forecast is at 1253, but once you run it through hold model for the same data you will get a forecast of 1162 look at this with the best RMSE and the combination of alpha is 0.65 and 65 percent beta is 1 and this is what the advantage of hold model the same problem we have discussed in exponential smoothing also simple exponential smoothing and now through hold model and here is the you know outcome the competitive analysis between hold model and simple exponential smoothing model and why people extend the hold use the hold model when the data has you know or extend the simple exponential smoothing model with trend data and use hold model here is the you know illustration. I believe you understood the difference between simple simple exponential smoothing model and hold model and in which case you should use exponential smoothing model and in which case you should extend the exponential smoothing model and use the hold model that is clear. So, today with that let us conclude the session of trend analysis and hold model. In the next session, we will extend this hold concept let me come back to the that particular slide where we started today here. So, if you remember in the beginning of today's session we talked about hold exponential smoothing model to discuss today. In the next class we will extend the hold exponential smoothing model to hold to inter exponential smoothing model or inters hold exponential smoothing model which takes care of seasonality and trend together. So, this combination only trend part we have discussed today. If the seasonality is also involved in the data in that case you need to use the inters hold model and this we will discuss in the next session. Thank you.