 Hello everyone, welcome to the session of time series analysis. Today, I will discuss winter's hold method. In the previous session, we discussed different type of exponential smoothing model like simple exponential smoothing model, then hold model where you you know consider the trend part if the data has a uptrend or downtrend then you can extend the exponential smoothing model by considering hold model. Today we will extend the hold model by considering the seasonality also. That means if the time series data or past historical data behave like the seasonality as well as the trend included then which model to bring or which model to use to make the forecast for the future. In India generally you know there are four type of quarters say quarter one, quarter two, quarter three, quarter four or you can say you know say summer, winter, monsoon, this type of autumn, this type of you know different type of season can also be divided. Generally it could be quarter wise seasonality or monthly seasonality. In that case how to use the seasonality index then different type of you know quarter leverage method or simple average method or say you know normalization process and to make the forecast using seasonality of the data or using seasonality index that we have also discussed. But that was restricted only for seasonal data and the seasonality. So, there is no such you know uptrend or downtrend involved in the data only seasonality over a period of time which will be repeated. The pattern of the seasonality will be repeated over a period of time that we have discussed in the previous session. Today we will not consider only the seasonality and the basic models of you know say quarter leverage method or say you know normalization process we will not follow that. We will extend the hold model. Remember the hold model in that model we consider the extension of simple exponential smoothing model where you consider all the past period of your series and then you extend that by including the smoothing component in a decay manner and then we have bought the trend factor that is called with double exponential series model that is called hold model where you will have one the level value and another is the trend value. If you put together you will get the hold model which is better than exponential smoothing model and it capture the uptrend or downtrend effectively. So, that is the hold model. But also we have discussed that hold model has a you know limitation that it only capture the trend part but it does not capture the seasonality pattern. So, that means if the data has both seasonality and trend together so hold model fails even exponential smoothing models also fails. So, in that case you have to extend the hold model winter which is student of hold who has extended the concept of hold or the formula of hold by integrating another concept called seasonality. So, in winter hold model you will have three components one is the level then trend and then seasonality. Remember as I mentioned earlier it is like a another time series model you have the past data and that you have to analyze the behavior the components of the data and you have to forecast for the future. The logic remains same the underlying theory remains same that it is a time series data and you have to make a forecast. But we are using exponential smoothing model but with a triple exponential smoothing that means one will be the level series another is the base value look at the base value the level component represents the level components represent the underlying average value of the time series the base level value like the stairs example I have talked about. And then trend component which will be a you know may be uptrend or downtrend depending on increasing or down decreasing type of data and then the third component which is coming addition to the you know hold model or say you know addition to the hold model these two are nothing but the hold model and this one is the simple exponential smoothing model say you know put together and then if you extend that by adding the like seasonal component you will get the winter hold model. Examples can be like this suppose you have a data look at the seasonality look at the uptrend also. So, here you know you can see the data pattern which is following to some extent seasonality as well as the uptrend. So, every time every year you will see the peak. So, high sales and the low sales it will be repeated the pattern will be repeated we all know the pattern will be repeated, but and it is known pattern to some extent, but the sales are going up companies performing well. So, therefore, there will be uptrend also or may be downtrend also in that case which model to select the answer is the winter hold model or hold winter whichever you can say that. So, let us understand the detail of winter hold model today before I go to that since it is a actually extension of hold model it is mandatory or it is important for us or it will be easier for us to recall the formula of hold model because it is just extension of hold model. So, therefore, we should remember the hold model what we have discussed in the session of hold method analysis. So, remember in the hold model we had you know double expansion smoothing. So, one is the level and another is the trend hold has segregated the level part the stairs example that I had given level part and the trend stairs you are going up through stairs as well as the you know trend is increasing also like size of the stairs also increasing. So, ultimately you are going up as well as the stair size is changing. So, this way you can think about the hold model I will not repeat that just I am summarizing that as recap of hold model. So, here is the level value the base value which is nothing but the weighted combination of the actual plus. So, alpha say 50 percent of actual plus 50 percent of the previous period intermediate level and trend which we noted as t minus 1 if you calculate l t. So, if you calculate l 5 t minus 1 is the intermediate value l 4 and t 4. So, using that we have calculated the level value similarly it is nothing but the weighted combination of actual data and the forecasted this is nothing but the forecast actually if you recall that if you recall the exponential model through that it is easier to discuss the hold model. So, I have shown that also like you know how from x exponential smoothing model how you got the hold model like this part additional part I have discussed that then trend part we have added as a separate component that is called the segregated like you know they have segregated. So, trend is nothing but the weighted beta say beta component smoothing constant for trend beta of the level gap the stairs gap you can see plus 1 minus beta weighted combination of the previous period trend. So, that combination is nothing but a new trend if you add these two level plus trend you will get the forecast for the forthcoming period this is what the hold model. Now, we are extending that by adding seasonality right trend component will be as it is level component will be as it is little load of differences will be there remember here in this level value you had in the hold model you had y t right the actual data here you will see the differences only only there you will find the differences because you have a seasonality. So, actual y t data you cannot use in your level. So, what is the difference in especially or extension you are doing in hold model let us see here what is the difference between hold and winter hold if you can summarize these are the three statements you can say that it is extension of hold model by adding the smooth component. So, you will add one more component smooth value will come here here I will show you in the next slide. So, one more component calculation will come and one more series will add that is called smooth part. So, level trend and smooth both three component put together you will define your winter method then it allows both the trend and seasonality to allow both trend and seasonality pattern to take into account in your new model extended model and then it is in the computing forecast we add the equation of seasonality as an index. Let us see how that index been calculated as a additional component and how that will be integrated to the level and trend together and we make a revised forecast by taking into account of seasonality or seasonal component. Let us go now let us summarize the initial information that we need now three parameter in hold model if you go back to hold model you had two parameter right let us say alpha and beta. Now we are adding winter so in winter like you know seasonal component you are adding so you will have one more component called gamma. So, three components will come into the picture here here it is I have mentioned three parameter one for say level base value another for trend which is represented beta smooth component right beta for trend and then gamma for seasonal component these three will optimize because initially we will assume them initialization of the parameter are required initially to start the execution the iteration in excel I have shown you the hold model also winter also I will show you. So, initially you need the initial assumption alpha beta gamma once you know that initial value or you assign the initial value then once you start the iteration over a period of time you will optimize the alpha beta and gamma the best combination of alpha beta gamma can be optimized through excel solver I will show you or in python you can do it that also. So, now I will illustrate them first model and then the excel illustration then based on this initial value of alpha beta gamma you start the initial level this is alpha beta gamma the parameters the smoothing components the alpha beta gamma the ranges of them are lies between 0 to 1 remember the excel illustration that hold model you can understand what I am trying to say actually now. So, this initial value you will put, but later we will optimize then we using this initial alpha beta gamma you start calculating your initial level three equations we will get I will show you in the next slides, but in hold you have two equation and then the final forecast formula, but here you will have three formula level trend and component level trend and signality initial value for them also you have to calculate remember in hold model same as it is in hold model first you initialize the value of level and then trend initial value you assume that how to do that that I have also discussed initial assumption of level plot strain using alpha and beta as initial assumption and then once you start your actual iteration of hold model automatically your new level and trend will be calculated in iterative manner and at the end once you will minimize your error or rmsc you will see what is the best alpha beta and the overall calculation of level and final forecast you find. So, similar here also we will calculate the initial value level trend and signality three component with the with the value of alpha beta gamma as initial value and then we will make the forecast that forecast is the intermediate forecast. Then once you repeat the process the iteration using winter method or winter hold method effectively over a period of time you will get a new iteration new level new trend new season and the alpha beta gamma will also be optimized what initially you have assumed that will go new or revised or the best optimum alpha beta gamma will replace this initial alpha beta gamma and the once you will find the optimum alpha beta gamma based on the optimization or based on the minimum rmsc you consider that as the best combination of the parameter and the corresponding final level trend and the signality and make the forecast. Let us see how this you know calculations are being done here through winter hold model. So, first as it is level so here remember the hold model I will show you here I will put the hold model so that you can you know see a competitive analysis also what additional part has been added by winter by extending the hold model. So, that you will also get to know first let us understand the three steps also like three component level. So, level here remember it is as it is as it is hold model only difference here in hold model you had y t here, but now here you have written y t by s t minus m s is nothing, but the seasonal index right seasonal index which is nothing, but say 0 to 1 say say maybe it may it may go beyond 1 also because if it is a 4 quarter say then in that case you know it can be you know seasonality index we have I have shown you in that session you may remember the index if it is a say 0.9 that means, 10 percent less than the average quarterly sales right if it is a say 1.15 that means, 15 percent extra sale you are coming in a particular quarter. So, this way it may reach by total sum of this four index should be you know four right we all know right. So, here it one index for that particular pre-ord say. So, actual data by index here you are dividing by the index that value you have to consider in your level calculation, but in hold model you have taken the direct data because you have not considered the seasonal factor there, but here you are adding the seasonal factors. So, therefore, actual you have to remove the seasonality you have to decenalize the data that means, if the data have a seasonality you decenalize the data by dividing the actual value by the index. So, if this value will come down here. So, this value we are calculating here rather than this actual value remember that this actual value we are not taking here you are taking this decenalized value here because you are dividing the data by the index. So, if it is a down value low sale your when you subtract by the index you will it will go up actually because here for low cell the index will be less than 1. So, effectively this value will go up. So, this value we will have to consider here. So, this is the only difference in level calculation or the base calculation in winter method. Rest formula of level is same as it is like hold model. Here you can see here also look at here. So, here you had the hold model actual data the previous period level and trend weightage are same the forecast value it is nothing, but y t hat actually if you remember the exponential and then hold and then now winter hold. So, this value but we are considering the previous level plus trend, but this value we have replaced with this right. So, this is the level. Now if you see the trend look at the trend calculation now the trend is same as it is hold model there is no changes in the calculation of trend look at here the trend model of trend formula of winter method like beta. So, 50 percent of the level gap the stairs gap as I mean as I talked about the intermediate levels that which are you are calculating in the previous step plus 50 percent say or whatever the weight you will finalize for trend of previous trend. So, previous trend you are not taking fully say 50 percent of that and 50 percent of the level value the actual basal. So, that combination is your trend same as it is look at same as it is hold model no changes here. Now seasonality new components are coming new additional information has come because of seasonality aspects of winter method or time series data. So, here you have to calculate the index also because that will be repeated manner now right it will go back again and it will come you have to calculate it and again you have to calculate this iterative I will show you in excel illustration you have to calculate all three one by one in every iteration and the corresponding forecast also. Now remember how we have calculate the seasonality here this is nothing but gamma is the weight here also you are taking weighted every here every time whether level trend and seasonality you are effectively calculating the weighted combination right of actual or previous period say trend and the level gap previous period forecast for level and the current actual value by dividing by the seasonal index. So, this is your level now for seasonality seasonal also how we will calculate the index this index right this is it will lie between say 0 to 1 or maybe closer to 1 nearby plus minus 1 say not 0 to 1 it may be plus as I mentioned said 30 percent extra it can be 30 percent less can be the weightage of that particular index it is nothing but gamma say 50 percent say of this is nothing but you are calculating ST now ST now nothing but you know it is nothing but yt by your say LT that means level value. So, this is nothing but your seasonal value. So, just a same calculations from here also from here also you calculate the actual by seasonality index is nothing but your base value decennalized data LT which is your level now you are calculating index. So, how you calculate the index actual by the level value the decennalized value actual by the decennalized value like here if you come back here look at here what I told about if this is your actual look at here if this is your actual here this one and this is your decennalized value if you take actual by decennalized value you will get the index actual by index if you can calculate the decennalized value L value in that case if you divide by actual by L L you will get the index other way. So, the same logic. So, that they have done here actual by the level value the base value or decennalized value you can say is nothing but your index, but you are not taking that as it is you are taking 50 percent of that plus 50 percent weighted combination may not be 50 percent here I will show the examples that I have about the gamma is may be say 16 percent or something whatever say 10 percent it can be. So, not a matter. So, whatever if gamma is say 40 percent then 1 minus gamma will be 60 percent. So, this actual by level this index say 50 percent of that you are taking plus 50 percent of the previous period say decennal index. So, that index say and the current value index. So, that weighted combination is your decennal index you might say sir what is m here m is nothing but the number of period m is say 4 period if you take the monthly data then it will be 12, but if you take say quarterly data then it will be 4 m equals to 4. So, wherever you are suppose you are if t is 6 then you take t say you know t 6 minus m the corresponding m you will get through the data and corresponding the previous period index you know as intermediate calculations that index you take 50 percent of that say second quarter to if it is second quarter then corresponding here second quarter you take are you getting my point. So, corresponding previous year on year data index you have to call then you take that weighted combination as your index you will see the calculations once I will open the excel or the data elastation you will get to know that. Now, suppose you got the level trend and seasonal value. So, all are you know base value in interactive suppose you are intermediate steps of a of a iteration three values level trend and seasonal index you have calculated right s t is the index remember simple index. So, once you calculate all three you have to make the forecast for that particular period right you have calculated level value you have calculated say trend value say trend value if you add these two you will get the forecast right forecast for corresponding period right next period this is the intermediate level and trend this is what the hold model but here you are using winter method that means the seasonal components also you have to include. So, level plus trend are nothing but the base two value level plus trend now you multiply with the index with the index look at here. So, this is your forecast for the next period. Now, you might say there is a this formula and this formula are not same. So, what is the different aspects are being included in the final formula I will elaborate, but for to understand the process the overall mindset that level you have calculated trend you have calculated add these two and multiply with the index because this is this is your decentralized forecast level plus trend for the new period, but you have to multiply the index also like as I told you in the you know in one of the session that you know if you have the forecast say you digitalize the data you get the trend line and you make the forecast using trend line, but this forecast is not the final forecast what you have to do in this particular period you have to make the forecast right here you have to make the forecast suppose you have a four quarters say four quarter. Now, these are the actual data and you have prepared the digitalized data and you have made a trend line suppose trend line forecast basic say and this forecast is not the final forecast. That means, LT plus TT say level plus trend value is not the final forecast you got level plus trend look at the trend you found and forecast you have made, but that is not the final you have to multiply the index that index calculated here seasonal index that also you have to multiply. How come suppose here it is trend line so this index say it is been low so less than 1 so it will come down again say then it will suppose like this like this. So, this way this pattern will again be bringing back. So, you have it because the pattern will be repeated that is the concept of seasonality. So, once you multiply the index your trend forecast level plus trend value the base forecast will be rearranged as per the weightage of the index as per the weightage of the index that we have calculated this is what the forecast throw in term method remember how effectively he has developed this level plus trend multiply by the index effectively you are again bringing back the digitalized value with the trend line into your pattern of forecast as per the seasonal weightage or the index. Now, let us see how you might say that sir this particular formula look at the last this formula let me open a highlight point you will get to know say this formula and this formula that I have shown are different. So, this is nothing but here it is actually nothing means 1. So, it is age so how many next period forthcoming period you want to forecast that is been noted by age look at here age is another number of periods ahead you want to forecast. So, 2 period 3 period 4 period minimum 4 period you need to make forecast right minimum 4 period 1 period 2 period 3 period 4 period at least 1 year forecast you need to make because it is a quarterly data. So, therefore, age will be on a minimum 4. So, 1 2 3 4 1 means next year first quarter right done 2 means if age is 2 then next year second quarter if age is 3 next year third quarter like as it is trend will be so strong like hold model remember in hold model I told you like this plus we can do it like you know this LT plus MT remember some additional period right additional like you know M or you can say age say it will match with this particular formula. So, suppose age that means additional couple of period you can forecast because trade will be so strong here also the trend that you have calculated it will be so strong that couple of forthcoming period you can forecast not only 1 period. So, therefore, if like hold model if age is 2 here also that means you are forecasting with the same level and trend and the seasonality index you are calculating the second period 3 means age 3 means 4 third period age 4 means 4 period you can increase that or extend that also, but there might be a chance of less accuracy, but at least 4 period forecast you can do for the forthcoming year you can make it for 2 years forecast also using this formula it is if you have a significant amount of data within this age value you can make couple of forecast also it is very strong model. So, now this is what age and M is nothing, but the number of you know period that is a 4 period number of period of the season cycle say so 4 period quarter on quarter to quarter 3 quarter 4. So, this way you can use the formula and you can illustrate the winter hold model.