Hello everyone, welcome to the course of business forecasting. Today, we will discuss the details of ARIMA process. In the previous session, we discussed the concept of ACF and PACF. These two concepts we will use today to illustrate the ARIMA process or say ARIMA process. Now, let us start first with the basic model of ARIMA that is called autoregressive process. So, in general in ARIMA, there are four process. One is called the AR process autoregressive process. Today, we will illustrate that first and then the next model is called moving average process. This moving average is not same as moving average of time series data. This is a different concept of ARIMA. So, then we will study the moving average process of ARIMA. These two are independent model AR process and MA process. Both are part of ARIMA. And then we will extend the concept of AR and MA into ARMA process, autoregressive moving average process. This is also you know combination of AR and MA. How it is been developed? That also we will understand. And then we will go to the ultimum model that is called ARIMA, autoregressive integrated moving average process or model. Remember all four models that you need to study as a part of ARIMA required SCF and PSF, autocorrelation function and partial autocorrelation function. While it is required that I have discussed in the last session, today we will also illustrate it because today we are going to discuss the ARIMA models. So, let us start with first model that is called autoregressive process. Now, when it comes to the autoregressive process or AR model, remember we are focusing on time series. Our data are time series data and in time series data you need to use regression. And you do not have any independent variable. Remember in regression what we do? In regression, in regression we use independent variable and dependent variable right. We have the data sets and we put the line or feed the line say alpha plus beta x right. This way we develop our regression analysis. And we also find the correlation coefficient among the data between the independent variable and dependent variable and illustrate the concept of regression. This is what basic regression or you can extend that to multiple regression also. When you have a more than one independent variable say you know if you have one more variable say you can feed another regression another multiple regression model where say you know you will have another variable say.  Now, we are not focusing on regression, but we will use the regression concept in time series. That means, when you have a time series data time series data say y1, y2, y3, y4, y5, y6, dot, dot, dot, dot, dot, say. With this data you need to use regression. Therefore, we call it as a auto regression. It can be 1 lakh time series data that means AR model of order 1. It can be AR model of order 2 or lack 2. It can be AR model of order 3 and further. The question here is that how the mechanism works, why we call it as auto regressive model and how regression comes into the picture of AR process of Arima. Remember AR model is the most important model of Arima process. If you understand the AR process actually major majority of the understanding of Arima process are done. So, we need to concentrate on understanding the AR process effectively. Now, as I mentioned, we are not going to discuss the regression at all here, but the concept of regression because we do not have independent variable here. It is a time series data and you need to use AR model or say Arima model. Now, the concept of regression need to use here to fit a AR model of time series data. On time series data remember. Now, one understanding I can share with you. Suppose you have a data of say time series say you know any say temperature data or say you know some demand data or whatever which to some extent are stationary and you need to use the AR process. Remember one assumption that you know generally AR process or MA process are being effectively used for stationary data. If the data are non-stationary then can we use the Arima model or AR process? We can use there is a concept called you know differencing process the integrated concept will come. We will discuss that under Arima model the Altima model when the data will be non-stationary and how to convert non-stationary data into stationary data and then how can you call the AR process MA process or ARMA process. So, for the time being our assumption is that data are stationary and we will be using AR process or MA process or ARMA process. Non-stationary part we will discuss later. Now, when you have a stationary time series data say and you need to use AR process or AR model, what is the mechanism to develop it? Suppose you have a time series data and you need to fit the AR process. Look at the general formula first understand the basic formula. So, here is the general formula of auto regressive model. Your auto data past lag look at the past lag suppose you want to fit say Y5 say your Y5 will be say look at the formula here it will be alpha plus say you know beta of Y4. So, this is what your formula say if you consider one lag time series AR process like AR process of order 1. So, the concept is how it is been developed. In general formula is that if you have a p lag last p lags here I have shown you on up to 3 lag or 3 pure. Lag means it is a older pure like you know today temperature, day before yesterday's temperature, day before yesterday's temperature 2, 3 days back data. So, these are called lag or time series data old time series data and that you need to use to fit your future forecast or fit your AR process.  So, let us see suppose here you have a general model which consider say up to order p and these are the simple illustration of 3 different initial model. Now, how these models are developed? Let us understand through a illustrative example and the concept also. Here you can see like the previous model that I have explained. So, here you can see we have taken only one order AR process only one order only like today stock price or say today's temperature or today's gold price will depend on yesterday only. For the time being look at here only this model we will be trying to illustrate first. Then you can extend up to 2 order AR process, 3 order AR process and further you can also extend up to p order. But now let us understand with a basic AR model of order 1. Let us understand that. So, p denotes the order of a AR model. Look at here p denotes the order of a AR process. So, that p you need to find once you understand the AR process next question will be what is the order of the model that will be finalized that you need to identify also that we will discuss later. Now, let us discuss this with some numerical example. So, that we can understand the process of AR in a effective manner. So, we have taken the same TCS data like the way we have illustrated the SCF and PSCF concept. So, same TCS data of last 3 months we have taken say and these are the data closing price of October 2 December of 2023 say. So, this data will be used to illustrate the AR process right. Now, let us come here how the AR process works. Look at the screen now. First as I mentioned suppose you have the time series data right time series data say Y this is the stock price say YT you can say. So, Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, Y10 dot dot dot you have the data set. Suppose up to Y10 we have taken suppose here we have taken the almost 90 days data 3 months data, but suppose here I have listed this data here right here you can see total 3 months data we have taken. Now, what you do you consider the lag 1 order lag right we are discussing 1 order lag now of AR process. So, what you do you copy the data you can see the excel also here the screen sort I have kept here for your quick understanding copy the data and paste 1 row below for the time being understand. And then suppose we will write YT minus 1 say this is what yesterday data say yesterday data right. Suppose for your understanding this is today's data and you copy and paste them just 1 row below look at now just it is 1 row below like this suppose suppose you have say 10 pure data though we have 3 months data for the time being say. Now, you have pasted you can do that right in excel you can do it just, but 1 row below and then you delete this row and the last row what happen if suppose you have a sufficient amount of data. Now, you can observe here that you have a set of data of YT and YT minus 1 that means today this is all today and yesterday everywhere where you are suppose you are here wherever you are suppose you are at Y7 if this is your today your yesterday Y6 right you can see the data your yesterday was like this you can imagine this. So, this way you have copied and pasted the all the data in the second column by 1 row below what the advantage we will get from here. Since we need to run the regression since we need to run the regression and computer does not understand whether it is a same data time series data or it is a different independent variable data. Now, what we will do we will ask the computer or ask the you know excel that draw or in python whatever software you will have to use to run the regression that please run the regression for this data as X and this data as say Y. So, now, excel has independent variable and dependent variable because excel does not understand that it is your same data right. So, excel only know that this is one set where I am being instructed as you can see the independent data sets and this as a dependent data sets. So, now, once you consider these two data like all yesterday data which are in second column as X independent variable and consider the same data it is a trick and it is a same data as your original data as your Y or say dependent variable it is become a regression. You run your regression you will get a regression line say Y t equals to alpha plus beta Y t minus 1 this is the representative right or you can write X also this is what yesterday and this is your today. So, the general structure now you found a regression model, but this regression model is based on all yesterday data. Now, you have fit a regression this regression is nothing but AR model of order 1 simple this model is called AR model of order 1. What it is called regression, but not regression it, is auto regression because here your data are auto data that is it simple regression you are doing simple regression. But, since your auto data you have consider therefore, we call it as auto regression model. But how many pass like we have consider how many pass data you have consider your regression model? It is a regression model of order 1 simple linear regression. Therefore, will call it as a AR model of order 1. So, order of the model is also been given here. So, AR 1 this is what simply AR process and this is the basic concept of AR process here you can see we have developed that we have taken the data and 1 lakh below 1 row below the data we have copied then we have considered this as independent variable this as a dependent variable this is your dependent variable and this is your independent variable right. Now you consider then as your regression to your regression formula and you run the model you will get your AR model of order 1 come here look at if you run the model you will get if you run the excel here I have shown you if you run the excel you will get the regression model because excel would not understand that it is the same data auto data excel will consider that it is a independent variable it is a dependent variable and excel will come out with the prediction and here is the prediction look at this is your you know intercept intercept and the slope if you consider then you will find your AR process regression model or you can say auto regression model of order 1 because you have taken only 1 order only yesterday look at here this is what your AR model AR process that is it. Now you can calculate the error of your model also like you know you can calculate the RMSE total error and then the mean square error and then the square root of it you will get the RMSE also for your safety purpose you can keep it how strong your model is but generally you know this type of ARIMA models are been measured either through RMSE or through AIC ACHIC information criteria. So, for the time being we are not focusing on the AIC calculations we will be using the RMSE model that our RMSE is this and this model predicts this forecast right. Now the question here is that this is the AR process simply regression you have done, but you have considered the same data with a trick of considering them as your independent variable like these are called dummy independent variable you do not had any x and y you do not had any x and y, but you have considered your regression in a tricky manner by using the same data therefore, we call it as auto regression with your past lag. Now the question here is that the question here is that ok it is fine that we have considered the auto regression we understood the AR model of what are one, but it is a stock price right or say temperature or say gold price or say crude oil price whatever wherever you want to use say you know the AR process or ARIMA process why you have considered why we have considered only one lag it can be more than one lag also right why it will be dependent only on yesterday stock price of TCS or any company may not be dependent only on yesterday it might have a dependence your day before yesterday also 2 days back also. Now let us think about that the stock price of a company will not only fully dependent on yesterday or one lag back it might depend on 2 days back data day before yesterday also yes we can do that also in that case what you do you copy the data this is a notation capital Y right T minus 2 2 days back data you just copy the main data and paste them in the second column in the third column from here. So, here it is Y 1 here it is Y 2 here it is Y 3 here it is Y 4 here it is Y 5 Y 6 Y 7 Y 8 Y 9 Y 10 say suppose you have the data you can do this also you can create one more column here and you can feed the data.  Now you have a sample data like this consider this as your today let me open the pane again consider this as your today this is your today look at here this is this data is your today data this data is your yesterday data and this data is your day before yesterday yesterday. So, what happens now since we come up to the observation that the stock price or temperature or crude oil price will not dependent only on the previous day data or previous period data or time period it might depend on day before yesterday one 2 days back data also. So, in that case we are considering that my stock price since we are taken example we have taken example of say TCS stock price. So, let us discuss the concept through stock price itself otherwise you can extend that concept to any other practical example as per your project or as per your requirement right or the industrial data. So, now let us think about that. So, you have a stock price of actual data and then your yesterday data these this become second row second column has become your all yesterday and then the third column has become of has become it is a next variable second independent variable here it can say this is X 1 and this is X 2 and this is your Y say, but it is auto data all are auto data, but now you you consider them as your second independent variable because you we assume that my stock price will depend on not only on yesterday it may depend on day before yesterday also. Now, you have a sample same sample data since you have a large amount of data initial 2 row if you delete and the last 2 row if you delete it will not have much impact. So, you have a good amount of data now you consider this look at this this look at this one this data you consider as independent variable and ask Excel to run the multiple regression now multiple regression now right. So, two variables two independent variable will come what are them dummy to independent variable yesterday and day before yesterday right. So, this is your today yesterday and day before yesterday right. So, you consider this as one independent variable and second last column as your second independent and first middle column as your first independent of so, you have two independent variable say X 1 and and x 2 right and you have the y. You run the regression, multiple regression will get the output also. In that case, your regression model will be will not be like this. In that case, your regression model will be y t hat equals to alpha say. So, new alpha it will not be same alpha a value will change beta 1 y t minus 1 plus beta 2 y t minus 2 because you have two independent variable now because simple regression multiple regression you are doing it, but effectively your auto data. This is called AR process, this is called AR model of order 2. So, you got to know the AR model of order 2 also. This is also auto regressive model because auto data you have considered. This is what the AR process of order 1, AR process of order 2, you can extend the concept if you think that no my stock price will not depend on not only yesterday and day before yesterday, it might depend on 2 days back data also. So, in that case you can fit the multiple regression, we call it is auto regression model of order 3 also. Here I have mentioned all them. Here you can see now AR model of order 1, AR model of order 2, AR model of order 3. So, here you have taken 3 days data, old data lakh, 3 lakhs yesterday, day before yesterday 2 days back data. If you have fit your model, now you will this alpha will not be the same it will change the parameters value will change, but this is general structure that you can fit a AR model of order 1, 2, 3 depend on your requirement and the company's data and your industrial requirement. You can fit the AR process, but remember all this we have done for stationary data. Now, the question is that we can fit the AR model using the same data through regression, but it is auto regression by considering them as a dummy independent. If it is of order 1, order 2, order 3, this way you can fit it. I will show you in excel also and you can develop it. Look at here. So, this structure here you can see the how to consider the lack of order 1 and then how to fit it. Let us complete this process also then I will come back to the order selection of AR process. Let us complete this. So, come back to excel. Here you can see let us complete the AR process detail understanding. Here you can see we have the same data like 3 month data of TCS closing price here. We have listed here I have you can if you like you know unhide this you will get all 3 month data. Now, just we have copied the data and pasted 1 probe below. Look at here. We have copied the data and just 1 look at 3, 5, 1, 3. Look at here. Let me use my plane here. So, this data we have copied here and we have just pasted the data. So, this is what the first lack data we have copied. Now, what you do? How to fit the regression and find the AR model? Let us see also. Simple regression we will need to use. So, y input, y input will be your this data from second row onwards you have to consider this much and then your x input will be from here. This is your y 1 actually then y 2, y 3 up to you know just do not consider the last data because you have copied the data 1 row below and you need a pair of data to fit a regression and then we have not selected the level like the tag or the level caption. So, we will not select the level output directly will run and output data suppose we will put say here say suppose here right we will find the forecast here look at the intercept and slope. So, this intercept and slope may be little changes might be there this intercept and slope unity which as alpha and beta this will be your alpha this will be your beta and you can fit the line y t equals to alpha plus beta y t minus 1 because y t minus is your first a lack. So, this is your AR model AR model of order 1 that is it. Now, you can extend the data to order AR model of order 2 AR model of order 3 also this way you can also develop your AR process clear. Now, here also you can see whether the p is significant or not look at the p is quite good significant. So, you can use the AR model here for this type of this data also. So, we could see that the p is quite good and very less and it is significant. So, you can fit the AR process. So, this concept also order 1 for this type this data set. Now, come back to the main model and suppose if you look at this data and if you unhide the data say for this time being look at here same output we found earlier before I bought this excel 2 the session look at intercept and slope and you can fit the forecast also now the question is that how we will fit the forecast here as I mentioned look at the formula here this formula we will be using here say. So, you can see it is nothing, but let me write it here again it is nothing, but equals to alpha intercept plus slope into the previous period. What is your previous period? Suppose you want to forecast say you know for the next day this is stock price this is this is this will be your say next day. So, you can like y t minus 1 previous day lag forecast. Now, next you want to forecast suppose next day again this plus slope into this is your now yesterday now you just dragging the data formula is your next forecast. So, you can drag it whenever you have data in the middle you can also cross verify whether your forecast is right or wrong that RMS we have calculated you have taken the square of error the error difference and then we have taken the square of the M and then we have considered the RMSC also for the sake of understanding the M and if you take the square root of it you will get the RMSC this is the M and if you take the square root of it you will get the square root of this you will get your RMSC value. So, this is what I have mentioned in PPT. So, this is the overall process of error, error process of order 1 lag 1 with forecast with regression model forecast value as well as the error part RMSC part. Now, if you extend it to error model of order 2 order 3 you just copy the data just copy this data and paste it with 1 row below here suppose here with 1 row again create another column and just paste there you will be able to get say just copy this data and paste 1 row below from here say this value. So, now you have look at this is say lag 2 say lag 2. So, now you start from here you delete this data you delete sorry you delete these data sets and the below data also and you have now a sample data look at here sample data of say 3 months of PCA almost you know just only 2 periods data you have removed. So, not a matter this data you now take and carry for what your multiple regression of order 2 because you have considered yesterday and there if you yesterday as your 2 independent in that case this will be your lag 1 as it is this will be your lag 2 lag 2. So, you consider these 2 data sets this data sets and this data sets this is your x 1 this is your x 2 and this is your y. You feed the data with this you feed your regression model multiple regression will feed will get the error model of order 2. In that case your forecast will be alpha plus beta 1 say pi t minus 1 like this lag data formula will come in their structure with new input data and corresponding intercept here you will get one more variable right. The slope plus beta 2 into say x 2 here it is y t minus 2 say they be very yesterday data error process of order 2 is done also. This is what the detail of AR model. Now, let us come back to PPT again. Now, the question here is that very important part ok we understood the AR model in detail, but what would be the final order of your model? How many pass lag you would like to consider in your model? Will it be 1 lag, 2 lag, 3 lag? How will identify it? How will optimize it? Or how will finalize it? This is the most crucial part of AR process or AREMA process. Let us understand the concept of SCF and PSF. In the last session we have discussed detail of it, but we will recall that here to calculate or to finalize the p value the order of AR process. We understood the AR process. Now, remember if you fit a AR model of order 1, this is a regression right. You will find the correlation between the yesterday and everywhere yesterday. This is your AR model of order 1. You will find the correlation coefficient also say between these data like let me write here also. Suppose y and y t minus 1 right. This is your actual data y 2, y 3, y 4, y 5, y 6 dot dot dot. Here it is y 1, y 2, y 3, y 4, y 5, y 6 dot dot dot. And then you have taken the data and suppose you have fit your AR model of order 1 say alpha y t equals to alpha plus beta y t minus 1 say. And you will find the correlation coefficient also in excel you can calculate the correlation coefficient in detail we have discussed in the previous sessions. Suppose you will find the correlation coefficient say 0.8 between the yesterday and today's data between this data and that data. You can create the formula of correlation coefficient and you excel you will find it. Now, the question here is that these correlation coefficient are not simple correlation coefficient. This is auto correlation coefficient. Remember we have discussed detail in the last session with the same TCS data. This is called auto correlation coefficient. I think we found 90 percent around remember or 95 something 95 percent dependency are there. So, like today's closing price will have a big impact of tomorrow's opening price of TCS. So, it will be dependent heavily. Now, this called this data this correlation coefficient are called the auto correlation coefficient simple correlation formula, but we will call it as auto correlation coefficient. What you do?  Detail we have discussed. So, this is your time lag with say 0 with say 1, 2, 3, 4, 5 like this. This is your SCF. You draw it, create your correlogram so that you can understand how to draw your how to finalize your P value, the order of air process. Very crucial discussion you know concentrated here. So, now with the same data if you consider the same data if you do not paste with the one row below with if you paste with the same row and if you create a calculate your correlation coefficient will be 1 right with the same data correlation coefficient will be always 1 with same data with 0 lag same data. So, here it will be 1 say 1 we are not interested about that what we are interested here is that let me draw this here it is 1 say. So, now 1 say now we are focusing on our the auto correlation with 1 lag like this for this model. We found say from the previous session you can refer or you can use excel and you can calculate the auto correlation as say 0.95 suppose or 90 percent say.  This is what your SCF the correlogram you are drawing. Now, if you extend the model air model of order 2 suppose this you may find a correlation coefficient also between today and day before yesterday we have calculate remember we have you know calculated that the auto correlation of order 1 order 2 etcetera of lags. Suppose, if you consider the 2 days lag like stock price will depend on day before yesterday also in that case what will be a auto correlation coefficient right. In that case it will not be the as much as 0.95 or 90 percent it will be less early if I remember suppose it was say 90 percent right earlier as a 90 then 90 percent or 86 percent whatever then suppose you draw that 86 or 90 whatever suppose you found here this is suppose a 0.86 suppose if I remember 0.86 or something suppose that you put here this is what the correlation between today stock price and day before yesterday stock price 2 days lag. So, P 2 for P 2 for P 2 you found say the correlation value auto correlation value. Now, if you consider the 2 days back data in your regression auto regression model and in that case you want to calculate the correlation coefficient or auto correlation coefficient between we are using correlation, but it is a effectively auto correlation because it is auto data time series data no regression. Now, if you consider calculate the auto correlation between today and 2 days back data you will get it remember that calculation process simple auto correlation you can find in that case suppose it was say 0.7 suppose I forgot it suppose 0.7 you note down it you draw the graph here. 4 days back data you can consider like this this graph is called a CA function we have discussed it all in the previous session. Now, the question here is that now how will select the P from this graph? How many past period you want to consider and feed your final model here? What is your final model of time series data? And you should conclude that this is a AR process detail understanding and the P selection are done. So, we can wind up the AR process understanding no not done yet. The problem here is that you cannot select your P or the order of your final AR model from the SCF graph you have to select it from the PACF graph this is very interesting part time lags say this is PSCF remember we have discussed detail of PSCF for the TCS data itself. Why it is required? Because you know PSCF is nothing but the partial auto correlation for the partial auto correlation here you remember you when you consider say stock price of same data same time parameter or same variable say there is a big dependency right like stock price of today has a high dependency on yesterday stock price of today might have a dependence on debuffer yesterday also, but since it is auto data yesterday and debuffer yesterday has a correlation. So, therefore this direct correlation you cannot take here because there is a like you know yesterday was a today sometimes it was a today and debuffer yesterday was a yesterday also because if you go back the data the logic will be like this. So, therefore sometimes you have to understand that for time series data you cannot select the direct SCF value in your calculation process. The reason here is that in time series data your interdependency of past lags are there, but this is simple multiple regression right though it is auto regression but it is simple multiple regression look at the second formula it is a multiple regression of order 2, third one it is a multiple regression of order 3. So, when you run a multiple regression your minimum concept is that there should not be multi-colinearity there should not be any multi-colinearity we have discussed detail in multiple regression model. So, here you are getting multi-colinearity though you are fitting regression or say auto regression or multiple regression, but multi-colinearity are coming because yesterday has a dependency or debuffer yesterday debuffer yesterday has a dependence on today's back data also, but that all of them we are considering to calculate your today's stock price. So, if you want to consider actual regression this is a multiple regression you are doing multiple regression, but since we are doing a basic regression therefore you need to find you need to maintain the guidelines the mandatory requirements of multiple regression also. What is the mandatory requirement? The multi-colinearity should not be there among the data multi-colinearity should not be there among the data, but here you have a multi-colinearity because it says auto data huge dependency are there because yesterday was dependent on a debuffer yesterday. So, therefore, since you have a multi-colinearity involvement in the time series data in air process therefore, you cannot select the order of your model from direct PSF the dependency this is the dependency right correlation auto correlation here you can see 95 percent here is 86 percent here is 70 percent too much of dependency are there, but it is a fake dependency because interdependent like you know multi-colinearity involved here you take the actual dependency actual relationship actual linear relationship the linear relationship and that can be calculated from PSF partial correlation actual impact remember refer the last session that we have discussed about it. So, now in this case what will happen here like I can give you one more example suppose you know you need to take a decision where your all decision past decision depends on your father's opinion say and you can fit a regression this simple regression linear regression it is fine, but now when you consider your suppose you will have to consider your mother's opinion in your decision making also. Now, you have a two independent variable one your father's opinion another your mother's opinion and you need to you need to fit your multiple regression like dependency relationship among your opinion based on your father's opinion and mother's opinion you can fit your multiple regression, but in case if you for the sake of illustration in case you find that your mother's actually depends on your father's opinion. So, how all decision depends on your Father's opinion in that case there is a multicollinearity so you cannot take the full opinion of your mother for father's opinion you can take, but for mother's opinion you cannot take full opinion you need to take the partial opinion that is called or to some extent the partial correlation coefficient or in multiple regression why you cannot take the entire opinion of independent variables in case they are correlated. Same concept we need to use here also in time series data when you have considered the past lakhs you make sure that you consider the partial relationship among the partial impact of it not the full because there is interdependency among yesterday and day before yesterday. So they were sometimes yesterday today and yesterday. So therefore, that direct impact you cannot take here right. So here what we will do you will calculate the first for the 0 lakh for the 0 lakh we are not discussing say 0 lakh we are not discussing we are considering only the first lakhs. Say let us let me delete that also anyway since we have drawn here we will keep as it is we are not discussing that for 1 lakh we will start is it is like yesterday it is say 0.95 say we will take it as it is like yesterday data and this and today. Now day before yesterday impact you want to consider say here or here whatever here model of order 2 or order 3 we are just fitting the PSA graph then we will select the P. Now remember all this we are being for P selection only the how many past lakh you want to consider look at the climax it will come now. Now when you go to the day before yesterday you have to take partial opinion how to calculate the PSA we have discussed you know the detail calculation you can refer that suppose we found the PSA here earlier case you can see the second day like you know correlation autocorrelation was 0.86. Now when you when you take remove the impact of say intermediate Y t minus 1 from Y t minus 2 and Y t it will be like you know your partial impact of yes day before yesterday with current data like day before yesterday stock price the impact of that on today stock price will have a very less say point say 0.2 and then if you go back further older data older lack and in that case your impact may be 0.01, may be 0.02 like this immediately it will fall this is what the PSCF then the concept here is that you draw your band line how many past lakh you want to consider a band line you create your threshold and if you find that your clearly visible standing line it can be negatively correlated also not a matter for example you will get to know suppose here you see how many clearly visible standing line you can see from PSA graph not from SCA graph from your PSA graph. Suppose here we found two line let me put a different color suppose here we found two line the first one you do not need to calculate because it is auto data so lag 1 and lag 2 you found with clearly visible standing line so here P equals to 2 so you fit your model AR model of order 2 that is it because you found only two clearly visible standing line who is having a clear impact the past lag this is the lag yesterday data and their impact of today and this is the day before yesterday and their impact on today stock price. But after that you can see the impact is very less because PSA value has very less closer to 0 and it is falling in between your band line or say threshold line epsilon interval so you can exclude them you do not need to include that. So, therefore, in AR process the P selection the order of AR selection of AR model should be selected from the should be read from the PSA graph not from the SCF graph because these correlation are not correct these are autocorrelation and multicorrelation involved AR therefore you cannot take that data and fit your AR model in that case you might find a infinite number of data you might need to like in a huge amount of data 10 pre-order 20 pre-order 30 pre-order because there will be an exponential decay of data and how many pre-order you want to consider this is a wrong calculation you cannot consider it as your wrong not wrong calculation this is a wrong way of selecting your auto regression model. So, you have to select your AR model final AR model by reading your PSAF graph you will always 99 percent case you will see that in case your model the data follow AR process we have not discussed the MA and ARMA detail now we are considering on AR you will find that most of those times your PSAF graph will fall after 1 or 2 period or maybe maximum 3 period. So, there you stop after that sudden fall look at here SCF will have exponential decay and first P terms of PSAF will have a significant difference from 0 outside the parallel line outside the parallel line after that there will be sudden fall sudden fall will be there because partial impact after if you go back to the old data like 10 days ago what happened with the TCS price that might not have a much impact of today's stock price maybe couple of period will have a huge impact. So, you do not need to go to the 10th period of older data I need to fit your model you can see the you know movement of the data, but overall in order to fit your ARMA model you need to take couple of period and to see the clear impact for certain trading etcetera it will have a great impact of AR process. So, this P selection of AR process has to be taken from PSAF not from SCF because PSAF gives the actual correlation of your previous lacks the correlation how it is impacted or how it is been explained with the past data. So, that you should select from your PSAF graph not from the SCF graph, but the picture the SCF and PSAF put together are called the correlogram and AR model should be read from the PSAF not from SCF and PSAF will have exponential decay in general and PSAF will have a cutoff sudden cutoff sudden fall closer to 0. So, then you select clearly visible line you will be able to see. So, here you can see one example say AR model suppose we will consider this model say AR model and this one AR model these two we will discuss later we will discuss later. Now, just focus this first model first graph. So, here you can see both SCF and PSAF I have drawn in the same graph.   So, the white line the PSAF is just only one standing line you can see only one standing line is here look at this on the white one after that look at closure to 0 look at closure to 0 look at closure to 0 closure to 0 in between the you know threshold line. So, therefore, in this graph both we have plotted in the same graph. So, therefore, you can see from this if the data follow this type of pattern the time series data and you want to use some ARIMA model in that case you select the AR model of order 1 because PSAF has only one cutoff only 1 lakh the actual data with 100 percent you know correlation we are not plotted here we have plotted only the lag 1 onwards. So, you have you can see 50 percent impact after that closure to 0. So, you consider only AR model of order 1 and look at the SCF has a exponential decay look at the second graph here also you can see SCF this also AR of order 1 this also AR model of order 1 you might say that sir can you give an example of AR model of order 2. Yes you can draw a graph and you may draw you may get a model like in that case you are suppose your PSCF will be like this say and after that closure to 0 and SCF will have a PSCF will be like this say and SCF will have a exponential decay SCF will have a exponential decay. So, in that case you select AR model of order 2 because here you can see 2 clearly visible standing line this way it can be negative way also if this is a negative standing line not a matter because data could be in different manner also like you know like oxygenality pattern also etcetera. So, in that case sales might change quarter to quarter or month to month basis. So, in that case your you can extend the data with negative line also, but clearly visible line should be there which will be outside of our band line like clear partial correlation impact should be there that is it and this is this is what is the this standing line these are nothing, but the partial correlation correlation among the data how much it is impacted by the older data that is it that value we are calculating and from there you select your P may be order 1 order 2 maximum order 2 3 people take after that it does not go generally 1 or 2 it is sufficient may not go to larger data with time series model. This is what the SCF and PSA breeding and AR model. Now, let us take a break after the break we will discuss the MA process.


Last year, I announced the end to our combat mission in Iraq, and today we've removed more than 100,000 troops. Now that the wars in Iraq and Afghanistan have officially ended, the United States is facing yet another threat to global security, with the rise of the Islamic State of Iraq in Syria, Republican and Democratic lawmakers, and the White House are in the middle of a contention on what the United States' role should be in fighting global terrorism. More than 2,000 coalition air strikes have pounded these terrorists.  A week after President Obama submitted a war authorization built to Congress, lawmakers in Capitol Hill are cautious to give the President a blank check in fears of another combat war in this region that has become the graveyard of American hopes for stability and democracy in the region. From James Foley to Peter Kassek, from Stephen Sotloff to Kaden Mueller, as ISIS and its affiliates from all over the region and discriminately behead its hostages, critics of the Presidents like Senators John McCain and Lindsey Graham have denounced his foreign policy and strategy in containing the organization's expansion. Some have called for deployment of combat troops, however the President and his allies have insisted that this is not an option, at least for now. Meanwhile, after the Charlie Hebdo attacks in Paris that caused the lives of 12 victims and the hostage incidents in Sydney and Denmark, the United States compelled to confront the growing threat of Islamophobia. After the murder of three Muslim students in Chapel Hill, North Carolina, UC Berkeley's Muslim Student Association held a candlelight vigil in solidarity with the families of the victims. We sat down with Dr. Newsome, a counterterrorism analyst at the University of California, Berkeley to examine the motivations behind ISIS Crusade and what the National Coalition's counter-tactics should be. There are lots of vectors for attacks on the home end and these attacks can not involve hard borders at all, so cyber attacks are an example, but not just cyber attacks that begin and end in cyberspace, also the use of social media to radicalize persons already within the United States. Dr. Bosnian, a professor of Near Eastern Studies in the same institution, points to the Bush administration and its handling of the Iraq War and how it led to the rise of the Islamic State. The core of it is that we had a functioning state in Iraq and we invaded Iraq in 2003. Thus we collapsed a functioning state and created a political vacuum in an area that is highly diverse and highly contentious. With the threats of ISIS growing on all fronts, countries all over the world struggle to contain its expansion. Meanwhile, the livelihood of millions of Muslim immigrants are kept in the dark, as their day-to-day interactions have been compromised. Join us next week on the second part of this CalTV News Special as we dive into the growing threats of Islamophobia among Muslim Americans. My name is Carl David, until next time.


Welcome to the building Alphonse Ariens, one of the locations of Saxion, University of Applied Sciences. Here, education is situated next to companies and housing facilities for international students. Within all Saxion locations, educational facilities can be found. But this building also houses the research group's lightweight structures, XR Lab, Saxion Centre for Entrepreneurship and the research group Mechatronics. In this video, we will show you around the facilities of the research group Mechatronics, which will be used as the main location for the master's degree programme, Robotic Systems Engineering. Here, we have everything at hand needed for the applied research and education on autonomous robotic systems and smart industrial robots, offices, lecture rooms, places to discuss ideas, a variety of lab environments and places to relax form the perfect combination for the master's degree programme. The smart industry field lab T-Valley will be joining the location once the construction is finalised after summer 2021. A large portion of the educational activities of the students will take place here in this building. We are here in the research and engineering lab and just around the corner, there is the Students Lab where we can organise inspiring lectures but also allow students to work on smaller projects in smaller groups. Besides the general lecture room, students will use indoor and outdoor testing facilities of the Research Lab Mechatronics. In addition, we have a mechanical and an electrical workshop where students can work on realised systems and use the 3D printing facilities for rapid prototyping. State-of-the-art robotic manipulators are available to facilitate innovative solutions in various automation challenges. Think of automated material handling applications, pick and place applications, smart logistics and human-robot collaboration. These systems will provide hands-on experience in advanced control themes related to control and manipulation and provide a platform for assignments around smart industrial systems. In the vision lab, cameras, illumination, imaging optics and optical filters are available to provide robotic systems with eyes needed for a variety of applications. Quality inspection, object recognition and shape matching are themes relevant for the next step in smart industrial systems. At this facility, students will encounter and explore the latest state-of-the-art in all aspects of computer vision. The OptiTrack facility is used in research and education related to autonomous ground and aerial robots. The motion capture system allows a ground truth for the accurate localisation algorithms that students develop using onboard sensors. In this lab, recent developments in localisation and navigation are put to the test in the context of sensing and perception. Moreover, different trajectory tracking and interaction controllers can be tested in the context of the activities dedicated to control and manipulation. Testing the controllers and localisation algorithms of the autonomous robotic systems in an outdoor setting is possible in the outside drone dome. Here, autonomous systems are exposed to the harsher outside conditions where the applicability of ground and aerial robots can be evaluated. Using flexible localisation tools, positioning of autonomous systems outside can be monitored for the understanding of system behaviour. Because the students are within walking distance from the researchers, engineers and the professors of the research group Micatronics, we believe that there will be a lot of interaction between researchers and students. This will make the central space in the audience-plane building the beating heart of the learning community around the master's degree programme. Students have a lot of access to nice facilities outside Saxion. We are a founder of two innovation networks. One is T-Valley, which is focused on Megatronics and Robotics. They have a lot of nice experience centre, of course, production environments where they can really develop their systems in real practice. The other one is Space 53, which is the drone innovation cluster, which is situated at the former airport of Twente. And there we have another outdoor testing facilities, especially for unmanned systems, to test those systems in real environment. We have created a lot of synergy with those two innovation networks. Engineers are really in our lab, busy with the development projects we do together. And with that also they bring in hardware. And we have also, of course, a yearly conference, where it's very interesting for students to join. And we have competence groups, where we have focus groups on special topics where students can involve as well. We thank you for your visit to the facilities of the research group Micatronics and the master's degree programme Robotics Systems Engineering. We hope to welcome you to our labs in the future.


The weird circle. In this cave, by the restless sea, we are met to call from out the past stories, strange, and weird. Bellkeeper, pull the bell, so all weird circle. The trial from murder. That's the title of tonight's weird circle story brought to you by Ogden's. From the pen of the noted author, Charles Dickens, the trial from murder provides excellent story material for a smooth radio adaptation. Speaking of smoothness and adaptability, you'll find Ogden's fine cut, a smooth roll your own cigarette tobacco. And if you look for a product that's completely adaptable to the smoker's taste, well it's simple to make smoking satisfaction come true with Ogden's. Yes, on all counts, Ogden's provides genuine smoking enjoyment. You'll find Ogden's easy to roll, delightful to smoke. Yes, easy to roll, delightful to smoke. And now the Ogden's Playhouse presents the trial from murder by Charles Dickens. Out of the past, phantoms of a world gone by speak again the immortal tale, the trial from murder. Here we are, Mr. Trelawney, new town jail. Ah, gloomy old place it is, sir. Watch your step getting out. Thank you, Derek.  Then you can see the hanging. Oh no, Derek, I didn't come down here to watch the hanging of Peter Cook. Then begging your pardon, sir, I don't understand. Well, it's hard to explain. It's as though some strange force has urged me to come here today. I've felt it ever since the trial ended. Well, sir, I for one never did believe all that rubbish they were whispering about you during the trial. Supernatural powers, Mr. Trelawney can read the witnesses' minds. Why, there never was a more practical man than you, sir. Form into the jury, I tried my best to be fair, Derek. But I can't think back on the trial without a shudder.  Oh, look, sir, the flag is up. Your Peter Cook is dead. Yes. And suddenly a great weight seems lifted from my soul. Strange. Why, I feel happy for the first time since it all began. Robert Blackwell has just come out to the prison gate, sir. I think he's coming over here. Even a great lawyer like Blackwell couldn't stop you from hanging Peter Cook. Oh, this is a fortunate meeting. I've long wanted to talk to you. It's Mr. Trelawney, isn't it? Yes, Mr. Blackwell, I'm at your service. Even though you'll never persuade me Peter Cook wasn't guilty of murder. He was, sir. He was. Confessed at the last minute. I was completely taken in by a scoundrel. Disagreeable business, the whole thing. Would you care to join me for a stroll? I'd be delighted, Mr. Blackwell. That will be all, Derek. I'll be home for dinner. Yes, sir. I suppose there are many questions in your mind. Mr. Trelawney, from the moment you appeared in the jury box and my client leaped to his feet screaming that you must be challenged, I felt you were the key to the whole mysterious business. I was. Through no fault of my own. In a way, I was a puppet moved about by forces that can't be explained in any logical way. I've long wanted to tell the whole story. Just what I was hoping for, my friend. I've long wanted to hear it. I warn you, Mr. Blackwell, you won't believe me. You'll laugh at me. Call me crazy. Not at all, Mr. Trelawney. Now, here's a quiet little part. Let's rest on this bench while we chat. All right, sir. As I was saying, Mr. Trelawney, I have every reason to believe you implicitly. Then you have more faith in me than I had in myself. Do you know that after the trial, I went to a specialist? And it was with relief that I heard him say that I was sound and body and mind. Excellent. And now a confession from the counsel for the defense.  That you are a well-known bank director, respected and admired. A man of character and integrity. Not given to imagination and fancy. Oh, I'm flattered. But was that sufficient proof? Perhaps not. However, on four different occasions during the trial, I was startled by a sense of someone standing at my elbow. A weird sensation. The sort of a chilly presence that made the hair stiffen at the back of my neck. You may recall that during my summation, I lost my place and glanced quickly in back of me. You certainly looked startled. There was no one there, of course. But there was. It was John Talbot. John Talbot? The dead man? The man murdered by Peter Cook. I saw him frequently during the trial. But my good man, such things don't happen.  But it was as much a fact as two and two make four. And at least, Mr. Blackwell, you've given me a motive for telling my strange story. First, let me tell you that I don't follow crime stories in the daily papers. And it wasn't until much later that I discovered that this first incident took place before Peter Cook was even suspected or arrested. I was standing by the window in my apartment overlooking the park. Derek, my servant was in the room at the time. Shall I lay out your evening clothes, Mr. Trelawney? If you will, Derek. But first, take a look out this window. Something strange going on. What is it, sir? Do you see that man approaching from Berkley Street? The one with his collar turned up. Yes, sir. He seems to be in a bit of a hurry. What's the way he keeps glancing in back of him? That man thinks he's being followed. I'd say he acts a bit frightened, sir. You're right, Derek. And now I see the reason. Good heavens. What a peculiar-looking chap following him. No wonder he wants to get away from him. Where, sir? Stand over here. You can see him now, Derek. No hat, his hair blowing in the wind. He has the whitest, most unhealthy-looking face I think I ever saw. I don't see him, sir. Well, you must be blind, Derek. He's practically in front of the window now. Look, look. Oh, he sees us.  I don't like the look in his eyes, Derek. He must be a madman. Doesn't he look mad to you? I'm sorry, sir, but I don't see a soul. What's that? I don't see anyone there, sir. You must be crazy, Derek. He's moving away now, but looking back over his shoulder and still pointing. Are you quite sure you're feeling well, Mr. Trelawney? If you're trying to be annoying, Derek, you're succeeding admirably. Is this your idea of a joke? No, sir. I'm very sorry, sir. May I draw the blinds? Please do. Let's have no more of this nonsense. Serves me right for not minding my own business. Yes, sir. What's that? I mean, no, sir. That is, shall I lay out your evening clothes? Yes. Yes, I'll change in the dressing room. It's warm there. Derek, did you leave the dressing room door ajar? No, sir. I closed it tight to keep the heat in. Well, it's ajar now. Wait a minute. I have a feeling there's someone in there. That's impossible, sir. You'd have to go through this room to get there, and we'd have seen anybody who did so. Yes. Yes, you're right, of course. Look, look, and remember. Derek, did you hear something? No, sir. What is it? There's someone in that room. Open the door wide. Yes, sir. There. You see, Mr. Trelawney, there's nobody there. If I may say so, sir, your image... What's the matter, Derek? I... I don't know, sir. All of a sudden, I felt cored. Sort of a chill came over me. You... you didn't see anyone in there when I opened the door, did you, sir? I... I thought I did, just for a flash. It was the white-faced man I saw on the street a moment ago. And that's the truth, as you recall it, looking back? I'm certain of it. Moreover, Derek came to believe me. That chill of death that he felt convinced him. It's odd that neither one of us connected the incident in any way with the summons for jury duty that he brought to me at breakfast the next morning. As a busy man, I was merely annoyed with what I thought would be some dull civil case. But several days later, I arrived at the courthouse. I remembered so clearly the dim, gas-lit passageways to the court. I remember that black, heavy fog hanging like murky curtains outside the great windows. I think I moved as one in a dream until you and the prosecutor started examining the jurist. As I stepped into the box, I remember your client, Peter Cook, meeting you to his feet.  He'll hang me, sure. I would have failed trial. I demand that you challenge him. Please, please, Mr. Cook. Your Honor, I ask your forgiveness for this outburst. My client is unstrung. With your permission, however, I will examine the juror. Your name, sir. Hugh Trelawney. Do you know the defendant, Peter Cook? No, sir. Never saw him before? I believe I saw him in front of my home several days ago. He was being followed at the time. At least he looks like the man I have in mind. I'm sure I don't need to remind you that you're under oath, sir. Now, my client seems to believe that you are prejudiced against him in some way. Is that true? I know absolutely nothing about him. So far as I know, I've never spoken to him or read anything about the case. I have no personal interest in the matter whatsoever. Your Honor, my client, Peter Cook, seems to have been a bit hasty in his remarks. I feel he's made a mistake. I withdraw my challenge. And this incident, which you'll recall, Mr. Blackwell, was the first inkling I had of any connection between the scene in front of my home and the trial. Naturally, at that point, I felt a tension had been centered on me unfairly. It wasn't until the prosecution had started its case that I discovered the further link. The prosecutor was speaking at the time I recorded. The state submits this locket as exhibit A. I shall ask each juror to examine it carefully. There are several interesting points about it. First, it contains a picture of the deceased John Talbot, who the prosecution will seek to prove was foully murdered by the defendant Peter Cook. Now secondly, I'll ask you to notice that the case is bent, as though it was stepped on and crushed during a struggle. Later, we will bring witnesses to prove that this locket was found in the effects of Peter Cook carefully hidden from the sight of man. I, Mr. Foreman, will you take a moment to look at this locket and then pass it on to the next juror? Yes, sir. Thank you. The picture has been identified as that of John Talbot. Naturally, he was younger when the... I say, is there anything that matters, sir? This picture gave me quite a start. It looks very much like the man I saw following the defendant on the street in front of my house the other day. May I request that the juror confine himself to the examination of the prosecution's evidence? His remarks are irrelevant. The more so, since John Talbot died some three weeks ago. I begged the court's pardon. I was younger then, and my face was not then drained of blood. But quick, a glass of water, someone, the foreman is fated. No, no, no, no. I'm... I'm all right now. It was just for a moment. Let me have the locket again. Take the picture out. You're from the locket. Just a moment, Mr. Foreman. What are you doing? Stop that at once. You're destroying the evidence. I'm sorry, sir, but there's something on the back of this that I must see. There. Do you see what it is? No, Mr. Foreman, I do not. It's blood, sir. Dried human blood. In Charles Dickens, we meet a writer of many facets and strange parallels. An author with a faculty of applying his genius to all fields of plot from a tender, romantic turn to such a story as we hear tonight. The story of a juryman invested with incredible and supernatural powers of judgment. Friends, the judgment of the Canadian Roll Your Own Smoker's jury regarding the excellence of Ogden's fine-cut tobacco has been unanimous. Yes, ask the person who likes to roll his own cigarettes to name his choice of tobacco, and it's almost certain that his verdict will be Ogden's. You see, Ogden's has been pleasing the majority of Roll Your Own Smokers making new friends right down through the years. Try Ogden's. Taste the rich mellow goodness of tobaccos that are sun-ripened for absolute smoking satisfaction. You'll find Ogden's easy to roll, delightful to smoke. Yes, easy to roll, delightful to smoke. Back now to Hugh Trelawney and the trial for murder. Trelawney has found that of all those concerned in the trial, he alone can see and hear the ghost of the murdered man, John Talbot. As foreman of the jury, his questions have revealed important evidence against the defendant, Peter Cook. Right now, he's explaining to the prosecutor how this mysterious voice guided the course of the trial. And so you see, Mr. Blackwell, how the rumors of my so-called supernatural power started. How I was drawn into the horrible affair almost against my will. Even you look startled when I pride out the picture of poor dead John Talbot. Naturally, Mr. Trelawney, naturally. You seem to do it without any apparent reason. Of course the fact that the locket had been found among Peter Cook's things meant nothing. Everyone knew they had been friends, but finding dried blood on the back of the picture tied Peter Cook to the scene of the killing in the mind of all your fellow jurors. Miss, I see that now.  Now, this voice, at any time during the trial, did any of your fellow jurors see or hear the murdered man? There was no doubt that the murdered man, this ghost of mine, made his presence known to others and myself. Other jurors told me afterwards that they had awful dreams in which the figure appeared and told them that Peter Cook must die. It was this fact that kept me sane. As long as others felt some disturbance, I knew it was not my imagination leading me astray. For instance, nearly every night of the trial, I saw his ghastly figure at the tavern. But each night that I saw him, some juror would awake, screaming a fear to tell of dreaming of John Talbot. But as long as I was the only one who actually saw or heard him, I could not risk my reputation by speaking up.  Then you do believe my story, Mr. Blackwell? I do, sir, and I'll tell you why when you finish it. Well, it was after you started the defense of Peter Cook that I began to hear the voice more frequently. You, of course, recall your star witness, the young woman who lived in the same building as Peter Cook and John Talbot. I do. Her name was Alice Hawkins. She started out as an excellent character witness for my client. But you soon put a stop to that. I remember the afternoon when you first started questioning her. I heard the voice several times as the testimony went on. It seemed impossible that others could not hear it too. But I know now that no one did. Since she was your witness, I recall you were using a gentle, friendly method of questioning. Tell us again, my dear, how long you have known the defendant Peter Cook? Over two years, sir. And you say that in all that time, you never knew him to do an unkind act. Was he on good terms with that unfortunate man, John Talbot? Well, they were very friendly, sir. Of course they had little arguments, but it was all very friendly. But you never saw an open act of hostility. Is that correct? He tried to kill me on Easter Sunday. What was that? May I ask the jury to pay more strict attention, please? It suddenly seems very cold in here, Your Honor. Quiet, please. Now, in short, Miss Hawkins, you never saw signs of anything except friendship between the two men? Mr. Cook is a good, fine man, sir. She loves him. She would lie for him. Are you witness, Mr. Prosecutor? May I ask another question, Your Honor? I object, Your Honor. The foreman of the jury has constantly interrupted this one. Your question seems to have had an important bearing on this case so far, Mr. Trelawney. You may speak. Just two things I'm impelled to ask. First, isn't this young lady in love with the defendant? Second, didn't the defendant make an attempt on the life of John Talbot on Easter Sunday last? I object, Your Honor. Miss Hawkins, please answer these questions. And remember, you're under oath. Are you in love with Peter Cook? Well, what if I am? Is there anything wrong with that? Well, it makes you a rather doubtful character witness, I should say. And more important, what about this other attempt on John Talbot's life? That's a lie. What's a lie? Nobody tried to kill him. Well, what did happen then on Easter Sunday? He almost met with a bad accident. How did it happen? All three of us had gone to church. Afterwards, we decided to climb to the cathedral tower for the view. On the way down, John Talbot slipped and fell. If he hadn't caught all of the railing almost immediately, he would have... He would have plunged to his death. Is that correct? Yes, sir. But it was only an accident. I'm sure it was. And did the friendly relations you've mentioned continue after this accident? Well, not exactly. And why not? Tell us the reason. No, I won't. I won't tell you anything more. Oh, my dear girl, a man has died. Another man is on trial for his life. You must tell us everything you know, or you'll find yourself in a very unhappy position. Now, you've already hinted that friendly relations were strained between the two men after this time. Why? He accused Peter Cook of pushing him down the stairs. But he didn't do it. I'm sure he didn't do it. Well, you, Trelawney, you have set them on the path of truth. It was at this point, Mr. Trelawney, that I had my agents look into your background. Two simple questions from the foreman of the jury, and the prosecution starts to upset my whole defense. Naturally, I didn't suspect the murdered man was working at your elbow. Did he actually call you by name? That was the first time, Mr. Blackwell, and though I was familiar with that ghostly voice by then, it gave me a start, I can tell you. Nevertheless, I believe Peter Cook would be a free man today instead of dangling at the end of a rope in Newton jail if it hadn't been for the next day, Mr. Trelawney. Yes, I'm inclined to agree, Mr. Blackwell. Your suicide theory was built up ingeniously with the doctor's testimony. I know you believed it to have been a fact at the time, because there was a ring of sincerity in your voice. You will note, gentlemen of the jury, that everything Dr. Croydon has said points to suicide. John Talbot the deceased was an unhappy man. He had come mistakenly to believe that his friend, his best friend, had made an attempt on his life. He knew that he had lost the woman he loved to that friend. The good doctor has shown how the wound could have been self-inflicted by a man holding the razor in his left hand. We know John Talbot was left-handed. And in that case, it makes it possible that he did commit suicide. Your Honor, I beg of you to let me speak. Again? Your Honor, if the foreman of the jury has deeper knowledge of this case, I ask that he be made... Mr. Blackwell, I have no knowledge of the case other than what I've heard here in this room. But I am impelled to ask that young woman, Alice Hawkins, that she be recalled in question further. She can tell us whether or not it would have been possible for John Talbot to meet his death by his own hand. The clerk will recall the witness. This is most irregular, Your Honor. Your Honor, if it will further justice, I'm sure my learned colleague, Mr. Blackwell, will have no objection. I'll cross-examine after you finish with your witness, Mr. Blackwell. I have no questions, Mr. Prosecutor. I am confident she can throw no further light on the case. Very well, Mr. Blackwell. Now, Miss Hawkins, is it true that the deceased was left-handed? Yes, sir, he was. He wrote letters and performed all the regular little routines of life with his left hand? Yes, sir, he did. Well, then I failed to see what the foreman of the jury... In other words, Miss Hawkins, he could have easily cut his own throat with his left hand, couldn't he? Well, speak up, my girl. No, sir. It would have been impossible. You have done well, Your Honor. You have done well. You see, sir, it was this way. I don't think anyone knew about it but me. John Talbot was very sensitive about it. Although he was left-handed, there were many things he did only with his right hand. One day he told me he had an accident while a boy, and though he could still write and use his left hand, he could never lift it above his shoulder. And so, Mr. Blackwell, as you recall, the trial came to an end. The jury withdrew to the great room in the tavern to reach their verdict. I remember the strange feeling that came over me as the discussion went on, as though I was becoming less and less Hugh Trelawney, and more and more the voice of the murdered man. The little vestrimon, Mr. Jones, was the only one who was still voting, not guilty. I agree, Mr. Trelawney, it couldn't have been suicide, but would Peter Cook have killed him just to keep him silent? Peter Cook has been proven a man of violent hate. I know all that. I know that he was about to marry Alice Hawkins and that he had another wife in existence. I know that the dead man had threatened to expose him, but is that enough? Always keep in mind that John Talbot loved Alice Hawkins, too. When he found out that Cook was leading her on, his reaction would have been violent. His threat to expose Peter Cook would have caused strong hate between them. I still don't see the picture. Who blew out the candle? Light it quickly. I wish I were out of this business. Hugh Trelawney, talk to this man. That's better. Why couldn't they give us more than one candle? It frightened you, too, Mr. Trelawney. You're quite as a ghost. Listen carefully, Mr. Jones. I'll try to set the picture for you. Morning. It might have been the morning of the murder. Peter Cook came to my room. Peter Cook comes to John Talbot's room. He threatened me. He threatens Talbot's life unless he changes his mind about exposing him to Alice Hawkins. I loved her. He loves her deeply. To hear Peter Cook speak of her lightly infuriated him. He tells Peter Cook that he intends to reveal him as he is. A man already married who intends to wreck her life by this despicable trick.  The room is tense with hate. The fury of frustration burns in the eyes of Peter Cook. He sees the razor near a tanner, takes but a moment. With the strength and quickness born of hate, he moves. The morning sun looks in. Looks in on the lifeless body of John Talbot. It's uncanny. You might have been there yourself.  Yes, I can see it now. So clearly that it frightens me. Look, I'm shivering. The very coldness of death is in the air. Let us give the verdict quickly. Peter Cook is guilty. We filed back across the street into the old courthouse. Across the street in the rain, with a growl of thunder in our ears. The dim, gaslit courtroom was deserted except for the officials and the judge. I stood in the jury box, my heart pounding, my mind reeling. Your verdict, gentlemen. Guilty, your honor.   You are free. And my life has been perfectly natural and normal since I heard the words, You are free, you Trelawney. You are free. Strange, strange. And although Peter Cook never saw or heard this presence, he knew he was doomed from the day you stepped into the jury box. He knew he would be found guilty. He did, Mr. Trelawney. He did. In the confusion that followed the verdict, the judge and I were the only ones to hear what he had to say when sentence was pronounced. Did it have to do with my strange part in the trial? It convinced me that a supernatural force had been at work since the beginning. When the death sentence was passed upon him, Peter Cook said to the judge, Your honor, I knew I was a doomed man when the foreman of my jury came into the box. I knew he would never let me off because, before I was arrested, he somehow got to my bedside in a dream one night. Walk me and put a rope around my neck. From the time warn pages of the past, we have brought you the story, The Trial for Murder. Bell Keepers, Call the Bell. As we come to the end of another weird circle story brought to you by Ogdens, here's a reminder till we meet again at the Ogdens Playhouse one week hence. There's a full week ahead during which smoking enjoyment will make your working and leisure hours brighter. If you roll your own cigarettes, make sure of getting the most out of your smoking by turning to Ogdens. Yes, follow the rule of the smoker who knows a good thing. Make your choice, Ogdens, fine cut tobacco. You'll find Ogdens easy to roll, delightful to smoke. Yes, easy to roll, delightful to smoke. Next week at this time another weird circle story, The Werewolf by Frederick Marriott. Be sure to join us.  It's a delightful, smooth, pipeful every time.