So, last time what we had seen was the statistical model for linear regression, which is of this type where some values x i are the control variable or the input variable or the independent variable and given values of those x i, the values y i take on a value alpha plus beta x i plus some error term. The relationship of y and x is approximately linear given by this alpha plus beta x i, but there is some error because of random causes. So, if we plot data paired sets of data y and x, then we may be able to observe some sort of linear relationship and the first task of these linear regression model is to calibrate this type of regression equation where values of alpha and beta have to be determined. The problem is that because these error terms which you do not know I mean you do not know the so, these error terms e i are assumed to be mean 0 and variance sigma square. So, you do not know those you do not know that variance. So, what you observe are values of x i's and y i's. So, you can pretend that given x i value y i is as per this and we want to find the best value of alpha and beta which will fit this equation. So, what we do is we take the criterion of minimizing the sum of squared errors. So, given x i y i minus alpha plus beta x i is the error term because the e i is assumed to be mean 0. So, that you can remove. So, this is the pictorial representation that there is a line alpha plus beta x i underlying line alpha plus beta x i superimposed that on that are some error terms. So, alpha is the y intercept in this diagram and beta is the slope of this line. So, that is what has been determined. So, if the relationship between y i and y i and x is exactly linear then it all values y i will lie on this straight line alpha plus beta x i, but because of this error term each of the for a given x i the value y i will be some normally distributed random variable with mean alpha plus beta x i is that ok. So, it could be a little more than alpha plus beta x i or little less than alpha plus beta x i and the variance is sigma square. So, that is what we actually observe. So, given all those values that we observe we have to fit a linear expression. So, that is one way of looking at it which is more in the spirit of curve fitting. There is a statistical basis on the basis of I mean on the principle of maximum likelihood estimators which is also given in Ross's book which you can see. In the case of normally distributed errors these two estimates of the regression line the so called regression line turn out to be the same that is the least squares error criterion and the maximum likelihood estimate of this of this line alpha plus beta x i they turn out to be the same. So, we have derived this line alpha plus beta x i that means estimates a and b of alpha and beta using the principle of least square errors. So, the way that goes is you look at the observed value which is y i minus the predicted value as per the regression equation is alpha plus beta x i. So, if you fit values a and b for alpha and beta then the predicted value is a plus b x i. So, the difference in the predicted value and observed value is y i minus a i minus y i minus a minus b x i some of the squares of all these errors because you want to penalize both positive and negative deviations from this set I mean from this value. So, the values of a and b obtained by the minimizing this expression as a function of a and b they are called least square estimators of a and b of alpha and b. So, in this sum of square error term we had seen that the x i values and the corresponding y i values they are known this gives you a quadratic expression in a and b can be minimized sum of I mean two linear equations in two variables.  If you set derivatives partial derivatives with respect to a and b equal to 0 you get two linear equations in a and b which you can solve those are called the normal equations 1 and 2 in this slide they can be solved and you get these. So, the point is that for given values of x i because the a and b values are the error term e i is the normal random variable with mean 0 and variance sigma square b is which has the y terms. So, in this expression for b let us say the x i values are all known. So, in the summation each x i is known x bar which is the summation of x i divided by n if there n n observations that is also known and the denominator is also known x i minus x bar square that is like the variance of the x i is treated. So, it is like the standard deviation square. So, you treat the x i treat the x i values not as random variables but just as a set of values which you observe that has got a standard deviation. So, that is the term there, but for given values of x i the y i's are all random variables right each y i is equal to alpha plus beta x i plus e i square. So, for a for given values of x i these are random variables. So, actually this b b is a sum of normal random variables. So, actually b b is b is itself a random variable. So, it has got some some b in and variance and distribution. So, it can be shown that the expected value of a is actually alpha the parameter that we are trying to estimate and the expected value of b is actually beta the parameter that we are trying to estimate. So, the expressions for a and b if we treat x i as known and y i's are random variable then a and b are random variables with expectations equal to alpha and beta. So, you can treat the whole set of observations that you see as outcomes of an experiment. If the underlying model is y is equal to alpha plus beta x i plus plus e i then for a given x i the y i is a random variable because it is it has got this e i term and that is what you observe. So, for a for a given set of values you compute this.  So, here in when you when you compute when you compute b as per this expression for given x i and observed y i it is like the realization of a random variable is that ok, but before you do it what you observe y i is a random function of x i because of the error term. So, it has got a certain expectation it has got a certain variance and so on right. So, supposing I think of supposing I think of doing a regression analysis of a large population of people where I look at height versus weight. Now, this is clearly I mean an example where these are characteristics of a certain population it would be difficult to argue that you know height causes weight or weight causes height that would be a bit stretching the thing may be there is some common genetic characteristic which causes both height and weight. So, that that may be, but in any case we are able to observe two attributes of the population. So, we are trying to see if there is any relationship between height and weight of of an individual. So, if we sample several individuals we get we get several readings ok. So, now the question is if this is sort of commonsensical that the taller the person the heavier he or she may be, but you know you have all seen tall thin people and short fat people and also some short thin people and tall fat people and several medium people and all those things right. So, you will you will see a scatter, but if you if you just look at the cloud of points or on a two dimensional graph of the attributes you can say that there is some sort of correlation. So, actually if you if you compute the correlation coefficient between height and weight I believe it is approximately 0.4 or something like that. So, it is it is positively correlated that means you expect a taller person to be heavier, but it is not that you know if the if there are two people and one person is twice as tall as the other person then she is going to be twice as heavy as the other person. So, it is not like a linear relationship or something it is it is got some correlation, but it is not it is not exact and in any case the for a for a given height you you see a certain spread of weight. So, if you if you take any any height like like this height whatever it is you you see that there is a spread. So, for given height of whatever you see that there are some some light people and some heavier people. So, you see a certain spread because of the randomness inherent in the evolution I mean the growth of living beings ok. So, so what we want to so we can do the following thing we can plot this in such a way that the units of height are in some units say centimeters, but the scale is such that one standard deviation of the height unit is equal to one standard deviation of the weight unit in at least on this page. So, if I you know if I if I move one centimeter on this page it corresponds to some number of standard deviation of the height parameter and same thing on the weight parameter. So, then the I plot a 45 degree standard deviation line which says. So, there will be a certain mean there will be a certain mean height right and there will be people who are one standard deviation more than the mean height means people who are taller than average by one standard deviation and people who are heavier than average on one standard deviation. So, that line will be a 45 degree line is that ok you got what I am trying to do. So, this is the standard deviation line which says this is the mean height this is the mean weight of the population that I am observing and this is the point corresponding to mean height plus one standard deviation and mean weight plus one standard deviation and this is the line corresponding to mean weight minus one standard deviation and so on. So, plus minus k sigma for different values of k that line will be a 45 degree line as because of the scaling of the x axis and the y axis is that ok that is what I have just schematically represented here. Now, the question which we want is the following that given given a certain height guess I mean given a certain height what is the average weight of the person ok. So, given given a certain height. So, let us say I mean in centimeters I do not know what is what is the average height which people have you think I mean what is your height in centimeters I mean does anyone know their average height that does anyone know their height in centimeters 170 ok. So, let us say supposing that is the average height of this population. So, and the standard deviation of this group is something let us say 10 centimeters ok we can compute the standard deviation. So, we plot this thing which is the 45 degree line is 170 plus 10 centimeter on the x axis and the average weight may be whatever 62 kilo whatever it may be is on the y axis and it has got a standard deviation of let us say 4 kilo. So, we can plot plot that thing. So, now the question is if I give a height like 180 centimeters what is the average weight of people who are 180 centimeters tall ok. So, there will be a certain spread. So, if I take. So, this is say. So, somewhere here in the in the midpoint of this range is say 170 this is 180. So, I would I would have a certain range ok. So, what is the average of people in this range? Similarly, for people here what is the average of people in this range?  So, what you would see is that if I if I take a band here and ask what is the average of average weight of such people then you would see that if I go by this line that is likely to be an underestimate. And if I go by I mean if I if I am that is if I am below the mean height then if I go by this line then that would be a slight underestimated. If I go to the right of this that means if I look at the average height of people in a certain band here average weight of certain people in a certain band here then that would be a slight overestimate. So, actually if you if you if you go see you are you are looking at every person has got a discrete value of height or weight you know. So, if I if I ask what is the average height average weight of a person who is 180 kilos then you know I round it off and I say say I create a small vertical strip of people who are approximately 180 kilos say plus minus 0.5 kilos rounded off to the nearest kilo. And there will be some people in that band right. So, what is the average weight of such people? So, if I plot that curve that will be some straight I mean that will you know because of the nature of this curve that will be I mean because of the nature of this data that will be it will be something like this. So, the regression line is actually a smoothed version of this line. So, what comes out if you if you look at this scatter plot type of data it it actually you can you can see it that the the regression line actually will be something like see if I see data like this then I suspect that there is a linear relationship. That means, as one increases the other increases on average. So, I mean I may have some outlier here and here and things like that or here, but on average this is the cloud of of data. But if I look at so this is the 45 degree line if I look at this strip average of these people is likely to be above this line right. If I look at this strip the average of these people is likely to be below this line. So, actually the regression line is something like this ok. So, if you look at a set of data and if you sort of sketch these type of data and look at scatter plots of related things you will see that the regression line is is going to be something like this. And in the case of correlation equals 0 correlation equals 1 and correlation equals minus 1 you can see that this is the case, but for other values it. So, this this so the actual statement is something like this let me let me just go back to this for a minute. So, this is the typical shape of linearly related data that you have the American football type cluster data that is what we typically see. So, if you plot the line of x plus k sigma x that is the standard deviation of the x variable and y plus k sigma y then you call this the standard deviation line. So, if you scale it properly then that will be a 45 degree line just for convenience. This will pass through the center of the data this will pass through the x bar y bar point ok because that is k equal to 0. So, you can you can see that. So, can we use this line to predict the average value of y given given x ok. So, what you see is that when the correlation is positive that this average for y lies above the standard deviation line for x when x is below the standard deviation. So, for x k sigma below the standard deviation the average value of y will be above the standard deviation line and for when x is above the standard deviation the average value of y will be below the standard deviation ok. So, so this is what was observed.  So, this term regression it actually means to go back to ok. So, if you just look at English meaning of regression it is to go back to. So, what why is this term used in the context of curve fitting or statistical analysis of correlated data. So, he conducted in a series of observations measurements of father's height versus son's height ok. So, in the sample that he had this thing and I think he took quite a few many hundreds of observations and this is the summary of what he apparently concluded. The average height of father's around 68 inches average height of son's was around 69 inches. So, you know in one generation there was a some small increase in the in the height because of better nutrition whatever whatever whatever was the condition in that society. The standard deviation for both was around 27 2.7 inches and the correlation coefficient was positive 0.5 which means that on average taller fathers have taller sons because of genetics may be ok. But it is not but the but the key point was that fathers who are k sigma in this case sigma f that is the standard deviation of the father's heights and the standard deviation of the son's heights were both same. So, sigma f is equal to sigma s, but father's yeah this is what I am talking about. So, the average height of father's was 68 inches average height of son's was 69 inches for both the standard deviation is approximately 2.7 inches and the correlation coefficient was 0.5. So, the scatter plot data of this for many hundreds was was compute was gathered and computed and in this cloud of data this person saw that there is some sort of linear relationship. I mean he hypothesized that it is useful to have a linear relationship as an explanation between the two because it is simple to explain simple to manipulate and so on. And so, from this type of data it it it shows that taller fathers on average have taller taller sons, but fathers who are k sigma. So, let us say fathers who are sigma above the mean. So, the mean is 68 inches and so fathers who are approximately 70.7 inches that means who are taller than the mean by one sigma. Their sons were were not you know one sigma taller than their mean. Their sons were only r times taller than the mean and similarly for shorter if a father was one standard deviation below if a set of fathers was one standard deviation below the mean height for fathers. The sons were shorter on average, but they were not as short as as the fathers relatively speaking. So, they were only r times shorter than the mean of the their population the sons population. In this case you know the standard deviation for both is the same because you are measuring height only and the standard deviation did not change too much from generation to generation. The average height changed a little bit, but the standard deviation did not change. So, in this case you can actually sort of related, but even if any other things was measured like supposing you are measuring height versus weight and so the the right way to think about it is that if I look at correlated values of any any paired data the regression equation shows that if if the x variable is say one standard deviation above the mean of the x values the corresponding y variables their average will be only r times more than the mean of the y values, where r is the coefficient of correlation ok.  So, so what it says is that taller taller fathers have taller sons, but not as tall as as as much taller about their mean as the as the father. So, so Dalton Galton sorry not Dalton Galton term this this phenomenon as regression to mediocrity. So, taller fathers sons of tall fathers were taller than average, but not as much above average as their fathers shorter shorter sons of shorter fathers were also shorter than average, but not as short as not as much shorter as deviation from the mean as their fathers. So, they tended towards the average and and taller fathers had sons who tended towards the average. So, Galton term this term as going back to the mean or he called it regression to mediocrity or something. So, he was some aristocrat or whatever. So, he had some meaning some sociological type of explanation that society pulls you back to mediocrity or anyway he called it regression to mediocrity or regression to the mean is the is the phenomenon clear what I am talking about. So, you can read the explanation in in the book also and so the the other question which we talked about briefly in the last class which I want to just explain here is what to regress on what. So, we called x as the independent variable and y is the dependent variable. So, like supposing we are talking about height and weight data then what is independent and what is dependent actually both are both are probably dependent on on some gene or or environmental factor or or something on nutrition or whatever it is some combination of some unknown things we can make a guess, but right now we just want to look at the relationship between height and weight. So, we cannot say height causes weight or weight causes height we can we can very well regress either on a on the other any one of them we can regress on the other. So, the example of even in the example of linking sun's height to father's height maybe you can say sun's height is caused by father's height because of genetics, but you can regress the other way also and there you can certainly not argue that you know the father's height is caused by the sun's height that that would be a little stretching. So, remember that regression does not imply causality if if one of the variables in fact causes the other then we might be tempted to try that relationship first that means whatever is the causal variable we take as the independent variable and the the caused variable we take as a dependent variable, but otherwise you can regress anything on anything. The other reason we may want to regress one variable on the other is that we want to predict that one. So, if I want to predict the the the average weight of a group of people based on height measurements then I will take y as the independent variable which is high which is weight and x as the dependent variable independent variable which is height. So, if I want to predict one on the other then the one which I want to predict I take as the dependent variable. So, then I but I can I can do regression either way I can regress y on x or x on y. So, the if we if we follow this argument then so, this is y and this is x this is the sd line and this is the regression of y on x it sort of gives you the average values of y for a given x you can do the other way also and that will be this sort of line. So, this will actually give the the other type of thing that means, I I will I supposing for a given y I want to find out what is the value of x then you will see that this is a better explanation of the the midpoint of this line then this standard deviation line which is this one is that ok. So, you can actually if you if you look at this data. So, you can you can look at any typical exercise in in Ross's book or anywhere else and and actually try this you can try regression y on x and and x on y. So, suggested exercise is chapters I mean exercise 7 in chapter 9 which is the chapter on regression in Ross's book. So, there so, you can on any set of regression data you can you can you can construct a regression line for y given x that is called regressing y on x or x on y. So, it is actually paired set of data in in this example as in several other examples the the data are clearly sort of two related attributes from the same population there is no implication of causality. So, it is you know that so, you can try both regression and you can actually you will you will get coefficients alpha and beta in both the cases you can actually think about it and you can derive a simple relationship between what are the relationship between those alpha and beta coefficients that you get in when you regress y on x and x on y it will be some sort of reciprocal of each other. So, you can work it out actually ok. So, just let me complete this part which is the sort of technical part of it. So, this is all this I have put up. So, these are the sections which I would like you to read. So, the basic regression model is this. So, for for given x i y i is a normally distributed random variable with unknown parameters alpha beta and sigma square ok. So, now you can you can think about it as a hypothesis testing you can give confidence intervals you can you can go back to the framework of the statistical. So, if you have a statistical model like this then from the given data you can actually compute all the statistical quantities of interest. So, sigma square must be estimated from the given data and alpha beta are chosen to minimize some error term. So, the distribution. So, b b is the least square estimator that we have given a formula for b it is a lean it is an it is a linear combination of independent normal random variables with expected value beta and variance equal to this. So, if you see the variance of b it is sigma square and just depend on the x values and the n values. So, the. So, similarly a is also a normally distributed random variable with that mean alpha and the variance of a is that ok. So, this can be derived ok. So, the part that is remaining is the estimate of sigma. So, what we see is the sigma is see if sigma is large then what do you expect that the y i will be quite scattered around alpha plus beta x i. So, if I look at y minus a minus b x i then I will get that is an estimate of sigma. So, the sums of squares of residuals. So, the residual is for fixed value of a and b and x i I mean for fixed value of a and b the residual is y i minus a minus b x i. So, this term once I once I find a and b using those formulas then this term is an estimate of sigma square. The only thing is I have used the same expressions for estimating those mean values. So, I have to divide it by n minus 2 ok. So, the sums of squares of residuals divide by n minus 2 is actually turns out to be an unbiased estimator of sigma square which is the variance of the error term ok. So, this also can be can be derived. So, in summary a which is got from that formula actually the mathematics of it from the normal equations those two normal equations linear equations got by setting derivatives of the error terms equal to 0 with respect to a and b they are known parameters. Turn out to be such that you first compute b and based on that you compute a ok, but it is all a function of the data y and x and if you treat x as given and y as a random variable given that x then this is also a random variable and what you observe is a realization of the random variable ok. So, in summary a is an unbiased estimator of alpha b is an unbiased estimator of beta. What are a and b? They are from the quantities derived from the normal equations and the sums of squares of residuals. So, once a and b are determined then the sums of squares of residuals this SSR that divide by n minus 2 is an unbiased estimator of sigma square ok. So, that is the basic summary of the regression linear regression model and the associated statistics. So, in fact, we know the variances of these things also. So, we can actually give confidence intervals for these a and b and things like that. So, you know we can write a regression equation for anything this because this formulae are there. So, you can write you can construct a regression equation for anything, but how good is it for prediction? So, if I use it for prediction then I will get an estimate, but the spread of that estimate may be quite large it might be quite it might be so large that it is almost useful almost useless to predict. So, that depends on the data of course. So, you can compute those those confidence intervals for associated predictions. So, just to complete this thing the notation is you know if you if you write it in this sort of covariance like term S x y and the variance term S x is standard deviation square and S y y then we can actually do inference about regression. So, for example, one of the first questions you would ask is that is the dependent variable really correlated with the independent variable you know I see this cloud does it really mean anything or is it just you know x and y are just moving around randomly I mean they are just independent random parameters. So, is there really a connection? So, the hypothesis is is beta equal to 0 if beta is equal to 0 then y is just alpha plus some noise whatever x is y is. So, you know I may just have some some values which are you know just noise terms you know if I just see a just a some values like this. So, there is no there is nothing I mean if I if I tell you that x is this value then y has got a certain spread if I tell you x is this value then y has got a certain spread which seems almost the same as this. So, I do not get any more information about the likely values of y if I tell you what x is and maybe vice versa also. So, in this case there is no relationship between x and y. So, if I plot a linear regression then you know I am going to get a line like this this is the best fit. So, beta is equal to 0 slope is equal to 0 ok. So, that is one of the fundamental you know hypothesis regarding regression that is is beta equal to is beta equal to 0. So, we can pose this question in the language of hypothesis testing and take that as the null hypothesis and we can sort of work it through it turns out that I mean just going through the same whatever I said about the parameters b and the estimates of those things. So, actually it turns out that this quantity here which seems complicated, but basically it is I mean the main thing to note is that it is something that explicitly computable. So, square root of s x x x is the standard deviation of the x values b minus beta, beta is in this case 0. So, that is the hypothesis divide by sigma. So, sigma is estimated by that. So, this quantity has and divide by the denominator which is also a known quantity once you compute it. So, this quantity treated as a random variable has a t distribution with n minus 2 degrees of freedom. So, from the regression data if we treat x i as random variables one can construct a new random variable this one. It looks complicated, but it is a function of the known parameters which has a known distribution. So, using that known distribution we know what values it can take with what probabilities. So, we can test hypothesis. See in hypothesis testing what we did was to answer the questions of our interest we found a known random variable with a given distribution. So, that will have certain properties and. So, therefore, it can take on extreme values only with the small probability. So, then we are able to put type 1 errors, type 2 errors, level of significance. So, because of the randomness it is possible that this random variable can take on some extreme values you know very large values or very small value, but that is very unlikely. So, we put a bound on this value and if it lies in a certain interval then we come to some conclusion about the hypothesis. If it lies outside then we reject this hypothesis. So, that is what we do here. So, if h naught is true that is the null hypothesis that beta is equal to 0 then this random variable is the t distribution with n minus 2 degrees of freedom. So, then that gives rise to this test. So, compute this quantity n minus 2 s x x divided by s r square root of that times absolute value of b. So, given the data this left hand side in this test can be computed. And on the right hand side you have the critical region of the test which is t distribution with n minus 2 degrees of freedom and if gamma is the level of significance then gamma by 2 and if this test statistic is outside that then we reject the null hypothesis otherwise we accept it. So, it is the same logic as before only the expressions are little complicated, but it can be I mean. So, I would not expect you to derive this or something it is actually quite straight forward it follows the same logic as before and it just deals with normal random variables and t t distribution. So, this this can be derived. The point is that there is a well defined statistical test for accepting or rejecting the hypothesis that there is a relationship between the input and output variables. Last thing which I would like to say is I think this is the last one. That there is a there is a value called the r square value which is used in regression models which is the extent of fit. So, see I can I can have regression I can plot regression values for anything I can plot regression values for supposing these are the values then you know my regression line is like this. Yeah I can if I look at these two cases in this case increasing x values indicate increasing y value in both cases correlation is positive. In this case increasing x value explains the variation in the y value. It says the y value increases and to a very great extent the explanation is fully captured by the x variable. In this case or let let me just exaggerate it a little bit. In this case also if I if I increase the x value the y value also increases, but the increase in y value is not explained by the increase in x value to that extent. So, obviously you know there is a coefficient of determination which I mean this is a better regression fit than this right. So, what is the what is the quantitative way of saying that? So, it is the it is the the deviation sums of squares of deviation of the y values with respect to its mean minus the sum of squares of errors of the residual divide by the I mean just to get a standardized quantity that is called the r square value. So, this r square value you can show is between 0 and 1. So, if this if there is r square values is high close to 1 then we say that the regression fit is good. So, the coefficient of determination is good ok. So, this is just a quantitative test for how well the the the regression line explains the data it no causality and all that how how well does it explain the data ok. So, basically so y y increases as x increases, but how much of the increase of y is explained by increase of x this r square value will tell you that it is a value between 0 and 1 that that can be shown. So, this r square value is between 0 and 1 a value close to 1 indicates that the regression model provides a good explanation for the variability in the y values. A value close to 0 indicates that the model does not really explain the variability in y I mean the x values so, this r square actually it is in the example of the father and son heights business this r square value is 0.961 which is quite good. So, of course, the the square I mean you would you would suspect that this r square has something to do with the correlation coefficient r if I just view. So, forget about this random variable business if I just look at a set of value say say heights and a set of value say weights as just independent data sets. And if I compute the correlation between these two sets of data then I get the correlation coefficient r that is actually the square root of this r square only the sign has to be appropriately put because you know it could be positively correlated or negatively correlated for convenience this r square value is the sort of square of that. So, it so, correlation coefficient r is between minus 1 and 1 and this r square value is square of that. So, it is between 0 and 1 high value of r square indicates good fit of the regression model and low value of r square indicates that the regression model is not a very good one ok. So, of course, there are many caveats this this r is the the standard coefficient of correlation. So, it is the square root of r square actually. So, except for the sign indicating whether it is a positive correlation or negative correlation the sample correlation coefficient is equal to the square root of the coefficient of determination. And the sign of r if you want to sort of derive it from the data it is the same as that of b that slope coefficient you know we have we have we have calibrated the regression model as y is equal to a plus b x. So, if b is positive it means a positive correlation if b is negative it means a negative correlation. So, that that the sign of r is the same as that of b and the value of r is the is the square root of r square ok. So, now where should you use this regression model well whenever you see data like this you can use a regression model if you see data like this regression model is very great and probably somebody has cooked up the data this is very too good to be true, but you can use the regression model this one is also seems ok. But if I see data like this you know then I I would be a bit foolish to use a linear regression model to explain the relationship of y and x right it is not linear I mean it is clearly not linear it is some nice quadratic or something like that. So, linear regression model is is not really useful, but you know the thing is that you can you can still do it what will you get if you if you if you use a linear regression model here well you will get some line like this I mean it this this is the best fit straight line fitting this data. So, that is not very useful. So, for high values of x and for low values of x I mean it it just says that you know the values of y are clustered around this in some way. So, it does not tell you the qualitative nature of this behavior is that for some low values of x and for some high values of x the values of y dip down and there is an so it is a it is a quadratic relationship the regression model will not be able to capture that although technically you can you can do this and you can compute r square and you can compute a and b and all that you will probably get a low value of r square which should warn you that this thing, but the the point is that when you are using these models in in in practice please plot the data and look at it first before doing anything please plot it and look at it. So, later on if you use regression models anywhere the first step is to plot the data and and have a look because you can always put all this data and spreadsheets and you know software and and get get some coefficients and keep on doing it, but you should not lose the basic sense of what what is it that you are trying to do is that ok. So, all this data analysis and all that is fine, but please do not discard your common sense if something does not agree with your common sense chances are you know something is wrong. So, please take a look at it in a few cases your intuition may be wrong, but it is definitely worth a look. So, do not disregard your common sense no matter what the software tells you or the program tells you or whatever tells you do not disregard your common sense. So, no matter how many marks you get in this course or do not get in this course one thing which you have it use your common sense do not lose it.


You can't buy an online e-course on how to do this or how to do that. Pay that money, download the material, fill it out, pass the test, and then walk away anointed of God. The power of the Holy Ghost is found only in the presence of the Holy Ghost.


To truly enjoy sports, you must be strong. Walking into a gym, hearing the sound of the ball being dribbled, shoes squeaking, the whistle of a referee and the fans cheering, is a feeling that is unlike any. For me, what separated Haikyuu from other sport anime from the get-go was the very real and believable emotion that comes into playing sports at a high level. The incredible highs and the crushing lows that come with sport, the mental strength that comes with tirelessly repeating in action until it's second nature, and playing a sport for that one moment in which you fall in love with it. For me, Haikyuu encapsulates all these feelings with the foundation that is their masterful storytelling. Haikyuu follows the Karasuno volleyball team, which focuses on Hinata Shoyo, the main protagonist of the show. And while many games were played throughout three seasons, I will jump through what I felt were the most important moments in terms of emotion in Haikyuu. Beginning with the very first episode of the show, it really captures that feeling of a first game, whether it be a first game back or the first game ever. Immediately, Hinata takes us through the varying emotions of playing your very first game, from the pregame nerves to the warm-up, and then through the game. And through this first episode, they flash back through to Hinata's hard work to try and get members to simply join him on the court. This alone showcases his desire for simply wanting to be on the court. And then when we finally see him hit his first spike, it's blocked. But we see him brought back up when he actually scores the next spike. And then we see the bliss of spiking a ball or scoring a goal or hitting a three. It's all the same. Hinata is desperate to feel that high again. And when he loses, he's desperate to know what it feels like when you win. After they lose his first and last junior high school volleyball game, the truth about team sports is revealed to him quite bluntly. In order for your team to win, all the pieces must work together. All of your members on the court must be strong. In a team sport, there is no one person who can lead their team to victory. It is done with all six or five or eleven, no matter the sport, not one. In order to remain on the court. Jumping to the middle of the first season before Asahi rejoins the team, he does not want to come back to volleyball. There are certain defining moments in sport, and some include scoring a game-winning basket, spike, or a goal. But on the other hand, suffering a crushing defeat can truly define a team or a player. By the way, they bounce back. For Asahi, in his second year, the ace was constantly given sets against the iron wall that is that day tech. And Asahi would get his spikes blocked repeatedly throughout the game. Each block gutted his confidence to the point of him leaving the team. For Asahi, loss drove him to the point where he fell out of love with volleyball. Sport is arguably more mental than it is physical, and Haikyuu is able to properly present the two sides of it. Some people would have left volleyball behind to never return to it, but Asahi would come back to the team, displaying his passion for volleyball, and his growing mental fortitude. Oikawa is one of my favorite players in Haikyuu. Because of his contrast to the freakishly talented duo Hinata and Kageyama, he represents work ethic and hard work to a tee. He is an example of what happens when an individual who is not special, not incredibly athletic or insanely physically gifted, is committed and is passionate about being the best he can possibly be. He has studied matches, trained not only his body but his mind to be able to fight at a high level, to pick apart his opponents and to bring out the best of his teammates. He is comparable to the Kauai Leonards and the Steph Currys of sport, so yes, he is talented, but has worked intensely harder than the rest to get to where he is. And throughout this first encounter with Aoba Josai, we see Karasuno's great run come to an end, and the way the game ended would be on the duo's freakish quick being blocked. The team has felt lost before, but not when their team felt unstoppable. Karasuno surely felt as though they could never lose, and for this game to be ripped from their duo's hands, who would help bring the team this far, that pain is indescribable. Specifically for Hinata and Kageyama, the final play would get replayed in their minds over and over again, and they would feel guilty and responsible for their team's downfall. When you lose, you feel as though it's all on you. When you miss that final shot, your mind starts to create what if scenarios? What if you had hit it? What if you had made a shot earlier in the game? What if you made the right pass? But it's life. All of that guilt and that anger just builds. Like Hinata and Kageyama do, you need to find an outlet to release that anger. And like Karasuno's advisor would tell them, they have fallen, but only the strongest stand back up.  To play sports, you must be strong. Until the interhigh preliminaries, Karasuno goes to Tokyo to partake in several training camps with other schools. During that time, we have another low, when Kageyama and Hinata have their argument. All teams have moments whether it be on or off the court, when they must figure out in that space why a particular thing is not working. However, the argument should not be a focus, but rather what the team or players do after that argument is important. Hinata and Kageyama spent time practicing on their parts of the quick, in order to make it as unstoppable as possible. Which meant that Kageyama would have to put a specific spin on the set, and Hinata would have to be able to fight midair. And throughout these training camps, Karasuno won three games and suffered 63 losses. However, during the camp, they practiced new moves and sets that would become crucial in the future. To play sports, you must be strong. You have to suffer losses in order to win. Which brings us past the interhigh prelims. To the spring prefecture playoffs where Karasuno faces off against Aoba Josai for the second time. And let's take a look at that final play of the match. We have Karasuno's team feeling an incredible bliss. From losing against Aoba Josai in those exact same circumstances, that feeling of not only winning against an impossibly tough opponent, but winning against the very team that broke their spirits before, is a feeling unknown to them. All of their diving recieves, the blood of the captain, the sweat of the team and the literal tears that were placed on those courts, were all for that moment. With that, I want to point out the parallels in Season 1 and Season 2, specifically for Hinata and Kageyama, and how they defeated Aoba Josai in the exact same manner that they lost, a quick. And this time the duo would spread their wings and soar over their competition. This is where the mental strength comes into play. Sport is survival of the fittest. To move forward to remain on the court, you must become strong. It's good seeing the fruits of your labor and hard work tend into results and personal gain and wins, but on the other hand, that same hard work, the tacks that you have to put on your body, all for it to go down in one single play. Aoba Josai played their hearts out. They played as good as they could have possibly played, but on that day, it was simply not their day. Seeing their players hold back tears and then completely break down, again encapsulates the mental aspect of sports. It shows the intense amount of pressure that athletes put on themselves to perform. The blame that the ace put solely on himself. There is not one person that could feel exactly what he was feeling at that moment, but the other ten or so players wearing that jersey. And what hurts even more was that it was their final game together, as Oikawa and some of the others were seniors. And the final moment I want to look at is Tsukishima's passion during the third season. While they were having spring practices, Tsuki was often shown not wanting to put the extra effort in to try and better his skills and his reasoning being that he will never be the best, that there was always someone better than him. He then asks Bokuto, Ace of Fukurodani, why he plays, and he tells him about the specific moment where he got hooked onto volleyball. He then tells Tsuki that once that moment comes, he will truly love the sport. He will get hooked. In this game against Shiratorizawa, a fire was lit under the middle blocker. When his moment came, Tsuki rose to his competition and blocked the ace's powerful spike, and his passion for volleyball was announced. That's a moment I think that applies for everyone, and Haikyuu demonstrates that perfectly. Whether it be an amazing spike, a goal that you've been practicing, or having a very good game, these moments are often defining moments that one never forgets. It's a very precise moment that feels like you are stuck in between time and space, and that you are meant to do nothing else but to play. For me, Haikyuu's prowess lies not only in its amazing storytelling, but its ability to convey emotion and as an athlete, Haikyuu precisely points the exact same emotions I felt, like the players on their team. Haikyuu succeeds by making me feel like another player on the roster. I feel exhausted when I'm watching them walk into their fifth set, and it also broke my heart, like it was my team, who lost after being so close. Regardless of the sport you play, Haikyuu's ability to echo the same passion and heart that go into team sports, and to be able to convey the ups and downs of sport are some of the reasons why Haikyuu is one of my favorite shows. Haikyuu's messages of resiliency and growth don't apply only to sports, but to life. Only the strong get to remain on the court, that goes for any court. Build yourself to become the strongest person that you can possibly be, and you remain on the court forever.


So YouTube team keep it clean. What's going on? It's Aang Raven here with another video. This is a video that I figured was probably gonna make this year. I actually initially thought it was gonna happen last year, especially with them having drafted Rashad Bateman in Thailand Wallace. And then Miles Boykin having that injury at the beginning of the season. Like right before it started, I believe it was a hamstring injury. I thought the Ravens were going to move on last season, but they didn't. But you could tell by their actions, which spoke a lot louder than their words. You could tell that last year was it when it came to Miles Boykin. Reason I say that is because when he did get healthy, he recovered from his hamstring. He had recovered from, I think it was a pinky injury. It was either pinky or the index finger, either one. He recovered from both of those injuries and he was just on special teams. And then I think he might have called like, had maybe a pass go his way, maybe two at the most, but you could tell like that was it. They didn't hardly have him on offense anymore. And it just seemed like he had ran his course. Like that was a wrap. So with that being said, the Ravens are releasing them. They're releasing them, cutting them, waving them, whatever you want to call it. His time with the Baltimore Ravens is over. Now he was, of course, a third round pick in 2019 and there were flashes. There were flashes. Most of them came in 2019. There were a couple that came in 2020. We didn't really get to see much of him this past season. But I remember in 2019, first game of the season. First game of the season. I know everybody remembers the game with Lamar Jackson against the Browns from last season with Lamar Jackson. He's dropping back and back and back and back and back. He got your David and Clowney coming from one side, Miles Garrett coming from the other side. And he throws it up in the end zone. Mark Andrews comes down with it. But the very first time that he did that in the NFL, it was week one, 2019 against the Miami Dolphins with Lamar Jackson. He's dropping back, dropping back and going back and then he throws it, launches it, back at the end zone. Touchdown, Miles Boykin. And then one of the more memorable plays from Miles Boykin, his rookie year in 2019, it came against the Seattle Seahawks. I think it was either the very first play of the game or definitely the first drive against the Seahawks. Well, Lamar dropped back and launched it up. And Miles Boykin with a cornerback draped all over and came down with it. And then of course the probably the most memorable play for him in 2020 was against the Cowboys where he threw the Miles Boykin. He made one guy miss, had some good blocks in front of him and took it to the end zone. So there were these flashes with Miles Boykin and the Baltimore Ravens, but then there was some not so pleasant flashes. We remember the play against, I think it was the Washington football team where I think RG3 just threw him a bomb and he just, he wasn't ready for it. I think RG3 might have thrown it a little bit early. There was another play. I think it was against the Browns in week one of 2020 where Miles Boykin, it was a, they weren't going for a back shoulder. They were going for a comeback route in the end zone because the Ravens were like on a 10 yard line or something like that. They were super close. Lamar Jackson threw it to Miles Boykin anticipating him having turned around a bit earlier and the timing and the chemistry is just way off. And that was probably the theme when it came to Lamar Jackson and Miles Boykin, the chemistry just wasn't there. It wasn't there. And what I'm hoping is that moving forward, wherever Miles Boykin ends up landing, wherever he ends up going, whatever team he catches on with, hopefully this new team will utilize him the right way. Hopefully this new team will really maximize his potential and it won't just be potential. He won't just be known as a blocking wide receiver. No, no, no. Because offensive linemen are supposed to be known for their blocking. Now it's a good thing that he can block, but you got to use him more than just a blocker. And the Ravens just, they didn't do that. We always talk about it on here how with the Baltimore Ravens, for receivers especially, if you're not drafted in the first round, it's gonna be a long shot for you to make it. It's gonna be a long shot for you to have a successful career with the Baltimore Ravens. If you are not a first round receiver and even with a first round receiver, it's not even a given. Like you have Travis Taylor's, you have Mark Clayton's, you have Rashad Perryman's that they didn't make it. They didn't make it. But now you have a Hollywood Brown and you have a Rashad Bateman and they look to be well on their way. But still, even the first round is not a given. But then even the second round, ooh, it's, that's where it gets really shaky. Of course we know about Tory Smith, but other than that, what receiver did they draft in the second round? I don't even remember, but Tory Smith made it. But then you go to third round and after that, it's, no, it just, it doesn't happen. Why? Because they don't invest into those guys like that. They don't. They don't invest into those guys like that. So I do, when they third round pick, scary. Miles Boykin was the third round pick. You see that? James Proshay is a fifth or sixth round pick. Tyler Wallace, the fourth round pick. The investment's just, it's not there like that when it comes to those later round picks for the Ravens. So hopefully they can change that. Hopefully. I wish we could have got Keith Martin and T, I mean T Martin and Keith Williams, a little earlier. I wish we could have got them in 2018, 2019. I wish. Cause maybe the trajectory of Ravens receivers might have tilted on the right side a little bit earlier. A little bit earlier. But either way, with Miles Boykin, it was said, what is this? This April, it's April 15th, 3.48 PM. It was said about a month ago, well actually yeah, a little tiny bit more than a month ago. Maybe like a month in a couple of days. Cause I think it was right before free agency or maybe it was during free, either way. It was said a little over a month ago that the Ravens were trying to trade Miles Boykin. And when that report came out, we just knew it was a matter of time. I actually thought that something like this, whether he was traded or released or whatnot, I actually thought something like this would happen sooner, especially once that report came out. Because when reports like that come out, that means, all right, the team, they've been trying to deal this player, but they haven't gotten the right compensation form. So they doing one last effort, one last pitch to everybody. One last, all right, let's go for it. Let's put it out there to see, okay, what can we get for this player? What kind of draft picks, what kind of capital can we get for this player? It's the team giving one last effort to try to make it happen. So I thought that this would happen, like I said, a whole lot sooner than it is happening. But no. So the timing, the timing with this, it is a couple of weeks before the draft, it certainly is, but at the same time, ooh, time is a little tricky. Young guy, so that's good. The potential is there, you just gotta reach in, get it out, that's it. And I think with Boykin, I always said this, said this from jump, said that with Boykin, it's all about opportunity and it's about volume. It's about volume. Yeah, we know there's only one football that goes around, but it's about volume. You give him the looks, the targets, then he'll make something happen for you. It was not happening there with the Ravens though. It just wasn't, it wasn't. We keep talking about a big body receiving and all that. Yeah, it wasn't sitting right there, but just ain't giving the chance. Just ain't giving the chance. So I do respect that they are releasing him. Reason I say that is because you wouldn't want a Tyson Williams situation to where you know the guy just needs a shot and he could do his thing, but you just holding them down.  So I respect that they are releasing him.  But all right, cool. It is what it is. They releasing them now. All right, so hopefully again, wherever he goes next, he can go there and go do his thing. Wherever he goes next, hopefully he could take off. Wherever he goes next, hopefully it could be a more past heavy offense, that's really throwing the ball like that and he could take off. So we'll see, man. We'll see. Recent receivers that I can think that the Ravens drafted that they moved on, but Shaw Perryman, it certainly got better for him after the Ravens. He went to the Browns. He did his thing over there. He went to the Bucks. Yeah, I remember that. I think he was on the Lions for a little bit too, but he went to the Bucks and he did his thing. He wasn't getting all these eye-popping numbers, but his impact was better. And it was just really with the Browns. I think the Browns, that was his best stint. And he even burned the Ravens too. We all remember the game, week 17. We, it was week 17, the game to get into the playoffs. 2018, Lamar Jackson vs. Baker Mayfield, all or nothing. So he caught the Ravens slipping on that one. Who else? Tori Smith, he went on to Greener, well, not Greener Pastures, but he still did his thing, won another Super Bowl with the Eagles. He went to the 49ers first, went to the Eagles, went to the Panthers. I feel like he went to one more team too. That's all I can remember. Jordan Lashley, Jaleel Scott went to the Jets. That didn't work out. Jordan Lashley went, I think he went to the Raiders real quick, but it didn't work out. And I'm trying to think anybody else off the top of my head, but right now I just, I really can't. Nobody's really coming to me like that. So we wishing Miles Boykin the best. Y'all already know what time it was, man. Hopefully everything works out in his favor, and this can be his chance to get a real shot. Cause with a lot of players, I know a lot of fans, and that's not just with the Ravens. This can happen with any single team. A lot of fans could be like, oh, that guy's terrible.  This guy's bad. This guy can't play. He can't catch. He can't run. He can't block. He can't do this. He can't do that. He's not good. A lot of fans are, they quick to do that. Cause that's what fans do. It's normal. But so much is based on situation. So much is based on where you're at. If you're at the right place at the right time, or if you're at the wrong place at the wrong time. So much is based on that. And we've seen that. We see it with quarterbacks, wide receivers tight. We see it with all kinds of players. A player can be a beast. He can be a monster. They go to the wrong organization. And it's like, ooh, all that can change. Cause that organization, maybe not, they may not specialize in that particular position. They may not specialize in really maximizing that particular position. So that player goes to a team, but that team, they just, they can't do it. They don't do it. That's not what they do. So hopefully he goes to a team that does. Team Keep It Clean. I love y'all. I appreciate y'all. And I guess that's it. So just like my guy Miles Boykin is, I'm out, but I'm excited to see where he catches on. I can't wait, man. I love y'all. Shout out to Boykin. Shout out to y'all. We gone.


Hello friends. I'm Sanjay Gupta. I welcome you on Sanjay Gupta Tech School. In this video, I'm going to explain how you can check whether a number is having digits in ascending order or not. So here you can see two numbers are available in NBrival. So first is valid as per this caption and second is not valid. So now let's see how first is valid and second is invalid. So in first number if you see 1, 2, 3, 5, 7. So all the numbers are arranged in increasing order. So 1 after it, number 2 is available which is greater than 1, then V is greater than 2 and so on. Now after this, if we see this number 1, 2, 6, 3, 2. So here you can see 1, 2, 6, they are arranged in increasing order. But after 6, 3 and 2 are decreasing numbers. So as a whole this number is not arranged in ascending order. So if in a number all values are arranged in increasing order automatically. So that number is true as per our requirement and another one is false. So we need to print the output whether the number is arranged in ascending order or not. So now I'm going to write a solution for this. So I'm going to use these variables. So let's say I'm writing S1, S2 and I'm going to use a flag variable. Then printF. So this printF will display a message and enter a number. So we are going to read that number using scanF and that number will be stored inside and read it. Now with this and I need to check whether the number is arranged in ascending order or not. So let's say I'm taking this 1, 2, 3, 5, 7, 1, 2, 3, 5, 7. So I'm taking this as example. So our example is 1, 2, 3, 5, 7. So what I'm going to do, I'm going to first extract this number. So I'm moving from right to left. So if I have 7, so I need to check whether the next number that is previous to 7 is smaller to 7 or not. If it is smaller, then I need to check another number. Otherwise I can terminate the checking process. So first I'm going to identify that number with this S1 equals to n modulus 10. So n is 1, 2, 3, 5, 7. And if we apply modulus operation with 10, so always last visit will be received and that will be stored inside S1. Now I can write this n equals to n by 10. So if we divide any number by 10, so its last visit will be removed. So n is now having 1, 2, 3, 5, 7, 1. After this, I'm going to apply a loop which will check this n greater than 0. So if n is greater than 0, after doing this operation, if still n is greater than 0, then I'm going to write S2 equals to n modulus 10. So current value of n is 1, 2, 3, 5. So again we are applying modulus 10. So S2 will be having 5. So what we can do, here we can apply a if condition. So if S2 is greater than S1, if S2 is greater than S1, then we need to break the loop. Otherwise we can repeat the loop. So here I'm going to check this. If S2, S2 is the second number and S1 is the last number. So if this second last position is greater than last position, then we can break this loop. So here I'm going to use this flag. So if it is true, then flag will be 1 and I'm going to break this loop. If it doesn't happen, then I'm going to apply S1 equals to S2 and n equals to n by 10. And after completion of this loop, I have to check what is the value of f. So if f is 1, it means number is not aligned in ascending order. Otherwise it is. So after completion of this loop, I am writing if f equals to 0, then you can print ascending as you can print not in ascending order. So now I'm going to explain it completely. So here we identified S1. So S1 was 7 and was 1, 2, 3, 5 till here. Now check this condition. It is true. Then again check n modulus 10. So S2 will be having 5. Check this condition. S2 is greater than S1. No, S2 is 5. S1 is 7. So 5 is not greater than 7. So it will be 4. And we will be here. So I'm assigning S2 into S1. So S1 will be having 5 now. And n equals to n by 10. So this 5 will be removed and n will be having 1, 2, 3 only. So initially it was this, then it was this. And now it is 1, 2, 3. So every time we are removing last visit to this. So check this condition n is greater than 0 because it is 1, 2, 3. Perform this n modulus 10. So S2 will be having 3 now. S2 will be having 3 now. So check this 3 with 5. So S2 is 3. S1 is 5. So 3 greater than 5 again falls. We come here. S2 will be assigning to S1. So S1 will be 3 now. And perform n equals to n by 10. So now n will be 12. So 12 is greater than 0. Condition is true. Check this condition. S2 mod, sorry, n modulus 10. So 12 modulus 10. S2 will be having 2. Now check S2 with S1. So 2 greater than 3 falls. So again S2 will be assigned into S1 and will be divided by 10. So it will be 1. 1 is greater than 0, 2. 1 modulus 10. The meter will be 1. So S2 will be having now 1. Check this condition. Again it is false. So S1 will be having value of S2. So S1 will be having 1. Then again divide n by 10. So it will be 0. So this loop will not repeat. So you saw in this case, if we have this number, in this case, this if condition is always false. And if this condition is always false, then S will remain 0. Because S will convert into 1 when this condition will be true. But in that case, if condition is always false, so S will be 0. So this condition will be true and it will be assigned. Now if we take this example, 1, 2, 6, 3, 2. So in first time S1 will be having 2 as this number. And n will be having 1, 2, 6, 3. When we come here, so S2 will be having 3. Now check this. 3 greater than 2. So this is true. And f will be 1 and loop will be 7. And if f is 1, so it will not arrange in assignment. So this way you can check whether your number is having digits in assigning or not. So I hope you understood this logic, how I implemented it with the Anthropsy program. So if you want to watch more programming related videos, you can open my channel, go to playlist. And there you can find various programming related videos to watch them. And I hope whatever I explained in this video you understood. So thank you for watching this video.


Welcome everyone to the PhD in Historic Preservation virtual open house. We're delighted that you could all join for this and really appreciative of the time that you've devoted to thinking about the program. We're super excited about your interests in the program. We are here as a faculty to tell you a little bit about the program. So I'm Jorge Otero Pailos. I'm the director of the program and I'm joined by Erica Avrami. Hello. Who's a member of the PhD faculty and Professor Lola Ben-Alon. Hello everyone. Hi. And also Sarah Grace Godwin who is our assistant director of the preservation program. And so now you're all joining from different places and so on. So many of you have reached out and we've had some emails back and forth.  We're gonna, we will have time for questions where we have an hour or two for this. And we want to just familiarize you with the program and familiarize you a little bit with us as faculty and make plenty of time for you to ask all the questions that you would like. Now, the first question I have for you is can you see my shared screen? Everybody can see that. Okay, fantastic. Well, that's already half the battle. So let us dive right in. The PhD program in historic preservation is cited, is at Columbia and some of you might have been in the school. Some of you have never been here but this is the main quadrangle of Columbia University and within that this big blue arrow points to Avery Hall and this is the center, let's say of the school but the school of architecture, planning and preservation is actually spread over four buildings. And so the preservation PhD is really housed in the Preservation Technology Lab which is in Sherma Horn Hall, it's the building next to it. And the Preservation Technology Lab serves teaching and advanced research on the preservation of building materials through a constantly evolving myriad of analog and digital technologies. And as part of its mission, the lab encourages awareness of and creative approaches to preservation technology as a means to bring about meaningful cultural, social, political and ecological change in the built environment. And so we achieve this through facilitating interdisciplinary doctoral and master's level research. We also host visiting scholars, often they're either working on their PhD or they're postdoctoral scholars or professors that come in and spend some time with us. And we promote dialogues between leading preservationists, architects, artists, engineers, creative technologists and scientists in the lab. The lab is also the repository for an extensive collection of building materials and fragments of historic buildings that are used by faculty and students in teaching and research. And it's a collection that is searchable online and that you can have access to as well. The PhD program in preservation was launched in 2017 and is oriented towards the training of future historic preservation scholars. It is the first of its kind in the United States. And so it aims to, excuse me, sorry about that. It aims to expand the discipline's range of intellectual engagement and also to cultivate new paradigms of scholarly research, experimental practice, global action, policy and communication. So the doctoral program underscores a historical understanding of the disciplines evolving challenges and purposes. And we want to promote theoretical speculation, design speculation, policy speculation, material speculation on alternative modes of practice suited to deal with the ethical, technical, aesthetic and social problems of our time. So we are looking for candidates who can engage with critical and scholarly culture and become the next leaders of the discipline. Candidates are expected to conduct independent research with support from the preservation's faculties, wide range of expertise. And in that preservation research is really something that we want to support. So we are looking for different, for independent projects. You are all going to each in your applications propose a project, but we also want to be very intentional about the kinds of projects that we are soliciting this year. We admit one student per year and this particular year, we're putting out a call for that kind of, for a particular kind of researcher we're gonna tell you a little bit more about it in a second. So part of the research that you would be doing is taking advantage of all of the different components of the research capacities of the school. A major part of it is Avery Library. So that is another site where our PhD students are engaging deeply with our archives and it has an incredible collection of both original drawings and objects and also limited edition historical books going back centuries. And also very importantly, one of the connections of the preservation PhD is with the Natural Materials Lab, which is directed by Professor Lola Ben-Alon who serves on the PhD and preservation committee. So I will let Professor Ben-Alon tell you a little bit more about the lab and its intersections with the Preservation Technology Lab as well. Thank you, Jorge. So a few words about the Natural Materials Lab that is joining in conversation and collaboration with the Historic Preservation Lab. The lab was founded in 2020, for years ago to investigate raw, geogenic, biogenic, earth-based, fiber-based materials, their life cycles, supply chains and policy, but also fabrication techniques, mix designs and upscale possibilities. There are several research projects and teaching courses that are affiliated with the Natural Materials Lab with the goal to craft new ways to imagine and invent socially equitable and ecologically sustainable futures. And the work here converges several fields, including material design, material sciences from microscale to structural scale, geology, geography and material, geographies and supply chains, architectural design, civil engineering, including structural and thermal behaviors of materials and assemblies and also art and experimental design. There are approximately between 10 to 15 research students, interns and assistants at the lab each term. And the goal and idea is to have the Natural Materials Lab as a resource with our material testing and fabrication possibilities as part of the Historic Preservation PhD program. Yes, and back to you. I think this is a really good segue for us to talk a little bit about how the rest of the faculty fits into this. So the PhD faculty includes all the people that you see on the screen right now. And each of us has a different area of expertise and focus.  So I want to turn the microphone over to Erica who talked a little bit about her work as well. Hello, everyone. I'm Erica Abrami. I've been part of the GSAP faculty now for almost 10 years and very excited that there's such an interest in the PhD program. My research really looks at Historic Preservation through the lens of social justice and climate adaptation. And so it involves more social science-based research looking at community-engaged methodologies for how we adapt and promote resilience within communities, but also looking at qualitative and some quantitative analyses of how we think about distributive justice, how questions of historic buildings and relationship to climate create benefits, as well as sometimes challenges for disparate communities and how we as preservationists can address these challenges and on the horizon through public policy. I think on the screen here, it talks about a program that I've been running the Urban Heritage Sustainability and Social Inclusion Initiative, which has produced a series of volumes around some of these issues that I've just been discussing. But my work also intersects a bit with Lola. I'm at the moment in Ghana, where I'm teaching a studio where we've traveled with students to Ghana focused on earthen architecture. And we're looking at the role of this traditional and ubiquitous form of construction material and the ways in which systems of colonization, religious transfer and devaluing of earthen architecture has affected the ability for people to transmit this constructive culture across generations. So in that sense, my social science interests and my climate interests also intersect around earthen architecture. Right, thank you, Erica. And I can tell you a little bit about my work as well. So I was trained as an architect and a historian and I practice in preservation and experimental preservation, where I take artistic methodologies and technologies to really critically inquire about the aspects of preservation that might be overlooked by the larger, let's say institutional established authorized frameworks in the field. So very much as you've seen in the rest of the faculty, the critical approach that we each take varies differently depending on our methodologies, but they're all very complimentary. And in my practice, you know, I tend to be very hands-on, but also through the hands-on and design aspect of working on particular sites, trying to develop that process as an inquiry that raises questions that then can be leading to larger scholarly and more traditional archival, historical, theoretical research. So really bringing together theory and practice that way through a critical engagement and hands-on engagement with materials and materiality, with constituencies and communities that are associated with those materialities, because every site has a community that is related to it. So working very closely with the stakeholders and the stewards of sites to carry out that work of experimental preservation. Other members of our faculty that are not here today are Mabel Wilson and Lucia Ale. Mabel Wilson is an architect and a historian who leads the African and American African Diaspora Studies and is director of the Institute of Research in African American Studies. She also co-directs the Global Africa Lab. She has an active practice where she has also worked on memorials like the UVA Memorial to Enslaved People. She's also has an active installation practice where she currently has an installation at the Venice Art Biennial. She is also a scholar and has been writing actively on African American experience and how it is spatialized through architecture and has been, especially in the 19th and 20th century. And so very critically engaged in questions of memory and heritage through that lens. Lucia Ale is a historian and she is the director of the Buell Center for American Architecture. She is an expert on the history of international preservation institutions such as UNESCO and wrote a book called Designs of Destruction. So she comes at it more from a historical lens and unpacking the various intersections between the building of heritage institutions and various geopolitical realities especially of the post-war but also she gets into the interwar period as well. So that's to give you a sense of the faculty in the program, in the preservation program but at Columbia we also have a much larger PhD faculty that intersect in various ways with the work that our students in preservation do. So just to give you a very quick overview, we can talk about each of these people separately if you have questions. And then of course there's a very large historic preservation faculty that works with our students that works in the lab, some of them do and some of them are not working in the lab but with whom you will have also a conversation. They're also at Columbia various research centers and institutes and we certainly encourage you to look into these as you prepare your application because we are going to want to know what you wanna work on and who you would think might help you in that research best and what resources we have at Columbia that would help you carry that research out and these institutes and centers are certainly part of the capacity of the university to help you. Obviously we talked about the preservation technology in the natural materials labs as really the anchor points for that research. So we're gonna go into the outline of the program real quick. It is a fully funded program. So the program, you get a stipend, you get a, you get tuition exemption so you get paid essentially to come and work on that research here. The program is a five-year program of which the first two years are spent really developing your project, taking classes. You will be taking a preservation PhD colloquium once a year in those two years plus other doctoral research colloquia and seminars at GSAP. Your third year is a year that is spent on, and I don't know if I remember. Yeah, your third year is a year where you're gonna be taking the qualifying exam. So these are the exams that basically certify that you've taken all the requirements and that you have a command of the discipline and there's a major and a minor exam. And then in that third year as well, you present your dissertation proposal, your official dissertation proposal. Then over the following two years, you write and execute that dissertation. So that's the five years. So all this to say that we fully expect that you're coming in with a dissertation proposal now that is going to change after the two years you spend in our school and that on that third year, you're really going to fully flesh out what you try to do. So you can in your proposal, as you write your proposal for admission, you can say that you might see this research developing in this or that direction in relationship to this or that lab or institutional center or faculty member. We encourage you to identify what faculty members you would be interested to work with. And your advisor should come from the PhD and preservation faculty, but you don't have to choose an advisor until that third year. So you can work with everyone in the program and then at that third year, you're going to choose the advisor. These are just some of the general requirements. After that third year, you pass those exams, you get the master and fill master philosophy that allows you that makes you all but dissertation. That's when you go to those two final years of the program and there's also a language requirement. This is because many people are working internationally and there is that you have to have one extra language apart from English. So now that we've covered the requirements and we've covered the length of the program, we wanna tell you a little bit more about the kind of research proposals that we are encouraging this year. So all candidates have to have a background in historic preservation. So either a master's in historic preservation or have related knowledge for a number of years, either in the field or an academia that qualifies them as having a background in historic preservation. So that's number one, that's very important for everyone to consider that. And secondly, we again have one position and we are very intentional about the kind of research that we encourage every year. But that doesn't mean that if your research falls out of that, you cannot apply. You may also apply with a, let's say, different dissertation proposal, but just for you to be aware that this is what we're looking for this year and what we are really seeking in a PhD candidate. And so I'm gonna turn it over to Professor Ben-Alan to tell you a little bit more about that. Yeah, thank you Jorge. So this year, the PhD in Historic Preservation, at GSAP is especially seeking candidates who can propose a project that bridges experimental preservation technologies and material design research to develop a dissertation as a total project that addresses pressing contemporary environmental and social issues and challenges. We especially seek candidates who will utilize different methodologies from the range of facilities. We have from the fields of conservation, art, architecture, science and technology. Candidates will have the opportunity here to cultivate a scholarly dialogue across history and theory of preservation technology, but also building physics and social sciences. As you heard of all of the backgrounds of the professors who will be part of the program. A deep focus on natural, low-carbon and bio-based materials like earth, stone, et cetera, is highly desirable this year. And the research topic will ideally not only address environmental urgencies in the face of climate change, but will also critically interrogate societal impacts related to marginalized communities, resource scarcity and land use. We hope that the candidate will take advantage of access to a wide range of scholarly resources within and outside GSAP, including research faculty and facilities. Specifically, the preservation technology lab and the natural materials lab will provide valuable opportunities for hands on experience and access to testing tools for various materials, scales and experimental setups. This would include conducting research experiments and developing installations and demonstrations ranging from micro-scale to macro and building scales. Lastly, the ideal candidate for this program should, especially this year, have a background in preservation and proficiency in academic writing, conducting experimental setups and a solid understanding of how to integrate historical research with a critical perspective on fabrication and preservation. The program highly encourages and supports research collaborations with schools from other institutions, as Jorge showed in the slide with the different labs, centers and facilities in and outside GSAP. And we also encourage active participation in international conferences and workshops, of course. So these are the main points that we were hoping to emphasize this year for candidates who are interested in applying. Thank you, Lola. So why don't we turn it over to questions? So from those of you that are in attendance, it's a lot we just put on the table. So please go ahead, yeah. You can either just raise your hand with a Zoom or you can just jump right in. Hello. Oh, sorry. You should go, Kaya. Sorry. Thank you. My name is Kaya Kerner. It's very nice to be here. It's very nice to hear about the program. I've been following this program for a while, the work of Jorge, and it's really nice to hear work being done kind of in relation to materiality. I guess I'm curious if you could just summarize that once again. So it's really having to do with like prototyping, biomaterials and social aspects, the project focus. I can answer that, Jorge. So the project that proposed may range from different scales, geographies, localities and materials. Prosotypes can certainly part of an experimental setup and it can be offered in a range of scales. And I think what would be really interesting for us is to see how such prototypes or suggestions for prototypes can be contextualized within an argument or a context study or a certain topic that may include several case studies, of course, and then culminated through a combination of material prototyping and preservation and experimental preservation and can even conclude in a certain project or hands-on project or installation or integration with the community or a series of demonstrations. So it's very open. Thank you. So to add to that a little bit, clearly we want you to take advantage of the resources of the Preservation Technology Lab, the Natural Materials Lab and to think through how you use the materials through how you're going to engage with preservation through those resources and thinking about the design process of the design of materials, materials that are to be used for preservation as an inquiry, not simply as solving a problem but as a method of unpacking of opening up critical questions in the discipline of preservation. So part of what we're looking for when we read your applications is a sense that you understand the discourse of preservation. What is your position within the preservation field? How do you situate yourself in the whole of the preservation enterprise? That is a major part of what we're looking for. And then how is your intended engagement with materials and materiality a way of fully, more fully articulating and inquiring into what you feel is relevant and important and urgent in the field and what is missing from the field because a dissertation is a contribution to the field. So you need to also articulate to some degree why would you want to do this? What is missing? How has this not been covered already? So having a sense that you understand what came before you and you have a position vis-à-vis that in a sense of the urgency of your contribution is very important for us. So you should definitely articulate that. And of course, we want to know why you are the person, what you've done so far, your experiences, how you came to this point, how you came to this question, where are you in your life right now that makes it such that you would really benefit from this, but the discipline also needs you to do this work, right? That's something that your statement really should flesh out. Erica, do you want to add to that? I just need to, I think that you said it well, Jorge. I mean, part of what we are looking for is that capacity to be a self-directed researcher because a PhD program really is about learning how to become a better researcher as well as an educator. And so being invested in avenues of inquiry that whether they're strictly material or bridging across the social sciences, experimental design, materiality, scientific research, et cetera, your capacity to really think innovatively and creatively toward those kinds of interrogations. Other questions? Hello, my name is Amani. I have a pretty logistical question. So given the self-directed nature of the program and the relatively low amount of very specific required coursework, if an applicant were already enrolled in a program at GSAP and they had relevant coursework and they were able to be in the program, is there a way that that relevant coursework could apply towards the credits of the program and perhaps have the program take less than five years or would that either ages be unlikely given how relevant the coursework would need to be or is that just not allowed and you would need to start from zero with your credits? Yeah, you would need to start from zero is the short answer. We expect that everyone has a master's degree and so everyone will have taken courses that are relevant to the program or even very close to the requirements of the program but the requirements of the PhD colloquium which are the core requirements and the number of credits for the coursework are really meant to give you a chance to take full advantage of the luxury that it is to take this time to develop your research. So we want that time to let these ideas develop and you should take advantage of that. So it's really something that we encourage you to to take advantage of really. Am I named? Does that answer your question? Yes, it does. Thank you. Are you at GCEP now? I am not. Oh, you're not, okay. Anna? Yeah, hi everyone and thank you for this introduction. Yeah, I'm Anna Yu and now I'm working as a new media artist and yeah, working a lot on the cultural heritage part and I have worked for two years as a fellow at the Department of History of Art and Architecture at Harvard and I had a mark tool from Yale, so a little bit from my background and I want to see the potential focus of the PhD students that you are considering like to consider candidate who are interested to combine preservation, climate material, this art and media technology like other technologies besides material like this kind of photogram tree and putting all these things together and use art and media as a way for me to work. Lola, do you want to answer that? Yeah, that's a great question. Thank you, Anna. Of course, when we say materials and indicating the opportunities possible in the Preservation Technology Lab and the Natural Materials Lab, invite for hands-on, tactile work with materials in an experimental and empirical ways but it's not exclude, of course, other means of experimental preservation technologies that may include a range of other mediums including, of course, media, data and avenues towards other scientific inquiries of experimental preservation. So this can certainly be a really interesting and viable part of the proposal and then, again, going back to the three resources, the Avery Library as a place to argument your research and contextualize it in the history and theory of preservation technology and then the Preservation Technology Lab with the Material Library and the tools and equipment and the Natural Materials Lab with the strong emphasis on raw, low-carbon and material resiliency. Think about how these can be weaved together with your interest in media and art and design and definitely sounds like additional realms can be weaved within these. So, yeah. I'm gonna also share, for those of you that are here, I wanna share the page for our current PhD students. So you can see the kinds of projects that they've been working on and again, they're each very different. So that also gives you a sense of where the cohort and the people that will be in the program with you, your classmates, not classmates, but your cohort, yeah. Jorge, there's also a question in the chat. Oh, okay. Okay, so from Mohammed Hossain, could we research and work on urban heritage and urban historic preservation in this PhD program? I'd say yes, absolutely. And there could be ways in which you could examine the urban environment through a material lens and there's so many ways in which that could be articulated and we're very interested and also, I mean, that immediately raises the question of scaling. So when you're working on a particular research project about material design, questions of provenance, where are these materials coming from? How are these materials deployable in a contemporary setting at a large scale? Where would they be needed? And many of these questions of adapting to climate change of course intersect with these when you begin to think about urban scale. We encourage you to take a look at some of the work that Professor Avrami has done on this subject as well and the research initiatives that she's been leading and overlapping with these questions of the existing built environment. So these are new and innovative ways to think about the urban, the built environment that are not necessarily just limited to the heritage designation of a portion of that environment. Maybe Erika, you wanna chime in on this? Absolutely, so part of what we're looking at through a collaboration with the climate school, the existing built environment, Earth Network is how we oftentimes segregate the built environment as new construction and then this stuff called heritage or historic preservation. But there's this big gap in between of the existing built environment, lots of existing buildings, all of which and an era of climate change will need to be retrofitted and adapted in different ways. And communities may need to migrate as well. And so I think some of the things that Lola and I have talked about in the context of this Lola as part of this network is how do we look to the use of natural materials or low carbon materials oftentimes derived from vernacular or historic traditions and scale them up for retrofitting in low carbon ways. So that the embodied carbon involved in retrofitting buildings and adapting buildings is minimized. And so you could look at that at an individual kind of building level, you could look at that at a material level. But as Jorge said, we oftentimes still need to scale this up because just retrofitting or just adapting one building isn't going to get us to a place that meets the challenge of decarbonization. So thinking about ways in which we can implement both develop and implement practices across entire communities, for example, or phasing within communities or certain building typologies or within certain kinds of construction materials. All of those take us to a more systemic level that is represented or sort of a critical characteristic of the urban context that it represents a sort of systemic built environment question around historic preservation and its intersection with climate, as well as social justice. Okay. I see Mohammed Hussain has said thank you on the chat. So other questions. Okay, I see Jayeen. Thank you very much for your introduction. Gave me some basic information in addition to the information you presented, I'd like to know what the annual enrollment number is. So it is one. The annual enrollment number is one candidate. So we admit one person per year. You're welcome. Other questions. Feel free to just jump in. We have a couple more minutes. Why just one candidate that, well, that is what we have funding for. So we, these are fully funded positions. And so that's our, Erika, did you wanna jump in? I just wanted to add to that, that even though it's one candidate in this particular doctoral program, you are among other doctoral students within the graduate school of architecture and within the university writ large. And that there are opportunities to pursue inquiries beyond the architecture school. There are even opportunities to take courses within a network of universities associated, or in the geographic area of New York City. If there's a course that you're particularly interested in as a doctoral student, you're allowed to petition to take that course at NYU or at Rutgers. So while it's just one student, you have an opportunity to network with NEET and engage with lots of other doctoral students. Yeah, so you'll be in a real, it's a really amazing place because you have this very robust and very rich PhD research community at GSAP. I think it's quite unique, really, it's certainly in the United States, but I think everywhere that there is, that there's a very large cohort of PhD students and everybody really informs each other and participates in events together. And there's a very lively dialogue happening about the research in all of these various fields together at the PhD level. And of course, there's also the Italian Academy at Columbia University invites postdoctoral researchers in preservation every year. So these are senior figures in historic preservation that come in with whom our PhD students have dialogues or as a seminar that is held over there that our PhD students can participate in. So that's also part of all these institutions and centers and that we have at Columbia. And we're very fortunate to have a very robust preservation set of institutional capacities. And people, because ultimately it comes down to people. We'll take one last question from Anna. Could thesis be related to a specific real preservation project? And do you consider international topics involving other countries like China or the EU? Erica, do you want to take this? Could we, yeah. Certainly a dissertation could be associated with a specific site or a specific project. But what we want to encourage you to do is really be digging deeply into a set of research questions that allow you to contextualize a particular site within a broader theoretical inquiry and to allow a particular case, for example, to be generalizable in some way as a form of research. And so you can certainly focus on other countries. We have doctoral students right now who are looking at digital documentation histories in various parts of the world. Another who's looking at a post-disaster heritage recovery in various parts of the world. So I don't see that as an issue. But a project itself is not a dissertation, right? We want you to really be thinking about your dissertation as a substantive research effort. Thank you so much. Right. Well, I see Kaya has her hand up. Kaya, do you want to jump in real quick? Yes, I know you said of the last question, but I just wanted to just ask one more. In terms of having a background in historic preservation, if a candidate doesn't necessarily have, for example, a degree but has, I know you mentioned experience being kind of qualifying for the program. So for example, working on an excavation or having a background in heritage management, does that apply to that kind of requirement? Yes, that would, yeah. And that is spelled out in the page on the GSAPP web page on the PhD program that states the requirements for entry, the qualifications necessary for entry. So definitely if you have a previous degree in preservation or relevant experience in the field, then that qualifies you. Perfect, thank you. OK, well, we are very grateful that you all attended the virtual open house and very excited that you're interested in the PhD program. So thank you so much. We are available for questions as you developed your proposals and your applications. Please do reach out and let us know. We encourage you also, we didn't talk about it so much, but we encourage you to obtain good letters of recommendation from folks who have PhD backgrounds so they can speak to your ability as a PhD candidate. So consider that as you think about who you're going to ask to write letters of recommendation. But we're just really thrilled that you joined us and very excited to continue this conversation with each of you and to read your applications in due course. OK, well, thank you so much.