 Okay, so it's amazing to see from Steve's talk, you know, the complexity of modeling the MGO. But on the other hand, how far we've come, and I don't think, I think it's fair to say that we wouldn't have an S2S project now if it weren't for the advances in the models in being able to predict the MGO. And it's really the, it's, you could say it's the most important source of skill on the sub-seasonal time scale that we're able to model. And as Steve talked yesterday about the stratosphere, and that's also a potential one, the land surface, but the MGO is the one that, you know, the project has really been driven by. And then if you compare that with ENSO, ENSO modeling, people had quite good understanding and models of ENSO way back in the 1980s, and you could ask the question, well, how has ENSO prediction skill increased since then? And if you look in the models, actually, we don't have much increase in skill since those early days of ENSO prediction. That may be, that may be starting to change now, but I think there's more, there's also more to ENSO prediction and modeling than is in those early, early models of Cana Zebiac intermediate coupled models. But nonetheless, fair to say, I think that the understanding of ENSO is still, still in advance of what, of, of the MGO from, from what Steve was saying. Anyway, for this talk, I, I think I can go through it fairly quickly. And I, we thought it would just be useful to say something about tailoring of forecasts. I'm gonna, particularly talk about seasonal forecasts, but I'm just thinking about how one could also tailor information coming from sub-seasonal to seasonal forecasts for use in, for use in application. So if I, if I don't keep going, this is probably gonna keep turning off, right? So I'll talk about the, the workhorse for doing this regression models for tailoring and calibrating seasonal forecasts. Some, some examples and then quantile, I'll say a few more words about quantile regression that I mentioned briefly on Monday. And so, as I mentioned on Monday, this is the type of thing we have for seasonal forecasts, but in terms of the seasonal total of rainfall and in terms of, of teresile categories. And can, can we make this more useful for, for particular specific applications? And so we, we showed this one as well as being, one has two distributions here, one, one is the historical distribution of observations and forecast distribution that, that we're making of that. And, and the question would be, we, we talked about this in terms of seasonal total of rainfall, but what about if we want to make a, a forecast distribution of some other quantity that's more relevant to an application? So the way that this is normally done is using linear regression models. And this is, this is the case here for, say, given a, a set of GCM hind casts or, or other predictors. Could also be some antecedent sea surface temperatures, for example, and a set of observations, we can build a regression model between them. And so this is, this, in this equation, X of t is the predictor and Y of t is the predictant, but they don't necessarily have to be the same quantity. It doesn't have to be the seasonal total of, of rainfall that's the predictant. You could choose another predictant, and that's the essence of using linear regression models for making, making tailored forecasts that you can take your predictor being some, some variables from your GCM predictions. Could be the seasonal total of rainfall, or it could be the wind fields or some other, some other quantity from the GCM. And then predict whatever, whatever it is that you like, if you have some observations for it, it's the predictant. So the bias is, is B and you estimate A and B by minimizing the sum of squares or residual error term. And regression models trained on GCM hindcars versus historical data are called, called MOS correction sometimes, stands for model output statistics, so you may hear that MOS correction used to describe this process. And you can use general, generalized linear models can be used to, to go to, if, if your, if your observations are not distributed in a Gaussian way. And we'll, we'll see that application of that to the quantile regression. So the choice of predictors, as I said, they don't have to be the same quantity, the X doesn't have to be the same quantity as Y. You can use more than one, more than one predictor, then you'll have a multiple linear regression. That, that often in the past has, has led to one of the main pitfalls in this, that if you choose a lot of, if you use multiple linear regression and you choose a lot of predictors, then if you choose enough predictors, your residual error can be reduced to zero by, by including enough of these. And if you include as, as many as you have sample points in your time series, then you, then you can make a perfect fit. And that will be the, the classic phenomenon of, of overfitting. So the rule of thumb is that you need five or ten samples per predictor. And this is the, the essential point that if you want to train one of these models, you need lots of, lots of hind casts. You need long hind cast sets and observational data to go with them so that you can train the model. And the rule of thumb is you need five or ten of these per predictor. So if you had three, three predictors, you would need, in a seasonal sense, 30, 30 years of, of hind casts and 30 years of, of observation. So that then raises the question in the sub-seasonal case where we were seeing in S2S that often we, the hind cast sets are much shorter than we're used to dealing with for, for seasonal forecasts. And so that, that's a, that's a possible question. So the golden rule in doing this is that predictors also should be chosen from, from, from some physical considerations. That's especially, especially relevant when, when we're building statistical models where the, the X of t is not something coming from the, coming from the GCN, but it's anti-seed and sea surface temperatures. And you, you could, for example, or, or some, some observed, observed quantity. You want to make sure there's a physical relationship between, between X and Y or you get into problems. And also that the, the, when you're doing this, you need to estimate the skill from, from some independent data. So choice of predictant, it could be, it could be a, it could be a high, higher resolution than the GCN. So if you think of a GCN, lower, being lower resolution, 100 to 300 kilometers, say, if you chose station scale precipitation, that would yield a statistical downscaling. So this is also a method of choice for statistically downscaling seasonal forecasts. It could be a, a more user relevant variable like a reservoir inflow and I'll show, show an example of that. And so that's where we say that we will be tailoring this to a forecast. Or sometimes people say bespoke is all these clothing metaphors. So varieties of doing this, the, the, the equation I had before, a single predictor and single predictant is the univariate case or the multiple regression where I said you can get into problems if you have too many predictors. But one method of choice for dealing with this multiplicity of predictors, also if, if you think that these predictors may be not, not linearly, linearly independent is to use a principle components regression where you take, you calculate the, the EOS and principle components of the, the predictor set. And in that way you can reduce the dimension to just a few and they're, they're linearly independent of each other. So you don't have that problem of, of multiple co-linearity. And so that's the case. We, we call that principle components regression if we're using PCs of some field. So it could be a precipitation field, say coming from the, coming from a GCM over, over a region used to predict why at a single, single station that would be principle components regression. Or in the full multivariate case where we have multiple field, fields of predictors as well as fields of predictants, then we have a full multivariate pattern regression. And the, the method of choice for, for doing that is canonical correlation analysis. So we not only do an EOF analysis of the, the predictor field of the, say the GCM precipitation field and get the PCs of that, but we also do an EOF analysis of the observed field, say the station, station network and get the, the, the leading principle components of that. Now you need to do some truncation and, and choose the truncation limits, limits of that, of those in, in those, in that case when you do this kind of approach, you need to truncate to include only a few principle components on, in, in each case. And that, that is, is often done under, under cross, cross validation in, in maximizing some, some, some skill measure under cross validation. So this is a tool that's been developed at the, the IRI. Simon Mason is the, the person who's been really the one who's developed this tool almost single-handedly. It had some help, but that's really his, his tool. It was motivated by the experience in, in, in regional climate outlook for in Africa, where people were using statistical regression to, to make their climate outlook for, for forecasts. In which case they would often choose statistical predictors, sea surface temperatures, things like that. But often they, they were fishing for the predictors that, that gave them the best skill. And so these, these regressions that they derive were, were overfit. And it was giving forecasts that, that were, that were way over confident and weren't, weren't at all, weren't at all accurate in, in, in real time. So when you came to make a real forecast, it had no skill. But if you looked at, looked at the training period where it was overfit, apparently you had a lot of skills. So that was a case where, where where the, the, the uncertainty in the forecast was not being communicated in, at, at all. And so that's where, where Simon had developed this tool to, to apply canonical correlation analysis and principal components regression. Also in a, in a tool kit which would allow you to, to also document this, the skill under cross-validation using a whole slew of, of WMO recommended skill measures. So it's a Windows-based tool kit. I don't know, does anyone here use CPT? Ah, there's a, there's a few people. So it run, it runs under Windows. It's used, I think in, in, in many of the, the National Meteorological Services use this for their, for their seasonal forecasting around the world. And it's used in many of the, the regional climate outlook fora. I think it's become, it's become a method that's, that's actually been, been adopted by, by WMO as a recommended method for, for making, making regional scale, statistically downscale forecast at, at local scale. Typically it's been used with, with statistical predictors in those regional climate outlook fora. But there's an, there's a big push now to, to use the dynamical model output from the, the global producing centers, or the WMO lead center, or the, the North American multi-model ensemble. So to use predictors that are, that are, for example, precipitation fields from the GCM, or, or predicted SST fields from the GCM, there's another popular way of doing it. There's also, if you go to the, the IRI website, there's, there's also a, a Linux version of this. It doesn't have all the graphics, but there's a, there's a Fortran code's Linux, Linux version, which we call the, the batch version. So I'll show a little example here of tailoring seasonal forecast to reservoir inflow. And this is the, the reservoir from Manila in the Philippines, the Angap reservoir that supplies almost all of Manila's water. And the, the, the key inflow season here is the October, November, December season into the reservoir. And this is just illustrating that you, you can use in this case because it's so connected with El Nino. If you have El Nino, you have a drought of the Philippines. And because of the seasonality of El Nino, you get a, you, you can use the, the sea surface temperatures from the, the, the season before as an effective predictor to through, through this, through a linear regression model with inflows into the reservoir. And it's a case where it's difficult to actually improve on this if you use GCM information. If you use GCM forecast, you, you can, you can scarcely improve on that actually because it's the, the, the ENSO signal is so strong and it is so well, well represented by, by sea surface temperatures over the Nino, Nino region. And because you have that nice, that, that nice seasonality of, of El Nino that you can, you can use sea surface temperatures from the boreal summer, late boreal summer to full-cast October, November, December inflow. So just to look at that in a bit more, bit more detail. So what you need is a long historical time series of stream flow in this case. So it's contingent on having that record of the observational predictant quantity and then a matching historical time series of the predictor. So here it's September, actually September Nino's 3.4 SSD. But this could also be the, the precipitation averaged over October, November, December forecasted from the GCM initialized, say on the 1st of October or the 1st of September ensemble mean. So here's my two time series of September sea surface temperature anomaly in Nino's 3.4 region. And so we have data from October, from 1981 to 2006 for both time series for the inflow and the SST. So here's our Y and our X. And you can see that they're quite nicely inversely correlated. So that when you have La Nino, you have higher flow. And when you have El Nino, you have lower flow. And what we do then is to make a cross-validated hind cast of O and D inflow where, where we use the technique of cross-validation where we, we leave out the year to be forecast from the data to train the model. And then we make a prediction for the left out year. And that is what is, we call here Y hat. It's the one that's been estimated for each of the left out years. And that's plotted in the green line here along with what actually happened which was in, in, in the blue line there. So you can see that they, they, they mimic each other quite well. And this is, this is just a little schematic of, of that cross-validate and cross-validation technique. So to predict 19, in this case it's 1951. We, we use the training period from all the other years to predict 1952 likewise, etc. And then so we get predictions for, for these left out years that are, are clean and that they didn't see any of the, any of the training data. And then you correlate that with what actually happened in those years to get an estimate of the skill. So that's called cross-validation. Sometimes it's done if you think there might be some leakage actually across the, the years, you can leave out more years. So sometimes people leave out say a year either side or two years either side or so leave three out or leave five out is, is, is, is better if you have, if you have enough, if you have enough, if you have enough years. Another, an even better way of doing it is, is not to use any of the training period before the year you're, you're, sorry, after the year you're trying to predict. So to only use data up to the year you're trying to predict to, to mimic as much as possible what would happen in real time. And you can see that you, obviously you need to have a longer training period to do that because for predicting 1951 we don't have any years. We'd have to have years before that. We'd have to have a 30 year, 30 year period say before that to be able to do that sort of, that, that technique which is, which is the best one if you, if you have enough data. But you see that mentioning a time and time again that you need, you need long hind cast set and long historical records to be able to construct these models. So the, the, then you can look at the error residuals once you've fitted this model and these should be approximately normally distributed for this, for this to be, this, this, for the technique to be valid and you can see that well it's not too bad in this case. And just, just to mention that if you look at the, our Y time series, this is our, our inflows, you can see that it's not too badly skewed. So what you can do if, if this is very skewed, you can, you can transform it using, using a statistical transformation like a box, cocks transform or some, some, or some quantile, quantile empirical transformation to, to make it more, make it more Gaussian because it, this is, this is based, this linear regression based on, on Gaussian assumptions. So you can see that it's fairly well, these are fairly Gaussianally distributed. But if we came down to be looking in the, in the sub-seasonal context that a weekly rainfall, for example, weekly average rainfall, it might not be the case. And so I think there's a question as to whether or not this technique is, is how applicable it is on, on sub-seasonal scales. So we're talking about getting, getting, I'll show you something univariate. I mean, how do we get a, how do we get a, a PDF, how do we get a distribution out of that? So essentially what we're doing is there, we're predicting the ensemble, we're predicting the, the mean of the, the forecast distribution. So we assume a normal distribution with the mean given by the regression model. So we predict the mean, that's the only thing that we predict with this model is to predict the mean. So somehow, so we can get the shift of the distribution, but somehow we have to estimate the spread of the distribution to get the standard deviation of our, our Gaussian forecast distribution. And this is typically estimated from the spread of the errors of the past forecast. So the, the error residuals are from, from, from hind casts. And that, that's just just depicted in the slide. So the, if you fit your, your, your model under cross validation, these are all the hind casts. We can compare the, the hind casts in the green with the, the observations in the red. And for, for each one of those, there's a, there's a residual error. And we, if we take the, the, we calculate the variance of, of those errors, we can use that as the, the spread of our forecast distribution. And so we can get, we can get a, a PDF out of our regression model in this way. So that, that's typically how regression models are used to, to calibrate forecasts. You'll use some past performance of the model to get the, get the spread of the distribution. So if you're, you'll, if you're, if you're, if your model is, is very accurate, then you'll have a smaller spread in terms of those error residuals. If the error residuals are large, then you'll have a large spread. And you'll, you'll have a wide, wide distribution. So that was for reservoir, reservoir inflow. This is a case where it's been done for the Philippines for rice production. So rice production in the, the, this is the January to, January to June dry season. And this is all tabulated. You, you can find this on the web tabulated by, by the, I think, food and agricultural organization. You can find crop production statistics for, for countries around the world. And this is done for two cases here using regional statistics, regional crop productions on the, on the left and, and provincial production on the right. And so what we've done is for, we have these records for 2000, for, for 1980 through, through 2007. And we've used those rice productions as our Y variable. And then as our X variable, we've used the GCM predicted precipitation for the October to December season. I think that is initialized on, on, on June, June, June 1st. And so what's shown here is the anomaly correlation skill for those. And so you can see that in some, in some, for some, some of the regions we have a good, very good correlation skill of over 0.6. Many, many of the, many of the regions or provinces don't, don't have any, any, any skill. But you can see that there, there are some in which there's some, some reasonable skill. And again, this is through this, this, the, the impact of El Nino really on the, on the region. Now, I think this is leave one, leave one out. Yeah. Or it might have been leave three out. I forget. But it's not spitting the time series. This was the one that Vincent showed yesterday, onset date over, over the, over the maritime continent, where here our on, our Y variables is, is the onset date, onset date anomalies. And our X variables are the sea surface temperature over the, over the tropical Pacific and, and, and maritime continent region from July. And then at the bottom is the cross-validated anomaly correlation skill. And you can see that that, in that case, also getting up to some pretty high values of, of, of over 0.7. And we did that also with station data. And you, you see, you see some, some values of, I'd say, of the order of sort of 0.5 in, in the better regions over, over, over Java, southern, southern, and southern Borneo. Here's another example of predictans that, that could be something different from seasonal total. So what we've looked at here is some daily rainfall data from the India Met Department and calculated, as, as Vincent did yesterday, making this decomposition for the seasonal total with a number of rainy days times the mean intensity of rainfall. And this is anomaly correlation score of cross-validated regression with observed tropical Indo-Pacific SST 1901 to 2004. So there's no, actually there's no, there's no lead time here. This is using concurrent SST. So it's a method here of showing, well, what's the potential predictability from sea surface temperature in, in these three quantities. And as, as Vincent was showing yesterday, we, we see, we, we often see this where we have more spatial coherence in the rainfall, rainfall occurrence field than we do in the intensity field, just because of the kind of things that Steve was talking about, that, that tropical convection is, is a very noisy, is a very noisy phenomenon. So if you take a spatial area, you, you, the intensity on a particular day would vary greatly over, over, over the area, but it might be raining at some point in the day everywhere so that you have a smoother field for rainfall occurrence. And so it's a way to, in, in seasonal forecasting, it's a way to, it can be a way to filter the field, to filter out a, a more predictable part. And so a question here is, would that, could you actually do that for, for daily, on the, on the sub-seasonal scale if you took, if you took a weekly precip, if you look at the number of rainy days in the week, perhaps that is actually more predictable than the weekly average of rainfall itself. I'm going to skip over that because that was what Vincent was talking about yesterday and I, in the interest of time. I just wanted to show that, that we had, this thing about rainfall frequency versus rainfall amount. We had a, we had a workshop in Singapore and Ryzan was there in a, in the back there somewhere in, in, this, this was a while back in, in 2007 on, but that was also the thing that, that Ryzan is going to be talking about after the break. This was a workshop on seasonal, inter-seasonal climate prediction and applications way back then. And so maybe it's like a forerunner of S2S. And what, what each country did was to bring its own data, rainfall, daily rainfall, as well as daily rainfall. And then we made two predictants. One was the, the rainfall frequency or we call it the number of dry days. And the, another one was just the seasonal rainfall total. And each country used, used CPT to make, to make some downscaling from GCM forecast for March, April, May season. And then just plot the skill score on, on this, on the map. So we gathered all the country's outputs together on one map. And so the size of the circle is the anomaly correlation. So I think you probably can see it best in the color bar here. So the deeper blue is getting up to sort of point, point six, point seven, point eight. But what I want you to notice is that the ones on the right, where we're looking at rainfall frequency tend to have larger deeper blue circles than the ones on the, on the left. So you can especially see that over, over the Philippines and over Malaysia. Also, also over Thailand. So I think that's, that was quite, quite a remarkable result back, back way back then. So in the last five, 10 minutes, I just want to talk about what, another variant of, of regression that might be more applicable on, on the sub-seasonal timescale. So quanta regression. The, and I showed this result already on Monday, but I'm just going to show you a little bit about the background of this on, on the next couple of slides. So the ultimate goal of regression analysis is to, is to model, is to model the conditional distribution of the response variable, our, our Y variable, given a set of explanatory variables. So this is generally called distributional regression, because you want to have the whole distribution. You, you don't, you don't just want to get, get the, the, the mean of the distribution like we did in, in the Gaussian case. And then we had to estimate the spread in some other way. So this, this is quite general and there, there are, there, there are various parametric and non-parametric ways of that people have used to go about that. And I'm not going to, I'm not going to describe them. I'll just mention this quanta regression as being one example of this where the predict hand is, is a quantile of the forecast PDF. So we say we're, we're interested in the probability of exceedance of the median or the 80th percentile or something. We choose our quantile like we were talking about on Monday that we showed you that IRI mat room, which we call the flexible forecast where you can choose your, your quantile and get the probability of exceeding or not exceeding that. And in, in that case, we use a Gaussian methodology or transform Gaussian to get to, to that in the case of the IRI seasonal forecast. But this is a, a method that's been used, I think quite a bit in, in weather forecasting. It was introduced into the, into the weather forecasting community by, by Dan Wilkes, I think. I'm, I'm quoting from his paper here in 2000, 2007, 2008. And it's been used by people like Tom Hamill to look at medium range weather forecasts or, or TIGI, TIGI data. So in this case, a logistic regression is well suited to predicting a probability rather than, rather than the measurable physical quantity. And so that's what that looks like here. Here's the, here's the logit or the logistic regression where this is the, this is the log of the odds, right? This is the, the probability, P is the probability of something happening divided by the probability of it not happening. So that's the odds. And this is the log of the odds. And that is the, that is the predict and the thing that we want to predict. And we base that, that on some predictor, predictor variables, the, the f of x, so x is the predictor. So P is the, in this case, it's the probability of not exceeding a particular quantile. So that, that the, the equation is, is linear in, on the logistic or on the log odds scale, as you can see here. So here's an example of that from, from Dan Wilkes' paper, actually 2009. And what, what, what's shown in the paper is a forecast from the NCEP GFS model. And it's accumulated week two precipitation for particular places in the U.S. So this is a weekly average. I'm showing you a particular forecast, a weekly average for 28th November through the 2nd of December 2001 for Minneapolis. So as x, they take the GFS ensemble mean precipitation at, at the nearest grid point and they square root it. So because this is, we're getting down to shorter time scales, you can make, you can make it more, more Gaussian by, by transforming it in some way. So typical in rainfall is to do something like square root or cube root. So you think about some big, big extremes that get compressed down by, by doing that. And p is the probability of not exceeding various, various quantiles. And so this, what is shown in this graph is individual, individual regressions for different quantiles. And so I colored in the red one here is the median. This is the probability of not exceeding the median. So the cumulative probability up to the median. And what's plotted along the x-axis here is the, the ensemble mean precip from the, from the model. And this is all trained under cross validation as well. And so it's showing as, as, as the ensemble mean precipitation predicted by the model increases, then the probability of, of getting less, getting less than the median obviously is decreasing. So the curve is sloping down to, to the right. And the way that this has been done is using training data window of, of, of plus or minus 45 days around, around the forecast date to train, to train the regression model and get the, get the, the coefficients in this, this f of x. So what, what you see here, he did that so that the red one is doing that for the median, but then have also done it for some other quantiles for the 10th, 10th percentile, the, the 33rd percentile, the 60, 67th percentile, the 90th and the 95th. And what you see there is there's some, there's a problem because they, they're crossing over each other and we, we don't want that to happen. So doing individual regression is, is for different quantiles is, is not, is not well posed. And that's the sense of his paper. And what he introduced in 2009 was this, this methodology of extended logistic regression. So in normal case the lines may, may, may cross, but doing this will, will alleviate it by introducing this, this additional term here. So Q is the quantile. So it's just what we had before. It's the, the, the probability of not exceeding the quantile. And what we have here is some function of that quantile. So this, this specifies parallel functions of the predictors x whose intercepts increase monotonically with the increasing threshold Q. It's a, it's a way to ensure that those curves don't cross. And you can fit this in MATLAB using the GLM, GLM fit for example. And so here's the result from his paper of doing that. So the one on the right is what I showed before when you're just doing individual, individual quantile regressions for different quantiles. And then this is the result of, of introducing this additional term G of Q here where the, the functional form is, I think the square root of Q is what's, what's used for G of Q. And you can see now that we get, it's enforced these parallel, it's enforced these regression lines to be parallel. So, so it's well behaved. And this is the one that I showed, I showed this on Monday, a preliminary case from Nicola Vigo at, at IRI where he's applied, applied that formula using, using GLM fit here to the CFS we forecast over the, over the North America for the, this is showing July, August, September for the 1999, 2010 period from this is using S2S. It's actually using S2S data from the S2S database. And as I show, what I'm showing here is the reliability diagrams. He did this for, for the, the quantiles here are the, the 33rd and the 36th, the, and the 66th. But one interesting thing is you can train this on, on using teresile categories, but then calculate for any, any quantile you like. So, it's a way to get a flexible format forecast like we do have already for the seasonal forecast. But there's, there's, I think this method may be better suited to, to sub-seasonal where, where, where you're, where you're looking at weekly instead of seasonal. It, it's probably a good, a good technique for, for doing, for doing seasonal forecast as well actually. And so you can see that there's a fairly good reliability showing week one through week, week four in the different colors for the below normal, uh, near normal in the middle of the distribution and, and above normal we're starting to get, get a bit, a bit off at the higher predicted values for the above normal categories. So it's starting to be a bit, a bit uncolored, calibrated, but you see you're getting worse as you go from week one to, out to week four. And maybe it was still something we need to work on a bit there. At the moment we're using the square root of the, of the ensemble mean of the CFS forecast, which have just, have four ensemble members. And this is integrating over daily starts actually with that model all throughout the season. So we're not using, we're not using weekly starts, we're using all of the starts. And so they're, they're all overlapping with one another as well in the weekly, weekly averages. And it's done separately for every grid point on, in the domain. So I think there, there's techniques have been introduced for thinking about how you might also be able to pool over grid points and, and not, and not have, not be fitting this individually for, for neighboring points because you may also have an issue that this is all wavering around a lot on, on the grid scale and you, and you don't want that to happen. Yeah, that could well be the case. Yeah. Yes. Yeah. Yeah, we should, we should plot the, the histograms of the, the issuance of the different, different categories. And then you'd see that these ones are, are rarer than, than in the middle. Yeah. So it may be to do with more extreme events. So main points. Seasonal forecasts are sometimes tailored, expressing in terms of a predict hand of interest like rainfall frequency, monsoon onset date could be a drought probability, river flow, crop yield. This can also be a form of, a forecast calibration or, or, or, or downscaling. I should say this is using, using regression models according to the, the choice of your, your predict hand. So this would be a forecast calibration as well if you just were to choose the, the seasonal rainfall itself at, at, on, on your scale of interest. So it'll be a sort of combination of statistical downscaling or, or forecast calibration and forecast calibration as, if you just look at the seasonal rainfall itself, whereas we can, as I, as I showed the, the technique is flexible. So it allows you to choose the predict hand that you like. Regression models using GCM ensemble mean forecasts and, and antecedent climate conditions. And this could be some, could be some variables from, from the GCM. Usually Gaussian or transformed Gaussian. Most regression models are limited to this conditional mean as a function of the predictor. And you need to, to get the spread some other how, but this is a interesting technique of quantile regression using extended logistic regression which has been used in weather forecasting and that, that we're, we're looking at in, looking into this as a way of, of doing this for, for sub-seasonal forecasts. So I'll stop there. Thanks.