How is this computed in R? So far, so good, we get the same output. Now, what if we want to parallelise computations. Actually, it is possible. The program is now online …. Yes, I still want to get a better understanding of optimization routines, in R. Before looking at the quantile regression, let us compute the median, or the quantile, from a sample.

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To illustrate, consider a sample from a lognormal distribution,. The R code is now. Consider the following dataset, with rents of flat, in a major German city, as function of the surface, the year of construction, etc.

So use here. Here again, it seems to work quite well. We can use a different probability level, of course, and get a plot.

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Now that we understand how to run the optimization program with one covariate, why not try with two? For instance, let us see if we can explain the rent of a flat as a linear function of the surface and the age of the building. Results are quite different. I could not figure out what went wrong with the linear program. Not only coefficients are very different, but also predictions…. In the introduction, he came back briefly on a nice discussion we usually have in economics on the kind of model we should consider. It was about optimal control. In many applications, we start with a one period economy, then a two period economy, and pretend that we can extend it to n period economy.

And then, the continuous case can also be considered. A few years ago, I was working on sports game as an optimal effort startegy within in a game — fixed time. I asked a good friend of mine, Romuald , to help me on some technical parts of proofs, but he did not like so much my discrete-time model, and wanted to move to continuous time. And for now six years, we keep saying that someday we should get back to that paper….

He works with stochastic processes, I work with time series. Of course, we can find connections, but most of the time, the techniques are very different. So I wanted to illustrate that idea.

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The goal is to find the value of the maximum, numerically. And here, there are two very different strategies. For the first case, we can use the standard R function, and see how long it takes to use simulations to get an approximation of the maximum. So, indeed, it looks like computational time to find the maximum in a list of n elements is linear in n , i. And R code is faster than home-made code. But also, interestingly, using continus time based on analysis techniques can be much faster. So, sometimes, considering continuous time models can be much easier to solve, from a numerical perspective.

Eleventh post of our series on classification from scratch. Today, that should be the last one… unless I forgot something important. So today, we discuss boosting. I might start with a non-conventional introduction. And I am quite sure it has to do with my background in econometrics. This is an optimization problem.

And from a numerical perspective, optimization is solve using gradient descent this is why this technique is also called gradient boosting. And the gradient descent can be visualized like below. Two important comments here.

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First of all, the idea should be weird to any econometrician. Here it works because we consider simple non linear model. And actually, something that can be used is to add a shrinkage parameter. The idea of weak learners is extremely important here. But somehow, we should stop, someday. I said that I will not mention this part in this series of posts, maybe later on.

But heuristically, we should stop when we start to overfit. I will get back on that issue later one in this post, but again, those ideas should probably be dedicated to another series of posts.

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So here, we will somehow optimize knots locations. There is a package to do so. And just to illustrate, use a Gaussian regression here, not a classification we will do that later on. Consider the following dataset with only one covariate. Let us try something else. What if we consider at each step a regression tree, instead of a linear-by-parts regression that was considered with linear splines. This time, with those trees, it looks like not only we have a good model, but also a different model from the one we can get using a single regression tree.

There is clearly an impact of that shrinkage parameter.

It has to be small to get a good model. This is the idea of using weak learners to get a good prediction. It will be more complicated because residuals are usually not very informative in a classification. And it will be hard to shrink. In our initial discussion, the goal was to minimize a convex loss function. What we do here is related to gradient descent or Newton algorithm. Previously, we were learning from our errors. At each iteration, the residuals are computed and a weak model is fitted to these residuals. The the contribution of this weak model is used in a gradient descent optimization process.

Here things will be different, because from my understanding it is more difficult to play with residuals, because null residuals never exist in classifications. So we will add weights. Initially, all the observations will have the same weights. But iteratively, we ill change them. We will increase the weights of the wrongly predicted individuals and decrease the ones of the correctly predicted individuals. Somehow, we want to focus more on the difficult predictions. This algorithm is well described in wikipedia , so we will use it. And as previously, one can include some shrinkage. To visualize the convergence of the process, we will plot the total error on our dataset.

Here we face a classical problem in machine learning: we have a perfect model. With zero error. Liebezeit , ;. Widdleton Ed. Grieben , ;. Putnam's sons, ;. Avrom Bik, " R. Gershom G.