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In probability and statisticsthe quantile functionassociated with a probability distribution of a random variablespecifies the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability. It is also called the percent-point function or inverse cumulative distribution function.

In terms of the distribution function Fthe quantile function Q returns the value x such that. Here we capture the fact that the quantile function returns the minimum value of x from amongst all those values whose c. Note that the infimum function can be replaced by the minimum function, since the distribution function is right-continuous and weakly monotonically increasing.

The quantile is the unique function satisfying the Galois inequalities. If the function F is continuous and strictly monotonically increasing, then the inequalities can be replaced by equalities, and we have:. In general, even though the distribution function F may fail to possess a left or right inversethe quantile function Q behaves as an "almost sure left inverse" for the distribution function, in the sense that. The quartiles are therefore:.

Quantile functions are used in both statistical applications and Monte Carlo methods. The quantile function is one way of prescribing a probability distribution, and it is an alternative to the probability density function pdf or probability mass functionthe cumulative distribution function cdf and the characteristic function.

The quantile function, Qof a probability distribution is the inverse of its cumulative distribution function F. The derivative of the quantile function, namely the quantile density function, is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function. For statistical applications, users need to know key percentage points of a given distribution. Before the popularization of computers, it was not uncommon for books to have appendices with statistical tables sampling the quantile function.

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Monte-Carlo simulations employ quantile functions to produce non-uniform random or pseudorandom numbers for use in diverse types of simulation calculations. A sample from a given distribution may be obtained in principle by applying its quantile function to a sample from a uniform distribution. The demands, for example, of simulation methods in modern computational finance are focusing increasing attention on methods based on quantile functions, as they work well with multivariate techniques based on either copula or quasi-Monte-Carlo methods  and Monte Carlo methods in finance.

The evaluation of quantile functions often involves numerical methodsas the example of the exponential distribution above is one of the few distributions where a closed-form expression can be found others include the uniformthe Weibullthe Tukey lambda which includes the logistic and the log-logistic.

When the cdf itself has a closed-form expression, one can always use a numerical root-finding algorithm such as the bisection method to invert the cdf. Other algorithms to evaluate quantile functions are given in the Numerical Recipes series of books. Algorithms for common distributions are built into many statistical software packages. Quantile functions may also be characterized as solutions of non-linear ordinary and partial differential equations.

The ordinary differential equations for the cases of the normalStudentbeta and gamma distributions have been given and solved. The normal distribution is perhaps the most important case. Because the normal distribution is a location-scale familyits quantile function for arbitrary parameters can be derived from a simple transformation of the quantile function of the standard normal distribution, known as the probit function. Unfortunately, this function has no closed-form representation using basic algebraic functions; as a result, approximate representations are usually used.Suppose a real estate analyst wants to predict home prices from factors like home age and distance from job centers.

But what if they want to predict not just a single estimate, but also the likely range? This is called the prediction intervaland the general method for producing them is known as quantile regression. Just as regressions minimize the squared-error loss function to predict a single point estimate, quantile regressions minimize the quantile loss in predicting a certain quantile.

The most popular quantile is the median, or the 50th percentile, and in this case the quantile loss is simply the sum of absolute errors. Other quantiles could give endpoints of a prediction interval; for example a middlepercent range is defined by the 10th and 90th percentiles. The quantile loss differs depending on the evaluated quantile, such that more negative errors are penalized more for higher quantiles and more positive errors are penalized more for lower quantiles.

This graph shows how the quantile loss varies with the error, depending on the quantile. We can also look at this by quantile for under- and over-estimated predictions. The higher the quantile, the more the quantile loss function penalizes underestimates and the less it penalizes overestimates. And in Python code, where we can replace the branched logic with a maximum statement:.

This analysis will use the Boston housing datasetwhich contains observations representing towns in the Boston area. It includes 13 features alongside the target, the median value of owner-occupied homes. Quantile regression therefore is predicting the share of towns not homes with median home values below a value. I train the models on 80 percent and test on the remaining 20 percent. For easier visualization, the first set of models uses a single feature: AGEthe proportion of owner-occupied units built prior to As we might expect, towns with older homes have lower home values, though the relationship is noisy.

Although OLS predicts the mean rather than the median, we can still calculate prediction intervals from it based on standard errors and the inverse normal CDF:. This baseline approach produces linear and parallel quantiles centered around the mean which is predicted as the median.

A well-tuned model will show about 80 percent of dots in between the top and bottom lines. Linear models extend beyond the mean to the median and other quantiles.A loss function is a measure of how good a prediction model does in terms of being able to predict the expected outcome.

Think of loss function like undulating mountain and gradient descent is like sliding down the mountain to reach the bottommost point. There is not a single loss function that works for all kind of data.

It depends on a number of factors including the presence of outliers, choice of machine learning algorithm, time efficiency of gradient descent, ease of finding the derivatives and confidence of predictions. The purpose of this blog series is to learn about different losses and how each of them can help data scientists. Loss functions can be broadly categorized into 2 types: Classification and Regression Loss.

In future posts I cover loss functions in other categories. Please let me know in comments if I miss something. Also, all the codes and plots shown in this blog can be found in this notebook. Regression functions predict a quantity, and classification functions predict a label. MSE is the sum of squared distances between our target variable and predicted values. Below is a plot of an MSE function where the true target value isand the predicted values range betweento 10, MAE is the sum of absolute differences between our target and predicted variables.

So it measures the average magnitude of errors in a set of predictions, without considering their directions. In short, using the squared error is easier to solve, but using the absolute error is more robust to outliers.

Whenever we train a machine learning model, our goal is to find the point that minimizes loss function. Of course, both functions reach the minimum when the prediction is exactly equal to the true value. In the first case, the predictions are close to true values and the error has small variance among observations. In the second, there is one outlier observation, and the error is high.

What do we observe from this, and how can it help us to choose which loss function to use? In the 2nd case above, the model with RMSE as loss will be adjusted to minimize that single outlier case at the expense of other common examples, which will reduce its overall performance.

MAE loss is useful if the training data is corrupted with outliers i.

### Quantile LOESS – Combining a moving quantile window with LOESS (R function)

Intuitively, we can think about it like this: If we only had to give one prediction for all the observations that try to minimize MSE, then that prediction should be the mean of all target values. But if we try to minimize MAE, that prediction would be the median of all observations. We know that median is more robust to outliers than mean, which consequently makes MAE more robust to outliers than MSE.New to Quantile measures? We've got you covered. Learn more about how Quantile measures can help your child.

Find free resources associated with state standards appropriate for students based on their Quantile measure. With Quantile measures, state departments of education can demonstrate the value of state assessments to stakeholders throughout their states. Warning: It looks like JavaScript is currently disabled. Please be aware that some features of this website will not work as intended.

Find Math Activities Search by keyword, state, grade or Quantile measure for targeted, free math resources. Use Quantile Measures New to Quantile measures? What to do with a Quantile measure. Monitor Growth Is your child on track to graduate college- and career- ready?

Find out more with our growth planner. Find Your Lesson Want to find targeted resources to supplement classroom math lessons? Find Your Lesson. Get Resources for Students Find free resources associated with state standards appropriate for students based on their Quantile measure. Search the Math Skills Database. For State Departments of Education With Quantile measures, state departments of education can demonstrate the value of state assessments to stakeholders throughout their states.One area that Deep Learning has not explored extensively is the uncertainty in estimates.

Most Deep Learning frameworks currently focus on giving a best estimate as defined by a loss function.

### Deep Quantile Regression

Occasionally something beyond a point estimate is required to make a decision. This is where a distribution would be useful. Bayesian statistics lends itself to this problem really well since a distribution over the dataset is inferred.

However, Bayesian methods so far have been rather slow and would be expensive to apply to large datasets. As far as decision making goes, most people actually require quantiles as opposed to true uncertainty in an estimate. What would be interesting is for arguments sake the 10th and 90th percentile. Note that the uncertainty is different to quantiles in that I could request for a confidence interval on the 90th quantile.

This article will purely focus on inferring quantiles. In regression the most commonly used loss function is the mean squared error function. If we were to take the negative of this loss and exponentiate it, the result would correspond to the gaussian distribution.

The mode of this distribution the peak corresponds to the mean parameter. Hence, when we predict using a neural net that minimised this loss we are predicting the mean value of the output which may have been noisy in the training set. The loss in Quantile Regression for an individual data point is defined as:. The average loss over the entire dataset is shown below:. If we were to take the negative of the individual loss and exponentiate it, we get the distribution know as the Asymmetric Laplace distribution, shown below.

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The reason that this loss function works is that if we were to find the area under the graph to the left of zero it would be alpha, the required quantile. This loss function consistently estimates the median 50th percentileinstead of the mean. The forward model is no different to what you would have had when doing MSE regression. All that changes is the loss function. The following few lines defines the loss function defined in the section above. When it comes to compiling the neural network, just simply do:.Uncertainty estimation can be especially difficult when the data is heteroskedasticthat is, when the variance of the target variable changes across the value of the predictors.

The quantile loss can be used with most loss-based regression techniques to estimate predictive intervals by estimating the value of a certain quantile of the target variable at any point in feature-space. Both can be installed via pip:. There are some pretty unreasonable outliers: longitudes can only lie between andand latitudes between and The high-density area around Kennedy airport.

The dropoff location distribution looks pretty similar to the pickup locations distribution, except there are a higher density of dropoffs in residential areas within Brooklyn across the river to the east of Manhattan. We can also plot the difference between pickup and dropoff locations. Now we can plot the difference between the number of pickups and dropoffs.

## Quantile regression

Red areas in the plot below are locations where there are more pickups than dropoffs, and blue areas are locations where there are more dropoffs than pickups. Zooming in on La Guardia airport, we can see the separate departure more dropoffs and arrival more pickups areas. The departure dropoff area is on the top level, while arrival pickup area is on a lower level, leading to less accurate GPS signals, causing the more diffuse spread of the pickup location points.

How accurate can we be if we simply always predict the mean? What if we just use the distance of the trip as a predictor? This results in much better predictions than just guessing the mean. One thing we need to look out for when predicting uncertainty is heteroskedasticity! For example, a simple linear model. Indeed it looks like the true data is heteroskedastic - the variance of the fare amount appears to increase with the distance of the trip.

Side note: in the plot above of fares vs time, you can see the flat-rate fee to La Guardia and JFK airports undergo a price hike at the end of summer ! And then the price of the JFK flat-rate continue to creep up afterwards. You can also see that there was some sort of constant interval charge before that period, whereas after prices are more continuous. The variance also changes across pickup location!

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The fares are much more variable when the pickup is across the Hudson river in Jersey City and Hoboken than it is in, say, downtown Manhattan. Instead, we have to allow our estimate of uncertainty to vary across data-space.

One way to do this is to use a quantile regression. What if we now want to predict our uncertainty as to our estimate, instead of only coming up with a single best estimate? Predicting an estimate and the uncertainty as to that estimate seems especially challenging when that uncertainty varies with our independent variables.This method is useful when the need arise to fit robust and resistant Need to be verified a smoothed line for a quantile an example for such a case is provided at the end of this post.

If you wish to use the function in your own code, simply run inside your R console the following line:. From the abstract:. In recent years, a growing need has arisen in different fields, for the development of computational systems for automated analysis of large amounts of data high-throughput. Dealing with non-standard noise structure and outliers, that could have been detected and corrected in manual analysis, must now be built into the system with the aid of robust methods.

During progression a tracking system might lose track of the animal, inserting occasionally very large outliers into the data.

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During lingering, and even more so during arrests, outliers are rare, but the recording noise is large relative to the actual size of the movement. The statistical implications are that the two types of behavior require different degrees of smoothing and resistance. An additional complication is that the two interchange many times throughout a session. As a result, the statistical solution adopted needs not only to smooth the data, but also to recognize, adaptively, when there are arrests.

To the best of our knowledge, no single existing smoothing technique has yet been able to fulfill this dual task. If all we wanted to do was to perform moving average running average on the data, using R, we could simply use the rollmean function from the zoo package. But since we wanted also to allow quantile smoothing, we turned to use the rollapply function. Here is the R function that implements the LOESS smoothed repeated running quantile with implementation for using this with a simple implementation for using average instead of quantile :.

More on the math of the algorithm can be found in the original article. Here is the code to produce the above plot. Do you understand that there are two functions lowess and loess? The former is sort-of a predecessor but nobody but really old books still talks about it. Thanks Dirk, […] Regarding the loess!

I ended up deciding I would call it the way I did, but after reading what you wrote, I realized I made a mistake. I went through the article and corrected the lowess to loess, while also adding a paragraph for explain my reasoning.

The is not a robust procedure. For robust quantile smoothing look at the cobs package on CRAN.

8. Loss Functions for Regression and Classification

May I ask why do you think this procedure is not robust? Hi, Well lowess is not robust, there is a not too pathological counter example in the lowess examples I believe, and a thread from a couple years ago by Martin Maechler. But also intuitively the procedure you are proposing seems a little dodgy.

You take the full data, and throw away most of it to do a running smoother on the quantiles? Generally one would want to do some procedure that uses all the data, but weights things accordingly to get a smooth quantile.

Look at some of Paul Eilers papers on asymetric least squares approaches.