If you use a constant cutoff value then you will erroneously classify every data point as an outlier. To conclude, in high dimensions every data point is far away from the mean. Notice that the percentage of outliers is close to 2.5% independent of the dimension of the problem! By knowing the distribution of distances in high-dimensional MVN data, we are able to define a cutoff value that does not classify every point as an outlier. The following graph shows the percentage of simulated observations that are classified as outliers when you use this scheme. PctOutliers = sum(dist>cutoff)/N /* dist > statistical cutoff */ PctOutliers = j(nrow(dim),1) /* pct outliers for chi-square d cutoff */Ĭutoff = sqrt( quantile("chisquare", 0.975, d) ) /* dist^2 ~ chi-square */ The graph shows that the cutoff distance is greater than the minimum distance and that the two distances increase in tandem. The following graph shows a the 97.5th percentile as a function of the dimension d. A standard technique (which is used by the ROBUSTREG procedure to classify outliers) is to define an outlier to be an observation whose distance to the mean exceeds the 97.5th percentile. (This is discussed in the article " Testing data for multivariate normality.") Therefore, you can use quantiles of the chi-square distribution to define outliers. For multivariate normal data in d dimensions, you can show that the squared Mahalanobis distances are distributed like a chi-square distribution with d degrees of freedom. The resolution to this dilemma is to incorporate the number of dimensions into the definition of a cutoff value. How to define an outlier cutoff in high dimensions If you do, you will, as Peter said, conclude that in 50 dimensions every point is an outlier. In 150 dimensions, every point is more than 10 standardized units away! Consequently, you cannot define outliers a priori to be observations that are more than 5 units away from the mean. In 50 dimensions, every point of the multivariate normal distribution is more than 5 standardized units away from the origin. The graph shows that the minimum distance to the origin is a function of the dimension. The following graph shows the distance of the closest point to the origin for various dimensions. MinDist = min(dist) /* minimum distance to mu */ X = randnormal(N, mu, Sigma) /* X ~ MVN(mu, Sigma) */ Sigma = CompSym(d,1-rho,rho) /* get (d x d) correlation matrix */ MinDist = j(nrow(dim),1) /* minimum distance to center */ Load module=Mahalanobis /* or insert definition of module here */ * Helper function: return correlation matrix with "compound symmetry" structure: Computes the distance of the closest point to the origin.The Mahalanobis distance is a standardized distance that takes into account correlations between the variables. Uses the Mahalanobis module to compute the Mahalanobis distance between each point and the origin.Simulates a random sample from the MVN(0,Σ) distribution.The following SAS/IML program does the following: Let's see what happens for high-dimensional data. That sounds good, right? In one dimension, an observation from a normal distribution that is more than 5 standard deviations away from the mean is an extreme outlier. But how close is "close"? You might extrapolate from your knowledge of the univariate normal distribution and try to define an "outlier" to be any point whose distance from the origin is more than some constant, such as five standardized units. Suppose that you simulate 1,000 observations from a multivariate normal distribution (denoted MVN(0,Σ)) in d-dimensional space.īecause the density of the distribution is highest near the mean (in this case, the origin), most points are "close" to the mean. You can demonstrate this fact with a simulation. What he meant is that most points in a high-dimensional cloud of points are far away from the center of the cloud.ĭistances and outliers in high dimensions "when there are a bunch of dimensions, every data point is an outlier"Īnd remarked on the curse of dimensionality. After my post on detecting outliers in multivariate data in SAS by using the MCD method, Peter Flom commented
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While consulting for medical projects over the years and working on some, I collected hundreds of domain-specific libraries and frameworks built to help developers make medical applications. It's a self-hosted application which means it can be installed and hosted at private hosting for teams.ġ8 Frameworks, Libraries, and Projects for Building Medical Applications It's completely free and open-source software that comes with no restrictions to use or to modify. MicroDraw is a lightweight web-based collaborative annotation application for displaying and viewing large-scale (DeepZoom) images. The Human Protein Atlas is a Swedish-based project that started 2003Ĭollaboratively Annotate large-scale DeepZoom images with MicroDraw Some of my relatives who are studying medicine. I recommended the project already to my colleagues who are teaching at medical schools. I planned to write about this amazing project months ago, soon after I wrote about OpenSeadragon library. The Human Protein Atlas is What every Medical, Human Biology Students and Scientist Need It is released and hosted by IBM, with full instructions, video tutorials The project is started by Yuksel Temiz, an engineer and a designer from Switzerland, who later released it as an open-source for educational purposes. MicroscoPy is an open-source MICROSCOPE built using LEGO bricks, 3D-printing, Arduino and Raspberry Pi. MicroscoPy: Build Your Own Low-cost Digital Microscope using Raspberry Pi The project has detailed instructions on how to setup your Raspberry Pi It is an ideal solution for taking still images using a 12 MP HQ camera attached to a gemmological microscope. Microscope-PiCam is a free open-source web-based solution that acts as the soul of a Raspberry Pi based microscope. Microscope-PiCam: An Open-source Raspberry Pi Microscope It achieves steps well below 100 nanometers when driven with miniature stepper motors, and is stable to within a few microns over several days. This design for a 3D printed microscope stage uses plastic flexures, meaning its motion is free from friction and vibration. Optomechanics is a crucial part of any microscope when working at high magnification, it is absolutely crucial to keep the sample steady and to be able to bring it into focus precisely.Īccurate motion control is extremely difficult using printed mechanical parts, as good linear motion typically requires tight tolerances and a smooth surface finish. It comes with a sophisticated guide that helps anyone print the microscope with any 3D printer, an open-source software to start using the microscope which works on Windows, Linux, and macOS, and rich tutorials to make its usage smooth and fun. There are many different options for the optics, ranging from a webcam lens to a 100x, oil immersion objective. The OpenFlexure Microscope is a 3D printable microscope, including a precise mechanical stage to move the sample and focus the optics. Solo is also the first all-in-one drone to offer live streaming of HD video from your GoPro directly to your mobile device, as well as built-in HDMI for live broadcast in HD.Ĭablecam: Lock Solo onto a virtual cable between any two points in space. Its powerful and one-of-a-kind computer-assisted Smart Shots allow you to set up the exact shot you want, then let Solo execute the shot with the precision of a seasoned cinematographer and with a list of Smart Shots to choose from, the perfect shot is always just a few taps away. Solo’s intelligence makes it possible for beginners and pros alike to capture professional-grade shots from day one. By splitting up the work like this, Solo dramatically reduces the likelihood of mechanical failure in flight and opens up a world of possibility for adding advanced capabilities and features. The autopilot-traditionally a drone’s only nerve center-now works sort of like Solo’s “brain stem,” responsible solely for the basic rudiments of flight. They handle all of Solo’s high-level functions, like Smart Shots, HD video transmission and exclusive GoPro control features. You can think of these computers as Solo’s “frontal cortex”-the most advanced part of the brain. This processing power enables Solo to do amazing things that no personal drone has ever been able to do. What makes it smart? Not only does Solo have the world’s most advanced Pixhawk 2 autopilot system, but it also runs two 1 GHz computers, one on the copter and one in the controller, connected via a powerful dedicated Wi-Fi signal carried by 3DR Link. Solo delivers this promise: You get the shot, every time. In the end, we know people really just want great shots: They want access to professional tools that enable them to get those shots easily and consistently, without the need to spend years becoming a professional drone pilot. While this meant giving Solo smooth and powerful flight dynamics, it also demanded we make a smarter drone, one that could handle the flying so you can retake creative control and focus on getting great video. When we made Solo, we set out to create not just the best drone, but the best aerial photography experience imaginable. 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And it’s the only all-in-one drone to offer live video streaming in vivid HD from your GoPro® directly to your mobile device-at ranges up to half a mile. Solo is powered by two 1 GHz computers for the best and smoothest aerial photography experience imaginable, including many world-first technologies and a host of built-in safety and support features. With features like pushbutton flight and computer-assisted Smart Shots, Solo makes it easy for anyone to get professional aerial photos and video. The world’s smartest drone is also the easiest to fly. Paris Metro ticket machines can appear like the following: Tickets for the Paris RER can be purchased at any RER or Metro station ticket window or ticket vending machine. The maximum price of a RER ticket, regardless of distance, will be 5€. CDG Airport to central Paris (from Zone 5): 11.45€Īs of March 1, 2022, RER tickets (excluding airport routes) now have a price ceiling of 5€.Some example trip RER fares ( prices valid as of January 1, 2023): Paris RER ticket prices are determined station to station for travel beyond Zone 1 and range from 2.10€ to upwards of 14.10€. Paris RER tickets ( Billet Ile-de-France) are similar to the one shown here: Travel beyond Paris Zone 1 on the RER requires a special ticket: a Billet Ile-de-France. When taking a RER train strictly within central Paris Zone 1 (visible in the first RER Map), you can use basic Paris Metro tickets for travel on the RER. Paris RER trains use two different types of tickets depending on which part of the RER network you will be traveling. Here are some instructions on reading Paris RER train timetables. Paris RER schedules can be confusing to read. RER C Timetable (Eiffel Tower, Versailles, Orly Airport).RER A Timetable (Paris Disneyland, La Défense).at Paris Chatelet les Hallesįull Paris RER Timetables for download and print can be found here: Last RER train from Disney at 0:24 a.m.First RER A train leaves Disneyland at 5:13 a.m.Last RER A train leaves Paris Chatelet les Halles at 12:36 a.m.First RER A train leaves Paris Chatelet les Halles at 5:18 a.m.RER A Paris to Disneyland (Marne La Vallée station) Last RER B train leaves Paris Gare du Nord at 12:15 a.m.First RER B train leaves Paris Gare du Nord at 4:53 a.m.Last RER B train leaves CDG T2 at 11:50 p.m.First RER B train leaves CDG T2 at 4:50 a.m.Here are some examples of first and last Paris RER trains to popular destinations:ĬDG Airport Terminal 2 to Paris, Gare du Nord station Paris RER trains operate as a scheduled service, unlike the Paris Metro. Paris RER First & Last Trains – Schedules & Timetables combined Paris RER & Metro map showing lines, stations & interchanges between the Metro & RER.basic RER map showing 5 RER train lines with branching endpoint stations and the 5 fare zones for Paris and surrounding suburban area (Ile-de-France). The public transit authority of Paris ( RATP) offers multiple RER maps that can be downloaded here: |