Parallel coordinates

Parallel coordinates ( also | |- coordinates; parallel coordinate plot english, PCP ) are a method for the visualization of high-dimensional structures and multivariate data. In the right graph, the vertical lines indicate the axes of the coordinate system. Unlike in the scatter diagram in the two coordinate axes are disposed orthogonal to each other, here they extend in parallel and at equal distance. Each line from left to right corresponds to a data point and is represented by a polygon with vertices on the parallel axes. The position of the corner of the i-th axis corresponds to the i-th coordinate of the point.

History

Often the invention of parallel coordinates Maurice d' Ocagne in 1885 is attributed, however, this publication has except that the words in title has nothing to do with the same visualization technique, but merely describes a transform function on coordinate systems. There is also doubt already before 1885 representations of parallel coordinates, for example, by H. Gannett and FW Hewes in 1883 ( see link in the reference ). Almost 80 years later, the original idea of Alfred island mountain was used again in 1959. Since 1977 they have been systematically developed and popularized by him. Most commonly they are used in algorithms for collision avoidance in air traffic (1987 ), the data mining in image analysis method, in which optimization, process control, and intrusion detection in computers. The decisive factor for the successful application of parallel coordinates was Wegmans Article Hyperdimensional Data Analysis Using Parallel Coordinates from the year 1990.

Generalized parallel coordinates have been proposed by Moustafa and Wegman 2002 and 2006. Here, the Cartesian coordinate system by means of basic functions in a parameter space, and this then mapped to parallel coordinates. This allows a connection between generalized parallel coordinates, the Grand Tour and the Andrews curves produced.

Pros and Cons

The parallel coordinates have advantages and disadvantages:

• An increase in the dimension is merely the addition of other ( perpendicular ) axis.
• Since parallel coordinates map a higher-dimensional space onto a two-dimensional space, enters a loss of information. This can be measured with the aid of Parseval 's identity.
• With practice certain two-dimensional and also higher-dimensional structures in parallel coordinates are easy to recognize. The graph below shows various two-dimensional structures ( perfectly positively and negatively correlated data points, cluster, district and normally distributed data) once a scatter plot ( top) and parallel coordinates. There are patterns in parallel coordinates known for ( hyper) planes, curves, more smooth ( hyper) surfaces, similarity, convexity and non-directional surfaces. The point- line duality is an indication that the mathematical foundations of projective geometry originate.

For the visualization of high-dimensional data in the statistics, three important aspects must be considered: