roomthily
roomthily:

"This is Not a Data Visualization"
via The Auguries of Innocence: This Is Not a Data Visualization.
But visualizations are not the data. The data is not the sum of the experience. We’ve been inappropriately using data visualizations as the basis for statements and conclusions. We’re leaving out rigorous statistical analysis, and appropriate qualifiers such as confidence intervals. It’s exciting that we’ve become more and more a society of pattern-seekers. But it’s important that we don’t become lazy and cavalier with what we do with those observations. We also have to remember our audience, and how the audience puts into context what they see.

roomthily:

"This is Not a Data Visualization"

via The Auguries of Innocence: This Is Not a Data Visualization.

But visualizations are not the data. The data is not the sum of the experience. We’ve been inappropriately using data visualizations as the basis for statements and conclusions. We’re leaving out rigorous statistical analysis, and appropriate qualifiers such as confidence intervals. It’s exciting that we’ve become more and more a society of pattern-seekers. But it’s important that we don’t become lazy and cavalier with what we do with those observations. We also have to remember our audience, and how the audience puts into context what they see.

Il test di Turing superato? Non precisamente…

visualizingmath

ryanandmath:

spring-of-mathematics:

Integral and the difference between: The Riemann-Darboux approach & The Lebesgue approach.
Riemann-Darboux’s integration (in blue) and Lebesgue integration (in red).  To get some intuition about the different approaches to integration, let us imagine that it is desired to find a mountain’s volume (above sea level).
The Riemann-Darboux approach: Divide the base of the mountain into a grid of 1 meter squares. Measure the altitude of the mountain at the center of each square. The volume on a single grid square is approximately 1 m2 × (that square’s altitude), so the total volume is the sum of the altitudes.

The Lebesgue approach: Draw a contour map of the mountain, where adjacent contours are 1 meter of altitude apart. The volume of earth contained in a single contour is approximately 1 m × (that contour’s area), so the total volume is the sum of these areas.

Professor Gerald B.Folland  summarizes the difference between the Riemann and Lebesgue approaches thus: “to compute the Riemann integral of f, one partitions the domain [a, b] into subintervals”, while in the Lebesgue integral, “one is in effect partitioning the range of f “ 

( Gerald B.Folland (1999). Real analysis: Modern techniques and their applications. Pure and Applied Mathematics (New York) (Second ed.). New York: John Wiley & Sons Inc. xvi+386. )

See more at Integral on Wikipedia  - Riemann integral   - Lebesgue integration

Images, I shared at Integral - Contour line - Riemann sum - Serpentine wall at the University Virginia.

Another intuitive way of thinking of Lebesgue integrals (as described by Paul Montel): “I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.”

Also worth mentioning that you can integrate a wider variety of functions using the Lebesgue integral. Very general and very cool.