SOCR EduMaterials Activities RNG - Revision history

PriscillaChui: /* Applications */

2008-05-03T01:32:37Z

‎Applications

PriscillaChui: /* Exercises */

2008-05-03T01:32:18Z

‎Exercises

PriscillaChui: /* Background & Motivation */

2008-05-03T01:31:53Z

‎Background & Motivation

PriscillaChui: /* Goals */

2008-05-03T01:31:30Z

‎Goals

PriscillaChui: /* Summary */

2008-05-03T01:30:06Z

‎Summary

PriscillaChui: /* SOCR Educational Materials - Activities - SOCR Random Number Generation (RNG) Activity */

2008-05-01T17:22:57Z

‎ SOCR Educational Materials - Activities - SOCR Random Number Generation (RNG) Activity

IvoDinov at 00:57, 18 September 2007

2007-09-18T00:57:24Z

IvoDinov at 00:56, 18 September 2007

2007-09-18T00:56:47Z

IvoDinov at 00:53, 18 September 2007

2007-09-18T00:53:22Z

IvoDinov: /* Applications */

2007-03-06T07:38:05Z

‎Applications

@@ Line 32: / Line 32: @@
 <center>[[Image:SOCR_Activities_PowerTransformGraphing_Dinov_091707_Fig3.jpg|400px]]</center>
-===Applications===
+==Applications==
 The RNG background and motivation section clearly described some of the critical scientific and technological challenges that rely upon the existence of quality RNGs. Here we present the applications of the SOCR RNG's for various interactive activities and demonstrations.
 * [[SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs | Power-Transform Family Graphs]]

@@ Line 19: / Line 19: @@
 '''Where does this sampling-need come from?''' Random number generators have several important applications in statistical modeling, computer simulation, cryptography, etc. For example, data collection is often very expensive. Hence, to do appropriate inference on datasets of smaller sizes, we may consider simulating repeatedly from appropriate distributions, instead of using real observations. Another example of why are random number generators so important comes from cryptography. It is a commonly held misconception that every encryption method can be broken. Claude Shannon, Bell Labs, 1948, proved that the [http://en.wikipedia.org/wiki/One-time_pad one-time pad cipher] is unbreakable, provided the secret key is truly random and of length equal or greater than the length of the encoded message. [http://en.wikipedia.org/wiki/Monte_carlo_simulation Monte Carlo simulations] are also based on RNGs and are used for finding numerical solutions to (multi-dimensional) mathematical problems that cannot easily be solved exactly. For example, integration, differentiation, root-finding, etc.
-===Exercises===
+==Exercises==
 * '''Exercise 1''': Go to the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] and click on the '''Data Generation''' tab. Select 200 observations from the [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Distributions Generalized Beta Distribution], as shown on the image below. Choose this four-tuple for the parameters <math> \alpha=1.5; \beta=3; A=0; B=7</math>. Copy these 200 values in your mouse buffer (CNT-C) and paste them in the '''Data''' tab of the '''LineCharts --> PowerTransformHistogramChart''' under [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts]. Then ''Map'' this column to ''XYValue'' (under the '''MAP''' tab) and click '''Update_Chart'''. This will generate the histogram of the 200 observations. Indeed, this graph should look like a discrete analog of the [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Distributions Generalized Beta] density curve. You can see exactly what the [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Distributions Generalized Beta Distribution] looks like by going to [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] and selecting <math> Beta(\alpha=1.5; \beta=3; A=0; B=7)</math>.

@@ Line 10: / Line 10: @@
 * present applications of random number generation
-===Background & Motivation===
+==Background & Motivation==
 '''How many natural processes or phenomena in real life can you describe that have an exact mathematical close-form description and are completely deterministic?''' Arrival times to school each day? Motion of the Moon around the Earth? The computer CPU? The atomic clock? It is an unsettling paradox that all natural phenomena we observe are stochastic in nature. Yet, we do not know how to replicate any of them exactly. There are good computational strategies to approximate natural processes using analytical mathematical models; however, upon careful review one always finds out a deterministic pattern in all purely computationally generated processes.
@@ Line 18: / Line 18: @@
 '''Where does this sampling-need come from?''' Random number generators have several important applications in statistical modeling, computer simulation, cryptography, etc. For example, data collection is often very expensive. Hence, to do appropriate inference on datasets of smaller sizes, we may consider simulating repeatedly from appropriate distributions, instead of using real observations. Another example of why are random number generators so important comes from cryptography. It is a commonly held misconception that every encryption method can be broken. Claude Shannon, Bell Labs, 1948, proved that the [http://en.wikipedia.org/wiki/One-time_pad one-time pad cipher] is unbreakable, provided the secret key is truly random and of length equal or greater than the length of the encoded message. [http://en.wikipedia.org/wiki/Monte_carlo_simulation Monte Carlo simulations] are also based on RNGs and are used for finding numerical solutions to (multi-dimensional) mathematical problems that cannot easily be solved exactly. For example, integration, differentiation, root-finding, etc.
 ===Exercises===

@@ Line 4: / Line 4: @@
 This activity describes the need, general methods and SOCR utilities for random number generation and simulation. [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] allows interactive sampling from any [[About_pages_for_SOCR_Distributions | SOCR Distribution]]. The simulated data may then be easily copied and pasted in different SOCR [http://www.socr.ucla.edu/htmls/SOCR_Analyses.html Analyses] or [http://www.socr.ucla.edu/htmls/SOCR_Charts.html Graphing] tools for further interrogation.
-===Goals===
+==Goals==
 The aims of this activity are to:
 * motivate the need for robust random number generators

@@ Line 1: / Line 1: @@
 == [[SOCR_EduMaterials_Activities | SOCR Educational Materials - Activities]] - SOCR Random Number Generation (RNG) Activity ==
-=== Summary===
+== Summary==
 This activity describes the need, general methods and SOCR utilities for random number generation and simulation. [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] allows interactive sampling from any [[About_pages_for_SOCR_Distributions | SOCR Distribution]]. The simulated data may then be easily copied and pasted in different SOCR [http://www.socr.ucla.edu/htmls/SOCR_Analyses.html Analyses] or [http://www.socr.ucla.edu/htmls/SOCR_Charts.html Graphing] tools for further interrogation.

@@ Line 27: / Line 27: @@
 </center>
-* '''Exercise 2''': Let’s get some more simulated data: Go to [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] and generate 100 Cauchy Distributed variables. Copy these data in your mouse buffer (CNT-C). Of course, you may choose any other distribution. Next, paste (CNT-V) these 100 observations in [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] ('''Line-Charts -> Power Transform Chart'''). ''Map'' this column to ''XYValue'' (under the '''MAP''' tab) and click '''Update_Chart''' to see the index plot of this data in RED! <center>[[Image:SOCR_Activities_PowerTransformGraphing_Dinov_022007_Fig1.jpg|400px]]
+* '''Exercise 2''': Let’s get some more simulated data: Go to [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] and generate 100 Cauchy Distributed variables. Copy these data in your mouse buffer (CNT-C). Of course, you may choose any other distribution. Next, paste (CNT-V) these 100 observations in [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] ('''Line-Charts -> Power Transform Chart'''). ''Map'' this column to ''XYValue'' (under the '''MAP''' tab) and click '''Update_Chart''' to see the index plot of this data in RED!
 [[Image:SOCR_Activities_PowerTransformGraphing_Dinov_022007_Fig2.jpg|400px]]</center>
 Imagine now that we want to compare the bell-shape-looking [[About_pages_for_SOCR_Distributions | Cauchy]] and [[About_pages_for_SOCR_Distributions | Normal]] distributions. As pointed out above, we can generate 1,000 random Cauchy observations (using the '''Data Generation''' tab of the [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler]). Then we can fit the best possible Normal distribution to the histogram of these 1,000 Cauchy observations. As the figure below shows the best ([http://en.wikipedia.org/wiki/Maximum_likelihood maximum likelihood estimates]) for the mean and variance of the best normal fit are ''(-0.8529654842885793, 6355.713457104439)''. The measure of centrality (mean) is pretty accurately estimated (exact center of Cauchy is zero (Cauchy is symmetric), even though its mean is undefined). The variance is large, as Cauchy distribution has much heavier tails than Normal distribution -- hence it's ''Bell-Shapeness'' is '''misleading'''! The distribution insert on the right illustrates the exact Cauchy distribution, which you can see interactively in [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions].

@@ Line 34: / Line 34: @@
 The RGN background and motivation section clearly described some of the critical scientific and technological challenges that rely upon the existence of quality RNGs. Here we present the applications of the SOCR RNG's for various interactive activities and demonstrations.
 * [[SOCR_EduMaterials_Activities_PowerTransformFamily_Graphs | Power-Transform Family Graphs]]
-* [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_ModelerActivities_MixtureModel_1  | Mixture Model Experiment]
+* [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_ModelerActivities_MixtureModel_1   Mixture Model Experiment]