SOCR EduMaterials Activities PowerTransformFamily Graphs - Revision history

IvoDinov: /* Example */ linked to SOCR Charts

2009-10-17T04:16:36Z

‎Example: linked to SOCR Charts

IvoDinov: /* Example */

2009-10-17T04:11:16Z

‎Example

IvoDinov at 04:09, 17 October 2009

2009-10-17T04:09:53Z

IvoDinov: /* Example */ entered Earth planeraty distances data

2009-10-17T04:07:38Z

‎Example: entered Earth planeraty distances data

IvoDinov at 03:59, 17 October 2009

2009-10-17T03:59:09Z

PriscillaChui: /* Exercises */

2008-05-01T17:15:48Z

‎Exercises

IvoDinov: /* Background */

2007-11-06T20:07:08Z

‎Background

IvoDinov: /* Background */

2007-11-06T20:06:25Z

‎Background

IvoDinov: /* Background */

2007-11-06T20:06:04Z

‎Background

Jenny: Reverted edit of Zj0Ufx, changed back to last version by Jenny

2007-08-17T22:49:32Z

Reverted edit of Zj0Ufx, changed back to last version by Jenny

@@ Line 68: / Line 68: @@
 ==Example==
-The table below shows data for [http://answers.yahoo.com/question/index?qid=20090311060915AALBs4f planetary distances from the Sun] in terms of millions of miles (X) and [http://en.wikipedia.org/wiki/Light-year light-years] (Y). The X vs.Y scatterplot shows a positive association curving up.  Making an identify transformation of X (Transformed_X) and a power-transformation of Y for power-exponent=0.65 (Transformed_Y) linearizes the X vs. Y association, as shown on the image below.
+The table below shows data for [http://answers.yahoo.com/question/index?qid=20090311060915AALBs4f planetary distances from the Sun] in terms of millions of miles (X) and [http://en.wikipedia.org/wiki/Light-year light-years] (Y). The X vs.Y scatterplot shows a positive association curving up.  Making an identify transformation of X (Transformed_X) and a power-transformation of Y for power-exponent=0.65 (Transformed_Y) linearizes the X vs. Y association, as shown on the image below. Use the [http://socr.ucla.edu/htmls/SOCR_Charts.html SOCR PowerTransformScatterChartdemo applet].
 <center>

@@ Line 93: / Line 93: @@
 | Pluto || 3666 || 247.68 || 3666.0 || 35.97734496244632
 |}
 <center>[[Image:SOCR_Activities_PowerTransformGraphing_Dinov_101609_Fig12.png|400px]]</center>

@@ Line 94: / Line 94: @@
 |}
-<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig1.jpg|300px]]</center>
+<center>[[Image:SOCR_Activities_PowerTransformGraphing_Dinov_101609_Fig12.png|400px]]</center>
 <hr>

@@ Line 68: / Line 68: @@
 ==Example==
-The table below shows data for planetary distances from the Sun in terms of millions of miles (X) and light-years (Y). The X vs.Y scatterplot shows a positive association curving up.  Making an identify transformation of X (Transformed_X) and a power-transformation of Y for power-exponent=0.65 (Transformed_Y) linearizes the X vs. Y association, as shown on the image below.
+The table below shows data for [http://answers.yahoo.com/question/index?qid=20090311060915AALBs4f planetary distances from the Sun] in terms of millions of miles (X) and [http://en.wikipedia.org/wiki/Light-year light-years] (Y). The X vs.Y scatterplot shows a positive association curving up.  Making an identify transformation of X (Transformed_X) and a power-transformation of Y for power-exponent=0.65 (Transformed_Y) linearizes the X vs. Y association, as shown on the image below.
 <center>
@@ Line 93: / Line 93: @@
 | Pluto || 3666 || 247.68 || 3666.0 || 35.97734496244632
 |}
 <hr>

@@ Line 66: / Line 66: @@
 ** Try experimenting with other (real or simulated) data sets and different Power parameters (λ). What are the general effects of increasing/decreasing λ in any of these domains [-10;0], [0;1] and [1;10]?
 ** For each of the exercises (X-Y scatter-plot, QQ-Normal plot, Histogram plot and Time/Index plot) empirically study the effects of the power transform as a tool for normalizing the data. You can take samples of size 100 from Student’s T-distribution (low df) and determine appropriate levels of λ for which the transformed data is (visually) well approximated by a Normal Distribution.
 <hr>

@@ Line 19: / Line 19: @@
 ==Exercises==
-=== '''Exercise 1''': Power Transformation Family in a X-Y scatter Plot Setting===
+=== '''Exercise 1''': Power Transformation Family in a X-Y Scatter Plot Setting===
-* This exercise demonstrates the characteristics of the power-transform when applied independently to the two processes in an X-Y scatter plot setting. In this situation, one observed paired (X,Y) observations which are typically plotted X vs. Y in the 2D plane. We are interested in studying the effects of independently applying the power transforms to the X and Y processes. How and why would the corresponding scatter plot change as we vary the power parameters for X and Y?
+* This exercise demonstrates the characteristics of the power-transform when applied independently to the two processes in an X-Y scatter plot setting. In this situation, one observes paired (X,Y) values which are typically plotted X vs. Y in the 2D plane. We are interested in studying the effects of independently applying the power transforms to the X and Y processes. How and why would the corresponding scatter plot change as we vary the power parameters for X and Y?
-* First, point your browser to [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] and select the '''PowerTransformXYStatterChart''' (Line-Charts -> PowerTransformXYStatterChart). Then either use the default data provided for this chart, enter your own data (remember to '''MAP''' the data before your '''UPDATE''' the chart) or obtain SOCR simulated data from the '''Data-Generation''' tab of the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] (an example is shown later in Exercise 4). As shown on the image below, try changing the power parameters for the X and Y power-transforms and observe the graphical behavior of the transformed scatter-plot (blue points connected by a thin line) versus the native (original) data (red color points). We have applied a linear rescaling to the power-transform data to map it in the same space as the original data. This is done purely for visualization purposes, as without this rescaling it will be difficult to see the correspondence of the transformed and original data. Also note the changes of the numerical summaries for the transformed data (bottom text area) as you update the power parameters. What power parameters would you suggest that make the X-Y relation most linear?
+* First, point your browser to [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] and select the '''PowerTransformXYStatterChart''' (Line-Charts -> PowerTransformXYStatterChart). You may use the default data provided for this chart, enter your own data (remember to '''MAP''' the data before your '''UPDATE''' the chart), or obtain SOCR simulated data from the '''Data-Generation''' tab of the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] (an example is shown later in Exercise 4). As shown on the image below, try changing the power parameters for the X and Y power-transforms and observe the graphical behavior of the transformed scatter-plot (blue points connected by a thin line) versus the native (original) data (red color points). We have applied a linear rescaling to the power-transform data to map it in the same space as the original data. This is done purely for visualization purposes, as without this rescaling it will be difficult to see the correspondence of the transformed and original data. Also note the changes of the numerical summaries for the transformed data (bottom text area) as you update the power parameters. What power parameters would you suggest that make the X-Y relation most linear?
 <center>[[Image:SOCR_Activities_PowerTransformGraphing_Dinov_022007_Fig7.jpg|400px]]</center>
@@ Line 29: / Line 29: @@
 * The second exercise demonstrates the effects of the power-transform applied to data in a QQ-Normal plot setting. We are interested in studying the effects of power transforming the native (original) data on the quantiles, relative the Normal quantiles (i.e., QQ-Normal plot effects). How and why do you expect the QQ-Normal plot to change as we vary the power parameter?
-* Again go to [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] and select the '''PowerTransformQQNormalPlotChart''' (Line-Charts -> PowerTransformQQNormalPlotChart). You can use different data for this experiment - either use the default data provided with the QQ-Normal chart, enter your own data (remember to '''MAP''' the data before your '''UPDATE''' the chart) or obtain SOCR simulated data from the '''Data-Generation''' tab of the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] (an example is shown later in Exercise 4). Change the power-transform parameter (using the slider or the by typing int he text area) and observe the graphical behavior of the transformed data in the QQ-Normal plot (green points connected by a thin line) versus the plot of the native data (red color points). What power parameter would you suggest that make the (transformed) data quantiles similar to those of the Normal distribution? Why?
+* Again go to [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] and select the '''PowerTransformQQNormalPlotChart''' (Line-Charts -> PowerTransformQQNormalPlotChart). You can use different data for this experiment - either use the default data provided with the QQ-Normal chart, enter your own data (remember to '''MAP''' the data before your '''UPDATE''' the chart) or obtain SOCR simulated data from the '''Data-Generation''' tab of the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] (an example is shown later in Exercise 4). Change the power-transform parameter (using the slider or the by typing in the text area) and observe the graphical behavior of the transformed data in the QQ-Normal plot (green points connected by a thin line) versus the plot of the native data (red color points). What power parameter would you suggest that make the (transformed) data quantiles similar to those of the Normal distribution? Why?
 <center>[[Image:SOCR_Activities_PowerTransformGraphing_Dinov_022007_Fig8.jpg|400px]]</center>
@@ Line 35: / Line 35: @@
 * This exercise demonstrates the effects on the histogram distribution after applying the power-transform to the (observed or simulated) data. In this experiment, we want to see whether we can reduce the variance of a dataset and make its histogram more symmetric, unimodal and bell-shaped.
-* Again go to [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] and select the '''PowerTransformHistogramChart''' (Bar-Charts -> XYPlot -> PowerTransformHistogramChart). We will use SOCR simulated data from the '''Data-Generation''' tab of the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler], however you may chose to use the default data for this chart or enter your own data. The image below shows you the [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Distributions Generalized Beta Distribution] using [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions].
+* Again go to [http://www.socr.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] and select the '''PowerTransformHistogramChart''' (Bar-Charts -> XYPlot -> PowerTransformHistogramChart). We will use SOCR simulated data from the '''Data-Generation''' tab of the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler], however you may choose to use the default data for this chart or enter your own data. The image below shows you the [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Distributions Generalized Beta Distribution] using [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions].
 <center>[[Image:SOCR_Activities_PowerTransformGraphing_Dinov_022007_Fig9.jpg|400px]]</center>
-* Go to the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] and select 200 observations from the [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Distributions Generalized Beta Distribution] (<math>\alpha = 1.5; \beta = 3; A=0; B=7</math>), as shown on the image below. Copy these 200 values in your mouse buffer (CNT-C) and paste them in the '''Data''' tab of the '''PowerTransformHistogramChart'''. 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 analoge of the [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Distributions Generalized Beta] density curve above.
+* Go to the [http://www.socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] and select 200 observations from the [http://wiki.stat.ucla.edu/socr/index.php/About_pages_for_SOCR_Distributions Generalized Beta Distribution] (<math>\alpha = 1.5; \beta = 3; A=0; B=7</math>), as shown on the image below. Copy these 200 values in your mouse buffer (CNT-C) and paste them in the '''Data''' tab of the '''PowerTransformHistogramChart'''. 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 above.
 <center>[[Image:SOCR_Activities_PowerTransformGraphing_Dinov_022007_Fig10.jpg|400px]]</center>

@@ Line 11: / Line 11: @@
                  \left\{
                    \begin{matrix}
-                     {(y^{\lambda}-1)} / {\lambda} & \mathrm{for\ } \lambda \neq 0 \\ & \\
+                     {(y^{\lambda}-1) \over {\lambda} } & \mathrm{for\ } \lambda \neq 0 \\ & \\
                      \log{y} & \mathrm{for\ } \lambda = 0
                    \end{matrix}

@@ Line 7: / Line 7: @@
 The '''power transformation family''' is often used for transforming data for the purpose of making it more Normal-like. The power transformation is continuously varying with respect to the power parameter <math>\lambda</math> and defined, as continuous piece-wise function, for all <math>y>0</math> by
-<math>
+<center><math>
            y^{(\lambda)} =
                  \left\{
@@ Line 16: / Line 16: @@
                  \right.
 </math>
 ==Exercises==

@@ Line 6: / Line 6: @@
 == Background==
 The '''power transformation family''' is often used for transforming data for the purpose of making it more Normal-like. The power transformation is continuously varying with respect to the power parameter <math>\lambda</math> and defined, as continuous piece-wise function, for all <math>y>0</math> by
-<center><math>y^{(\lambda)} = \left \{ {(y^{\lambda}-1)} / {\lambda}, if \lambda \neq 0; and \log{y}, if \lambda = 0  \right\} </math> </center>
 ==Exercises==

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[http://davte.info/library/html/panasonic-cellulare.htm panasonic cellulare vs3] == [[SOCR_EduMaterials_Activities \| SOCR Educational Materials - Activities ]] - SOCR Power Transformation Family Graphing Activity ==	+	== [[SOCR_EduMaterials_Activities \| SOCR Educational Materials - Activities ]] - SOCR Power Transformation Family Graphing Activity ==

	== Summary ==		== Summary ==