SOCR EduMaterials ModelerActivities MixtureModel 1

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* '''Data Generation''': You typically have investigator-acquired data that you need to fit a model to. In this case we will generate the data by randomly sampling using the SOCR resource. Go to the SOCR [http://socr.stat.ucla.edu/htmls/SOCR_Modeler.html Modeler]  and select the '''Data Generation''' tab from the right panel.
* '''Data Generation''': You typically have investigator-acquired data that you need to fit a model to. In this case we will generate the data by randomly sampling using the SOCR resource. Go to the SOCR [http://socr.stat.ucla.edu/htmls/SOCR_Modeler.html Modeler]  and select the '''Data Generation''' tab from the right panel.
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<center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig1.jpg|400px]]</center>
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<center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig2.jpg|400px]]</center>
   
   
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Now, click the '''Raw Data''' check-box in the left panel, select '''Laplace Distribution''' (or any other distribution you want to sample data from), choose the '''sample-size''' to be 100 (keep the center, mu, at zero) and click '''Sample'''. Then go to the '''Data''' tab, in the right panel.
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**Now, click the '''Raw Data''' check-box in the left panel, select '''Laplace Distribution''' (or any other distribution you want to sample data from), choose the '''sample-size''' to be 100 (keep the center, mu, at zero) and click '''Sample'''. Then go to the '''Data''' tab, in the right panel. There you should see the 100 random Laplace observations stored as a column vector.
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** Next, go back to the '''Data Generation''' tab from the right panel and change the center of the Laplace distribution (set Mu=20, say). Click '''Sample''' again and you will see the list of randomly generated data in the '''Data''' tab expand to 200 (as you sampled another set of 100 random Laplace observations).
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* '''Exercise 2''': Now we'll learn how to use these Card experimental results to compute expectations, summarize the outcomes of a large numer of experiments and validate these '''random''' simulations with the corresponding theoretical probabilities.
 
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** First, select the [http://socr.stat.ucla.edu/htmls/SOCR_Experiments.html Card-Experiment]. Then we can run 10 (or more) experiments by clicking the <nowiki><RUN></nowiki> Button (<nowiki>>></nowiki>). The output may look like this (recall that the actual cards will be different each time you perform the experiment):
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* '''Exploratory Data Analysis (EDA)''': Go to the Data tab and select all observations in the data column (use CTR-A, or mouse-copy). Then open another web browser and go to SOCR [http://socr.stat.ucla.edu/htmls/SOCR_Charts.html Charts]. Choose '''HistogramChartDemo2''', say, clear the default data ('''Data''' tab) and paste in (CTR-V or mouse paste-in) the first column the 200 observations that you sampled in the Modeler (above). Then you need to '''map''' the values - go to the Mapping tab, select the first column where you pasted the data (C1) and click '''XValue'''. This will move the C1 column label from the right bin to the bottom-right bin. Finally, click Update Chart and go to the Graph tab to see your histogram of the 200 (bimodal) Laplace observations.
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig4.jpg|300px]]</center>
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<center>[[Image:SOCR_ModelerActivities_MixtureModelFit_Dinov_011707_Fig3.jpg|400px]]</center>
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** Now select with the mouse all outcome values in the eleven columns (recall that the first column is the index of the experiment, the remaining 2xN, in this case 10, columns represent the denomination (1<=Y<=13) and the suit (0<=Z<=3) of the cards in the randomly drawn hand). Save this selection (sky-blue highlight) in your mouse buffer, <nowiki><CNTR>+C, or <APPLE>+C</nowiki>. Open a new browser tab or window, envoke the [http://socr.stat.ucla.edu/htmls/SOCR_Charts.html SOCR Charts] and select the SOCR Area-Chart Demo:
 
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig5.jpg|300px]]</center>
 
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** Press the <nowiki><CLEAR></nowiki> Button to remove the default data and go to the <nowiki><DATA></nowiki> tab where you will click on the Bottom-Right (empty) cell and hit <nowiki><TAB></nowiki> key to open a new 11-th column. Finally Click on the Top-Left cell and press the <nowiki><PASTE></nowiki> Button to fill in the Card-Experiment results in to the data-table of the Area-Chart
 
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig9.jpg|300px]]</center>
 
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** Then y ou need to go to the <MAPPING> tab and map the Index column (1) onto '''Series''', and the rest of the columns should be mapped to '''Categories'''. Then click on <Update_Chart> Button and select the <GRAPH> Tab to see the visual depiction of the summary statistics for your results of the Card-Experiment.
 
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig8.jpg|250px]] [[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig11.jpg|250px]]</center>
 
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** The final result should look like this, where you can clearly identify the proportions, frequencies and distributions of all the denominations and suits of the randomly drawn hands of cards that you did in the Card-Experiment above.
 
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig7.jpg|300px]]</center>
 
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* '''Exercise 3''': Suppose we need to validate that a coin given to us is '''fair'''. We toss the coin 6 times independently and observe only one Head. If the coin was fair P(Head)=P(Tail)=0.5 and we would expect about 3 Heads and 3 Tails, right? Under these fair coin assumptions what is the ('''theoretical''') probability that only 1 Head is observed in 6 tosses? Use the [http://socr.stat.ucla.edu/htmls/SOCR_Experiments.html Binomial Coin Experiment] to:
 
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** Emprerically compute the odds (chances) of observing one Head in 6 fair-coin-tosses (run 100 experiments and record the number of them that contain exactly 1 Head);
 
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** Emperically estimate the Bias of the coin we have tested. Experiment with tossing 30 coins at a time. You should change the p-value=P(Head), run experiments and pick a value on the X-axis that the emperical distribution (red-histogram) peaks at. Perhaps you want this peak X value to be close to the observed 1-out-of-6 Head-count for the original test of the coin. Explain your findings!
 
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig2.jpg|300px]]</center>
 
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* '''Exercise 4''': In the SOCR [http://socr.stat.ucla.edu/htmls/SOCR_Experiments.html Ball and Urn Experiment] how does the distribution of the number of red balls (Y) depend on the sampling strategy (with or without replacement)? Do N, R and n also play roles?
 
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** Suppose N=100, n=5, R=30 and you run 1,000 experiments. What proportion of the 1,000 samples had zero or one red balls in them? Record this value.
 
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** Now run the Binomial Coin Experiment with n=5 and p= 0.3. Run the Binomial experiment 1,000 times? What is the proportion of observations that have zero or one head in them? Record this value also. How close is the proportion value you abtained before to this sample proportion value? Is there a reason to expect that these two quantities (coming from two distinct experiments and two different underlying probability models) should be similar? Explain.
 
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<center>[[Image:SOCR_Activities_CardCoinSampling_Dinov_092206_Fig3.jpg|300px]]</center>
 
<hr>
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Revision as of 17:35, 17 January 2007

SOCR Modeler Activities - SOCR Mixture Model Fitting Activity

This is a SOCR Activity that demonstrates random sampling and fitting of mixture models to data

  • Data Generation: You typically have investigator-acquired data that you need to fit a model to. In this case we will generate the data by randomly sampling using the SOCR resource. Go to the SOCR Modeler and select the Data Generation tab from the right panel.
    • Now, click the Raw Data check-box in the left panel, select Laplace Distribution (or any other distribution you want to sample data from), choose the sample-size to be 100 (keep the center, mu, at zero) and click Sample. Then go to the Data tab, in the right panel. There you should see the 100 random Laplace observations stored as a column vector.
    • Next, go back to the Data Generation tab from the right panel and change the center of the Laplace distribution (set Mu=20, say). Click Sample again and you will see the list of randomly generated data in the Data tab expand to 200 (as you sampled another set of 100 random Laplace observations).


  • Exploratory Data Analysis (EDA): Go to the Data tab and select all observations in the data column (use CTR-A, or mouse-copy). Then open another web browser and go to SOCR Charts. Choose HistogramChartDemo2, say, clear the default data (Data tab) and paste in (CTR-V or mouse paste-in) the first column the 200 observations that you sampled in the Modeler (above). Then you need to map the values - go to the Mapping tab, select the first column where you pasted the data (C1) and click XValue. This will move the C1 column label from the right bin to the bottom-right bin. Finally, click Update Chart and go to the Graph tab to see your histogram of the 200 (bimodal) Laplace observations.


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