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<h1>Random Walk</h1>
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<h2>Introduction</h2>
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<p>This lab features a random walk simulation that answers the following question: Given the radius of a circular room, how many random steps does it take to reach to reach the wall from the center? To explore this, we will look at the distribution of steps required and describe it in terms of a discrete probability distribution. Later on, we will look at the radius parameter and find relationships between it and the different summary statistics that we compute (mean, median, range, variance, skewness).</p>
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<p>The paper will start off with the methods used in this lab. The programming language R was used and a simulation was created to simulate a random walk according to the description above. It will also discuss the methods used to find the discrete probability distribution that best fits the data and how the relationship between the radii and the summary statistics were found using regression.</p>
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<p>After discussing the methods, the paper goes into the results. The discrete probability distribution of the data is revealed and the equations of the different summary statistics based on a regression of the radius is shared. </p>
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<h2>Methods</h2>
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<h3>Obtaining the distribution of steps required</h3>
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<p>The simulation written performs the following steps given the size of a room:</p>
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<ol>
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<li>Start off in the center of the room</li>
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<li>Calculate a random angle $\theta$</li>
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<li>Step in the X direction by $cos(\theta)$</li>
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<li>Step in the Y direction by $sin(\theta)$</li>
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<li>Repeat steps 2-4 until the wall is reached</li>
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</ol>
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<p>The number of steps required is then recorded into a vector and the simulation is performed 999 more times. This gives us the distribution of steps required to reach the wall given the room size.</p>
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<h3>Computing summary statistics</h3>
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<p>Given the distribution vector pertaining to the room size, we then find the following information:</p>
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<ul>
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<li>Measures of Central Tendency: Mean & Median</li>
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<li>Measures of Variation: Variance & Range</li>
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<li>Skewness & Shape of Histogram</li>
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</ul>
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<p>For every room size selected, a total of three trials of 1000 simulations each is conducted.</p>
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<h3>Fitting the data to a distribution</h3>
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<p>Since the simulation counts the steps required before reaching the walls of the room, it describes the process of a geometric distribution. To confirm the suspicion, the function <code>fitdistr</code> inside the package <code>MASS</code> was used to find the parameter (probability) of the geometric distribution to best fit the simulation data. In this case the simulation data used is of room size 100.</p>
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<p>The probability density histogram of the data is then shown with the overlay of the geometric distribution that best fit the data.</p>
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<p>Another more robust way to visually see the goodness of fit of the distribution is to use what is called a Quantile-Quantile Plot (Q-Q Plot), Using the quantiles from the theoretical distribution, we compare it to the quantiles of the simulated data and check to see that it follows the Q-Q line closely.</p>
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<h3>Finding relationships between the radii and the summary statistics</h3>
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<p>From the simulations performed, vectors have been saved containing the summary statistics for the different room sizes. Vectors are then created from these vectors as columns inside a data frame. Having this data frame allows us to use <code>ggplot</code> to graph the data using scatter plots. </p>
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<p><code>ggplot</code> was used as opposed to the conventional <code>base</code> package since it allows us to add layers to the graphs, which was used to show the geometric distribution on top of the density histogram in the previous section. <code>ggplot</code> is also aesthetically more pleasing as it decreases the margins and makes smart choices on the shadings of the axes on the graph. </p>
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<p>After plotting the data, regressions are then taken with respect to different polynomial degrees of radii and analyzed for best fit using the adjusted r squared value as a means for comparison. </p>
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<p>The adjusted r squared value is a good comparator since it tells us a proportion of how much the variation in the data is accounted for by the model.</p>
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<h2>Results</h2>
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<p>The shape of the distribution matches the geometric distribution as shown in Figure IX & X. Figure IX shows how the probability density histogram nicely fits the overlay of the geometric distribution with the fitted parameters. The description of the problem closely matches the description of the geometric probability distribution and both distributions are skewed right. These observations increase our confidence that the distribution is a good fit. This can further be verified with the Q-Q Plot. </p>
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<p>The Q-Q Plot in Figure X shows how the theoretical quantiles compare with the quantiles from the distribution. Since the observed sample quantiles start off in one side of the Q-Q line and ends up on the other side at the end, we can say that the data is not skewed with respect to the theoretical distribution. This along with the fact that the sample quantiles closely fit the Q-Q line supports the proposition that the geometric distribution is a good fit for the steps required in the random walk.</p>
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<p>Regressions were then performed for each summary statistic and the models that had the lowest degree polynomial with respect to radii and with relatively small standard error were chosen.</p>
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<p>For the mean, quadratic regression was performed and it produces the following model
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$$
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\mu(Steps) = -31.409 + 7.800(radii) + 0.837(radii^2)
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$$
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With this model, the adjusted $r^2$ value is 99.938%, which tells us that 99.938% of the variation in means is accounted for by the quadratic regression above. It's fit to the observed values can be seen in Figure XI.</p>
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<p>For the median, quadratic regression was performed and produces the following model
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$$
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\widetilde{Steps} = -19.525 + 5.096(radii) + 0.687(radii^2)
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$$
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The model above has an adjusted $r^2$ value of 99.957% which tells us that 99.957% of the variation in medians is accounted for by the regression above. It's fit to the observed values can be seen in Figure XIII.</p>
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<p>For the range, the quadratic regression performed creates the following model
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$$
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Range(Steps) = 63.873 - 9.073(radii) + 5.572(radii^2)
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$$
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The model has an adjusted $r^2$ value of 99.278% which tells us that 99.278% of the variation in the ranges of steps is accounted for by the regression above. It's fit to the observed values can be seen in Figure XV.</p>
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<p>For the variance, the cubic regression performed creates the following model
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$$
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\sigma^2(Steps) = -88471.866 + 32256.076(radii) - 2418.835(radii^2) + 61.650(radii^3)
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$$
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The model has an adjusted $r^2$ value of 99.779% which tells us that 99.779% of the variation in the variations of steps is accounted for by the regression above. It's fit to the observed values can be seen in Figure XVII</p>
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<p>Looking at the scatter plot of radii vs skewness of steps shown in Figure XVIII, there appears to be no relationship between the radii and skewness of steps. What the scatter plot suggests that the skewness is uniformly distributed.</p>
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<h2>Conclusions</h2>
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<p>In summation, the distribution of steps required to reach the end of the wall follow a geometric distribution. This was backed up using the probability density histogram and the Q-Q Plot.</p>
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<p>Most of the summary statistics follow a quadratic regression while the variation follows a cubic regression. Skewness are uniformly distributed across the different simulations.</p>
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<p>In other words, the mean, median, and range follow a function that is a second degree polynomial based on the size of the room ($r^2$) and the variation follows a function that is a third degree polynomial based on the size of the room $(r^3)$</p>
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<h2>Appendix</h2>
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<h3>Tables</h3>
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<h4>Table I, Summary Statistics for Room Radius of 2</h4>
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<table>
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<thead>
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<tr>
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<th>Trial</th>
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<th>Mean</th>
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<th>Median</th>
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<th>Variance</th>
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<th>Range</th>
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<th>Skewness</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>1</td>
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<td>5.827</td>
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<td>5.000</td>
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<td>13.471</td>
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<td>36.000</td>
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<td>2.446</td>
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</tr>
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<tr>
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<td>2</td>
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<td>5.761</td>
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<td>5.000</td>
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<td>10.635</td>
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<td>22.000</td>
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<td>1.780</td>
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</tr>
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<tr>
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<td>3</td>
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<td>5.805</td>
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<td>5.000</td>
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<td>11.430</td>
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<td>21.000</td>
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<td>1.882</td>
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</tr>
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</tbody>
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</table>
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<h4>Table II, Summary Statistics for Room Radius of 3</h4>
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<table>
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<thead>
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<tr>
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<th>Trial</th>
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<th>Mean</th>
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<th>Median</th>
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<th>Variance</th>
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<th>Range</th>
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<th>Skewness</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>1</td>
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<td>11.566</td>
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<td>9.000</td>
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<td>59.055</td>
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<td>60.000</td>
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<td>1.854</td>
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</tr>
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<tr>
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<td>2</td>
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<td>11.940</td>
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<td>10.000</td>
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<td>74.487</td>
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<td>68.000</td>
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<td>2.467</td>
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</tr>
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<tr>
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<td>3</td>
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<td>11.242</td>
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<td>9.000</td>
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<td>53.417</td>
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<td>48.000</td>
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<td>1.922</td>
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</tr>
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</tbody>
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</table>
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<h4>Table III, Summary Statistics for Room Radius of 4</h4>
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<table>
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<thead>
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<tr>
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<th>Trial</th>
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<th>Mean</th>
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<th>Median</th>
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<th>Variance</th>
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<th>Range</th>
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<th>Skewness</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>1</td>
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<td>19.306</td>
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<td>15.000</td>
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<td>171.688</td>
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<td>90.000</td>
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<td>1.721</td>
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</tr>
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<tr>
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<td>2</td>
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<td>18.127</td>
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<td>15.000</td>
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<td>143.284</td>
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<td>74.000</td>
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<td>1.712</td>
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</tr>
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<tr>
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<td>3</td>
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<td>18.993</td>
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<td>15.000</td>
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<td>180.892</td>
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<td>111.000</td>
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<td>2.089</td>
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</tr>
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</tbody>
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</table>
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<h4>Table IV, Summary Statistics for Room Radius of 5</h4>
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<table>
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<thead>
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<tr>
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<th>Trial</th>
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<th>Mean</th>
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<th>Median</th>
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<th>Variance</th>
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<th>Range</th>
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<th>Skewness</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>1</td>
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<td>29.272</td>
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<td>23.000</td>
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<td>405.817</td>
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<td>150.000</td>
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<td>1.737</td>
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</tr>
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<tr>
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<td>2</td>
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<td>28.382</td>
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<td>22.000</td>
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<td>402.801</td>
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<td>146.000</td>
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<td>1.862</td>
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</tr>
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<tr>
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<td>3</td>
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<td>27.891</td>
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<td>22.000</td>
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<td>390.221</td>
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<td>211.000</td>
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<td>2.308</td>
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</tr>
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</tbody>
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</table>
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<h4>Table V, Summary Statistics for Room Radius of 10</h4>
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<table>
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<thead>
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<tr>
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<th>Trial</th>
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<th>Mean</th>
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<th>Median</th>
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<th>Variance</th>
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<th>Range</th>
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<th>Skewness</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>1</td>
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<td>110.234</td>
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<td>87.000</td>
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<td>6070.976</td>
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<td>547.000</td>
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<td>1.780</td>
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</tr>
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<tr>
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<td>2</td>
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<td>109.594</td>
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<td>88.000</td>
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<td>5928.037</td>
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<td>542.000</td>
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<td>1.864</td>
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</tr>
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<tr>
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<td>3</td>
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<td>108.398</td>
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<td>86.000</td>
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<td>6469.467</td>
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<td>621.000</td>
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<td>2.191</td>
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</tr>
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</tbody>
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</table>
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<h4>Table VI, Summary Statistics for Room Radius of 25</h4>
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<table>
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<thead>
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<tr>
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<th>Trial</th>
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<th>Mean</th>
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<th>Median</th>
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<th>Variance</th>
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<th>Range</th>
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|
|
<th>Skewness</th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>1</td>
|
||
|
|
<td>623.260</td>
|
||
|
|
<td>492.000</td>
|
||
|
|
<td>197161.8</td>
|
||
|
|
<td>3262</td>
|
||
|
|
<td>1.965</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>2</td>
|
||
|
|
<td>637.890</td>
|
||
|
|
<td>511.000</td>
|
||
|
|
<td>212826.6</td>
|
||
|
|
<td>3394</td>
|
||
|
|
<td>2.009</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>3</td>
|
||
|
|
<td>653.661</td>
|
||
|
|
<td>528.5</td>
|
||
|
|
<td>211126.2</td>
|
||
|
|
<td>3694</td>
|
||
|
|
<td>1.841</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<h4>Table VII, Summary Statistics for Room Radius of 50</h4>
|
||
|
|
<table>
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th>Trial</th>
|
||
|
|
<th>Mean</th>
|
||
|
|
<th>Median</th>
|
||
|
|
<th>Variance</th>
|
||
|
|
<th>Range</th>
|
||
|
|
<th>Skewness</th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>1</td>
|
||
|
|
<td>2507.935</td>
|
||
|
|
<td>1939.0</td>
|
||
|
|
<td>3231980</td>
|
||
|
|
<td>11471</td>
|
||
|
|
<td>1.726</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>2</td>
|
||
|
|
<td>24675.615</td>
|
||
|
|
<td>1992.0</td>
|
||
|
|
<td>3248443</td>
|
||
|
|
<td>14150</td>
|
||
|
|
<td>2.060</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>3</td>
|
||
|
|
<td>2488.795</td>
|
||
|
|
<td>2001.5</td>
|
||
|
|
<td>3038915</td>
|
||
|
|
<td>14689</td>
|
||
|
|
<td>1.843</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<h4>Table VIII, Summary Statistics for Room Radius of 100</h4>
|
||
|
|
<table>
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th>Trial</th>
|
||
|
|
<th>Mean</th>
|
||
|
|
<th>Median</th>
|
||
|
|
<th>Variance</th>
|
||
|
|
<th>Range</th>
|
||
|
|
<th>Skewness</th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>1</td>
|
||
|
|
<td>8983.406</td>
|
||
|
|
<td>7257.0</td>
|
||
|
|
<td>38271895</td>
|
||
|
|
<td>52549</td>
|
||
|
|
<td>2.027</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>2</td>
|
||
|
|
<td>9372.917</td>
|
||
|
|
<td>7537.0</td>
|
||
|
|
<td>41658620</td>
|
||
|
|
<td>51810</td>
|
||
|
|
<td>1.867</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>3</td>
|
||
|
|
<td>8974.549</td>
|
||
|
|
<td>7294.5</td>
|
||
|
|
<td>41868765</td>
|
||
|
|
<td>60326</td>
|
||
|
|
<td>2.420</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<h3>Figures</h3>
|
||
|
|
<h4>Figure I, Histogram of Room Radius 2</h4>
|
||
|
|
<p><img src="data:image/png;base64,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
|
||
|
|
<h4>Figure II, Histogram of Room Radius 3</h4>
|
||
|
|
<p><img src="data:image/png;base64,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<h4>Figure III, Histogram of Radius 4</h4>
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<p><img src="data:image/png;base64,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
|
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<h4>Figure IV, Histogram of Radius 5</h4>
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<p><img src="data:image/png;base64,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<h4>Figure V, Histogram of Radius 10</h4>
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<p><img src="data:image/png;base64,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<h4>Figure VI, Histogram of Radius 25</h4>
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<p><img src="data:image/png;base64,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<h4>Figure VII, Histogram of Radius 50</h4>
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<p><img src="data:image/png;base64,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<h4>Figure VIII, Histogram of Radius 100</h4>
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<p><img src="data:image/png;base64,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<h4>Figure IX, Probability Density Histogram</h4>
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<p><img src="data:image/png;base64,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<h4>Figure X, QQ-Plot of Theoretical vs Sample Data</h4>
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<p><img src="data:image/png;base64,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<h4>Figure XI, Scatterplot of Radii vs Mean Steps</h4>
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<p><img src="data:image/png;base64,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<h4>Figure XII, Scatterplot with Quadratic Regression of Radii vs Mean Steps</h4>
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<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAABUAAAAPACAIAAAB7BESOAAAACXBIWXMAAB2HAAAdhwGP5fFlAAAgAElEQVR4nOzdd3xUVf7/8TuZ9J4QSEhDIEBCCE2QJr1KCyCoKBZQERRRmm1xf+7aRcEuriwKC+6KKJDQpUmXXnPoLZ000qdkZn5/3N35xpsQQtrMTV7PP3zcfO65kw8zk5j33HvP0VgsFgkAAAAAANg3B1s3AAAAAAAA7owADwAAAACAChDgAQAAAABQAQI8AAAAAAAqQIAHAAAAAEAFCPAAAAAAAKgAAR4AAAAAABUgwAMAAAAAoAIEeAAAAAAAVIAADwAAAACAChDgAQAAAABQAQI8AAAAAAAqQIAHAAAAAEAFCPAAAAAAAKgAAd5mBg8erLkTHx+fzp07jx8//u9//3tiYqJN+vzb3/4mN/PVV1+Vro8dO1aub9++vTJ1tVi2bFnpl+Dnn3+u5IEmk6lJkybWA3v06FGrfdoVnU63YsWKJ598Mjo6OiAgwNnZOSgoqFOnTrGxsUuXLs3JybF1gwAAAEB9QIC3a3l5ecePH//ll1/+3//7fy1btnz33Xdt3VGDs2rVqkqO/P333zMyMmq1Gfu0dOnS0NDQxx9/fPny5QkJCVlZWUajMT09/cSJE3FxcU8//XRQUND8+fNLSkrKPTw3Nzc3NzcvL6+O2wYAAABUx9HWDUDy9PT09fUtWzcajZmZmSaTyfrl/Pnz9Xr93//+97ptsEHbsGFDYWGhh4fHHUeuXr26DvqxN88///w333xj/VKr1YaGhgYFBaWnpyclJcmh3WAwvPvuuzt27NiyZYuXl1fpw00mk/zmDwoKSk1NrePmAQAAAHXhDLztTZs2LbE8aWlper3+ypUrs2bNcnZ2lge/995758+ft23Dsi5duowcOXLkyJEBAQGVqatUcXHx+vXr7zjMbDb/+uuvddCPXfn666+t6b1Tp04//fSTTqe7du3awYMHr169qtPp9u7dO2LECHnAgQMHZsyYYbtmAQAAANUjwNs1rVbbvHnzhQsXfvjhh3LFZDJ9//33tu1K9pe//CU+Pj4+Pr5Dhw6VqatOSEiIvFGZq+j37NmTnp5e+qh6T6/Xz58/X94eMGDAwYMHH3roIUfH/7uoR6vV9urVa/369QsXLpQry5cv37Rpkw16BQAAAOoFArw6vPTSS61atZK39+zZY9tmGoioqKi2bdtKkrRx48b8/PyKB1uvn3/wwQdrvTP7sHXrVnl2Oo1G8+9//9t6kUhZs2bNGj16tLy9ePHiOuoPAAAAqHcI8Oqg0Wg6d+4sb6elpdm2mYbjoYcekiRJp9NVfBW9xWKRr59v0qRJnz596qi5asjNzS0uLq7mgyQkJMgbUVFRTZo0qXhwbGysvHHixIlqft8qKy4uLiwstFgstmoAAAAAqCYCvGqEh4fLG05OTuUOOHv27Lx587p16xYUFOTi4tK0adOuXbtOmDAhLi7ObDZX/OC7du2aMmVKq1at3N3dAwICunTp8s4779y8ebOCQyZOnFjucnG3q9/OO++8I4/v1KlTBcOsq7uFhIQo/jnFxcWffvppbGxsu3btPDw8QkND+/TpM23atAsXLlSmgQpMmDBB3qj4Kvr9+/enpKRIkjRu3LjS15CXy2w2//zzz4888kjLli3d3d2DgoJ69eo1e/bsK1eu3LGfKrzE8+bNk5+3s2fPSpK0YsWKjh07+vr6uru7u7u7d+jQ4dFHHz1y5Mgdv3VZhYWF8kZlIvH999/fr1+/fv36tWvXTq68+eabGo3G+nSlpaXJffr5+ZU9/G6ftOTkZPnRnn32WUmSjh07NnToUF9fX09PTycnp9DQ0KFDh3733XcGg6GCnmvvfQUAAABUkQU2MmjQIPklmDt3bmXGP/zww/L4gQMHKnbpdLpnnnlGo9Hc7lWOiIg4efJkuQ+bn59vjakKAQEBGzZseOutt+Qvv/zyy9IHPvLII3J927ZtlanfTuk4dPny5dsNe+CBB+Qxr776aun6Dz/8EBgYWG7/Wq32+eefLykpqUwbpR9QPnzQoEEWiyU6OlqSJBcXl9zc3Nsd8vLLL1v/yWvXrpW3u3fvXnbk2bNnrVdSKDg4ODzzzDN6vb7cb1Hll3ju3LnygDNnzsyePbvcYzUazcyZM+/qWSr9RDk4OOzbt+9uD7feP6/g6+urGFmFJy0pKUke8Mwzz/z444+3+8wrMjLyxIkTt/vX1ez7CgAAAKg+zsCrg8lkOnr0qLzdt29fxd6nnnpqyZIlFotFkiRXV9fo6Oh+/fp16NDB29tbHnDp0qURI0YUFBQoDtTpdEOGDPn555/lLzUaTZs2bbp169a4cWNJkjIzMx988MEDBw7U3r9LkqRWrVpZ49maNWvKHZOTk7Nt2zZ5+4knnrDWV65c+dRTT8mzx2m12jZt2vTr1y8mJsbFxUWSJJPJ9PXXX98uKFaS/OmGXq+Pj48vd4DFYvnll18kSQoICOjXr18FD3X8+PHevXsfO3ZM/tLDw6Nbt26tW7d2cHCQJMlsNi9ZsmTYsGFlXyapGi+x1fLly+XJ5FxdXXv06PHkk0/26dNHPgFusVg+//xzayCvpO7du1s7j42NXbp0qXXJw8oYPXr04sWLv/rqK/lLb2/vxYsXL1682Drjnaw6T5okSSdPnpwyZYrRaJS/bNmyZadOndzd3eUvz507179//9OnTyuOqoP3FQAAAFAVtv38oCG7qzPwH3zwgTzY3d09LS2t9K7ff//d+mq++uqrOTk51l06nW7JkiXW2cXi4uIUD2s9PStJ0oABA65duybXTSbTr7/+qjgDWUtn4C0Wi3WO/V69epU7YMmSJfKArl27WosGg0H+oEGSpNjY2OvXr1t3paenWy8rcHFx0el0lezEUuYMvHzluSRJo0ePLne89QOOqVOnWiyW252BLygosE5DGB4evmHDBpPJJO/Ky8ubN2+enEglSZo1a5biW9TISyyfvR87dmzpSwnOnDnTpk0beUC7du0q/yzJnnvuudLvkMaNG0+dOnXt2rWlO6yYvFC8JElBQUFl91b5SbOegZc5OTm9//776enp1of95JNPXF1d5b0dOnQwGo3WY2vpfQUAAABUHwHeZu4Y4I1G4/Xr13fs2DF27Fh5pKOj48aNGxXD3nzzTXnvhAkTyn2c1157TR7wl7/8pXQ9KSnJegfy5MmTS2cYmRDCy8vLmoJqL8BfvXrVGjJTUlLKDhgyZEjZHqyxNjAw0GAwKA7R6XTWheiPHDlSyU4sZQK85X9X0Ts7O9+6davs+Dlz5sjjf/vtN8vtA/w777wj15s1a1Zuvl2+fLk8QKvVnjlzpvSuKr/Elj9/RvPwww+XPfbQoUPW7yvP8VZ5hYWFjz32mFSGg4NDly5dXnnllc2bNxcUFFTwCBUH+Co/aYoAv3z58rLHbtmyxXpLwrJly6z1WnpfAQAAANXHJfS29/HHH2vK4+Tk1KxZswEDBsgXlrdu3Xrbtm3WW8GtDAZDx44dO3bsOGXKlHIfv0uXLvKGfEmw1T//+U85Pvn5+X366adlZ1+LjIycNm1ajfwbK3bPPfd0795dkiSLxbJu3TrF3szMzB07dkiS5OzsPHHiRGs9IyND3oiIiCh7k7OLi8uHH344f/78+fPn3+4W6EqS56I3GAxxcXFl98oLyDVq1KiC6+fli67l7SVLlvj6+pYd8/jjjw8ePFgebF2UTlbll7g0Z2dn63UcpXXt2lW+1MJkMqWmpt7u8HK5u7uvWLHil19+GT58uFartdbNZvORI0c++uijYcOG+fn5DRw4cOXKlTqd7q4evJpPmlWXLl0ef/zxsvUhQ4aMGTNG3v7uu++s9Tp7XwEAAAB3iwCvDlqt9pVXXil3ibIPPvjg+PHjx48fHzZsWLnH3m7ZOess8bNnz7beSq3w0ksv1U1KsU7RV/Y2+F9//VX+oGHUqFH+/v7WeosWLeSNI0eO7N27t+xjTpky5e2333777bfbt29fnd4qmIv+8OHD169flyRp7NixFcw/f+rUKXma
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<h4>Figure XIII, Scatterplot of Radii vs Median Steps</h4>
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<p><img src="data:image/png;base64,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<h4>Figure XIV, Scatterplot with Quadratic Regression of Radii vs Median Steps</h4>
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<p><img src="data:image/png;base64,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<h4>Figure XV, Scatterplot of Radii vs Range of Steps</h4>
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<p><img src="data:image/png;base64,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<h4>Figure XVI, Scatterplot with Quadratic Regression of Radii vs Range of Steps</h4>
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<p><img src="data:image/png;base64,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<h4>Figure XVII, Scatterplot of Radii vs Variance of Steps</h4>
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<p><img src="data:image/png;base64,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<h4>Figure XVIII, Scatterplot with Cubic Regression of Radii vs Variance of Steps</h4>
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<p><img src="data:image/png;base64,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
|
||
|
|
<h4>Figure XIX, Scatterplot of Radii vs Skewness</h4>
|
||
|
|
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAABUAAAAPACAMAAADDuCPrAAADAFBMVEUAAAABAQECAgIDAwMEBAQFBQUGBgYHBwcICAgJCQkKCgoLCwsMDAwNDQ0ODg4PDw8QEBARERESEhITExMUFBQVFRUWFhYXFxcYGBgZGRkaGhobGxscHBwdHR0eHh4fHx8gICAhISEiIiIjIyMkJCQlJSUmJiYnJycoKCgpKSkqKiorKyssLCwtLS0uLi4vLy8wMDAxMTEyMjIzMzM0NDQ1NTU2NjY3Nzc4ODg5OTk6Ojo7Ozs8PDw9PT0+Pj4/Pz9AQEBBQUFCQkJDQ0NERERFRUVGRkZHR0dISEhJSUlKSkpLS0tMTExNTU1OTk5PT09QUFBRUVFSUlJTU1NUVFRVVVVWVlZXV1dYWFhZWVlaWlpbW1tcXFxdXV1eXl5fX19gYGBhYWFiYmJjY2NkZGRlZWVmZmZnZ2doaGhpaWlqampra2tsbGxtbW1ubm5vb29wcHBxcXFycnJzc3N0dHR1dXV2dnZ3d3d4eHh5eXl6enp7e3t8fHx9fX1+fn5/f3+AgICBgYGCgoKDg4OEhISFhYWGhoaHh4eIiIiJiYmKioqLi4uMjIyNjY2Ojo6Pj4+QkJCRkZGSkpKTk5OUlJSVlZWWlpaXl5eYmJiZmZmampqbm5ucnJydnZ2enp6fn5+goKChoaGioqKjo6OkpKSlpaWmpqanp6eoqKipqamqqqqrq6usrKytra2urq6vr6+wsLCxsbGysrKzs7O0tLS1tbW2tra3t7e4uLi5ubm6urq7u7u8vLy9vb2+vr6/v7/AwMDBwcHCwsLDw8PExMTFxcXGxsbHx8fIyMjJycnKysrLy8vMzMzNzc3Ozs7Pz8/Q0NDR0dHS0tLT09PU1NTV1dXW1tbX19fY2NjZ2dna2trb29vc3Nzd3d3e3t7f39/g4ODh4eHi4uLj4+Pk5OTl5eXm5ubn5+fo6Ojp6enq6urr6+vs7Ozt7e3u7u7v7+/w8PDx8fHy8vLz8/P09PT19fX29vb39/f4+Pj5+fn6+vr7+/v8/Pz9/f3+/v7////isF19AAAACXBIWXMAAB2HAAAdhwGP5fFlAAAgAElEQVR4nOzdCZwbdd3H8d/2PjjksgjILacKWBQQUcol17SUUqEtD1e5QQQ5FUSgIBRBUAEVKAiUQwQERFCRQxARATkEylFokVIo2e19d7vzTM5NttnvNv9/Zmfoft6v10Mmk0nyczv7eZJNMrEQAODEkh4AAD6tCCgAOCKgAOCIgAKAIwIKAI4IKAA4IqAA4IiAAoAjAgoAjggoADgioADgiIACgCMCCgCOCCgAOOqUgO5hrVbZbtiF/1vO611gdk329ACzv5Wf+lhw22FbrdFzwLaDx00vrupjK/neaiyqjAogTTo9oFk9L16+68UQ0HFrlKbode6S/LqUBrTaqOHMmbMSHQpAmc4K6Err5Q3onivCj5brevUP6AnZO+++wQ4b9sgu7DQ7tzKdAa06arPZ2gnPBaCkswJ6RnG5+b3TekVheHN5rlcM6MX77/9y+amza6MUbfe77KO55n/sFy0fllubyoBWH5WAAmnS6QGNXBX14OzluV4xoPWycDWz3RYVz/0sGuPh7EIaA9rOqAQUSJMkAtryBbOvL8/16h3QB80aprWeHWw2OHuaxoC2MyoBBdIkiYCGB5ttvDzXq3dALzPbquzsOLP1s6d1CejM+f63UaadUZcnoPPnttR1FADtSSSgZ5ptXlh87YyvDei19vYHPbC09eInjty07xoDx0wrBfSQwotHh7R5EWmM2balM781Wyd3I/OvGrx1v3V3Oe6tZeb4kdmWZWff2nXXfbOnpYAu3t/sc+/kFpfeffDGfQd8/bR385dEyT81v3Rj9Gz65/nFc812CsMzzF4Lb9vGrO+XRzzfeuNtbqD6YDWOel7hRfnPVL+LKWZHhy/u1cu6r7vX9aWn/+JOAPhJ6hHo7rmFhUc3FN+ps+krhQvnDC+sWfNPHQX07WirUqD2Kfxh9bcDCtfvfmJzmzmixnZ7ZtnxigFdMsxswITc4utfKdxIt6NzJbrZ7Mv5jQ+PVh6YX/yG2UX5gH6/sHXDKcXbbHsDVQerddTKgC5zF9mA3tGzsHKL4stt4k4A+EkioM2b5soT5opofbbedZtVotP15uRWLdgpG6LNd1gruuTbHQQ0jBJyRWFxelSO16PT8dlUbL7rl3pHp+e0mePNblGWxy2TkUJAl46ILn0tt+Y/q0fX7r/DZtH2Nig719Ropkzuoo2zac89R57fy+z5XEDPimbd6fBvZt9vdHNY/QaqDVbzqP/+9bVmq/z61zdVv4sooF/tE53bZLt+0X9Xe7XDOwHgJ4mAXmbW7+Pswt+zL8fPiBYW3hjF6MHchVGPbLfJUc7uyz1y0gEda7ZzYTF6av3V6GRxFN4h70cL06IHsr0XthnkuOxNrnXs/TMq1uYD2hI9tlw9/7Bt7hfM1v/T0jCcfWaUp9Oyq6Kn6PdkTz/MPZjLljp8LLqpltzADTY0+/b21zY3+2J7N7DsYC6jtv4NtMpdTMlepeel06ILr4xCus2SDu8EgJfODuiS9x8fatYj956c7B/6hhfWn2N2bvZ0SvQw7sj8x24mrNxhQCdF8ZqaX9wrv23U5AGLcysWrmn2QptB5o0qPOnd/qw/zy2tzQW05VizVQvbX2y2QSFct0aP37KPSs82Oyl7/i6zgWbXZRfPN/u/MF/8g/Mb/zvaeF47N7DsYC6jtga0yl3kAnprft1fGsxu6fBOAHhJ5KOcmz2ZX3/2tts+UtjknuwLIJELo+eexQ8rntlhQMMdzX6VW8j0sF5N+dspPiYdd955r4Rt3btv98IQPXcbvyC/LhfQ75qt/K/8+eZ1zB4tXmFPswuikycKr4qfZPYHs0Oyi980uyPMBbTXpMLG0YPmie3cwLKDuYxaCmi1u8gGdPviuuj/TX1jOe4EgIckAtr9xmXfZ3NNIaBRlcYU103p2WFArzLbK7fwG7Nh2dP/RE9Un5bDfHL3CVsUBvlsPkHZgGYfR/6usEV0G58vbX6j2a7RyeLo4XD2zw5fts0X97fPRUsLelu3xjAX0P2KG38tH9BqN7DsYC6jlgJa7S6yAb237Nazj4Y7vhMAzjr7s/Dr9s3mYPdFFZcveurHqxcC2t/s76X123UY0CkN1jP3RHa3wt9QF20aheOQ3zfqgT6689h1cyn/bfZcFNDc69tDCpdeZzaytOlzlstlOMTsrjCc0S0acy+zt3OPSXfMXhAF9PvFjXfMB7TaDSw7mMuopYBWu4tsQKeWVq5i9szy3gkAJ53+IlLL/06NnpmW3u7z36uP32uT3OEycgFdEJ1+WLre0A4Dmn3Eelt0Mq27rZX/y+kLa2Vvq2HLo+6eI2dqeSaItlvlozAb0OyL1tH/PZS/qPhuoaLca/S/MjsuDB/K/pHxJ2Y35t7lf2H2giigvyjeaCGgVW9g2cEcRi0FtNpdRAHt0/rYfrv8w9HlvRMAtUviVfizzNbMv2/+kS8Wfv83/XY+oI3R73rrW+pP7Tig15oNzZ8U3ugezjyj8MbH3od/oseKnvbbD8N8QNeduK/ZRvm/NJ7cJk4N2SpNMtss91rSpPCfuVePdjX7d3brKKC/Lt5iIaBVb6DKYLWPWgpotbuIArph61UGm42r7U4A1CiJgDY1ZD+8E7k5Wui961m3PTc9vD8f0MUVj0BP7jig0UPPvvPC8FtmL5XWtfz7gt37Z5Ox+sSKbV/ae+9jK1bsZ5b9KFIU0HXeDt+NTs7PrY4Cv9+vy+Xejrm52ZTw67ZeNGR/Wz9c2MfWyqW+SkCr30C1wWodtRTQandR+Qj0a7m/Oag7AeApkU8irZ3/Q2dT9Fs9qpDLQkDD1cr/BrpXxwHN3vZ94dRuxQ8KFS3556lREHerWPeWWf+KFWPMvhRmA9oje3S96Dl579znOMeaHb/s/4jvmY1f0MtG5Oea/HezQ3PrqwS0+g20N1gto5YCWu0uKv8GulrFD6vanQDwlEhA
|
||
|
|
<h3>R Code</h3>
|
||
|
|
<pre><code class="language-R">rm(list=ls())
|
||
|
|
library(moments)
|
||
|
|
library(ggplot2)
|
||
|
|
|
||
|
|
simulateRoom = function(r) {
|
||
|
|
n = 1000
|
||
|
|
results = numeric(n)
|
||
|
|
x = 0
|
||
|
|
y = 0
|
||
|
|
dist = 0
|
||
|
|
num = 0
|
||
|
|
for (i in 1:n) {
|
||
|
|
while(dist < r) {
|
||
|
|
deg = runif(1, min=0, max=360)
|
||
|
|
newx = cos(deg)
|
||
|
|
newy = sin(deg)
|
||
|
|
x = x + newx
|
||
|
|
y = y + newy
|
||
|
|
num = num+ 1
|
||
|
|
dist=(x^2 + y^2)^0.5
|
||
|
|
}
|
||
|
|
results[i] = num
|
||
|
|
x= 0
|
||
|
|
y = 0
|
||
|
|
dist = 0
|
||
|
|
num = 0
|
||
|
|
i = i + 1
|
||
|
|
}
|
||
|
|
return(results)
|
||
|
|
}
|
||
|
|
|
||
|
|
repetitions = 3
|
||
|
|
radii = c(2, 3, 4, 5, 10, 25, 50, 100)
|
||
|
|
|
||
|
|
median_stepcount = c()
|
||
|
|
mean_stepcount = c()
|
||
|
|
variance_stepcount = c()
|
||
|
|
range_stepcount = c()
|
||
|
|
skewness_stepcount = c()
|
||
|
|
for (r in radii) {
|
||
|
|
room_data = data.frame()
|
||
|
|
print(paste("Currently simulating r =", r))
|
||
|
|
for (i in 1:repetitions) {
|
||
|
|
simulation = simulateRoom(r)
|
||
|
|
|
||
|
|
## Summary Statistics
|
||
|
|
variance = var(simulation)
|
||
|
|
skewness_ = skewness(simulation)
|
||
|
|
range_data = range(simulation)
|
||
|
|
|
||
|
|
room_data_temp = data.frame(i, mean(simulation), median(simulation), variance, range_data[2] - range_data[1], skewness_)
|
||
|
|
names(room_data_temp) = c("Trial", "Mean", "Median", "Variance", "Range", "Skewness")
|
||
|
|
room_data = rbind(room_data, room_data_temp)
|
||
|
|
|
||
|
|
median_stepcount = c(median_stepcount, median(simulation))
|
||
|
|
mean_stepcount = c(mean_stepcount, mean(simulation))
|
||
|
|
variance_stepcount = c(variance_stepcount, variance)
|
||
|
|
range_stepcount = c(range_stepcount, range_data[2] - range_data[1])
|
||
|
|
skewness_stepcount = c(skewness_stepcount, skewness_)
|
||
|
|
|
||
|
|
simulation = data.frame(simulation)
|
||
|
|
print(ggplot(simulation, aes(x = simulation)) +
|
||
|
|
geom_histogram(color = 'black', fill = 'white') +
|
||
|
|
xlab("Steps") +
|
||
|
|
ylab("Count") +
|
||
|
|
ggtitle(paste("Distribution of r =", r)) +
|
||
|
|
theme_bw())
|
||
|
|
}
|
||
|
|
print(room_data)
|
||
|
|
}
|
||
|
|
|
||
|
|
## Show the probability distribution
|
||
|
|
library(MASS)
|
||
|
|
distribution = fitdistr(simulation$simulation, "geometric")
|
||
|
|
probability_simulation = data.frame(simulation,
|
||
|
|
rgeom(1:1000, distribution$estimate['prob']))
|
||
|
|
names(probability_simulation) = c("simulation", "geometric")
|
||
|
|
ggplot(data = probability_simulation, aes(x = simulation)) +
|
||
|
|
geom_histogram(aes(y = ..density..), color = 'black', fill = 'white') +
|
||
|
|
geom_density(aes(x = geometric), fill = '#0000AA', alpha = .5) +
|
||
|
|
xlab("Steps") +
|
||
|
|
ylab("Probability Density") +
|
||
|
|
ggtitle("Probability Density Histogram with Geometric Overlay") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Calculate QQ Plot of Geometric Distribution and Sample
|
||
|
|
yquantiles = quantile(simulation$simulation, c(0.25, 0.75))
|
||
|
|
xquantiles = qgeom(prob = distribution$estimate['prob'], c(0.25, 0.75))
|
||
|
|
slope = diff(yquantiles) / diff(xquantiles)
|
||
|
|
intercept = yquantiles[1] - slope * xquantiles[1]
|
||
|
|
ggplot() + aes(sample=simulation$simulation) +
|
||
|
|
stat_qq(distribution=qgeom, dparams = distribution$estimate['prob']) +
|
||
|
|
geom_abline(intercept=intercept, slope=slope) +
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ggtitle("QQ Plot of Geometric Distribution") +
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xlab("Theoretical Quantiles") +
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ylab("Sample Quantiles") +
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theme_bw()
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## Create a dataframe
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room_simulation_data = data.frame(as.vector(sapply(radii, function(x) { rep(x, repetitions)})), mean_stepcount, median_stepcount, range_stepcount, skewness_stepcount, variance_stepcount)
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names(room_simulation_data) = c("radii", "mean_stepcount", "median_stepcount", "range_stepcount", "skewness_stepcount", "variance_stepcount")
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######### ANALYSIS OF MEANS
|
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|
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ggplot(room_simulation_data, aes(x = radii, y = mean_stepcount)) +
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geom_point(shape = 21, size = 2) +
|
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xlab("Radii") +
|
||
|
|
ylab("Mean Steps") +
|
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|
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ggtitle("Radii vs Mean Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Quadratic Regression of Means
|
||
|
|
quadratic_mean_model = lm(mean_stepcount ~ poly(radii, 2, raw = T), data = room_simulation_data)
|
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|
|
quadratic_mean_model_summary = summary(quadratic_mean_model)
|
||
|
|
ggplot(room_simulation_data, aes(x = radii, y = mean_stepcount)) +
|
||
|
|
geom_point(shape = 21, size = 2) +
|
||
|
|
geom_smooth(method = "lm", formula = y ~ poly(x, 2, raw = T), se = T) +
|
||
|
|
xlab("Radii") +
|
||
|
|
ylab("Mean Steps") +
|
||
|
|
ggtitle("Radii vs Mean Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Analysis of Quadratic Regression of Means
|
||
|
|
quadratic_mean_coefficients = as.vector(quadratic_mean_model$coefficients)
|
||
|
|
print(paste("The quadratic regression equation is: Mean Steps =", quadratic_mean_coefficients[1], "+", quadratic_mean_coefficients[2], "radii", quadratic_mean_coefficients[3], "radii^2"))
|
||
|
|
print(paste(quadratic_mean_model_summary$adj.r.squared * 100, "% of the variation of the means is accounted for by the Quadratic Model", sep = ""))
|
||
|
|
|
||
|
|
######### ANALYSIS OF MEDIANS
|
||
|
|
ggplot(room_simulation_data, aes(x = radii, y = median_stepcount)) +
|
||
|
|
geom_point(shape = 21, size = 2) +
|
||
|
|
xlab("Radii") +
|
||
|
|
ylab("Median Steps") +
|
||
|
|
ggtitle("Radii vs Median Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Quadratic Regression of Medians
|
||
|
|
quadratic_median_model = lm(median_stepcount ~ poly(radii, 2, raw = T), data = room_simulation_data)
|
||
|
|
quadratic_median_model_summary = summary(quadratic_median_model)
|
||
|
|
ggplot(room_simulation_data, aes(x = radii, y = median_stepcount)) +
|
||
|
|
geom_point(shape = 21, size = 2) +
|
||
|
|
geom_smooth(method = "lm", formula = y ~ poly(x, 2, raw = T), se = T) +
|
||
|
|
xlab("Radii") +
|
||
|
|
ylab("Median Steps") +
|
||
|
|
ggtitle("Radii vs Median Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Analysis of Quadratic Regression of Medians
|
||
|
|
quadratic_median_coefficients = as.vector(quadratic_median_model$coefficients)
|
||
|
|
print(paste("The Quadratic Regression Equation is: Median Steps =", quadratic_median_coefficients[1], "+", quadratic_median_coefficients[2], "radii", quadratic_median_coefficients[3], "radii^2"))
|
||
|
|
print(paste(quadratic_median_model_summary$adj.r.squared * 100, "% of the variation of the medians is accounted for by the Quadratic Model", sep = ""))
|
||
|
|
|
||
|
|
######### ANALYSIS OF RANGES
|
||
|
|
ggplot(room_simulation_data, aes(x = radii, y = range_stepcount)) +
|
||
|
|
geom_point(shape = 21, size = 2) +
|
||
|
|
xlab("Radii") +
|
||
|
|
ylab("Range of Steps") +
|
||
|
|
ggtitle("Radii vs Range Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Quadratic Regression of Ranges
|
||
|
|
quadratic_range_model = lm(range_stepcount ~ poly(radii, 2, raw = T), data = room_simulation_data)
|
||
|
|
quadratic_range_model_summary = summary(quadratic_range_model)
|
||
|
|
ggplot(room_simulation_data, aes(x = radii, y = range_stepcount)) +
|
||
|
|
geom_point(shape = 21, size = 2) +
|
||
|
|
geom_smooth(method = "lm", formula = y ~ poly(x, 2, raw = T), se = T) +
|
||
|
|
xlab("Radii") +
|
||
|
|
ylab("Range of Steps") +
|
||
|
|
ggtitle("Radii vs Range of Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Analysis of Quadratic Regression of Ranges
|
||
|
|
quadratic_range_coefficients = as.vector(quadratic_range_model$coefficients)
|
||
|
|
print(paste("The Quadratic Regression Equation is: Range of Steps =", quadratic_range_coefficients[1], "+", quadratic_range_coefficients[2], "radii", quadratic_range_coefficients[3], "radii^2"))
|
||
|
|
print(paste(quadratic_range_model_summary$adj.r.squared * 100, "% of the variation of the ranges is accounted for by the Quadratic Model", sep = ""))
|
||
|
|
|
||
|
|
######### ANALYSIS OF SKEWNESS
|
||
|
|
ggplot(room_simulation_data, aes(x = radii, y = skewness_stepcount)) +
|
||
|
|
geom_point(shape = 21, size = 2) +
|
||
|
|
xlab("Radii") +
|
||
|
|
ylab("Skewness") +
|
||
|
|
ggtitle("Radii vs Skewness Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
######### ANALYSIS OF VARIANCES
|
||
|
|
ggplot(room_simulation_data, aes(x = radii, y = variance_stepcount)) +
|
||
|
|
geom_point(shape = 21, size = 2) +
|
||
|
|
xlab("Radii") +
|
||
|
|
ylab("Variance of Steps") +
|
||
|
|
ggtitle("Radii vs Variance Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Cubic Regression of Variance
|
||
|
|
cubic_variance_model = lm(variance_stepcount ~ poly(radii, 3, raw = T), data = room_simulation_data)
|
||
|
|
cubic_variance_model_summary = summary(cubic_variance_model)
|
||
|
|
ggplot(room_simulation_data, aes(x = radii, y = variance_stepcount)) +
|
||
|
|
geom_point(shape = 21, size = 2) +
|
||
|
|
geom_smooth(method = "lm", formula = y ~ poly(x, 3, raw = T), se = T) +
|
||
|
|
xlab("Radii") +
|
||
|
|
ylab("Variance of Steps") +
|
||
|
|
ggtitle("Radii vs Variance of Steps") +
|
||
|
|
theme_bw()
|
||
|
|
|
||
|
|
## Analysis of Cubic Regression
|
||
|
|
cubic_variance_coefficients = as.vector(cubic_variance_model$coefficients)
|
||
|
|
print(paste("The Cubic Regression Equation is: Variance of Steps =", cubic_variance_coefficients[1], "+", cubic_variance_coefficients[2], "radii", cubic_variance_coefficients[3], "radii^2", cubic_variance_coefficients[4], "radii^3"))
|
||
|
|
print(paste(cubic_variance_model_summary$adj.r.squared * 100, "% of the variation of the variations is accounted for by the Cubic Model", sep = ""))</code></pre>
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