7.
Substitute equation 7 into 1
8.
Substitute equation 7 into 2
9.
Subtract equation 8 multiplied by
and equation 9 multiplied by
10.
Substitute equation 10 and 7 into equation 3
11.
Substitute equation 10, 7 and 11 into equation 4
Substitute equation 11
Substitute
into equation 7
Substitute
into equation 11
Substitute
into equation 10
Another function that could model these points is an exponential.
Figure 2: Exponential Function: Applying Reflections and Translations
When an exponential is reflected on the x and y-axis, the shape is similar to the plotted points. It is then translated d-units up, where d would represent the horizontal asymptote. Since e, Euler’s number, is a transcendental constant, I defined a as e and added –k to create a similar graph.
Since we know that the d-value is 250,
Since this function has three variables, three points must be chosen. To determine the first point, the first four data points are averaged, then the next four points are averaged to find the second point and finally, the last three data points are averaged to find the third point.
Point 1:
=
Point 2:
=
Point3:
=
Subtract equation 2 from 1
Substitute
into equation 3
Substitute equation 4 and
into equation 1
46.469
Substitute
46.469 into equation 4
Figure 3: Exponential and Cubic Models
Both equations have their own limitations. The cubic function is not the best model because as
,
and as
,
the cubic function does not fit the range, but it fits the given data points well since the RMSE value for the cubic is 4.614. RMSE is the average magnitude of error; therefore a lower RMSE value is better. Also, for the exponential function as
,
and as
,
.
the exponential function is only a good model for the given data as well as predicting the heights in the future, but it only fits 2 of the data points. With an RMSE of 8.026, it is not as good of an indicator the selected data compared to the cubic.
A function that fits both the domain and range is a logistic. A Logistic Function can be defined simply as
. This is the optimal function to model this data because it creates an elongated s-curve.
Figure 4: Logistic Function
As
,
. This statement would be true considering how evolution has been occurring since humans first started to appear. Due to this, the height jumped would be increasing at an increasing rate up until evolution has finished or until the point where technology cannot further increase the result at an increasing the rate. After this point, the height jumped would be increasing, but at a decreasing rate.
As
,
, where a is the maximum height which humans can pass, but will approach it. Performing a regression with the 3 averaged points (1942, 200), (1962, 217.75) and (1976, 227.33) the following function was obtained:
Figure 5: Logistic Function
All of the functions were then graphed on logger pro
Figure 6: Cubic, Logistic and Exponential Models
All three functions present noticeably different graphs where, the logistic is in between.
To find the points for 1940 and 1944, a graph with a dip is necessary. Since World War 2 occurred throughout 1940 and 1944, the expected values would be lower than 1936 and 1948 due to rationing of food, casualties, loss of practice etc. which would thereby produce a dip. Therefore, the function
was defined and solved for using regression on Logger Pro. The following graph was obtained.
Figure 7:
Figure 7.1: Co-efficient
With an RMSE value of 3.93, it is a good model to evaluate points within this data set. Therefore, the height which may have been achieved in 1940 is 200cm and 198cm in 1944. Since 1984 is close to the data set, one can again use the sin function to solve for its predicted value since it has the lowest RMSE.
in 1984 the height which may be achieved will be 237cm. However, to predict the winning height for 2016, the logistic curve or the exponential curve would be used because they are a better indicator for predicting future values since they fit the range.
in 2016 the predicted height according to the exponential is 239cm and according to the logistic, it is 244cm.
Given the additional data below, a new scatter plot was created,
Figure 8: Logistic Model containing all data points Figure 8.1: Logistic Model Window
From 1896 to 2008, the height jumped has increased due to better equipment, training, technology etc. The scatter plot clearly shows an elongated “s-curve,” where its goes from increasing with an increasing rate of change to increasing with a decrease in rate of change. From 1896 to 1932, there was a slow increase. From 1948-1988 the rate of change was greater than before. Finally, from 1988 to 2008 there was a slight increase. Since, the logistic curve was plotted for the original data, it only took into account a small portion thereby producing a function which does not fit the overall model. To equate for these new points, changes must be made to the function. To create the s-curve the horizontal asymptotes could be set to 110 and 240
a vertical translation of 110 would be necessary. The value of ‘a’ would go down to 140. Also, it must then be horizontally compressed
b will go down as well since it’s negative.
This model can be applied to real-life situations where it can also be represented by a logistic function.
Figure 9: Release of Methane
The graph above displays methane emissions in parts per billion from 1978 to 2010. The concept is the exact same where the domain is
when B.C is considered negative. The range is also is
, where a is the maximum limit. It is impossible to produce less than 0 methane parts per billion and is impossible to produce over a certain value of a. Therefore, this graph would have two asymptotes, one a 0 and one at a. Also, in the future it is assumed that we will decrease or try to maintain a maximum level of methane since it is a leading cause in global warming. The models created were effective in visualizing the trend as well as predicting future results. Due to anomalies and factors affecting a person’s ability to jump therefore an exact model cannot be created to represent the data, only a relationship was made. An increase in time leads to an increase in height where it approaches a maximum height of “a.” This relationship held true for the specific and general case, but there may be variations. This investigation demonstrated how a portion of the whole can only determine the relationship, not the function. Therefore, one must make modifications to create a function for the general case. Understanding models involves finding the pattern. Moreover, by solving for the pattern, the general trend can therefore be established.
Works Cited
"High Jump World Records." Rob's Home of Sports, Fitness, Nutrition and Science. N.p., n.d. Web. 23 Feb. 2012. <http://www.topendsports.com/sport/athletics/record-high-jump.htm>.
"Logistic Equation -- from Wolfram MathWorld." Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. N.p., n.d. Web. 22 Feb. 2012. <http://mathworld.wolfram.com/LogisticEquation.html>.
"Logistic function - Wikipedia, the free encyclopedia." Wikipedia, the free encyclopedia. N.p., n.d. Web. 22 Feb. 2012. <http://en.wikipedia.org/wiki/Logistic_function>.
"Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE)." Eumetcal. N.p., n.d. Web. 23 Feb. 2012. <http://www.eumetcal.org/resources/ukmeteocal/verification/www/english/msg/ver_cont_var/uos3/uos3_ko1.htm>.
"NOAA/ESRL Global Monitoring Division - THE NOAA ANNUAL GREENHOUSE GAS INDEX (AGGI)." NOAA Earth System Research Laboratory. N.p., n.d. Web. 24 Feb. 2012. <http://www.esrl.noaa.gov/gmd/aggi/>.