20-5= 15
15 x 2 = 30
b being the period, ill substitute 30 into the equation of sin x = 2π÷b.
2π÷ 30 = 0.2094
B= 0.2094
C= The phase shift of the function is C÷ B. A positive phase shift will mean the graph moves to the right while a negative phase shift means the graph moves to the left.
The maximum points of y=sin x are at 2nπ + π÷2. So I’ll try to locate that point on the data. The minimum points are at 2nπ + 3π÷2 and the x intercepts are nπ.
First, we need to find the difference between the maximum (20) and minimum (5) values, 20-5=15
15÷2=7.5
The half of the difference is 7.5, then we add to the minimum (5)
7.5+5=12.5
12.5 is the period end in a normalized sine curve. The whole period is 30 in a complete sine curve, so the offset is 30- 12.5 which is 17.5.
We then need to multiply 17.5 by the stretch (B) to normalize the curve, which gives us C.
20-5 = 15
15÷2 = 12.5
12.5 + 5 = 17.5
17.5 x 0.2094 = 3.664
C = 3.664
D= the amount of vertical displacement which shows how the function graph moves up by using the midpoint formula
Middle point of graph is calculated by (Maximum + Minimum) ÷ 2
(21.65+15.20)÷2
D=18.425
Next to check the accuracy between my calculated values and the actual values of the graph modeled from the data, I will plot both my model and the data’s model onto a graph.
RMSE or the root means square error is a measure of the difference between the values predicted by a model (the graph in this case) and the actual values from the thing being modeled. RMSE is a good indicator of the accuracy and the individual differences between two models.
To check the values I got I used a Regression Curve fit to check.
A=3.123
B=0.2178
C= 3.970
D= 18.43
RMSE= 0.970936
The Regression Curve fit graph with the original graph from data
After I looked at the Regression Curve RMSE, I refined my model to
A=3.129
B=0.184
C=3.96
D=18.49
By tweaking certain decimal placed to reduce the RMSE.
Now the RMSE to the original graph is 0.305792
The tweaked Regression Curve Fit graph and the original graph from data
- On a new set of axes, draw your model function and the original graph. Comment on any differences. Refine your model if necessary
Next I use Curve Fit to check the RMSE of the values I worked out
RMSE: 0.188044
Since the RMSE is so low, it shows that my model function is extremely close to the original model. However there are still minor disrecepencies due to manual calculation and the rounding off decimal points.
Even though I tweaked the Regression Curve Fit graph, the values I work out has a lower RMSE than 0.305792 (of the tweaked Regression Curve Fit graph).
The graph modeled from the values I worked out and the graph modeled from original data
- Use technology to find another function that models the data. On a new set of axes, draw your model functions and the function you found using technology. Comment on any differences.
Before I chose another function that models the data I had two equations in mind. Either the Cubic, Quartic or Quintric. In order to see which equation models my graph more accurately I would look at the RMSE
Cubic graph
The RMSE is 0.0786418. This is already very accurate.
Quartic graph
The RMSE is 0.04564, this is very accurate and smaller then the Cubic graph’s RMSE.
Quintric
RMSE: 0.04725, this number is bigger then the Quartic graph but smaller then the Cubic graph, so I chose to use Quartic
Quartic is the most accurate graph, therefore I will use it for the next question. Some possible limitations in using the Quartic function is that it is only a best fit for age 2-20. If new data was presented, or we used this function to try and predict data outside of the data provided there may be errors. This is because the quartic function is only best fit for the current data provided, therefore we cannot solely rely on the quartic function for predicting values outside of the data provided.
If we zoom out, we will see the quartic graph dropping a lot after age 30. This proves this Quadratic may be the current best fit, but it’s unreliable if we look outside of our current age range.
- Use your model to estimate the BMI of a 30-year-old woman in the US. Discuss the reasonableness of your answer
From the pervious question I discovered that the Quartic is the most accurate compared to the graph modeled from the original data, so we should start by using the Quartic function
Quartic graph compared with origin
The co-ordinates are (29.95, 18.31)
Which means the BMI estimate for 30 year old women are around 18.38
This is unreasonable because this BMI of 30 year old women is much lower then the BMI of an age 20 or 25 year old woman.
30 year old women BMI: 18.38
20-25 year old women BMI: 20-21 (we can tell this by placing cursor on the line from the age of 20-30)
The BMI of these 30 year old women resembles that of an 11-12 year old girl.
30 year old women BMI: 18.38
11 year old girl BMI: 18.18
12 year old girl BMI: 18.70
If the BMI is calculated through taking the weight and dividing it by the square of the height, either a 30 year old women’s weight suddenly decrease a lot or the height increase a lot during age 20-30 (which is impossible due to fact that girls finish growth and reach maturity around 14).
If the quartic graph does not fit, then I will look at other graphs which can show a realistic BMI for 30 year old women
After looking at the sine, cubic, quartic, quintric graphs I noticed there is a deep decline in the BMI from age 20-30 so I had to look for a new function.
From clicking the equations available in the Curve Fit window I found that the Gaussian function shows a realistic trend from age 20-30.
From the Gaussian graph we can tell a 30 year old women’s BMI is 22.21, this is slightly higher then the 20 year olds BMI which is 21.65. It’s reasonable to assume that a 30 year old women’s BMI will be slightly higher then a 20 year old’s BMI.
So I decided that the Gaussian graph shows a reasonable estimate for a 30 year old’s BMI.
- User the Internet to find BMI data for females from another country. Does your model also fit this data? If not, what changes would you need to make? Discuss any limitations to your model
I found a graph of Bahraini girl and their BMI and age. I will now place this data into a graph form and compare it to the sine function.
Source:
To be able to compare the data of this table with my sine graph I will need to work out the sine function.
y= A sin(Bx+C)+D
D: Change vertical shift
Maximum = 22.5 (age 17-17.9)
Minimum = 15.2 (age 7-7.9)
(22.5+15.2)÷2 = 18.85
D= 18.85
A: Change in amplitude
22.5 – 18.85= 3.65
A= 3.65
B: Change horizontal scaling
Maximum: (17+17.9)÷2=17.45
Minimum: (7+7.9)÷2=7.45
17.45-7.45=10
10 x 2 = 20
2π÷20 = 0.3142
B= 0.3142
C: Change in horizontal shift
Maximum: 17.45
Minimum: 7.45
17.45-7.45=10
10÷2=5
5+5=10
10 x 0.3142= 3.142
C = 3.142
Finally
A= 3.65
B= 0.3142
C= 3.142
D= 18.85
y= 3.65 sin(0.3142x+3.142)+ 18.85
RME to original sine function: 2.03322
There are many limitations to the data of the BMI of the Bahrani girls. The first one being the age is a range which really decreases the accuracy of my manual calculation. As I am forced to take the average between the lower bound and upper bound of an age range. Second the age only goes up to 18-18.9 and not 20 like the other data.
There could also be an anomaly in the data presented this is because the maximum point it 22.5 for an age 17-17.9 while all the other girls are around 21.1-21.9 from age 14-18.9. This could explain the height difference of the graph for the Bahrani girls.
However apart from this anomaly the general shape of the curve B greatly matches the curve of graph A especially from age 2-6 when both graphs are touching.
Changes I will make will be to not take into account the anomaly 22.5 and see what happens.
D: Change vertical shift
Maximum = 22.5 (age 17-17.9) if I don’t use 22.5 I’ll use the second highest number 21.9
Minimum = 15.2 (age 7-7.9)
(21.9+15.2)/2 = 18.55
D= 18.55
A: Change in amplitude
21.9 – 18.85= 3.05
A= 3.05
This changes the function to:
y= 3.05 sin(0.3142x+3.142)+ 18.55
The RMSE becomes 1.717 which is smaller then the original Bahrani graph’s 2.03322 which prove my deduction is correct.
Still the sine graph produced from the data is not close enough and I am not satisfied with the results, so I will look at other functions and try to find one that fits more.
To do this I first have to model the data into a graph. As you can see, it is not a perfect curve. This is due to a few limitations. Firstly the ages are in ranges (which mean I have to add the two ages and dived by 2 to produce one value for the age). Also the BMI is a mean, which means it only shows the average which can result in this graph. In order to compare this with the graph made from the original BMI data, I must first find a function to model this graph.
I will try Cubic, Quintic, Quartic and Gaussian because these are the previous functions I have used. All I have to do is click Curve Fit and the computer automatically creates a graph that best fits using the equation I have selected.
First I tried Cubic with Curve Fit line. The RMSE is 0.595558
Red circles show the differences between Bahrani girl graph and Curve Fit line
Quartic graph with Curve Fit line
RMSE: 0.624147
Quintic with Curve Fit line
RMSE: 0.667095
Gaussian with Curve Fit Line
RMSE: 0.594122
After looking at all these equations, we can now tell indeed all the functions that fit the original BMI data fits the Bahrani women data as well.
However I also noticed that after 20 or 30, the graphs for the Bahrani girls started to drop significantly. This shows that the equations are not reasonable; the reason for the graph dropping may be insufficient data. Or because the data is the average.
Even with this weakness in the data, I chose to use the Gaussian graph of the Bahrani women to compare with the original data graph because the RMSE for the Gaussian graph is the lowest.
Gaussian graph values:
A: 7.102
B: 17.46
C: 5.595
D: 15.04
The Gaussian graph compared with the original data graph
RMSE: 0.795802
Conclusion
From looking at all these functions modeled from the Bahrani girl’s graph, we can tell it fits with the original BMI data for women.
After completing the task, I have found a flaw using the Gaussian function. Since it is named the ‘bell shape’ graph, after age 30 it drops back down to a straight line. This is extremely unreasonable, as women’s BMI will not drop significantly after age 20 to 15.1, which is equivalent to the BMI of a 3 year old.
However due to the limited data provided for the Bahraini girl, and I’ve tried every single function, there is no solution to that issue. However this can be solved by gaining more data on Bahrani women from age 19-30.
However from accuracy point of view a RMSE of 0.795802 when comparing the Gaussian graph and the original BMI of women shows the similarities. Based on my findings, the function that fits best is Gaussian, even though when it drops down to a constant of 15.1 it is still more reasonable then the Quartic graph which drops eventually to a 14.3. Dropping to 15.1 on the Gaussian graph and staying at that is reasonable if the BMI was slightly higher, this is because after age 30, a women’s BMI should not fluctuate much, if at all.
