The purpose of this portfolio is to find, by using different functions, the model that best fits China´s population between the years of 1950 to 2008. In order to reach the aim of the portfolio I will use functions like logarithmic function which is expressed by f(x)=a ln(x)+b. Quadratic function is one of the form f(x) = ax2+bx+c. Linear function it’s given by y=mx+b. However, according to the function I choose, I will modify it to pursue a better fitting relation and a model that not only best fits
The following table shows the population of China from 1950 to 1995.
Years | Population in millions |
1950 | 554.8 |
1955 | 609 |
1960 | 657.7 |
1965 | 729.2 |
1970 | 830.7 |
1975 | 927.8 |
1980 | 998.9 |
1985 | 1070 |
1990 | 1155.3 |1995 | 1220.5 |
According to the data shown I can define the variables and the parameters. The variables of the information shown are the Years (x-axis) and the Population in millions (y-axis). In the other side, the parameters of the data are the range of years (1950-1995).
NOTE: in all the graphs, the independent variable (years) and the dependent variable (population in million), will be thesame. Al though, for each graph there will be different constant variables (gradient, y-intercept, etc)
Now, I’m going to show all the graphs done in excel to show the relation between years and the population in millions of China by using different function.
NOTE: in all the graphs, the independent variable (years) and the dependent variable (population in million), will be the same. Although, for each graph there will be different constant variables (gradient, y-intercept, etc)
According to the graph, using the linear function, it is evinced that in the relation between the years and the population in millions the independent variable is (x-axis) and the dependent variable is population in millions (y-axis), the constant values will be the gradient (15.495) and they-intercept (-29688) and these constants are the parameters. Even it seems that the linear function is a very good way to relate the information, I can see that within the graph there are some limitations: the trend line doesn’t pass perfectly through all the points. The R2 (0.99479) is near 1 but not exact. The y-intercept is negative and the population of a country cannot be negative. Then, as thisfunction doesn’t relate the information in a 100% I will use other functions to relate them in a better way.
Now, I am going to see what happens if we use a quadratic function applying it to the growth of population in China.
The quadratic function it is expressed by f(x)=ax2+bx+c the constant variables would be (a) 0,0355, (b) -124.61 and (c) or y-intercept 108486so this could explain the parameter of the function. The quadratic function seems to fit in a better way than the linear function as it has a R2 value closer to 1 than the linear function making it a best fitting function for the model. I can see that within the graph there are some limitations: the trend line doesn’t pass perfectly through all the points. The R2 (0.9955) is near 1 but not exact.And the biggest limitation is that the shape of a quadratic function is a parabola then the population will be increasing infinitely and in real life this can’t happen as there will be points that population will decrease. Then, as this function doesn’t relate the information in a 100% correct I will use other functions to relate them in a better way. It must be take in account that the quadraticfunction is better than the linear function in this case.
Now, we would see what happens if we use a polynomial of 3 degrees applying it to the growth of population in China.
Polynomial of 3 degrees:
The polynomial of 3 degrees function it is expressed by f(x)=ax3+bx2+cx+d the constant variables would be (a) -0.0057, (b) 33.551, (c) -66242 and (d) or y-intercept 4E+07 so this could...