In order to find the unknowns a, b, c and d, 4 years have been chosen (3, 4, 5 and 6) to substitute into x in the above function. These values have been chosen because they are consecutive years that lie in the middle of the graph. The y value would be substituted with the total mass of that corresponding year. The parameter e would be substituted with the y-intercept where x=0, y=426.8. Therefore 426.8 will be substituted into e.
Table 3: This shows the chosen values of the rearranged years, x, the total mass, y, and e, the y-intercept at 426.8 that have been substituted into the equation.
Table 4: This shows the rearranged years and the simplified equation of table 3.
These equations are rearranged in to a matrix to solve the unknown coefficients:
When the matrix is solved the coefficients a, b, c and d are:
Therefore when the values are substituted into the template function, the equation can be rewritten as:
By using the equation above, a graph is drawn below:
Graph 3: This shows the function model acquired from solving the matrix equation.
Judging from the function-modelled graph, the model seems to fit quite well onto the graph because the line passes through most of the points. However, there are limitations to this graph because it cannot be used to predict the values for the next few years. As seen in the original graph, the total mass decreases right after the 8th year. The total mass might have been affected by environmental factors such as pollution, which would contribute to the decreasing population in the sea. This would have an impact on the fish caught in certain years because if there were less fish available, the amount caught would also decrease.
Graph 4: This shows a computer generated linear function and the function model.
The linear function model is not as accurate as the quartic function model because it does not pass through as many points as opposed to the other one. From the graph, the linear model does not pass through the points between the years 3-6 whereas the quartic model is more consistent as it passes through most of the points throughout the graph. Nevertheless, the linear function model has an advantage because it could estimate the total mass in the upcoming years provided that the total mass each year does not decrease.
In order to model the next set of values in the 2nd section, another function model will need to be considered because as seen in the quartic model, the graph keeps increasing. On the other hand, the total mass of the original graph plummets after the 8th year. This indicates that a quartic function is not suitable for the 2nd section.
Similar to what was done earlier, the second table of values would have to be rearranged where in the year 1989, x=0, and a y-intercept would occur. Therefore, at x=0, the y-intercept is 450.5.
Table 5: This shows the rearranged years of the 2nd section starting with 1989 where x=0.
Graph 5: This shows the second section of the original graph with the rearranged years.
A quartic function model is not suitable for this graph because the variables are not enough to accurately model this section of the graph. By looking at this graph, there are two inflection points; (years 2 and 6) and most of the other points are further spread apart when compared to the 1st section. Hence, a quintic equation with 6 variables would be able to accurately fit the points because the variables are responsible for the curving of the graph. Below is an example of a quintic function:
The x variable represents the year and the variable f(x) represents the total mass (tonnes). The parameters in this function is f, which is the y-intercept and the coefficients a, b, c, d and e are responsible for the curving of the graph.
By replicating what was done earlier in order to find the unknowns a, b, c, d and e, another 4 years have been chosen (2, 3, 4, 5 and 6) to substitute into x in the above function. This time 2 and 6 have been chosen because they are the inflection points. 3, 4 and 5 have been chosen because they lie in between these two points. The y value would be substituted with the total mass of that corresponding year. The parameter f would be substituted with the y-intercept which is 450.5 when x=0.
Table 6: This shows the chosen values of the rearranged years, x, the total mass, y, and f, the y-intercept at 450.5 that have been substituted into the equation.
Table 7: This shows the rearranged years and the simplified equation of table 6.
These equations are rearranged into a matrix to solve the unknown coefficients:
When the matrix is solved the coefficients for a, b, c, d and e are:
Therefore when you substitute these values into the template function the equation can be rewritten as:
By using the equation above, a graph is drawn below:
Graph 6: This shows the function model for the 2nd section acquired from solving the matrix equation.
As seen from the graph above, the quintic equation seems to fit the essential points of the graph. It passes through the y-intercept, and when x=8, and the line is also fairly close to the points at years 3 and 4. Although the model fits quite well, it has limitations; the total mass of fish caught each year is highly unpredictable because the figures can vary due to factors that cannot be controlled by humans. Therefore, this model can only be applied to these points because it would not fit other points if the values vary too much.
For the 3rd set of values, a quintic function would be used to model the behaviour of the graph. Similar to what was done in the first two processes, the 3rd table of values would need to be rearranged where at 1998, x=0 and the y-intercept is 487.2.
Table 8: This shows the rearranged years of the 3rd section starting with 1998 where x=0.
Graph 7: This shows the third section of the graph with the rearranged years.
Next, the x and y values for the years 2, 3, 4, 5 and 6 would be substituted into the template function and the parameter f would be substituted with the y-intercept, 487.2.
Table 9: This shows the chosen values of the rearranged years, x, the total mass, y, and f, the y-intercept at 487.2 that have been substituted into the equation.
Table 10: This shows the rearranged years and the simplified equation of table 9.
These equations are rearranged into a matrix to solve the unknown coefficients:
When the matrix is solved the coefficients a, b, c, d and e are:
3s.f
Therefore when the values are substituted into the template equation, the equation can be rewritten as:
Graph 7: This shows the function model for the 3rd section acquired from solving the matrix equation.
This function model does not pass through any of the data points between the years 0 and 8. It is not very accurate however, the model attempts to model the shape of the graph. It seems that this is the closest that a model could attempt to fit the data points because there is no clear trend for these data points. The total mass increases and decreases every year therefore, the graph does not have any apparent trends. There are limitations in this model; since the values are inconsistent it would be difficult to come up with a perfect model that could fit these data points.
Table 11: This shows the total mass of fish, in thousands of tonnes from fish farms.
The values above will be split into 2 graphs so that the quartic model that was developed earlier could be used to determine whether it fits these data. The years 1980-2000 would be in 1 graph and the years 2001-2006 would be in another graph. The reason for splitting the graph along these years is because between 1980-2000 the total mass increases. Between 2001-2002 the total mass decreases and then increases in 2003. From 2003 to 2006, the total mass increases continuously.
Graph 8: This shows the quartic function model for the first graph between the years 1980-2000.
The quartic model that was developed earlier does not fit the new data; it does not connect any of the points. The overall trend of the graph between year 0 and 20 seems to be increasing and the points form a curving shape. Between the years 0 and 14, the total mass increases at a slower rate and from year 15 onwards, the total mass increases at a faster pace because the gradient is steeper.
Graph 9: This shows a computer generated cubic function and the quartic model for the first graph.
The computer generated cubic model is much more accurate than the quartic model because it fits most of the data points. The curve of the cubic model overlaps most of the data points and it essentially matches the curve that had been formulated by the points. Hence, this model is more suitable for modelling the first part of data table.
Graph 10: This shows the quartic function model for the second graph between the years 2003-2006.
The quartic model that was developed earlier does not fit the new data; it does not connect any of the points. The data points form a curve and the total mass seem to increase at a constant rate from the 1st year onwards.
Graph 11: This shows a computer generated cubic function and the quartic model for the second graph.
The computer generated cubic model passes through most of the points hence, it is accurate when compared to the analytically developed quartic model. Since the cubic model fits the first and the second graph it can be concluded that the cubic model is suitable for modelling the data from the table above.
The overall trend in the first model is rather inaccurate because it does not pass through any of the points. The model does not suggest any possible values for the total mass and it cannot be used to predict future trends. The second model on the other hand is much more accurate and it would be suitable to use for predicting future trends. The model could also be used to compare the theoretical mass and the actual mass in a given year. In doing so, this would suggest whether the model is accurate and judging from the two graphs, the model seems fairly accurate because the line lies fairly close to the data points.
Conclusion:
In conclusion, both models could be used to predict future trends in both types of fishing provided that there are not external factors that might influence the total mass caught. The quartic and the quintic model are fairly accurate in modelling the total mass of fish caught in the sea. Both of the models could be used to predict future trends theoretically. However, these models have limitations because they cannot predict the exact values since the number of fish available each year varies. There may be more fish available in the sea in some years whereas in other years, this may decrease because the fish might have died due to pollution or other forms of human activity. The computer generated cubic model is more suitable when used to predict possible future trends because it is more accurate in modelling the total mass of fish from the fish farms. Since the fish are bred in fish farms, it is possible for people to control the total mass that is caught thus, indicating that the model could predict future trends to a degree of accuracy. By looking at the table of values, the total mass increases continuously every year because it has a lower chance of being affected by environmental or external factors that may decrease the mass of fish produced. Humans can control the number of fish in fish farms; since the total mass increased every year except for 2001 and 2002, it can be assumed that there is a high demand for fish by consumers. The possible reason for the decrease in numbers during the years 2001 and 2002 are probably due to the lower demand by consumers or the errors made by the breeders that resulted in the death of the fish.
