Two friends, Gregor and Ilya have completed their studies recently on IT and business respectively. They are interested to open a part time business in consultancy. Thus, to start their business, they need to rent an office space which is very costly and they think that this cost will affect the success they will receive from their project.
Tus, to avoid that, they have identified three different strategies with the help of which they can set up the business. In the first strategy, they will rent a location which will be fairly expensive in the district where they will start their business and where the potential customers of their business will be located. Despite of the expensive office space, they are expecting a good profit from their business when the market is favorable. On the other hand, in an unfavorable market, when the business will be less, they are expecting to have a loss from the business. Their second strategy is to rent an office in the neighboring suburb, where the office will be available at a cheaper rate. The third strategy thought by them is not to set up a business at all.
According to the estimates made by Gregor and Ilya, the profits and the losses that they can make in a favorable and an unfavorable market for the three types of strategies are expressed in the following table:
Table 1.1: Estimated Profit or Loss
|
Favourable |
Unfavourable |
|
|
Strategy 1 |
$20,000 |
$16,000 |
|
Strategy 2 |
$15,000 |
$6,000 |
|
Strategy 3 |
$0 |
$0 |
|
Table 1.2: Approach by Ilya |
|||
|
Favourable |
Unfavourable |
Best Profit / Loss |
|
|
Strategy 1 |
$20,000 |
$16,000 |
$20,000 |
|
Strategy 2 |
$15,000 |
$6,000 |
$15,000 |
|
Strategy 3 |
$0 |
$0 |
$0 |
Gregor is the kind of person who is conservative and has an adverse nature towards risk taking. Thus, Gregor’s approach will be to minimize the minimum profit that can be earned from the three strategies. Thus, it can be seen from table 1.3 that Gregor would choose strategy 3 and will not be interested in setting up the business at all.
|
Table 1.3: Approach by Gregor |
|||
|
Favourable |
Unfavourable |
Least Profit / Loss |
|
|
Strategy 1 |
$20,000 |
$16,000 |
$16,000 |
|
Strategy 2 |
$15,000 |
$6,000 |
$6,000 |
|
Strategy 3 |
$0 |
$0 |
$0 |
It has been observed that the the chance of having a favorable market is 55 percent. Thus, from table 1.4, it can be seen clearly that considering the greatest expected values that can be earned from the business, they will be choosing strategy 2 to set up the business.
|
Table 1.4: Expected Value |
|||
|
Favourable |
Unfavourable |
Expected Value |
|
|
Strategy 1 |
$11,000 |
$7,200 |
$3,800 |
|
Strategy 2 |
$8,250 |
$2,700 |
$5,550 |
|
Strategy 3 |
$0 |
$0 |
$0 |
The plot showing the expected values of returns from strategy 1 and strategy 2 is given in figure 1.1 when the probability of favorable market varies from 0 to 1.
iii) It can be seen from the graph that both strategy 1 and strategy 2 face loses in the expected returns when the probability of favorable market is less than 0.28. Thus, Strategy 3 will be selected. Thus, Strategy 3 will be selected when 0 ≤ P ≤ 0.28
Let the number of TV ads be x1, radio ads be x2, billboard ads be x3 and newspaper ads be x4.
Minimize Z = 960 x1 + 480 x2 + 600 x3 + 120 x4
Subject to the constraints:
x1 ≤ 10
x2 ≤ 10
x3 ≤ 10
x4 ≤ 10
x1 ≥ 10
x2 ≥ 10
x1 + x2 ≥ 6
960 x1 – 600 x3 – 120 x4 ≥ 0
And the non-negativity constraints x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0
Input Variable Area:
The regression model to predict the selling price of the house when area of the house is the independent variable is given by:
Selling Price = – 34301.5987 + (62.96 * Area)
The coefficient of determination for this model is (0.7952 ^ 2) = 0.6323. This indicates that 63.23 percent of the variability in the selling price can be explained by area of the house. It can also be seen by comparing figures 4.1 and 4.2 that when the test mode is 10-fold cross-validation, the training errors such as the relative absolute error and the root relative squared error are higher than the test mode of 15-fold cross-validation.
When area = 2000 ft2, selling price = – 34301.5987 + (62.96 * 2000) = 91618.4
Input Variable Bedrooms:
The regression model to predict the selling price of the house when number of bedrooms in the house is the independent variable is given by:
Selling Price = 648.6487 + (35168.9189 * Number of Bedrooms)
The coefficient of determination for this model is (0.5047 ^ 2) = 0.2547. This indicates that 25.47 percent of the variability in the selling price can be explained by the number of bedrooms in the house.
When number of bedrooms = 3, selling price = 648.6487 + (35168.9189 * 3) = 106155
The regression model to predict the selling price of the house when age of the house is the independent variable is given by:
Selling Price = 141448.2518 + (- 2256.7296 * Age)
The coefficient of determination for this model is (0.8629 ^ 2) = 0.7446. This indicates that 74.46 percent of the variability in the selling price can be explained by the age of the house.
When age of the house = 24 years, selling price = 141448.2518 + (- 2256.7296 * 24) = 87286.7
Thus, it can be seen from the models that most of the variability in the selling prices of the houses can be explained by the age of the house. Thus, the third model indicating age of the house as the input variable is the model with the most reliable prediction.
The regression model to predict the selling price of the house when area of the house and number of bedrooms in the house are the independent variables is given by:
Selling Price = -26129.5 + (76.1268 * Area) + (-12403.1 * Bedrooms)
The coefficient of determination for this model is (0.8616 ^ 2) = 0.7423. This indicates that 74.23 percent of the variability in the selling price can be explained by area and number of bedrooms in the house.
Input Variables Area and Age:
The regression model to predict the selling price of the house when area of the house and age of the house are the independent variables is given by:
Selling Price = 69793.9387 + (27.0743 * Area) + (-1554.9387 * Age)
The coefficient of determination for this model is (0.8798 ^ 2) = 0.774. This indicates that 77.4 percent of the variability in the selling price can be explained by area and age of the house.
Input Variables Age and Bedrooms:
The regression model to predict the selling price of the house when number of bedrooms in the house and age of the house are the independent variables is given by:
Selling Price = 99495.77 + (12389 * Bedrooms) + (-1985.53 * Age)
The coefficient of determination for this model is (0.9309 ^ 2) = 0.8665. This indicates that 86.65 percent of the variability in the selling price can be explained by number of bedrooms and age of the house.
Input Variables Age, Area and Bedrooms:
The regression model to predict the selling price of the house when area of the house, number of bedrooms in the house and age of the house are the independent variables is given by:
Selling Price = 70181.01 + (25.1505 * Area) + (-1574.39 * Age) + (1389.257 * Bedrooms)
The coefficient of determination for this model is (0.9407 ^ 2) = 0.8848. This indicates that 88.48 percent of the variability in the selling price can be explained by area, bedrooms and age of the house.
The model involving all the input variables is the best predictor model as it can explain the highest amount of variability in the selling prices. This model is even better than the best model that has been identified in part 1 as this model can predict more accurate selling price than the identified best model in part 1.
The correlation coefficient obtained as a result of the multilayer perceptron is 0.9399. This indicates that the coefficient of variation is (0.9339 ^ 2) = 0.9396. Thus, the explanation of the variability in the selling prices is a lot higher than all the models developed previously. The training error is also less and found to be 26.819 percent. Hence, it can be said that the multilayer perceptron is a better prediction model to develop than the regression model.
Logistic Regression
Naïve Bayes Model
Lift Chart for Logistic Regression Model
Lift Chart for Naïve Bayes Model
Conclusion
It can be seen from the ROC areas estimated from both the logistic regression model (0.888) and from the Naïve Bayes Model (0.845), that the area under the ROC curve is higher for the Logistic Classifier. Thus, Logistic classifier is considered as a better classifier than a Bayes Classifier.
Further, comparing the two lift charts, it can be seen clearly that the lift chart for the logistic classifier has a higher lift than that of the Naïve Bayes classifier. Thus, from here also it can be concluded that the Logistic classifier is a better classifier than Naïve Bayes classifier.
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