Parameter Derivation from Data Points

Computer Science Level 4

Two variables, x x and y y , are related in the following way:

y = α x β + σ e − γ x \large{y = \alpha x^{\beta} + \sigma e^{-\gamma x}}

Parameters α , β , σ \alpha, \beta, \sigma , and γ \gamma are real numbers that lie within the range ( 0 , 3 ) (0,3) . The symbol e e denotes Euler's number .

The 10 ( x , y ) (x,y) pairs below correspond to a particular set of α , β , σ \alpha, \beta, \sigma , and γ \gamma . y y -coordinates are approximated to 6 decimal places.

Determine α + β + σ + γ \alpha + \beta + \sigma + \gamma (to 1 decimal place).

x y
1 2.745561
2 3.503581
3 4.218876
4 4.903586
5 5.554965
6 6.172822
7 6.759282
8 7.317432
9 7.850492
10 8.361436


The answer is 4.7.

Steven Chase by Steven Chase , 37, USA

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1 solution

Hasmik Garyaka
Oct 24, 2017 
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import math
def approximation():
    table={1:2.745561,2:3.503581,3:4.218876,4:4.903586,5:5.554965,6:6.172822,7:6.759282,8:7.317432,9:7.850492,10:8.361436
}
    for a in range(0,30):
        alpha=a/10
        for b in range(0,30):
            beta=b/10
            for c in range(0,30):
                gamma=c/10
                for d in range(0,30):
                    delta=d/10
                    for x in range(1,11):
                        y=alpha*pow(x,beta)+delta*math.exp(-gamma*x)
                        if y<table[x]-0.01 or y>table[x]+0.01:
                            break
                    else:
                        print(alpha,beta,gamma,delta)

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