Nonlinear equation params. estimation

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Description:

From $n$ non-linear equations with $x_{1},..., x_n$ unknown variables a multivariate system of nonlinear equations with dimensional compatibility can be expressed:

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The solution is found when the difference is 0 (10 iterations approx.)

From Scipy: Non-linear solvers fsolve is used to estimate $f(x)$ distribution.

The dataset Cumulative Distribution is:

$F(X)$ = ${Pr}(a \leq X \leq b)$:

Note: Analytical tests should be performed to validate $f(x)$ & the obtained params $x_i,x_2,...,x_n:$

$$F(X) = \int_{a}^{b} f(x) dx$$

$\therefore$ the Expectancy can be expressed as $E[Y]$:

$$E [Y] = \hat{F}^{N} = \int_{a}^{b} x f(x) dx$$

And $\hat{F}^{N}$ can be modelled with samplings from $N$ random variables $X \sim U(a,b)$.

Beta Distribution

• $X\sim\beta(\alpha, \beta)$ with $f(x_{i}, f_{x_{i}}):$
  

x	f(x)	F(x)
40.00	inf	0.000000
40.01	1.040696	0.019154
40.02	0.758528	0.027915
40.03	0.630507	0.034797
40.04	0.553063	0.040688
...	...	...
48.95	0.247013	0.981838
48.96	0.265219	0.984395
48.97	0.290729	0.987167
48.98	0.330971	0.990257
48.99	0.413234	0.993916

Triangular Distribution

• $X\sim\text{T}(a,b,c)$:

x	f(x)	F(x)
0	40.000000	0.000000
1	40.090909	0.005102
2	40.181818	0.010203
3	40.272727	0.015305
4	40.363636	0.020406
...	...	...
95	48.636364	0.997085
96	48.727273	0.998360
97	48.818182	0.999271
98	48.909091	0.999818
99	49.000000	1.000000

• Montecarlo Estimation:
  

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if $V$ is the volume of an integral

$$V = \int_{a}^{b} f(x) dx$$

and it follows a uniform prob. density function $f(x)$ $X \sim U(a,b)$ it can be modelled with np.random.uniform.

$f(x) = \frac{1}{b-a}$ if $a \leq x \leq b$
$0$ otherwise

Nevertheless if the distribution isn't known it must be obtained from data or modelled with another method.

From where we've got by the definition the expectancy:

$$E[X] = \int_{a}^{b} x f(x) dx$$

Then, the mean of $x_{i}$ in $V$ is:

$$F = \frac{V}{N} \sum_{i=0}^{N-1} f\left(x_i\right)$$

if $N = N-1$ the equation can be rewritten from the solution of the integral as: $$F = \frac{b-a}{N-1} \sum_{i=0}^N f\left(x_i\right)$$

But Montecarlo estimations for $N$ random variables $X \sim U(a,b)$ must include the error $\xi$.
$\therefore$, the estimator can be expressed with the solution of the integral for all possible errors $\xi \in$ $(0,1]$ $\rightarrow$ $a = b$ :

$$\hat{F}^{N} = \frac{b-a}{N-1} \sum_{i=0}^{N} f\bigg[a + \xi(b-a)\bigg]$$

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Non-Linear Solvers

Newton Raphson Taylor-Series Jacobian

scipy.optimize.fsolve

scipy.stats.beta $X\sim\beta(\alpha, \beta)$

scipy.stats.gamma $X\sim\Gamma(\alpha, \beta)$

scipy.stats.triang $X\sim\text{T}(a,b,c)$

Monte Carlo Estimation