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url = 'https://apmonitor.com/che263/uploads/Main/stats_data.txt'

url = 'https://apmonitor.com/che263/uploads/Main/stats_data.txt'

url = 'https://apmonitor.com/che263/uploads/Main/stats_data.txt'

url = 'https://apmonitor.com/che263/uploads/Main/stats_data.txt'

   import pip
    pip.main(['install','uncertainties'])

   import pip
    pip.main(['install','uncertainties'])

m = 400

i1 = np.linspace(a3L,a3U*2,m)

plt.xlabel('a')
plt.ylabel('b')

         LineWidth=3,label='Joint Confidence Region')

plt.title('Parameter Statistics (y = a exp(b*x) + c)')

         LineWidth=3,label='Joint Confidence Region')

!!!! Joint Confidence Regions

(:toggle hide conf_int button show="Nonlinear Regression Solution":)

x = data['x'].values[0:20]
y = data['y'].values[0:20]

CS = plt.contour(a_grid,b_grid,sse,[5,20,50,100,200,500],\

(:toggle hide nonlinear button show="Nonlinear Regression Solution":)
(:div id=nonlinear:)

{$\frac{SSE(\theta)-SSE(\theta*)}{SSE(\theta*)}\le \frac{p}{n-p} F\left(\alpha,p,n-p\right)$}

with p=2 (number of parameters), n=100 (number of measurements), {`\theta`}=[a,b] (parameters), {`\theta*`} as the optimal parameters, SSE as the sum of squared errors, and the F statistic that has 3 arguments (alpha=confidence level, degrees of freedom 1, and degrees of freedom 2). For many problems, this creates a multi-dimensional nonlinear confidence region. In the case of 2 parameters, the nonlinear confidence region is a 2-dimensional space.

(:toggle hide linear button show="Linear Regression Solution":)
(:div id=linear:)

Data regression is adjusts parameter values to best fit the data. Parameter uncertainty and the predicted outcome uncertainty is important for qualifying the degree of confidence in the solution. This tutorial shows how to perform a statistical analysis with Python.

plt.scatter(x, y, s=3)

plt.plot(px, nom, c='black')

plt.plot(px, nom - 1.96 * std, c='orange')

plt.plot(px, lpb, 'k--')

data = pd.read_csv(url+file)

import matplotlib.pyplot as plt
import uncertainties.unumpy as unp

data = pd.read_csv(url)

url = 'https://apmonitor.com/che263/uploads/Main/'
file = 'stats_data.txt'
data = pd.read_csv(url+file)

!!!! Parameter Confidence

The nonlinear confidence interval is shown by producing a contour plot of the SSE objective function with variations in the two parameters. The optimal solution is shown at the center of the plot and the objective function becomes worse (higher) away from the optimal solution.

{$y = a \exp{b\, x} + c$}

{$\frac{SSE(\theta)-SSE(\theta^*)}{SSE(\theta^*)}\le \frac{p}{n-p} F\left(\alpha,p,n-p\right)$}

with p=2 (number of parameters), n=100 (number of measurements), {`\theta`}=[a,b] (parameters), {`\theta^*`} as the optimal parameters, SSE as the sum of squared errors, and the F statistic that has 3 arguments (alpha=confidence level, degrees of freedom 1, and degrees of freedom 2). For many problems, this creates a multi-dimensional nonlinear confidence region. In the case of 2 parameters, the nonlinear confidence region is a 2-dimensional space.

Use the data at the link below to create a linear and nonlinear regression.

(:toggle hide stats button show="Python Solution":)
(:div id=stats:)

!!!! Nonlinear Parameter Confidence Region

-> Attach:sse_contour.png