Regression Statistics with Python

Main.PythonRegressionStatistics History

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June 21, 2020, at 04:17 AM by 136.36.211.159 -
Deleted lines 564-582:

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February 14, 2019, at 10:23 PM by 10.37.111.106 -
Changed lines 37-38 from:
    import pip
    pip.main(['install','uncertainties'])
to:
    try:
        from pip import main as pipmain
    except:
        from pip._internal import main as pipmain
    pipmain(['install','uncertainties'])
Changed lines 154-155 from:
    import pip
    pip.main(['install','uncertainties'])
to:
    try:
        from pip import main as pipmain
    except:
        from pip._internal import main as pipmain
    pipmain(['install','uncertainties'])
March 21, 2018, at 11:53 PM by 10.37.35.33 -
Changed lines 439-440 from:

s_c = np.sqrt(pcov[2,2])

to:
  1. #####################################
  2. Single Variable Confidence Interval
  3. #####################################
Added lines 455-491:
  1. #####################################
  2. Linear Joint Confidence Interval
  3. thanks to Trent Okeson
  4. #####################################

eigenval, eigenvec = np.linalg.eig(pcov) max_eigenval = np.argmax(eigenval) largest_eigenvec = eigenvec[:, max_eigenval] largest_eigenval = eigenval[max_eigenval]

if max_eigenval == 1:

    smallest_eigenval = eigenval[0]
    smallest_eigenvec = eigenvec[:, 0]

else:

    smallest_eigenval = eigenval[1]
    smallest_eigenvec = eigenvec[:, 1]

angle = np.arctan2(largest_eigenvec[1], largest_eigenvec[0]) if angle < 0:

    angle += 2*np.pi

chisquare_val = 2.4477 theta_grid = np.linspace(0, 2*np.pi) phi = angle a_rot = chisquare_val*np.sqrt(largest_eigenval) b_rot = chisquare_val*np.sqrt(smallest_eigenval)

ellipse_x_r = a_rot*np.cos(theta_grid) ellipse_y_r = b_rot*np.sin(theta_grid)

R =

r_ellipse = np.stack((ellipse_x_r, ellipse_y_r)).T.dot(R) final_ellipse = r_ellipse + popt[None, 0:2]

  1. #####################################
  2. Nonlinear Joint Confidence Interval
  3. #####################################
Changed lines 494-496 from:

m = 400

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

to:

m = 200

i1 = np.linspace(a3L,a3U*2.0,m)

Changed lines 526-527 from:

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

to:

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

Added lines 538-539:

plt.plot(final_ellipse[:, 0], final_ellipse[:, 1], 'k-', label='Linear Joint Confidence Region')

Changed lines 541-542 from:
          LineWidth=3,label='Joint Confidence Region')
to:
          LineWidth=4,          label='Nonlinear Joint Confidence Region')
Changed line 546 from:

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

to:

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

March 21, 2018, at 11:44 PM by 10.37.35.33 -
Added lines 284-286:
  1. #####################################
  2. Single Variable Confidence Interval
  3. #####################################
Added lines 299-335:
  1. #####################################
  2. Linear Joint Confidence Interval
  3. thanks to Trent Okeson
  4. #####################################

eigenval, eigenvec = np.linalg.eig(pcov) max_eigenval = np.argmax(eigenval) largest_eigenvec = eigenvec[:, max_eigenval] largest_eigenval = eigenval[max_eigenval]

if max_eigenval == 1:

    smallest_eigenval = eigenval[0]
    smallest_eigenvec = eigenvec[:, 0]

else:

    smallest_eigenval = eigenval[1]
    smallest_eigenvec = eigenvec[:, 1]

angle = np.arctan2(largest_eigenvec[1], largest_eigenvec[0]) if angle < 0:

    angle += 2*np.pi

chisquare_val = 2.4477 theta_grid = np.linspace(0, 2*np.pi) phi = angle a_rot = chisquare_val*np.sqrt(largest_eigenval) b_rot = chisquare_val*np.sqrt(smallest_eigenval)

ellipse_x_r = a_rot*np.cos(theta_grid) ellipse_y_r = b_rot*np.sin(theta_grid)

R =

r_ellipse = np.stack((ellipse_x_r, ellipse_y_r)).T.dot(R) final_ellipse = r_ellipse + popt[None, :]

  1. #####################################
  2. Nonlinear Joint Confidence Interval
  3. #####################################
Added lines 382-383:

plt.plot(final_ellipse[:, 0], final_ellipse[:, 1], 'k--', label='Linear Joint Confidence Region')

Changed lines 385-386 from:
          LineWidth=3,label='Joint Confidence Region')
to:
          LineWidth=4,          label='Nonlinear Joint Confidence Region')
March 17, 2018, at 03:38 PM by 45.56.3.173 -
Added lines 247-250:

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/qOV8QaOzWXA" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe> (:htmlend:)

March 17, 2018, at 05:12 AM by 45.56.3.173 -
Changed lines 244-245 from:

Joint Confidence Regions

to:

Joint Confidence Region - Linear Regression

Added lines 357-465:

Joint Confidence Region - Nonlinear Regression

(:toggle hide conf_int_nonlinear button show="Joint Confidence Solution":) (:div id=conf_int_nonlinear:) (:source lang=python:) import numpy as np import scipy as sp from scipy import stats from scipy.optimize import curve_fit import matplotlib.pyplot as plt import pandas as pd

  1. import data

url = 'https://apmonitor.com/che263/uploads/Main/stats_data.txt' data = pd.read_csv(url) x = data['x'].values y = data['y'].values

def f(x, a, b, c):

    return a * np.exp(b*x) + c
  1. perform regression

popt, pcov = curve_fit(f, x, y)

  1. retrieve parameter values

a = popt[0] b = popt[1] c = popt[2]

  1. calculate parameter stdev

s_a = np.sqrt(pcov[0,0]) s_b = np.sqrt(pcov[1,1]) s_c = np.sqrt(pcov[2,2])

  1. calculate 95% confidence interval

aL = a-1.96*s_a aU = a+1.96*s_a bL = b-1.96*s_b bU = b+1.96*s_b

  1. calculate 3sigma interval

a3L = a-3.0*s_a a3U = a+3.0*s_a b3L = b-3.0*s_b b3U = b+3.0*s_b

  1. Generate contour plot of SSE ratio vs. Parameters
  2. meshgrid is +/- change in the objective value

m = 400

i1 = np.linspace(a3L,a3U*2,m) i2 = np.linspace(b3L,b3U,m) a_grid, b_grid = np.meshgrid(i1, i2) n = len(y) # number of data points p = 2 # number of parameters

  1. sum of squared errors

sse = np.empty((m,m)) for i in range(m):

    for j in range(m):
        at = a_grid[i,j]
        bt = b_grid[i,j]
        sse[i,j] = np.sum((y-f(x,at,bt,c))**2)
  1. normalize to the optimal solution

best_sse = np.sum((y-f(x,a,b,c))**2) fsse = (sse - best_sse) / best_sse

  1. compute f-statistic for the f-test

alpha = 0.05 # alpha, confidence

             # alpha=0.05 is 95% confidence

fstat = sp.stats.f.isf(alpha,p,(n-p)) flim = fstat * p / (n-p) obj_lim = flim * best_sse + best_sse

  1. Create a contour plot

plt.figure() CS = plt.contour(a_grid,b_grid,sse,[1,2,5,10], colors='gray') plt.clabel(CS, inline=1, fontsize=10) plt.xlabel('a') plt.ylabel('b')

  1. solid line to show confidence region

CS = plt.contour(a_grid,b_grid,sse,[obj_lim], colors='orange',linewidths=[2.0]) plt.clabel(CS, inline=1, fontsize=10)

  1. 2 dummy plots (gray,orange) for legend

plt.plot([aL,aL],[bL,bL],c='gray', LineWidth=3,label='Objective') plt.plot([aL,aU,aU,aL,aL],[bL,bL,bU,bU,bL],c='blue', LineWidth=3,label='Parameter Confidence Region') plt.plot([aL,aL],[bL,bL],c='orange', LineWidth=3,label='Joint Confidence Region') plt.scatter(a,b,s=10,c='red',marker='o', label='Optimal Solution')

plt.title('Parameter Statistics (y = a exp(b*x) + c)') plt.legend(loc=1)

  1. Save the figure as a PNG

plt.savefig('contour.png') plt.show() (:sourceend:) (:divend:)

March 17, 2018, at 03:51 AM by 45.56.3.173 -
Changed line 250 from:

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

to:

(:toggle hide conf_int button show="Joint Confidence Solution":)

Changed lines 263-265 from:

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

to:

x = data['x'].values y = data['y'].values

Changed line 323 from:

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

to:

CS = plt.contour(a_grid,b_grid,sse,[1,2,5,10],\

March 17, 2018, at 03:00 AM by 45.56.3.173 -
Changed lines 250-251 from:

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

to:

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

March 17, 2018, at 03:00 AM by 45.56.3.173 -
Added lines 249-350:

(:toggle hide nonlinear button show="Nonlinear Regression Solution":) (:div id=nonlinear:) (:source lang=python:) import numpy as np import scipy as sp from scipy import stats from scipy.optimize import curve_fit import matplotlib.pyplot as plt import pandas as pd

  1. import data

url = 'https://apmonitor.com/che263/uploads/Main/stats_data.txt' data = pd.read_csv(url) x = data['x'].values[0:20] y = data['y'].values[0:20]

def f(x, a, b):

    return a * x + b
  1. perform regression

popt, pcov = curve_fit(f, x, y)

  1. retrieve parameter values

a = popt[0] b = popt[1]

  1. calculate parameter stdev

s_a = np.sqrt(pcov[0,0]) s_b = np.sqrt(pcov[1,1])

  1. calculate 95% confidence interval

aL = a-1.96*s_a aU = a+1.96*s_a bL = b-1.96*s_b bU = b+1.96*s_b

  1. calculate 3sigma interval

a3L = a-3.0*s_a a3U = a+3.0*s_a b3L = b-3.0*s_b b3U = b+3.0*s_b

  1. Generate contour plot of SSE ratio vs. Parameters
  2. meshgrid is +/- change in the objective value

m = 200

i1 = np.linspace(a3L,a3U,m) i2 = np.linspace(b3L,b3U,m) a_grid, b_grid = np.meshgrid(i1, i2) n = len(y) # number of data points p = 2 # number of parameters

  1. sum of squared errors

sse = np.empty((m,m)) for i in range(m):

    for j in range(m):
        at = a_grid[i,j]
        bt = b_grid[i,j]
        sse[i,j] = np.sum((y-f(x,at,bt))**2)
  1. normalize to the optimal solution

best_sse = np.sum((y-f(x,a,b))**2) fsse = (sse - best_sse) / best_sse

  1. compute f-statistic for the f-test

alpha = 0.05 # alpha, confidence

             # alpha=0.05 is 95% confidence

fstat = sp.stats.f.isf(alpha,p,(n-p)) flim = fstat * p / (n-p) obj_lim = flim * best_sse + best_sse

  1. Create a contour plot

plt.figure() CS = plt.contour(a_grid,b_grid,sse,[5,20,50,100,200,500], colors='gray') plt.clabel(CS, inline=1, fontsize=10) plt.xlabel('Slope (a)') plt.ylabel('Intercept (b)')

  1. solid line to show confidence region

CS = plt.contour(a_grid,b_grid,sse,[obj_lim], colors='orange',linewidths=[2.0]) plt.clabel(CS, inline=1, fontsize=10)

  1. 2 dummy plots (gray,orange) for legend

plt.plot([aL,aL],[bL,bL],c='gray', LineWidth=3,label='Objective') plt.plot([aL,aU,aU,aL,aL],[bL,bL,bU,bU,bL],c='blue', LineWidth=3,label='Parameter Confidence Region') plt.plot([aL,aL],[bL,bL],c='orange', LineWidth=3,label='Joint Confidence Region') plt.scatter(a,b,s=10,c='red',marker='o', label='Optimal Solution')

plt.title('Parameter Statistics (y = a x + b)') plt.legend(loc=1)

  1. Save the figure as a PNG

plt.savefig('contour.png') plt.show() (:sourceend:) (:divend:)

March 17, 2018, at 01:43 AM by 45.56.3.173 -
Changed lines 252-254 from:

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

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.

to:

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

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.

March 16, 2018, at 08:01 PM by 45.56.3.173 -
Added lines 6-9:

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/r4pgGD1kpYM" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe> (:htmlend:)

March 16, 2018, at 07:38 PM by 45.56.3.173 -
Added lines 130-131:
March 16, 2018, at 07:37 PM by 45.56.3.173 -
Changed lines 131-132 from:

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

to:

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

March 16, 2018, at 07:37 PM by 45.56.3.173 -
Deleted line 23:

import uncertainties as unc

Added line 31:
    import uncertainties as unc
Changed lines 36-37 from:
to:
    import uncertainties as unc
Added lines 130-236:

(:toggle hide linear button show="Linear Regression Solution":) (:div id=linear:) (:source lang=python:) import numpy as np from scipy.optimize import curve_fit import matplotlib.pyplot as plt from scipy import stats import pandas as pd

  1. pip install uncertainties, if needed

try:

    import uncertainties.unumpy as unp
    import uncertainties as unc

except:

    import pip
    pip.main(['install','uncertainties'])
    import uncertainties.unumpy as unp
    import uncertainties as unc
  1. import data

url = 'https://apmonitor.com/che263/uploads/Main/stats_data.txt' data = pd.read_csv(url) x = data['x'].values y = data['y'].values n = len(y)

def f(x, a, b, c):

    return a * np.exp(b*x) + c

popt, pcov = curve_fit(f, x, y)

  1. retrieve parameter values

a = popt[0] b = popt[1] c = popt[2] print('Optimal Values') print('a: ' + str(a)) print('b: ' + str(b)) print('c: ' + str(c))

  1. compute r^2

r2 = 1.0-(sum((y-f(x,a,b,c))**2)/((n-1.0)*np.var(y,ddof=1))) print('R^2: ' + str(r2))

  1. calculate parameter confidence interval

a,b,c = unc.correlated_values(popt, pcov) print('Uncertainty') print('a: ' + str(a)) print('b: ' + str(b)) print('c: ' + str(c))

  1. plot data

plt.scatter(x, y, s=3, label='Data')

  1. calculate regression confidence interval

px = np.linspace(14, 24, 100) py = a*unp.exp(b*px)+c nom = unp.nominal_values(py) std = unp.std_devs(py)

def predband(x, xd, yd, p, func, conf=0.95):

    # x = requested points
    # xd = x data
    # yd = y data
    # p = parameters
    # func = function name
    alpha = 1.0 - conf    # significance
    N = xd.size          # data sample size
    var_n = len(p)  # number of parameters
    # Quantile of Student's t distribution for p=(1-alpha/2)
    q = stats.t.ppf(1.0 - alpha / 2.0, N - var_n)
    # Stdev of an individual measurement
    se = np.sqrt(1. / (N - var_n) *                  np.sum((yd - func(xd, *p)) ** 2))
    # Auxiliary definitions
    sx = (x - xd.mean()) ** 2
    sxd = np.sum((xd - xd.mean()) ** 2)
    # Predicted values (best-fit model)
    yp = func(x, *p)
    # Prediction band
    dy = q * se * np.sqrt(1.0+ (1.0/N) + (sx/sxd))
    # Upper & lower prediction bands.
    lpb, upb = yp - dy, yp + dy
    return lpb, upb

lpb, upb = predband(px, x, y, popt, f, conf=0.95)

  1. plot the regression

plt.plot(px, nom, c='black', label='y=a exp(b x) + c')

  1. uncertainty lines (95% confidence)

plt.plot(px, nom - 1.96 * std, c='orange', label='95% Confidence Region') plt.plot(px, nom + 1.96 * std, c='orange')

  1. prediction band (95% confidence)

plt.plot(px, lpb, 'k--',label='95% Prediction Band') plt.plot(px, upb, 'k--') plt.ylabel('y') plt.xlabel('x') plt.legend(loc='best')

  1. save and show figure

plt.savefig('regression.png') plt.show() (:sourceend:) (:divend:)

March 16, 2018, at 07:06 PM by 45.56.3.173 -
Changed line 5 from:

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.

to:

Regression is an optimization method for adjusting parameter values so that a correlation best fits data. Parameter uncertainty and the predicted uncertainty is important for qualifying the confidence in the solution. This tutorial shows how to perform a statistical analysis with Python for both linear and nonlinear regression.

March 16, 2018, at 06:14 PM by 45.56.3.173 -
Added lines 6-7:

Linear Regression

March 16, 2018, at 06:11 PM by 45.56.3.173 -
Changed lines 65-66 from:

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

to:

plt.scatter(x, y, s=3, label='Data')

Changed lines 101-102 from:

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

to:

plt.plot(px, nom, c='black', label='y=a x + b')

Changed lines 104-105 from:

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

to:

plt.plot(px, nom - 1.96 * std, c='orange', label='95% Confidence Region')

Changed line 108 from:

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

to:

plt.plot(px, lpb, 'k--',label='95% Prediction Band')

Added line 112:

plt.legend(loc='best')

March 16, 2018, at 05:35 PM by 45.56.3.173 -
Changed line 37 from:

data = pd.read_csv(url+file)

to:

data = pd.read_csv(url)

March 16, 2018, at 05:32 PM by 45.56.3.173 -
Changed lines 23-24 from:

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

to:

import matplotlib.pyplot as plt

Added lines 26-33:
  1. pip install uncertainties, if needed

try:

    import uncertainties.unumpy as unp

except:

    import pip
    pip.main(['install','uncertainties'])
    import uncertainties.unumpy as unp
March 16, 2018, at 05:30 PM by 45.56.3.173 -
Changed line 30 from:

data = pd.read_csv(url)

to:

data = pd.read_csv(url+file)

Added lines 102-103:

plt.ylabel('y') plt.xlabel('x')

March 16, 2018, at 05:28 PM by 45.56.3.173 -
Changed lines 29-31 from:

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

to:

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

March 16, 2018, at 05:28 PM by 45.56.3.173 -
Changed lines 20-106 from:
to:

import numpy as np from scipy.optimize import curve_fit import uncertainties as unc import matplotlib.pyplot as plt import uncertainties.unumpy as unp from scipy import stats import pandas as pd

  1. import data

url = 'https://apmonitor.com/che263/uploads/Main/' file = 'stats_data.txt' data = pd.read_csv(url+file) x = data['x'].values y = data['y'].values n = len(y)

def f(x, a, b):

    return a * x + b

popt, pcov = curve_fit(f, x, y)

  1. retrieve parameter values

a = popt[0] b = popt[1] print('Optimal Values') print('a: ' + str(a)) print('b: ' + str(b))

  1. compute r^2

r2 = 1.0-(sum((y-f(x,a,b))**2)/((n-1.0)*np.var(y,ddof=1))) print('R^2: ' + str(r2))

  1. calculate parameter confidence interval

a,b = unc.correlated_values(popt, pcov) print('Uncertainty') print('a: ' + str(a)) print('b: ' + str(b))

  1. plot data

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

  1. calculate regression confidence interval

px = np.linspace(14, 24, 100) py = a*px+b nom = unp.nominal_values(py) std = unp.std_devs(py)

def predband(x, xd, yd, p, func, conf=0.95):

    # x = requested points
    # xd = x data
    # yd = y data
    # p = parameters
    # func = function name
    alpha = 1.0 - conf    # significance
    N = xd.size          # data sample size
    var_n = len(p)  # number of parameters
    # Quantile of Student's t distribution for p=(1-alpha/2)
    q = stats.t.ppf(1.0 - alpha / 2.0, N - var_n)
    # Stdev of an individual measurement
    se = np.sqrt(1. / (N - var_n) *                  np.sum((yd - func(xd, *p)) ** 2))
    # Auxiliary definitions
    sx = (x - xd.mean()) ** 2
    sxd = np.sum((xd - xd.mean()) ** 2)
    # Predicted values (best-fit model)
    yp = func(x, *p)
    # Prediction band
    dy = q * se * np.sqrt(1.0+ (1.0/N) + (sx/sxd))
    # Upper & lower prediction bands.
    lpb, upb = yp - dy, yp + dy
    return lpb, upb

lpb, upb = predband(px, x, y, popt, f, conf=0.95)

  1. plot the regression

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

  1. uncertainty lines (95% confidence)

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

  1. prediction band (95% confidence)

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

  1. save and show figure

plt.savefig('regression.png') plt.show()

March 16, 2018, at 04:25 PM by 45.56.3.173 -
Changed lines 32-34 from:

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.

to:

Joint Confidence Regions

The joint confidence region 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.

March 16, 2018, at 04:24 PM by 45.56.3.173 -
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$$y = a \exp{b\, x} + c$$

to:

$$y = a \exp{\left(b\, x\right)} + c$$

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

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.

to:

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

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.

March 16, 2018, at 04:23 PM by 45.56.3.173 -
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Use the data at the link below to create a linear and nonlinear regression.

to:

Create a linear model with unknown coefficients a (slope) and b (intercept). Fit the model to the data by minimizing the sum of squared errors between the predicted and measured y values.

$$y = a \, x + b$$

Show the linear regression with 95% confidence bands and 95% prediction bands. The confidence band is the confidence region for the correlation equation. The prediction band is the region that contains approximately 95% of the measurements. If another measurement is taken, there is a 95% chance that it falls within the prediction band. Use the following data set that is a table of comma separated values.

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

to:

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

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Nonlinear Parameter Confidence Region

to:

Nonlinear Regression

Create a nonlinear regression with unknown coefficients a, b, and c.

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

Similar to the linear regression, fit the model to the data by minimizing the sum of squared errors between the predicted and measured y values. Show the nonlinear regression with 95% confidence bands and 95% prediction bands.

Parameter Confidence

March 16, 2018, at 03:47 PM by 45.56.3.173 -
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to:
March 16, 2018, at 03:46 PM by 45.56.3.173 -
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Nonlinear Parameter Confidence Region

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.

The parameter confidence region is determined with an F-test that specifies an upper limit to the deviation from the optimal solution

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

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.


March 16, 2018, at 03:01 PM by 45.56.3.173 -
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(:title Regression Statistics with Python:) (:keywords data regression, uncertainty, Python, unumpy, linear regression, prediction band, confidence interval:) (:description Statistics for confidence interval and prediction band from a linear or nonlinear regression. The uncertainties package is used in Python to generate the confidence intervals.:)

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.

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

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

(:sourceend:) (:divend:)

Linear and nonlinear regression tutorials can also available in Python, Excel, and Matlab. A multivariate nonlinear regression case with multiple factors is available with example data for energy prices in Python. Click on the appropriate link for additional information.


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