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%matplotlib inline
import numpy as np
import matplotlib.pylab as plt



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import scipy.optimize
import pandas as pd

What does the following code do?



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d = pd.read_csv("data/dataset_0.csv")
fig, ax = plt.subplots()
ax.plot(d.x,d.y,'o')

What does the following code do?



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def linear(x,a,b):
    return a + b*x

What does the following code do?



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def linear(x,a,b):
    return a + b*x

def linear_r(param,x,y):
    return linear(x,param[0],param[1]) - y

What does the following code do?



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def linear_r(param,x,y):                     # copied from previous cell
    return linear(x,param[0],param[1]) - y   # copied from previous cell

param_guesses = [1,1]
fit = scipy.optimize.least_squares(linear_r,param_guesses,
                                   args=(d.x,d.y))
fit_a = fit.x[0]
fit_b = fit.x[1]
sum_of_square_residuals = fit.cost

What does the following code do?

What the heck is linspace?
What are we plotting?



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x_range = np.linspace(np.min(d.x),np.max(d.x),100)

fig, ax = plt.subplots()
ax.plot(d.x,d.y,"o")
ax.plot(x_range,linear(x_range,fit_a,fit_b))

Put together



In [7]:

    
def linear(x,a,b):
    """Linear model of x using a (slope) and b (intercept)"""
    return a + b*x

def linear_r(param,x,y):
    """Residuals function for linear"""
    return linear(x,param[0],param[1]) - y

fig, ax = plt.subplots()

# Read data
d = pd.read_csv("data/dataset_0.csv")
ax.plot(d.x,d.y,'o')

# Perform regression
param_guesses = [1,1]
fit = scipy.optimize.least_squares(linear_r,param_guesses,args=(d.x,d.y))
fit_a = fit.x[0]
fit_b = fit.x[1]
sum_of_square_residuals = fit.cost

# Plot result
x_range = np.linspace(np.min(d.x),np.max(d.x),100)
ax.plot(x_range,linear(x_range,fit_a,fit_b))

fit









    Out[7]:





 active_mask: array([0., 0.])
        cost: 4.261679630541931
         fun: array([ 1.10967611,  0.37884357,  0.7262888 ,  0.19610326, -0.07363268,
        0.87402953, -0.04205402,  0.34759648, -0.25938691,  0.03667523,
       -0.17507308, -0.68958901, -0.08836881, -0.34453537, -0.55445953,
       -0.28176203, -0.30205479, -0.53989842,  0.02746354, -0.86197952,
       -0.2592627 , -0.24275086, -0.42436478, -0.32125392,  0.05872358,
       -0.29706174, -0.14169496, -0.89286187,  0.2943461 , -0.30268024,
       -0.46564262,  0.41435537,  0.40923643, -0.0323235 ,  0.25418016,
        0.81366676,  0.28895853,  0.0650563 ,  0.60869764,  0.33681213,
        0.35198181])
        grad: array([ 1.17124844e-08, -2.48689958e-14])
         jac: array([[ 1.        ,  0.        ],
       [ 1.        ,  0.25      ],
       [ 1.        ,  0.5       ],
       [ 1.        ,  0.75      ],
       [ 1.00000001,  1.        ],
       [ 1.        ,  1.25      ],
       [ 1.        ,  1.5       ],
       [ 0.99999999,  1.75      ],
       [ 1.        ,  2.        ],
       [ 1.00000001,  2.25      ],
       [ 1.        ,  2.5       ],
       [ 1.00000001,  2.75      ],
       [ 0.99999999,  3.        ],
       [ 0.99999999,  3.25      ],
       [ 0.99999999,  3.5       ],
       [ 0.99999999,  3.75      ],
       [ 0.99999999,  4.        ],
       [ 0.99999999,  4.25      ],
       [ 1.00000001,  4.5       ],
       [ 1.00000001,  4.75      ],
       [ 1.00000001,  5.        ],
       [ 1.00000001,  5.25      ],
       [ 1.00000001,  5.5       ],
       [ 0.99999999,  5.75      ],
       [ 0.99999999,  6.        ],
       [ 0.99999999,  6.25      ],
       [ 0.99999999,  6.5       ],
       [ 0.99999999,  6.75      ],
       [ 0.99999999,  7.        ],
       [ 0.99999999,  7.25      ],
       [ 1.00000001,  7.5       ],
       [ 1.00000001,  7.75      ],
       [ 1.00000001,  8.        ],
       [ 1.00000001,  8.25      ],
       [ 1.00000001,  8.5       ],
       [ 0.99999999,  8.75      ],
       [ 1.00000001,  9.        ],
       [ 1.00000001,  9.25      ],
       [ 1.00000001,  9.5       ],
       [ 1.00000001,  9.75      ],
       [ 1.00000001, 10.        ]])
     message: 'Both `ftol` and `xtol` termination conditions are satisfied.'
        nfev: 3
        njev: 3
  optimality: 1.1712484371262377e-08
      status: 4
     success: True
           x: array([1.13006323, 0.31994236])

For your assigned model:

Write a function and residuals function
Estimate the parameters of the model
Plot the data and the model on the same graph
Write the SSR and number of parameters on the board
If you finish early: plot the residuals and decide if you like your model
If you're still waiting: try to figure out which model best fits dataset_1.csv

$y = a \Big ( \frac{bx}{1 + bx} \Big )$

$y = a \Big ( \frac{bx^{c}}{1 + bx^{c}} \Big )$

$y = a(1 - e^{-bx})$

$y = a + bx^{2} + cx^{3}$

$y = a + bx^{2} + cx^{3} + dx^{4}$

$y = asin(bx + c)$

$y = aln(x + b)$

$y = aln(bx + c)$



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