How do I plot the Mean Squared Error from created functions?

Question

I'm trying to plot the mean squared error for different theta values.

This is my df called tips:

tips = array([ 1.01,  1.66,  3.5 ,  3.31,  3.61,  4.71,  2.  ,  3.12,  1.96,
            3.23,  1.71,  5.  ,  1.57,  3.  ,  3.02,  3.92,  1.67,  3.71,
            3.5 ,  3.35,  4.08,  2.75,  2.23,  7.58,  3.18,  2.34,  2.  ,
            2.  ,  4.3 ,  3.  ,  1.45,  2.5 ,  3.  ,  2.45,  3.27,  3.6 ,
            2.  ,  3.07,  2.31,  5.  ,  2.24,  2.54,  3.06,  1.32,  5.6 ,
            3.  ,  5.  ,  6.  ,  2.05,  3.  ,  2.5 ,  2.6 ,  5.2 ,  1.56,
            4.34,  3.51,  3.  ,  1.5 ,  1.76,  6.73,  3.21,  2.  ,  1.98,
            3.76,  2.64,  3.15,  2.47,  1.  ,  2.01,  2.09,  1.97,  3.  ,
            3.14,  5.  ,  2.2 ,  1.25,  3.08,  4.  ,  3.  ,  2.71,  3.  ,
            3.4 ,  1.83,  5.  ,  2.03,  5.17,  2.  ,  4.  ,  5.85,  3.  ,
            3.  ,  3.5 ,  1.  ,  4.3 ,  3.25,  4.73,  4.  ,  1.5 ,  3.  ,
            1.5 ,  2.5 ,  3.  ,  2.5 ,  3.48,  4.08,  1.64,  4.06,  4.29,
            3.76,  4.  ,  3.  ,  1.  ,  4.  ,  2.55,  4.  ,  3.5 ,  5.07,
            1.5 ,  1.8 ,  2.92,  2.31,  1.68,  2.5 ,  2.  ,  2.52,  4.2 ,
            1.48,  2.  ,  2.  ,  2.18,  1.5 ,  2.83,  1.5 ,  2.  ,  3.25,
            1.25,  2.  ,  2.  ,  2.  ,  2.75,  3.5 ,  6.7 ,  5.  ,  5.  ,
            2.3 ,  1.5 ,  1.36,  1.63,  1.73,  2.  ,  2.5 ,  2.  ,  2.74,
            2.  ,  2.  ,  5.14,  5.  ,  3.75,  2.61,  2.  ,  3.5 ,  2.5 ,
            2.  ,  2.  ,  3.  ,  3.48,  2.24,  4.5 ,  1.61,  2.  , 10.  ,
            3.16,  5.15,  3.18,  4.  ,  3.11,  2.  ,  2.  ,  4.  ,  3.55,
            3.68,  5.65,  3.5 ,  6.5 ,  3.  ,  5.  ,  3.5 ,  2.  ,  3.5 ,
            4.  ,  1.5 ,  4.19,  2.56,  2.02,  4.  ,  1.44,  2.  ,  5.  ,
            2.  ,  2.  ,  4.  ,  2.01,  2.  ,  2.5 ,  4.  ,  3.23,  3.41,
            3.  ,  2.03,  2.23,  2.  ,  5.16,  9.  ,  2.5 ,  6.5 ,  1.1 ,
            3.  ,  1.5 ,  1.44,  3.09,  2.2 ,  3.48,  1.92,  3.  ,  1.58,
            2.5 ,  2.  ,  3.  ,  2.72,  2.88,  2.  ,  3.  ,  3.39,  1.47,
            3.  ,  1.25,  1.  ,  1.17,  4.67,  5.92,  2.  ,  2.  ,  1.75,
            3.  ])

This is my squared loss function:

def squared_loss(y_obs, theta):
"""
Calculate the squared loss of the observed data and a summary statistic.

Parameters
------------
y_obs: an observed value
theta : some constant representing a summary statistic

Returns
------------
The squared loss between the observation and the summary statistic.
"""
return (y_obs - theta) ** 2

This is my Mean Squared Error Function:

def mean_squared_error(theta, data):
    
    return sum(squared_loss(data, theta)) / len(data)

This is the problem: In the cell below plot the mean squared error for different theta values. Note that theta_values are given. Make sure to label the axes on your plot. Remember to use the tips variable we defined earlier.

theta_values = np.linspace(0, 6, 100)

plt.plot(mean_squared_error(theta_values, tips))

This gives me the following: ValueError: operands could not be broadcast together with shapes (244,) (100,)

If the plot is correct, the observed minimization point should be 3. Does anyone know what I can do to get my plot to show up? I was thinking something like a for loop but not really sure.

Thanks!

Edit: Trying 244 in theta_values, though theta_values should be a given and not be touched.

Answer 1

The error comes because you are trying to broadcast 2 arrays of different sizes using your mean squared error function. Your tips df has size 244, and when you created your theta values array you set it to 100 equally spaced values between 0 and 6, resulting in a size of 100.

By using

theta_values = np.linspace(0, 6, 244)

you will create a theta_values variable with 244 values which will properly map to the tips dataframe and not cause an issue when calculating your MSE.

EDIT: To accommodate OP update, assuming the plot is meant to be squared error (SE) vs theta. The entire code to compute is shown below; along with the output plot. Reminder what is plotted is the squared error (i.e. error between y_true (assumed to be tips) and y_pred (assumed to be theta) squared) versus theta. The output does seem to show less fluctuation around 3, (as suggested by OP) but more clarification from OP is required.

import numpy as np
import matplotlib.pyplot as plt

tips = np.array([ 1.01,  1.66,  3.5 ,  3.31,  3.61,  4.71,  2.  ,  3.12,  1.96,
               3.23,  1.71,  5.  ,  1.57,  3.  ,  3.02,  3.92,  1.67,  3.71,
               3.5 ,  3.35,  4.08,  2.75,  2.23,  7.58,  3.18,  2.34,  2.  ,
               2.  ,  4.3 ,  3.  ,  1.45,  2.5 ,  3.  ,  2.45,  3.27,  3.6 ,
               2.  ,  3.07,  2.31,  5.  ,  2.24,  2.54,  3.06,  1.32,  5.6 ,
               3.  ,  5.  ,  6.  ,  2.05,  3.  ,  2.5 ,  2.6 ,  5.2 ,  1.56,
               4.34,  3.51,  3.  ,  1.5 ,  1.76,  6.73,  3.21,  2.  ,  1.98,
               3.76,  2.64,  3.15,  2.47,  1.  ,  2.01,  2.09,  1.97,  3.  ,
               3.14,  5.  ,  2.2 ,  1.25,  3.08,  4.  ,  3.  ,  2.71,  3.  ,
               3.4 ,  1.83,  5.  ,  2.03,  5.17,  2.  ,  4.  ,  5.85,  3.  ,
               3.  ,  3.5 ,  1.  ,  4.3 ,  3.25,  4.73,  4.  ,  1.5 ,  3.  ,
               1.5 ,  2.5 ,  3.  ,  2.5 ,  3.48,  4.08,  1.64,  4.06,  4.29,
               3.76,  4.  ,  3.  ,  1.  ,  4.  ,  2.55,  4.  ,  3.5 ,  5.07,
               1.5 ,  1.8 ,  2.92,  2.31,  1.68,  2.5 ,  2.  ,  2.52,  4.2 ,
               1.48,  2.  ,  2.  ,  2.18,  1.5 ,  2.83,  1.5 ,  2.  ,  3.25,
               1.25,  2.  ,  2.  ,  2.  ,  2.75,  3.5 ,  6.7 ,  5.  ,  5.  ,
               2.3 ,  1.5 ,  1.36,  1.63,  1.73,  2.  ,  2.5 ,  2.  ,  2.74,
               2.  ,  2.  ,  5.14,  5.  ,  3.75,  2.61,  2.  ,  3.5 ,  2.5 ,
               2.  ,  2.  ,  3.  ,  3.48,  2.24,  4.5 ,  1.61,  2.  , 10.  ,
               3.16,  5.15,  3.18,  4.  ,  3.11,  2.  ,  2.  ,  4.  ,  3.55,
               3.68,  5.65,  3.5 ,  6.5 ,  3.  ,  5.  ,  3.5 ,  2.  ,  3.5 ,
               4.  ,  1.5 ,  4.19,  2.56,  2.02,  4.  ,  1.44,  2.  ,  5.  ,
               2.  ,  2.  ,  4.  ,  2.01,  2.  ,  2.5 ,  4.  ,  3.23,  3.41,
               3.  ,  2.03,  2.23,  2.  ,  5.16,  9.  ,  2.5 ,  6.5 ,  1.1 ,
               3.  ,  1.5 ,  1.44,  3.09,  2.2 ,  3.48,  1.92,  3.  ,  1.58,
               2.5 ,  2.  ,  3.  ,  2.72,  2.88,  2.  ,  3.  ,  3.39,  1.47,
               3.  ,  1.25,  1.  ,  1.17,  4.67,  5.92,  2.  ,  2.  ,  1.75,
               3.  ])

theta_values = np.linspace(0, 6, 244)


def sqr_err(y_true, y_pred):
    """

    :param y_true: true values of y
    :param y_pred: predicted values of y
    :return: array of lenght original data containing mean squared error for each predictions
    """
    if len(y_true) != len(y_pred):
        raise IndexError("Mismathced array sizes, you inputted arrays with sizes {} and {}".format(len(y_true),
                                                                                                  len(y_pred)))
    else:
        length = len(y_true)

    sqrerror_out = [(y_pred[i]-y_true[i])**2 for i in range(length)]

    return np.array(sqrerror_out)


theta_value = np.linspace(0, 6, 244)

Squared_error = sqr_err(tips, theta_value)

plt.figure()
plt.plot(theta_values, Squared_error)
plt.xlabel('Theta Values')
plt.ylabel('Squared Error')
plt.show()