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I'm trying Sklearn's RANSAC algorithm implementation to produce a simple linear regression fit with built-in outlier detection/rejection (code below). This is what my raw data looks like:

enter image description here

Even using max_trials=10000000 (the Maximum number of iterations for random sample selection) the estimated coefficients (slope, intercept) and the R^2 & RMSE statistics vary quite a bit with each new run of the same exact raw data.

I would've assumed the results converged if one gave it enough iterations, but apparently this is not true. Are 10 million iterations not enough? Is my data too noisy? Am I missing some parameter I should fix in RANSACRegressor or is this just the nature of the RANSAC algorithm?

import numpy as np
from sklearn import linear_model
from sklearn.metrics import r2_score, mean_squared_error

# Data
x = [-0.099000000000000005, 0.094, 1.3129999999999999, 0.096000000000000002, 0.088999999999999996, -0.002, 0.11, 2.1440000000000001, -0.0040000000000000001, 0.245, 0.66600000000000004, 1.0029999999999999, 0.71899999999999997, 1.9359999999999999, 0.098000000000000004, 0.39400000000000002, 0.35699999999999998, 1.284, -0.027, 0.123, 1.837, 0.85299999999999998, 1.0669999999999999, 0.126, 0.078, -0.001, 0.318, 0.39900000000000002, 1.4259999999999999, 1.4970000000000001, 1.0860000000000001, -0.10000000000000001, 1.296, 0.44700000000000001, 0.23599999999999999, 0.26300000000000001, 1.5589999999999999, 1.1200000000000001, 1.2390000000000001, 1.0920000000000001, 1.1779999999999999, 1.915, -0.10299999999999999, 0.156, 0.53200000000000003, 1.871, -1.2170000000000001, 0.38]
y = [-4.5064700000000002, -3.8593700000000002, -3.5898699999999977, -3.7799699999999987, -3.7595699999999983, -3.8206699999999998, -3.7413699999999981, -3.0731699999999975, -3.7439699999999974, -3.6922700000000006, -3.7251700000000003, -3.6609700000000007, -3.8754699999999982, -3.1616699999999973, -3.8635699999999993, -3.8146700000000013, -3.6913699999999992, -3.5147699999999986, -3.852369999999997, -3.8125699999999991, -0.58567000000000036, -3.81447, -3.6871699999999983, -3.6793700000000005, -3.8035699999999988, -3.8506699999999991, -3.6703700000000019, -3.7025699999999979, -2.6628699999999981, -3.5626700000000007, -3.0997699999999959, 1.2492300000000007, -3.7071510000000014, -3.9486509999999981, -3.7250509999999988, -3.9179509999999986, -3.5861509999999974, -3.8082509999999985, -3.7271509999999992, -3.780851000000002, -3.6115509999999986, -3.4124880000000033, -3.8527880000000003, -3.7375880000000006, -3.8264880000000012, -3.4370879999999993, -4.0668880000000005, -3.8614880000000014]
# Proper shape
x, y = np.array(x).reshape(-1, 1), np.array(y).reshape(-1, 1)

# Fit linear model with RANSAC algorithm
ransac = linear_model.RANSACRegressor(min_samples=2, max_trials=10000000)

# Run regression 10 times with the same raw data
for _ in range(10):
    ransac.fit(x, y)

    # Estimated coefficients
    m, c = float(ransac.estimator_.coef_), float(ransac.estimator_.intercept_)
    print("\nm={:.3f}, c={:.3f}".format(m, c))

    # R^2 & RMSE statistics
    inlier_mask = ransac.inlier_mask_
    x_accpt, y_accpt = x[inlier_mask], y[inlier_mask]
    y_predict = m * x_accpt + c
    R2 = r2_score(y_accpt, y_predict)
    RMSE = np.sqrt(mean_squared_error(y_accpt, y_predict))
    print("R^2: {:.3f}, RMSE: {:.3f}".format(R2, RMSE))
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migrated from stackoverflow.com Jan 25 '18 at 17:53

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Here are some observations:

Increasing min_sample makes the model much more stable. I believe the model will find it difficult to decide on what is an outlier when only sampling from from a minimum of two different samples.

I dropped this into a for-loop to iterate over some min_samples and this real steadied the model. Here it is:

import numpy as np
from sklearn import linear_model
from sklearn.metrics import r2_score, mean_squared_error

# Data
x = [-0.099000000000000005, 0.094, 1.3129999999999999, 0.096000000000000002, 0.088999999999999996, -0.002, 0.11, 2.1440000000000001, -0.0040000000000000001, 0.245, 0.66600000000000004, 1.0029999999999999, 0.71899999999999997, 1.9359999999999999, 0.098000000000000004, 0.39400000000000002, 0.35699999999999998, 1.284, -0.027, 0.123, 1.837, 0.85299999999999998, 1.0669999999999999, 0.126, 0.078, -0.001, 0.318, 0.39900000000000002, 1.4259999999999999, 1.4970000000000001, 1.0860000000000001, -0.10000000000000001, 1.296, 0.44700000000000001, 0.23599999999999999, 0.26300000000000001, 1.5589999999999999, 1.1200000000000001, 1.2390000000000001, 1.0920000000000001, 1.1779999999999999, 1.915, -0.10299999999999999, 0.156, 0.53200000000000003, 1.871, -1.2170000000000001, 0.38] * 10
y = [-4.5064700000000002, -3.8593700000000002, -3.5898699999999977, -3.7799699999999987, -3.7595699999999983, -3.8206699999999998, -3.7413699999999981, -3.0731699999999975, -3.7439699999999974, -3.6922700000000006, -3.7251700000000003, -3.6609700000000007, -3.8754699999999982, -3.1616699999999973, -3.8635699999999993, -3.8146700000000013, -3.6913699999999992, -3.5147699999999986, -3.852369999999997, -3.8125699999999991, -0.58567000000000036, -3.81447, -3.6871699999999983, -3.6793700000000005, -3.8035699999999988, -3.8506699999999991, -3.6703700000000019, -3.7025699999999979, -2.6628699999999981, -3.5626700000000007, -3.0997699999999959, 1.2492300000000007, -3.7071510000000014, -3.9486509999999981, -3.7250509999999988, -3.9179509999999986, -3.5861509999999974, -3.8082509999999985, -3.7271509999999992, -3.780851000000002, -3.6115509999999986, -3.4124880000000033, -3.8527880000000003, -3.7375880000000006, -3.8264880000000012, -3.4370879999999993, -4.0668880000000005, -3.8614880000000014]*10
# Proper shape
x, y = np.array(x).reshape(-1, 1), np.array(y).reshape(-1, 1)

# Fit linear model with RANSAC algorithm

# Run regression 10 times with the same raw data
for n in range(2, 25):
    RMSE = []

    ransac = linear_model.RANSACRegressor(min_samples=n, max_trials=10000000)


    for _ in range(10):
        ransac.fit(x, y)

        # Estimated coefficients
        m, c = float(ransac.estimator_.coef_), float(ransac.estimator_.intercept_)
#         print("\nm={:.3f}, c={:.3f}".format(m, c))

        # R^2 & RMSE statistics
        inlier_mask = ransac.inlier_mask_
        x_accpt, y_accpt = x[inlier_mask], y[inlier_mask]
        y_predict = m * x_accpt + c
        R2 = r2_score(y_accpt, y_predict)
        RMSE.append(np.sqrt(mean_squared_error(y_accpt, y_predict)))
#         print("R^2: {:.3f}, RMSE: {:.3f}".format(R2, RMSE))
    print('min_samples: %d' % n, 'std: %f' % np.std(RMSE))

and the output,

min_samples: 2 std: 0.003821
min_samples: 3 std: 0.003895
min_samples: 4 std: 0.002477
min_samples: 5 std: 0.002746
min_samples: 6 std: 0.003013
min_samples: 7 std: 0.002673
min_samples: 8 std: 0.002763
min_samples: 9 std: 0.001543
min_samples: 10 std: 0.001367
min_samples: 11 std: 0.001390
min_samples: 12 std: 0.001390
min_samples: 13 std: 0.001486
min_samples: 14 std: 0.001390
min_samples: 15 std: 0.001214
min_samples: 16 std: 0.001390
min_samples: 17 std: 0.000000
min_samples: 18 std: 0.000910
min_samples: 19 std: 0.000000

and a graph of the trend, x is min_samples and y the standard deviation of the RMSE.

Model stability vs min_sample

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  • $\begingroup$ I too realized this was perhaps a better question for CV and I flagged it so that a moderator would move it over there, it never happened though. I run a few tests setting the min_samples parameter to a range of values but it doesn't seem to help much. The values themselves keep varying even when I use up to 90% of the data to produce the fit. Thank you anyway! $\endgroup$ – Gabriel Jan 25 '18 at 11:14
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    $\begingroup$ I have flagged it also - we will see. I would like to see a more stats/technical answer to your question! $\endgroup$ – josh Jan 25 '18 at 11:30
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You are missing the random seed parameter - RANSAC uses random numbers to select the samples to use for the iterations.

So what you are looking for is:

ransac = linear_model.RANSACRegressor(min_samples=n, max_trials=10000000, random_state= num)

Where num is an integer of your choosing, you can trial as many as you like in a loop and pick the best one as well.

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  • $\begingroup$ If I understand correctly that would fix the output result, but then I'd have to pick some number to initialize the regressor and I'm back to square one: how do I pick this number? $\endgroup$ – Gabriel Jul 13 '18 at 14:45

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