confidence and prediction intervals with StatsModels

问题

I do this linear regression with StatsModels:

import numpy as np
import statsmodels.api as sm
from statsmodels.sandbox.regression.predstd import wls_prediction_std

n = 100

x = np.linspace(0, 10, n)
e = np.random.normal(size=n)
y = 1 + 0.5*x + 2*e
X = sm.add_constant(x)

re = sm.OLS(y, X).fit()
print(re.summary())

prstd, iv_l, iv_u = wls_prediction_std(re)

My questions are, iv_l and iv_u are the upper and lower confidence intervals or prediction intervals?

How I get others?

I need the confidence and prediction intervals for all points, to do a plot.

回答1:

update see the second answer which is more recent. Some of the models and results classes have now a get_prediction method that provides additional information including prediction intervals and/or confidence intervals for the predicted mean.

old answer:

iv_l and iv_u give you the limits of the prediction interval for each point.

Prediction interval is the confidence interval for an observation and includes the estimate of the error.

I think, confidence interval for the mean prediction is not yet available in statsmodels. (Actually, the confidence interval for the fitted values is hiding inside the summary_table of influence_outlier, but I need to verify this.)

Proper prediction methods for statsmodels are on the TODO list.

Addition

Confidence intervals are there for OLS but the access is a bit clumsy.

To be included after running your script:

from statsmodels.stats.outliers_influence import summary_table

st, data, ss2 = summary_table(re, alpha=0.05)

fittedvalues = data[:, 2]
predict_mean_se  = data[:, 3]
predict_mean_ci_low, predict_mean_ci_upp = data[:, 4:6].T
predict_ci_low, predict_ci_upp = data[:, 6:8].T

# Check we got the right things
print np.max(np.abs(re.fittedvalues - fittedvalues))
print np.max(np.abs(iv_l - predict_ci_low))
print np.max(np.abs(iv_u - predict_ci_upp))

plt.plot(x, y, 'o')
plt.plot(x, fittedvalues, '-', lw=2)
plt.plot(x, predict_ci_low, 'r--', lw=2)
plt.plot(x, predict_ci_upp, 'r--', lw=2)
plt.plot(x, predict_mean_ci_low, 'r--', lw=2)
plt.plot(x, predict_mean_ci_upp, 'r--', lw=2)
plt.show()

This should give the same results as SAS, http://jpktd.blogspot.ca/2012/01/nice-thing-about-seeing-zeros.html

回答2:

For test data you can try to use the following.

predictions = result.get_prediction(out_of_sample_df)
predictions.summary_frame(alpha=0.05)

I found the summary_frame() method buried here and you can find the get_prediction() method here. You can change the significance level of the confidence interval and prediction interval by modifying the "alpha" parameter.

I am posting this here because this was the first post that comes up when looking for a solution for confidence & prediction intervals – even though this concerns itself with test data rather.

Here's a function to take a model, new data, and an arbitrary quantile, using this approach:

def ols_quantile(m, X, q):
  # m: OLS model.
  # X: X matrix.
  # q: Quantile.
  #
  # Set alpha based on q.
  a = q * 2
  if q > 0.5:
    a = 2 * (1 - q)
  predictions = m.get_prediction(X)
  frame = predictions.summary_frame(alpha=a)
  if q > 0.5:
    return frame.obs_ci_upper
  return frame.obs_ci_lower

回答3:

You can get the prediction intervals by using LRPI() class from the Ipython notebook in my repo (https://github.com/shahejokarian/regression-prediction-interval).

You need to set the t value to get the desired confidence interval for the prediction values, otherwise the default is 95% conf. interval.

The LRPI class uses sklearn.linear_model's LinearRegression , numpy and pandas libraries.

There is an example shown in the notebook too.

回答4:

summary_frame and summary_table work well when you need exact results for a single quantile, but don't vectorize well. This will provide a normal approximation of the prediction interval (not confidence interval) and works for a vector of quantiles:

def ols_quantile(m, X, q):
  # m: Statsmodels OLS model.
  # X: X matrix of data to predict.
  # q: Quantile.
  #
  from scipy.stats import norm
  mean_pred = m.predict(X)
  se = np.sqrt(m.scale)
  return mean_pred + norm.ppf(q) * se

回答5:

You can calculate them based on results given by statsmodel and the normality assumptions.

Here is an example for OLS and CI for the mean value:

import statsmodels.api as sm
import numpy as np
from scipy import stats

#Significance level:
sl = 0.05
#Evaluate mean value at a required point x0. Here, at the point (0.0,2.0) for N_model=2:
x0 = np.asarray([1.0, 0.0, 2.0])# If you have no constant in your model, remove the first 1.0. For more dimensions, add the desired values.

#Get an OLS model based on output y and the prepared vector X (as in your notation):
model = sm.OLS(endog = y, exog = X )
results = model.fit()
#Get two-tailed t-values:
(t_minus, t_plus) = stats.t.interval(alpha = (1.0 - sl), df =  len(results.resid) - len(x0) )
y_value_at_x0 = np.dot(results.params, x0)
lower_bound = y_value_at_x0 + t_minus*np.sqrt(results.mse_resid*( np.dot(np.dot(x0.T,results.normalized_cov_params),x0) ))
upper_bound = y_value_at_x0 +  t_plus*np.sqrt(results.mse_resid*( np.dot(np.dot(x0.T,results.normalized_cov_params),x0) ))

You can wrap a nice function around this with input results, point x0 and significance level sl.

I am unsure now if you can use this for WLS() since there are extra things happening there.

Ref: Ch3 in [D.C. Montgomery and E.A. Peck. “Introduction to Linear Regression Analysis.” 4th. Ed., Wiley, 1992].

来源：https://stackoverflow.com/questions/17559408/confidence-and-prediction-intervals-with-statsmodels

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