Documentation#
PYCSE#
Module containing useful scientific and engineering functions.
Linear regression
Nonlinear regression.
Differential equation solvers.
See http://kitchingroup.cheme.cmu.edu/pycse
Copyright 2020, John Kitchin (see accompanying license files for details).
- pycse.PYCSE.Rsquared(y, Y)#
Return R^2, or coefficient of determination.
y is a 1d array of observations. Y is a 1d array of predictions from a model.
- Returns:
- The R^2 value for the fit.
- pycse.PYCSE.bic(x, y, model, popt)#
Compute the Bayesian information criterion (BIC).
- Parameters:
- modelfunction(x, …) returns prediction for y
- poptoptimal parameters
- yarray, known y-values
- Returns:
- pycse.PYCSE.ivp(f, tspan, y0, *args, **kwargs)#
Solve an ODE initial value problem.
- Parameters:
- ffunction
- callable y’(x, y) = f(x, y)
- tspanarray
- The x points you want the solution at. The first and last points are used in
- tspan in solve_ivp.
- y0array
- Initial conditions
- *argstype
- arbitrary positional arguments to pass to solve_ivp
- **kwargstype arbitrary kwargs to pass to solve_ivp.
- max_step is set to be the min diff of tspan. dense_output is set to True.
- t_eval is set to the array specified in tspan.
- Returns:
- solution from solve_ivp
- pycse.PYCSE.lbic(X, y, popt)#
Compute the Bayesian information criterion for a linear model.
- Returns:
- BICfloat
- pycse.PYCSE.nlinfit(model, x, y, p0, alpha=0.05, **kwargs)#
Nonlinear regression with confidence intervals.
- Parameters:
- modelfunction f(x, p0, p1, …) = y
- xarray of the independent data
- yarray of the dependent data
- p0array of the initial guess of the parameters
- alpha100*(1 - alpha) is the confidence interval
i.e. alpha = 0.05 is 95% confidence
- kwargs are passed to curve_fit.
- Returns:
- [p, pint, SE]
p is an array of the fitted parameters pint is an array of confidence intervals SE is an array of standard errors for the parameters.
- pycse.PYCSE.nlpredict(X, y, model, loss, popt, xnew, alpha=0.05, ub=1e-05, ef=1.05)#
Prediction error for a nonlinear fit.
- Parameters:
- modelmodel function with signature model(x, …)
- lossloss function the model was fitted with loss(…)
- poptthe optimized paramters
- xnewx-values to predict at
- alphaconfidence level, 95% = 0.05
- ubupper bound for smallest allowed Hessian eigenvalue
- efeigenvalue factor for scaling Hessian
- This function uses numdifftools for the Hessian and Jacobian.
- See https://en.wikipedia.org/wiki/Prediction_interval#Unknown_mean,_unknown_variance
- Returns:
- y, yint, se
- ypredicted values
- yintprediction interval at alpha confidence interval
- sestandard error of prediction
- pycse.PYCSE.polyfit(x, y, deg, alpha=0.05, *args, **kwargs)#
Least squares polynomial fit with parameter confidence intervals.
- Parameters:
- xarray_like, shape (M,)
x-coordinates of the M sample points
(x[i], y[i]).- yarray_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column.
- degint
Degree of the fitting polynomial
- *args and **kwargs are passed to regress.
- Returns:
- [b, bint, se]
- b is a vector of the fitted parameters
- bint is a 2D array of confidence intervals
- se is an array of standard error for each parameter.
- pycse.PYCSE.predict(X, y, pars, XX, alpha=0.05, ub=1e-05, ef=1.05)#
Prediction interval for linear regression.
Based on the delta method.
- Parameters:
- Xknown x-value array, one row for each y-point
- yknown y-value array
- parsfitted parameters
- XXx-value array to make predictions for
- alphaconfidence level, 95% = 0.05
- ubupper bound for smallest allowed Hessian eigenvalue
- efeigenvalue factor for scaling Hessian
- See See https://en.wikipedia.org/wiki/Prediction_interval#Unknown_mean,_unknown_variance
- Returns
- y, yint, pred_se
- ythe predicted values
- yint: confidence interval
- pred_se: std error on predictions.
- pycse.PYCSE.regress(A, y, alpha=0.05, *args, **kwargs)#
Linear least squares regression with confidence intervals.
Solve the matrix equation (A p = y) for p.
The confidence intervals account for sample size using a student T multiplier.
This code is derived from the descriptions at http://www.weibull.com/DOEWeb/confidence_intervals_in_multiple_linear_regression.htm and http://www.weibull.com/DOEWeb/estimating_regression_models_using_least_squares.htm
- Parameters:
- Aa matrix of function values in columns, e.g.
A = np.column_stack([T**0, T**1, T**2, T**3, T**4])
- ya vector of values you want to fit
- alpha100*(1 - alpha) confidence level
- args and kwargs are passed to np.linalg.lstsq
- Returns:
- [b, bint, se]
- b is a vector of the fitted parameters
- bint is an array of confidence intervals. The ith row is for the ith parameter.
- se is an array of standard error for each parameter.
pycse.utils#
Provides utility functions in pycse.
Fuzzy comparisons for float numbers.
An ignore exception decorator
A handy function to read a google sheet.
- pycse.utils.feq(x, y, epsilon=np.float64(2.220446049250313e-16))#
Fuzzy equals.
x == y with tolerance
- pycse.utils.fge(x, y, epsilon=np.float64(2.220446049250313e-16))#
Fuzzy greater than or equal to .
x >= y with tolerance
- pycse.utils.fgt(x, y, epsilon=np.float64(2.220446049250313e-16))#
Fuzzy greater than.
x > y with tolerance
- pycse.utils.fle(x, y, epsilon=np.float64(2.220446049250313e-16))#
Fuzzy less than or equal to.
x <= y with tolerance
- pycse.utils.flt(x, y, epsilon=np.float64(2.220446049250313e-16))#
Fuzzy less than.
x < y with tolerance
- pycse.utils.ignore_exception(*exceptions)#
Ignore exceptions on decorated function.
>>> with ignore_exception(ZeroDivisionError): ... print(1/0)
- pycse.utils.read_gsheet(url, *args, **kwargs)#
Return a dataframe for the Google Sheet at url.
args and kwargs are passed to pd.read_csv The url should be viewable by anyone with the link.