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FitLinear and FitQuad
Classes are used for fitting a linear a quadratic trend on O-C diagram caused by mass transfer. It is also possible to calculate O-C values from observed times of minima (O ≡ tO). The class FitLinear could also be used for removing the linear trend in the data and then fit the only quadratic trend, see Example.
Functions and parameters of both classes are nearly identical. In the case, that some function is only in one function, it is highlighted.
For initialization these classes, it is necessary to give observed times of minima (O)
and linear ephemeris of minima, i. e. time of reference minimum and period. If the values
of O-C are already calculated, they could be given as the next parameter. If not, they
will be automatically calculated from given times. These values could be immediately
used in other calculations, they are stored in variable oc. It is also possible to give errors
or uncertainties of times of minima (and also O-Cs). Eventually, if there are given only
weights of individual points, it is possible to enter error as an inverted value of weight.
| Parameter | Type | Description |
|---|---|---|
| t | numpy.array or list | Observed times of minima (O). |
| t0 | float | Time of reference minimum. |
| P | float | Period of the system. |
| oc | numpy.array or list, optional | Calculated values of O-Cs in days. |
| err | numpy.array or list, optional | Errors of O-C values or times O. |
The basic function for fitting linear trend in O-C diagram. This function is only
available in a class FitLinear. New values of linear ephemeris are written to variable params;
their uncertainties are in variable params_err. The new times (C) are also
calculated stored in variable tC and new values of O-C stored in variable new_oc.
| Parameter | Type | Description |
|---|---|---|
| Output | numpy.array | New values of O-C calculated on the basis of new linear ephemeris in days. |
The basic function to fit the quadratic trend in the O-C diagram. This function is
only available in a class FitQuad. New values of Q and linear ephemeris are written
to variable params; their uncertainties are in variable params_err. The new times (C) are
also calculated stored in variable tC and new values of O-C stored in variable new_oc.
| Parameter | Type | Description |
|---|---|---|
| Output | numpy.array | New values of O-C for calculated Q and linear ephemeris in days. |
The function for fitting linear / quadratic trend in O-C diagrams using robust regression.
It is suitable to use this function in cases if there are a lot of outlier points in the
data. New values of fitted parameters are written to variable params; their uncertainties
are in variable params_err. The new times (C) are also calculated stored in variable tC
and new values of O-C stored in variable new_oc.
| Parameter | Type | Description |
|---|---|---|
| n iter | integer, optional | Number of iterations. |
| Output | numpy.array | New values of O-C. |
OCFit.FitLinear.FitMCMC(n_iter , limits, steps, fit_params = None, burn = 0, binn = 1, visible = True, db = None)
OCFit.FitQuad.FitMCMC(n_iter , limits, steps, fit_params = None, burn = 0, binn = 1, visible = True, db = None)
The function for fitting linear / quadratic trend in O-C diagram using Monte Carlo
(MC) method. Running of this function is possible only if the initial values of fitted
parameters were calculated. They could be obtained from fitting using function FitLinear
/ FitQuad or FitRobust. New values of fitted parameters are written to variable params;
their uncertainties are in variable params_err. The new times (C) are also calculated stored
in variable tC and new values of O-C stored in variable new_oc.
More detail description of the Monte Carlo method is in a part about the class OCFit.
| Parameter | Type | Description |
|---|---|---|
| n_iter | float | Number of MC steps. |
| limits | dictionary | Limits for fitted parameters. |
| steps | dictionary | The steps for fitted parameters. |
| fit_params | list | List of fitted parameters. Default value ['P', 't0'] or ['Q', 'P', 't0']. |
| burn | float, optional | Number of MC steps which will be reject from calculation. |
| binn | float, optional | The size of one block for binning. |
| db | string, optional | Name of file for saving MC sampling. |
| visible | boolean, optional | Display progress of fitting and summary of results from pymc. |
| Output | numpy.array | New values of O-C. |
Function for detail analysis of MC sampling obtained by FitMCMC. It will be generated a plot with a trace of fitting and common plot with trace and histogram for each of the fitted parameters. An also, the file with brief information about sampling, obtained values and errors of the parameter will be created. It is also possible to plot figures for Geweke analysis (details in Geweke (1992)).
Few common figures for all fitted parameters will be generated, too. There are histograms, deviances, correlation plots and 2D histograms displaying the confidence regions. For 2D histograms, the function ConfInt is used, everyone else is generated by the class TraceAnalysis from PyAstronomy. The last output is a table with correlation coefficients between fitted parameters.
| Parameter | Type | Description |
|---|---|---|
| db | string | Name of file with saved MC sampling. |
| eps | boolean, optional | Save figures also in eps format. |
| geweke | boolean, optional | Geweke analysis. |
Calculation of epoch of all observed minima. This function is called during the
initialization of the class. Obtained epochs are saved to variable epoch.
| Parameter | Type | Description |
|---|---|---|
| Output | numpy.array | Calculated epochs. |
Calculation of phase curve on the basis of given linear ephemeris. The final phase curve could be display on a figure.
| Parameter | Type | Description |
|---|---|---|
| P | float | Period. |
| t0 | float | Time of reference minimum. |
| plot | boolean, optional | Plot also a figure or only calculated phase curve. |
| Output | numpy.array | Phases and values of O-C. |
Calculation of errors of input data on a basis of values of fitted parameters. This function could not be used before fitting the data. It could be done using the function FitLinear / FitQuad or FitRobust. For all point, the same error will be assigned. The size of error will be set that the value of reduced squared error chi2r will be equal to 1.
The function is good to use if there is the only linear / quadratic trend in the data. Subsequently, during the repeated fitting better values of errors of parameters are obtained. It is necessary to use this function before fitting using the MC method by function FitMCMC if the errors of input data were not given during initialization.
| Parameter | Type | Description |
|---|---|---|
| Output | numpy.array | Calculated errors. |
Re-normalization of given errors of input data on a basis of values of fitted parameters. This function could not be used before fitting the data. It could be done using function FitLinear / FitQuad or FitRobust. The size of error will be set that the value of reduced squared error chi2r will be equal to 1. For plotting the figure, original values of errors are used.
The function is good to use if there is the only linear / quadratic trend in the data. Subsequently, during the repeated fitting better values of errors of parameters are obtained. It is useful to use this function before fitting using the MC method by function FitMCMC if the errors of input data are wrong calculated (under- or over-estimated).
| Parameter | Type | Description |
|---|---|---|
| Output | numpy.array | Calculated errors. |
Adding the weights and normalization of errors on a basis of values of fitted parameters. This function could not be used before fitting the data. It could be done using function FitLinear / FitQuad or FitRobust. The size of error will be set that the value of reduced squared error chi2r will be equal to 1.
The function is good to use if there is the only linear / quadratic trend in the data. Subsequently, during the repeated fitting better values of errors of parameters are obtained. It is necessary to use this function before fitting using the MC method by function FitMCMC.
| Parameter | Type | Description |
|---|---|---|
| w | numpy.array | Weights of points. |
| Output | numpy.array | Calculated errors. |
Summary of values and errors of fitted parameters. The output could be saved to file or written on screen.
| Parameter | Type | Description |
|---|---|---|
| name | string, optional | Name of output file. |
OCFit.FitLinear.Plot(name = None, no_plot = 0, no_plot_err = 0, eps = False, oc_min = True, time_type = 'JD', offset = 2400000, trans = True, title = None, epoch = False, min_type = False, weight = None, trans_weight = False, bw = False, double_ax = False, fig_size = None)
OCFit.FitQuad.Plot(name = None, no_plot = 0, no_plot_err = 0, eps = False, oc_min = True, time_type = 'JD', offset = 2400000, trans = True, title = None, epoch = False, min_type = False, weight = None, trans_weight = False, bw = False, double_ax = False, fig_size = None)
Plotting original values of O-C and linear / quadratic fit. It is possible to save the figure
to png file (eventually also to eps) or only display it on the screen. Outlier points or points
with big error could not be plotted. Values of O-C could be displayed in minutes or in
days. It is possible to set different type of Julian date displayed on x-label. The date could
be transformed by subtraction of given offset. If the date was already transformed before,
it is necessary to disable transformation of x-values (trans = False) but the value of the used
offset is suitable to set due to the label on the x-axis. It is also possible to use epochs on the x-axis or also have two axes with Julian dates and epochs together. In the case, if the values
of O-C of primary and also secondary minima are available, it is possible to distinguish
it using the parameter min_type = True. Primary minima are displayed be full circle and
the secondary by an empty circle. If the weights of points are available, they could be shown
on the figures by entering them to weight. Weights are divided into these categories: 0-3,
3-5, 5-8, 8-10. In the case, that the weights are not in this interval, it is necessary to
transform them by setting trans_weight = True. If the errors of data points were set
up during initialization of the class and now the weights are given to weight, there will be
plotted figure on according to given weights without error bars.
| Parameter | Type | Description |
|---|---|---|
| name | string, optional | Name of file (without extension) where the figure will be saved. |
| no_plot | integer, optional | Number of outlier points which will not be plotted. |
| no_plot_err | integer, optional | Number of points with the biggest errors which will not be plotted. |
| eps | boolean, optional | Save figure also to eps file. (It is necessary to set the name to variable name!) |
| oc_min | boolean, optional | Values of O-C display in minutes, not in days. |
| time_type | string, optional | Type of Julian date (JD, HJD, BJD). |
| offset | float, optional | Value of offset of dates due to shortening values on x axis. |
| trans | boolean, optional | Transform x values by the offset. |
| title | string, optional | Title of the figure. |
| epoch | boolean, optional | Axis x in epochs. |
| min_type | boolean, optional | Distinguishing primary and secondary minima. |
| weight | numpy.array, optional | Weights of points. |
| trans_weight | boolean, optional | Transformation of weights to the range 0 – 10. |
| bw | boolean, optional | Black and white figure. |
| double_ax | boolean, optional | Double x axis - dates + epochs. |
| fig_size | tuple, optional | Custom figure size - e.g. (12,6). |
OCFit.FitLinear.PlotRes(name = None, no_plot = 0, no_plot_err = 0, eps = False, oc_min = True, time_type = 'JD', offset = 2400000, trans = True, title = None, epoch = False, min_type = False, weight = None, trans_weight = False, bw = False, double_ax = False, fig_size = None)
OCFit.FitQuad.PlotRes(name = None, no_plot = 0, no_plot_err = 0, eps = False, oc_min = True, time_type = 'JD', offset = 2400000, trans = True, title = None, epoch = False, min_type = False, weight = None, trans_weight = False, bw = False, double_ax = False, fig_size = None)
Plotting the new values of O-C calculated for values of fitted parameters. It is
possible to save figure to png file (eventually also to eps) or only display it on the screen.
Outlier points or points with big error could not be plotted. Values of O-C could be
displayed in minutes or in days. It is possible to set different type of Julian date displayed
on x-label. The date could be transformed by subtraction of given offset. If the date was
already transformed before, it is necessary to disable transformation of x-values (trans = False) but the value of the used offset is suitable to set due to the label on the x-axis. It is
also possible to use epochs on the x-axis or also have two axes with Julian dates and epochs
together. In the case, if the values of O-C of primary and also secondary minima are
available, it is possible to distinguish it using the parameter min_type = True. Primary
minima are displayed be full circle and the secondary by an empty circle. If the weights
of points are available, they could be shown on the figures by entering them to weight.
Weights are divided into these categories: 0-3, 3-5, 5-8, 8-10. In the case, that the weights
are not in this interval, it is necessary to transform them by setting trans_weight = True.
If the errors of data points were set up during initialization of the class and now the
weights are given to weight, there will be plotted figure on according to given weights
without error bars.
| Parameter | Type | Description |
|---|---|---|
| name | string, optional | Name of file (without extension) where the figure will be saved. |
| no_plot | integer, optional | Number of outlier points which will not be plotted. |
| no_plot_err | integer, optional | Number of points with the biggest errors which will not be plotted. |
| eps | boolean, optional | Save figure also to eps file. (It is necessary to set the name to variable name!) |
| oc_min | boolean, optional | Values of O-C display in minutes, not in days. |
| time_type | string, optional | Type of Julian date (JD, HJD, BJD). |
| offset | float, optional | Value of offset of dates due to shortening values on x axis. |
| trans | boolean, optional | Transform x values by the offset. |
| title | string, optional | Title of the figure. |
| epoch | boolean, optional | Axis x in epochs. |
| min_type | boolean, optional | Distinguishing primary and secondary minima. |
| weight | numpy.array, optional | Weights of points. |
| trans_weight | boolean, optional | Transformation of weights to the range 0 – 10. |
| bw | boolean, optional | Black and white figure. |
| double_ax | boolean, optional | Double x axis - dates + epochs. |
| fig_size | tuple, optional | Custom figure size - e.g. (12,6). |
Saving original values of O-C calculated from given linear ephemeris to file. The data are in columns: observed times of minima, epochs, values of O-C, errors (if given) or weight (if given).
| Parameter | Type | Description |
|---|---|---|
| name | string | Name of output file. |
| weight | numpy.array, optional | Weights of data points. |
Saving residue (new values of O-C) calculated from values of fitted parameters to file. The data are in columns: observed times of minima, epochs, values of O-C, errors (if given) or weight (if given).
| Parameter | Type | Description |
|---|---|---|
| name | string | Name of output file. |
| weight | numpy.array, optional | Weights of data points. |
Example of work with both classes with simulated data. Data from the file is simply to load
using function numpy.loadtxt.
from OCFit import FitQuad, FitLinear
import numpy as np
#generate data
E = np.arange(0, 100, 1) #epochs
#simulation of observed times
t = 1e-5*E**2+15*E + 1540.4 + np.random.normal(scale = 0.01, size = E.shape)
err = 0.01*np.ones(E.shape) #errors of 'observed' times
#usage of FitLinear to first estimation of linear ephemeris
#initialization using estiamtion (original values) of linear ephemeris
#for long time range of observation, the value of period has to be close
#to the right value, otherwise the primary and secondary minims could be mixed
lin = FitLinear(t, 1540, 15.01, err = err)
oc = lin.oc #O-C calculated from original ephemeris
lin.FitLinear() #linear fit
lin.FitRobust() #fitting using robust regression
lin.Summary() #summary of parameters
err = lin.CorrectErr() #re-normalization of errors
oc = lin.FitRobust() #fitting using robust regression
lin.Summary() #summary of parameters
#plotting figures
#plot of original values of O-C with fit, without transformation of x axis
lin.Plot(trans = False)
#plot of residual O-C, without transformation of x axis
lin.PlotRes(trans = False)
#usage of class FitQuad
#as estimation of linear ephemeris the resault from FitLinear are used
#new values of O-C are only fitted
quad = FitQuad(t, lin.t0, lin.P, oc = oc, err = err)
oc = quad.oc #O-C calculated from original ephemeris
quad.FitQuad() #quadratic fit
quad.FitRobust() #fitting using robust regression
quad.Summary() #summary of parameters
#plotting figures
#plot of original values of O-C with fit, without transformation of x axis
quad.Plot(trans = False)
#plot of residual O-C, without transformation of x axis
quad.PlotRes(trans = False)