Available classes

Currently, there are two classes (Analysis and PlotAll) available in the predictr package. I will continue to add new classes in near future.

Analysis

Analysis contains all necessary methods for the Weibull analysis.

Default arguments and values

This table provides information on alle arguments that are passed to the Analysis class.

Parameter	default value	type	description
df	None	list of floats	List of failures
ds	None	list of floats	List of suspensions (right-censored only)
bounds	None	str	Confidence bounce method to be used in mle() or mrr()
bounds_type	None	str	Setting for the bounds: either two-sided or one-sided
show	False	bool	If True, the Weibull probability plot will be plotted
bcm	None	str	Defines the bias-correction method in mle()
cl	0.9	float	Sets the confidence level when bounds are used
bs_size	5000	int	Number of bootstrap samples
est_type	'median'	str	Sets the statistic to compute from the bootstrap samples
plot_style	'ggplot'	str	Choose a style according to your needs. See matplotlib style references for more available styles.
unit	'-'	str	Unit of failures and suspensions, e.g. 's', 'ms', 'no. of cycle' etc.
x_label	'Time to Failure'	string	Label for the x-axis
y_label	'Unreliability'	string	Label for the y-axis
xy_fontsize	12	float	fontsize for the axes label and ticks
legend_fontsize	9	float	Fontsize for the legend
plot_title	'Weibull Probability Plot'	string	Title for the plot
plot_title_fontsize	12	float	Fontsize of the plot title
fig_size	(6, 7)	tuple of floats	Sets figure width and height in inches: (width, height)
save	False	boolean	the beta and eta length of lists.
plot_ranks	True	boolean	If True, median ranks will be plotted.
show_legend	True	boolean	If True, the legend will be plotted
kwarg: path		string	Path defines the directory and format of the figure E.g. r'var/user/.../test.pdf'

Important:

• df = None will raise an error. There has to be at least one failure.

Parameter estimation methods

One can either use the Maximum Likelihood Estimation or Median Rank Regression.

Maximum likelihood estimation (MLE):

from predictr import Analysis
prototype_a = Analysis(...) # create an instance
prototype_a.mle() # use instance methods

Median Rank Regression (MRR)

from predictr import Analysis
prototype_a = Analysis(...) # create an instance
prototype_a.mrr() # use instance methods

Bias-correction methods

Since parameter estimation methods are only asymptotically unbiased (sample sizes -> "infinity"), bias-correction methods are useful when you have only a few failures. These methods correct the Weibull shape and scale parameter. The following table provides possible configurations. Bias-corrections for mrr() are not supported, yet.

Bias-correction method	mle()	mrr()	argument value	config.	statistic
C4 aka 'reduced bias adjustment'	x	-	'c4'	-	-
Hirose and Ross method	x	-	'hrbu'	-	-
Non-parametric Bootstrap correction	x	-	'np_bs'	bs_size	'mean', 'median', 'trimmed_mean'
Parametric Bootstrap correction	x	-	'p_bs'	bs_size	'mean', 'median', 'trimmed_mean'

Confidence bounds methods

Analysis supports nearly all state of the art confidence bounds methods.

confidence bounds	mle()	mrr()	uncensored data	censored data	bounds_type	argument value
Beta-Binomial Bounds	-	x	x	x	'2s', '1sl', '1su'	'bbb'
Monte-Carlo Pivotal Bounds	-	x	x	x	'2s', '1sl', '1su'	'mcpb'
Non-Parametric Bootstrap Bounds	x	x	x	x	'2s', '1sl', '1su'	'npbb'
Parametric Bootstrap Bounds	x	x	x	x	'2s', '1sl', '1su'	'pbb'
Fisher Bounds	x	-	x	x	'2s', '1sl', '1su'	'fb'
Likelihood Ratio Bounds	x	-	x	x	'2s', '1sl', '1su'	'lrb'

Important:

• mle() and mrr() support only specific confidence bounds methods. For instance, you can't use Beta-Binomial Bounds with mle(). This will also raise an error. Use the table above to check, whether a combination of parameter estimation and confidence bounds method is supported.
• '2s': two-sided confidence bounds, '1su': upper confidence bounds, '1sl': lower confidence bounds. If Beta-Binomial Bounds are used, the lower bound represents the lower percentile bound at a specific time ((pctl) is added in the plot legend). If Fisher Bounds are used, the lower bound represents the lower time bound at a specific percentile.

Examples

Maximum Likelihood Estimation (MLE)

Uncensored sample

Example:

failures = [0.4508831,  0.68564703, 0.76826143, 0.88231395, 1.48287253, 1.62876357]
prototype_a = Analysis(df=failures, bounds='fb',show=True)
prototype_a.mle()

Censored sample

Example:

failures = [0.4508831,  0.68564703, 0.76826143, 0.88231395, 1.48287253, 1.62876357]
suspensions = [1.9, 2.0, 2.0]
prototype_a = Analysis(df=failures, ds=suspensions, bounds='lrb',show=True)
prototype_a.mle()

Median Rank Regression (MRR)

Uncensored sample

Example:

failures = [0.4508831,  0.68564703, 0.76826143, 0.88231395, 1.48287253, 1.62876357]
prototype_a = Analysis(df=failures, bounds='bbb',show=True)
prototype_a.mrr()

Censored sample

Example:

failures = [0.4508831,  0.68564703, 0.76826143, 0.88231395, 1.48287253, 1.62876357]
suspensions = [1.9, 2.0, 2.0]
prototype_a = Analysis(df=failures, ds=suspensions, bounds='mcpb',show=True)
prototype_a.mrr()

Bias-corrections

As already mentioned, only mle() support bias-corrections. The samples in these examples are drawn from a two-parameter Weibull distribution with a shape parameter of 2.0 and a scale parameter of 1.0.

Uncensored sample

It is appearent that the estimates of beta and eta are now closer to the ground truth values. The dotted grey line in the plot is the "biased" MLE line, the bia-corrected line is blue. The legend contains all needed information.

failures = [0.4508831,  0.68564703, 0.76826143, 0.88231395, 1.48287253, 1.62876357]
prototype_a = Analysis(df=failures, bounds='fb', show=True, bcm='c4')
prototype_a.mle()

The estimates can for the Weibull parameters can be compared directly, since they are available as attributes

print(f'biased beta: {prototype_a.beta:4f} --> bias-corrected beta: {prototype_a.beta_c4:4f}')
>>> biased beta: 2.511134 --> bias-corrected beta: 2.108248

Censored sample

The data is type II right-censored.

failures = [0.38760099164906514, 0.5867052007217437, 0.5878056753744406, 0.602290402929083, 0.6754829518358306, 0.7520219855697948]
suspensions = [0.7520219855697948, 0.7520219855697948]
prototype_a = Analysis(df=failures, ds=suspensions, bounds='lrb', show=True, bcm='hrbu')
prototype_a.mle()

Modifying the Weibull plot

Axes labels and title

You can modify the axes label, plot title and the fontsize. Also, you can save the plot by setting save=True and path='your/own/directory/example.pdf'.

failures = [0.4508831,  0.68564703, 0.76826143, 0.88231395, 1.48287253, 1.62876357]
prototype_a = Analysis(df=failures, bounds='fb',show=True, plot_title='New Project', x_label='No. of Cycles', unit='10^3', y_label='Unreliability: 1-R', xy_fontsize=12, save=True, path=r'var/user/test.pdf')
prototype_a.mle()

Figure size, plot legend and median rank markers

You can customize the fontsize that is being used in the plot legend. If you don't want a legend set show_legend=False. By default, the markers for the median ranks will be plotted. Set plot_ranks=False if you don't want median rank markers in your plot. The figure size can be modified with fig_size=(width, height). Width and height set the figure size in inches.

failures = [0.4508831,  0.68564703, 0.76826143, 0.88231395, 1.48287253, 1.62876357]
prototype_a = Analysis(df=failures, bounds='fb',show=True, show_legend=True, legend_fontsize=10, show_ranks=False, fig_size=(7, 7))
prototype_a.mle()

PlotAll

PlotAll plots class objects from Analysis in one figure. Currently, only data from mle() is supported. Theoretically, you can plot as many objects as you like -> provide a list of colors as a kwarg in PlotAll(objects, **kwargs).mult_weibull(). For now, six colors are supported by default, but you can pass an infinit amount of colors to the mult_weibull() method.

Available methods:

Methods	Description
mult_weibull()	Plots multiple Analysis class instances in one Weibull plot
contour_plot()	Plots contour plots when likelihood ratio bounds are used in Analysis
weibull_pdf()	Plots one or more Weibull probability density functions. Axes are completely customizable.
simple_weibull()	Plots the Weibull probability plot for a given pair of beta and eta. If failures and/or suspensions are given, the median ranks are plotted as well.

Default Arguments of each method

Most of the arguments are either self explanatory or already defined in default arguments and values

Methods	Default arguments
mult_weibull()	x_label='Time To Failure', y_label='Unreliability', plot_title='Weibull Probability Plot', xy_fontsize=12, plot_title_fontsize=12, fig_size=(6, 7), x_bounds=None, plot_ranks=True, save=False, linestyle=None, legend_fontsize=9, **kwargs
contour_plot()	show_legend=True, save=False, **kwargs
weibull_pdf()	beta=None, eta=None, linestyle=None, labels = None,x_label = None, y_label=None, xy_fontsize=10, legend_fontsize=8, plot_title='Weibull PDF', plot_title_fontsize=12, x_bounds=None, fig_size=None, color=None, save=False, plot_style='ggplot', **kwargs
simple_weibull()	beta, eta, unit='-', x_label = 'Time to Failure', y_label = 'Unreliability', xy_fontsize=12, plot_title_fontsize=12, plot_title='Weibull Probability Plot', fig_size=(6, 7), show_legend=True, legend_fontsize=9, save=False, df=None, ds=None, **kwargs

Parameter(s)	default value	type	description
df	None	list of floats	List of failures
ds	None	list of floats	List of suspensions (right-censored only)
plot_style	'ggplot'	str	Choose a style according to your needs. See matplotlib style references for more available styles.
unit	'-'	str	Unit of failures and suspensions, e.g. 's', 'ms', 'no. of cycle' etc.
x_label	depends on method	string	Label for the x-axis
y_label	depends on method	string	Label for the y-axis
labels		string	List containing the labels for the plot legend in weibull_pdf()
xy_fontsize	12	float	fontsize for the axes label and ticks
legend_fontsize	9	float	Fontsize for the legend
plot_title	'Weibull Probability Plot'	string	Title for the plot
plot_title_fontsize	12	float	Fontsize of the plot title
fig_size	(6, 7)	tuple of floats	Sets figure width and height in inches: (width, height)
save	False	boolean	If True, the plot is saved according to the path (kwargs)
plot_ranks	True	boolean	If True, median ranks will be plotted.
show_legend	True	boolean	If True, the legend will be plotted
weibull_pdf: beta, eta	None, None	list of floats or None	Attributes from Analysis object. Pairs of beta and eta values to be plotted. Each parameter pair must have the same index value.
linestyle	['-', '--', ':', '-.']	list of strings	Defines the linestyle(s) in the plot. Must be greater or equal to the length of beta ans eta lists
color	None	list of strings	List containing the colormap for the plotted lines. Length of list must be equal to the beta and eta length of lists or the number of Analysis objects.
x_bounds		list of floats	Sets x-axis boundaries: [start, stop] or [start, end, steps inbetween], respectively.
simple_weibull:beta, eta		float	Weibull parameter pair which will be plotted
kwarg: path		string	Path defines the directory and format of the figure E.g. r'var/user/.../test.pdf'

mult_weibull()

Both with two-sided bounds - default colors

from predictr import Analysis, PlotAll

# Create new objects, e.g. name them prototype_a and prototype_b
failures_a = [0.30481336314657737, 0.5793918872111126, 0.633217732127894, 0.7576700925659532,
              0.8394342818048925, 0.9118100898948334, 1.0110147142055477, 1.0180126386295232,
              1.3201853093496474, 1.492172669340363]
prototype_a = Analysis(df=failures_a, bounds='lrb', bounds_type='2s')
prototype_a.mle()

failures_b = [1.8506941739639076, 2.2685555679846954, 2.380993183650987, 2.642404955035375,
              2.777082863078587, 2.89527127055147, 2.9099992138728927, 3.1425481097241,
              3.3758727398694406, 3.8274990886889997]
prototype_b = Analysis(df=failures_b, bounds='pbb', bounds_type='2s')
prototype_b.mle()

# Create dictionary with Analysis objects
# Keys will be used in figure legend. Name them as you please.
objects = {'proto_a': prototype_a, 'proto_b': prototype_b}

# Use mult_weibull() method
PlotAll(objects).mult_weibull()

One object with a one-sided lower bound, the other one has two-sided bounds - default colors

You can plot every bounds_type ('2s', '1sl', '1su') and combine them.

from predictr import Analysis, PlotAll

# Create new objects, e.g. name them prototype_a and prototype_b
failures_a = [0.30481336314657737, 0.5793918872111126, 0.633217732127894, 0.7576700925659532,
              0.8394342818048925, 0.9118100898948334, 1.0110147142055477, 1.0180126386295232,
              1.3201853093496474, 1.492172669340363]
prototype_a = Analysis(df=failures_a, bounds='fb', bounds_type='1sl')
prototype_a.mle()

failures_b = [1.8506941739639076, 2.2685555679846954, 2.380993183650987, 2.642404955035375,
              2.777082863078587, 2.89527127055147, 2.9099992138728927, 3.1425481097241,
              3.3758727398694406, 3.8274990886889997]
prototype_b = Analysis(df=failures_b, bounds='npbb', bounds_type='2s')
prototype_b.mle()

# Create dictionary with Analysis objects
# Keys will be used in figure legend. Name them as you please.
objects = {'proto_a': prototype_a, 'proto_b': prototype_b}

# Use mult_weibull() method
# Set plot_ranks=True, if you want to plot the median rank markers
PlotAll(objects).mult_weibull(plot_ranks=False)

Three objects - custom colors

from predictr import Analysis, PlotAll

# Create new objects
failures_a = [0.30481336314657737, 0.5793918872111126, 0.633217732127894, 0.7576700925659532,
              0.8394342818048925, 0.9118100898948334, 1.0110147142055477, 1.0180126386295232,
              1.3201853093496474, 1.492172669340363]
prototype_a = Analysis(df=failures_a, bounds='fb', bounds_type='2s')
prototype_a.mle()

failures_b = [1.8506941739639076, 2.2685555679846954, 2.380993183650987, 2.642404955035375,
              2.777082863078587, 2.89527127055147, 2.9099992138728927, 3.1425481097241,
              3.3758727398694406, 3.8274990886889997]
prototype_b = Analysis(df=failures_b, bounds='npbb', bounds_type='2s')
prototype_b.mle()

failures_c = [0.04675399107295282, 0.31260891592041457, 0.32121232576015757, 0.6013488316204837,
              0.7755159796641791, 0.8994041575114923, 0.956417788622185, 1.1967354178170764,
              1.6115311492838604, 2.1120891587523793]
prototype_c = Analysis(df=failures_c, bounds='pbb', bounds_type='2s')
prototype_c.mle()

objects = {'proto_a': prototype_a, 'proto_b': prototype_b, 'secret': prototype_c}
# Create list with custom colors and pass to the instance method
colors = ['green', 'red', 'blue']
PlotAll(objects).mult_weibull(plot_ranks=False, color=colors)

contour_plot()

contour_plot() only works for likelihood ratio bounds. Hence, you have to use bounds='lrb' in the Analysis class. This method supports all bounds types and all confidence levels. You can pass as many objects as you want to.

Plot a single Analysis object

from predictr import Analysis, PlotAll

# Create new objects
failures_a = [0.30481336314657737, 0.5793918872111126, 0.633217732127894, 0.7576700925659532,
              0.8394342818048925, 0.9118100898948334, 1.0110147142055477, 1.0180126386295232,
              1.3201853093496474, 1.492172669340363]
prototype_a = Analysis(df=failures_a, bounds='lrb', bounds_type='2s')
prototype_a.mle()

objects = {'initial design': prototype_a}
PlotAll(objects).contour_plot()

Plot a multiple Analysis objects

failures_a = [0.30481336314657737, 0.5793918872111126, 0.633217732127894, 0.7576700925659532,
              0.8394342818048925, 0.9118100898948334, 1.0110147142055477, 1.0180126386295232,
              1.3201853093496474, 1.492172669340363]
prototype_a = Analysis(df=failures_a, bounds='lrb', bounds_type='2s')
prototype_a.mle()

failures_c = [0.04675399107295282, 0.31260891592041457, 0.32121232576015757, 0.6013488316204837,
              0.7755159796641791, 0.8994041575114923, 0.956417788622185, 1.1967354178170764,
              1.6115311492838604, 2.1120891587523793]
prototype_c = Analysis(df=failures_c, bounds='lrb', bcm = 'hrbu', bounds_type='2s')
prototype_c.mle()

# Create dictionary with Analysis objects
# Keys will be used in figure legend. Name them as you please.
objects = {'initial design': prototype_a, 'final design': prototype_c}
PlotAll(objects).contour_plot()

weibull_pdf()

This method plots one or more Weibull probability density functions. Axes are completely customizable.

Arguments: weibull_pdf(self, beta=None, eta=None, linestyle=None, labels = None, x_label = None, y_label=None, xy_fontsize=10, legend_fontsize=8, plot_title='Weibull PDF', plot_title_fontsize=12, x_bounds=None, fig_size=None, color=None, save=False, plot_style='ggplot', kwargs)

from predictr import Analysis, PlotAll

# Use analysis for the parameter estimation
failures1 = [3, 3, 3, 3, 3, 3, 4, 4, 9]
failures2 = [3, 3, 5, 6, 6, 4, 9]
failures3 = [5, 6, 6, 6, 7, 9]

a = Analysis(df=failures1, bounds='lrb', bounds_type='2s', show = False, unit= 'min')
a.mle()

b = Analysis(df=failures1, ds = failures2, bounds='fb', bounds_type='2s', show = False, unit= 'min')
b.mle()

c = Analysis(df=failures3, bounds='lrb', bcm='hrbu', bounds_type='2s', show = False, unit= 'min')
c.mle()

# Use weibull_pdf method in PlotAll to plot the Weibull pdfs
# beta contains the Weibull shape parameters, which were estimated using Analysis class. Do the same for the Weibull scale parameter eta.
# Cusomize the path directory in order to use this code
PlotAll().weibull_pdf(beta = [a.beta, b.beta, c.beta], eta = [a.eta, b.eta, c.eta],
                      linestyle=['-', '--', ':'], labels = ['A', 'B', 'C'],
                x_bounds=[0, 20, 100], plot_title = 'Comparison of three Prototypes',
                x_label='Time to Failure', y_label='Density Function',
                save=True, color=['black', 'black', 'black'], path=r'/your/custom/path/test.pdf')

simple_weibull()

This method plots the Weibull probability plot for a given pair of beta and eta. If failures and/or suspenions are given, the median ranks are plotted as well.

from predictr import Analysis, PlotAll

# If save=True, you must set the path argument, e.g. path=r'/your/custom/path/test.pdf'
PlotAll().simple_weibull(beta =2.0, eta=1, show_legend=True, x_label='Cycles until failure', plot_title='Simple Weibull')