Work log

Lab Notebook[edit]

Interpreting Exponentiated Weibull Models[edit]

Now I'm going to work on interpreting Exponentiated Weibull models and I'm going to tabulate the frequency of qualitatively different distributions by wiki.

The Exponentiated Weibull has 3 parameters. Two are shape parameters ( $\alpha >0$ and $\gamma >0$ ) and one is a scale parameter ( $\lambda >0$ ). The major qualitative distinctions in interpreting the model are in terms of the shape parameters.

According to this analysis of the Exponentiated Weibull:

If $\alpha =1$ and $\gamma =1$ then we have an exponential distribution with parameter $\lambda$ .
If $\alpha =1$ we have a Weibull distribution.

In this case the failure rate is always increasing (positive ageing) if $\gamma >1$ and always decreasing (negative ageing) if $\gamma <1$ .

If $\gamma =1$ then we have a exponentiated exponential distribution and the failure rate may not be monotonic.

In this case, and if $\alpha >0$ then the failure rate increases when $0<x<\lambda$ .
On the other hand if $\alpha <0$ then the failure rate decreases when $x>\lambda$ .

If $\gamma >1$ and $\alpha >1$ then we have positive ageing (the failure rate is increasing).
If $\gamma <1$ and $\alpha <1$ then we have negative ageing (the failure rate is decreasing).
If the two shape parameters have opposite signs then interpreting the model may require closer inspection of hazard and/or survival curves.

Inconveniently, it looks like almost all of the time we have $\alpha >1$ and $\gamma <1$ .

expweib_tab = table[table.model =='exponweib'].copy().reset_index()
expweib_tab['a'] = expweib_tab.params.apply(lambda r: r[0])
expweib_tab['c'] = expweib_tab.params.apply(lambda r: r[1])
expweib_tab['scale'] = expweib_tab.params.apply(lambda r: r[3])
expweib_tab['a_ge_1'] = expweib_tab.a >= 1 
expweib_tab['c_ge_1'] = expweib_tab.c >= 1 

expweib_tab = expweib_tab.drop(['level_0','index'],1)
pd.crosstab(expweib_tab['a_ge_1'],expweib_tab['c_ge_1'])

Note that in the code a = $\alpha$ and c = $\gamma$ .

$\gamma \geq 1$	False	True
$a\geq 1$
False	0	1
True	241	0

So, inconveniently, I don't know what we can say qualitatively about reading times just from looking at the parameter estimates.

Next I'm going to plot hazard functions for a handful of wikis to see if there is anything we can say in general about these parameters from the distributions.