Hello,
I have a hierarchical regression model that is predicting the residuals from a standard time series forecasting model. Time is not a component within this regression model however.
I have what looks to be three clusters of data…
- Large negative tail values
- Values that hover around 0.
- Large positive tail values.
See below for my ppc.
I’d like to try a mixture model (for the 1st time) and I’m a little lost on this tutorial: Gaussian Mixture Model — PyMC example gallery
What is the initval and where is the author getting them? Do I necessarily need a transform argument in there? I’ve never had to use this before in a model so I’m not sure what it is doing.
For context, below is my current model:
def make_next_level_hierarchy_variable(name, mu, alpha, beta, dims=None):
with modelcontext(None):
sigma_ = pm.Gamma(f'{name}_sigma', alpha=alpha, beta=beta)
offset = pm.Normal(f'{name}_offset', dims=dims)
return pm.Deterministic(f'{name}', mu + sigma_ * offset, dims=dims)
with pm.Model(coords=coords) as constant_model:
#Data that does not change
cat_to_bl_map = pm.Data('cat_to_bl_map', cat_to_bl_idx, mutable=False)
subcat_to_cat_map = pm.Data('subcat_to_cat_map', subcat_to_cat_idx, mutable=False)
ic_to_subcat_map = pm.Data('ic_to_subcat_map', ic_to_subcat_idx, mutable=False)
ic_to_item_map = pm.Data('ic_to_item_map', ic_to_item_idx, mutable = False)
#Data that does change
pm_loc_idx = pm.Data('loc_idx', location_idx, mutable = True)
pm_item_idx = pm.Data('item_idx', item_idx, mutable=True)
pm_time_idx = pm.Data('time_idx', time_idx, mutable=True)
observed_eaches = pm.Data('observed_eaches', df_train.residual_corrected, mutable=True)
promo_ = pm.Data('promotion', promo_idx, mutable = True)
cann_ = pm.Data('cannibalization', cann_idx, mutable = True)
promo_cann_ = pm.Data('promo_cann', promo_cann_idx, mutable=True)
month_2_ = pm.Data('month_2', month_2_idx, mutable = True)
month_3_ = pm.Data('month_3', month_3_idx ,mutable = True)
month_4_ = pm.Data('month_4', month_4_idx ,mutable = True)
month_5_ = pm.Data('month_5', month_5_idx ,mutable = True)
month_6_ = pm.Data('month_6', month_6_idx ,mutable = True)
month_7_ = pm.Data('month_7', month_7_idx ,mutable = True)
month_8_ = pm.Data('month_8', month_8_idx ,mutable = True)
month_9_ = pm.Data('month_9', month_9_idx ,mutable = True)
month_10_ = pm.Data('month_10', month_10_idx,mutable = True)
month_11_ = pm.Data('month_11', month_11_idx ,mutable = True)
month_12_ = pm.Data('month_12', month_12_idx ,mutable = True)
loc_intercept = pm.Normal('loc_intercept', mu = 0, sigma = 1, dims = ['location'])
loc_bl = utility_functions.make_next_level_hierarchy_variable(name='loc_bl', mu=loc_intercept, alpha=2, beta=1, dims=['business_line', 'location'])
loc_cat = utility_functions.make_next_level_hierarchy_variable(name='loc_cat', mu=loc_bl[cat_to_bl_map], alpha=2, beta=1, dims=['category', 'location'])
loc_subcat = utility_functions.make_next_level_hierarchy_variable(name='loc_subcat', mu=loc_cat[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory', 'location'])
loc_ic = utility_functions.make_next_level_hierarchy_variable(name='loc_ic', mu=loc_subcat[ic_to_subcat_map], alpha=2, beta=1, dims=['ic', 'location'])
loc_item = utility_functions.make_next_level_hierarchy_variable(name='loc_item', mu=loc_ic[ic_to_item_map], alpha=2, beta=1, dims=['item', 'location'])
mu_promo = pm.Normal('mu_promo', mu = 0, sigma = 1)
bl_promo = utility_functions.make_next_level_hierarchy_variable(name='bl_promo', mu=mu_promo, alpha=2, beta=1, dims=['business_line'])
cat_promo = utility_functions.make_next_level_hierarchy_variable(name='cat_promo', mu=bl_promo[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_promo = utility_functions.make_next_level_hierarchy_variable(name='subcat_promo', mu=cat_promo[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_promo = utility_functions.make_next_level_hierarchy_variable(name='ic_promo', mu=subcat_promo[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_promo = utility_functions.make_next_level_hierarchy_variable(name='item_promo', mu=ic_promo[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_cann = pm.Normal('mu_cann', mu = 0, sigma = 1)
bl_cann = utility_functions.make_next_level_hierarchy_variable(name='bl_cann', mu=mu_cann, alpha=2, beta=1, dims=['business_line'])
cat_cann = utility_functions.make_next_level_hierarchy_variable(name='cat_cann', mu=bl_cann[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_cann = utility_functions.make_next_level_hierarchy_variable(name='subcat_cann', mu=cat_cann[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_cann = utility_functions.make_next_level_hierarchy_variable(name='ic_cann', mu=subcat_cann[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_cann = utility_functions.make_next_level_hierarchy_variable(name='item_cann', mu=ic_cann[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_promo_cann = pm.Normal('mu_promo_cann', mu = 0, sigma = 1)
bl_promo_cann = utility_functions.make_next_level_hierarchy_variable(name='bl_promo_cann', mu=mu_promo_cann, alpha=2, beta=1, dims=['business_line'])
cat_promo_cann = utility_functions.make_next_level_hierarchy_variable(name='cat_promo_cann', mu=bl_promo_cann[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_promo_cann = utility_functions.make_next_level_hierarchy_variable(name='subcat_promo_cann', mu=cat_promo_cann[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_promo_cann = utility_functions.make_next_level_hierarchy_variable(name='ic_promo_cann', mu=subcat_promo_cann[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_promo_cann = utility_functions.make_next_level_hierarchy_variable(name='item_promo_cann', mu=ic_promo_cann[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month2 = pm.Normal('mu_month2', mu = 0, sigma = 1)
bl_month2 = utility_functions.make_next_level_hierarchy_variable(name='bl_month2', mu=mu_month2, alpha=2, beta=1, dims=['business_line'])
cat_month2 = utility_functions.make_next_level_hierarchy_variable(name='cat_month2', mu=bl_month2[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month2 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month2', mu=cat_month2[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month2 = utility_functions.make_next_level_hierarchy_variable(name='ic_month2', mu=subcat_month2[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month2 = utility_functions.make_next_level_hierarchy_variable(name='item_month2', mu=ic_month2[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month3 = pm.Normal('mu_month3', mu = 0, sigma = 1)
bl_month3 = utility_functions.make_next_level_hierarchy_variable(name='bl_month3', mu=mu_month3, alpha=2, beta=1, dims=['business_line'])
cat_month3 = utility_functions.make_next_level_hierarchy_variable(name='cat_month3', mu=bl_month3[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month3 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month3', mu=cat_month3[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month3 = utility_functions.make_next_level_hierarchy_variable(name='ic_month3', mu=subcat_month3[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month3 = utility_functions.make_next_level_hierarchy_variable(name='item_month3', mu=ic_month3[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month4 = pm.Normal('mu_month4', mu = 0, sigma = 1)
bl_month4 = utility_functions.make_next_level_hierarchy_variable(name='bl_month4', mu=mu_month4, alpha=2, beta=1, dims=['business_line'])
cat_month4 = utility_functions.make_next_level_hierarchy_variable(name='cat_month4', mu=bl_month4[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month4 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month4', mu=cat_month4[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month4 = utility_functions.make_next_level_hierarchy_variable(name='ic_month4', mu=subcat_month4[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month4 = utility_functions.make_next_level_hierarchy_variable(name='item_month4', mu=ic_month4[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month5 = pm.Normal('mu_month5', mu = 0, sigma = 1)
bl_month5 = utility_functions.make_next_level_hierarchy_variable(name='bl_month5', mu=mu_month5, alpha=2, beta=1, dims=['business_line'])
cat_month5 = utility_functions.make_next_level_hierarchy_variable(name='cat_month5', mu=bl_month5[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month5 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month5', mu=cat_month5[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month5 = utility_functions.make_next_level_hierarchy_variable(name='ic_month5', mu=subcat_month5[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month5 = utility_functions.make_next_level_hierarchy_variable(name='item_month5', mu=ic_month5[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month6 = pm.Normal('mu_month6', mu = 0, sigma = 1)
bl_month6 = utility_functions.make_next_level_hierarchy_variable(name='bl_month6', mu=mu_month6, alpha=2, beta=1, dims=['business_line'])
cat_month6 = utility_functions.make_next_level_hierarchy_variable(name='cat_month6', mu=bl_month6[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month6 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month6', mu=cat_month6[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month6 = utility_functions.make_next_level_hierarchy_variable(name='ic_month6', mu=subcat_month6[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month6 = utility_functions.make_next_level_hierarchy_variable(name='item_month6', mu=ic_month6[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month7 = pm.Normal('mu_month7', mu = 0, sigma = 1)
bl_month7 = utility_functions.make_next_level_hierarchy_variable(name='bl_month7', mu=mu_month7, alpha=2, beta=1, dims=['business_line'])
cat_month7 = utility_functions.make_next_level_hierarchy_variable(name='cat_month7', mu=bl_month7[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month7 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month7', mu=cat_month7[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month7 = utility_functions.make_next_level_hierarchy_variable(name='ic_month7', mu=subcat_month7[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month7 = utility_functions.make_next_level_hierarchy_variable(name='item_month7', mu=ic_month7[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month8 = pm.Normal('mu_month8', mu = 0, sigma = 1)
bl_month8 = utility_functions.make_next_level_hierarchy_variable(name='bl_month8', mu=mu_month8, alpha=2, beta=1, dims=['business_line'])
cat_month8 = utility_functions.make_next_level_hierarchy_variable(name='cat_month8', mu=bl_month8[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month8 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month8', mu=cat_month8[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month8 = utility_functions.make_next_level_hierarchy_variable(name='ic_month8', mu=subcat_month8[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month8 = utility_functions.make_next_level_hierarchy_variable(name='item_month8', mu=ic_month8[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month9 = pm.Normal('mu_month9', mu = 0, sigma = 1)
bl_month9 = utility_functions.make_next_level_hierarchy_variable(name='bl_month9', mu=mu_month9, alpha=2, beta=1, dims=['business_line'])
cat_month9 = utility_functions.make_next_level_hierarchy_variable(name='cat_month9', mu=bl_month9[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month9 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month9', mu=cat_month9[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month9 = utility_functions.make_next_level_hierarchy_variable(name='ic_month9', mu=subcat_month9[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month9 = utility_functions.make_next_level_hierarchy_variable(name='item_month9', mu=ic_month9[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month10 = pm.Normal('mu_month10', mu = 0, sigma = 1)
bl_month10 = utility_functions.make_next_level_hierarchy_variable(name='bl_month10', mu=mu_month10, alpha=2, beta=1, dims=['business_line'])
cat_month10 = utility_functions.make_next_level_hierarchy_variable(name='cat_month10', mu=bl_month10[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month10 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month10', mu=cat_month10[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month10 = utility_functions.make_next_level_hierarchy_variable(name='ic_month10', mu=subcat_month10[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month10 = utility_functions.make_next_level_hierarchy_variable(name='item_month10', mu=ic_month10[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month11 = pm.Normal('mu_month11', mu = 0, sigma = 1)
bl_month11 = utility_functions.make_next_level_hierarchy_variable(name='bl_month11', mu=mu_month11, alpha=2, beta=1, dims=['business_line'])
cat_month11 = utility_functions.make_next_level_hierarchy_variable(name='cat_month11', mu=bl_month11[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month11 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month11', mu=cat_month11[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month11 = utility_functions.make_next_level_hierarchy_variable(name='ic_month11', mu=subcat_month11[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month11 = utility_functions.make_next_level_hierarchy_variable(name='item_month11', mu=ic_month11[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu_month12 = pm.Normal('mu_month12', mu = 0, sigma = 1)
bl_month12 = utility_functions.make_next_level_hierarchy_variable(name='bl_month12', mu=mu_month12, alpha=2, beta=1, dims=['business_line'])
cat_month12 = utility_functions.make_next_level_hierarchy_variable(name='cat_month12', mu=bl_month12[cat_to_bl_map], alpha=2, beta=1, dims=['category'])
subcat_month12 = utility_functions.make_next_level_hierarchy_variable(name='subcat_month12', mu=cat_month12[subcat_to_cat_map], alpha=2, beta=1, dims=['subcategory'])
ic_month12 = utility_functions.make_next_level_hierarchy_variable(name='ic_month12', mu=subcat_month12[ic_to_subcat_map], alpha=2, beta=1, dims=['ic'])
item_month12 = utility_functions.make_next_level_hierarchy_variable(name='item_month12', mu=ic_month12[ic_to_item_map], alpha=2, beta=1, dims=['item'])
mu = (loc_item[pm_item_idx, pm_loc_idx] + item_month2[pm_item_idx]*month_2_
+ item_month3[pm_item_idx]*month_3_ + item_month4[pm_item_idx]*month_4_ + item_month5[pm_item_idx]*month_5_ + item_month6[pm_item_idx]*month_6_
+ item_month7[pm_item_idx]*month_7_ + item_month8[pm_item_idx]*month_8_ + item_month9[pm_item_idx]*month_9_ + item_month10[pm_item_idx]*month_10_
+ item_month11[pm_item_idx]*month_11_ + item_month12[pm_item_idx]*month_12_
+ item_promo[pm_item_idx]*promo_ + item_cann[pm_item_idx]*cann_ + item_promo_cann[pm_item_idx]*promo_cann_)
sigma = pm.Uniform('sigma',lower = 1, upper = 20 )
eaches = pm.Normal('predicted_eaches',
mu=mu,
# nu= 3,
sigma=sigma,
observed=observed_eaches,)