patsy中的交互效果，patsy.dmatrices为":“和"+”或"*“提供重复列

Question

我有一个列的dataframe，我打算将这两个列作为分类变量来处理。

第一列是country，它的值有SGP、AUS、MYS等。第二列是一天中的时间，它的值为24小时格式，如00、11、14、15等，事件是一个具有1/0标志的二进制变量。我明白要对它们进行分类，我需要在进行Logistic回归之前使用patsy。这个，我使用dmatrices构建。

Usecase :只考虑和time_day的交互效应(以及其他属性，如“操作系统”)

f= 'event_int ~ time_day:country'
y,X = patsy.dmatrices(f, df, return_type='dataframe')
X.columns
Index([u'Intercept', u'country[T.HKG]', u'country[T.IDN]', u'country[T.IND]', u'country[T.MYS]', u'country[T.NZL]', u'country[T.PHL]', u'country[T.SGP]', u'time_day[T.02]:country[AUS]', u'time_day[T.03]:country[AUS]', u'time_day[T.04]:country[AUS]', u'time_day[T.05]:country[AUS]', u'time_day[T.06]:country[AUS]', u'time_day[T.07]:country[AUS]', u'time_day[T.08]:country[AUS]', u'time_day[T.09]:country[AUS]', u'time_day[T.10]:country[AUS]', u'time_day[T.11]:country[AUS]', u'time_day[T.12]:country[AUS]', u'time_day[T.NA]:country[AUS]', u'time_day[T.02]:country[HKG]', u'time_day[T.03]:country[HKG]', u'time_day[T.04]:country[HKG]', u'time_day[T.05]:country[HKG]', u'time_day[T.06]:country[HKG]', u'time_day[T.07]:country[HKG]', u'time_day[T.08]:country[HKG]', u'time_day[T.09]:country[HKG]', u'time_day[T.10]:country[HKG]', u'time_day[T.11]:country[HKG]', u'time_day[T.12]:country[HKG]', u'time_day[T.NA]:country[HKG]', u'time_day[T.02]:country[IDN]', u'time_day[T.03]:country[IDN]', u'time_day[T.04]:country[IDN]', u'time_day[T.05]:country[IDN]', u'time_day[T.06]:country[IDN]', u'time_day[T.07]:country[IDN]', u'time_day[T.08]:country[IDN]', u'time_day[T.09]:country[IDN]', u'time_day[T.10]:country[IDN]', u'time_day[T.11]:country[IDN]', u'time_day[T.12]:country[IDN]', u'time_day[T.NA]:country[IDN]', u'time_day[T.02]:country[IND]', u'time_day[T.03]:country[IND]', u'time_day[T.04]:country[IND]', u'time_day[T.05]:country[IND]', u'time_day[T.06]:country[IND]', u'time_day[T.07]:country[IND]', u'time_day[T.08]:country[IND]', u'time_day[T.09]:country[IND]', u'time_day[T.10]:country[IND]', u'time_day[T.11]:country[IND]', u'time_day[T.12]:country[IND]', u'time_day[T.NA]:country[IND]', u'time_day[T.02]:country[MYS]', u'time_day[T.03]:country[MYS]', u'time_day[T.04]:country[MYS]', u'time_day[T.05]:country[MYS]', u'time_day[T.06]:country[MYS]', u'time_day[T.07]:country[MYS]', u'time_day[T.08]:country[MYS]', u'time_day[T.09]:country[MYS]', u'time_day[T.10]:country[MYS]', u'time_day[T.11]:country[MYS]', u'time_day[T.12]:country[MYS]', u'time_day[T.NA]:country[MYS]', u'time_day[T.02]:country[NZL]', u'time_day[T.03]:country[NZL]', u'time_day[T.04]:country[NZL]', u'time_day[T.05]:country[NZL]', u'time_day[T.06]:country[NZL]', u'time_day[T.07]:country[NZL]', u'time_day[T.08]:country[NZL]', u'time_day[T.09]:country[NZL]', u'time_day[T.10]:country[NZL]', u'time_day[T.11]:country[NZL]', u'time_day[T.12]:country[NZL]', u'time_day[T.NA]:country[NZL]', u'time_day[T.02]:country[PHL]', u'time_day[T.03]:country[PHL]', u'time_day[T.04]:country[PHL]', u'time_day[T.05]:country[PHL]', u'time_day[T.06]:country[PHL]', u'time_day[T.07]:country[PHL]', u'time_day[T.08]:country[PHL]', u'time_day[T.09]:country[PHL]', u'time_day[T.10]:country[PHL]', u'time_day[T.11]:country[PHL]', u'time_day[T.12]:country[PHL]', u'time_day[T.NA]:country[PHL]', u'time_day[T.02]:country[SGP]', u'time_day[T.03]:country[SGP]', u'time_day[T.04]:country[SGP]', u'time_day[T.05]:country[SGP]', u'time_day[T.06]:country[SGP]', u'time_day[T.07]:country[SGP]', u'time_day[T.08]:country[SGP]', u'time_day[T.09]:country[SGP]', ...], dtype='object')

我希望看到只列名与两个国家& time_day，但事实并非如此。我可以通过指定X = X.ix[:,range(7,len(X.columns))]、来手动获取子集，但这意味着对每个dataset进行HARDCODING。

我的理解是A*B与A:B的不同之处在于它没有列出A+B有趣的东西，尽管我没有看到A，即仅在上面的输出中看到time_day的分类值。

此外，当我执行以下操作时，要显式地将"country“单独从"X”dataframe中排除在外，这是不起作用的，而且我得到了与上面相同的输出。

f='event_int ~ time_day:country-country'
y,X = patsy.dmatrices(f, df, return_type='dataframe')
X.columns
Index([u'Intercept', u'country[T.HKG]', u'country[T.IDN]', u'country[T.IND]', u'country[T.MYS]', u'country[T.NZL]', u'country[T.PHL]', u'country[T.SGP]', u'time_day[T.02]:country[AUS]', u'time_day[T.03]:country[AUS]', u'time_day[T.04]:country[AUS]', u'time_day[T.05]:country[AUS]', u'time_day[T.06]:country[AUS]', u'time_day[T.07]:country[AUS]', u'time_day[T.08]:country[AUS]', u'time_day[T.09]:country[AUS]', u'time_day[T.10]:country[AUS]', u'time_day[T.11]:country[AUS]', u'time_day[T.12]:country[AUS]', u'time_day[T.NA]:country[AUS]', u'time_day[T.02]:country[HKG]', u'time_day[T.03]:country[HKG]', u'time_day[T.04]:country[HKG]', u'time_day[T.05]:country[HKG]', u'time_day[T.06]:country[HKG]', u'time_day[T.07]:country[HKG]', u'time_day[T.08]:country[HKG]', u'time_day[T.09]:country[HKG]', u'time_day[T.10]:country[HKG]', u'time_day[T.11]:country[HKG]', u'time_day[T.12]:country[HKG]', u'time_day[T.NA]:country[HKG]', u'time_day[T.02]:country[IDN]', u'time_day[T.03]:country[IDN]', u'time_day[T.04]:country[IDN]', u'time_day[T.05]:country[IDN]', u'time_day[T.06]:country[IDN]', u'time_day[T.07]:country[IDN]', u'time_day[T.08]:country[IDN]', u'time_day[T.09]:country[IDN]', u'time_day[T.10]:country[IDN]', u'time_day[T.11]:country[IDN]', u'time_day[T.12]:country[IDN]', u'time_day[T.NA]:country[IDN]', u'time_day[T.02]:country[IND]', u'time_day[T.03]:country[IND]', u'time_day[T.04]:country[IND]', u'time_day[T.05]:country[IND]', u'time_day[T.06]:country[IND]', u'time_day[T.07]:country[IND]', u'time_day[T.08]:country[IND]', u'time_day[T.09]:country[IND]', u'time_day[T.10]:country[IND]', u'time_day[T.11]:country[IND]', u'time_day[T.12]:country[IND]', u'time_day[T.NA]:country[IND]', u'time_day[T.02]:country[MYS]', u'time_day[T.03]:country[MYS]', u'time_day[T.04]:country[MYS]', u'time_day[T.05]:country[MYS]', u'time_day[T.06]:country[MYS]', u'time_day[T.07]:country[MYS]', u'time_day[T.08]:country[MYS]', u'time_day[T.09]:country[MYS]', u'time_day[T.10]:country[MYS]', u'time_day[T.11]:country[MYS]', u'time_day[T.12]:country[MYS]', u'time_day[T.NA]:country[MYS]', u'time_day[T.02]:country[NZL]', u'time_day[T.03]:country[NZL]', u'time_day[T.04]:country[NZL]', u'time_day[T.05]:country[NZL]', u'time_day[T.06]:country[NZL]', u'time_day[T.07]:country[NZL]', u'time_day[T.08]:country[NZL]', u'time_day[T.09]:country[NZL]', u'time_day[T.10]:country[NZL]', u'time_day[T.11]:country[NZL]', u'time_day[T.12]:country[NZL]', u'time_day[T.NA]:country[NZL]', u'time_day[T.02]:country[PHL]', u'time_day[T.03]:country[PHL]', u'time_day[T.04]:country[PHL]', u'time_day[T.05]:country[PHL]', u'time_day[T.06]:country[PHL]', u'time_day[T.07]:country[PHL]', u'time_day[T.08]:country[PHL]', u'time_day[T.09]:country[PHL]', u'time_day[T.10]:country[PHL]', u'time_day[T.11]:country[PHL]', u'time_day[T.12]:country[PHL]', u'time_day[T.NA]:country[PHL]', u'time_day[T.02]:country[SGP]', u'time_day[T.03]:country[SGP]', u'time_day[T.04]:country[SGP]', u'time_day[T.05]:country[SGP]', u'time_day[T.06]:country[SGP]', u'time_day[T.07]:country[SGP]', u'time_day[T.08]:country[SGP]', u'time_day[T.09]:country[SGP]', ...], dtype='object')

--这让我觉得":“是"*”的一种简化形式，因为它错过了一个范畴变量。，我认为它无法理解两者都是绝对变量？

f='event_int ~ time_day*country'
y,X = patsy.dmatrices(f, df, return_type='dataframe')
X.columns
Index([u'Intercept', u'time_day[T.02]', u'time_day[T.03]', u'time_day[T.04]', u'time_day[T.05]', u'time_day[T.06]', u'time_day[T.07]', u'time_day[T.08]', u'time_day[T.09]', u'time_day[T.10]', u'time_day[T.11]', u'time_day[T.12]', u'time_day[T.NA]', u'country[T.HKG]', u'country[T.IDN]', u'country[T.IND]', u'country[T.MYS]', u'country[T.NZL]', u'country[T.PHL]', u'country[T.SGP]', u'time_day[T.02]:country[T.HKG]', u'time_day[T.03]:country[T.HKG]', u'time_day[T.04]:country[T.HKG]', u'time_day[T.05]:country[T.HKG]', u'time_day[T.06]:country[T.HKG]', u'time_day[T.07]:country[T.HKG]', u'time_day[T.08]:country[T.HKG]', u'time_day[T.09]:country[T.HKG]', u'time_day[T.10]:country[T.HKG]', u'time_day[T.11]:country[T.HKG]', u'time_day[T.12]:country[T.HKG]', u'time_day[T.NA]:country[T.HKG]', u'time_day[T.02]:country[T.IDN]', u'time_day[T.03]:country[T.IDN]', u'time_day[T.04]:country[T.IDN]', u'time_day[T.05]:country[T.IDN]', u'time_day[T.06]:country[T.IDN]', u'time_day[T.07]:country[T.IDN]', u'time_day[T.08]:country[T.IDN]', u'time_day[T.09]:country[T.IDN]', u'time_day[T.10]:country[T.IDN]', u'time_day[T.11]:country[T.IDN]', u'time_day[T.12]:country[T.IDN]', u'time_day[T.NA]:country[T.IDN]', u'time_day[T.02]:country[T.IND]', u'time_day[T.03]:country[T.IND]', u'time_day[T.04]:country[T.IND]', u'time_day[T.05]:country[T.IND]', u'time_day[T.06]:country[T.IND]', u'time_day[T.07]:country[T.IND]', u'time_day[T.08]:country[T.IND]', u'time_day[T.09]:country[T.IND]', u'time_day[T.10]:country[T.IND]', u'time_day[T.11]:country[T.IND]', u'time_day[T.12]:country[T.IND]', u'time_day[T.NA]:country[T.IND]', u'time_day[T.02]:country[T.MYS]', u'time_day[T.03]:country[T.MYS]', u'time_day[T.04]:country[T.MYS]', u'time_day[T.05]:country[T.MYS]', u'time_day[T.06]:country[T.MYS]', u'time_day[T.07]:country[T.MYS]', u'time_day[T.08]:country[T.MYS]', u'time_day[T.09]:country[T.MYS]', u'time_day[T.10]:country[T.MYS]', u'time_day[T.11]:country[T.MYS]', u'time_day[T.12]:country[T.MYS]', u'time_day[T.NA]:country[T.MYS]', u'time_day[T.02]:country[T.NZL]', u'time_day[T.03]:country[T.NZL]', u'time_day[T.04]:country[T.NZL]', u'time_day[T.05]:country[T.NZL]', u'time_day[T.06]:country[T.NZL]', u'time_day[T.07]:country[T.NZL]', u'time_day[T.08]:country[T.NZL]', u'time_day[T.09]:country[T.NZL]', u'time_day[T.10]:country[T.NZL]', u'time_day[T.11]:country[T.NZL]', u'time_day[T.12]:country[T.NZL]', u'time_day[T.NA]:country[T.NZL]', u'time_day[T.02]:country[T.PHL]', u'time_day[T.03]:country[T.PHL]', u'time_day[T.04]:country[T.PHL]', u'time_day[T.05]:country[T.PHL]', u'time_day[T.06]:country[T.PHL]', u'time_day[T.07]:country[T.PHL]', u'time_day[T.08]:country[T.PHL]', u'time_day[T.09]:country[T.PHL]', u'time_day[T.10]:country[T.PHL]', u'time_day[T.11]:country[T.PHL]', u'time_day[T.12]:country[T.PHL]', u'time_day[T.NA]:country[T.PHL]', u'time_day[T.02]:country[T.SGP]', u'time_day[T.03]:country[T.SGP]', u'time_day[T.04]:country[T.SGP]', u'time_day[T.05]:country[T.SGP]', u'time_day[T.06]:country[T.SGP]', u'time_day[T.07]:country[T.SGP]', u'time_day[T.08]:country[T.SGP]', u'time_day[T.09]:country[T.SGP]', ...], dtype='object')

--如果我显式地将它们声明为“绝对”的vars，我将得到以下-:

f='event_int ~ C(time_day):C(country)'
y,X = patsy.dmatrices(f, df, return_type='dataframe')
X.columns
Index([u'Intercept', u'C(country)[T.HKG]', u'C(country)[T.IDN]', u'C(country)[T.IND]', u'C(country)[T.MYS]', u'C(country)[T.NZL]', u'C(country)[T.PHL]', u'C(country)[T.SGP]', u'C(time_day)[T.02]:C(country)[AUS]', u'C(time_day)[T.03]:C(country)[AUS]', u'C(time_day)[T.04]:C(country)[AUS]', u'C(time_day)[T.05]:C(country)[AUS]', u'C(time_day)[T.06]:C(country)[AUS]', u'C(time_day)[T.07]:C(country)[AUS]', u'C(time_day)[T.08]:C(country)[AUS]', u'C(time_day)[T.09]:C(country)[AUS]', u'C(time_day)[T.10]:C(country)[AUS]', u'C(time_day)[T.11]:C(country)[AUS]', u'C(time_day)[T.12]:C(country)[AUS]', u'C(time_day)[T.NA]:C(country)[AUS]', u'C(time_day)[T.02]:C(country)[HKG]', u'C(time_day)[T.03]:C(country)[HKG]', u'C(time_day)[T.04]:C(country)[HKG]', u'C(time_day)[T.05]:C(country)[HKG]', u'C(time_day)[T.06]:C(country)[HKG]', u'C(time_day)[T.07]:C(country)[HKG]', u'C(time_day)[T.08]:C(country)[HKG]', u'C(time_day)[T.09]:C(country)[HKG]', u'C(time_day)[T.10]:C(country)[HKG]', u'C(time_day)[T.11]:C(country)[HKG]', u'C(time_day)[T.12]:C(country)[HKG]', u'C(time_day)[T.NA]:C(country)[HKG]', u'C(time_day)[T.02]:C(country)[IDN]', u'C(time_day)[T.03]:C(country)[IDN]', u'C(time_day)[T.04]:C(country)[IDN]', u'C(time_day)[T.05]:C(country)[IDN]', u'C(time_day)[T.06]:C(country)[IDN]', u'C(time_day)[T.07]:C(country)[IDN]', u'C(time_day)[T.08]:C(country)[IDN]', u'C(time_day)[T.09]:C(country)[IDN]', u'C(time_day)[T.10]:C(country)[IDN]', u'C(time_day)[T.11]:C(country)[IDN]', u'C(time_day)[T.12]:C(country)[IDN]', u'C(time_day)[T.NA]:C(country)[IDN]', u'C(time_day)[T.02]:C(country)[IND]', u'C(time_day)[T.03]:C(country)[IND]', u'C(time_day)[T.04]:C(country)[IND]', u'C(time_day)[T.05]:C(country)[IND]', u'C(time_day)[T.06]:C(country)[IND]', u'C(time_day)[T.07]:C(country)[IND]', u'C(time_day)[T.08]:C(country)[IND]', u'C(time_day)[T.09]:C(country)[IND]', u'C(time_day)[T.10]:C(country)[IND]', u'C(time_day)[T.11]:C(country)[IND]', u'C(time_day)[T.12]:C(country)[IND]', u'C(time_day)[T.NA]:C(country)[IND]', u'C(time_day)[T.02]:C(country)[MYS]', u'C(time_day)[T.03]:C(country)[MYS]', u'C(time_day)[T.04]:C(country)[MYS]', u'C(time_day)[T.05]:C(country)[MYS]', u'C(time_day)[T.06]:C(country)[MYS]', u'C(time_day)[T.07]:C(country)[MYS]', u'C(time_day)[T.08]:C(country)[MYS]', u'C(time_day)[T.09]:C(country)[MYS]', u'C(time_day)[T.10]:C(country)[MYS]', u'C(time_day)[T.11]:C(country)[MYS]', u'C(time_day)[T.12]:C(country)[MYS]', u'C(time_day)[T.NA]:C(country)[MYS]', u'C(time_day)[T.02]:C(country)[NZL]', u'C(time_day)[T.03]:C(country)[NZL]', u'C(time_day)[T.04]:C(country)[NZL]', u'C(time_day)[T.05]:C(country)[NZL]', u'C(time_day)[T.06]:C(country)[NZL]', u'C(time_day)[T.07]:C(country)[NZL]', u'C(time_day)[T.08]:C(country)[NZL]', u'C(time_day)[T.09]:C(country)[NZL]', u'C(time_day)[T.10]:C(country)[NZL]', u'C(time_day)[T.11]:C(country)[NZL]', u'C(time_day)[T.12]:C(country)[NZL]', u'C(time_day)[T.NA]:C(country)[NZL]', u'C(time_day)[T.02]:C(country)[PHL]', u'C(time_day)[T.03]:C(country)[PHL]', u'C(time_day)[T.04]:C(country)[PHL]', u'C(time_day)[T.05]:C(country)[PHL]', u'C(time_day)[T.06]:C(country)[PHL]', u'C(time_day)[T.07]:C(country)[PHL]', u'C(time_day)[T.08]:C(country)[PHL]', u'C(time_day)[T.09]:C(country)[PHL]', u'C(time_day)[T.10]:C(country)[PHL]', u'C(time_day)[T.11]:C(country)[PHL]', u'C(time_day)[T.12]:C(country)[PHL]', u'C(time_day)[T.NA]:C(country)[PHL]', u'C(time_day)[T.02]:C(country)[SGP]', u'C(time_day)[T.03]:C(country)[SGP]', u'C(time_day)[T.04]:C(country)[SGP]', u'C(time_day)[T.05]:C(country)[SGP]', u'C(time_day)[T.06]:C(country)[SGP]', u'C(time_day)[T.07]:C(country)[SGP]', u'C(time_day)[T.08]:C(country)[SGP]', u'C(time_day)[T.09]:C(country)[SGP]', ...], dtype='object')

问题:

1.如何只包含交互效果&对于这些变量不包括其他任何内容？

2.为什么在第二种情况下将-country排除在外？

相关:Statsmodels formula API (patsy): How to exclude a subset of interaction components?

编辑了基于@Nathaniel j.smith以下-:的答案来排序自己的疑难解答

f2='event_int ~ country:time_day'
y2,X2 = patsy.dmatrices(f2, df, return_type='dataframe')
X2.design_info.term_names
['Intercept', 'country:time_day']

f1='event_int ~ country:time_day-1'
y1,X1 = patsy.dmatrices(f1, df, return_type='dataframe')
X1.design_info.term_names
['country:time_day']

Answer 1

简单回答:试试event_int ~ -1 + time_day:country

较长的答覆：

首先要理解的是，patsy如何决定构建一个设计矩阵有两个不同的阶段。首先，它确定要包含哪些术语。术语是像a或a:b这样的东西。( a和b在a:b中称为a；术语a包含一个单独的因素，也被拼写为a。)找出存在的术语需要扩展和简化您给出的公式，直到您有了一个只使用+和:的表达式。a*b扩展为a + b + a:b等。减法是在这个阶段发生的操作：a + b - a简化为简单的b。所以a*b - a扩展到a + b + a:b - a，简化为b + a:b，但是a:b - a和a:b是一样的，因为没有a可以减去，所以- a被忽略了。这就是为什么编写time_day:country - country和编写time_day:country是一样的。

然后在第二阶段，一旦patsy决定了包含哪些术语，就必须决定如何对这些术语进行编码。这是你遇到麻烦的阶段。

一般规则是，patsy遍历每个包含绝对因素的术语，并计算出它可以使用的一组列，这些列将使模型足够灵活，可以包括指定的交互，但不会与已经添加的任何术语重复使用。

在这种情况下，您的麻烦是由patsy默认添加的截取术语引起的：event_int ~ time_day:country被解释为event_int ~ 1 + time_day:country。这告诉patsy，您希望有一个列表示单独的截获项，然后是包含交互的第二组列--但是这些列与拦截不重叠。显然的虚拟编码方法是time_day和country都是冗余的(共线的)与拦截，所以patsy找到了一个有点复杂的方案，没有这个属性。如果您删除拦截，那么您告诉patsy，它可以继续使用简单的方案，所以它做到了。

关于patsy如何选择编码方案的详细说明如下：http://patsy.readthedocs.org/en/latest/formulas.html#redundancy-and-categorical-factors

手册部分的第一部分可能有太多的数学，但是如果你向下滚动，有一些很好的图表，可以让它更清楚地说明发生了什么(并为数学提供了一些上下文)。如果您搜索y ~ 1 + a:b，您将看到一个图表，它具体显示了当您键入event_int ~ time_day:country时所处的情况。如果您搜索y ~ 1 + a + b + a:b，您将看到event_int ~ time_day*country案例中正在发生的事情的图片。

除了查看X.columns之外，查看X.design_info.term_names和X.design_info.term_slices也很有用，它们向您展示了patsy认为存在哪些“术语”以及它们对应的列。(a和a:b是术语，每个术语都生成多个列。)y ~ 1 + a:b图中的粗轮廓表示，在这种情况下，单项a:b生成两组列:一组列编码b和处理编码，另一组列编码虚拟编码b和处理编码a的成对产品。

最后，解释输出的两个技巧：(1)您可以确定patsy实际上是将这些因素看作是绝对的，因为列名看起来类似于varname[something involving the var's value]。数值因素要么类似于varname，要么(在少有的情况下，将2d矩阵作为预测器传递) varname[column index]。(2)注意country[T.HKG]和country[HKG]的区别-前者表明patsy使用的是降低级别的“处理”编码来避免冗余，而后者则表示简单的虚拟编码。当然，就单个列而言，这些列是相同的，但从概念上讲，差异是非常重要的-- T.模式意味着它删除了其中一个列(注意没有country[T.AUS])，所以像您考虑的那样对列进行细分将不会奏效！

希望这能有所帮助！

Answer 2

看起来我们需要删除常量以避免引用编码。

    >>> f0 = 'event_int ~ C(time_day):C(country) - 1'
    >>> y,X0 = patsy.dmatrices(f0, df, return_type='dataframe')
    >>> print len(X0.columns)
    12
    >>> print X0.columns
    Index([u'C(time_day)[1]:C(country)[a]', u'C(time_day)[2]:C(country)[a]', 
           u'C(time_day)[3]:C(country)[a]', u'C(time_day)[4]:C(country)[a]', 
           u'C(time_day)[1]:C(country)[b]', u'C(time_day)[2]:C(country)[b]', 
           u'C(time_day)[3]:C(country)[b]', u'C(time_day)[4]:C(country)[b]', 
           u'C(time_day)[1]:C(country)[c]', u'C(time_day)[2]:C(country)[c]', 
           u'C(time_day)[3]:C(country)[c]', u'C(time_day)[4]:C(country)[c]'],
           dtype=object)

这是基于一个简单的例子，有3个国家和4个时间段：

import numpy as np
import pandas as pd
import patsy

time = np.tile([1, 2, 3, 4], 3)
country = np.repeat(['a', 'b', 'c'], 4)
event_int = np.random.randint(0, 2, size=len(time))

df = pd.DataFrame({'event_int':event_int, 'time_day':time, 'country':country})