Advanced Usage¶
This section covers more advanced modelx concepts and techniques that are not covered by the earlier sections.
Contents
Advanced Concepts¶
In this section, more concepts we haven’t yet covered are introduced. Some of them are demonstrated by examples in the following section.
Space Members¶
Spaces can contain cells and other spaces. In fact, spaces have 3 kinds of
their “members”. You can get those members as if they are attributes
of the containing spaces,
by attribute access(.
) expression.
 Cells
As we have seen in the previous example, spaces contain cells, and the cells belong to spaces. One cells must belong to one ane only one space.
The
cells
property of Space returns a dictionary of all the cells associated with their names. (Sub)spaces
As previously mentioned, spaces can be created in another space. Spaces in another space are called subspaces of the containing space. There 2 kinds of subspaces, static subspaces and dynamic subspace.
Static subspaces are those that are created manually, just like those created in models. There is no difference between spaces created directly under a model and static spaces created under a space, except for their parents being different types.
You can create a static subspace by calling
new_space
method of their parents:model, space = new_model(), new_space() space.new_space('Subspace1') @defcells def foo(): return 123
You can get a subspace as an attribute of the parent space, or by accessing the parent space’s
spaces
property:>>> space.spaces['Subspace1'].foo() 123 >>> space.Subspace1.foo() 123
The other kind of subspaces, is dynamic subspaces. Unlike static suspaces, dynamic subspaces can only be created in spaces, but not directly in models.
Dynamic spaces are parametrized spaces that are created onthefly when requested through call(
()
) or subscript([]
) operation on their parent spaces. We’ll explore more on dynamic spaces in the next section, in conjunction with space inheritance by going through an example. References
Often times you want access from cells formulas in a space to other objects than cells or subspaces in the same space. References are names bound to arbitrary objects that are accessible from within the same space:
model, space = new_model(), new_space() @defcells def bar(): return 2 * n
bar
cells above refers ton
, which has not yet been defined. Withoutn
being defined, callingbar
will raise an error. To define a referencen
, you can simply assign a value ton
attribute ofspace
:>>> space.n = 3 >>> bar() 6
The
refs
property of space returns a mapping of reference names to their objects:>>> list(space.refs.keys()) ['__builtins__', 'n', '_self']
Be default,
__builtins__
and_self
are defined in any space. In fact,__builtins__
is defined by default as a “global” reference in the model. Global references are names accessible from any space in a model. Other than the default reference, you can define your own, by simply assigning a value as an attribute of the model:>>> model.z = 4 >>> list(model.refs.keys()) ['z', '__builtins__'] >>> list(space.refs.keys()) ['z', '__builtins__', 'n', '_self']
__builtins__
points to Python builtin module. It is defined to allow cells formulas to use builtin functions._self
points to the space itself. This allows cells formulas to get access to its parent space.
As mentioned earlier, formulas of cells are evaluated in the namespace that is associated with their parent spaces.
The namespace of a space is a mapping of names to the space members. As explained in the previous section, space members are either cells of the space, or subspaces of the space or references accessible from the space.
The table below breaks down all the members in the namespace by its types and subtypes.
cells 
self cells 
Cells defined in or overridden in the space 
derived cells 
Cells inherited from one of the base spaces 

spaces 
self spaces 
Subspace defined in or overridden in the space 
derived spaces 
Subspace inherited from one of the base spaces 

references (refs) 
self references 
References defined in or overridden in the space 
derived references 
References inherited from one of the base spaces 

global references 
Global references defined in the parent model 

local references 
Only 

parameters 
(Only in dynamic spaces) Space parameters 
Each type of the members has “self” members and “derived” members. Those distinctions stem from space inheritance explained in the next section.
Space Inheritance¶
Space inheritance is a concept analogous to class inheritance in objectoriented programming languages. By making full use of space inheritance, you can organize multiple spaces sharing similar features into an inheritance tree of spaces, minimizing duplicated formula definitions, keeping your model organized and transparent while maintaining model integrity.
Inheritance lets one space use(inherit) other spaces, as base spaces. The inheriting space is called a derived space of the base spaces. The cells in the base spaces are copied automatically in the derived space. In the derived space, formulas of cells from base spaces can be overridden. You can also add cells to the derived space, that do not exist in any of the base spaces.
A space can have multiple base spaces. This is called multiple inheritance. The base spaces can have their base spaces, and derivedbase relationships between spaces make up a directional graph of dependency. In case of multiple inheritance, we need a way to order base spaces to determine the priority of base spaces. modelx uses the same algorithm as Python for ordering bases, which is, C3 superclass linearization algorithm (a.k.a C3 Method Resolution Order or MRO). The links below are provided in case you want to know more about C3 MRO.
More complex example¶
Let’s see how inheritance works by a simple code of
pricing life insurance policies.
First, we are goint to create a very simple life model as a space and name it
Life
.
Then we’ll populate the space with cells that calculate the number of death
and remaining lives by age.
Then to price a term life policy, we will derive a TermLife
space from
the Life
space, and add some cells to calculate death benefits
paid to the insured, and their present value.
Next, we want to model an endowment policy. Since the endowment policy
pays out a maturity benefit in addition to the death benefits covered by the
term life policy, we derive a Endowment
space from TermLife
,
and make a residual change to the benefits
formula.
Creating the Life space¶
Below is a mathematical representation of the life model we’ll
build as a Life
space.
where, \(l(x)\) denotes the number of lives at age x, \(d(x)\) denotes the number of death occurring between the age x and age x + 1, \(q\) denotes the annual mortality rate (for simplicity, we’ll assume a constant mortality rate of 0.003 for all ages for the moment.) One letter names like l, d, q would be too short for real world practices, but we use them here just for simplicity, as they often appear in classic actuarial textbooks. Yet another simplification is, we set the starting age of x at 50, just to get output shorter. As long as we use a constant mortality age, it shouldn’t affect the results whether the starting age is 0 or 50. Below the modelx code for this life model:
model, life = new_model(), new_space('Life')
def l(x):
if x == x0:
return 100000
else:
return l(x  1)  d(x  1)
def d(x):
return l(x) * q
def q():
return 0.003
l, d, q = defcells(l, d, q)
life.x0 = 50
The second to last line of the code above has the same effect as putting
@defcells
decorator on top of each of the 3 function definitions.
This line creates 3 new cells
from the 3 functions in the Life
space, and rebind names l
, d
,
q
to the 3 cells in the current scope.
You must have noticed that l(x)
formula is referring
to the name x0
, which is not defined yet.
The last line is for defining x0
as the issue age
in the Life
model and assigning a value to it.
To examine the space, we can check values of the cells in Life
as below:
>>> l(60)
97040.17769489168
>>> life.frame
l d q
x
50.0 100000.000000 300.000000 NaN
51.0 99700.000000 299.100000 NaN
52.0 99400.900000 298.202700 NaN
53.0 99102.697300 297.308092 NaN
54.0 98805.389208 296.416168 NaN
55.0 98508.973040 295.526919 NaN
56.0 98213.446121 294.640338 NaN
57.0 97918.805783 293.756417 NaN
58.0 97625.049366 292.875148 NaN
59.0 97332.174218 291.996523 NaN
60.0 97040.177695 NaN NaN
NaN NaN NaN 0.003
Deriving the Term Life space¶
Next, we’ll see how we can extend this space to represent a term life policy.
To simplify things, here we focus on one policy with the sum
assured of 1 (in whatever unit of currency).
With this assumption, if we define benefits(x)
as the expected value at
issue of benefits paid between the age x and x + 1, then it should
equate to the probability of death between age x and x + 1, of the
insured at the point of issue. In a math expression, this should be written:
where \(l(x)\) and \(d(x)\) are the same definition from the preceding example, and \(x0\) denotes the issue age of the policy. And further we define the present value of benefits at age x as:
n
denotes the policy term in years, and disc_rate
denotes the
discounting rate for the present value calculation.
Continued from the previous code, we are going to derive the TermLife
space
from the Life
space, to add the benefits and present value calculations.
term_life = model.new_space(name='TermLife', bases=life)
@defcells
def benefits(x):
if x < x0 + n:
return d(x) / l(x0)
if x <= x0 + n:
return 0
@defcells
def pv_benefits(x):
if x < x0:
return 0
elif x <= x0 + n:
return benefits(x) + pv_benefits(x + 1) / (1 + disc_rate)
else:
return 0
The first line in the sample above creates TermLife
space derived
from the Life
space, by passing the Life
space as bases
parameter
to the new_space
method of the model. The TermLife
space at this point
has the same cells as its sole base space Life
space.
The following 2 cells definitions (2 function definitions with defcells
decorators), are for adding the cells that did not exist in Life
space. The formulas are referring to the names
that are not defined yet. Those are n
, disc_rate
.
We need to define those in the TermLife
space.
The reference x0
is inherited from the Life
space.
term_life.n = 10
term_life.disc_rate = 0
You get the following results by examining the TermLife
space (The
order of the columns in the DataFrame may be different on your screen).:
>>> term_life.pv_benefits(50)
0.02959822305108317
>>> term_life.frame
d q l pv_benefits benefits
x
50.0 300.000000 NaN 100000.000000 0.029598 0.003000
51.0 299.100000 NaN 99700.000000 0.026598 0.002991
52.0 298.202700 NaN 99400.900000 0.023607 0.002982
53.0 297.308092 NaN 99102.697300 0.020625 0.002973
54.0 296.416168 NaN 98805.389208 0.017652 0.002964
55.0 295.526919 NaN 98508.973040 0.014688 0.002955
56.0 294.640338 NaN 98213.446121 0.011733 0.002946
57.0 293.756417 NaN 97918.805783 0.008786 0.002938
58.0 292.875148 NaN 97625.049366 0.005849 0.002929
59.0 291.996523 NaN 97332.174218 0.002920 0.002920
60.0 NaN NaN NaN 0.000000 0.000000
61.0 NaN NaN NaN 0.000000 NaN
NaN NaN 0.003 NaN NaN NaN
You can see that the values of l
, d
, q
cells are the same
as those in Life
space, as Life
and LifeTerm
have exactly
the same formulas for those cells, but be aware that
those cells are not shared between the base and derived spaces.
Unlike class inheritance in OOP languages, space inheritance is in terms of
space instances(or objects), not classes,
so cells are copied from the base spaces to derived space
upon creating the derived space.
Deriving the Endowment space¶
We’re going to create another space to test overriding inherited cells.
We will derive Endowment
space from LifeTerm
space. The diagram
below shows the relationships of the 3 spaces considered here.
A space from which an arrow originates is derived from the space the
arrow points to.
The endowment policy pays out the maturity benefit of 1
at the end of its policy term.
We have defined benefits
cells as the expected value of benefits,
so in addition to the death benefits considered in LifeTerm
space,
we’ll add the maturity benefit by overriding the benefits
definition
in Endowment
space. In reality, the insured will not get both death
and maturity benefits, but here we are considering an probabilistic model,
so the benefits would be the sum of expected value of death and maturity
benefits:
endowment = model.new_space(name='Endowment', bases=term_life)
@defcells
def benefits(x):
if x < x0 + n:
return d(x) / l(x0)
elif x == x0 + n:
return l(x) / l(x0)
else:
return 0
And the same operations on the Endowment
space produces the following
results:
>>> endowment.pv_benefits(50)
1.0
>>> endowment.frame
pv_benefits benefits l q d
x
50.0 1.000000 0.003000 100000.000000 NaN 300.000000
51.0 0.997000 0.002991 99700.000000 NaN 299.100000
52.0 0.994009 0.002982 99400.900000 NaN 298.202700
53.0 0.991027 0.002973 99102.697300 NaN 297.308092
54.0 0.988054 0.002964 98805.389208 NaN 296.416168
55.0 0.985090 0.002955 98508.973040 NaN 295.526919
56.0 0.982134 0.002946 98213.446121 NaN 294.640338
57.0 0.979188 0.002938 97918.805783 NaN 293.756417
58.0 0.976250 0.002929 97625.049366 NaN 292.875148
59.0 0.973322 0.002920 97332.174218 NaN 291.996523
60.0 0.970402 0.970402 97040.177695 NaN NaN
61.0 0.000000 NaN NaN NaN NaN
NaN NaN NaN NaN 0.003 NaN
You can see pv_benefits
for all ages and benefits
for age 60
show values different from TermLife
as we overrode benefits
.
pv_benefits(50)
being 1 is not surprising. The disc_rate
set to 1 in TermLife
space is also inherited to the Endowment
space.
The discounting rate of benefits being 1 means by taking the
present value of the benefits, we are simply taking the sum of
all expected values of future benefits, which must equates to 1,
because the insured gets 1 by 100% chance.
Dynamic spaces¶
In many situations, you want to apply a set of calculations in a space, or a tree of spaces, to different data sets. You can achieve that by applying the space inheritance on dynamic spaces.
Dynamic spaces are parametrized spaces that are created onthefly when
requested through call(()
) or subscript([]
) operation on their parent
spaces.
To define dynamic spaces in a parent space,
you create the space with a parameter function whose signature is
used to define space parameters. The parameter function should return,
if any, a mapping of parameter names to their arguments,
to be pass on to the new_space
method, when the dynamic spaces
are created.
To see how this works, let’s continue with the previous example.
In the last example, we manually set the issue age x0
of the policy
to 50, and the policy term n
to 10.
We’ll extend this example and create policies as dynamic spaces with
with different policy attributes.
Assume we have 3 term life polices with the following attributes:
Policy ID 
Issue Age 
Policy Term 

1 
50 
10 
2 
60 
15 
3 
70 
5 
We’ll create this sample data as a nested list:
data = [[1, 50, 10], [2, 60, 15], [3, 70, 5]]
The diagram shows the design of the model we are going to create.
The lines with a filled diamond shape on one end indicates that
Policy
model is the parent space of the 3 dynamic spaces, Policy1
,
Policy2
, Policy3
, each of which represents
each of the 3 policies above.
While Policy
is the parent space of the 3 dynamic space,
it is also the base space of them.
Policy
space inherits its members from Term
model, and in turn
Policy
is inherited by the 3 dynamic spaces.
This inheritance is represented by the unfilled arrowhead next the
filled diamond.
Below is a script to extend the model as we designed above.
def params(policy_id):
return {'name': 'Policy%s' % policy_id,
'bases': _self}
policy = model.new_space(name='Policy', bases=term_life, formula=params)
policy.data = data
@defcells
def x0():
return data[policy_id  1][1]
@defcells
def n():
return data[policy_id  1][2]
The params
function is passed to the constructor of the Policy
space
as the argument of formula
parameter. The signature of params
func
is used to determine the parameter of the dynamic spaces,
and the returned dictionary is passed to the new_space
as arguments when
the dynamic spaces are created.
params
is called when you create the dynamic subspaces of
Policy
, by calling the nthe element of Policy
.
params
is evaluated in the Policy
’s namespace. _self
is a spacial reference that points to Policy
.
The parameter policy_id
becomes available within the namespace of each
dynamic space.
In each of the dynamic spaces, the values of x0
and n
are
taken from data
for each policy:
>>> policy(1).pv_benefits(50)
0.02959822305108317
>>> policy(2).pv_benefits(60)
0.04406717516109439
>>> policy(3).pv_benefits(70)
0.014910269595243001
>>> policy(3).frame
n x0 d benefits l pv_benefits q
x
NaN 5.0 70.0 NaN NaN NaN NaN 0.003
70.0 NaN NaN 300.000000 0.003000 100000.000000 0.014910 NaN
71.0 NaN NaN 299.100000 0.002991 99700.000000 0.011910 NaN
72.0 NaN NaN 298.202700 0.002982 99400.900000 0.008919 NaN
73.0 NaN NaN 297.308092 0.002973 99102.697300 0.005937 NaN
74.0 NaN NaN 296.416168 0.002964 98805.389208 0.002964 NaN
75.0 NaN NaN NaN 0.000000 NaN 0.000000 NaN
76.0 NaN NaN NaN NaN NaN 0.000000 NaN
>>> policy.spaces
{'Policy1': <Space Policy[1] in Model1>,
'Policy2': <Space Policy[2] in Model1>,
'Policy3': <Space Policy[3] in Model1>}
Dynamic spaces of a base space are not passed on to the derived spaces.
When a space is derived from a base space that has dynamically created
subspaces, those dynamically created subspaces themselves are not passed
on to the derived spaces. Instead, the parameter function of the base
space is inherited, so subspaces of the derived space are created upon
call(using ()
) or subscript (using []
) operators
the derived space with arguments.
Reading Excel files¶
You can read data stored in an Excel file into newly created cells.
Space has two methods new_cells_from_excel
and new_space_from_excel
.
new_space_from_excel
is also available on Model.
You need to have Openpyxl package available in your Python environment
to use these methods.
new_cells_from_excel
method reads values from a range in an Excel file,
creates cells and populates them with the values in the range.
new_space_from_excel
methods reads values from a range in an Excel file,
creates a space, and in that space, creates
dynamic spaces using one or some of the index rows and/or columns
as space parameters, and creates cells in the dynamics spaces populating
them with the values in the range.
Refer to the modelx reference for concrete description of those methods.
Exporting to Pandas objects¶
If you have Pandas installed in your Python environment, you can export values
of cells to Pandas’ DataFrame or Series objects.
Spaces have frame
property, which generates a DataFrame
object whose columns are cells names, and whose indexes are
cells parameters. Multiple cells in a space may have different
sets of parameters. Generated