An statistical accumulator allows one to perform incremental calculations. For instance, for computing the arithmetic mean of a stream of numbers given at arbitrary times one could make an object which keeps track of the current number of items given, n and their sum, sum. When one requests the mean, the object simply returns sum/n.
An accumulator like this allows you to compute incrementally in the sense that, when given a new number, you don't need to recompute the entire sum and count.
Similar accumulators can be written for other statistics (cf. boost library for a C++ implementation).
How would you implement accumulators in Python? The code I came up with is:
class Accumulator(object):
"""
Used to accumulate the arithmetic mean of a stream of
numbers. This implementation does not allow to remove items
already accumulated, but it could easily be modified to do
so. also, other statistics could be accumulated.
"""
def __init__(self):
# upon initialization, the numnber of items currently
# accumulated (_n) and the total sum of the items acumulated
# (_sum) are set to zero because nothing has been accumulated
# yet.
self._n = 0
self._sum = 0.0
def add(self, item):
# the 'add' is used to add an item to this accumulator
try:
# try to convert the item to a float. If you are
# successful, add the float to the current sum and
# increase the number of accumulated items
self._sum += float(item)
self._n += 1
except ValueError:
# if you fail to convert the item to a float, simply
# ignore the exception (pass on it and do nothing)
pass
#property
def mean(self):
# the property 'mean' returns the current mean accumulated in
# the object
if self._n > 0:
# if you have more than zero items accumulated, then return
# their artithmetic average
return self._sum / self._n
else:
# if you have no items accumulated, return None (you could
# also raise an exception)
return None
# using the object:
# Create an instance of the object "Accumulator"
my_accumulator = Accumulator()
print my_accumulator.mean
# prints None because there are no items accumulated
# add one (a number)
my_accumulator.add(1)
print my_accumulator.mean
# prints 1.0
# add two (a string - it will be converted to a float)
my_accumulator.add('2')
print my_accumulator.mean
# prints 1.5
# add a 'NA' (will be ignored because it cannot be converted to float)
my_accumulator.add('NA')
print my_accumulator.mean
# prints 1.5 (notice that it ignored the 'NA')
Interesting design questions arise:
How to make the accumulator
thread-safe?
How to safely remove
items?
How to architect in a way
that allows other statistics to be
plugged in easily (a factory for statistics)
For a generalized, threadsafe higher-level function, you could use something like the following in combination with the Queue.Queue class and some other bits:
from Queue import Empty
def Accumulator(f, q, storage):
"""Yields successive values of `f` over the accumulation of `q`.
`f` should take a single iterable as its parameter.
`q` is a Queue.Queue or derivative.
`storage` is a persistent sequence that provides an `append` method.
`collections.deque` may be particularly useful, but a `list` is quite acceptable.
>>> from Queue import Queue
>>> from collections import deque
>>> from threading import Thread
>>> def mean(it):
... vals = tuple(it)
... return sum(it) / len(it)
>>> value_queue = Queue()
>>> LastThreeAverage = Accumulator(mean, value_queue, deque((), 3))
>>> def add_to_queue(it, queue):
... for value in it:
... value_queue.put(value)
>>> putting_thread = Thread(target=add_to_queue,
... args=(range(0, 12, 2), value_queue))
>>> putting_thread.start()
>>> list(LastThreeAverage)
[0, 1, 2, 4, 6, 8]
"""
try:
while True:
storage.append(q.get(timeout=0.1))
q.task_done()
yield f(storage)
except Empty:
pass
This generator function evades most of its purported responsibility by delegating it to other entities:
It relies on Queue.Queue to supply its source elements in a thread-safe manner
A collections.deque object can be passed in as the value of the storage parameter; this provides, among other things, a convenient way to only use the last n (in this case 3) values
The function itself (in this case mean) is passed as a parameter. This will result in less-than-optimally efficient code in some cases, but is readily applied to all sorts of situations.
Note that there is a possibility of the accumulator timing out if your producer thread takes longer than 0.1 seconds per value. This is easily remedied by passing a longer timeout or by removing the timeout parameter entirely. In the latter case the function will block indefinitely at the end of the queue; this usage makes more sense in a case where it's being used in a sub thread (usually a daemon thread). Of course you can also parametrize the arguments that are passed to q.get as a fourth argument to Accumulator.
If you want to communicate end of queue, i.e. that there are no more values to come, from the producer thread (here putting_thread), you can pass and check for a sentinel value or use some other method. There is more info in this thread; I opted to write a subclass of Queue.Queue called CloseableQueue that provides a close method.
There are various other ways you could customize the behaviour of such a function, for example by limiting the queue size; this is just an example of usage.
edit
As mentioned above, this loses some efficiency because of the necessity of recalculation and also, I think, doesn't really answer your question.
A generator function can also accept values through its send method. So you can write a mean generator function like
def meangen():
"""Yields the accumulated mean of sent values.
>>> g = meangen()
>>> g.send(None) # Initialize the generator
>>> g.send(4)
4.0
>>> g.send(10)
7.0
>>> g.send(-2)
4.0
"""
sum = yield(None)
count = 1
while True:
sum += yield(sum / float(count))
count += 1
Here the yield expression is both bringing values —the arguments to send— into the function, while simultaneously passing the calculated values out as the return value of send.
You can pass the generator returned by a call to that function to a more optimizable accumulator generator function like this one:
def EfficientAccumulator(g, q):
"""Similar to Accumulator but sends values to a generator `g`.
>>> from Queue import Queue
>>> from threading import Thread
>>> value_queue = Queue()
>>> g = meangen()
>>> g.send(None)
>>> mean_accumulator = EfficientAccumulator(g, value_queue)
>>> def add_to_queue(it, queue):
... for value in it:
... value_queue.put(value)
>>> putting_thread = Thread(target=add_to_queue,
... args=(range(0, 12, 2), value_queue))
>>> putting_thread.start()
>>> list(mean_accumulator)
[0.0, 1.0, 2.0, 3.0, 4.0, 5.0]
"""
try:
while True:
yield(g.send(q.get(timeout=0.1)))
q.task_done()
except Empty:
pass
If I were doing this in Python, there are two things I would do differently:
Separate out the functionality of each accumulator.
Not use #property in any way you did.
For the first one, I would likely want to come up with an API for performing an accumulation, perhaps something like:
def add(self, num) # add a number
def compute(self) # compute the value of the accumulator
Then I would create a AccumulatorRegistry that holds onto these accumulators, and allows the user to call actions and add to all of them. The code may look like:
class Accumulators(object):
_accumulator_library = {}
def __init__(self):
self.accumulator_library = {}
for key, value in Accumulators._accumulator_library.items():
self.accumulator_library[key] = value()
#staticmethod
def register(name, accumulator):
Accumulators._accumulator_library[name] = accumulator
def add(self, num):
for accumulator in self.accumulator_library.values():
accumulator.add(num)
def compute(self, name):
self.accumulator_library[name].compute()
#staticmethod
def register_decorator(name):
def _inner(cls):
Accumulators.register(name, cls)
return cls
#Accumulators.register_decorator("Mean")
class Mean(object):
def __init__(self):
self.total = 0
self.count = 0
def add(self, num):
self.count += 1
self.total += num
def compute(self):
return self.total / float(self.count)
I should probably speak to your thread-safe question. Python's GIL protects you from a lot of threading issues. There are a few things you may way to do to protect yourself though:
If these objects are localized to one thread, use threading.local
If not, you can wrap the operations in a lock, using the with context syntax to deal with holding the lock for you.
Related
I have a problem where I need to produce something which is naturally computed recursively, but where I also need to be able to interrogate the intermediate steps in the recursion if needed.
I know I can do this by passing and mutating a list or similar structure. However, this looks ugly to me and I'm sure there must be a neater way, e.g. using generators. What I would ideally love to be able to do is something like:
intermediate_results = [f(x) for x in range(T)]
final_result = intermediate_results[T-1]
in an efficient way. While my solution is not performance critical, I can't justify the massive amount of redundant effort in that first line. It looks to me like a generator would be perfect for this except for the fact that f is fundamentally much more suited to recursion in my case (which at least in my mind is the complete opposite of a generator, but maybe I'm just not thinking far enough outside of the box).
Is there a neat Pythonic way of doing something like this that I just don't know about, or do I just need to just capitulate and pollute my function f by passing it an intermediate_results list which I then mutate as a side-effect?
I have a generic solution for you using a decorator. We create a Memoize class which stores the results of previous times the function is executed (including in recursive calls). If the arguments given have already been seen, the cached versions are used to quickly lookup the result.
The custom class has the benefit over an lru_cache in that you can see the results.
from functools import wraps
class Memoize:
def __init__(self):
self.store = {}
def save(self, fun):
#wraps(fun)
def wrapper(*args):
if args not in self.store:
self.store[args] = fun(*args)
return self.store[args]
return wrapper
m = Memoize()
#m.save
def fibo(n):
if n <= 0: return 0
elif n == 1: return 1
else: return fibo(n-1) + fibo(n-2)
Then after running different things you can see what the cache contains. When you run future function calls, m.store will be used as a lookup so calculation doesn't need to be redone.
>>> f(8)
21
>>> m.store
{(1,): 1,
(0,): 0,
(2,): 1,
(3,): 2,
(4,): 3,
(5,): 5,
(6,): 8,
(7,): 13,
(8,): 21}
You could modify the save function to use the name of the function and the args as the key, so that multiple function results can be stored in the same Memoize class.
You can use your existing solution that makes many "redundant" calls to f, but employ the use of function caching to save the results to previous calls to f.
In other words, when f(x1) is called, it's input arguments and corresponding return values are saved, and the next time it is called, the result is simply pulled from the cache
see functools.lru_cache for the standard library solution to this
ie:
from functools import lru_cache
#lru_cache
intermediate_results = [f(x) for x in range(T)]
final_result = intermediate_results[T-1]
Note, however, f must be a pure function (no side-effects, 1-to-1 mapping) for this to work properly
Having considered your comments, I'll now try to give another perspective on the problem.
So, let's consider a concrete example:
def f(x):
a = 2
return g(x) + a if x != 0 else 0
def g(x):
b = 1
return h(x) - b
def h(x):
c = 1/2
return f(x-1)*(1+c)
I
First of all, it should be mentioned that (in our particular case) the algorithm has form of: f(x) = p(f(x - 1)) for some p. It follows that f(x) = p^x(f(0)) = p^x(0). That means we should just apply p to 0 x times to get the desired result, which can be done in an iterative process, so this can be written without recursion. Though I believe that your real case is much harder. Moreover, it would be too boring and uninformative to stop here)
II
Generally speaking, we can divide all possible solutions into two groups: the ones that require refactoring (i.e. rewriting functions f, g, h) and the ones that do not. I have little to offer from the latter one (and I don't think anyone can). Consider the following, however:
def fk(x, k):
a = 2
return k(gk(x, k) + a if x != 0 else 0)
def gk(x, k):
b = 1
return k(hk(x, k) - b)
def hk(x, k):
c = 1/2
return k(fk(x-1, k)*(1+c))
def printret(x):
print(x)
return x
f(4, printret) # see what happens
Inspired by continuation-passing style, but that's totally not it.
What's the point? It's something between your idea of passing a list to write down all the computations and memoizing. This k carries additional behavior with it, such as printing or writing to list (you can make a function that writes to some list, why not?). But if you look carefully you'll see that it lefts inner code of these functions practically untouched (only input and output to function are affected), so one can produce a decorator associated with a function like printret that does essentially the same thing for f, g, h.
Pros: no need to modify code, much more flexible than passing a list, no additional work (like in memoizing).
Cons: Impure (printing or modifying sth), not so flexible as we would like.
III
Now let's see how modifying function bodies can help. Don't be afraid of what's written below, take your time and play with that thing a little.
class Logger:
def __init__(self, lst, cur_val):
self.lst = lst
self.cur_val = cur_val
def bind(self, f):
res = f(self.cur_val)
return Logger([self.cur_val] + res.lst + self.lst, res.cur_val)
def __repr__(self):
return "Logger( " + repr({'value' : self.cur_val,'lst' : self.lst}) + " )"
def unit(x):
return Logger([], x)
# you can also play with lala
def lala(x):
if x <= 0:
return unit(1)
else:
return lala(x - 1).bind(lambda y: unit(2*y))
def f(x):
a = 2
if x == 0:
return unit(0)
else:
return g(x).bind(lambda y: unit(y + a))
def g(x):
b = 1
return h(x).bind(lambda y: unit(y - b))
def h(x):
c = 1/2
return f(x-1).bind(lambda y: unit(y*(1+c)))
f(4) # see for yourself
Logger is called a monad. I'm not very familiar with this concept myself, but I guess I'm doing everything right) f, g, h are functions that take a number and return a Logger instance. Logger's bind takes in a function (like f) and returns Logger with new value (computed by f) and updated 'logs'. The key point - as I see it - is the ability to do whatever we want with collected functions in the order the resulting value was calculated.
Afterword
I'm not at all some kind of 'guru' of functional programming, I believe I'm missing a lot of things here. But what I've understood is that functional programming is about inversing the flow of the program. That's why, for instance, I totally agree with your opinion about generators being opposed to functional programming. When we use generator gen in, say, function func, we yield values one by one to func and func does sth with them in e.g. a loop. The functional approach would be to make gen a function taking func as a parameter and make func perform computations on 'yielded' values. It's like gen and func exchanged their places. So the flow is inversed! And there are plenty of other ways of inversing the flow. Monads are one of them.
itertools islice gets a generator, start value and stop value. it will give you the elements between the start value and stop value as a generator. if islice is not clear you can check the docs here https://docs.python.org/3/library/itertools.html
intermediate_result = map(f, range(T))
final_result = next(itertools.islice(intermediate_result, start=T-1, stop=T))
Update: I've started a thread on python-ideas to propose additional syntax or a stdlib function for this purpose (i.e. specifying the first value sent by yield from). So far 0 replies... :/
How do I intercept the first yielded value of a subgenerator but delegate the rest of the iteration to the latter using yield from?
For example, suppose we have an arbitrary bidirectional generator subgen, and we want to wrap this in another generator gen. The purpose of gen is to intercept the first yielded value of subgen and delegate the rest of the generation—including sent values, thrown exceptions, .close(), etc.—to the sub-generator.
The first thing that might come to mind could be this:
def gen():
g = subgen()
first = next(g)
# do something with first...
yield "intercepted"
# delegate the rest
yield from g
But this is wrong, because when the caller .sends something back to the generator after getting the first value, it will end up as the value of the yield "intercepted" expression, which is ignored, and instead g will receive None as the first .send value, as part of the semantics of yield from.
So we might think to do this:
def gen():
g = subgen()
first = next(g)
# do something with first...
received = yield "intercepted"
g.send(received)
# delegate the rest
yield from g
But what we've done here is just moving the problem back by one step: as soon as we call g.send(received), the generator resumes its execution and doesn't stop until it reaches the next yield statement, whose value becomes the return value of the .send call. So we'd also have to intercept that and re-send it. And then send that, and that again, and so on... So this won't do.
Basically, what I'm asking for is a yield from with a way to customize what the first value sent to the generator is:
def gen():
g = subgen()
first = next(g)
# do something with first...
received = yield "intercepted"
# delegate the rest
yield from g start with received # pseudocode; not valid Python
...but without having to re-implement all of the semantics of yield from myself. That is, the laborious and poorly maintainable solution would be:
def adaptor(generator, init_send_value=None):
send = init_send_value
try:
while True:
send = yield generator.send(send)
except StopIteration as e:
return e.value
which is basically a bad re-implementation of yield from (it's missing handling of throw, close, etc.). Ideally I would like something more elegant and less redundant.
If you're trying to implement this generator wrapper as a generator function using yield from, then your question basically boils down to whether it is possible to specify the first value sent to the "yielded from" generator. Which it is not.
If you look at the formal specification of the yield from expression in PEP 380, you can see why. The specification contains a (surprisingly complex) piece of sample code that behaves the same as a yield from expression. The first few lines are:
_i = iter(EXPR)
try:
_y = next(_i)
except StopIteration as _e:
_r = _e.value
else:
...
You can see that the first thing that is done to the iterator is to call next() on it, which is basically equivalent to .send(None). There is no way to skip that step and your generator will always receive another None whenever yield from is used.
The solution I've come up with is to implement the generator protocol using a class instead of a generator function:
class Intercept:
def __init__(self, generator):
self._generator = generator
self._intercepted = False
def __next__(self):
return self.send(None)
def send(self, value):
yielded_value = self._generator.send(value)
# Intercept the first value yielded by the wrapped generator and
# replace it with a different value.
if not self._intercepted:
self._intercepted = True
print(f'Intercepted value: {yielded_value}')
yielded_value = 'intercepted'
return yielded_value
def throw(self, type, *args):
return self._generator.throw(type, *args)
def close(self):
self._generator.close()
__next__(), send(), throw(), close() are described in the Python Reference Manual.
The class wraps the generator passed to it when created will mimic its behavior. The only thing it changes is that the first value yielded by the generator is replaced by a different value before it is returned to the caller.
We can test the behavior with an example generator f() which yields two values and a function main() which sends values into the generator until the generator terminates:
def f():
y = yield 'first'
print(f'f(): {y}')
y = yield 'second'
print(f'f(): {y}')
def main():
value_to_send = 0
gen = f()
try:
x = gen.send(None)
while True:
print(f'main(): {x}')
# Send incrementing integers to the generator.
value_to_send += 1
x = gen.send(value_to_send)
except StopIteration:
print('main(): StopIteration')
main()
When ran, this example will produce the following output, showing which values arrive in the generator and which are returned by the generator:
main(): first
f(): 1
main(): second
f(): 2
main(): StopIteration
Wrapping the generator f() by changing the statement gen = f() to gen = Intercept(f()), produces the following output, showing that the first yielded value has been replaced:
Intercepted value: first
main(): intercepted
f(): 1
main(): second
f(): 2
As all other calls to any of the generator API are forwarded directly to the wrapped generator, it should behave equivalently to the wrapped generator itself.
If I understand the question, I think this works? Meaning, I ran this script and it did what I expected, which was to print all but the first line of the input file. But as long as the generator passed as the argument to the skip_first function can be iterator over, it should work.
def skip_first(thing):
_first = True
for _result in thing:
if _first:
_ first = False
continue
yield _result
inp = open("/var/tmp/test.txt")
for line in skip_first(inp):
print(line, end="")
I have found some nice examples (here, here) of implementing SICP-like streams in Python. But I am still not sure how to handle an example like the integral found in SICP 3.5.3 "Streams as signals."
The Scheme code found there is
(define (integral integrand initial-value dt)
(define int
(cons-stream initial-value
(add-streams (scale-stream integrand dt)
int)))
int)
What is tricky about this one is that the returned stream int is defined in terms of itself (i.e., the stream int is used in the definition of the stream int).
I believe Python could have something similarly expressive and succinct... but not sure how. So my question is, what is an analogous stream-y construct in Python? (What I mean by a stream is the subject of 3.5 in SICP, but briefly, a construct (like a Python generator) that returns successive elements of a sequence of indefinite length, and can be combined and processed with operations such as add-streams and scale-stream that respect streams' lazy character.)
There are two ways to read your question. The first is simply: How do you use Stream constructs, perhaps the ones from your second link, but with a recursive definition? That can be done, though it is a little clumsy in Python.
In Python you can represent looped data structures but not directly. You can't write:
l = [l]
but you can write:
l = [None]
l[0] = l
Similarly you can't write:
def integral(integrand,initial_value,dt):
int_rec = cons_stream(initial_value,
add_streams(scale_stream(integrand,dt),
int_rec))
return int_rec
but you can write:
def integral(integrand,initial_value,dt):
placeholder = Stream(initial_value,lambda : None)
int_rec = cons_stream(initial_value,
add_streams(scale_stream(integrand,dt),
placeholder))
placeholder._compute_rest = lambda:int_rec
return int_rec
Note that we need to clumsily pre-compute the first element of placeholder and then only fix up the recursion for the rest of the stream. But this does all work (alongside appropriate definitions of all the rest of the code - I'll stick it all at the bottom of this answer).
However, the second part of your question seems to be asking how to do this naturally in Python. You ask for an "analogous stream-y construct in Python". Clearly the answer to that is exactly the generator. The generator naturally provides the lazy evaluation of the stream concept. It differs by not being naturally expressed recursively but then Python does not support that as well as Scheme, as we will see.
In other words, the strict stream concept can be expressed in Python (as in the link and above) but the idiomatic way to do it is to use generators.
It is more or less possible to replicate the Scheme example by a kind of direct mechanical transformation of stream to generator (but avoiding the built-in int):
def integral_rec(integrand,initial_value,dt):
def int_rec():
for x in cons_stream(initial_value,
add_streams(scale_stream(integrand,dt),int_rec())):
yield x
for x in int_rec():
yield x
def cons_stream(a,b):
yield a
for x in b:
yield x
def add_streams(a,b):
while True:
yield next(a) + next(b)
def scale_stream(a,b):
for x in a:
yield x * b
The only tricky thing here is to realise that you need to eagerly call the recursive use of int_rec as an argument to add_streams. Calling it doesn't start it yielding values - it just creates the generator ready to yield them lazily when needed.
This works nicely for small integrands, though it's not very pythonic. The Scheme version works by optimising the tail recursion - the Python version will exceed the max stack depth if your integrand is too long. So this is not really appropriate in Python.
A direct and natural pythonic version would look something like this, I think:
def integral(integrand,initial_value,dt):
value = initial_value
yield value
for x in integrand:
value += dt * x
yield value
This works efficiently and correctly treats the integrand lazily as a "stream". However, it uses iteration rather than recursion to unpack the integrand iterable, which is more the Python way.
In moving to natural Python I have also removed the stream combination functions - for example, replaced add_streams with +=. But we could still use them if we wanted a sort of halfway house version:
def accum(initial_value,a):
value = initial_value
yield value
for x in a:
value += x
yield value
def integral_hybrid(integrand,initial_value,dt):
for x in accum(initial_value,scale_stream(integrand,dt)):
yield x
This hybrid version uses the stream combinations from the Scheme and avoids only the tail recursion. This is still pythonic and python includes various other nice ways to work with iterables in the itertools module. They all "respect streams' lazy character" as you ask.
Finally here is all the code for the first recursive stream example, much of it taken from the Berkeley reference:
class Stream(object):
"""A lazily computed recursive list."""
def __init__(self, first, compute_rest, empty=False):
self.first = first
self._compute_rest = compute_rest
self.empty = empty
self._rest = None
self._computed = False
#property
def rest(self):
"""Return the rest of the stream, computing it if necessary."""
assert not self.empty, 'Empty streams have no rest.'
if not self._computed:
self._rest = self._compute_rest()
self._computed = True
return self._rest
def __repr__(self):
if self.empty:
return '<empty stream>'
return 'Stream({0}, <compute_rest>)'.format(repr(self.first))
Stream.empty = Stream(None, None, True)
def cons_stream(a,b):
return Stream(a,lambda : b)
def add_streams(a,b):
if a.empty or b.empty:
return Stream.empty
def compute_rest():
return add_streams(a.rest,b.rest)
return Stream(a.first+b.first,compute_rest)
def scale_stream(a,scale):
if a.empty:
return Stream.empty
def compute_rest():
return scale_stream(a.rest,scale)
return Stream(a.first*scale,compute_rest)
def make_integer_stream(first=1):
def compute_rest():
return make_integer_stream(first+1)
return Stream(first, compute_rest)
def truncate_stream(s, k):
if s.empty or k == 0:
return Stream.empty
def compute_rest():
return truncate_stream(s.rest, k-1)
return Stream(s.first, compute_rest)
def stream_to_list(s):
r = []
while not s.empty:
r.append(s.first)
s = s.rest
return r
def integral(integrand,initial_value,dt):
placeholder = Stream(initial_value,lambda : None)
int_rec = cons_stream(initial_value,
add_streams(scale_stream(integrand,dt),
placeholder))
placeholder._compute_rest = lambda:int_rec
return int_rec
a = truncate_stream(make_integer_stream(),5)
print(stream_to_list(integral(a,8,.5)))
i would like to perform a calculation using python, where the current value (i) of the equation is based on the previous value of the equation (i-1), which is really easy to do in a spreadsheet but i would rather learn to code it
i have noticed that there is loads of information on finding the previous value from a list, but i don't have a list i need to create it! my equation is shown below.
h=(2*b)-h[i-1]
can anyone give me tell me a method to do this ?
i tried this sort of thing, but that will not work as when i try to do the equation i'm calling a value i haven't created yet, if i set h=0 then i get an error that i am out of index range
i = 1
for i in range(1, len(b)):
h=[]
h=(2*b)-h[i-1]
x+=1
h = [b[0]]
for val in b[1:]:
h.append(2 * val - h[-1]) # As you add to h, you keep up with its tail
for large b list (brr, one-letter identifier), to avoid creating large slice
from itertools import islice # For big list it will keep code less wasteful
for val in islice(b, 1, None):
....
As pointed out by #pad, you simply need to handle the base case of receiving the first sample.
However, your equation makes no use of i other than to retrieve the previous result. It's looking more like a running filter than something which needs to maintain a list of past values (with an array which might never stop growing).
If that is the case, and you only ever want the most recent value,then you might want to go with a generator instead.
def gen():
def eqn(b):
eqn.h = 2*b - eqn.h
return eqn.h
eqn.h = 0
return eqn
And then use thus
>>> f = gen()
>>> f(2)
4
>>> f(3)
2
>>> f(2)
0
>>>
The same effect could be acheived with a true generator using yield and send.
First of, do you need all the intermediate values? That is, do you want a list h from 0 to i? Or do you just want h[i]?
If you just need the i-th value you could us recursion:
def get_h(i):
if i>0:
return (2*b) - get_h(i-1)
else:
return h_0
But be aware that this will not work for large i, as it will exceed the maximum recursion depth. (Thanks for pointing this out kdopen) In that case a simple for-loop or a generator is better.
Even better is to use a (mathematically) closed form of the equation (for your example that is possible, it might not be in other cases):
def get_h(i):
if i%2 == 0:
return h_0
else:
return (2*b)-h_0
In both cases h_0 is the initial value that you start out with.
h = []
for i in range(len(b)):
if i>0:
h.append(2*b - h[i-1])
else:
# handle i=0 case here
You are successively applying a function (equation) to the result of a previous application of that function - the process needs a seed to start it. Your result looks like this [seed, f(seed), f(f(seed)), f(f(f(seed)), ...]. This concept is function composition. You can create a generalized function that will do this for any sequence of functions, in Python functions are first class objects and can be passed around just like any other object. If you need to preserve the intermediate results use a generator.
def composition(functions, x):
""" yields f(x), f(f(x)), f(f(f(x)) ....
for each f in functions
functions is an iterable of callables taking one argument
"""
for f in functions:
x = f(x)
yield x
Your specs require a seed and a constant,
seed = 0
b = 10
The equation/function,
def f(x, b = b):
return 2*b - x
f is applied b times.
functions = [f]*b
Usage
print list(composition(functions, seed))
If the intermediate results are not needed composition can be redefined as
def composition(functions, x):
""" Returns f(x), g(f(x)), h(g(f(x)) ....
for each function in functions
functions is an iterable of callables taking one argument
"""
for f in functions:
x = f(x)
return x
print composition(functions, seed)
Or more generally, with no limitations on call signature:
def compose(funcs):
'''Return a callable composed of successive application of functions
funcs is an iterable producing callables
for [f, g, h] returns f(g(h(*args, **kwargs)))
'''
def outer(f, g):
def inner(*args, **kwargs):
return f(g(*args, **kwargs))
return inner
return reduce(outer, funcs)
def plus2(x):
return x + 2
def times2(x):
return x * 2
def mod16(x):
return x % 16
funcs = (mod16, plus2, times2)
eq = compose(funcs) # mod16(plus2(times2(x)))
print eq(15)
While the process definition appears to be recursive, I resisted the temptation so I could stay out of maximum recursion depth hades.
I got curious, searched SO for function composition and, of course, there are numerous relavent Q&A's.
What is lazy evaluation in Python?
One website said :
In Python 3.x the range() function returns a special range object which computes elements of the list on demand (lazy or deferred evaluation):
>>> r = range(10)
>>> print(r)
range(0, 10)
>>> print(r[3])
3
What is meant by this?
The object returned by range() (or xrange() in Python2.x) is known as a lazy iterable.
Instead of storing the entire range, [0,1,2,..,9], in memory, the generator stores a definition for (i=0; i<10; i+=1) and computes the next value only when needed (AKA lazy-evaluation).
Essentially, a generator allows you to return a list like structure, but here are some differences:
A list stores all elements when it is created. A generator generates the next element when it is needed.
A list can be iterated over as much as you need, a generator can only be iterated over exactly once.
A list can get elements by index, a generator cannot -- it only generates values once, from start to end.
A generator can be created in two ways:
(1) Very similar to a list comprehension:
# this is a list, create all 5000000 x/2 values immediately, uses []
lis = [x/2 for x in range(5000000)]
# this is a generator, creates each x/2 value only when it is needed, uses ()
gen = (x/2 for x in range(5000000))
(2) As a function, using yield to return the next value:
# this is also a generator, it will run until a yield occurs, and return that result.
# on the next call it picks up where it left off and continues until a yield occurs...
def divby2(n):
num = 0
while num < n:
yield num/2
num += 1
# same as (x/2 for x in range(5000000))
print divby2(5000000)
Note: Even though range(5000000) is a generator in Python3.x, [x/2 for x in range(5000000)] is still a list. range(...) does it's job and generates x one at a time, but the entire list of x/2 values will be computed when this list is create.
In a nutshell, lazy evaluation means that the object is evaluated when it is needed, not when it is created.
In Python 2, range will return a list - this means that if you give it a large number, it will calculate the range and return at the time of creation:
>>> i = range(100)
>>> type(i)
<type 'list'>
In Python 3, however you get a special range object:
>>> i = range(100)
>>> type(i)
<class 'range'>
Only when you consume it, will it actually be evaluated - in other words, it will only return the numbers in the range when you actually need them.
A github repo named python patterns and wikipedia tell us what lazy evaluation is.
Delays the eval of an expr until its value is needed and avoids repeated evals.
range in python3 is not a complete lazy evaluation, because it doesn't avoid repeated eval.
A more classic example for lazy evaluation is cached_property:
import functools
class cached_property(object):
def __init__(self, function):
self.function = function
functools.update_wrapper(self, function)
def __get__(self, obj, type_):
if obj is None:
return self
val = self.function(obj)
obj.__dict__[self.function.__name__] = val
return val
The cached_property(a.k.a lazy_property) is a decorator which convert a func into a lazy evaluation property. The first time property accessed, the func is called to get result and then the value is used the next time you access the property.
eg:
class LogHandler:
def __init__(self, file_path):
self.file_path = file_path
#cached_property
def load_log_file(self):
with open(self.file_path) as f:
# the file is to big that I have to cost 2s to read all file
return f.read()
log_handler = LogHandler('./sys.log')
# only the first time call will cost 2s.
print(log_handler.load_log_file)
# return value is cached to the log_handler obj.
print(log_handler.load_log_file)
To use a proper word, a python generator object like range are more like designed through call_by_need pattern, rather than lazy evaluation