Samplitude (s8e) is a statistical distributions command line tool

## Project description

# Samplitude

CLI generation and plotting of random variables:

```
$ samplitude "sin(0.31415) | sample(6) | round | cli"
0.0
0.309
0.588
0.809
0.951
1.0
```

The word *samplitude* is a portmanteau of *sample* and *amplitude*. This
project also started as an étude, hence should be pronounced *sampl-étude*.

`samplitude`

is a chain starting with a *generator*, followed by zero or more
*filters*, followed by a consumer. Most generators are infinite (with the
exception of `range`

and `lists`

and possibly `stdin`

). Some of the filters can
turn infinite generators into finite generators (like `sample`

and `gobble`

),
and some filters can turn finite generators into infinite generators, such as
`choice`

.

*Consumers* are filters that necessarily flush the input; `list`

, `cli`

,
`json`

, `unique`

, and the plotting tools, `hist`

, `scatter`

and `line`

are
examples of consumers. The `list`

consumer is a Jinja2 built-in, and other
Jinja2 consumers are `sum`

, `min`

, and `max`

:

```
samplitude "sin(0.31415) | sample(5) | round | max | cli"
0.951
```

For simplicity, **s8e** is an alias for samplitude.

## Generators

In addition to the standard `range`

function, we support infinite generators

`exponential(lambd)`

:`lambd`

is 1.0 divided by the desired mean.`uniform(a, b)`

: Get a random number in the range`[a, b)`

or`[a, b]`

depending on rounding.`gauss(mu, sigma)`

:`mu`

is the mean, and`sigma`

is the standard deviation.`normal(mu, sigma)`

: as above`lognormal(mu, sigma)`

: as above`triangular(low, high)`

: Continuous distribution bounded by given lower and upper limits, and having a given mode value in-between.`beta(alpha, beta)`

: Conditions on the parameters are`alpha > 0`

and`beta > 0`

. Returned values range between 0 and 1.`gamma(alpha, beta)`

: as above`weibull(alpha, beta)`

:`alpha`

is the scale parameter and`beta`

is the shape parameter.`pareto(alpha)`

: Pareto distribution.`alpha`

is the shape parameter.`vonmises(mu, kappa)`

:`mu`

is the mean angle, expressed in radians between 0 and`2*pi`

, and`kappa`

is the concentration parameter, which must be greater than or equal to zero. If kappa is equal to zero, this distribution reduces to a uniform random angle over the range 0 to`2*pi`

.

Provided that you have installed the `scipy.stats`

package, the

`pert(low, peak, high)`

distribution is supported.

We have a special infinite generator (filter) that works on finite generators:

`choice`

,

whose behaviour is explained below.

For input from files, either use `words`

with a specified environment variable
`DICTIONARY`

, or pipe through

`stdin()`

which reads from `stdin`

.

If the file is a csv file, there is a `csv`

generator that reads a csv file with
Pandas and outputs the first column (if nothing else is specified). Specify the
column with either an integer index or a column name:

```
>>> samplitude "csv('iris.csv', 'virginica') | counter | cli"
0 50
1 50
2 50
```

For other files, we have the `file`

generator:

```
>>> s8e "file('iris.csv') | sample(1) | cli"
150,4,setosa,versicolor,virginica
```

Finally, we have `combinations`

and `permutations`

that are inherited from
itertools and behave exactly like those.

```
>>> s8e "'ABC' | permutations | cli"
```

However, the output of this is rather non-UNIXy, with the abstractions leaking through:

```
>>> s8e "'HT' | permutations | cli"
('H', 'T')
('T', 'H')
```

So to get a better output, we can use an *elementwise join* `elt_join`

:

```
>>> s8e "'HT' | permutations | elt_join | cli"
H T
T H
```

which also takes a seperator as argument:

```
>>> s8e "'HT' | permutations | elt_join(';') | cli"
H;T
T;H
```

This is already supported by Jinja's `map`

function (notice the strings around `join`

):

```
>>> s8e "'HT' | permutations | map('join', ';') | cli"
H;T
T;H
```

We can thus count the number of permutations of a set of size 10:

```
>>> s8e "range(10) | permutations | len"
3628800
```

The `product`

generator takes two generators and computes a cross-product of
these. In addition,

## A warning about infinity

All generators are (potentially) infinite generators, and must be sampled with
`sample(n)`

before consuming!

## Usage and installation

Install with

```
pip install samplitude
```

or to get bleeding release,

```
pip install git+https://github.com/pgdr/samplitude
```

### Examples

This is pure Jinja2:

```
>>> samplitude "range(5) | list"
[0, 1, 2, 3, 4]
```

However, to get a more UNIXy output, we use `cli`

instead of `list`

:

```
>>> s8e "range(5) | cli"
0
1
2
3
4
```

To limit the output, we use `sample(n)`

:

```
>>> s8e "range(1000) | sample(5) | cli"
0
1
2
3
4
```

That isn't very helpful on the `range`

generator, which is already finite, but
is much more helpful on an infinite generator. The above example is probably
better written as

```
>>> s8e "count() | sample(5) | cli"
0
1
2
3
4
```

However, much more interesting are the infinite random generators, such as the
`uniform`

generator:

```
>>> s8e "uniform(0, 5) | sample(5) | cli"
3.3900198868059235
1.2002767137709318
0.40999391897569126
1.9394585953696264
4.37327472704115
```

We can round the output in case we don't need as many digits (note that `round`

is a generator as well and can be placed on either side of `sample`

):

```
>>> s8e "uniform(0, 5) | round(2) | sample(5) | cli"
4.98
4.42
2.05
2.29
3.34
```

### Selection and modifications

The `sample`

behavior is equivalent to the `head`

program, or from languages
such as Haskell. The `head`

alias is supported:

```
>>> samplitude "uniform(0, 5) | round(2) | head(5) | cli"
4.58
4.33
1.87
2.09
4.8
```

`drop`

is also available:

```
>>> s8e "uniform(0, 5) | round(2) | drop(2) | head(3) | cli"
1.87
2.09
4.8
```

To **shift** and **scale** distributions, we can use the `shift(s)`

and
`scale(s)`

filters.
To get a Poisson distribution process starting at 15, we can run

```
>>> s8e "poisson(4) | shift(15) | sample(5) |cli"
18
21
19
22
17
```

or to get the Poisson point process (exponential distribution),

```
>>> s8e "exponential(4) | round | shift(15) | sample(5) |cli"
16.405
15.54
15.132
15.153
15.275
```

Both `shift`

and `scale`

work on generators, so to add `sin(0.1)`

and
`sin(0.2)`

, we can run

```
>>> s8e "sin(0.1) | shift(sin(0.2)) | sample(10) | cli"
```

### Choices and other operations

Using `choice`

with a finite generator gives an infinite generator that chooses
from the provided generator:

```
>>> samplitude "range(0, 11, 2) | choice | sample(6) | cli"
8
0
8
10
4
6
```

Jinja2 supports more generic lists, e.g., lists of strings. Hence, we can write

```
>>> s8e "['win', 'draw', 'loss'] | choice | sample(6) | sort | cli"
draw
draw
loss
loss
loss
win
```

... and as in Python, strings are also iterable:

```
>>> s8e "'HT' | cli"
H
T
```

... so we can flip six coins with

```
>>> s8e "'HT' | choice | sample(6) | cli"
H
T
T
H
H
H
```

We can flip 100 coins and count the output with `counter`

(which is
`collections.Counter`

)

```
>>> s8e "'HT' | choice | sample(100) | counter | cli"
H 47
T 53
```

The `sort`

functionality works as expected on a `Counter`

object (a
`dict`

type), so if we want the output sorted by key, we can run

```
>>> s8e "range(1,7) | choice | sample(100) | counter | sort | elt_join | cli" 42 # seed=42
1 17
2 21
3 12
4 21
5 13
6 16
```

There is a minor hack to sort by value, namely by `swap`

-ing the Counter twice:

```
>>> s8e "range(1,7) | choice | sample(100) |
counter | swap | sort | swap | elt_join | cli" 42 # seed=42
3 12
5 13
6 16
1 17
2 21
4 21
```

The `swap`

filter does an element-wise reverse, with element-wise reverse
defined on a dictionary as a list of `(value, key)`

for each key-value pair in
the dictionary.

So, to get the three most common anagram strings, we can run

```
>>> s8e "words() | map('sort') | counter | swap | sort(reverse=True) |
swap | sample(3) | map('first') | elt_join('') | cli"
aeprs
acerst
opst
```

Using `stdin()`

as a generator, we can pipe into `samplitude`

. Beware that
`stdin()`

flushes the input, hence `stdin`

(currently) does not work with
infinite input streams.

```
>>> ls | samplitude "stdin() | choice | sample(1) | cli"
some_file
```

Then, if we ever wanted to shuffle `ls`

we can run

```
>>> ls | samplitude "stdin() | shuffle | cli"
some_file
```

```
>>> cat FILE | samplitude "stdin() | cli"
# NOOP; cats FILE
```

### The fun powder plot

For fun, if you have installed `matplotlib`

, we support plotting, `hist`

being
the most useful.

```
>>> samplitude "normal(100, 5) | sample(1000) | hist"
```

An exponential distribution can be plotted with `exponential(lamba)`

. Note that
the `cli`

output must be the last filter in the chain, as that is a command-line
utility only:

```
>>> s8e "normal(100, 5) | sample(1000) | hist | cli"
```

To **repress output after plotting**, you can use the `gobble`

filter to empty
the pipe:

```
>>> s8e "normal(100, 5) | sample(1000) | hist | gobble"
```

The
`pert`

distribution
takes inputs `low`

, `peak`

, and `high`

:

```
>>> s8e "pert(10, 50, 90) | sample(100000) | hist(100) | gobble"
```

Although `hist`

is the most useful, one could imaging running `s8e`

on
timeseries, where a `line`

plot makes most sense:

```
>>> s8e "sin(22/700) | sample(200) | line"
```

The scatter function can also be used, but requires that the input stream is a
stream of pairs, which can be obtained either by the `product`

generator, or via
the `pair`

or `counter`

filter:

```
s8e "normal(100, 10) | sample(10**5) | round(0) | counter | scatter"
```

### Fourier

A fourier transform is offered as a filter `fft`

:

```
>>> samplitude "sin(0.1) | shift(sin(0.2)) | sample(1000) | fft | line | gobble"
```

## Your own filter

If you use Samplitude programmatically, you can register your own filter by sending a dictionary

```
{'name1' : filter1,
'name2' : filter2,
#...,
'namen' : filtern,
}
```

to the `samplitude`

function.

### Example: secretary problem

Suppose you want to emulate the secretary problem ...

#### Intermezzo: The problem

For those not familiar, you are a boss, Alice, who wants to hire a new secretary Bob. Suppose you want to hire the tallest Bob of all your candidates, but the candidates arrive in a stream, and you know only the number of candidates. For each candidate, you have to accept (hire) or reject the candidate. Once you have rejected a candidate, you cannot undo the decision.

The solution to this problem is to look at the first `n/e`

(`e~2.71828`

being
the Euler constant) candidates, and thereafter accept the first candidate taller
than all of the `n/e`

first candidates.

#### A Samplitude solution

Let `normal(170, 10)`

be the candidate generator, and let `n=100`

. We create a
filter `secretary`

that takes a stream and an integer (`n`

) and picks according
to the solution. In order to be able to assess the quality of the solution
later, the filter must forward the entire list of candidates; hence we annotate
the one we choose with `(c, False)`

for a candidate we rejected, and `(c, True)`

denotes the candidate we accepted.

```
def secretary(gen, n):
import math
explore = int(n / math.e)
target = -float('inf')
i = 0
# explore the first n/e candidates
for c in gen:
target = max(c, target)
yield (c, False)
i += 1
if i == explore:
break
_ok = lambda c, i, found: ((i == n-1 and not found)
or (c > target and not found))
have_hired = False
for c in gen:
status = _ok(c, i, have_hired)
have_hired = have_hired or status
yield c, status
i += 1
if i == n:
return
```

Now, to emulate the secretary problem with Samplitude:

```
from samplitude import samplitude as s8e
# insert above secretary function
n = 100
filters = {'secretary': secretary}
solution = s8e('normal(170, 10) | secretary(%d) | list' % n, filters=filters)
solution = eval(solution) # Samplitude returns an eval-able string
cands = map(lambda x: x[0], solution)
opt = [s[0] for s in solution if s[1]][0]
# the next line prints in which position the candidate is
print(1+sorted(cands, reverse=True).index(opt), '/', n)
```

In about 67% of the cases we can expect to get one of the top candidates, whereas the remaining 33% of the cases will be uniformly distributed. Running 100k runs with a population of size 1000 reveals the structure.

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