# Computing Statistics

Often, one wants to compile statistics on what is going on in the optimization. The `Statistics` are able to compile such data on arbitrary attributes of any designated object. To do that, one needs to register the desired statistic functions inside the stats object using the exact same syntax as in the toolbox.

    stats = tools.Statistics(key=lambda ind: ind.fitness.values)

The statistics object is created using a key as first argument. This key must be supplied a function that will later be applied to the data on which the statistics are computed. The previous code sample uses the `fitness.values` attribute of each element.

    stats.register("avg", numpy.mean)
    stats.register("std", numpy.std)
    stats.register("min", numpy.min)
    stats.register("max", numpy.max)

The statistical functions are now registered. The `register` function expects an alias as first argument and a function operating on vectors as second argument. Any subsequent argument is passed to the function when called. The creation of the statistics object is now complete.

## Predefined Algorithms

When using a predefined algorithm such as `eaSimple()`, `eaMuPlusLambda()`, `eaMuCommaLambda()`, or `eaGenerateUpdate()`, the statistics object previously created can be given as argument to the algorithm.

    pop, logbook = algorithms.eaSimple(pop, toolbox, cxpb=0.5, mutpb=0.2, ngen=0,
                                       stats=stats, verbose=True)

Statistics will automatically be computed on the population every generation. The verbose argument prints the statistics on screen while the optimization takes place. Once the algorithm returns, the final population and a `Logbook` are returned. See the next section or the `Logbook` documentation for more information.

## Writing Your Own Algorithm

When writing your own algorithm, including statistics is very simple. One only needs to compile the statistics on the desired object. For example, compiling the statistics on a given population is done by calling the `compile()` method.

    record = stats.compile(pop)

The argument to the compile function must be an iterable of elements on which the key will be called. Here, our population (`pop`) contains individuals. The statistics object will call the key function on every individual to retrieve their `fitness.values` attribute. The resulting array of values is finally given to each statistic function and the result is put into the `record` dictionary under the key associated with the function. Printing the record reveals its nature.

    >>> print(record)
    {'std': 4.96, 'max': 63.0, 'avg': 50.2, 'min': 39.0}

How to save and pretty print the statistics is shown in the next section.

## Multi-objective Statistics

As statistics are computed directly on the values with numpy function, all the objectives are combined together by the default behaviour of numpy. Thus, one needs to specify the axis on which to operate. This is achieved by giving the axis as an additional argument to the register function.

    stats = tools.Statistics(key=lambda ind: ind.fitness.values)
    stats.register("avg", numpy.mean, axis=0)
    stats.register("std", numpy.std, axis=0)
    stats.register("min", numpy.min, axis=0)
    stats.register("max", numpy.max, axis=0)

One can always specify the axis even in the case of single objective. The only effect is to produce a different output, as the objects are numpy arrays.

    >>> print(record)
    {'std': array([ 4.96]), 'max': array([ 63.]), 'avg': array([ 50.2]),
    'min': array([ 39.])}

## Multiple Statistics

It is also possible to compute statistics on different attributes of the population individuals. For instance, it is quite common in genetic programming to have statistics on the height of the trees in addition to their fitness. One can combine multiple `Statistics` objects in a `MultiStatistics`.

    stats_fit = tools.Statistics(key=lambda ind: ind.fitness.values)
    stats_size = tools.Statistics(key=len)
    mstats = tools.MultiStatistics(fitness=stats_fit, size=stats_size)

Two statistics objects are created in the same way as before. The second object will retrieve the size of the individuals by calling `len()` on each of them. Once created, the statistics objects are given to a MultiStatistics one, where the arguments are given using keywords. These keywords will serve to identify the different statistics. The statistical functions can be registered only once in the multi-statistics, as shown below, or individually in each statistics.

    mstats.register("avg", numpy.mean)
    mstats.register("std", numpy.std)
    mstats.register("min", numpy.min)
    mstats.register("max", numpy.max)

The multi-statistics object can be given to an algorithm or they can be compiled using the exact same procedure as the simple statistics.

    record = mstats.compile(pop)

This time the `record` is a dictionary of dictionaries. The first level contains the keywords under which the statistics objects have been registered and the second level is similar to the previous simple statistics object.

    >>> print(record)
    {'fitness': {'std': 1.64, 'max': 6.86, 'avg': 1.71, 'min': 0.166},
    'size': {'std': 1.89, 'max': 7, 'avg': 4.54, 'min': 3}}

# Logging Data

Once the data is produced by the statistics (or multi-statistics), one can save it for further use in a `Logbook`. The logbook is intended to be a chronological sequence of entries (as dictionaries). It is directly compliant with the type of data returned by the statistics objects, but not limited to this data. In fact, anything can be incorporated in an entry of the logbook.

    logbook = tools.Logbook()
    logbook.record(gen=0, evals=30, **record)

The `record()` method takes a variable number of arguments, each of which is data to be recorded. In the last example, we saved the generation, the number of evaluations and everything contained in the `record` produced by a statistics object using the star magic. All records will be kept in the logbook until its destruction.

After a number of records, one may want to retrieve the information contained in the logbook.

    gen, avg = logbook.select("gen", "avg")

The `select()` method provides a way to retrieve all the information associated with a keyword in all records. This method takes a variable number of string arguments, which are the keywords used in the record or statistics object. Here, we retrieved the generation and the average fitness using a single call to select.

A logbook is a picklable object (as long as all inserted objects are picklable) providing a very nice way to save the statistics of an evolution on disk.

    import pickle
    	pickle.dump(logbook, lb_file)

Note

Every algorithm returns a logbook containing the statistics for every generation and the number of evaluations for the whole evolution.

## Printing to Screen

A logbook can be printed to screen or file. Its `__str__()` method returns a header of each key inserted in the first record and the complete logbook for each of these keys. The rows are in chronological order of insertion while the columns are in an undefined order. The easiest way to specify an order is to set the `header` attribute to a list of strings specifying the order of the columns.

    logbook.header = "gen", "avg", "spam"

The result is:

    >>> print(logbook)
    gen   avg      spam
    0     [ 50.2]

A column name containing no entry in a specific record will be left blank as for the `spam` column in the last example.

A logbook also contains a stream property returning only the yet unprinted entries.

    >>> print(logbook.stream)
    gen   avg      spam
    0     [ 50.2]
    >>> logbook.record(gen=1, evals=15, **record)
    >>> print(logbook.stream)
    1     [ 50.2]

## Dealing with Multi-statistics

The logbook is able to cope with the dictionary of dictionaries return by the `MultiStatistics` object. In fact, it will log the data in `chapters` for each sub dictionary contained in the record. Thus, a multi record can be used exactly as a record.

    logbook = tools.Logbook()
    logbook.record(gen=0, evals=30, **record)

One difference is the column ordering, where we can specify an order for the chapters and their content as follows:

    logbook.header = "gen", "evals", "fitness", "size"
    logbook.chapters["fitness"].header = "min", "avg", "max"
    logbook.chapters["size"].header = "min", "avg", "max"

The resulting output is:

    >>> print(logbook)
                         fitness                 size
                -------------------------  ---------------
    gen   evals min      avg      max      min   avg   max
    0     30    0.165572 1.71136  6.85956  3     4.54  7

Retrieving the data is also done through the chapters.

    gen = logbook.select("gen")
    fit_mins = logbook.chapters["fitness"].select("min")
    size_avgs = logbook.chapters["size"].select("avg")

The generations, minimum fitness and average size are obtained, chronologically ordered. If some data is not available, a `None` appears in the vector.

## Some Plotting Sugar

One of the most common operations when an optimization is finished is to plot the data during the evolution. The `Logbook` allows to do this very efficiently. Using the select method, one can retrieve the desired data and plot it using matplotlib.

    gen = logbook.select("gen")
    fit_mins = logbook.chapters["fitness"].select("min")
    size_avgs = logbook.chapters["size"].select("avg")

    import matplotlib.pyplot as plt

    fig, ax1 = plt.subplots()
    line1 = ax1.plot(gen, fit_mins, "b-", label="Minimum Fitness")
    ax1.set_xlabel("Generation")
    ax1.set_ylabel("Fitness", color="b")
    for tl in ax1.get_yticklabels():
        tl.set_color("b")

    ax2 = ax1.twinx()
    line2 = ax2.plot(gen, size_avgs, "r-", label="Average Size")
    ax2.set_ylabel("Size", color="r")
    for tl in ax2.get_yticklabels():
        tl.set_color("r")

    lns = line1 + line2
    labs = [l.get_label() for l in lns]
    ax1.legend(lns, labs, loc="center right")

    plt.show()

When added to the symbolic regression example, it gives the following graphic: