<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 The Containers tutorial shows examples of each of the container types in HoloViews, and it is useful to look at the description of each type there, as you work through this tutorial.  
This tutorial shows you how to combine the various container types, in order to build data structures that can contain all of the data that you want to visualize or analyze, in an extremely flexible way.  For instance, you may have a large set of measurements of different types of data (numerical, image, textual notations, etc.) from different experiments done on different days, with various different parameter values associated with each one.  HoloViews can store all of this data together, which will allow you to select just the right bit of data "on the fly" for any particular analysis or visualization, by indexing, slicing, selecting, and sampling in this data structure.
To illustrate the full functionality provided, we will create an example of the maximally nested object structure currently possible with HoloViews
Step1:  This code produces what looks like a relatively simple animation of two side-by-side figures, but is actually a deeply nested data structure
Step2:  The structure of this object can be seen in the repr()
Step3:  Nesting hierarchy <a id='NestingHierarchy'></a>
To help us understand this structure, here is a schematic for us to refer to as we unpack this object, level by level
Step4:  In the examples below, we will unpack this data structure using attribute access (explained in the Introductory tutorial) as well as indexing and slicing (explained in the Sampling Data tutorial).
GridSpace Level
Elements within a Layout, such as the GridSpace in this example, are reached via attribute access
Step5:  HoloMap Level
This GridSpace consists of nine HoloMaps arranged in a two-dimensional space.  Let's now select one of these HoloMap objects, by indexing to retrieve the one at [Amplitude,Power] [0.5,1.0], i.e. the lowest amplitude and power
Step6:  As shown in the schematic above, a HoloMap contains many elements with associated keys. In this example, these keys are indexed with a dimension Frequency, which is why the Frequency varies when you play the animation here.
Overlay Level
The repr() showed us that the HoloMap is composed of Overlay objects, six in this case (giving six frames to the animation above).  Let us access one of these elements, i.e. one frame of the animation above, by indexing to retrieve an Overlay associated with the key with a Frequency of 1.0
Step7:  NdOverlay Level
As the repr() shows, the Overlay contains a Points object and an NdOverlay object.  We can access either one of these using the attribute access supported by Overlay
Step8:  Curve Level
The NdOverlay is so named because it is an overlay of items indexed by dimensions, unlike the regular attribute-access overlay types.  In this case it is indexed by Phase, with four values.  If we index to select one of these values, we will get an individual Curve, e.g. the one with zero phase
Step9:  Data Level
At this point, we have reached the end of the HoloViews objects; below this object is only the raw data as a Numpy array
Step10:  Actually, HoloViews will let you go even further down, accessing data inside the Numpy array using the continuous (floating-point) coordinate systems declared in HoloViews. E.g. here we can ask for a single datapoint, such as the value at x=5.2
Step11:  Indexing into 1D Elements like Curve and higher-dimensional but regularly gridded Elements like Image, Surface, and HeatMap will return the nearest defined value (i.e., the results "snap" to the nearest data item)
Step12:  For other Element types, such as Points, snapping is not supported and thus indexing down into the .data array will be less useful, because it will only succeed for a perfect floating-point match on the key dimensions.  In those cases, you can still use all of the access methods provided by the numpy array itself, via .data, e.g. .data[52], but note that such native operations force you to use the native indexing scheme of the array, i.e. integer access starting at zero, not the more convenient and semantically meaningful continuous coordinate systems we provide through HoloViews.
Indexing using .select
The curve displayed immediately above shows the final, deepest Element access possible in HoloViews for this object
Step13:  The second form demonstrates HoloViews' deep indexing feature, which allows indexes to cross nested container boundaries. The above is as far as we can index before reaching a heterogeneous type (the Overlay), where we need to use attribute access. Here is the more explicit method of indexing down to a curve, using .select to specify dimensions by name instead of bracket-based indexing by position
<ASSISTANT_TASK:>
Python Code:
import numpy as np
import holoviews as hv
hv.notebook_extension()
np.random.seed(10)
def sine_curve(phase, freq, amp, power, samples=102):
    xvals = [0.1* i for i in range(samples)]
    return [(x, amp*np.sin(phase+freq*x)**power) for x in xvals]
phases =      [0, np.pi/2, np.pi, 3*np.pi/2]
powers =      [1,2,3]
amplitudes =  [0.5,0.75, 1.0]
frequencies = [0.5, 0.75, 1.0, 1.25, 1.5, 1.75]
gridspace = hv.GridSpace(kdims=['Amplitude', 'Power'], group='Parameters', label='Sines')
for power in powers:
    for amplitude in amplitudes:
        holomap = hv.HoloMap(kdims=['Frequency'])
        for frequency in frequencies:
            sines = {phase : hv.Curve(sine_curve(phase, frequency, amplitude, power))
                     for phase in phases}
            ndoverlay = hv.NdOverlay(sines , kdims=['Phase']).relabel(group='Phases',
                                                                      label='Sines', depth=1)
            overlay = ndoverlay * hv.Points([(i,0) for i in range(0,10)], group='Markers', label='Dots')
            holomap[frequency] = overlay
        gridspace[amplitude, power] = holomap
penguins = hv.RGB.load_image('../assets/penguins.png').relabel(group="Family", label="Penguin")
layout = gridspace + penguins
Explanation: The Containers tutorial shows examples of each of the container types in HoloViews, and it is useful to look at the description of each type there, as you work through this tutorial.  
This tutorial shows you how to combine the various container types, in order to build data structures that can contain all of the data that you want to visualize or analyze, in an extremely flexible way.  For instance, you may have a large set of measurements of different types of data (numerical, image, textual notations, etc.) from different experiments done on different days, with various different parameter values associated with each one.  HoloViews can store all of this data together, which will allow you to select just the right bit of data "on the fly" for any particular analysis or visualization, by indexing, slicing, selecting, and sampling in this data structure.
To illustrate the full functionality provided, we will create an example of the maximally nested object structure currently possible with HoloViews:
End of explanation
layout
Explanation: This code produces what looks like a relatively simple animation of two side-by-side figures, but is actually a deeply nested data structure:
End of explanation
print(repr(layout))
Explanation: The structure of this object can be seen in the repr():
End of explanation
print(repr(layout))
Explanation: Nesting hierarchy <a id='NestingHierarchy'></a>
To help us understand this structure, here is a schematic for us to refer to as we unpack this object, level by level:
<center><img src="http://assets.holoviews.org/nesting-diagram.png"></center>
Everything that is displayable in HoloViews has this same basic structure, although any of the levels can be omitted in simpler cases, and many different Element types (not containers) can be substituted for any other.  
Since HoloViews 1.3.0, you are allowed to build data-structures that violate this hierarchy (e.g., you can put Layout objects into HoloMaps) but the resulting object cannot be displayed. Instead, you will be prompted with a message to call the collate method. Using the collate method will allow you to generate the appropriate object that correctly obeys the hierarchy shown above, so that it can be displayed.
As shown in the diagram, there are three different types of container involved:
Basic Element: elementary HoloViews object containing raw data in an external format like Numpy or pandas.
Homogenous container (UniformNdMapping): collections of Elements or other HoloViews components that are all the same type.  These are indexed using array-style key access with values sorted along some dimension(s), e.g. [0.50] or ["a",7.6].
Heterogenous container (AttrTree): collections of data of different types, e.g. different types of Element.  These are accessed by categories using attributes, e.g. .Parameters.Sines, which does not assume any ordering of a dimension.
We will now go through each of the containers of these different types, at each level.
Layout Level
Above, we have already viewed the highest level of our data structure as a Layout. Here is the repr of entire Layout object, which reflects all the levels shown in the diagram:
End of explanation
layout.Parameters.Sines
Explanation: In the examples below, we will unpack this data structure using attribute access (explained in the Introductory tutorial) as well as indexing and slicing (explained in the Sampling Data tutorial).
GridSpace Level
Elements within a Layout, such as the GridSpace in this example, are reached via attribute access:
End of explanation
layout.Parameters.Sines[0.5, 1]
Explanation: HoloMap Level
This GridSpace consists of nine HoloMaps arranged in a two-dimensional space.  Let's now select one of these HoloMap objects, by indexing to retrieve the one at [Amplitude,Power] [0.5,1.0], i.e. the lowest amplitude and power:
End of explanation
layout.Parameters.Sines[0.5, 1][1.0]
Explanation: As shown in the schematic above, a HoloMap contains many elements with associated keys. In this example, these keys are indexed with a dimension Frequency, which is why the Frequency varies when you play the animation here.
Overlay Level
The repr() showed us that the HoloMap is composed of Overlay objects, six in this case (giving six frames to the animation above).  Let us access one of these elements, i.e. one frame of the animation above, by indexing to retrieve an Overlay associated with the key with a Frequency of 1.0:
End of explanation
(layout.Parameters.Sines[0.5, 1][1].Phases.Sines +
 layout.Parameters.Sines[0.5, 1][1].Markers.Dots)
Explanation: NdOverlay Level
As the repr() shows, the Overlay contains a Points object and an NdOverlay object.  We can access either one of these using the attribute access supported by Overlay:
End of explanation
l=layout.Parameters.Sines[0.5, 1][1].Phases.Sines[0.0]
l
repr(l)
Explanation: Curve Level
The NdOverlay is so named because it is an overlay of items indexed by dimensions, unlike the regular attribute-access overlay types.  In this case it is indexed by Phase, with four values.  If we index to select one of these values, we will get an individual Curve, e.g. the one with zero phase:
End of explanation
type(layout.Parameters.Sines[0.5, 1][1].Phases.Sines[0.0].data)
Explanation: Data Level
At this point, we have reached the end of the HoloViews objects; below this object is only the raw data as a Numpy array:
End of explanation
layout.Parameters.Sines[0.5, 1][1].Phases.Sines[0.0][5.2]
Explanation: Actually, HoloViews will let you go even further down, accessing data inside the Numpy array using the continuous (floating-point) coordinate systems declared in HoloViews. E.g. here we can ask for a single datapoint, such as the value at x=5.2:
End of explanation
layout.Parameters.Sines[0.5, 1][1].Phases.Sines[0.0][5.23], layout.Parameters.Sines[0.5, 1][1].Phases.Sines[0.0][5.27]
Explanation: Indexing into 1D Elements like Curve and higher-dimensional but regularly gridded Elements like Image, Surface, and HeatMap will return the nearest defined value (i.e., the results "snap" to the nearest data item):
End of explanation
o1 = layout.Parameters.Sines.select(Amplitude=0.5, Power=1.0).select(Frequency=1.0)
o2 = layout.Parameters.Sines.select(Amplitude=0.5, Power=1.0, Frequency=1.0)
o1 + o2
Explanation: For other Element types, such as Points, snapping is not supported and thus indexing down into the .data array will be less useful, because it will only succeed for a perfect floating-point match on the key dimensions.  In those cases, you can still use all of the access methods provided by the numpy array itself, via .data, e.g. .data[52], but note that such native operations force you to use the native indexing scheme of the array, i.e. integer access starting at zero, not the more convenient and semantically meaningful continuous coordinate systems we provide through HoloViews.
Indexing using .select
The curve displayed immediately above shows the final, deepest Element access possible in HoloViews for this object:
python
layout.Parameters.Sines[0.5, 1][1].Phases.Sines[0.0]
This is the curve with an amplitude of 0.5, raised to a power of 1.0 with frequency of 1.0 and 0 phase. These are all the numbers, in order, used in the access shown above.
The .select method is a more explicit way to use key access, with both of these equivalent to each other:
End of explanation
layout.Parameters.Sines.select(Amplitude=0.5,Power=1.0, 
                               Frequency=1.0).Phases.Sines.select(Phase=0.0)
Explanation: The second form demonstrates HoloViews' deep indexing feature, which allows indexes to cross nested container boundaries. The above is as far as we can index before reaching a heterogeneous type (the Overlay), where we need to use attribute access. Here is the more explicit method of indexing down to a curve, using .select to specify dimensions by name instead of bracket-based indexing by position:
End of explanation
<END_TASK>