### Install HyperdimensionalComputing.jl in Package Mode

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/README.md

Install the package by entering Julia's package mode (pressing ']') and then adding the repository URL.

```julia
]add https://github.com/MichielStock/HyperdimensionalComputing.jl#main
```

--------------------------------

### Install HyperdimensionalComputing.jl using Pkg.jl

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/README.md

Install the package using the Pkg.jl manager by providing the repository URL.

```julia
using Pkg; Pkg.add(url = "https://github.com/MichielStock/HyperdimensionalComputing.jl")
```

--------------------------------

### Associative Property Example 2

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Another demonstration of associative properties, showing that USTATES * DOL * MEXICO is approximately equal to KRO (Swedish Krona).

```julia
USTATES * DOL * MEXICO ≈ KRO
```

--------------------------------

### Associative Property Example 3

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

A further example of associative properties, indicating that USTATES * DOL * MEXICO is approximately equal to DOL (US Dollar).

```julia
USTATES * DOL * MEXICO ≈ DOL
```

--------------------------------

### Associative Property Example 1

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Illustrates an associative property where multiplying USTATES by DOL and then by MEXICO is approximately equal to multiplying MEXICO by PES.

```julia
USTATES * DOL * MEXICO ≈ PES
```

--------------------------------

### Calculate Similarity Between Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/README.md

Shows how to compute the similarity between two hypervectors using the 'similarity' function. The default behavior uses the best metric for the hypervector type. Examples include comparing two distinct hypervectors and comparing a hypervector with a combined one.

```julia
julia> a = GradedBipolarHV(10);

julia> b = GradedBipolarHV(10);

julia> c = a + b;

julia> similarity(a, b)
0.7857595020921353

julia> similarity(a, c)
0.9241024788902716

julia> similarity(b, c)
0.9295606930584789
```

--------------------------------

### Initialize Animal and Sound Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Create bipolar hypervectors for different animals and their corresponding sounds. These will be used to build an associative memory.

```julia
dog_hv = BipolarHDV()
cat_hv = BipolarHDV()
cow_hv = BipolarHDV()
animals = [dog_hv, cat_hv, cow_hv]
```

```julia
bark_hv = BipolarHDV()
meow_hv = BipolarHDV()
moo_hv  = BipolarHDV()
sounds = [bark_hv, meow_hv, moo_hv]
```

--------------------------------

### Initialize Character to Hypervector Mapping

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Create a dictionary mapping each character of the alphabet (and space) to a unique bipolar hypervector. This is the first step in encoding text sequences.

```julia
char2hv = Dict(c => BipolarHDV() for c in 'a':'z')
char2hv[' '] = BipolarHDV()
```

--------------------------------

### Load Handcalcs Package

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Loads the Handcalcs package for documentation generation. This is typically hidden in final documentation.

```julia
using Handcalcs #hide
```

--------------------------------

### Create Binary, Bipolar, and Ternary Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/README.md

Demonstrates the creation of different types of hypervectors: BinaryHV (default size 10,000), BipolarHV with a specified size, and TernaryHV from an existing vector.

```julia
using HyperdimensionalComputing

julia> x = BinaryHV() # default length is 10,000
10000-element BinaryHV:
 1
 0
 0
 0
 0
 1
 1
 0
 ⋮
 1
 0
 1
 0
 1
 0
 0
 0

julia> y = BipolarHV(6) # different size
6-element BipolarHV:
  1
 -1
  1
  1
 -1
  1

julia> z = TernaryHV([1, 1, -1, 0, 0, 0, 1, 1, -1, 0]) # From a vector
10-element TernaryHV:
  1
  1
 -1
  0
  0
  0
  1
  1
 -1
  0
```

--------------------------------

### Load HyperdimensionalComputing Package

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Loads the HyperdimensionalComputing package. This is a prerequisite for using HDC functionalities.

```julia
using HyperdimensionalComputing

ρ(hv::AbstractHDV, n::Int=1) = Π(hv,n) #hide
```

--------------------------------

### Create Associative Memory for Animal Sounds

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Combine animal and sound hypervectors using binding and bundling to create an associative memory. This allows for querying relationships between entities.

```julia
memory = (dog_hv * bark_hv) + (cat_hv * meow_hv) + (cow_hv * moo_hv);
nothing #hide
```

--------------------------------

### Permutation Dissimilarity (using @handcalcs)

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Demonstrates the dissimilarity between a hypervector and its permuted version (h₁ != ρ(h₁)) using '@handcalcs'. This is typically hidden.

```julia
@handcalcs h₁ != ρ(h₁) # hide
```

--------------------------------

### Find Equivalent of Dollar in Mexico

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Demonstrates finding the hypervector corresponding to Mexico's peso by multiplying the dollar hypervector with the USTATES-MEXICO bundle.

```julia
DOL * F_UM ≈ PES
```

--------------------------------

### Bundle, Bind, and Shift Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/README.md

Illustrates hypervector operations: bundling (creating a similar vector from multiple vectors), binding (creating a dissimilar vector), and shifting (cyclically shifting elements). Overloaded operators '+' for bundle and '*' for bind are shown, along with 'ρ' as an alias for 'shift'.

```julia
julia> x, y, z = GradedHV(10), GradedHV(10), GradedHV(10);

julia> bundle([x,y,z])
10-element GradedHV{Float64}:
 0.24160752324192117
 0.0004620105852614061
 0.21122703146393468
 0.8806160209097325
 0.748086047467331
 0.29234791431258145
 0.2804922831134219
 0.18556141274368268
 0.08208462507331278
 0.9015873952761569

julia> x + y + z
10-element GradedHV{Float64}:
 0.24160752324192117
 0.0004620105852614061
 0.21122703146393468
 0.8806160209097325
 0.748086047467331
 0.29234791431258145
 0.2804922831134219
 0.18556141274368268
 0.08208462507331278
 0.9015873952761569

julia> bind([x, y, z])
10-element GradedHV{Float64}:
 0.5061620120454368
 0.26070792597812487
 0.513025014409134
 0.4717013369896599
 0.49961155414024105
 0.46309236900588013
 0.5006006610486007
 0.533210817253628
 0.529186380757248
 0.46622954535393824

julia> x * y * z
10-element GradedHV{Float64}:
 0.5061620120454368
 0.26070792597812487
 0.513025014409134
 0.4717013369896599
 0.49961155414024105
 0.46309236900588013
 0.5006006610486007
 0.533210817253628
 0.529186380757248
 0.46622954535393824

julia> shift(x, 2)
10-element GradedHV{Float64}:
 0.6572388961520694
 0.38592896770130286
 0.5732316268033676
 0.1280407117618328
 0.13125571831454053
 0.40139428175287556
 0.5286213065298765
 0.1038774285737532
 0.43575817327464744
 0.32479919832749704

julia> ρ(x, 2)
10-element GradedHV{Float64}:
 0.6572388961520694
 0.38592896770130286
 0.5732316268033676
 0.1280407117618328
 0.13125571831454053
 0.40139428175287556
 0.5286213065298765
 0.1038774285737532
 0.43575817327464744
 0.32479919832749704

julia> shift!(x, 2)
10-element Vector{Float64}:
 0.6572388961520694
 0.38592896770130286
 0.5732316268033676
 0.1280407117618328
 0.13125571831454053
 0.40139428175287556
 0.5286213065298765
 0.1038774285737532
 0.43575817327464744
 0.32479919832749704

julia> x
10-element GradedHV{Float64}:
 0.6572388961520694
 0.38592896770130286
 0.5732316268033676
 0.1280407117618328
 0.13125571831454053
 0.40139428175287556
 0.5286213065298765
 0.1038774285737532
 0.43575817327464744
 0.32479919832749704
```

--------------------------------

### Permute Hypervector (using @handcalcs)

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Demonstrates the permutation operation on a single hypervector using the '@handcalcs' macro for symbolic representation. This is typically hidden.

```julia
@handcalcs h₁ # hide
```

--------------------------------

### Query Animal from Sound in Associative Memory

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Determine which animal corresponds to a given sound (e.g., "moo") by querying the associative memory. This demonstrates reverse lookup capabilities.

```julia
findmax(hv -> similarity(memory * moo_hv, hv), animals)
```

--------------------------------

### Create Multiple Bipolar Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Initializes three bipolar hypervectors of 8 dimensions each for subsequent operations.

```julia
h₁ = BipolarHDV(8)
h₂ = BipolarHDV(8)
h₃ = BipolarHDV(8);
nothing #hide
```

--------------------------------

### Compare Bundles of Country Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Compares different bundle combinations of Sweden, USA, and Mexico hypervectors to explore associative relationships.

```julia
F_UM = USTATES * MEXICO
F_SU = SWEDEN * USTATES
F_SM = SWEDEN * MEXICO

F_SU * F_UM ≈ F_SM
```

--------------------------------

### Query Animal Sound from Associative Memory

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Search the associative memory for the sound associated with a specific animal (e.g., dog) by comparing the bound hypervector with the sound hypervectors.

```julia
findmax(hv -> similarity(memory * dog_hv, hv), sounds)
```

--------------------------------

### Declare Concept Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Initializes bipolar hypervectors for abstract concepts like COUNTRY, CAPITAL, and MONEY.

```julia
COUNTRY = BipolarHDV()
CAPITAL = BipolarHDV()
MONEY = BipolarHDV()
```

--------------------------------

### Search for a Subsequence (N-gram) in Encoded Phrases

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Encode a query phrase (e.g., "crazy") using the same n-gram method and then find the most similar hypervector among the encoded phrases. This demonstrates sequence retrieval.

```julia
query = ngrams("crazy", char2hv)
findmax(h -> similarity(query, h), phrases_hvs)
```

--------------------------------

### Calculate Similarity Using Broadcasting

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Apply the `similarity` function element-wise between a reference hypervector and a vector of hypervectors using Julia's broadcasting syntax.

```julia
similarity.(Ref(h₁), [h₁, h₂, h₃])
```

--------------------------------

### Bind Hypervectors using bind

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Combines multiple hypervectors into a single hypervector that is dissimilar to its constituents using the 'bind' function.

```julia
bind([h₁, h₂, h₃])
```

--------------------------------

### Permutation Dissimilarity with shift amounts

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Compares a hypervector with its permuted versions shifted by different amounts (1, 2, and 3 positions) to show dissimilarity.

```julia
h₁ != ρ(h₁, 1) != ρ(h₁, 2) != ρ(h₁, 3)
```

--------------------------------

### Permute Hypervector (using ρ and @handcalcs)

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Shows the symbolic representation of a permuted hypervector (ρ(h₁)) using the '@handcalcs' macro. This is typically hidden.

```julia
@handcalcs ρ(h₁) # hide
```

--------------------------------

### Define Similarity and Equality for BipolarHDV

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Defines custom similarity and approximate equality functions for BipolarHDV types, crucial for comparing hypervectors.

```julia
using HyperdimensionalComputing
using LinearAlgebra #hide

HyperdimensionalComputing.similarity(u::T,v::T) where T<:BipolarHDV = dot(u, v) / (norm(u) * norm(v)) #hide
HyperdimensionalComputing.isapprox(u::T, v::T) where T<:BipolarHDV = similarity(u, v) > 3/sqrt(length(v)) #hide
HyperdimensionalComputing.hash(u::AbstractHDV) = hash(u.v, hash(typeof(v)))
HyperdimensionalComputing.isequal(u::T, v::T) where T<:AbstractHDV = hash(u) == hash(v)
```

--------------------------------

### Create Bipolar Hypervector from AbstractVector

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Creates a bipolar hypervector directly from a provided AbstractVector of random bipolar values.

```julia
BipolarHDV(rand([-1,1], 8))
```

--------------------------------

### Bundle USTATES and MEXICO

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Creates a bundle (multiplication) of the USTATES and MEXICO hypervectors to represent their combined associations.

```julia
F_UM = USTATES * MEXICO
```

--------------------------------

### Compare Country Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Compares the holistic hypervectors for USA and Mexico using the defined approximate equality.

```julia
USTATES ≈ MEXICO
```

--------------------------------

### Bind Hypervectors using '*' operator

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Combines multiple hypervectors using the overloaded '*' operator, resulting in a hypervector dissimilar to its constituents.

```julia
h₁ * h₂ * h₃
```

--------------------------------

### Create Default Bipolar Hypervector

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Creates a random bipolar hypervector with the default dimensionality of 10,000.

```julia
BipolarHDV()
```

--------------------------------

### Bundle Hypervectors using '+' operator

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Combines multiple hypervectors using the overloaded '+' operator, resulting in a hypervector similar to its constituents.

```julia
h₁ + h₂ + h₃
```

--------------------------------

### Bundle Hypervectors using aggregate

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Combines multiple hypervectors into a single hypervector that is similar to its constituents using the 'aggregate' function.

```julia
aggregate([h₁, h₂, h₃])
```

--------------------------------

### Represent Countries with Holistic Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Creates holistic hypervectors for USA and Mexico by combining concept hypervectors with specific entity hypervectors.

```julia
USA = BipolarHDV()
MEX = BipolarHDV()

WDC = BipolarHDV()
MXC = BipolarHDV()

DOL = BipolarHDV()
PES = BipolarHDV()

USTATES = (COUNTRY * USA) + (CAPITAL * WDC) + (MONEY * DOL)
MEXICO  = (COUNTRY * MEX) + (CAPITAL * MXC) + (MONEY * PES)
```

--------------------------------

### Create Bipolar Hypervector with Specific Dimensionality

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Creates a random bipolar hypervector with a specified number of dimensions.

```julia
BipolarHDV(8)
```

--------------------------------

### Encode Phrases using N-grams

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Define and apply an n-gram encoding function to convert phrases into hypervectors. This function uses bundling and shifting to capture sequential information.

```julia
phrases = [
	"the quick brown fox jumps over the lazy dog",
	"the slick grown box bumps under the hazy fog",
	"the thick known cox dumps inter the crazy cog",
	"the brick shown pox lumps enter the glazy jog",
	"the stick blown sox pumps winter the blazy log"
]

ngrams(p, d) = aggregate([d[p[i]] + Π(d[p[i+1]], 1) + Π(d[p[i+2]], 2) for i in 1:length(p)-2])

phrases_hvs = [ngrams(p, char2hv) for p in phrases]
```

--------------------------------

### Calculate Similarity With a Vector of Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Compare a single hypervector against a vector of hypervectors using the `similarity` function. This is useful for finding the most similar hypervector in a collection.

```julia
similarity(h₁, h₁)
```

--------------------------------

### Permute Hypervector with shift amount

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Applies a permutation (circular shift) to a hypervector by one position using the 'ρ' function.

```julia
ρ(h₁, 1)
```

--------------------------------

### Calculate Similarity Between Two Hypervectors

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/introduction-to-hdc.md

Use the `similarity` function to compare two hypervectors. This is a fundamental operation for retrieving information from HDC representations.

```julia
similarity(h₁, h₂)
```

--------------------------------

### Represent Sweden

Source: https://github.com/kermit-ugent/hyperdimensionalcomputing.jl/blob/main/docs/src/examples/whats-the-dollar-of-mexico.md

Creates a holistic hypervector for Sweden, similar to how USA and Mexico were represented.

```julia
SWE = BipolarHDV()
STO = BipolarHDV()
KRO = BipolarHDV()

SWEDEN = (COUNTRY * SWE) + (CAPITAL * STO) + (MONEY * KRO)
```

=== COMPLETE CONTENT === This response contains all available snippets from this library. No additional content exists. Do not make further requests.