Update readme, convenient syntax, and fixed some bugs

README.md

@@ -2,3 +2,73 @@
 
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+
+This package allows you to easily normalize an array over any dimensions.
+It also provides a bunch of normalization methods, such as the z-score, sigmoid, robust, mixed, and NaN-safe normalizations.
+
+## Usage
+
+Each normalization method is a subtype of `AbstractNormalization`. Instances of a normalization, including any parameters (such as the mean of a dataset) are stored in a variable of the `AbstractNormalization` type.
+For example, to normalize a 2D array using the `ZScore` normalization method (or any other `<: AbstractNormalization`) over all dimensions:
+```
+X = rand(100, 100)
+N = ZScore(X) # A normalization fit to X, NOT the normalized array
+N = ZScore()(X) # An alternative to the line above
+Y = N(X) # The normalized array
+Z = N(rand(100, 100)) # Apply a normalization with parameters fit to X on a new array
+```
+
+There is also an alternative, preferred, syntax:
+```
+using Statistics
+N = fit(ZScore, X)
+Y = normalize(X, N)
+normalize!(X, N) # In place, writing over X
+```
+
+A normalization can also be reversed:
+```
+_X = denormalize(X, N) # Apply the inverse normalization
+denormalize!(X, N) # Or do the inverse in place
+```
+
+Both syntaxes allow you to specify the dimensions to normalize over. For example, to normalize each 2D slice (i.e. iterating over the 3rd dimension) of a 3D array:
+```
+X = rand(100, 100, 10)
+N = fit(ZScore, X; dims=[1, 2])
+normalize!(X, N) # Each [1, 2] slice is normalized independently
+all(std(X; dims=[1, 2]) .≈ 1) # true
+```
+
+## Normalization methods
+
+Any of these normalizations will work in place of `ZScore` in the examples above:
+| Normalization | Formula | Description |
+|--|--| -- |
+| `ZScore` | $(x - \mu)/\sigma$ | Subtract the mean and scale by the standard deviation (aka standardisation) |
+| `Sigmoid` | $(1 + \exp(-\frac{x-\mu}{\sigma}))^{-1}$ | Map to the interval $[-1, 1]$ by applying a sigmoid transformation |
+| `MinMax` | $(x-\inf{x})/(\sup{x}-\inf{x})$ | Scale to the unit interval |
+
+
+### Robust normalizations
+This package also defines robust versions of any normalization methods that have $\mu$ (the mean) and $\sigma$ (the standard deviation) parameters. 
+`Robust` including `RobustZScore` and `RobustSigmoid`, use the `median` and `iqr/1.35` rather than the `mean` and `std` for a normalization that is less sensitive to outliers.
+There are also `Mixed` methods, such as `MixedZScore` and `MixedSigmoid`, that default to the `Robust` versions but use the regular parameters (`mean` and `std`) if the `iqr` is 0.
+
+### NaN-safe normalizations
+
+If the input array contains any `NaN` values, the normalizations given above will fit with `NaN` parameters and return `NaN` arrays. To circumvent this, any normalization can be made '`NaN`-safe', meaning it ignores `NaN` values in the input array. Using the `ZScore` example:
+```
+N = nansafe(ZScore)
+fit!(N, X)
+Y = N(X)
+```
+
+### New normalizations
+
+Finally, there is also a macro to define your own normalization (honestly you could just make the `struct` directly). For example, the `ZScore` is defined as:
+```
+@_Normalization ZScore (mean, std)  (x, 𝜇, 𝜎) -> x .= (x .- 𝜇)./𝜎  #=
+                                 =# (y, 𝜇, 𝜎) -> y .= y.*𝜎 .+ 𝜇
+```
+Here, the first argument is a name for the normalization, the second is a tuple of parameter functions, the third is vectorised function of an array `x` and any parameters, and the fourth is a function for the inverse transformation.