Objectives

Loading Fama and French data

#r "nuget: FSharp.Data, 5.0.2"
#r "nuget: NovaSBE.Finance, 0.5.0"
#r "nuget: FSharp.Stats, 0.5.0"
#r "nuget: Plotly.NET, 3.*"
#r "nuget: Plotly.NET.Interactive, 3.*"

open System
open FSharp.Data
open FSharp.Stats
open Plotly.NET
open NovaSBE.Finance.French

let ff3 = getFF3 Frequency.Daily

Some volatility functions.

First the input/output data types.

type ReturnObs = { Date: DateTime; Return: float}

type VolatilityPrediction = 
    { /// First date the prediction is valid for
      Date: DateTime
      /// The volatility prediction
      PredictedVol: float }

Some test data with increasing dates and random returns.

let seed = 99
Random.SetSampleGenerator(Random.RandThreadSafe(seed))

// Let's start with this sample of returns
let rnorm =  Distributions.Continuous.Normal.Init 0.01 1.0
let testData = 
    [| for i=1 to 100 do 
        { Date = DateTime(2010, 1, 1).AddDays(float i); 
          Return = rnorm.Sample() }
    |]

The first example we'll use is an expanding window of volatility.

/// <summary>Calculates sample average expanding window volatility.</summary>
let expandingWindowVol (data: array<ReturnObs>) =
    [| for i=2 to data.Length-1 do 
        let sd = data[..i-1] |> Array.map (fun x -> x.Return) |> stDev
        { VolatilityPrediction.Date = data[i].Date; PredictedVol = sd }
    |]

Let's try it with our test data.

let expandingWindowVolsTest = expandingWindowVol testData

First few observations.

expandingWindowVolsTest[..3]

Last few observations.

expandingWindowVolsTest[expandingWindowVolsTest.Length-4..]

Rolling window volatility.

/// <summary>Calculates realized volatity of a return observation 
/// over a rolling window of past observations.</summary>
/// <param name="width">The window width for calculating realized vol.</param>
/// <param name="data">The input data</param>
/// <returns>A prediction and date the prediction is valid for.</returns>
let rollingVol (width: int) (data: array<ReturnObs>) =
    data
    |> Array.sortBy (fun x -> x.Date)
    |> Array.windowed (width + 1)
    |> Array.map (fun window ->
        let sd =
            window 
            |> Array.take width 
            |> Array.map (fun x -> x.Return) 
            |> stDev
        let last = window |> Array.last
        { VolatilityPrediction.Date = last.Date; PredictedVol = sd })

Calculated using our test data

let rollingVolTest = rollingVol 22 testData

First few observations.

rollingVolTest[..3]

Last few observations.

rollingVolTest[rollingVolTest.Length-4..]

Exponential weight volatility.

/// <summary>Calculates realized volatity of a return observation using an exponential weight.</summary>
/// <param name="width">The window width for calculating realized vol.</param>
/// <param name="lambda">The exponential weight</param>
/// <param name="data">The input data</param>
/// <returns>A prediction and date the prediction is valid for.</returns>
let expRealizedVol (width: int) (lambda: float) (data: array<ReturnObs>) =
    data
    |> Array.sortByDescending (fun x -> x.Date)
    |> Array.windowed (width + 1)
    |> Array.Parallel.map (fun window ->
        let mu = window[1..] |> Array.averageBy (fun x -> x.Return)
        let mutable acc = 0.0
        for t = 1 to width do 
            let w = (1.0 - lambda)*lambda**(float t - 1.0)
            acc <- acc + w * (window[t].Return - mu)**2.0
        { VolatilityPrediction.Date = window[0].Date; PredictedVol = sqrt acc })
    |> Array.rev

Calculated using our test data.

let expRealizedVolTest = expRealizedVol 50 0.94 testData

First few observations.

expRealizedVolTest[..3]

Last few observations.

expRealizedVolTest[expRealizedVolTest.Length-4..]

Now a plot of all of them together.

let combinedTestData = 
    [ expandingWindowVolsTest, "expanding window"
      rollingVolTest, "realized"
      expRealizedVolTest, "exp realized" ]

[ for (vols, name) in combinedTestData do
    let dataForChart = vols |> Array.map (fun x -> x.Date, x.PredictedVol)
    Chart.Line(dataForChart,Name=name) ]
|> Chart.combine

Return predictions

type ReturnPrediction = { Date: DateTime; PredictedReturn: float }

let avgReturnAccumulator (xs: array<ReturnObs>) =
    let mutable acc = 0.0
    [| for i=0 to (xs.Length-2) do
        acc <- acc + xs[i].Return
        let avgReturn = acc / float (i + 1)
        { Date = xs[i+1].Date; PredictedReturn = avgReturn } |]

Test it.

testData
|> avgReturnAccumulator
|> Array.map (fun x -> x.Date, x.PredictedReturn)
|> Chart.Line

Combining predictions

Now form portfolios based on these predictions.

type VolAndReturnPrediction = { Date: DateTime; PredictedVol: float; PredictedReturn: float }

let combinePredictions (predReturns: seq<ReturnPrediction>) (predVols: seq<VolatilityPrediction>) =
    let predVols = 
        predVols 
        |> Seq.map (fun x -> x.Date, x.PredictedVol) 
        |> Map
    [ for retObs in predReturns do 
        if predVols.ContainsKey retObs.Date then
            { Date = retObs.Date 
              PredictedVol = predVols[retObs.Date]
              PredictedReturn = retObs.PredictedReturn } ]

Now let's have a function that creates returns off of that.

let managedPortfolio gamma predVols predReturns (xs: seq<ReturnObs>) =
    let preds = 
        combinePredictions predReturns predVols
        |> Seq.map (fun x -> x.Date, x)
        |> Map
    [| for x in xs do 
        if preds.ContainsKey x.Date then
            let pred = preds[x.Date]
            let w = pred.PredictedReturn / (gamma* pred.PredictedVol ** 2.0)
            { Date = x.Date
              Return = w * x.Return } |]

Let's compare results for our 3 volatility predictions.

First, expected returns.

let retObs = ff3 |> Array.map (fun x -> { Date = x.Date; Return = x.MktRf })

let expectedRet = 
    retObs
    |> avgReturnAccumulator

Plot it.

expectedRet
|> Array.map (fun x -> x.Date, x.PredictedReturn)
|> Chart.Line

Optimal portfolios using our expected returns and expected vols.

let expandingVols = expandingWindowVol retObs
let rollingVols = rollingVol 22 retObs
let exponentialVols = expRealizedVol 500 0.94 retObs

The portfolios.

let expandingVolPort  = managedPortfolio 3.0 expandingVols expectedRet retObs
let rollingVolPort = managedPortfolio 3.0 rollingVols expectedRet retObs
let exponentialVolPort = managedPortfolio 3.0 exponentialVols expectedRet retObs

Let's plot these.

let retAsTuple (xs: array<ReturnObs>) = xs |> Array.map (fun x -> x.Date, x.Return)

let cumulativeReturns (xs: array<DateTime * float>) =
    let xs = xs |> Array.sortBy (fun (dt, r) -> dt)
    let mutable accRet = 0.0
    [| for (dt, r) in xs do
        let cr = (1.0 + accRet) * (1.0 + r) - 1.0
        let broke = -1.0 + epsilon
        accRet <- max broke cr
        (dt, accRet) |]

open Plotly.NET.LayoutObjects

let plotCumulativeReturns name (xs: array<DateTime * float>) =
    let xsFromOne = 
        xs
        |> Array.map (fun (dt, r) -> dt, 1.0 + r)
    Chart.Line(xsFromOne, Name=name)
    |> Chart.withYAxisStyle(AxisType = StyleParam.AxisType.Log)

[ expandingVolPort, "expandingVol"
  rollingVolPort, "rollingVol"
  exponentialVolPort, "exponentialVol" ]
|> List.map (fun (port, name) ->
    port
    |> retAsTuple
    |> cumulativeReturns
    |> plotCumulativeReturns name)
|> Chart.combine
|> Chart.show

expandingVolPort
|> retAsTuple
|> cumulativeReturns
|> plotCumulativeReturns "expandingVol"
|> Chart.show

namespace System

Multiple items
namespace FSharp

--------------------
namespace Microsoft.FSharp

Multiple items
namespace FSharp.Data

--------------------
namespace Microsoft.FSharp.Data

namespace FSharp.Stats

namespace Plotly

namespace Plotly.NET

namespace NovaSBE

namespace NovaSBE.Finance

module French from NovaSBE.Finance

val ff3: FF3Obs array

val getFF3: frequency: Frequency -> FF3Obs array

type Frequency = | Daily | Monthly

union case Frequency.Daily: Frequency

type ReturnObs = { Date: DateTime Return: float }

Multiple items
[<Struct>] type DateTime = new: year: int * month: int * day: int -> unit + 16 overloads member Add: value: TimeSpan -> DateTime member AddDays: value: float -> DateTime member AddHours: value: float -> DateTime member AddMicroseconds: value: float -> DateTime member AddMilliseconds: value: float -> DateTime member AddMinutes: value: float -> DateTime member AddMonths: months: int -> DateTime member AddSeconds: value: float -> DateTime member AddTicks: value: int64 -> DateTime ...
<summary>Represents an instant in time, typically expressed as a date and time of day.</summary>

--------------------
DateTime ()
   (+0 other overloads)
DateTime(ticks: int64) : DateTime
   (+0 other overloads)
DateTime(ticks: int64, kind: DateTimeKind) : DateTime
   (+0 other overloads)
DateTime(date: DateOnly, time: TimeOnly) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int) : DateTime
   (+0 other overloads)
DateTime(date: DateOnly, time: TimeOnly, kind: DateTimeKind) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int, calendar: Globalization.Calendar) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int, hour: int, minute: int, second: int) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int, hour: int, minute: int, second: int, kind: DateTimeKind) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int, hour: int, minute: int, second: int, calendar: Globalization.Calendar) : DateTime
   (+0 other overloads)

Multiple items
val float: value: 'T -> float (requires member op_Explicit)

--------------------
type float = Double

--------------------
type float<'Measure> = float

type VolatilityPrediction = { Date: DateTime PredictedVol: float }

val seed: int

Multiple items
type Random = new: unit -> unit + 1 overload member GetItems<'T> : choices: ReadOnlySpan<'T> * length: int -> 'T array + 2 overloads member Next: unit -> int + 2 overloads member NextBytes: buffer: byte array -> unit + 1 overload member NextDouble: unit -> float member NextInt64: unit -> int64 + 2 overloads member NextSingle: unit -> float32 member Shuffle<'T> : values: Span<'T> -> unit + 1 overload static member Shared: Random
<summary>Represents a pseudo-random number generator, which is an algorithm that produces a sequence of numbers that meet certain statistical requirements for randomness.</summary>

--------------------
Random() : Random
Random(Seed: int) : Random

val SetSampleGenerator: rg: Random.IRandom -> unit
<summary> Sets the random number generator used for sampling. </summary>

Multiple items
type RandThreadSafe = interface IRandom new: unit -> RandThreadSafe + 1 overload val mutable rnd: ThreadLocal<Random>

--------------------
new: unit -> Random.RandThreadSafe
new: n: int -> Random.RandThreadSafe

val rnorm: Distributions.ContinuousDistribution<float,float>

namespace FSharp.Stats.Distributions

namespace FSharp.Stats.Distributions.Continuous

type Normal = static member CDF: mu: float -> sigma: float -> x: float -> float static member CheckParam: mu: float -> sigma: float -> unit static member Estimate: samples: float seq -> ContinuousDistribution<float,float> static member Init: mu: float -> sigma: float -> ContinuousDistribution<float,float> static member InvCDF: mu: float -> sigma: float -> p: float -> float static member Mean: mu: float -> sigma: float -> float static member Mode: mu: float -> sigma: float -> float static member PDF: mu: float -> sigma: float -> x: float -> float static member Sample: mu: float -> sigma: float -> float static member SampleUnchecked: mu: float -> sigma: float -> float ...
<summary> Normal distribution. </summary>

static member Distributions.Continuous.Normal.Init: mu: float -> sigma: float -> Distributions.ContinuousDistribution<float,float>

val testData: ReturnObs array

val i: int

abstract Distributions.ContinuousDistribution.Sample: unit -> 'b

val expandingWindowVol: data: ReturnObs array -> VolatilityPrediction array
<summary>Calculates sample average expanding window volatility.</summary>

val data: ReturnObs array

type 'T array = 'T array

property Array.Length: int with get
<summary>Gets the total number of elements in all the dimensions of the <see cref="T:System.Array" />.</summary>
<exception cref="T:System.OverflowException">The array is multidimensional and contains more than <see cref="F:System.Int32.MaxValue">Int32.MaxValue</see> elements.</exception>
<returns>The total number of elements in all the dimensions of the <see cref="T:System.Array" />; zero if there are no elements in the array.</returns>

val sd: float

Multiple items
type Array = new: unit -> Array static member geomspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float array static member linspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float array

--------------------
new: unit -> Array

val map: mapping: ('T -> 'U) -> array: 'T array -> 'U array

val x: ReturnObs

ReturnObs.Return: float

val stDev: items: 'T seq -> 'U (requires member (-) and member Zero and member DivideByInt and member (+) and member ( * ) and member (+) and member (/) and member Sqrt)
<summary> Computes the sample standard deviation </summary>
<param name="items">The input sequence.</param>
<remarks>Returns NaN if data is empty or if any entry is NaN.</remarks>
<returns>standard deviation of a sample (Bessel's correction by N-1)</returns>

val expandingWindowVolsTest: VolatilityPrediction array

val rollingVol: width: int -> data: ReturnObs array -> VolatilityPrediction array
<summary>Calculates realized volatity of a return observation
over a rolling window of past observations.</summary>
<param name="width">The window width for calculating realized vol.</param>
<param name="data">The input data</param>
<returns>A prediction and date the prediction is valid for.</returns>

val width: int

Multiple items
val int: value: 'T -> int (requires member op_Explicit)

--------------------
type int = int32

--------------------
type int<'Measure> = int

val sortBy: projection: ('T -> 'Key) -> array: 'T array -> 'T array (requires comparison)

ReturnObs.Date: DateTime

val windowed: windowSize: int -> array: 'T array -> 'T array array

val window: ReturnObs array

val take: count: int -> array: 'T array -> 'T array

val last: ReturnObs

val last: array: 'T array -> 'T

val rollingVolTest: VolatilityPrediction array

val expRealizedVol: width: int -> lambda: float -> data: ReturnObs array -> VolatilityPrediction array
<summary>Calculates realized volatity of a return observation using an exponential weight.</summary>
<param name="width">The window width for calculating realized vol.</param>
<param name="lambda">The exponential weight</param>
<param name="data">The input data</param>
<returns>A prediction and date the prediction is valid for.</returns>

val lambda: float

val sortByDescending: projection: ('T -> 'Key) -> array: 'T array -> 'T array (requires comparison)

module Parallel from Microsoft.FSharp.Collections.ArrayModule

val mu: float

val averageBy: projection: ('T -> 'U) -> array: 'T array -> 'U (requires member (+) and member DivideByInt and member Zero)

val mutable acc: float

val t: int

val w: float

val sqrt: value: 'T -> 'U (requires member Sqrt)

val rev: array: 'T array -> 'T array

val expRealizedVolTest: VolatilityPrediction array

val combinedTestData: (VolatilityPrediction array * string) list

val vols: VolatilityPrediction array

val name: string

val dataForChart: (DateTime * float) array

val x: VolatilityPrediction

VolatilityPrediction.Date: DateTime
First date the prediction is valid for

VolatilityPrediction.PredictedVol: float
The volatility prediction

type Chart = static member AnnotatedHeatmap: zData: #('a1 seq) seq * annotationText: #(string seq) seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?X: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?XGap: int * [<Optional; DefaultParameterValue ((null :> obj))>] ?Y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?YGap: int * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a5 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a5 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorBar: ColorBar * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ReverseScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ZSmooth: SmoothAlg * [<Optional; DefaultParameterValue ((null :> obj))>] ?Transpose: bool * [<Optional; DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Optional; DefaultParameterValue ((false :> obj))>] ?ReverseYAxis: bool * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a1 :> IConvertible and 'a5 :> IConvertible) + 1 overload static member Area: x: #IConvertible seq * y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowMarkers: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerSymbol: MarkerSymbol * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiMarkerSymbol: MarkerSymbol seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineDash: DrawingStyle * [<Optional; DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?StackGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?Orientation: Orientation * [<Optional; DefaultParameterValue ((null :> obj))>] ?GroupNorm: GroupNorm * [<Optional; DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Optional; DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible) + 1 overload static member Bar: values: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Keys: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerPatternShape: PatternShape * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiMarkerPatternShape: PatternShape seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerPattern: Pattern * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?Base: #IConvertible * [<Optional; DefaultParameterValue ((null :> obj))>] ?Width: 'a4 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiWidth: 'a4 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible and 'a4 :> IConvertible) + 1 overload static member BoxPlot: [<Optional; DefaultParameterValue ((null :> obj))>] ?X: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?WhiskerWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?BoxPoints: BoxPoints * [<Optional; DefaultParameterValue ((null :> obj))>] ?BoxMean: BoxMean * [<Optional; DefaultParameterValue ((null :> obj))>] ?Jitter: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?PointPos: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Orientation: Orientation * [<Optional; DefaultParameterValue ((null :> obj))>] ?OutlineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?OutlineWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Outline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?AlignmentGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?OffsetGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?Notched: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?NotchWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?QuartileMethod: QuartileMethod * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible) + 2 overloads static member Bubble: x: #IConvertible seq * y: #IConvertible seq * sizes: int seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerSymbol: MarkerSymbol * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiMarkerSymbol: MarkerSymbol seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineDash: DrawingStyle * [<Optional; DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?StackGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?Orientation: Orientation * [<Optional; DefaultParameterValue ((null :> obj))>] ?GroupNorm: GroupNorm * [<Optional; DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible) + 1 overload static member Candlestick: ``open`` : #IConvertible seq * high: #IConvertible seq * low: #IConvertible seq * close: #IConvertible seq * x: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a5 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a5 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?IncreasingColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?Increasing: FinanceMarker * [<Optional; DefaultParameterValue ((null :> obj))>] ?DecreasingColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?Decreasing: FinanceMarker * [<Optional; DefaultParameterValue ((null :> obj))>] ?WhiskerWidth: float * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a5 :> IConvertible) + 1 overload static member Column: values: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Keys: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerPatternShape: PatternShape * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiMarkerPatternShape: PatternShape seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerPattern: Pattern * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?Base: #IConvertible * [<Optional; DefaultParameterValue ((null :> obj))>] ?Width: 'a4 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiWidth: 'a4 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible and 'a4 :> IConvertible) + 1 overload static member Contour: zData: #('a1 seq) seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?X: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a4 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a4 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorBar: ColorBar * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ReverseScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Transpose: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLineDash: DrawingStyle * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLineSmoothing: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLine: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContoursColoring: ContourColoring * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContoursOperation: ConstraintOperation * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContoursType: ContourType * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowContourLabels: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLabelFont: Font * [<Optional; DefaultParameterValue ((null :> obj))>] ?Contours: Contours * [<Optional; DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?NContours: int * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a1 :> IConvertible and 'a4 :> IConvertible) static member Funnel: x: #IConvertible seq * y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Width: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Offset: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Orientation: Orientation * [<Optional; DefaultParameterValue ((null :> obj))>] ?AlignmentGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?OffsetGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextInfo: TextInfo * [<Optional; DefaultParameterValue ((null :> obj))>] ?ConnectorLineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?ConnectorLineStyle: DrawingStyle * [<Optional; DefaultParameterValue ((null :> obj))>] ?ConnectorFillColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?ConnectorLine: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?Connector: FunnelConnector * [<Optional; DefaultParameterValue ((null :> obj))>] ?InsideTextFont: Font * [<Optional; DefaultParameterValue ((null :> obj))>] ?OutsideTextFont: Font * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible) static member Heatmap: zData: #('a1 seq) seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?X: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?XGap: int * [<Optional; DefaultParameterValue ((null :> obj))>] ?Y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?YGap: int * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a4 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a4 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorBar: ColorBar * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ReverseScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ZSmooth: SmoothAlg * [<Optional; DefaultParameterValue ((null :> obj))>] ?Transpose: bool * [<Optional; DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Optional; DefaultParameterValue ((false :> obj))>] ?ReverseYAxis: bool * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a1 :> IConvertible and 'a4 :> IConvertible) + 1 overload ...

static member Chart.Line: xy: (#IConvertible * #IConvertible) seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowMarkers: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Name: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Text: 'c * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiText: 'c seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TextPosition: StyleParam.TextPosition * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: StyleParam.TextPosition seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: StyleParam.Colorscale * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerSymbol: StyleParam.MarkerSymbol * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiMarkerSymbol: StyleParam.MarkerSymbol seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Marker: TraceObjects.Marker * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColorScale: StyleParam.Colorscale * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineWidth: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineDash: StyleParam.DrawingStyle * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?StackGroup: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Orientation: StyleParam.Orientation * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?GroupNorm: StyleParam.GroupNorm * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Fill: StyleParam.Fill * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart.GenericChart (requires 'c :> IConvertible)
static member Chart.Line: x: #IConvertible seq * y: #IConvertible seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowMarkers: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Name: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TextPosition: StyleParam.TextPosition * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: StyleParam.TextPosition seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: StyleParam.Colorscale * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerSymbol: StyleParam.MarkerSymbol * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiMarkerSymbol: StyleParam.MarkerSymbol seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Marker: TraceObjects.Marker * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColorScale: StyleParam.Colorscale * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineWidth: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineDash: StyleParam.DrawingStyle * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?StackGroup: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Orientation: StyleParam.Orientation * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?GroupNorm: StyleParam.GroupNorm * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Fill: StyleParam.Fill * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart.GenericChart (requires 'a2 :> IConvertible)

static member Chart.combine: gCharts: GenericChart.GenericChart seq -> GenericChart.GenericChart

type ReturnPrediction = { Date: DateTime PredictedReturn: float }

val avgReturnAccumulator: xs: ReturnObs array -> ReturnPrediction array

val xs: ReturnObs array

val avgReturn: float

val x: ReturnPrediction

ReturnPrediction.Date: DateTime

ReturnPrediction.PredictedReturn: float

type VolAndReturnPrediction = { Date: DateTime PredictedVol: float PredictedReturn: float }

val combinePredictions: predReturns: ReturnPrediction seq -> predVols: VolatilityPrediction seq -> VolAndReturnPrediction list

val predReturns: ReturnPrediction seq

Multiple items
val seq: sequence: 'T seq -> 'T seq

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type 'T seq = Collections.Generic.IEnumerable<'T>

val predVols: VolatilityPrediction seq

val predVols: Map<DateTime,float>

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module Seq from FSharp.Stats
<summary> Module to compute common statistical measure </summary>

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module Seq from Microsoft.FSharp.Collections

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type Seq = new: unit -> Seq static member geomspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float seq static member linspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float seq

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new: unit -> Seq

val map: mapping: ('T -> 'U) -> source: 'T seq -> 'U seq

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module Map from FSharp.Stats
<summary> Module to strore specialised computations on maps </summary>

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module Map from Microsoft.FSharp.Collections

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type Map<'Key,'Value (requires comparison)> = interface IReadOnlyDictionary<'Key,'Value> interface IReadOnlyCollection<KeyValuePair<'Key,'Value>> interface IEnumerable interface IStructuralEquatable interface IComparable interface IEnumerable<KeyValuePair<'Key,'Value>> interface ICollection<KeyValuePair<'Key,'Value>> interface IDictionary<'Key,'Value> new: elements: ('Key * 'Value) seq -> Map<'Key,'Value> member Add: key: 'Key * value: 'Value -> Map<'Key,'Value> ...

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new: elements: ('Key * 'Value) seq -> Map<'Key,'Value>

val retObs: ReturnPrediction

member Map.ContainsKey: key: 'Key -> bool

val managedPortfolio: gamma: float -> predVols: VolatilityPrediction seq -> predReturns: ReturnPrediction seq -> xs: ReturnObs seq -> ReturnObs array

val gamma: float

val xs: ReturnObs seq

val preds: Map<DateTime,VolAndReturnPrediction>

val x: VolAndReturnPrediction

VolAndReturnPrediction.Date: DateTime

val pred: VolAndReturnPrediction

VolAndReturnPrediction.PredictedReturn: float

VolAndReturnPrediction.PredictedVol: float

val retObs: ReturnObs array

val x: FF3Obs

FF3Obs.Date: DateTime

FF3Obs.MktRf: float

val expectedRet: ReturnPrediction array

val expandingVols: VolatilityPrediction array

val rollingVols: VolatilityPrediction array

val exponentialVols: VolatilityPrediction array

val expandingVolPort: ReturnObs array

val rollingVolPort: ReturnObs array

val exponentialVolPort: ReturnObs array

val retAsTuple: xs: ReturnObs array -> (DateTime * float) array

val cumulativeReturns: xs: (DateTime * float) array -> (DateTime * float) array

val xs: (DateTime * float) array

val dt: DateTime

val r: float

val mutable accRet: float

val cr: float

val broke: float

val epsilon: float
<summary> Double-precision machine round-off error. </summary>
<remarks> This value is actually different from Double.Epsilon. It is defined as 1.11022302462515654042E-16. </remarks>

val max: e1: 'T -> e2: 'T -> 'T (requires comparison)

namespace Plotly.NET.LayoutObjects

val plotCumulativeReturns: name: string -> xs: (DateTime * float) array -> GenericChart.GenericChart

val xsFromOne: (DateTime * float) array

static member Chart.withYAxisStyle: [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TitleText: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TitleFont: Font * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TitleStandoff: int * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Title: Title * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Color: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?AxisType: StyleParam.AxisType * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MinMax: (#IConvertible * #IConvertible) * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Mirror: StyleParam.Mirror * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowSpikes: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?SpikeColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?SpikeThickness: int * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowLine: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowGrid: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?GridColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ZeroLine: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ZeroLineColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Anchor: StyleParam.LinearAxisId * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Side: StyleParam.Side * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Overlaying: StyleParam.LinearAxisId * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Domain: (float * float) * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Position: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?CategoryOrder: StyleParam.CategoryOrder * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?CategoryArray: #IConvertible seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?RangeSlider: RangeSlider * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?RangeSelector: RangeSelector * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?BackgroundColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowBackground: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Id: StyleParam.SubPlotId -> (GenericChart.GenericChart -> GenericChart.GenericChart)

module StyleParam from Plotly.NET

type AxisType = | Auto | Linear | Log | Date | Category | MultiCategory member Convert: unit -> obj override ToString: unit -> string static member convert: (AxisType -> obj) static member toString: (AxisType -> string)
<summary> Sets the axis type. By default (Auto), plotly attempts to determined the axis type by looking into the data of the traces that referenced the axis in question. </summary>

union case StyleParam.AxisType.Log: StyleParam.AxisType

Multiple items
module List from FSharp.Stats
<summary> Module to compute common statistical measure on list </summary>

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module List from Microsoft.FSharp.Collections

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type List = new: unit -> List static member geomspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float list static member linspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float list

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type List<'T> = | op_Nil | op_ColonColon of Head: 'T * Tail: 'T list interface IReadOnlyList<'T> interface IReadOnlyCollection<'T> interface IEnumerable interface IEnumerable<'T> member GetReverseIndex: rank: int * offset: int -> int member GetSlice: startIndex: int option * endIndex: int option -> 'T list static member Cons: head: 'T * tail: 'T list -> 'T list member Head: 'T member IsEmpty: bool member Item: index: int -> 'T with get ...

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new: unit -> List

val map: mapping: ('T -> 'U) -> list: 'T list -> 'U list

val port: ReturnObs array

static member Chart.show: ch: GenericChart.GenericChart -> unit