# Was sind eigentlich stetige Zufallsvariablen?
#
# von
#
# Jonas Alexander Lache <jonas.lache(at)ruhr-uni-bochum.de>
# Daniel Meißner <daniel.meissner-i4k(at)ruhr-uni-bochum.de>
#
# Dieses Werk ist lizenziert unter einer Creative Commons
# Namensnennung-Weitergabe unter gleichen Bedingungen 4.0 International
# Lizenz. Um eine Kopie der Lizenz zu erhalten, besuchen Sie
# http://creativecommons.org/licenses/by-sa/4.0/.
#
# SPDX-License-Identifier: CC-BY-SA-4.0

from manim import *

import numpy as np
import matplotlib.pyplot as plt

ORCA_BLUE = "#002b44"
ORCA_RED = "#e61300"
ORCA_GREEN = "#4cb011"
ORCA_GREY = "#f5f5f5"
ORCA_WHITE = "#ffffff"

config.background_color = ORCA_WHITE


def histogram_to_file(data):
    filename = "histogram.svg"
    fig, ax = plt.subplots()
    ax.hist(data, color=ORCA_BLUE, density=True, bins="sturges")
    x = np.linspace(45, 48)
    ax.set_xlabel("Gewicht in Gramm")
    fig.savefig(filename, bbox_inches="tight", pad_inches=0.05)
    return filename

def scene_heading(title):
    heading = Tex(
        r"\textsf{{\bfseries {}}}".format(title),
        color=ORCA_BLUE,
        font_size=56,
        tex_environment="flushleft"
    )
    heading.to_edge(UL)
    return heading

class Definition(Scene):
    def construct(self):
        heading = scene_heading("Stetige Zufallsvariablen")
        self.play(Create(heading))
        self.wait(5.5)

        formulas = MathTex(
            "P(X = 2) = 0",
            r"P(X \in [2 - \epsilon, 2 + \epsilon])",
            r"= \int_{2-\epsilon}^{2+\epsilon} f(x) \:dx",
            color=ORCA_BLUE
        )
        formulas.arrange(DOWN)
        self.play(Create(formulas[0]), run_time=1)
        self.wait(6)
        formulas[1].next_to(formulas[0], DOWN)
        self.play(Create(formulas[1]), run_time=1)
        self.wait(4)
        formulas[2].next_to(formulas[1], RIGHT)
        self.play(Create(formulas[2]))
        self.wait(3)
        # Dichtefunktion
        dichte = Tex(
            r"{\sffamily \(f\) nennen wir \textbf{Dichtefunktion}}",
            color=ORCA_BLUE
        )
        dichte.next_to(formulas[1], 3*DOWN)
        self.play(Create(dichte), run_time=1)
        self.wait(0.5)
        self.play(FadeOut(formulas, dichte))

        definition = Tex(
            r"{\sffamily Eine Zufallsvariable \(X\) hat eine stetige Verteilung,}\\",
            r"{\sffamily falls eine messbare, nichtnegative Funktion \(f\) existiert,}\\",
            r"{\sffamily sodass\\}",
            color=ORCA_BLUE,
            tex_environment="flushleft"
        )
        interval_probability = MathTex(
            r"{{ P(a \le X \le b) }} = {{ \int_a^b f(x) \:dx }}",
            color = ORCA_BLUE
        )
        interval_probability.next_to(definition, DOWN)
        self.play(Create(definition[:2]), run_time=5)
        self.wait(1.5)
        self.play(Create(definition[2]), run_time=0.5)
        # self.wait()
        self.play(Create(interval_probability[0]), run_time=2)
        self.wait(2)
        self.play(Create(interval_probability[1:]), run_time=1)
        self.wait(8)
        self.play(FadeOut(definition, interval_probability, heading))
        self.wait(0.5)

class Modelling(Scene):
    def construct(self):
        heading = scene_heading("Verteilung der Packungsgewichte")
        self.play(Create(heading))
        subheading = Tex(r"\textsf{Modellbildung}", color=ORCA_BLUE)
        subheading.next_to(heading, DOWN).align_to(heading, LEFT)
        self.play(Create(subheading))
        self.wait(5.5)

        data = np.array([47.84,
                         45.09,
                         46.46,
                         45.66,
                         46.38,
                         45.73,
                         45.80,
                         46.47,
                         46.17,
                         46.65,
                         45.99,
                         46.32,
                         46.10,
                         46.47,
                         48.01,
                         46.87,
                         46.56,
                         46.24,
                         46.82,
                         46.32,
                         46.16,
                         46.46,
                         46.74,
                         47.24])

        histogram = SVGMobject(histogram_to_file(data), height=5)
        (histogram.next_to(subheading, DOWN, buff=0.3)
         .shift(-histogram.get_center() * np.array([1, 0, 0])))
        self.play(Create(histogram))
        self.wait(1)

        self.play(histogram.animate.scale_to_fit_width(5))
        self.play(histogram.animate.to_edge(RIGHT, buff=2))

        model = Tex(
            r"\textsf{Modellannahme}\\",
            r"\(X \sim \mathcal{N}(\mu, \sigma^2)\)",
            color=ORCA_BLUE,
            tex_environment="flushleft"
        )
        model.to_edge(LEFT, buff=2)
        model.align_to(histogram, UP)
        self.play(Create(model))
        self.wait(11.5)

        estimated_parameters = Tex(
            r"\textsf{Geschätzte Parameter}\\",
            r"""\(\mu = {0:.1f}\)\\
            \(\sigma = {1:.2f}\)""".format(data.mean(), data.std()),
            color=ORCA_BLUE,
            tex_environment="flushleft"
        )
        estimated_parameters.next_to(model, DOWN, buff=1)
        estimated_parameters.align_to(model, LEFT)
        self.play(Create(estimated_parameters))
        self.wait(2)
        self.play(FadeOut(histogram))
        self.play(estimated_parameters.animate.to_edge(RIGHT, buff=2))
        self.play(estimated_parameters.animate.align_to(model, UP))
        model_probability = MathTex(
            r"P(44 \le X \le 45) =",
            r"""\int_{44}^{45} \frac{1}{\sqrt{2\pi\sigma^2}}
            e^{-\frac{(x - \mu)^2}{2\sigma^2}} dx""",
            color=ORCA_BLUE
        )
        model_probability.next_to(estimated_parameters, DOWN, buff=1)
        model_probability.align_to(model, LEFT)
        self.play(Create(model_probability[0], run_time=3))
        self.wait(3.5)
        self.play(Create(model_probability[1], run_time=2))
        self.wait(9.5)
        self.play(FadeOut(subheading,model,estimated_parameters,model_probability))
        self.wait(0.5)

class DensityProperties(Scene):
    def construct(self):
        heading = scene_heading("Eigenschaften der Dichtefunktion")
        self.play(Create(heading))
        text = Tex(
            r"""\sffamily {Eine Dichtefunktion \(f\colon\mathbb{R}\to
            \mathbb{R}\) erfüllt zwei Eigenschaften:}""",
            color=ORCA_BLUE,
            tex_environment="flushleft"
        )
        formulas = MathTex(
            r"f(x) \ge 0",
            r"\int_{-\infty}^\infty f(x) \:dx = 1",
            color=ORCA_BLUE)
        formulas.center()
        formulas.arrange(DOWN)
        formulas[1].next_to(formulas[0], DOWN)
        text.next_to(formulas, UP, buff=1)
        text.align_to(heading, LEFT)
        self.play(Create(text))
        self.wait(2)
        self.play(Create(formulas[0]), run_time=1)
        self.wait(1.5)
        self.play(Create(formulas[1]), run_time=1)
        self.wait(1.2)
        self.play(FadeOut(heading,text,formulas))

class DistributionFunction(Scene):
    def construct(self):
        heading = scene_heading("Verteilung der Packungsgewichte")
        self.add(heading)
        self.wait(7)
        subheading = Tex(r"\textsf{Die Verteilungsfunktion}", color=ORCA_BLUE)
        subheading.next_to(heading, DOWN).align_to(heading, LEFT)
        self.play(Create(subheading))
        self.wait(3)
        text = Tex(
            r"""{\sffamily Sei \(X\) eine stetige oder diskrete
            Zufallsvariable.}\\""",
            color=ORCA_BLUE
        )
        formula = MathTex(r"F(x) := P(X \le x)", color=ORCA_BLUE)
        (formula.next_to(text, DOWN)
         .shift(-formula.get_center() * np.array([1, 0, 0])))
        self.play(Create(text))
        self.wait(5)
        self.play(Create(formula), run_time=1)
        self.wait(6.5)
        self.play(FadeOut(text, formula))

        probability = Tex(
            r"""{\sffamily Wie wahrscheinlich ist es,\\
            dass eine Packung weniger als 45 Gramm wiegt?}\\\vspace{0.5ex}""",
            r"\textsf{Antwort:}\\",
            r"\(P(X \le 45) = F(45) \approx 4.6\%\)",
            color=ORCA_BLUE,
            tex_environment="flushleft"
        )
        self.play(FadeIn(probability[0]))
        self.wait(4)
        (
            probability[1].next_to(probability[0], DOWN, buff=1)
            .align_to(probability[0], LEFT)
        )
        self.play(FadeIn(probability[1]))
        self.wait()
        self.play(Create(
            probability[2].next_to(probability[1], DOWN).shift(5*RIGHT)
        ))
        self.wait(3.5)
        self.play(FadeOut(heading, subheading, probability))
        self.wait(0.5)

class FromDistributionToDensity(Scene):
    def construct(self):
        heading = scene_heading(
            "Zusammenhang Dichte und Verteilungsfunktion"
        )
        self.play(Create(heading))
        self.wait(9)

        texts = Tex(
            r"\textsf{Von Dichtefunktion zur Verteilungsfunktion}",
            r"\textsf{und umgekehrt}",
            color=ORCA_BLUE
        )
        formulas = MathTex(
            r"F(x) = P(X \le x) = \int_{-\infty}^x f(u) du",
            r"f(x) = F'(x)",
            color=ORCA_BLUE
        )

        self.play(Create(texts[0].next_to(heading, DOWN, buff=0.8)))
        self.wait(3)
        self.play(Create(
            formulas[0].next_to(texts[0], DOWN)
            .shift(-formulas[0].get_center() * np.array([1, 0, 0])))
        )
        self.wait(2)
        self.play(Create(
            texts[1].next_to(formulas[0], DOWN)
            .align_to(texts[0], LEFT)
        ))
        self.wait(2)
        self.play(Create(
            formulas[1].next_to(texts[1], 2*DOWN)
            .shift(-formulas[1].get_center() * np.array([1, 0, 0])))
        )
        self.wait()