Fitting a "hinged" piecewise linear function

.gitignore

.ipynb_checkpoints/
.venv/
__pycache__/

README.md

Fitting a "hinged" piecewise linear function

Run Notebook Server

python -m pip install --upgrade pip virtualenv
python -m virtualenv --python=python3.7 .venv
.venv/bin/python -m pip install --requirement requirements.txt
.venv/bin/jupyter notebook

Requirements

To update

python -m pip install --upgrade pip-tools  # For `pip-compile`
pip-compile \
  --generate-hashes \
  --upgrade \
  --output-file=requirements.txt \
  requirements.txt.in

find-split.ipynb

nb_helpers.py

import matplotlib.pyplot as plt
import numpy as np
import seaborn


def new_ax():
    figure = plt.figure(figsize=(9, 6), dpi=80)
    return figure.gca()


def plot_exact(h, a, b, c, x_start, x_end):
    if not x_start < h < x_end:
        raise ValueError("Invalid interval", x_start, h, x_end)

    ax = new_ax()
    x_vals = np.linspace(x_start, x_end, 1025)
    delta_x = x_vals - h
    y_vals = a + delta_x * np.where(delta_x <= 0.0, b, c)
    ax.plot(x_vals, y_vals)
    ax.plot([h], [a], marker="o", color="black")


def noisy_data(x_vals, h, a, b, c, seed):
    delta_x = x_vals - h
    y_vals_exact = a + delta_x * np.where(delta_x <= 0.0, b, c)

    # Add some noise.
    random_state = np.random.RandomState(seed=seed)
    noise = random_state.randn(len(y_vals_exact))
    y_vals = y_vals_exact * (1.0 + noise / 8.0)
    return y_vals_exact, y_vals


def plot_noisy_data(h, a, x_vals, y_vals_exact, y_vals):
    ax = new_ax()
    ax.plot(
        x_vals,
        y_vals_exact,
        marker="o",
        alpha=0.5,
        color="black",
        linestyle="dashed",
    )
    ax.plot(x_vals, y_vals, marker="o")
    ax.plot([h], [a], marker="o", color="black")
    ax.set_xlabel("x", family="monospace")
    ax.set_ylabel("y", rotation=0, family="monospace")


def plot_residuals(guesses, residuals_by_guess, annotated_x):
    ax = new_ax()
    ax.plot(guesses, residuals_by_guess)

    annotated_y = []
    for x_val in annotated_x:
        nearest_guess = np.argmin(abs(guesses - x_val))
        annotated_y.append(residuals_by_guess[nearest_guess])

    ax.plot(
        annotated_x, annotated_y, marker="o", linestyle="None", color="black"
    )
    ax.set_xlabel("guess(x_hinge)", family="monospace")
    ax.set_ylabel("Residual")


def fit_coeffs(x_vals, y_vals, x_hinge):
    num_vals = len(x_vals)
    delta_x = x_vals - x_hinge
    # y = A + B CHI(x <= x_hinge) (x - x_hinge) + C CHI(x > x_hinge) (x - x_hinge)
    b_vals = np.where(delta_x <= 0.0, delta_x, 0.0)
    c_vals = np.where(delta_x > 0.0, delta_x, 0.0)
    a_vals = np.ones_like(b_vals)

    lhs_mat = np.vstack([a_vals, b_vals, c_vals]).T
    rhs_mat = y_vals[:, np.newaxis]

    # coeffs, residuals, rank, singular_values
    coeffs, residuals, _, _ = np.linalg.lstsq(lhs_mat, rhs_mat, rcond=None)
    # Unpack the values
    a, b, c = coeffs
    residual, = residuals  # Assumes y_vals is 1D
    return a, b, c, residual


def find_hinge(x_vals, y_vals, num_guesses=65):
    # NOTE: This assumes the hinge happens **within** the data.
    min_residual = float("inf")
    x_hinge_at_min = None

    sorted_x = sorted(x_vals)
    x_extrema = (sorted_x[0], sorted_x[-1])
    for interval_start, interval_end in zip(sorted_x, sorted_x[1:]):
        guesses = np.linspace(interval_start, interval_end, num_guesses)
        for guess in guesses:
            # NOTE: Since the hinge happens **within** the data, leave out
            #       the endpoints.
            if guess in x_extrema:
                continue

            _, _, _, residual = fit_coeffs(x_vals, y_vals, guess)
            if residual >= min_residual:
                continue

            min_residual = residual
            x_hinge_at_min = guess

    if x_hinge_at_min is None:
        raise ValueError("No minimum was found")

    a, b, c, _ = fit_coeffs(x_vals, y_vals, x_hinge_at_min)
    return x_hinge_at_min, a, b, c


def describe_line(h, a, b, c):
    # Cast from ``np.float64`` to ``float`` to avoid "shape" issues and other
    # __str__ / __format__ differences.
    h = float(h)
    a = float(a)
    b = float(b)
    c = float(c)

    if h <= 0 or a <= 0 or b <= 0 or c <= 0:
        raise ValueError(
            "describe_line only written for positive values", h, a, b, c
        )

    return "\n".join(
        [
            "y =",
            r"\begin{cases}",
            f"{a} + {b}(x - {h}), & \\text{{if }} x \leq {h} \\\\",
            f"{a} + {c}(x - {h}), & \\text{{if }} x > {h}",
            r"\end{cases}",
        ]
    )


def plot_best_fit(h, a, b, c, x_vals, y_vals):
    fit_x = np.sort(np.append(x_vals, h))
    delta_x = fit_x - h
    fit_y = a + delta_x * np.where(delta_x <= 0.0, b, c)

    ax = new_ax()
    ax.plot(fit_x, fit_y, alpha=0.5, color="black", linestyle="dashed")
    ax.plot([h], [a], marker="o", color="black")
    ax.plot(x_vals, y_vals, marker="o", linestyle="None")
    ax.set_xlabel("x", family="monospace")
    ax.set_ylabel("y", rotation=0, family="monospace")

requirements.txt

#
# This file is autogenerated by pip-compile
# To update, run:
#
#    pip-compile --generate-hashes --output-file=requirements.txt requirements.txt.in
#
appnope==0.1.0 \
    --hash=sha256:5b26757dc6f79a3b7dc9fab95359328d5747fcb2409d331ea66d0272b90ab2a0 \
    --hash=sha256:8b995ffe925347a2138d7ac0fe77155e4311a0ea6d6da4f5128fe4b3cbe5ed71 \
    # via ipykernel, ipython
attrs==19.3.0 \
    --hash=sha256:08a96c641c3a74e44eb59afb61a24f2cb9f4d7188748e76ba4bb5edfa3cb7d1c \
    --hash=sha256:f7b7ce16570fe9965acd6d30101a28f62fb4a7f9e926b3bbc9b61f8b04247e72 \
    # via jsonschema
backcall==0.1.0 \
    --hash=sha256:38ecd85be2c1e78f77fd91700c76e14667dc21e2713b63876c0eb901196e01e4 \
    --hash=sha256:bbbf4b1e5cd2bdb08f915895b51081c041bac22394fdfcfdfbe9f14b77c08bf2 \
    # via ipython
bleach==3.1.0 \
    --hash=sha256:213336e49e102af26d9cde77dd2d0397afabc5a6bf2fed985dc35b5d1e285a16 \
    --hash=sha256:3fdf7f77adcf649c9911387df51254b813185e32b2c6619f690b593a617e19fa \
    # via nbconvert
cycler==0.10.0 \
    --hash=sha256:1d8a5ae1ff6c5cf9b93e8811e581232ad8920aeec647c37316ceac982b08cb2d \
    --hash=sha256:cd7b2d1018258d7247a71425e9f26463dfb444d411c39569972f4ce586b0c9d8 \
    # via matplotlib
decorator==4.4.1 \
    --hash=sha256:54c38050039232e1db4ad7375cfce6748d7b41c29e95a081c8a6d2c30364a2ce \
    --hash=sha256:5d19b92a3c8f7f101c8dd86afd86b0f061a8ce4540ab8cd401fa2542756bce6d \
    # via ipython, traitlets
defusedxml==0.6.0 \
    --hash=sha256:6687150770438374ab581bb7a1b327a847dd9c5749e396102de3fad4e8a3ef93 \
    --hash=sha256:f684034d135af4c6cbb949b8a4d2ed61634515257a67299e5f940fbaa34377f5 \
    # via nbconvert
entrypoints==0.3 \
    --hash=sha256:589f874b313739ad35be6e0cd7efde2a4e9b6fea91edcc34e58ecbb8dbe56d19 \
    --hash=sha256:c70dd71abe5a8c85e55e12c19bd91ccfeec11a6e99044204511f9ed547d48451 \
    # via nbconvert
importlib-metadata==1.3.0 \
    --hash=sha256:073a852570f92da5f744a3472af1b61e28e9f78ccf0c9117658dc32b15de7b45 \
    --hash=sha256:d95141fbfa7ef2ec65cfd945e2af7e5a6ddbd7c8d9a25e66ff3be8e3daf9f60f \
    # via jsonschema
ipykernel==5.1.3 \
    --hash=sha256:1a7def9c986f1ee018c1138d16951932d4c9d4da01dad45f9d34e9899565a22f \
    --hash=sha256:b368ad13edb71fa2db367a01e755a925d7f75ed5e09fbd3f06c85e7a8ef108a8 \
    # via ipywidgets, jupyter, jupyter-console, notebook, qtconsole
ipython-genutils==0.2.0 \
    --hash=sha256:72dd37233799e619666c9f639a9da83c34013a73e8bbc79a7a6348d93c61fab8 \
    --hash=sha256:eb2e116e75ecef9d4d228fdc66af54269afa26ab4463042e33785b887c628ba8 \
    # via nbformat, notebook, qtconsole, traitlets
ipython==7.11.1 \
    --hash=sha256:0f4bcf18293fb666df8511feec0403bdb7e061a5842ea6e88a3177b0ceb34ead \
    --hash=sha256:387686dd7fc9caf29d2fddcf3116c4b07a11d9025701d220c589a430b0171d8a \
    # via ipykernel, ipywidgets, jupyter-console
ipywidgets==7.5.1 \
    --hash=sha256:13ffeca438e0c0f91ae583dc22f50379b9d6b28390ac7be8b757140e9a771516 \
    --hash=sha256:e945f6e02854a74994c596d9db83444a1850c01648f1574adf144fbbabe05c97 \
    # via jupyter
jedi==0.15.2 \
    --hash=sha256:1349c1e8c107095a55386628bb3b2a79422f3a2cab8381e34ce19909e0cf5064 \
    --hash=sha256:e909527104a903606dd63bea6e8e888833f0ef087057829b89a18364a856f807 \
    # via ipython
jinja2==2.10.3 \
    --hash=sha256:74320bb91f31270f9551d46522e33af46a80c3d619f4a4bf42b3164d30b5911f \
    --hash=sha256:9fe95f19286cfefaa917656583d020be14e7859c6b0252588391e47db34527de \
    # via nbconvert, notebook
jsonschema==3.2.0 \
    --hash=sha256:4e5b3cf8216f577bee9ce139cbe72eca3ea4f292ec60928ff24758ce626cd163 \
    --hash=sha256:c8a85b28d377cc7737e46e2d9f2b4f44ee3c0e1deac6bf46ddefc7187d30797a \
    # via nbformat
jupyter-client==5.3.4 \
    --hash=sha256:60e6faec1031d63df57f1cc671ed673dced0ed420f4377ea33db37b1c188b910 \
    --hash=sha256:d0c077c9aaa4432ad485e7733e4d91e48f87b4f4bab7d283d42bb24cbbba0a0f \
    # via ipykernel, jupyter-console, notebook, qtconsole
jupyter-console==6.0.0 \
    --hash=sha256:308ce876354924fb6c540b41d5d6d08acfc946984bf0c97777c1ddcb42e0b2f5 \
    --hash=sha256:cc80a97a5c389cbd30252ffb5ce7cefd4b66bde98219edd16bf5cb6f84bb3568 \
    # via jupyter
jupyter-core==4.6.1 \
    --hash=sha256:464769f7387d7a62a2403d067f1ddc616655b7f77f5d810c0dd62cb54bfd0fb9 \
    --hash=sha256:a183e0ec2e8f6adddf62b0a3fc6a2237e3e0056d381e536d3e7c7ecc3067e244 \
    # via jupyter-client, nbconvert, nbformat, notebook, qtconsole
jupyter==1.0.0 \
    --hash=sha256:3e1f86076bbb7c8c207829390305a2b1fe836d471ed54be66a3b8c41e7f46cc7 \
    --hash=sha256:5b290f93b98ffbc21c0c7e749f054b3267782166d72fa5e3ed1ed4eaf34a2b78 \
    --hash=sha256:d9dc4b3318f310e34c82951ea5d6683f67bed7def4b259fafbfe4f1beb1d8e5f
kiwisolver==1.1.0 \
    --hash=sha256:05b5b061e09f60f56244adc885c4a7867da25ca387376b02c1efc29cc16bcd0f \
    --hash=sha256:210d8c39d01758d76c2b9a693567e1657ec661229bc32eac30761fa79b2474b0 \
    --hash=sha256:26f4fbd6f5e1dabff70a9ba0d2c4bd30761086454aa30dddc5b52764ee4852b7 \
    --hash=sha256:3b15d56a9cd40c52d7ab763ff0bc700edbb4e1a298dc43715ecccd605002cf11 \
    --hash=sha256:3b2378ad387f49cbb328205bda569b9f87288d6bc1bf4cd683c34523a2341efe \
    --hash=sha256:400599c0fe58d21522cae0e8b22318e09d9729451b17ee61ba8e1e7c0346565c \
    --hash=sha256:47b8cb81a7d18dbaf4fed6a61c3cecdb5adec7b4ac292bddb0d016d57e8507d5 \
    --hash=sha256:53eaed412477c836e1b9522c19858a8557d6e595077830146182225613b11a75 \
    --hash=sha256:58e626e1f7dfbb620d08d457325a4cdac65d1809680009f46bf41eaf74ad0187 \
    --hash=sha256:5a52e1b006bfa5be04fe4debbcdd2688432a9af4b207a3f429c74ad625022641 \
    --hash=sha256:5c7ca4e449ac9f99b3b9d4693debb1d6d237d1542dd6a56b3305fe8a9620f883 \
    --hash=sha256:682e54f0ce8f45981878756d7203fd01e188cc6c8b2c5e2cf03675390b4534d5 \
    --hash=sha256:76275ee077772c8dde04fb6c5bc24b91af1bb3e7f4816fd1852f1495a64dad93 \
    --hash=sha256:79bfb2f0bd7cbf9ea256612c9523367e5ec51d7cd616ae20ca2c90f575d839a2 \
    --hash=sha256:7f4dd50874177d2bb060d74769210f3bce1af87a8c7cf5b37d032ebf94f0aca3 \
    --hash=sha256:8944a16020c07b682df861207b7e0efcd2f46c7488619cb55f65882279119389 \
    --hash=sha256:8aa7009437640beb2768bfd06da049bad0df85f47ff18426261acecd1cf00897 \
    --hash=sha256:9105ce82dcc32c73eb53a04c869b6a4bc756b43e4385f76ea7943e827f529e4d \
    --hash=sha256:933df612c453928f1c6faa9236161a1d999a26cd40abf1dc5d7ebbc6dbfb8fca \
    --hash=sha256:939f36f21a8c571686eb491acfffa9c7f1ac345087281b412d63ea39ca14ec4a \
    --hash=sha256:9491578147849b93e70d7c1d23cb1229458f71fc79c51d52dce0809b2ca44eea \
    --hash=sha256:9733b7f64bd9f807832d673355f79703f81f0b3e52bfce420fc00d8cb28c6a6c \
    --hash=sha256:a02f6c3e229d0b7220bd74600e9351e18bc0c361b05f29adae0d10599ae0e326 \
    --hash=sha256:a0c0a9f06872330d0dd31b45607197caab3c22777600e88031bfe66799e70bb0 \
    --hash=sha256:aa716b9122307c50686356cfb47bfbc66541868078d0c801341df31dca1232a9 \
    --hash=sha256:acc4df99308111585121db217681f1ce0eecb48d3a828a2f9bbf9773f4937e9e \
    --hash=sha256:b64916959e4ae0ac78af7c3e8cef4becee0c0e9694ad477b4c6b3a536de6a544 \
    --hash=sha256:d22702cadb86b6fcba0e6b907d9f84a312db9cd6934ee728144ce3018e715ee1 \
    --hash=sha256:d3fcf0819dc3fea58be1fd1ca390851bdb719a549850e708ed858503ff25d995 \
    --hash=sha256:d52e3b1868a4e8fd18b5cb15055c76820df514e26aa84cc02f593d99fef6707f \
    --hash=sha256:db1a5d3cc4ae943d674718d6c47d2d82488ddd94b93b9e12d24aabdbfe48caee \
    --hash=sha256:e3a21a720791712ed721c7b95d433e036134de6f18c77dbe96119eaf7aa08004 \
    --hash=sha256:e8bf074363ce2babeb4764d94f8e65efd22e6a7c74860a4f05a6947afc020ff2 \
    --hash=sha256:f16814a4a96dc04bf1da7d53ee8d5b1d6decfc1a92a63349bb15d37b6a263dd9 \
    --hash=sha256:f2b22153870ca5cf2ab9c940d7bc38e8e9089fa0f7e5856ea195e1cf4ff43d5a \
    --hash=sha256:f790f8b3dff3d53453de6a7b7ddd173d2e020fb160baff578d578065b108a05f \
    --hash=sha256:fe51b79da0062f8e9d49ed0182a626a7dc7a0cbca0328f612c6ee5e4711c81e4 \
    # via matplotlib
markupsafe==1.1.1 \
    --hash=sha256:00bc623926325b26bb9605ae9eae8a215691f33cae5df11ca5424f06f2d1f473 \
    --hash=sha256:09027a7803a62ca78792ad89403b1b7a73a01c8cb65909cd876f7fcebd79b161 \
    --hash=sha256:09c4b7f37d6c648cb13f9230d847adf22f8171b1ccc4d5682398e77f40309235 \
    --hash=sha256:1027c282dad077d0bae18be6794e6b6b8c91d58ed8a8d89a89d59693b9131db5 \
    --hash=sha256:24982cc2533820871eba85ba648cd53d8623687ff11cbb805be4ff7b4c971aff \
    --hash=sha256:29872e92839765e546828bb7754a68c418d927cd064fd4708fab9fe9c8bb116b \
    --hash=sha256:43a55c2930bbc139570ac2452adf3d70cdbb3cfe5912c71cdce1c2c6bbd9c5d1 \
    --hash=sha256:46c99d2de99945ec5cb54f23c8cd5689f6d7177305ebff350a58ce5f8de1669e \
    --hash=sha256:500d4957e52ddc3351cabf489e79c91c17f6e0899158447047588650b5e69183 \
    --hash=sha256:535f6fc4d397c1563d08b88e485c3496cf5784e927af890fb3c3aac7f933ec66 \
    --hash=sha256:62fe6c95e3ec8a7fad637b7f3d372c15ec1caa01ab47926cfdf7a75b40e0eac1 \
    --hash=sha256:6dd73240d2af64df90aa7c4e7481e23825ea70af4b4922f8ede5b9e35f78a3b1 \
    --hash=sha256:717ba8fe3ae9cc0006d7c451f0bb265ee07739daf76355d06366154ee68d221e \
    --hash=sha256:79855e1c5b8da654cf486b830bd42c06e8780cea587384cf6545b7d9ac013a0b \
    --hash=sha256:7c1699dfe0cf8ff607dbdcc1e9b9af1755371f92a68f706051cc8c37d447c905 \
    --hash=sha256:88e5fcfb52ee7b911e8bb6d6aa2fd21fbecc674eadd44118a9cc3863f938e735 \
    --hash=sha256:8defac2f2ccd6805ebf65f5eeb132adcf2ab57aa11fdf4c0dd5169a004710e7d \
    --hash=sha256:98c7086708b163d425c67c7a91bad6e466bb99d797aa64f965e9d25c12111a5e \
    --hash=sha256:9add70b36c5666a2ed02b43b335fe19002ee5235efd4b8a89bfcf9005bebac0d \
    --hash=sha256:9bf40443012702a1d2070043cb6291650a0841ece432556f784f004937f0f32c \
    --hash=sha256:ade5e387d2ad0d7ebf59146cc00c8044acbd863725f887353a10df825fc8ae21 \
    --hash=sha256:b00c1de48212e4cc9603895652c5c410df699856a2853135b3967591e4beebc2 \
    --hash=sha256:b1282f8c00509d99fef04d8ba936b156d419be841854fe901d8ae224c59f0be5 \
    --hash=sha256:b2051432115498d3562c084a49bba65d97cf251f5a331c64a12ee7e04dacc51b \
    --hash=sha256:ba59edeaa2fc6114428f1637ffff42da1e311e29382d81b339c1817d37ec93c6 \
    --hash=sha256:c8716a48d94b06bb3b2524c2b77e055fb313aeb4ea620c8dd03a105574ba704f \
    --hash=sha256:cd5df75523866410809ca100dc9681e301e3c27567cf498077e8551b6d20e42f \
    --hash=sha256:e249096428b3ae81b08327a63a485ad0878de3fb939049038579ac0ef61e17e7 \
    # via jinja2
matplotlib==3.1.2 \
    --hash=sha256:08ccc8922eb4792b91c652d3e6d46b1c99073f1284d1b6705155643e8046463a \
    --hash=sha256:161dcd807c0c3232f4dcd4a12a382d52004a498174cbfafd40646106c5bcdcc8 \
    --hash=sha256:1f9e885bfa1b148d16f82a6672d043ecf11197f6c71ae222d0546db706e52eb2 \
    --hash=sha256:2d6ab54015a7c0d727c33e36f85f5c5e4172059efdd067f7527f6e5d16ad01aa \
    --hash=sha256:5d2e408a2813abf664bd79431107543ecb449136912eb55bb312317edecf597e \
    --hash=sha256:61c8b740a008218eb604de518eb411c4953db0cb725dd0b32adf8a81771cab9e \
    --hash=sha256:80f10af8378fccc136da40ea6aa4a920767476cdfb3241acb93ef4f0465dbf57 \
    --hash=sha256:819d4860315468b482f38f1afe45a5437f60f03eaede495d5ff89f2eeac89500 \
    --hash=sha256:8cc0e44905c2c8fda5637cad6f311eb9517017515a034247ab93d0cf99f8bb7a \
    --hash=sha256:8e8e2c2fe3d873108735c6ee9884e6f36f467df4a143136209cff303b183bada \
    --hash=sha256:98c2ffeab8b79a4e3a0af5dd9939f92980eb6e3fec10f7f313df5f35a84dacab \
    --hash=sha256:d59bb0e82002ac49f4152963f8a1079e66794a4f454457fd2f0dcc7bf0797d30 \
    --hash=sha256:ee59b7bb9eb75932fe3787e54e61c99b628155b0cedc907864f24723ba55b309
mistune==0.8.4 \
    --hash=sha256:59a3429db53c50b5c6bcc8a07f8848cb00d7dc8bdb431a4ab41920d201d4756e \
    --hash=sha256:88a1051873018da288eee8538d476dffe1262495144b33ecb586c4ab266bb8d4 \
    # via nbconvert
more-itertools==8.0.2 \
    --hash=sha256:b84b238cce0d9adad5ed87e745778d20a3f8487d0f0cb8b8a586816c7496458d \
    --hash=sha256:c833ef592a0324bcc6a60e48440da07645063c453880c9477ceb22490aec1564 \
    # via zipp
nbconvert==5.6.1 \
    --hash=sha256:21fb48e700b43e82ba0e3142421a659d7739b65568cc832a13976a77be16b523 \
    --hash=sha256:f0d6ec03875f96df45aa13e21fd9b8450c42d7e1830418cccc008c0df725fcee \
    # via jupyter, notebook
nbformat==4.4.0 \
    --hash=sha256:b9a0dbdbd45bb034f4f8893cafd6f652ea08c8c1674ba83f2dc55d3955743b0b \
    --hash=sha256:f7494ef0df60766b7cabe0a3651556345a963b74dbc16bc7c18479041170d402 \
    # via ipywidgets, nbconvert, notebook
notebook==6.0.2 \
    --hash=sha256:399a4411e171170173344761e7fd4491a3625659881f76ce47c50231ed714d9b \
    --hash=sha256:f67d76a68b1074a91693e95dea903ea01fd02be7c9fac5a4b870b8475caed805 \
    # via jupyter, widgetsnbextension
numpy==1.18.0 \
    --hash=sha256:03bbde29ac8fba860bb2c53a1525b3604a9b60417855ac3119d89868ec6041c3 \
    --hash=sha256:1baefd1fb4695e7f2e305467dbd876d765e6edd30c522894df76f8301efaee36 \
    --hash=sha256:1c35fb1131362e6090d30286cfda52ddd42e69d3e2bf1fea190a0fad83ea3a18 \
    --hash=sha256:3c68c827689ca0ca713dba598335073ce0966850ec0b30715527dce4ecd84055 \
    --hash=sha256:443ab93fc35b31f01db8704681eb2fd82f3a1b2fa08eed2dd0e71f1f57423d4a \
    --hash=sha256:56710a756c5009af9f35b91a22790701420406d9ac24cf6b652b0e22cfbbb7ff \
    --hash=sha256:62506e9e4d2a39c87984f081a2651d4282a1d706b1a82fe9d50a559bb58e705a \
    --hash=sha256:6f8113c8dbfc192b58996ee77333696469ea121d1c44ea429d8fd266e4c6be51 \
    --hash=sha256:712f0c32555132f4b641b918bdb1fd3c692909ae916a233ce7f50eac2de87e37 \
    --hash=sha256:854f6ed4fa91fa6da5d764558804ba5b0f43a51e5fe9fc4fdc93270b052f188a \
    --hash=sha256:88c5ccbc4cadf39f32193a5ef22e3f84674418a9fd877c63322917ae8f295a56 \
    --hash=sha256:905cd6fa6ac14654a6a32b21fad34670e97881d832e24a3ca32e19b455edb4a8 \
    --hash=sha256:9d6de2ad782aae68f7ed0e0e616477fbf693d6d7cc5f0f1505833ff12f84a673 \
    --hash=sha256:a30f5c3e1b1b5d16ec1f03f4df28e08b8a7529d8c920bbed657f4fde61f1fbcd \
    --hash=sha256:a9d72d9abaf65628f0f31bbb573b7d9304e43b1e6bbae43149c17737a42764c4 \
    --hash=sha256:ac3cf835c334fcc6b74dc4e630f9b5ff7b4c43f7fb2a7813208d95d4e10b5623 \
    --hash=sha256:b091e5d4cbbe79f0e8b6b6b522346e54a282eadb06e3fd761e9b6fafc2ca91ad \
    --hash=sha256:cc070fc43a494e42732d6ae2f6621db040611c1dde64762a40c8418023af56d7 \
    --hash=sha256:e1080e37c090534adb2dd7ae1c59ee883e5d8c3e63d2a4d43c20ee348d0459c5 \
    --hash=sha256:f084d513de729ff10cd72a1f80db468cff464fedb1ef2fea030221a0f62d7ff4 \
    --hash=sha256:f6a7421da632fc01e8a3ecd19c3f7350258d82501a646747664bae9c6a87c731
pandas==0.25.3 \
    --hash=sha256:00dff3a8e337f5ed7ad295d98a31821d3d0fe7792da82d78d7fd79b89c03ea9d \
    --hash=sha256:22361b1597c8c2ffd697aa9bf85423afa9e1fcfa6b1ea821054a244d5f24d75e \
    --hash=sha256:255920e63850dc512ce356233081098554d641ba99c3767dde9e9f35630f994b \
    --hash=sha256:26382aab9c119735908d94d2c5c08020a4a0a82969b7e5eefb92f902b3b30ad7 \
    --hash=sha256:33970f4cacdd9a0ddb8f21e151bfb9f178afb7c36eb7c25b9094c02876f385c2 \
    --hash=sha256:4545467a637e0e1393f7d05d61dace89689ad6d6f66f267f86fff737b702cce9 \
    --hash=sha256:52da74df8a9c9a103af0a72c9d5fdc8e0183a90884278db7f386b5692a2220a4 \
    --hash=sha256:61741f5aeb252f39c3031d11405305b6d10ce663c53bc3112705d7ad66c013d0 \
    --hash=sha256:6a3ac2c87e4e32a969921d1428525f09462770c349147aa8e9ab95f88c71ec71 \
    --hash=sha256:7458c48e3d15b8aaa7d575be60e1e4dd70348efcd9376656b72fecd55c59a4c3 \
    --hash=sha256:78bf638993219311377ce9836b3dc05f627a666d0dbc8cec37c0ff3c9ada673b \
    --hash=sha256:8153705d6545fd9eb6dd2bc79301bff08825d2e2f716d5dced48daafc2d0b81f \
    --hash=sha256:975c461accd14e89d71772e89108a050fa824c0b87a67d34cedf245f6681fc17 \
    --hash=sha256:9962957a27bfb70ab64103d0a7b42fa59c642fb4ed4cb75d0227b7bb9228535d \
    --hash=sha256:adc3d3a3f9e59a38d923e90e20c4922fc62d1e5a03d083440468c6d8f3f1ae0a \
    --hash=sha256:bbe3eb765a0b1e578833d243e2814b60c825b7fdbf4cdfe8e8aae8a08ed56ecf \
    --hash=sha256:df8864824b1fe488cf778c3650ee59c3a0d8f42e53707de167ba6b4f7d35f133 \
    --hash=sha256:e45055c30a608076e31a9fcd780a956ed3b1fa20db61561b8d88b79259f526f7 \
    --hash=sha256:ee50c2142cdcf41995655d499a157d0a812fce55c97d9aad13bc1eef837ed36c \
    # via seaborn
pandocfilters==1.4.2 \
    --hash=sha256:b3dd70e169bb5449e6bc6ff96aea89c5eea8c5f6ab5e207fc2f521a2cf4a0da9 \
    # via nbconvert
parso==0.5.2 \
    --hash=sha256:55cf25df1a35fd88b878715874d2c4dc1ad3f0eebd1e0266a67e1f55efccfbe1 \
    --hash=sha256:5c1f7791de6bd5dbbeac8db0ef5594b36799de198b3f7f7014643b0c5536b9d3 \
    # via jedi
pexpect==4.7.0 \
    --hash=sha256:2094eefdfcf37a1fdbfb9aa090862c1a4878e5c7e0e7e7088bdb511c558e5cd1 \
    --hash=sha256:9e2c1fd0e6ee3a49b28f95d4b33bc389c89b20af6a1255906e90ff1262ce62eb \
    # via ipython
pickleshare==0.7.5 \
    --hash=sha256:87683d47965c1da65cdacaf31c8441d12b8044cdec9aca500cd78fc2c683afca \
    --hash=sha256:9649af414d74d4df115d5d718f82acb59c9d418196b7b4290ed47a12ce62df56 \
    # via ipython
prometheus-client==0.7.1 \
    --hash=sha256:71cd24a2b3eb335cb800c7159f423df1bd4dcd5171b234be15e3f31ec9f622da \
    # via notebook
prompt-toolkit==2.0.10 \
    --hash=sha256:46642344ce457641f28fc9d1c9ca939b63dadf8df128b86f1b9860e59c73a5e4 \
    --hash=sha256:e7f8af9e3d70f514373bf41aa51bc33af12a6db3f71461ea47fea985defb2c31 \
    --hash=sha256:f15af68f66e664eaa559d4ac8a928111eebd5feda0c11738b5998045224829db \
    # via ipython, jupyter-console
ptyprocess==0.6.0 \
    --hash=sha256:923f299cc5ad920c68f2bc0bc98b75b9f838b93b599941a6b63ddbc2476394c0 \
    --hash=sha256:d7cc528d76e76342423ca640335bd3633420dc1366f258cb31d05e865ef5ca1f \
    # via pexpect, terminado
pygments==2.5.2 \
    --hash=sha256:2a3fe295e54a20164a9df49c75fa58526d3be48e14aceba6d6b1e8ac0bfd6f1b \
    --hash=sha256:98c8aa5a9f778fcd1026a17361ddaf7330d1b7c62ae97c3bb0ae73e0b9b6b0fe \
    # via ipython, jupyter-console, nbconvert, qtconsole
pyparsing==2.4.6 \
    --hash=sha256:4c830582a84fb022400b85429791bc551f1f4871c33f23e44f353119e92f969f \
    --hash=sha256:c342dccb5250c08d45fd6f8b4a559613ca603b57498511740e65cd11a2e7dcec \
    # via matplotlib
pyrsistent==0.15.6 \
    --hash=sha256:f3b280d030afb652f79d67c5586157c5c1355c9a58dfc7940566e28d28f3df1b \
    # via jsonschema
python-dateutil==2.8.1 \
    --hash=sha256:73ebfe9dbf22e832286dafa60473e4cd239f8592f699aa5adaf10050e6e1823c \
    --hash=sha256:75bb3f31ea686f1197762692a9ee6a7550b59fc6ca3a1f4b5d7e32fb98e2da2a \
    # via jupyter-client, matplotlib, pandas
pytz==2019.3 \
    --hash=sha256:1c557d7d0e871de1f5ccd5833f60fb2550652da6be2693c1e02300743d21500d \
    --hash=sha256:b02c06db6cf09c12dd25137e563b31700d3b80fcc4ad23abb7a315f2789819be \
    # via pandas
pyzmq==18.1.1 \
    --hash=sha256:01b588911714a6696283de3904f564c550c9e12e8b4995e173f1011755e01086 \
    --hash=sha256:0573b9790aa26faff33fba40f25763657271d26f64bffb55a957a3d4165d6098 \
    --hash=sha256:0fa82b9fc3334478be95a5566f35f23109f763d1669bb762e3871a8fa2a4a037 \
    --hash=sha256:1e59b7b19396f26e360f41411a5d4603356d18871049cd7790f1a7d18f65fb2c \
    --hash=sha256:2a294b4f44201bb21acc2c1a17ff87fbe57b82060b10ddb00ac03e57f3d7fcfa \
    --hash=sha256:355b38d7dd6f884b8ee9771f59036bcd178d98539680c4f87e7ceb2c6fd057b6 \
    --hash=sha256:4b73d20aec63933bbda7957e30add233289d86d92a0bb9feb3f4746376f33527 \
    --hash=sha256:4ec47f2b50bdb97df58f1697470e5c58c3c5109289a623e30baf293481ff0166 \
    --hash=sha256:5541dc8cad3a8486d58bbed076cb113b65b5dd6b91eb94fb3e38a3d1d3022f20 \
    --hash=sha256:6fca7d11310430e751f9832257866a122edf9d7b635305c5d8c51f74a5174d3d \
    --hash=sha256:7369656f89878455a5bcd5d56ca961884f5d096268f71c0750fc33d6732a25e5 \
    --hash=sha256:75d73ee7ca4b289a2a2dfe0e6bd8f854979fc13b3fe4ebc19381be3b04e37a4a \
    --hash=sha256:80c928d5adcfa12346b08d31360988d843b54b94154575cccd628f1fe91446bc \
    --hash=sha256:83ce18b133dc7e6789f64cb994e7376c5aa6b4aeced993048bf1d7f9a0fe6d3a \
    --hash=sha256:8b8498ceee33a7023deb2f3db907ca41d6940321e282297327a9be41e3983792 \
    --hash=sha256:8c69a6cbfa94da29a34f6b16193e7c15f5d3220cb772d6d17425ff3faa063a6d \
    --hash=sha256:8ff946b20d13a99dc5c21cb76f4b8b253eeddf3eceab4218df8825b0c65ab23d \
    --hash=sha256:972d723a36ab6a60b7806faa5c18aa3c080b7d046c407e816a1d8673989e2485 \
    --hash=sha256:a6c9c42bbdba3f9c73aedbb7671815af1943ae8073e532c2b66efb72f39f4165 \
    --hash=sha256:aa3872f2ebfc5f9692ef8957fe69abe92d905a029c0608e45ebfcd451ad30ab5 \
    --hash=sha256:cf08435b14684f7f2ca2df32c9df38a79cdc17c20dc461927789216cb43d8363 \
    --hash=sha256:d30db4566177a6205ed1badb8dbbac3c043e91b12a2db5ef9171b318c5641b75 \
    --hash=sha256:d5ac84f38575a601ab20c1878818ffe0d09eb51d6cb8511b636da46d0fd8949a \
    --hash=sha256:e37f22eb4bfbf69cd462c7000616e03b0cdc1b65f2d99334acad36ea0e4ddf6b \
    --hash=sha256:e6549dd80de7b23b637f586217a4280facd14ac01e9410a037a13854a6977299 \
    --hash=sha256:ed6205ca0de035f252baa0fd26fdd2bc8a8f633f92f89ca866fd423ff26c6f25 \
    --hash=sha256:efdde21febb9b5d7a8e0b87ea2549d7e00fda1936459cfb27fb6fca0c36af6c1 \
    --hash=sha256:f4e72646bfe79ff3adbf1314906bbd2d67ef9ccc71a3a98b8b2ccbcca0ab7bec \
    # via jupyter-client, notebook
qtconsole==4.6.0 \
    --hash=sha256:4de25b8895957d23ceacf2526b6f0a76da4e60e60115611930d387c853f3cb08 \
    --hash=sha256:654f423662e7dfe6a9b26fac8ec76aedcf742c339909ac49f1f0c1a1b744bcd1 \
    # via jupyter
scipy==1.4.1 \
    --hash=sha256:00af72998a46c25bdb5824d2b729e7dabec0c765f9deb0b504f928591f5ff9d4 \
    --hash=sha256:0902a620a381f101e184a958459b36d3ee50f5effd186db76e131cbefcbb96f7 \
    --hash=sha256:1e3190466d669d658233e8a583b854f6386dd62d655539b77b3fa25bfb2abb70 \
    --hash=sha256:2cce3f9847a1a51019e8c5b47620da93950e58ebc611f13e0d11f4980ca5fecb \
    --hash=sha256:3092857f36b690a321a662fe5496cb816a7f4eecd875e1d36793d92d3f884073 \
    --hash=sha256:386086e2972ed2db17cebf88610aab7d7f6e2c0ca30042dc9a89cf18dcc363fa \
    --hash=sha256:71eb180f22c49066f25d6df16f8709f215723317cc951d99e54dc88020ea57be \
    --hash=sha256:770254a280d741dd3436919d47e35712fb081a6ff8bafc0f319382b954b77802 \
    --hash=sha256:787cc50cab3020a865640aba3485e9fbd161d4d3b0d03a967df1a2881320512d \
    --hash=sha256:8a07760d5c7f3a92e440ad3aedcc98891e915ce857664282ae3c0220f3301eb6 \
    --hash=sha256:8d3bc3993b8e4be7eade6dcc6fd59a412d96d3a33fa42b0fa45dc9e24495ede9 \
    --hash=sha256:9508a7c628a165c2c835f2497837bf6ac80eb25291055f56c129df3c943cbaf8 \
    --hash=sha256:a144811318853a23d32a07bc7fd5561ff0cac5da643d96ed94a4ffe967d89672 \
    --hash=sha256:a1aae70d52d0b074d8121333bc807a485f9f1e6a69742010b33780df2e60cfe0 \
    --hash=sha256:a2d6df9eb074af7f08866598e4ef068a2b310d98f87dc23bd1b90ec7bdcec802 \
    --hash=sha256:bb517872058a1f087c4528e7429b4a44533a902644987e7b2fe35ecc223bc408 \
    --hash=sha256:c5cac0c0387272ee0e789e94a570ac51deb01c796b37fb2aad1fb13f85e2f97d \
    --hash=sha256:cc971a82ea1170e677443108703a2ec9ff0f70752258d0e9f5433d00dda01f59 \
    --hash=sha256:dba8306f6da99e37ea08c08fef6e274b5bf8567bb094d1dbe86a20e532aca088 \
    --hash=sha256:dc60bb302f48acf6da8ca4444cfa17d52c63c5415302a9ee77b3b21618090521 \
    --hash=sha256:dee1bbf3a6c8f73b6b218cb28eed8dd13347ea2f87d572ce19b289d6fd3fbc59 \
    # via seaborn
seaborn==0.9.0 \
    --hash=sha256:42e627b24e849c2d3bbfd059e00005f6afbc4a76e4895baf44ae23fe8a4b09a5 \
    --hash=sha256:76c83f794ca320fb6b23a7c6192d5e185a5fcf4758966a0c0a54baee46d41e2f
send2trash==1.5.0 \
    --hash=sha256:60001cc07d707fe247c94f74ca6ac0d3255aabcb930529690897ca2a39db28b2 \
    --hash=sha256:f1691922577b6fa12821234aeb57599d887c4900b9ca537948d2dac34aea888b \
    # via notebook
six==1.13.0 \
    --hash=sha256:1f1b7d42e254082a9db6279deae68afb421ceba6158efa6131de7b3003ee93fd \
    --hash=sha256:30f610279e8b2578cab6db20741130331735c781b56053c59c4076da27f06b66 \
    # via bleach, cycler, jsonschema, prompt-toolkit, pyrsistent, python-dateutil, traitlets
terminado==0.8.3 \
    --hash=sha256:4804a774f802306a7d9af7322193c5390f1da0abb429e082a10ef1d46e6fb2c2 \
    --hash=sha256:a43dcb3e353bc680dd0783b1d9c3fc28d529f190bc54ba9a229f72fe6e7a54d7 \
    # via notebook
testpath==0.4.4 \
    --hash=sha256:60e0a3261c149755f4399a1fff7d37523179a70fdc3abdf78de9fc2604aeec7e \
    --hash=sha256:bfcf9411ef4bf3db7579063e0546938b1edda3d69f4e1fb8756991f5951f85d4 \
    # via nbconvert
tornado==6.0.3 \
    --hash=sha256:349884248c36801afa19e342a77cc4458caca694b0eda633f5878e458a44cb2c \
    --hash=sha256:398e0d35e086ba38a0427c3b37f4337327231942e731edaa6e9fd1865bbd6f60 \
    --hash=sha256:4e73ef678b1a859f0cb29e1d895526a20ea64b5ffd510a2307b5998c7df24281 \
    --hash=sha256:559bce3d31484b665259f50cd94c5c28b961b09315ccd838f284687245f416e5 \
    --hash=sha256:abbe53a39734ef4aba061fca54e30c6b4639d3e1f59653f0da37a0003de148c7 \
    --hash=sha256:c845db36ba616912074c5b1ee897f8e0124df269468f25e4fe21fe72f6edd7a9 \
    --hash=sha256:c9399267c926a4e7c418baa5cbe91c7d1cf362d505a1ef898fde44a07c9dd8a5 \
    # via ipykernel, jupyter-client, notebook, terminado
traitlets==4.3.3 \
    --hash=sha256:70b4c6a1d9019d7b4f6846832288f86998aa3b9207c6821f3578a6a6a467fe44 \
    --hash=sha256:d023ee369ddd2763310e4c3eae1ff649689440d4ae59d7485eb4cfbbe3e359f7 \
    # via ipykernel, ipython, ipywidgets, jupyter-client, jupyter-core, nbconvert, nbformat, notebook, qtconsole
wcwidth==0.1.8 \
    --hash=sha256:8fd29383f539be45b20bd4df0dc29c20ba48654a41e661925e612311e9f3c603 \
    # via prompt-toolkit
webencodings==0.5.1 \
    --hash=sha256:a0af1213f3c2226497a97e2b3aa01a7e4bee4f403f95be16fc9acd2947514a78 \
    --hash=sha256:b36a1c245f2d304965eb4e0a82848379241dc04b865afcc4aab16748587e1923 \
    # via bleach
widgetsnbextension==3.5.1 \
    --hash=sha256:079f87d87270bce047512400efd70238820751a11d2d8cb137a5a5bdbaf255c7 \
    --hash=sha256:bd314f8ceb488571a5ffea6cc5b9fc6cba0adaf88a9d2386b93a489751938bcd \
    # via ipywidgets
zipp==0.6.0 \
    --hash=sha256:3718b1cbcd963c7d4c5511a8240812904164b7f381b647143a89d3b98f9bcd8e \
    --hash=sha256:f06903e9f1f43b12d371004b4ac7b06ab39a44adc747266928ae6debfa7b3335 \
    # via importlib-metadata

# WARNING: The following packages were not pinned, but pip requires them to be
# pinned when the requirements file includes hashes. Consider using the --allow-unsafe flag.
# setuptools

requirements.txt.in

matplotlib
numpy
jupyter
seaborn