1.3 MiB
1.3 MiB
Algorytm najszybszego spadku dla regresji wielomianowej.
Zakładamy, że dysponujemy zbiorem składającym się z dwóch cech (x i y). Modelujemy zależność y od x za pomocą funkcji wielomianowej. Celem projektu jest implementacja metody najszybszego spadku dla tego problemu. Zakładamy kwadratową funkcję straty. Implementacja powinna umożliwiać podanie stopnia wielomianu, który ma być użyty do modelowania. Implementacja powinna zwracać wektor oszacowanych parametrów oraz pokazywać wizualnie zmiany wartości funkcji straty wraz z postępem uczenia.
Inches for Height and Pound for Weight
Import i preprocessing danych
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
np.set_printoptions(suppress=True)
degree = 2
X_plot = np.linspace(0, 1, 1000)
initial_theta = np.matrix([0] * (degree + 1)).reshape(degree + 1, 1)
data = pd.read_csv('weight-height.csv')[["Height", "Weight"]]
data = data.dropna()
data_matrix = np.matrix(data)
m, n_plus_1 = data_matrix.shape
n = n_plus_1 - 1
X = (np.ones((m, 1)))
for i in range(1, degree + 1):
Xn = np.power(data_matrix[:, 0:n], i)
Xn /= np.amax(Xn, axis=0)
X = np.concatenate((X, Xn), axis=1)
X = np.matrix(X).reshape(m, degree * n + 1)
Y = np.matrix(data_matrix[:, -1])
Metody do regresji wielomianowej
def polynomial_regression(theta, x):
return sum(theta * np.power(x, i) for i, theta in enumerate(theta.tolist()))
def mean_squared_error(theta, X, Y):
J = 1.0 / (2.0 * m) * ((X * theta - Y).T * (X * theta - Y))
return J.item()
def gradient(theta, X, Y):
return 1.0 / len(Y) * (X.T * (X * theta - Y))
def gradient_descent(X, Y, theta, gradient = gradient, cost_function = mean_squared_error, alpha=0.1, eps=10**-5, max_steps = 10000000000):
cost = cost_function(theta, X, Y)
logs = [[cost, theta]]
for i in range(max_steps):
theta = theta - alpha * gradient(theta, X, Y)
next_cost = cost_function(theta, X, Y)
logs.append([next_cost, theta])
if abs(cost - next_cost) <= eps:
break
cost = next_cost
return theta, logs
#print(mean_squared_error([1,2,1,1],[1,2,43,1]))
#mean_squared_error(polynomial_regression(initial_theta, X), Y)
final_theta, logs = gradient_descent(X, Y, initial_theta)
print(logs)
[[13546.918787280789, matrix([[0], [0], [0]])], [8345.193627588793, matrix([[16.14403568], [13.70734956], [11.67579352]])], [5189.87054682073, matrix([[28.69527371], [24.38844647], [20.79400838]])], [3275.841980034606, matrix([[38.44834831], [32.7126086 ], [27.9202962 ]])], [2114.750558200912, matrix([[46.02214336], [39.20111989], [33.49520265]])], [1410.3713664195284, matrix([[51.89866338], [44.25997121], [37.86183734]])], [983.0220536345228, matrix([[56.45330726], [48.20535989], [41.28742387]])], [723.7119655620031, matrix([[59.97844643], [51.28354945], [43.98008363]])], [566.3301978366925, matrix([[62.70178813], [53.68633698], [46.10190754]])], [470.7755903354751, matrix([[64.80067727], [55.56309958], [47.77913665]])], [412.72381924673124, matrix([[66.41323513], [57.03017637], [49.11009111]])], [377.4203805034337, matrix([[67.64703455], [58.17817563], [50.17134547]])], [355.9155745086213, matrix([[68.5858563 ], [59.07766566], [51.02253821]])], [342.78074551310306, matrix([[69.29495126], [59.78360709], [51.71011747]])], [334.72301164982593, matrix([[69.82513867], [60.33880483], [52.2702584 ]])], [329.74499609880246, matrix([[70.21599802], [60.77659656], [52.73113527]])], [326.63514180358277, matrix([[70.49835491], [61.12294665], [53.11469099]])], [324.6585495966253, matrix([[70.69621725], [61.39807686], [53.43801534]])], [323.36944862405903, matrix([[70.82828303], [61.6177365 ], [53.71441834]])], [322.4974457688022, matrix([[70.90911481], [61.79419159], [53.95426622]])], [321.87852750067395, matrix([[70.95005431], [61.93699524], [54.16563249]])], [321.41320738809407, matrix([[70.95993479], [62.0535877 ], [54.35480504]])], [321.0411394640971, matrix([[70.94563583], [62.14976365], [54.52668109]])], [320.7257193525429, matrix([[70.91251545], [62.23003615], [54.6850747 ]])], [320.4447437106345, matrix([[70.86474662], [62.29792009], [54.8329564 ]])], [320.1847444889924, matrix([[70.80557936], [62.356153 ], [54.97263973]])], [319.9375522407513, matrix([[70.73754489], [62.40686692], [55.10592653]])], [319.69821151398736, matrix([[70.66261456], [62.45172236], [55.23422015]])], [319.46371638282716, matrix([[70.58232375], [62.49201255], [55.35861355]])], [319.23224345341146, matrix([[70.49786822], [62.52874474], [55.47995796]])], [319.00268662520443, matrix([[70.41017927], [62.56270336], [55.59891637]])], [318.7743748892694, matrix([[70.31998223], [62.59449937], [55.71600511]])], [318.5469011520213, matrix([[70.22784197], [62.62460861], [55.83162635]])], [318.32001840415444, matrix([[70.13419846], [62.65340167], [55.9460933 ]])], [318.09357673940707, matrix([[70.03939434], [62.68116717], [56.0596499 ]])], [317.8674851517765, matrix([[69.94369646], [62.70812991], [56.17248614]])], [317.64168836270113, matrix([[69.84731264], [62.73446497], [56.28475001]])], [317.4161527650302, matrix([[69.75040475], [62.7603087 ], [56.39655676]])], [317.1908578969991, matrix([[69.65309885], [62.78576731], [56.50799616]])], [316.96579127056555, matrix([[69.55549309], [62.81092349], [56.61913814]])], [316.7409452344107, matrix([[69.45766395], [62.83584168], [56.73003715]])], [316.51631507111836, matrix([[69.35967094], [62.86057204], [56.84073562]])], [316.291897842972, matrix([[69.26156045], [62.88515366], [56.95126658]])], [316.0676916918459, matrix([[69.16336858], [62.90961697], [57.06165579]])], [315.84369541453685, matrix([[69.06512345], [62.93398567], [57.1719233 ]])], [315.6199082051712, matrix([[68.96684694], [62.95827823], [57.28208472]])], [315.39632949895605, matrix([[68.8685561 ], [62.98250903], [57.39215224]])], [315.17295887740215, matrix([[68.77026421], [63.00668927], [57.50213533]])], [314.94979601083503, matrix([[68.6719816 ], [63.03082767], [57.61204139]])], [314.72684062352306, matrix([[68.57371631], [63.05493104], [57.72187618]])], [314.5040924725258, matrix([[68.47547459], [63.07900466], [57.83164421]])], [314.28155133486314, matrix([[68.37726132], [63.10305267], [57.94134896]])], [314.0592169997324, matrix([[68.27908029], [63.12707827], [58.05099319]])], [313.83708926378836, matrix([[68.18093443], [63.15108397], [58.16057901]])], [313.6151679282785, matrix([[68.08282604], [63.17507172], [58.27010811]])], [313.3934527973068, matrix([[67.9847569 ], [63.19904304], [58.37958179]])], [313.17194367678, matrix([[67.88672839], [63.22299912], [58.48900106]])], [312.95064037376767, matrix([[67.78874158], [63.24694088], [58.59836673]])], [312.7295426961153, matrix([[67.6907973 ], [63.27086904], [58.70767943]])], [312.5086504522082, matrix([[67.59289618], [63.29478417], [58.81693965]])], [312.28796345082947, matrix([[67.49503872], [63.3186867 ], [58.92614778]])], [312.0674815010726, matrix([[67.39722531], [63.34257699], [59.03530414]])], [311.84720441228944, matrix([[67.29945623], [63.36645528], [59.14440896]])], [311.6271319940578, matrix([[67.20173171], [63.39032181], [59.25346245]])], [311.40726405616215, matrix([[67.10405192], [63.41417672], [59.36246476]])], [311.18760040858166, 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matrix([[65.74126445], [63.74696772], [60.88316903]])], [308.1336538449585, matrix([[65.64426074], [63.77065564], [60.99141202]])], [307.9170318720284, matrix([[65.54730207], [63.79433258], [61.09960467]])], [307.7006111733808, matrix([[65.45038844], [63.81799856], [61.20774702]])], [307.48439156200163, matrix([[65.35351981], [63.84165358], [61.31583909]])], [307.2683728510506, matrix([[65.25669617], [63.86529765], [61.4238809 ]])], [307.05255485386084, matrix([[65.1599175 ], [63.88893077], [61.53187247]])], [306.83693738393913, matrix([[65.06318377], [63.91255295], [61.63981383]])], [306.6215202549654, matrix([[64.96649498], [63.9361642 ], [61.747705 ]])], [306.4063032807928, matrix([[64.8698511 ], [63.95976451], [61.85554602]])], [306.19128627544745, matrix([[64.7732521 ], [63.9833539 ], [61.96333689]])], [305.9764690531282, matrix([[64.67669798], [64.00693238], [62.07107765]])], [305.76185142820657, matrix([[64.5801887 ], [64.03049993], [62.17876831]])], [305.54743321522653, 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Metoda gradientu prostego
Metody do wykresów
def plot_polynomial_regression(theta):
fig = plt.figure(figsize=(10,5))
Y_plot = polynomial_regression(theta, X_plot).tolist()
chart = fig.add_subplot()
chart.plot(X[:,1], Y ,"go")
chart.plot(X_plot, Y_plot, color="red", lw=2, label=f"degree {len(theta)}")
plt.show()
#plot_polynomial_regression(initial_theta)
plot_polynomial_regression(final_theta)
Wyniki za pomocą gotowej biblioteki
from sklearn.preprocessing import PolynomialFeatures, StandardScaler
from sklearn.pipeline import make_pipeline
from sklearn.linear_model import Ridge, LinearRegression
model = make_pipeline(PolynomialFeatures(degree=degree, include_bias=True),
LinearRegression())
model.fit(data[["Height"]],Y)
Pipeline(steps=[('polynomialfeatures', PolynomialFeatures()), ('linearregression', LinearRegression())])
X_plot_2 = np.linspace(-50, 100, 1000)
Y_plot_2 = model.predict([[x] for x in X_plot_2])
fig = plt.figure(figsize=(10,5))
chart = fig.add_subplot()
chart.plot(data["Height"], Y ,"go")
chart.plot(X_plot_2, Y_plot_2, color="red", lw=2, label=f"degree {degree}")
degree
2