Sequences, Time Series and Prediction in Tessorflow quizs on Coursera (一）

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第一周知识点总结：
（1）几种常见的时间序列
–趋势序列（trend）：朝一个特定方向变化

import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
from tensorflow import keras

def plot_series(time, series, format="-", start=0, end=None, label=None):
    plt.plot(time[start:end], series[start:end], format, label=label)
    plt.xlabel("Time")
    plt.ylabel("Value")
    if label:
        plt.legend(fontsize=14)
    plt.grid(True)

def trend(time, slope=0):
    return slope * time
time = np.arange(4 * 365 + 1)
baseline = 10
series = trend(time, 0.1)

plt.figure(figsize=(10, 6))
plot_series(time, series)
plt.show()

–季节序列：周期性变化

相关代码：

def seasonal_pattern(season_time):
    """Just an arbitrary pattern, you can change it if you wish"""
    return np.where(season_time < 0.4,
                    np.cos(season_time * 2 * np.pi),
                    1 / np.exp(3 * season_time))

def seasonality(time, period, amplitude=1, phase=0):
    """Repeats the same pattern at each period"""
    season_time = ((time + phase) % period) / period
    return amplitude * seasonal_pattern(season_time)
baseline = 10
amplitude = 40
series = seasonality(time, period=365, amplitude=amplitude)

plt.figure(figsize=(10, 6))
plot_series(time, series)
plt.show()

slope = 0.05
series = baseline + trend(time, slope) + seasonality(time, period=365, amplitude=amplitude)

plt.figure(figsize=(10, 6))
plot_series(time, series)
plt.show()

–噪声序列：无法预测的白噪声随机值

def white_noise(time, noise_level=1, seed=None):
    rnd = np.random.RandomState(seed)
    return rnd.randn(len(time)) * noise_level
noise_level = 5
noise = white_noise(time, noise_level, seed=42)

plt.figure(figsize=(10, 6))
plot_series(time, noise)
plt.show()

往时间序列里加噪声

series += noise

plt.figure(figsize=(10, 6))
plot_series(time, series)
plt.show()

–Autocorrelation: auto correlated time series. Namely it correlates with a delayed copy of itself often called a lag.

–现实生活中的时间序列一般被认为是上面这四种时间序列的组合，即“Trend + Seasonality + Autocorrelation + Noise”

下面的代码就是讲上面几种时间序列进行组合

split_time = 1000
time_train = time[:split_time]
x_train = series[:split_time]
time_valid = time[split_time:]
x_valid = series[split_time:]

def autocorrelation(time, amplitude, seed=None):
    rnd = np.random.RandomState(seed)
    φ1 = 0.5
    φ2 = -0.1
    ar = rnd.randn(len(time) + 50)
    ar[:50] = 100
    for step in range(50, len(time) + 50):
        ar[step] += φ1 * ar[step - 50]
        ar[step] += φ2 * ar[step - 33]
    return ar[50:] * amplitude
def autocorrelation(time, amplitude, seed=None):
    rnd = np.random.RandomState(seed)
    φ = 0.8
    ar = rnd.randn(len(time) + 1)
    for step in range(1, len(time) + 1):
        ar[step] += φ * ar[step - 1]
    return ar[1:] * amplitude
series = autocorrelation(time, 10, seed=42)
plot_series(time[:200], series[:200])
plt.show()

series = autocorrelation(time, 10, seed=42) + trend(time, 2)
plot_series(time[:200], series[:200])
plt.show()

series = autocorrelation(time, 10, seed=42) + seasonality(time, period=50, amplitude=150) + trend(time, 2)
plot_series(time[:200], series[:200])
plt.show()

series = autocorrelation(time, 10, seed=42) + seasonality(time, period=50, amplitude=150) + trend(time, 2)
series2 = autocorrelation(time, 5, seed=42) + seasonality(time, period=50, amplitude=2) + trend(time, -1) + 550
series[200:] = series2[200:]
#series += noise(time, 30)
plot_series(time[:300], series[:300])
plt.show()

def impulses(time, num_impulses, amplitude=1, seed=None):
    rnd = np.random.RandomState(seed)
    impulse_indices = rnd.randint(len(time), size=10)
    series = np.zeros(len(time))
    for index in impulse_indices:
        series[index] += rnd.rand() * amplitude
    return series 

series = impulses(time, 10, seed=42)
plot_series(time, series)
plt.show()

def autocorrelation(source, φs):
    ar = source.copy()
    max_lag = len(φs)
    for step, value in enumerate(source):
        for lag, φ in φs.items():
            if step - lag > 0:
              ar[step] += φ * ar[step - lag]
    return ar
signal = impulses(time, 10, seed=42)
series = autocorrelation(signal, {
    
    1: 0.99})
plot_series(time, series)
plt.plot(time, signal, "k-")
plt.show()

signal = impulses(time, 10, seed=42)
series = autocorrelation(signal, {
    
    1: 0.70, 50: 0.2})
plot_series(time, series)
plt.plot(time, signal, "k-")
plt.show()

series_diff1 = series[1:] - series[:-1]
plot_series(time[1:], series_diff1)

—Non-Stationary Time Series:发生某个事件后，序列与过去的时间发生很大变化的时间序列，像股票走势。

对于这样的序列，我们一般只需要后面的时间数据进行训练，以后前面的数据对我们后续的预测没有多大作用

–天真预测（naive forecast）：取序列的最后一个值，并且假设后面的值与它相同，我们一般用天真预测值作为时间序列模型的基线（baseline）指标。

时间序列模型中的一些性能指标：mse和mae
第一周测验：第二题选D