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如何利用python進(jìn)行時(shí)間序列分析

 更新時(shí)間:2020年08月04日 10:43:54   作者:大熊貓?zhí)陨? 
這篇文章主要介紹了如何利用python進(jìn)行時(shí)間序列分析,文中通過(guò)示例代碼介紹的非常詳細(xì),對(duì)大家的學(xué)習(xí)或者工作具有一定的參考學(xué)習(xí)價(jià)值,需要的朋友們下面隨著小編來(lái)一起學(xué)習(xí)學(xué)習(xí)吧

題記:畢業(yè)一年多天天coding,好久沒(méi)寫paper了。在這動(dòng)蕩的日子里,也希望寫點(diǎn)東西讓自己靜一靜。恰好前段時(shí)間用python做了一點(diǎn)時(shí)間序列方面的東西,有一丁點(diǎn)心得體會(huì)想和大家分享下。在此也要特別感謝顧志耐和散沙,讓我喜歡上了python。

什么是時(shí)間序列

時(shí)間序列簡(jiǎn)單的說(shuō)就是各時(shí)間點(diǎn)上形成的數(shù)值序列,時(shí)間序列分析就是通過(guò)觀察歷史數(shù)據(jù)預(yù)測(cè)未來(lái)的值。在這里需要強(qiáng)調(diào)一點(diǎn)的是,時(shí)間序列分析并不是關(guān)于時(shí)間的回歸,它主要是研究自身的變化規(guī)律的(這里不考慮含外生變量的時(shí)間序列)。

為什么用python

  用兩個(gè)字總結(jié)“情懷”,愛(ài)屋及烏,個(gè)人比較喜歡python,就用python擼了。能做時(shí)間序列的軟件很多,SAS、R、SPSS、Eviews甚至matlab等等,實(shí)際工作中應(yīng)用得比較多的應(yīng)該還是SAS和R,前者推薦王燕寫的《應(yīng)用時(shí)間序列分析》,后者推薦“基于R語(yǔ)言的時(shí)間序列建模完整教程”這篇博文(翻譯版)。python作為科學(xué)計(jì)算的利器,當(dāng)然也有相關(guān)分析的包:statsmodels中tsa模塊,當(dāng)然這個(gè)包和SAS、R是比不了,但是python有另一個(gè)神器:pandas!pandas在時(shí)間序列上的應(yīng)用,能簡(jiǎn)化我們很多的工作。

環(huán)境配置

  python推薦直接裝Anaconda,它集成了許多科學(xué)計(jì)算包,有一些包自己手動(dòng)去裝還是挺費(fèi)勁的。statsmodels需要自己去安裝,這里我推薦使用0.6的穩(wěn)定版,0.7及其以上的版本能在github上找到,該版本在安裝時(shí)會(huì)用C編譯好,所以修改底層的一些代碼將不會(huì)起作用。

時(shí)間序列分析

1.基本模型

  自回歸移動(dòng)平均模型(ARMA(p,q))是時(shí)間序列中最為重要的模型之一,它主要由兩部分組成: AR代表p階自回歸過(guò)程,MA代表q階移動(dòng)平均過(guò)程,其公式如下:

   依據(jù)模型的形式、特性及自相關(guān)和偏自相關(guān)函數(shù)的特征,總結(jié)如下:

在時(shí)間序列中,ARIMA模型是在ARMA模型的基礎(chǔ)上多了差分的操作。

2.pandas時(shí)間序列操作

大熊貓真的很可愛(ài),這里簡(jiǎn)單介紹一下它在時(shí)間序列上的可愛(ài)之處。和許多時(shí)間序列分析一樣,本文同樣使用航空乘客數(shù)據(jù)(AirPassengers.csv)作為樣例。

數(shù)據(jù)讀?。?/p>

# -*- coding:utf-8 -*-
import numpy as np
import pandas as pdfrom datetime import datetimeimport matplotlib.pylab as plt
# 讀取數(shù)據(jù),pd.read_csv默認(rèn)生成DataFrame對(duì)象,需將其轉(zhuǎn)換成Series對(duì)象df = pd.read_csv('AirPassengers.csv', encoding='utf-8', index_col='date')df.index = pd.to_datetime(df.index) # 將字符串索引轉(zhuǎn)換成時(shí)間索引ts = df['x'] # 生成pd.Series對(duì)象# 查看數(shù)據(jù)格式ts.head()ts.head().index

查看某日的值既可以使用字符串作為索引,又可以直接使用時(shí)間對(duì)象作為索引

復(fù)制代碼 代碼如下:
ts['1949-01-01']ts[datetime(1949,1,1)]

兩者的返回值都是第一個(gè)序列值:112

如果要查看某一年的數(shù)據(jù),pandas也能非常方便的實(shí)現(xiàn)

ts['1949']

切片操作:

ts['1949-1' : '1949-6']

注意時(shí)間索引的切片操作起點(diǎn)和尾部都是包含的,這點(diǎn)與數(shù)值索引有所不同

pandas還有很多方便的時(shí)間序列函數(shù),在后面的實(shí)際應(yīng)用中在進(jìn)行說(shuō)明。

3. 平穩(wěn)性檢驗(yàn)

我們知道序列平穩(wěn)性是進(jìn)行時(shí)間序列分析的前提條件,很多人都會(huì)有疑問(wèn),為什么要滿足平穩(wěn)性的要求呢?在大數(shù)定理和中心定理中要求樣本同分布(這里同分布等價(jià)于時(shí)間序列中的平穩(wěn)性),而我們的建模過(guò)程中有很多都是建立在大數(shù)定理和中心極限定理的前提條件下的,如果它不滿足,得到的許多結(jié)論都是不可靠的。以虛假回歸為例,當(dāng)響應(yīng)變量和輸入變量都平穩(wěn)時(shí),我們用t統(tǒng)計(jì)量檢驗(yàn)標(biāo)準(zhǔn)化系數(shù)的顯著性。而當(dāng)響應(yīng)變量和輸入變量不平穩(wěn)時(shí),其標(biāo)準(zhǔn)化系數(shù)不在滿足t分布,這時(shí)再用t檢驗(yàn)來(lái)進(jìn)行顯著性分析,導(dǎo)致拒絕原假設(shè)的概率增加,即容易犯第一類錯(cuò)誤,從而得出錯(cuò)誤的結(jié)論。

平穩(wěn)時(shí)間序列有兩種定義:嚴(yán)平穩(wěn)和寬平穩(wěn)

嚴(yán)平穩(wěn)顧名思義,是一種條件非??量痰钠椒€(wěn)性,它要求序列隨著時(shí)間的推移,其統(tǒng)計(jì)性質(zhì)保持不變。對(duì)于任意的τ,其聯(lián)合概率密度函數(shù)滿足:

嚴(yán)平穩(wěn)的條件只是理論上的存在,現(xiàn)實(shí)中用得比較多的是寬平穩(wěn)的條件。

寬平穩(wěn)也叫弱平穩(wěn)或者二階平穩(wěn)(均值和方差平穩(wěn)),它應(yīng)滿足:

  • 常數(shù)均值
  • 常數(shù)方差
  • 常數(shù)自協(xié)方差

平穩(wěn)性檢驗(yàn):觀察法和單位根檢驗(yàn)法

基于此,我寫了一個(gè)名為test_stationarity的統(tǒng)計(jì)性檢驗(yàn)?zāi)K,以便將某些統(tǒng)計(jì)檢驗(yàn)結(jié)果更加直觀的展現(xiàn)出來(lái)。

# -*- coding:utf-8 -*-
from statsmodels.tsa.stattools import adfuller
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
# 移動(dòng)平均圖
def draw_trend(timeSeries, size):
 f = plt.figure(facecolor='white')
 # 對(duì)size個(gè)數(shù)據(jù)進(jìn)行移動(dòng)平均
 rol_mean = timeSeries.rolling(window=size).mean()
 # 對(duì)size個(gè)數(shù)據(jù)進(jìn)行加權(quán)移動(dòng)平均
 rol_weighted_mean = pd.ewma(timeSeries, span=size)

 timeSeries.plot(color='blue', label='Original')
 rolmean.plot(color='red', label='Rolling Mean')
 rol_weighted_mean.plot(color='black', label='Weighted Rolling Mean')
 plt.legend(loc='best')
 plt.title('Rolling Mean')
 plt.show()

def draw_ts(timeSeries): f = plt.figure(facecolor='white')
 timeSeries.plot(color='blue')
 plt.show()

'''  Unit Root Test
 The null hypothesis of the Augmented Dickey-Fuller is that there is a unit
 root, with the alternative that there is no unit root. That is to say the
 bigger the p-value the more reason we assert that there is a unit root
'''
def testStationarity(ts):
 dftest = adfuller(ts)
 # 對(duì)上述函數(shù)求得的值進(jìn)行語(yǔ)義描述
 dfoutput = pd.Series(dftest[0:4], index=['Test Statistic','p-value','#Lags Used','Number of Observations Used'])
 for key,value in dftest[4].items():
  dfoutput['Critical Value (%s)'%key] = value
 return dfoutput

# 自相關(guān)和偏相關(guān)圖,默認(rèn)階數(shù)為31階
def draw_acf_pacf(ts, lags=31):
 f = plt.figure(facecolor='white')
 ax1 = f.add_subplot(211)
 plot_acf(ts, lags=31, ax=ax1)
 ax2 = f.add_subplot(212)
 plot_pacf(ts, lags=31, ax=ax2)
 plt.show()

觀察法,通俗的說(shuō)就是通過(guò)觀察序列的趨勢(shì)圖與相關(guān)圖是否隨著時(shí)間的變化呈現(xiàn)出某種規(guī)律。所謂的規(guī)律就是時(shí)間序列經(jīng)常提到的周期性因素,現(xiàn)實(shí)中遇到得比較多的是線性周期成分,這類周期成分可以采用差分或者移動(dòng)平均來(lái)解決,而對(duì)于非線性周期成分的處理相對(duì)比較復(fù)雜,需要采用某些分解的方法。下圖為航空數(shù)據(jù)的線性圖,可以明顯的看出它具有年周期成分和長(zhǎng)期趨勢(shì)成分。平穩(wěn)序列的自相關(guān)系數(shù)會(huì)快速衰減,下面的自相關(guān)圖并不能體現(xiàn)出該特征,所以我們有理由相信該序列是不平穩(wěn)的。

單位根檢驗(yàn):ADF是一種常用的單位根檢驗(yàn)方法,他的原假設(shè)為序列具有單位根,即非平穩(wěn),對(duì)于一個(gè)平穩(wěn)的時(shí)序數(shù)據(jù),就需要在給定的置信水平上顯著,拒絕原假設(shè)。ADF只是單位根檢驗(yàn)的方法之一,如果想采用其他檢驗(yàn)方法,可以安裝第三方包arch,里面提供了更加全面的單位根檢驗(yàn)方法,個(gè)人還是比較鐘情ADF檢驗(yàn)。以下為檢驗(yàn)結(jié)果,其p值大于0.99,說(shuō)明并不能拒絕原假設(shè)。

3. 平穩(wěn)性處理

由前面的分析可知,該序列是不平穩(wěn)的,然而平穩(wěn)性是時(shí)間序列分析的前提條件,故我們需要對(duì)不平穩(wěn)的序列進(jìn)行處理將其轉(zhuǎn)換成平穩(wěn)的序列。

a. 對(duì)數(shù)變換

對(duì)數(shù)變換主要是為了減小數(shù)據(jù)的振動(dòng)幅度,使其線性規(guī)律更加明顯(我是這么理解的時(shí)間序列模型大部分都是線性的,為了盡量降低非線性的因素,需要對(duì)其進(jìn)行預(yù)處理,也許我理解的不對(duì))。對(duì)數(shù)變換相當(dāng)于增加了一個(gè)懲罰機(jī)制,數(shù)據(jù)越大其懲罰越大,數(shù)據(jù)越小懲罰越小。這里強(qiáng)調(diào)一下,變換的序列需要滿足大于0,小于0的數(shù)據(jù)不存在對(duì)數(shù)變換。

ts_log = np.log(ts)
test_stationarity.draw_ts(ts_log)

b. 平滑法

根據(jù)平滑技術(shù)的不同,平滑法具體分為移動(dòng)平均法和指數(shù)平均法。

移動(dòng)平均即利用一定時(shí)間間隔內(nèi)的平均值作為某一期的估計(jì)值,而指數(shù)平均則是用變權(quán)的方法來(lái)計(jì)算均值

test_stationarity.draw_trend(ts_log, 12)

從上圖可以發(fā)現(xiàn)窗口為12的移動(dòng)平均能較好的剔除年周期性因素,而指數(shù)平均法是對(duì)周期內(nèi)的數(shù)據(jù)進(jìn)行了加權(quán),能在一定程度上減小年周期因素,但并不能完全剔除,如要完全剔除可以進(jìn)一步進(jìn)行差分操作。

c. 差分

時(shí)間序列最常用來(lái)剔除周期性因素的方法當(dāng)屬差分了,它主要是對(duì)等周期間隔的數(shù)據(jù)進(jìn)行線性求減。前面我們說(shuō)過(guò),ARIMA模型相對(duì)ARMA模型,僅多了差分操作,ARIMA模型幾乎是所有時(shí)間序列軟件都支持的,差分的實(shí)現(xiàn)與還原都非常方便。而statsmodel中,對(duì)差分的支持不是很好,它不支持高階和多階差分,為什么不支持,這里引用作者的說(shuō)法:

作者大概的意思是說(shuō)預(yù)測(cè)方法中并沒(méi)有解決高于2階的差分,有沒(méi)有感覺(jué)很牽強(qiáng),不過(guò)沒(méi)關(guān)系,我們有pandas。我們可以先用pandas將序列差分好,然后在對(duì)差分好的序列進(jìn)行ARIMA擬合,只不過(guò)這樣后面會(huì)多了一步人工還原的工作。

diff_12 = ts_log.diff(12)
diff_12.dropna(inplace=True)
diff_12_1 = diff_12.diff(1)
diff_12_1.dropna(inplace=True)
test_stationarity.testStationarity(diff_12_1)

從上面的統(tǒng)計(jì)檢驗(yàn)結(jié)果可以看出,經(jīng)過(guò)12階差分和1階差分后,該序列滿足平穩(wěn)性的要求了。

d. 分解

所謂分解就是將時(shí)序數(shù)據(jù)分離成不同的成分。statsmodels使用的X-11分解過(guò)程,它主要將時(shí)序數(shù)據(jù)分離成長(zhǎng)期趨勢(shì)、季節(jié)趨勢(shì)和隨機(jī)成分。與其它統(tǒng)計(jì)軟件一樣,statsmodels也支持兩類分解模型,加法模型和乘法模型,這里我只實(shí)現(xiàn)加法,乘法只需將model的參數(shù)設(shè)置為"multiplicative"即可。

from statsmodels.tsa.seasonal import seasonal_decompose
decomposition = seasonal_decompose(ts_log, model="additive")

trend = decomposition.trend
seasonal = decomposition.seasonal
residual = decomposition.resid

得到不同的分解成分后,就可以使用時(shí)間序列模型對(duì)各個(gè)成分進(jìn)行擬合,當(dāng)然也可以選擇其他預(yù)測(cè)方法。我曾經(jīng)用過(guò)小波對(duì)時(shí)序數(shù)據(jù)進(jìn)行過(guò)分解,然后分別采用時(shí)間序列擬合,效果還不錯(cuò)。由于我對(duì)小波的理解不是很好,只能簡(jiǎn)單的調(diào)用接口,如果有誰(shuí)對(duì)小波、傅里葉、卡爾曼理解得比較透,可以將時(shí)序數(shù)據(jù)進(jìn)行更加準(zhǔn)確的分解,由于分解后的時(shí)序數(shù)據(jù)避免了他們?cè)诮r(shí)的交叉影響,所以我相信它將有助于預(yù)測(cè)準(zhǔn)確性的提高。

4. 模型識(shí)別

在前面的分析可知,該序列具有明顯的年周期與長(zhǎng)期成分。對(duì)于年周期成分我們使用窗口為12的移動(dòng)平進(jìn)行處理,對(duì)于長(zhǎng)期趨勢(shì)成分我們采用1階差分來(lái)進(jìn)行處理。

rol_mean = ts_log.rolling(window=12).mean()
rol_mean.dropna(inplace=True)
ts_diff_1 = rol_mean.diff(1)
ts_diff_1.dropna(inplace=True)
test_stationarity.testStationarity(ts_diff_1)

觀察其統(tǒng)計(jì)量發(fā)現(xiàn)該序列在置信水平為95%的區(qū)間下并不顯著,我們對(duì)其進(jìn)行再次一階差分。再次差分后的序列其自相關(guān)具有快速衰減的特點(diǎn),t統(tǒng)計(jì)量在99%的置信水平下是顯著的,這里我不再做詳細(xì)說(shuō)明。

ts_diff_2 = ts_diff_1.diff(1)
ts_diff_2.dropna(inplace=True)

數(shù)據(jù)平穩(wěn)后,需要對(duì)模型定階,即確定p、q的階數(shù)。觀察上圖,發(fā)現(xiàn)自相關(guān)和偏相系數(shù)都存在拖尾的特點(diǎn),并且他們都具有明顯的一階相關(guān)性,所以我們?cè)O(shè)定p=1, q=1。下面就可以使用ARMA模型進(jìn)行數(shù)據(jù)擬合了。這里我不使用ARIMA(ts_diff_1, order=(1, 1, 1))進(jìn)行擬合,是因?yàn)楹胁罘植僮鲿r(shí),預(yù)測(cè)結(jié)果還原老出問(wèn)題,至今還沒(méi)弄明白。

from statsmodels.tsa.arima_model import ARMA
model = ARMA(ts_diff_2, order=(1, 1)) 
result_arma = model.fit( disp=-1, method='css')

5. 樣本擬合

模型擬合完后,我們就可以對(duì)其進(jìn)行預(yù)測(cè)了。由于ARMA擬合的是經(jīng)過(guò)相關(guān)預(yù)處理后的數(shù)據(jù),故其預(yù)測(cè)值需要通過(guò)相關(guān)逆變換進(jìn)行還原。

predict_ts = result_arma.predict()
# 一階差分還原diff_shift_ts = ts_diff_1.shift(1)diff_recover_1 = predict_ts.add(diff_shift_ts)# 再次一階差分還原
rol_shift_ts = rol_mean.shift(1)
diff_recover = diff_recover_1.add(rol_shift_ts)
# 移動(dòng)平均還原
rol_sum = ts_log.rolling(window=11).sum()
rol_recover = diff_recover*12 - rol_sum.shift(1)
# 對(duì)數(shù)還原
log_recover = np.exp(rol_recover)
log_recover.dropna(inplace=True)

我們使用均方根誤差(RMSE)來(lái)評(píng)估模型樣本內(nèi)擬合的好壞。利用該準(zhǔn)則進(jìn)行判別時(shí),需要剔除“非預(yù)測(cè)”數(shù)據(jù)的影響。

ts = ts[log_recover.index] # 過(guò)濾沒(méi)有預(yù)測(cè)的記錄plt.figure(facecolor='white')
log_recover.plot(color='blue', label='Predict')
ts.plot(color='red', label='Original')
plt.legend(loc='best')
plt.title('RMSE: %.4f'% np.sqrt(sum((log_recover-ts)**2)/ts.size))
plt.show()

觀察上圖的擬合效果,均方根誤差為11.8828,感覺(jué)還過(guò)得去。

6.完善ARIMA模型

前面提到statsmodels里面的ARIMA模塊不支持高階差分,我們的做法是將差分分離出來(lái),但是這樣會(huì)多了一步人工還原的操作?;谏鲜鰡?wèn)題,我將差分過(guò)程進(jìn)行了封裝,使序列能按照指定的差分列表依次進(jìn)行差分,并相應(yīng)的構(gòu)造了一個(gè)還原的方法,實(shí)現(xiàn)差分序列的自動(dòng)還原。

# 差分操作
def diff_ts(ts, d):
 global shift_ts_list
 # 動(dòng)態(tài)預(yù)測(cè)第二日的值時(shí)所需要的差分序列
 global last_data_shift_list
 shift_ts_list = []
 last_data_shift_list = []
 tmp_ts = ts
 for i in d:
  last_data_shift_list.append(tmp_ts[-i])
  print last_data_shift_list
  shift_ts = tmp_ts.shift(i)
  shift_ts_list.append(shift_ts)
  tmp_ts = tmp_ts - shift_ts
 tmp_ts.dropna(inplace=True)
 return tmp_ts

# 還原操作
def predict_diff_recover(predict_value, d):
 if isinstance(predict_value, float):
  tmp_data = predict_value
  for i in range(len(d)):
   tmp_data = tmp_data + last_data_shift_list[-i-1]
 elif isinstance(predict_value, np.ndarray):
  tmp_data = predict_value[0]
  for i in range(len(d)):
   tmp_data = tmp_data + last_data_shift_list[-i-1]
 else:
  tmp_data = predict_value
  for i in range(len(d)):
   try:
    tmp_data = tmp_data.add(shift_ts_list[-i-1])
   except:
    raise ValueError('What you input is not pd.Series type!')
  tmp_data.dropna(inplace=True)
 return tmp_data

現(xiàn)在我們直接使用差分的方法進(jìn)行數(shù)據(jù)處理,并以同樣的過(guò)程進(jìn)行數(shù)據(jù)預(yù)測(cè)與還原。

diffed_ts = diff_ts(ts_log, d=[12, 1])
model = arima_model(diffed_ts)
model.certain_model(1, 1)
predict_ts = model.properModel.predict()
diff_recover_ts = predict_diff_recover(predict_ts, d=[12, 1])
log_recover = np.exp(diff_recover_ts)

是不是發(fā)現(xiàn)這里的預(yù)測(cè)結(jié)果和上一篇的使用12階移動(dòng)平均的預(yù)測(cè)結(jié)果一模一樣。這是因?yàn)?2階移動(dòng)平均加上一階差分與直接12階差分是等價(jià)的關(guān)系,后者是前者數(shù)值的12倍,這個(gè)應(yīng)該不難推導(dǎo)。

對(duì)于個(gè)數(shù)不多的時(shí)序數(shù)據(jù),我們可以通過(guò)觀察自相關(guān)圖和偏相關(guān)圖來(lái)進(jìn)行模型識(shí)別,倘若我們要分析的時(shí)序數(shù)據(jù)量較多,例如要預(yù)測(cè)每只股票的走勢(shì),我們就不可能逐個(gè)去調(diào)參了。這時(shí)我們可以依據(jù)BIC準(zhǔn)則識(shí)別模型的p, q值,通常認(rèn)為BIC值越小的模型相對(duì)更優(yōu)。這里我簡(jiǎn)單介紹一下BIC準(zhǔn)則,它綜合考慮了殘差大小和自變量的個(gè)數(shù),殘差越小BIC值越小,自變量個(gè)數(shù)越多BIC值越大。個(gè)人覺(jué)得BIC準(zhǔn)則就是對(duì)模型過(guò)擬合設(shè)定了一個(gè)標(biāo)準(zhǔn)(過(guò)擬合這東西應(yīng)該以辯證的眼光看待)。

def proper_model(data_ts, maxLag):
 init_bic = sys.maxint
 init_p = 0
 init_q = 0
 init_properModel = None
 for p in np.arange(maxLag):
  for q in np.arange(maxLag):
   model = ARMA(data_ts, order=(p, q))
   try:
    results_ARMA = model.fit(disp=-1, method='css')
   except:
    continue
   bic = results_ARMA.bic
   if bic < init_bic:
    init_p = p
    init_q = q
    init_properModel = results_ARMA
    init_bic = bic
 return init_bic, init_p, init_q, init_properModel

相對(duì)最優(yōu)參數(shù)識(shí)別結(jié)果:BIC: -1090.44209358 p: 0 q: 1 ,RMSE:11.8817198331。我們發(fā)現(xiàn)模型自動(dòng)識(shí)別的參數(shù)要比我手動(dòng)選取的參數(shù)更優(yōu)。

7.滾動(dòng)預(yù)測(cè)

所謂滾動(dòng)預(yù)測(cè)是指通過(guò)添加最新的數(shù)據(jù)預(yù)測(cè)第二天的值。對(duì)于一個(gè)穩(wěn)定的預(yù)測(cè)模型,不需要每天都去擬合,我們可以給他設(shè)定一個(gè)閥值,例如每周擬合一次,該期間只需通過(guò)添加最新的數(shù)據(jù)實(shí)現(xiàn)滾動(dòng)預(yù)測(cè)即可。基于此我編寫了一個(gè)名為arima_model的類,主要包含模型自動(dòng)識(shí)別方法,滾動(dòng)預(yù)測(cè)的功能,詳細(xì)代碼可以查看附錄。數(shù)據(jù)的動(dòng)態(tài)添加:

from dateutil.relativedelta import relativedeltadef _add_new_data(ts, dat, type='day'):
if type == 'day':
  new_index = ts.index[-1] + relativedelta(days=1)
 elif type == 'month':
  new_index = ts.index[-1] + relativedelta(months=1)
 ts[new_index] = dat

def add_today_data(model, ts, data, d, type='day'):
 _add_new_data(ts, data, type) # 為原始序列添加數(shù)據(jù)
 # 為滯后序列添加新值
 d_ts = diff_ts(ts, d)
 model.add_today_data(d_ts[-1], type)

def forecast_next_day_data(model, type='day'):
 if model == None:
  raise ValueError('No model fit before')
 fc = model.forecast_next_day_value(type)
 return predict_diff_recover(fc, [12, 1])

現(xiàn)在我們就可以使用滾動(dòng)預(yù)測(cè)的方法向外預(yù)測(cè)了,取1957年之前的數(shù)據(jù)作為訓(xùn)練數(shù)據(jù),其后的數(shù)據(jù)作為測(cè)試,并設(shè)定模型每第七天就會(huì)重新擬合一次。這里的diffed_ts對(duì)象會(huì)隨著add_today_data方法自動(dòng)添加數(shù)據(jù),這是由于它與add_today_data方法中的d_ts指向的同一對(duì)象,該對(duì)象會(huì)動(dòng)態(tài)的添加數(shù)據(jù)。

ts_train = ts_log[:'1956-12']
ts_test = ts_log['1957-1':]

diffed_ts = diff_ts(ts_train, [12, 1])
forecast_list = []
for i, dta in enumerate(ts_test):
 if i%7 == 0:
  model = arima_model(diffed_ts)
  model.certain_model(1, 1)
 forecast_data = forecast_next_day_data(model, type='month')
 forecast_list.append(forecast_data)
 add_today_data(model, ts_train, dta, [12, 1], type='month')

predict_ts = pd.Series(data=forecast_list, index=ts['1957-1':].index)log_recover = np.exp(predict_ts)original_ts = ts['1957-1':]

動(dòng)態(tài)預(yù)測(cè)的均方根誤差為:14.6479,與前面樣本內(nèi)擬合的均方根誤差相差不大,說(shuō)明模型并沒(méi)有過(guò)擬合,并且整體預(yù)測(cè)效果都較好。

8. 模型序列化

在進(jìn)行動(dòng)態(tài)預(yù)測(cè)時(shí),我們不希望將整個(gè)模型一直在內(nèi)存中運(yùn)行,而是希望有新的數(shù)據(jù)到來(lái)時(shí)才啟動(dòng)該模型。這時(shí)我們就應(yīng)該把整個(gè)模型從內(nèi)存導(dǎo)出到硬盤中,而序列化正好能滿足該要求。序列化最常用的就是使用json模塊了,但是它是時(shí)間對(duì)象支持得不是很好,有人對(duì)json模塊進(jìn)行了拓展以使得支持時(shí)間對(duì)象,這里我們不采用該方法,我們使用pickle模塊,它和json的接口基本相同,有興趣的可以去查看一下。我在實(shí)際應(yīng)用中是采用的面向?qū)ο蟮木幊?,它的序列化主要是將類的屬性序列化即可,而在面向過(guò)程的編程中,模型序列化需要將需要序列化的對(duì)象公有化,這樣會(huì)使得對(duì)前面函數(shù)的參數(shù)改動(dòng)較大,我不在詳細(xì)闡述該過(guò)程。

總結(jié)

與其它統(tǒng)計(jì)語(yǔ)言相比,python在統(tǒng)計(jì)分析這塊還顯得不那么“專業(yè)”。隨著numpy、pandas、scipy、sklearn、gensim、statsmodels等包的推動(dòng),我相信也祝愿python在數(shù)據(jù)分析這塊越來(lái)越好。與SAS和R相比,python的時(shí)間序列模塊還不是很成熟,我這里僅起到拋磚引玉的作用,希望各位能人志士能貢獻(xiàn)自己的力量,使其更加完善。實(shí)際應(yīng)用中我全是面向過(guò)程來(lái)編寫的,為了闡述方便,我用面向過(guò)程重新羅列了一遍,實(shí)在感覺(jué)很不方便。原本打算分三篇來(lái)寫的,還有一部分實(shí)際應(yīng)用的部分,不打算再寫了,還請(qǐng)大家原諒。實(shí)際應(yīng)用主要是具體問(wèn)題具體分析,這當(dāng)中第一步就是要查詢問(wèn)題,這步花的時(shí)間往往會(huì)比較多,然后再是解決問(wèn)題。以我前面項(xiàng)目遇到的問(wèn)題為例,當(dāng)時(shí)遇到了以下幾個(gè)典型的問(wèn)題:1.周期長(zhǎng)度不恒定的周期成分,例如每月的1號(hào)具有周期性,但每月1號(hào)與1號(hào)之間的時(shí)間間隔是不相等的;2.含有缺失值以及含有記錄為0的情況無(wú)法進(jìn)行對(duì)數(shù)變換;3.節(jié)假日的影響等等。

附錄

# -*-coding:utf-8-*-
import pandas as pd
import numpy as np
from statsmodels.tsa.arima_model import ARMA
import sys
from dateutil.relativedelta import relativedelta
from copy import deepcopy
import matplotlib.pyplot as plt

class arima_model:

 def __init__(self, ts, maxLag=9):
  self.data_ts = ts
  self.resid_ts = None
  self.predict_ts = None
  self.maxLag = maxLag
  self.p = maxLag
  self.q = maxLag
  self.properModel = None
  self.bic = sys.maxint

 # 計(jì)算最優(yōu)ARIMA模型,將相關(guān)結(jié)果賦給相應(yīng)屬性
 def get_proper_model(self):
  self._proper_model()
  self.predict_ts = deepcopy(self.properModel.predict())
  self.resid_ts = deepcopy(self.properModel.resid)

 # 對(duì)于給定范圍內(nèi)的p,q計(jì)算擬合得最好的arima模型,這里是對(duì)差分好的數(shù)據(jù)進(jìn)行擬合,故差分恒為0
 def _proper_model(self):
  for p in np.arange(self.maxLag):
   for q in np.arange(self.maxLag):
    # print p,q,self.bic
    model = ARMA(self.data_ts, order=(p, q))
    try:
     results_ARMA = model.fit(disp=-1, method='css')
    except:
     continue
    bic = results_ARMA.bic
    # print 'bic:',bic,'self.bic:',self.bic
    if bic < self.bic:
     self.p = p
     self.q = q
     self.properModel = results_ARMA
     self.bic = bic
     self.resid_ts = deepcopy(self.properModel.resid)
     self.predict_ts = self.properModel.predict()

 # 參數(shù)確定模型
 def certain_model(self, p, q):
   model = ARMA(self.data_ts, order=(p, q))
   try:
    self.properModel = model.fit( disp=-1, method='css')
    self.p = p
    self.q = q
    self.bic = self.properModel.bic
    self.predict_ts = self.properModel.predict()
    self.resid_ts = deepcopy(self.properModel.resid)
   except:
    print 'You can not fit the model with this parameter p,q, ' \
      'please use the get_proper_model method to get the best model'

 # 預(yù)測(cè)第二日的值
 def forecast_next_day_value(self, type='day'):
  # 我修改了statsmodels包中arima_model的源代碼,添加了constant屬性,需要先運(yùn)行forecast方法,為constant賦值
  self.properModel.forecast()
  if self.data_ts.index[-1] != self.resid_ts.index[-1]:
   raise ValueError('''The index is different in data_ts and resid_ts, please add new data to data_ts.
   If you just want to forecast the next day data without add the real next day data to data_ts,
   please run the predict method which arima_model included itself''')
  if not self.properModel:
   raise ValueError('The arima model have not computed, please run the proper_model method before')
  para = self.properModel.params

  # print self.properModel.params
  if self.p == 0: # It will get all the value series with setting self.data_ts[-self.p:] when p is zero
   ma_value = self.resid_ts[-self.q:]
   values = ma_value.reindex(index=ma_value.index[::-1])
  elif self.q == 0:
   ar_value = self.data_ts[-self.p:]
   values = ar_value.reindex(index=ar_value.index[::-1])
  else:
   ar_value = self.data_ts[-self.p:]
   ar_value = ar_value.reindex(index=ar_value.index[::-1])
   ma_value = self.resid_ts[-self.q:]
   ma_value = ma_value.reindex(index=ma_value.index[::-1])
   values = ar_value.append(ma_value)

  predict_value = np.dot(para[1:], values) + self.properModel.constant[0]
  self._add_new_data(self.predict_ts, predict_value, type)
  return predict_value

 # 動(dòng)態(tài)添加數(shù)據(jù)函數(shù),針對(duì)索引是月份和日分別進(jìn)行處理
 def _add_new_data(self, ts, dat, type='day'):
  if type == 'day':
   new_index = ts.index[-1] + relativedelta(days=1)
  elif type == 'month':
   new_index = ts.index[-1] + relativedelta(months=1)
  ts[new_index] = dat

 def add_today_data(self, dat, type='day'):
  self._add_new_data(self.data_ts, dat, type)
  if self.data_ts.index[-1] != self.predict_ts.index[-1]:
   raise ValueError('You must use the forecast_next_day_value method forecast the value of today before')
  self._add_new_data(self.resid_ts, self.data_ts[-1] - self.predict_ts[-1], type)

if __name__ == '__main__':
 df = pd.read_csv('AirPassengers.csv', encoding='utf-8', index_col='date')
 df.index = pd.to_datetime(df.index)
 ts = df['x']

 # 數(shù)據(jù)預(yù)處理
 ts_log = np.log(ts)
 rol_mean = ts_log.rolling(window=12).mean()
 rol_mean.dropna(inplace=True)
 ts_diff_1 = rol_mean.diff(1)
 ts_diff_1.dropna(inplace=True)
 ts_diff_2 = ts_diff_1.diff(1)
 ts_diff_2.dropna(inplace=True)

 # 模型擬合
 model = arima_model(ts_diff_2)
 # 這里使用模型參數(shù)自動(dòng)識(shí)別
 model.get_proper_model()
 print 'bic:', model.bic, 'p:', model.p, 'q:', model.q
 print model.properModel.forecast()[0]
 print model.forecast_next_day_value(type='month')

 # 預(yù)測(cè)結(jié)果還原
 predict_ts = model.properModel.predict()
 diff_shift_ts = ts_diff_1.shift(1)
 diff_recover_1 = predict_ts.add(diff_shift_ts)
 rol_shift_ts = rol_mean.shift(1)
 diff_recover = diff_recover_1.add(rol_shift_ts)
 rol_sum = ts_log.rolling(window=11).sum()
 rol_recover = diff_recover*12 - rol_sum.shift(1)
 log_recover = np.exp(rol_recover)
 log_recover.dropna(inplace=True)

 # 預(yù)測(cè)結(jié)果作圖
 ts = ts[log_recover.index]
 plt.figure(facecolor='white')
 log_recover.plot(color='blue', label='Predict')
 ts.plot(color='red', label='Original')
 plt.legend(loc='best')
 plt.title('RMSE: %.4f'% np.sqrt(sum((log_recover-ts)**2)/ts.size))
 plt.show()

修改的arima_model代碼

# Note: The information criteria add 1 to the number of parameters
#  whenever the model has an AR or MA term since, in principle,
#  the variance could be treated as a free parameter and restricted
#  This code does not allow this, but it adds consistency with other
#  packages such as gretl and X12-ARIMA
 
from __future__ import absolute_import
from statsmodels.compat.python import string_types, range
# for 2to3 with extensions
 
from datetime import datetime
 
import numpy as np
from scipy import optimize
from scipy.stats import t, norm
from scipy.signal import lfilter
from numpy import dot, log, zeros, pi
from numpy.linalg import inv
 
from statsmodels.tools.decorators import (cache_readonly,
           resettable_cache)
import statsmodels.tsa.base.tsa_model as tsbase
import statsmodels.base.wrapper as wrap
from statsmodels.regression.linear_model import yule_walker, GLS
from statsmodels.tsa.tsatools import (lagmat, add_trend,
          _ar_transparams, _ar_invtransparams,
          _ma_transparams, _ma_invtransparams,
          unintegrate, unintegrate_levels)
from statsmodels.tsa.vector_ar import util
from statsmodels.tsa.ar_model import AR
from statsmodels.tsa.arima_process import arma2ma
from statsmodels.tools.numdiff import approx_hess_cs, approx_fprime_cs
from statsmodels.tsa.base.datetools import _index_date
from statsmodels.tsa.kalmanf import KalmanFilter
 
_armax_notes = """
 
  Notes
  -----
  If exogenous variables are given, then the model that is fit is
 
  .. math::
 
   \\phi(L)(y_t - X_t\\beta) = \\theta(L)\epsilon_t
 
  where :math:`\\phi` and :math:`\\theta` are polynomials in the lag
  operator, :math:`L`. This is the regression model with ARMA errors,
  or ARMAX model. This specification is used, whether or not the model
  is fit using conditional sum of square or maximum-likelihood, using
  the `method` argument in
  :meth:`statsmodels.tsa.arima_model.%(Model)s.fit`. Therefore, for
  now, `css` and `mle` refer to estimation methods only. This may
  change for the case of the `css` model in future versions.
"""
 
_arma_params = """\
 endog : array-like
  The endogenous variable.
 order : iterable
  The (p,q) order of the model for the number of AR parameters,
  differences, and MA parameters to use.
 exog : array-like, optional
  An optional arry of exogenous variables. This should *not* include a
  constant or trend. You can specify this in the `fit` method."""
 
_arma_model = "Autoregressive Moving Average ARMA(p,q) Model"
 
_arima_model = "Autoregressive Integrated Moving Average ARIMA(p,d,q) Model"
 
_arima_params = """\
 endog : array-like
  The endogenous variable.
 order : iterable
  The (p,d,q) order of the model for the number of AR parameters,
  differences, and MA parameters to use.
 exog : array-like, optional
  An optional arry of exogenous variables. This should *not* include a
  constant or trend. You can specify this in the `fit` method."""
 
_predict_notes = """
  Notes
  -----
  Use the results predict method instead.
"""
 
_results_notes = """
  Notes
  -----
  It is recommended to use dates with the time-series models, as the
  below will probably make clear. However, if ARIMA is used without
  dates and/or `start` and `end` are given as indices, then these
  indices are in terms of the *original*, undifferenced series. Ie.,
  given some undifferenced observations::
 
   1970Q1, 1
   1970Q2, 1.5
   1970Q3, 1.25
   1970Q4, 2.25
   1971Q1, 1.2
   1971Q2, 4.1
 
  1970Q1 is observation 0 in the original series. However, if we fit an
  ARIMA(p,1,q) model then we lose this first observation through
  differencing. Therefore, the first observation we can forecast (if
  using exact MLE) is index 1. In the differenced series this is index
  0, but we refer to it as 1 from the original series.
"""
 
_predict = """
  %(Model)s model in-sample and out-of-sample prediction
 
  Parameters
  ----------
  %(params)s
  start : int, str, or datetime
   Zero-indexed observation number at which to start forecasting, ie.,
   the first forecast is start. Can also be a date string to
   parse or a datetime type.
  end : int, str, or datetime
   Zero-indexed observation number at which to end forecasting, ie.,
   the first forecast is start. Can also be a date string to
   parse or a datetime type. However, if the dates index does not
   have a fixed frequency, end must be an integer index if you
   want out of sample prediction.
  exog : array-like, optional
   If the model is an ARMAX and out-of-sample forecasting is
   requested, exog must be given. Note that you'll need to pass
   `k_ar` additional lags for any exogenous variables. E.g., if you
   fit an ARMAX(2, q) model and want to predict 5 steps, you need 7
   observations to do this.
  dynamic : bool, optional
   The `dynamic` keyword affects in-sample prediction. If dynamic
   is False, then the in-sample lagged values are used for
   prediction. If `dynamic` is True, then in-sample forecasts are
   used in place of lagged dependent variables. The first forecasted
   value is `start`.
  %(extra_params)s
 
  Returns
  -------
  %(returns)s
  %(extra_section)s
"""
 
_predict_returns = """predict : array
   The predicted values.
 
"""
 
_arma_predict = _predict % {"Model" : "ARMA",
       "params" : """
   params : array-like
   The fitted parameters of the model.""",
       "extra_params" : "",
       "returns" : _predict_returns,
       "extra_section" : _predict_notes}
 
_arma_results_predict = _predict % {"Model" : "ARMA", "params" : "",
         "extra_params" : "",
         "returns" : _predict_returns,
         "extra_section" : _results_notes}
 
_arima_predict = _predict % {"Model" : "ARIMA",
        "params" : """params : array-like
   The fitted parameters of the model.""",
        "extra_params" : """typ : str {'linear', 'levels'}
 
   - 'linear' : Linear prediction in terms of the differenced
    endogenous variables.
   - 'levels' : Predict the levels of the original endogenous
    variables.\n""", "returns" : _predict_returns,
        "extra_section" : _predict_notes}
 
_arima_results_predict = _predict % {"Model" : "ARIMA",
          "params" : "",
          "extra_params" :
          """typ : str {'linear', 'levels'}
 
   - 'linear' : Linear prediction in terms of the differenced
    endogenous variables.
   - 'levels' : Predict the levels of the original endogenous
    variables.\n""",
          "returns" : _predict_returns,
          "extra_section" : _results_notes}
 
_arima_plot_predict_example = """  Examples
  --------
  >>> import statsmodels.api as sm
  >>> import matplotlib.pyplot as plt
  >>> import pandas as pd
  >>>
  >>> dta = sm.datasets.sunspots.load_pandas().data[['SUNACTIVITY']]
  >>> dta.index = pd.DatetimeIndex(start='1700', end='2009', freq='A')
  >>> res = sm.tsa.ARMA(dta, (3, 0)).fit()
  >>> fig, ax = plt.subplots()
  >>> ax = dta.ix['1950':].plot(ax=ax)
  >>> fig = res.plot_predict('1990', '2012', dynamic=True, ax=ax,
  ...      plot_insample=False)
  >>> plt.show()
 
  .. plot:: plots/arma_predict_plot.py
"""
 
_plot_predict = ("""
  Plot forecasts
      """ + '\n'.join(_predict.split('\n')[2:])) % {
      "params" : "",
       "extra_params" : """alpha : float, optional
   The confidence intervals for the forecasts are (1 - alpha)%
  plot_insample : bool, optional
   Whether to plot the in-sample series. Default is True.
  ax : matplotlib.Axes, optional
   Existing axes to plot with.""",
      "returns" : """fig : matplotlib.Figure
   The plotted Figure instance""",
      "extra_section" : ('\n' + _arima_plot_predict_example +
           '\n' + _results_notes)
      }
 
_arima_plot_predict = ("""
  Plot forecasts
      """ + '\n'.join(_predict.split('\n')[2:])) % {
      "params" : "",
       "extra_params" : """alpha : float, optional
   The confidence intervals for the forecasts are (1 - alpha)%
  plot_insample : bool, optional
   Whether to plot the in-sample series. Default is True.
  ax : matplotlib.Axes, optional
   Existing axes to plot with.""",
      "returns" : """fig : matplotlib.Figure
   The plotted Figure instance""",
    "extra_section" : ('\n' + _arima_plot_predict_example +
         '\n' +
         '\n'.join(_results_notes.split('\n')[:3]) +
        ("""
  This is hard-coded to only allow plotting of the forecasts in levels.
""") +
        '\n'.join(_results_notes.split('\n')[3:]))
      }
 
 
def cumsum_n(x, n):
 if n:
  n -= 1
  x = np.cumsum(x)
  return cumsum_n(x, n)
 else:
  return x
 
 
def _check_arima_start(start, k_ar, k_diff, method, dynamic):
 if start < 0:
  raise ValueError("The start index %d of the original series "
       "has been differenced away" % start)
 elif (dynamic or 'mle' not in method) and start < k_ar:
  raise ValueError("Start must be >= k_ar for conditional MLE "
       "or dynamic forecast. Got %d" % start)
 
 
def _get_predict_out_of_sample(endog, p, q, k_trend, k_exog, start, errors,
        trendparam, exparams, arparams, maparams, steps,
        method, exog=None):
 """
 Returns endog, resid, mu of appropriate length for out of sample
 prediction.
 """
 if q:
  resid = np.zeros(q)
  if start and 'mle' in method or (start == p and not start == 0):
   resid[:q] = errors[start-q:start]
  elif start:
   resid[:q] = errors[start-q-p:start-p]
  else:
   resid[:q] = errors[-q:]
 else:
  resid = None
 
 y = endog
 if k_trend == 1:
  # use expectation not constant
  if k_exog > 0:
   #TODO: technically should only hold for MLE not
   # conditional model. See #274.
   # ensure 2-d for conformability
   if np.ndim(exog) == 1 and k_exog == 1:
    # have a 1d series of observations -> 2d
    exog = exog[:, None]
   elif np.ndim(exog) == 1:
    # should have a 1d row of exog -> 2d
    if len(exog) != k_exog:
     raise ValueError("1d exog given and len(exog) != k_exog")
    exog = exog[None, :]
   X = lagmat(np.dot(exog, exparams), p, original='in', trim='both')
   mu = trendparam * (1 - arparams.sum())
   # arparams were reversed in unpack for ease later
   mu = mu + (np.r_[1, -arparams[::-1]] * X).sum(1)[:, None]
  else:
   mu = trendparam * (1 - arparams.sum())
   mu = np.array([mu]*steps)
 elif k_exog > 0:
  X = np.dot(exog, exparams)
  #NOTE: you shouldn't have to give in-sample exog!
  X = lagmat(X, p, original='in', trim='both')
  mu = (np.r_[1, -arparams[::-1]] * X).sum(1)[:, None]
 else:
  mu = np.zeros(steps)
 
 endog = np.zeros(p + steps - 1)
 
 if p and start:
  endog[:p] = y[start-p:start]
 elif p:
  endog[:p] = y[-p:]
 
 return endog, resid, mu
 
 
def _arma_predict_out_of_sample(params, steps, errors, p, q, k_trend, k_exog,
        endog, exog=None, start=0, method='mle'):
 (trendparam, exparams,
  arparams, maparams) = _unpack_params(params, (p, q), k_trend,
           k_exog, reverse=True)
 # print 'params:',params
 # print 'arparams:',arparams,'maparams:',maparams
 endog, resid, mu = _get_predict_out_of_sample(endog, p, q, k_trend, k_exog,
             start, errors, trendparam,
             exparams, arparams,
             maparams, steps, method,
             exog)
# print 'mu[-1]:',mu[-1], 'mu[0]:',mu[0]
 forecast = np.zeros(steps)
 if steps == 1:
  if q:
   return mu[0] + np.dot(arparams, endog[:p]) + np.dot(maparams,
                resid[:q]), mu[0]
  else:
   return mu[0] + np.dot(arparams, endog[:p]), mu[0]
 
 if q:
  i = 0 # if q == 1
 else:
  i = -1
 
 for i in range(min(q, steps - 1)):
  fcast = (mu[i] + np.dot(arparams, endog[i:i + p]) +
     np.dot(maparams[:q - i], resid[i:i + q]))
  forecast[i] = fcast
  endog[i+p] = fcast
 
 for i in range(i + 1, steps - 1):
  fcast = mu[i] + np.dot(arparams, endog[i:i+p])
  forecast[i] = fcast
  endog[i+p] = fcast
 
 #need to do one more without updating endog
 forecast[-1] = mu[-1] + np.dot(arparams, endog[steps - 1:])
 return forecast, mu[-1] #Modified by me, the former is return forecast
 
 
def _arma_predict_in_sample(start, end, endog, resid, k_ar, method):
 """
 Pre- and in-sample fitting for ARMA.
 """
 if 'mle' in method:
  fittedvalues = endog - resid # get them all then trim
 else:
  fittedvalues = endog[k_ar:] - resid
 
 fv_start = start
 if 'mle' not in method:
  fv_start -= k_ar # start is in terms of endog index
 fv_end = min(len(fittedvalues), end + 1)
 return fittedvalues[fv_start:fv_end]
 
 
def _validate(start, k_ar, k_diff, dates, method):
 if isinstance(start, (string_types, datetime)):
  start = _index_date(start, dates)
  start -= k_diff
 if 'mle' not in method and start < k_ar - k_diff:
  raise ValueError("Start must be >= k_ar for conditional "
       "MLE or dynamic forecast. Got %s" % start)
 
 return start
 
 
def _unpack_params(params, order, k_trend, k_exog, reverse=False):
 p, q = order
 k = k_trend + k_exog
 maparams = params[k+p:]
 arparams = params[k:k+p]
 trend = params[:k_trend]
 exparams = params[k_trend:k]
 if reverse:
  return trend, exparams, arparams[::-1], maparams[::-1]
 return trend, exparams, arparams, maparams
 
 
def _unpack_order(order):
 k_ar, k_ma, k = order
 k_lags = max(k_ar, k_ma+1)
 return k_ar, k_ma, order, k_lags
 
 
def _make_arma_names(data, k_trend, order, exog_names):
 k_ar, k_ma = order
 exog_names = exog_names or []
 ar_lag_names = util.make_lag_names([data.ynames], k_ar, 0)
 ar_lag_names = [''.join(('ar.', i)) for i in ar_lag_names]
 ma_lag_names = util.make_lag_names([data.ynames], k_ma, 0)
 ma_lag_names = [''.join(('ma.', i)) for i in ma_lag_names]
 trend_name = util.make_lag_names('', 0, k_trend)
 exog_names = trend_name + exog_names + ar_lag_names + ma_lag_names
 return exog_names
 
 
def _make_arma_exog(endog, exog, trend):
 k_trend = 1 # overwritten if no constant
 if exog is None and trend == 'c': # constant only
  exog = np.ones((len(endog), 1))
 elif exog is not None and trend == 'c': # constant plus exogenous
  exog = add_trend(exog, trend='c', prepend=True)
 elif exog is not None and trend == 'nc':
  # make sure it's not holding constant from last run
  if exog.var() == 0:
   exog = None
  k_trend = 0
 if trend == 'nc':
  k_trend = 0
 return k_trend, exog
 
 
def _check_estimable(nobs, n_params):
 if nobs <= n_params:
  raise ValueError("Insufficient degrees of freedom to estimate")
 
 
class ARMA(tsbase.TimeSeriesModel):
 
 __doc__ = tsbase._tsa_doc % {"model" : _arma_model,
         "params" : _arma_params, "extra_params" : "",
         "extra_sections" : _armax_notes %
         {"Model" : "ARMA"}}
 
 def __init__(self, endog, order, exog=None, dates=None, freq=None,
     missing='none'):
  super(ARMA, self).__init__(endog, exog, dates, freq, missing=missing)
  exog = self.data.exog # get it after it's gone through processing
  _check_estimable(len(self.endog), sum(order))
  self.k_ar = k_ar = order[0]
  self.k_ma = k_ma = order[1]
  self.k_lags = max(k_ar, k_ma+1)
  self.constant = 0 #Added by me
  if exog is not None:
   if exog.ndim == 1:
    exog = exog[:, None]
   k_exog = exog.shape[1] # number of exog. variables excl. const
  else:
   k_exog = 0
  self.k_exog = k_exog
 
 def _fit_start_params_hr(self, order):
  """
  Get starting parameters for fit.
 
  Parameters
  ----------
  order : iterable
   (p,q,k) - AR lags, MA lags, and number of exogenous variables
   including the constant.
 
  Returns
  -------
  start_params : array
   A first guess at the starting parameters.
 
  Notes
  -----
  If necessary, fits an AR process with the laglength selected according
  to best BIC. Obtain the residuals. Then fit an ARMA(p,q) model via
  OLS using these residuals for a first approximation. Uses a separate
  OLS regression to find the coefficients of exogenous variables.
 
  References
  ----------
  Hannan, E.J. and Rissanen, J. 1982. "Recursive estimation of mixed
   autoregressive-moving average order." `Biometrika`. 69.1.
  """
  p, q, k = order
  start_params = zeros((p+q+k))
  endog = self.endog.copy() # copy because overwritten
  exog = self.exog
  if k != 0:
   ols_params = GLS(endog, exog).fit().params
   start_params[:k] = ols_params
   endog -= np.dot(exog, ols_params).squeeze()
  if q != 0:
   if p != 0:
    # make sure we don't run into small data problems in AR fit
    nobs = len(endog)
    maxlag = int(round(12*(nobs/100.)**(1/4.)))
    if maxlag >= nobs:
     maxlag = nobs - 1
    armod = AR(endog).fit(ic='bic', trend='nc', maxlag=maxlag)
    arcoefs_tmp = armod.params
    p_tmp = armod.k_ar
    # it's possible in small samples that optimal lag-order
    # doesn't leave enough obs. No consistent way to fix.
    if p_tmp + q >= len(endog):
     raise ValueError("Proper starting parameters cannot"
          " be found for this order with this "
          "number of observations. Use the "
          "start_params argument.")
    resid = endog[p_tmp:] - np.dot(lagmat(endog, p_tmp,
              trim='both'),
            arcoefs_tmp)
    if p < p_tmp + q:
     endog_start = p_tmp + q - p
     resid_start = 0
    else:
     endog_start = 0
     resid_start = p - p_tmp - q
    lag_endog = lagmat(endog, p, 'both')[endog_start:]
    lag_resid = lagmat(resid, q, 'both')[resid_start:]
    # stack ar lags and resids
    X = np.column_stack((lag_endog, lag_resid))
    coefs = GLS(endog[max(p_tmp + q, p):], X).fit().params
    start_params[k:k+p+q] = coefs
   else:
    start_params[k+p:k+p+q] = yule_walker(endog, order=q)[0]
  if q == 0 and p != 0:
   arcoefs = yule_walker(endog, order=p)[0]
   start_params[k:k+p] = arcoefs
 
  # check AR coefficients
  if p and not np.all(np.abs(np.roots(np.r_[1, -start_params[k:k + p]]
           )) < 1):
   raise ValueError("The computed initial AR coefficients are not "
        "stationary\nYou should induce stationarity, "
        "choose a different model order, or you can\n"
        "pass your own start_params.")
  # check MA coefficients
  elif q and not np.all(np.abs(np.roots(np.r_[1, start_params[k + p:]]
            )) < 1):
   return np.zeros(len(start_params)) #modified by me
   raise ValueError("The computed initial MA coefficients are not "
        "invertible\nYou should induce invertibility, "
        "choose a different model order, or you can\n"
        "pass your own start_params.")
 
  # check MA coefficients
  # print start_params
  return start_params
 
 def _fit_start_params(self, order, method):
  if method != 'css-mle': # use Hannan-Rissanen to get start params
   start_params = self._fit_start_params_hr(order)
  else: # use CSS to get start params
   func = lambda params: -self.loglike_css(params)
   #start_params = [.1]*(k_ar+k_ma+k_exog) # different one for k?
   start_params = self._fit_start_params_hr(order)
   if self.transparams:
    start_params = self._invtransparams(start_params)
   bounds = [(None,)*2]*sum(order)
   mlefit = optimize.fmin_l_bfgs_b(func, start_params,
           approx_grad=True, m=12,
           pgtol=1e-7, factr=1e3,
           bounds=bounds, iprint=-1)
   start_params = self._transparams(mlefit[0])
  return start_params
 
 def score(self, params):
  """
  Compute the score function at params.
 
  Notes
  -----
  This is a numerical approximation.
  """
  return approx_fprime_cs(params, self.loglike, args=(False,))
 
 def hessian(self, params):
  """
  Compute the Hessian at params,
 
  Notes
  -----
  This is a numerical approximation.
  """
  return approx_hess_cs(params, self.loglike, args=(False,))
 
 def _transparams(self, params):
  """
  Transforms params to induce stationarity/invertability.
 
  Reference
  ---------
  Jones(1980)
  """
  k_ar, k_ma = self.k_ar, self.k_ma
  k = self.k_exog + self.k_trend
  newparams = np.zeros_like(params)
 
  # just copy exogenous parameters
  if k != 0:
   newparams[:k] = params[:k]
 
  # AR Coeffs
  if k_ar != 0:
   newparams[k:k+k_ar] = _ar_transparams(params[k:k+k_ar].copy())
 
  # MA Coeffs
  if k_ma != 0:
   newparams[k+k_ar:] = _ma_transparams(params[k+k_ar:].copy())
  return newparams
 
 def _invtransparams(self, start_params):
  """
  Inverse of the Jones reparameterization
  """
  k_ar, k_ma = self.k_ar, self.k_ma
  k = self.k_exog + self.k_trend
  newparams = start_params.copy()
  arcoefs = newparams[k:k+k_ar]
  macoefs = newparams[k+k_ar:]
  # AR coeffs
  if k_ar != 0:
   newparams[k:k+k_ar] = _ar_invtransparams(arcoefs)
 
  # MA coeffs
  if k_ma != 0:
   newparams[k+k_ar:k+k_ar+k_ma] = _ma_invtransparams(macoefs)
  return newparams
 
 def _get_predict_start(self, start, dynamic):
  # do some defaults
  method = getattr(self, 'method', 'mle')
  k_ar = getattr(self, 'k_ar', 0)
  k_diff = getattr(self, 'k_diff', 0)
  if start is None:
   if 'mle' in method and not dynamic:
    start = 0
   else:
    start = k_ar
   self._set_predict_start_date(start) # else it's done in super
  elif isinstance(start, int):
   start = super(ARMA, self)._get_predict_start(start)
  else: # should be on a date
   #elif 'mle' not in method or dynamic: # should be on a date
   start = _validate(start, k_ar, k_diff, self.data.dates,
        method)
   start = super(ARMA, self)._get_predict_start(start)
  _check_arima_start(start, k_ar, k_diff, method, dynamic)
  return start
 
 def _get_predict_end(self, end, dynamic=False):
  # pass through so predict works for ARIMA and ARMA
  return super(ARMA, self)._get_predict_end(end)
 
 def geterrors(self, params):
  """
  Get the errors of the ARMA process.
 
  Parameters
  ----------
  params : array-like
   The fitted ARMA parameters
  order : array-like
   3 item iterable, with the number of AR, MA, and exogenous
   parameters, including the trend
  """
 
  #start = self._get_predict_start(start) # will be an index of a date
  #end, out_of_sample = self._get_predict_end(end)
  params = np.asarray(params)
  k_ar, k_ma = self.k_ar, self.k_ma
  k = self.k_exog + self.k_trend
 
  method = getattr(self, 'method', 'mle')
  if 'mle' in method: # use KalmanFilter to get errors
   (y, k, nobs, k_ar, k_ma, k_lags, newparams, Z_mat, m, R_mat,
    T_mat, paramsdtype) = KalmanFilter._init_kalman_state(params,
                 self)
 
   errors = KalmanFilter.geterrors(y, k, k_ar, k_ma, k_lags, nobs,
           Z_mat, m, R_mat, T_mat,
           paramsdtype)
   if isinstance(errors, tuple):
    errors = errors[0] # non-cython version returns a tuple
  else: # use scipy.signal.lfilter
   y = self.endog.copy()
   k = self.k_exog + self.k_trend
   if k > 0:
    y -= dot(self.exog, params[:k])
 
   k_ar = self.k_ar
   k_ma = self.k_ma
 
   (trendparams, exparams,
    arparams, maparams) = _unpack_params(params, (k_ar, k_ma),
             self.k_trend, self.k_exog,
             reverse=False)
   b, a = np.r_[1, -arparams], np.r_[1, maparams]
   zi = zeros((max(k_ar, k_ma)))
   for i in range(k_ar):
    zi[i] = sum(-b[:i+1][::-1]*y[:i+1])
   e = lfilter(b, a, y, zi=zi)
   errors = e[0][k_ar:]
  return errors.squeeze()
 
 def predict(self, params, start=None, end=None, exog=None, dynamic=False):
  method = getattr(self, 'method', 'mle') # don't assume fit
  #params = np.asarray(params)
 
  # will return an index of a date
  start = self._get_predict_start(start, dynamic)
  end, out_of_sample = self._get_predict_end(end, dynamic)
  if out_of_sample and (exog is None and self.k_exog > 0):
   raise ValueError("You must provide exog for ARMAX")
 
  endog = self.endog
  resid = self.geterrors(params)
  k_ar = self.k_ar
 
  if out_of_sample != 0 and self.k_exog > 0:
   if self.k_exog == 1 and exog.ndim == 1:
    exog = exog[:, None]
    # we need the last k_ar exog for the lag-polynomial
   if self.k_exog > 0 and k_ar > 0:
    # need the last k_ar exog for the lag-polynomial
    exog = np.vstack((self.exog[-k_ar:, self.k_trend:], exog))
 
  if dynamic:
   #TODO: now that predict does dynamic in-sample it should
   # also return error estimates and confidence intervals
   # but how? len(endog) is not tot_obs
   out_of_sample += end - start + 1
   pr, ct = _arma_predict_out_of_sample(params, out_of_sample, resid,
            k_ar, self.k_ma, self.k_trend,
            self.k_exog, endog, exog,
            start, method)
   self.constant = ct
   return pr
 
  predictedvalues = _arma_predict_in_sample(start, end, endog, resid,
             k_ar, method)
  if out_of_sample:
   forecastvalues, ct = _arma_predict_out_of_sample(params, out_of_sample,
               resid, k_ar,
               self.k_ma,
               self.k_trend,
               self.k_exog, endog,
               exog, method=method)
   self.constant = ct
   predictedvalues = np.r_[predictedvalues, forecastvalues]
  return predictedvalues
 predict.__doc__ = _arma_predict
 
 def loglike(self, params, set_sigma2=True):
  """
  Compute the log-likelihood for ARMA(p,q) model
 
  Notes
  -----
  Likelihood used depends on the method set in fit
  """
  method = self.method
  if method in ['mle', 'css-mle']:
   return self.loglike_kalman(params, set_sigma2)
  elif method == 'css':
   return self.loglike_css(params, set_sigma2)
  else:
   raise ValueError("Method %s not understood" % method)
 
 def loglike_kalman(self, params, set_sigma2=True):
  """
  Compute exact loglikelihood for ARMA(p,q) model by the Kalman Filter.
  """
  return KalmanFilter.loglike(params, self, set_sigma2)
 
 def loglike_css(self, params, set_sigma2=True):
  """
  Conditional Sum of Squares likelihood function.
  """
  k_ar = self.k_ar
  k_ma = self.k_ma
  k = self.k_exog + self.k_trend
  y = self.endog.copy().astype(params.dtype)
  nobs = self.nobs
  # how to handle if empty?
  if self.transparams:
   newparams = self._transparams(params)
  else:
   newparams = params
  if k > 0:
   y -= dot(self.exog, newparams[:k])
  # the order of p determines how many zeros errors to set for lfilter
  b, a = np.r_[1, -newparams[k:k + k_ar]], np.r_[1, newparams[k + k_ar:]]
  zi = np.zeros((max(k_ar, k_ma)), dtype=params.dtype)
  for i in range(k_ar):
   zi[i] = sum(-b[:i + 1][::-1] * y[:i + 1])
  errors = lfilter(b, a, y, zi=zi)[0][k_ar:]
 
  ssr = np.dot(errors, errors)
  sigma2 = ssr/nobs
  if set_sigma2:
   self.sigma2 = sigma2
  llf = -nobs/2.*(log(2*pi) + log(sigma2)) - ssr/(2*sigma2)
  return llf
 
 def fit(self, start_params=None, trend='c', method="css-mle",
   transparams=True, solver='lbfgs', maxiter=50, full_output=1,
   disp=5, callback=None, **kwargs):
  """
  Fits ARMA(p,q) model using exact maximum likelihood via Kalman filter.
 
  Parameters
  ----------
  start_params : array-like, optional
   Starting parameters for ARMA(p,q). If None, the default is given
   by ARMA._fit_start_params. See there for more information.
  transparams : bool, optional
   Whehter or not to transform the parameters to ensure stationarity.
   Uses the transformation suggested in Jones (1980). If False,
   no checking for stationarity or invertibility is done.
  method : str {'css-mle','mle','css'}
   This is the loglikelihood to maximize. If "css-mle", the
   conditional sum of squares likelihood is maximized and its values
   are used as starting values for the computation of the exact
   likelihood via the Kalman filter. If "mle", the exact likelihood
   is maximized via the Kalman Filter. If "css" the conditional sum
   of squares likelihood is maximized. All three methods use
   `start_params` as starting parameters. See above for more
   information.
  trend : str {'c','nc'}
   Whether to include a constant or not. 'c' includes constant,
   'nc' no constant.
  solver : str or None, optional
   Solver to be used. The default is 'lbfgs' (limited memory
   Broyden-Fletcher-Goldfarb-Shanno). Other choices are 'bfgs',
   'newton' (Newton-Raphson), 'nm' (Nelder-Mead), 'cg' -
   (conjugate gradient), 'ncg' (non-conjugate gradient), and
   'powell'. By default, the limited memory BFGS uses m=12 to
   approximate the Hessian, projected gradient tolerance of 1e-8 and
   factr = 1e2. You can change these by using kwargs.
  maxiter : int, optional
   The maximum number of function evaluations. Default is 50.
  tol : float
   The convergence tolerance. Default is 1e-08.
  full_output : bool, optional
   If True, all output from solver will be available in
   the Results object's mle_retvals attribute. Output is dependent
   on the solver. See Notes for more information.
  disp : bool, optional
   If True, convergence information is printed. For the default
   l_bfgs_b solver, disp controls the frequency of the output during
   the iterations. disp < 0 means no output in this case.
  callback : function, optional
   Called after each iteration as callback(xk) where xk is the current
   parameter vector.
  kwargs
   See Notes for keyword arguments that can be passed to fit.
 
  Returns
  -------
  statsmodels.tsa.arima_model.ARMAResults class
 
  See also
  --------
  statsmodels.base.model.LikelihoodModel.fit : for more information
   on using the solvers.
  ARMAResults : results class returned by fit
 
  Notes
  ------
  If fit by 'mle', it is assumed for the Kalman Filter that the initial
  unkown state is zero, and that the inital variance is
  P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
  r, order = 'F')
 
  """
  k_ar = self.k_ar
  k_ma = self.k_ma
 
  # enforce invertibility
  self.transparams = transparams
 
  endog, exog = self.endog, self.exog
  k_exog = self.k_exog
  self.nobs = len(endog) # this is overwritten if method is 'css'
 
  # (re)set trend and handle exogenous variables
  # always pass original exog
  k_trend, exog = _make_arma_exog(endog, self.exog, trend)
 
  # Check has something to estimate
  if k_ar == 0 and k_ma == 0 and k_trend == 0 and k_exog == 0:
   raise ValueError("Estimation requires the inclusion of least one "
       "AR term, MA term, a constant or an exogenous "
       "variable.")
 
  # check again now that we know the trend
  _check_estimable(len(endog), k_ar + k_ma + k_exog + k_trend)
 
  self.k_trend = k_trend
  self.exog = exog # overwrites original exog from __init__
 
  # (re)set names for this model
  self.exog_names = _make_arma_names(self.data, k_trend, (k_ar, k_ma),
           self.exog_names)
  k = k_trend + k_exog
 
  # choose objective function
  if k_ma == 0 and k_ar == 0:
   method = "css" # Always CSS when no AR or MA terms
 
  self.method = method = method.lower()
 
  # adjust nobs for css
  if method == 'css':
   self.nobs = len(self.endog) - k_ar
 
  if start_params is not None:
   start_params = np.asarray(start_params)
 
  else: # estimate starting parameters
   start_params = self._fit_start_params((k_ar, k_ma, k), method)
 
  if transparams: # transform initial parameters to ensure invertibility
   start_params = self._invtransparams(start_params)
 
  if solver == 'lbfgs':
   kwargs.setdefault('pgtol', 1e-8)
   kwargs.setdefault('factr', 1e2)
   kwargs.setdefault('m', 12)
   kwargs.setdefault('approx_grad', True)
  mlefit = super(ARMA, self).fit(start_params, method=solver,
          maxiter=maxiter,
          full_output=full_output, disp=disp,
          callback=callback, **kwargs)
  params = mlefit.params
 
  if transparams: # transform parameters back
   params = self._transparams(params)
 
  self.transparams = False # so methods don't expect transf.
 
  normalized_cov_params = None # TODO: fix this
  armafit = ARMAResults(self, params, normalized_cov_params)
  armafit.mle_retvals = mlefit.mle_retvals
  armafit.mle_settings = mlefit.mle_settings
  armafit.mlefit = mlefit
  return ARMAResultsWrapper(armafit)
 
 
#NOTE: the length of endog changes when we give a difference to fit
#so model methods are not the same on unfit models as fit ones
#starting to think that order of model should be put in instantiation...
class ARIMA(ARMA):
 
 __doc__ = tsbase._tsa_doc % {"model" : _arima_model,
         "params" : _arima_params, "extra_params" : "",
         "extra_sections" : _armax_notes %
         {"Model" : "ARIMA"}}
 
 def __new__(cls, endog, order, exog=None, dates=None, freq=None,
    missing='none'):
  p, d, q = order
  if d == 0: # then we just use an ARMA model
   return ARMA(endog, (p, q), exog, dates, freq, missing)
  else:
   mod = super(ARIMA, cls).__new__(cls)
   mod.__init__(endog, order, exog, dates, freq, missing)
   return mod
 
 def __init__(self, endog, order, exog=None, dates=None, freq=None,
     missing='none'):
  p, d, q = order
  if d > 2:
   #NOTE: to make more general, need to address the d == 2 stuff
   # in the predict method
   raise ValueError("d > 2 is not supported")
  super(ARIMA, self).__init__(endog, (p, q), exog, dates, freq, missing)
  self.k_diff = d
  self._first_unintegrate = unintegrate_levels(self.endog[:d], d)
  self.endog = np.diff(self.endog, n=d)
  #NOTE: will check in ARMA but check again since differenced now
  _check_estimable(len(self.endog), p+q)
  if exog is not None:
   self.exog = self.exog[d:]
  if d == 1:
   self.data.ynames = 'D.' + self.endog_names
  else:
   self.data.ynames = 'D{0:d}.'.format(d) + self.endog_names
  # what about exog, should we difference it automatically before
  # super call?
 
 def _get_predict_start(self, start, dynamic):
  """
  """
  #TODO: remove all these getattr and move order specification to
  # class constructor
  k_diff = getattr(self, 'k_diff', 0)
  method = getattr(self, 'method', 'mle')
  k_ar = getattr(self, 'k_ar', 0)
  if start is None:
   if 'mle' in method and not dynamic:
    start = 0
   else:
    start = k_ar
  elif isinstance(start, int):
    start -= k_diff
    try: # catch when given an integer outside of dates index
     start = super(ARIMA, self)._get_predict_start(start,
                 dynamic)
    except IndexError:
     raise ValueError("start must be in series. "
          "got %d" % (start + k_diff))
  else: # received a date
   start = _validate(start, k_ar, k_diff, self.data.dates,
        method)
   start = super(ARIMA, self)._get_predict_start(start, dynamic)
  # reset date for k_diff adjustment
  self._set_predict_start_date(start + k_diff)
  return start
 
 def _get_predict_end(self, end, dynamic=False):
  """
  Returns last index to be forecast of the differenced array.
  Handling of inclusiveness should be done in the predict function.
  """
  end, out_of_sample = super(ARIMA, self)._get_predict_end(end, dynamic)
  if 'mle' not in self.method and not dynamic:
   end -= self.k_ar
 
  return end - self.k_diff, out_of_sample
 
 def fit(self, start_params=None, trend='c', method="css-mle",
   transparams=True, solver='lbfgs', maxiter=50, full_output=1,
   disp=5, callback=None, **kwargs):
  """
  Fits ARIMA(p,d,q) model by exact maximum likelihood via Kalman filter.
 
  Parameters
  ----------
  start_params : array-like, optional
   Starting parameters for ARMA(p,q). If None, the default is given
   by ARMA._fit_start_params. See there for more information.
  transparams : bool, optional
   Whehter or not to transform the parameters to ensure stationarity.
   Uses the transformation suggested in Jones (1980). If False,
   no checking for stationarity or invertibility is done.
  method : str {'css-mle','mle','css'}
   This is the loglikelihood to maximize. If "css-mle", the
   conditional sum of squares likelihood is maximized and its values
   are used as starting values for the computation of the exact
   likelihood via the Kalman filter. If "mle", the exact likelihood
   is maximized via the Kalman Filter. If "css" the conditional sum
   of squares likelihood is maximized. All three methods use
   `start_params` as starting parameters. See above for more
   information.
  trend : str {'c','nc'}
   Whether to include a constant or not. 'c' includes constant,
   'nc' no constant.
  solver : str or None, optional
   Solver to be used. The default is 'lbfgs' (limited memory
   Broyden-Fletcher-Goldfarb-Shanno). Other choices are 'bfgs',
   'newton' (Newton-Raphson), 'nm' (Nelder-Mead), 'cg' -
   (conjugate gradient), 'ncg' (non-conjugate gradient), and
   'powell'. By default, the limited memory BFGS uses m=12 to
   approximate the Hessian, projected gradient tolerance of 1e-8 and
   factr = 1e2. You can change these by using kwargs.
  maxiter : int, optional
   The maximum number of function evaluations. Default is 50.
  tol : float
   The convergence tolerance. Default is 1e-08.
  full_output : bool, optional
   If True, all output from solver will be available in
   the Results object's mle_retvals attribute. Output is dependent
   on the solver. See Notes for more information.
  disp : bool, optional
   If True, convergence information is printed. For the default
   l_bfgs_b solver, disp controls the frequency of the output during
   the iterations. disp < 0 means no output in this case.
  callback : function, optional
   Called after each iteration as callback(xk) where xk is the current
   parameter vector.
  kwargs
   See Notes for keyword arguments that can be passed to fit.
 
  Returns
  -------
  `statsmodels.tsa.arima.ARIMAResults` class
 
  See also
  --------
  statsmodels.base.model.LikelihoodModel.fit : for more information
   on using the solvers.
  ARIMAResults : results class returned by fit
 
  Notes
  ------
  If fit by 'mle', it is assumed for the Kalman Filter that the initial
  unkown state is zero, and that the inital variance is
  P = dot(inv(identity(m**2)-kron(T,T)),dot(R,R.T).ravel('F')).reshape(r,
  r, order = 'F')
 
  """
  arima_fit = super(ARIMA, self).fit(start_params, trend,
           method, transparams, solver,
           maxiter, full_output, disp,
           callback, **kwargs)
  normalized_cov_params = None # TODO: fix this?
  arima_fit = ARIMAResults(self, arima_fit._results.params,
         normalized_cov_params)
  arima_fit.k_diff = self.k_diff
  return ARIMAResultsWrapper(arima_fit)
 
 def predict(self, params, start=None, end=None, exog=None, typ='linear',
    dynamic=False):
  # go ahead and convert to an index for easier checking
  if isinstance(start, (string_types, datetime)):
   start = _index_date(start, self.data.dates)
  if typ == 'linear':
   if not dynamic or (start != self.k_ar + self.k_diff and
        start is not None):
    return super(ARIMA, self).predict(params, start, end, exog,
             dynamic)
   else:
    # need to assume pre-sample residuals are zero
    # do this by a hack
    q = self.k_ma
    self.k_ma = 0
    predictedvalues = super(ARIMA, self).predict(params, start,
                end, exog,
                dynamic)
    self.k_ma = q
    return predictedvalues
  elif typ == 'levels':
   endog = self.data.endog
   if not dynamic:
    predict = super(ARIMA, self).predict(params, start, end,
              dynamic)
 
    start = self._get_predict_start(start, dynamic)
    end, out_of_sample = self._get_predict_end(end)
    d = self.k_diff
    if 'mle' in self.method:
     start += d - 1 # for case where d == 2
     end += d - 1
     # add each predicted diff to lagged endog
     if out_of_sample:
      fv = predict[:-out_of_sample] + endog[start:end+1]
      if d == 2: #TODO: make a general solution to this
       fv += np.diff(endog[start - 1:end + 1])
      levels = unintegrate_levels(endog[-d:], d)
      fv = np.r_[fv,
         unintegrate(predict[-out_of_sample:],
            levels)[d:]]
     else:
      fv = predict + endog[start:end + 1]
      if d == 2:
       fv += np.diff(endog[start - 1:end + 1])
    else:
     k_ar = self.k_ar
     if out_of_sample:
      fv = (predict[:-out_of_sample] +
        endog[max(start, self.k_ar-1):end+k_ar+1])
      if d == 2:
       fv += np.diff(endog[start - 1:end + 1])
      levels = unintegrate_levels(endog[-d:], d)
      fv = np.r_[fv,
         unintegrate(predict[-out_of_sample:],
            levels)[d:]]
     else:
      fv = predict + endog[max(start, k_ar):end+k_ar+1]
      if d == 2:
       fv += np.diff(endog[start - 1:end + 1])
   else:
    #IFF we need to use pre-sample values assume pre-sample
    # residuals are zero, do this by a hack
    if start == self.k_ar + self.k_diff or start is None:
     # do the first k_diff+1 separately
     p = self.k_ar
     q = self.k_ma
     k_exog = self.k_exog
     k_trend = self.k_trend
     k_diff = self.k_diff
     (trendparam, exparams,
      arparams, maparams) = _unpack_params(params, (p, q),
               k_trend,
               k_exog,
               reverse=True)
     # this is the hack
     self.k_ma = 0
 
     predict = super(ARIMA, self).predict(params, start, end,
               exog, dynamic)
     if not start:
      start = self._get_predict_start(start, dynamic)
      start += k_diff
     self.k_ma = q
     return endog[start-1] + np.cumsum(predict)
    else:
     predict = super(ARIMA, self).predict(params, start, end,
               exog, dynamic)
     return endog[start-1] + np.cumsum(predict)
   return fv
 
  else: # pragma : no cover
   raise ValueError("typ %s not understood" % typ)
 
 predict.__doc__ = _arima_predict
 
 
class ARMAResults(tsbase.TimeSeriesModelResults):
 """
 Class to hold results from fitting an ARMA model.
 
 Parameters
 ----------
 model : ARMA instance
  The fitted model instance
 params : array
  Fitted parameters
 normalized_cov_params : array, optional
  The normalized variance covariance matrix
 scale : float, optional
  Optional argument to scale the variance covariance matrix.
 
 Returns
 --------
 **Attributes**
 
 aic : float
  Akaike Information Criterion
  :math:`-2*llf+2* df_model`
  where `df_model` includes all AR parameters, MA parameters, constant
  terms parameters on constant terms and the variance.
 arparams : array
  The parameters associated with the AR coefficients in the model.
 arroots : array
  The roots of the AR coefficients are the solution to
  (1 - arparams[0]*z - arparams[1]*z**2 -...- arparams[p-1]*z**k_ar) = 0
  Stability requires that the roots in modulus lie outside the unit
  circle.
 bic : float
  Bayes Information Criterion
  -2*llf + log(nobs)*df_model
  Where if the model is fit using conditional sum of squares, the
  number of observations `nobs` does not include the `p` pre-sample
  observations.
 bse : array
  The standard errors of the parameters. These are computed using the
  numerical Hessian.
 df_model : array
  The model degrees of freedom = `k_exog` + `k_trend` + `k_ar` + `k_ma`
 df_resid : array
  The residual degrees of freedom = `nobs` - `df_model`
 fittedvalues : array
  The predicted values of the model.
 hqic : float
  Hannan-Quinn Information Criterion
  -2*llf + 2*(`df_model`)*log(log(nobs))
  Like `bic` if the model is fit using conditional sum of squares then
  the `k_ar` pre-sample observations are not counted in `nobs`.
 k_ar : int
  The number of AR coefficients in the model.
 k_exog : int
  The number of exogenous variables included in the model. Does not
  include the constant.
 k_ma : int
  The number of MA coefficients.
 k_trend : int
  This is 0 for no constant or 1 if a constant is included.
 llf : float
  The value of the log-likelihood function evaluated at `params`.
 maparams : array
  The value of the moving average coefficients.
 maroots : array
  The roots of the MA coefficients are the solution to
  (1 + maparams[0]*z + maparams[1]*z**2 + ... + maparams[q-1]*z**q) = 0
  Stability requires that the roots in modules lie outside the unit
  circle.
 model : ARMA instance
  A reference to the model that was fit.
 nobs : float
  The number of observations used to fit the model. If the model is fit
  using exact maximum likelihood this is equal to the total number of
  observations, `n_totobs`. If the model is fit using conditional
  maximum likelihood this is equal to `n_totobs` - `k_ar`.
 n_totobs : float
  The total number of observations for `endog`. This includes all
  observations, even pre-sample values if the model is fit using `css`.
 params : array
  The parameters of the model. The order of variables is the trend
  coefficients and the `k_exog` exognous coefficients, then the
  `k_ar` AR coefficients, and finally the `k_ma` MA coefficients.
 pvalues : array
  The p-values associated with the t-values of the coefficients. Note
  that the coefficients are assumed to have a Student's T distribution.
 resid : array
  The model residuals. If the model is fit using 'mle' then the
  residuals are created via the Kalman Filter. If the model is fit
  using 'css' then the residuals are obtained via `scipy.signal.lfilter`
  adjusted such that the first `k_ma` residuals are zero. These zero
  residuals are not returned.
 scale : float
  This is currently set to 1.0 and not used by the model or its results.
 sigma2 : float
  The variance of the residuals. If the model is fit by 'css',
  sigma2 = ssr/nobs, where ssr is the sum of squared residuals. If
  the model is fit by 'mle', then sigma2 = 1/nobs * sum(v**2 / F)
  where v is the one-step forecast error and F is the forecast error
  variance. See `nobs` for the difference in definitions depending on the
  fit.
 """
 _cache = {}
 
 #TODO: use this for docstring when we fix nobs issue
 
 def __init__(self, model, params, normalized_cov_params=None, scale=1.):
  super(ARMAResults, self).__init__(model, params, normalized_cov_params,
           scale)
  self.sigma2 = model.sigma2
  nobs = model.nobs
  self.nobs = nobs
  k_exog = model.k_exog
  self.k_exog = k_exog
  k_trend = model.k_trend
  self.k_trend = k_trend
  k_ar = model.k_ar
  self.k_ar = k_ar
  self.n_totobs = len(model.endog)
  k_ma = model.k_ma
  self.k_ma = k_ma
  df_model = k_exog + k_trend + k_ar + k_ma
  self._ic_df_model = df_model + 1
  self.df_model = df_model
  self.df_resid = self.nobs - df_model
  self._cache = resettable_cache()
  self.constant = 0 #Added by me
 
 @cache_readonly
 def arroots(self):
  return np.roots(np.r_[1, -self.arparams])**-1
 
 @cache_readonly
 def maroots(self):
  return np.roots(np.r_[1, self.maparams])**-1
 
 @cache_readonly
 def arfreq(self):
  r"""
  Returns the frequency of the AR roots.
 
  This is the solution, x, to z = abs(z)*exp(2j*np.pi*x) where z are the
  roots.
  """
  z = self.arroots
  if not z.size:
   return
  return np.arctan2(z.imag, z.real) / (2*pi)
 
 @cache_readonly
 def mafreq(self):
  r"""
  Returns the frequency of the MA roots.
 
  This is the solution, x, to z = abs(z)*exp(2j*np.pi*x) where z are the
  roots.
  """
  z = self.maroots
  if not z.size:
   return
  return np.arctan2(z.imag, z.real) / (2*pi)
 
 @cache_readonly
 def arparams(self):
  k = self.k_exog + self.k_trend
  return self.params[k:k+self.k_ar]
 
 @cache_readonly
 def maparams(self):
  k = self.k_exog + self.k_trend
  k_ar = self.k_ar
  return self.params[k+k_ar:]
 
 @cache_readonly
 def llf(self):
  return self.model.loglike(self.params)
 
 @cache_readonly
 def bse(self):
  params = self.params
  hess = self.model.hessian(params)
  if len(params) == 1: # can't take an inverse, ensure 1d
   return np.sqrt(-1./hess[0])
  return np.sqrt(np.diag(-inv(hess)))
 
 def cov_params(self): # add scale argument?
  params = self.params
  hess = self.model.hessian(params)
  return -inv(hess)
 
 @cache_readonly
 def aic(self):
  return -2 * self.llf + 2 * self._ic_df_model
 
 @cache_readonly
 def bic(self):
  nobs = self.nobs
  return -2 * self.llf + np.log(nobs) * self._ic_df_model
 
 @cache_readonly
 def hqic(self):
  nobs = self.nobs
  return -2 * self.llf + 2 * np.log(np.log(nobs)) * self._ic_df_model
 
 @cache_readonly
 def fittedvalues(self):
  model = self.model
  endog = model.endog.copy()
  k_ar = self.k_ar
  exog = model.exog # this is a copy
  if exog is not None:
   if model.method == "css" and k_ar > 0:
    exog = exog[k_ar:]
  if model.method == "css" and k_ar > 0:
   endog = endog[k_ar:]
  fv = endog - self.resid
  # add deterministic part back in
  #k = self.k_exog + self.k_trend
  #TODO: this needs to be commented out for MLE with constant
  #if k != 0:
  # fv += dot(exog, self.params[:k])
  return fv
 
 @cache_readonly
 def resid(self):
  return self.model.geterrors(self.params)
 
 @cache_readonly
 def pvalues(self):
 #TODO: same for conditional and unconditional?
  df_resid = self.df_resid
  return t.sf(np.abs(self.tvalues), df_resid) * 2
 
 def predict(self, start=None, end=None, exog=None, dynamic=False):
  return self.model.predict(self.params, start, end, exog, dynamic)
 predict.__doc__ = _arma_results_predict
 
 def _forecast_error(self, steps):
  sigma2 = self.sigma2
  ma_rep = arma2ma(np.r_[1, -self.arparams],
       np.r_[1, self.maparams], nobs=steps)
 
  fcasterr = np.sqrt(sigma2 * np.cumsum(ma_rep**2))
  return fcasterr
 
 def _forecast_conf_int(self, forecast, fcasterr, alpha):
  const = norm.ppf(1 - alpha / 2.)
  conf_int = np.c_[forecast - const * fcasterr,
       forecast + const * fcasterr]
 
  return conf_int
 
 def forecast(self, steps=1, exog=None, alpha=.05):
  """
  Out-of-sample forecasts
 
  Parameters
  ----------
  steps : int
   The number of out of sample forecasts from the end of the
   sample.
  exog : array
   If the model is an ARMAX, you must provide out of sample
   values for the exogenous variables. This should not include
   the constant.
  alpha : float
   The confidence intervals for the forecasts are (1 - alpha) %
 
  Returns
  -------
  forecast : array
   Array of out of sample forecasts
  stderr : array
   Array of the standard error of the forecasts.
  conf_int : array
   2d array of the confidence interval for the forecast
  """
  if exog is not None:
   #TODO: make a convenience function for this. we're using the
   # pattern elsewhere in the codebase
   exog = np.asarray(exog)
   if self.k_exog == 1 and exog.ndim == 1:
    exog = exog[:, None]
   elif exog.ndim == 1:
    if len(exog) != self.k_exog:
     raise ValueError("1d exog given and len(exog) != k_exog")
    exog = exog[None, :]
   if exog.shape[0] != steps:
    raise ValueError("new exog needed for each step")
   # prepend in-sample exog observations
   exog = np.vstack((self.model.exog[-self.k_ar:, self.k_trend:],
        exog))
 
  forecast, ct = _arma_predict_out_of_sample(self.params,
            steps, self.resid, self.k_ar,
            self.k_ma, self.k_trend,
            self.k_exog, self.model.endog,
            exog, method=self.model.method)
  self.constant = ct
 
  # compute the standard errors
  fcasterr = self._forecast_error(steps)
  conf_int = self._forecast_conf_int(forecast, fcasterr, alpha)
 
  return forecast, fcasterr, conf_int
 
 def summary(self, alpha=.05):
  """Summarize the Model
 
  Parameters
  ----------
  alpha : float, optional
   Significance level for the confidence intervals.
 
  Returns
  -------
  smry : Summary instance
   This holds the summary table and text, which can be printed or
   converted to various output formats.
 
  See Also
  --------
  statsmodels.iolib.summary.Summary
  """
  from statsmodels.iolib.summary import Summary
  model = self.model
  title = model.__class__.__name__ + ' Model Results'
  method = model.method
  # get sample TODO: make better sample machinery for estimation
  k_diff = getattr(self, 'k_diff', 0)
  if 'mle' in method:
   start = k_diff
  else:
   start = k_diff + self.k_ar
  if self.data.dates is not None:
   dates = self.data.dates
   sample = [dates[start].strftime('%m-%d-%Y')]
   sample += ['- ' + dates[-1].strftime('%m-%d-%Y')]
  else:
   sample = str(start) + ' - ' + str(len(self.data.orig_endog))
 
  k_ar, k_ma = self.k_ar, self.k_ma
  if not k_diff:
   order = str((k_ar, k_ma))
  else:
   order = str((k_ar, k_diff, k_ma))
  top_left = [('Dep. Variable:', None),
     ('Model:', [model.__class__.__name__ + order]),
     ('Method:', [method]),
     ('Date:', None),
     ('Time:', None),
     ('Sample:', [sample[0]]),
     ('', [sample[1]])
     ]
 
  top_right = [
      ('No. Observations:', [str(len(self.model.endog))]),
      ('Log Likelihood', ["%#5.3f" % self.llf]),
      ('S.D. of innovations', ["%#5.3f" % self.sigma2**.5]),
      ('AIC', ["%#5.3f" % self.aic]),
      ('BIC', ["%#5.3f" % self.bic]),
      ('HQIC', ["%#5.3f" % self.hqic])]
 
  smry = Summary()
  smry.add_table_2cols(self, gleft=top_left, gright=top_right,
        title=title)
  smry.add_table_params(self, alpha=alpha, use_t=False)
 
  # Make the roots table
  from statsmodels.iolib.table import SimpleTable
 
  if k_ma and k_ar:
   arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
   mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
   stubs = arstubs + mastubs
   roots = np.r_[self.arroots, self.maroots]
   freq = np.r_[self.arfreq, self.mafreq]
  elif k_ma:
   mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
   stubs = mastubs
   roots = self.maroots
   freq = self.mafreq
  elif k_ar:
   arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
   stubs = arstubs
   roots = self.arroots
   freq = self.arfreq
  else: # 0,0 model
   stubs = []
  if len(stubs): # not 0, 0
   modulus = np.abs(roots)
   data = np.column_stack((roots.real, roots.imag, modulus, freq))
   roots_table = SimpleTable(data,
          headers=['   Real',
            '   Imaginary',
            '   Modulus',
            '  Frequency'],
          title="Roots",
          stubs=stubs,
          data_fmts=["%17.4f", "%+17.4fj",
             "%17.4f", "%17.4f"])
 
   smry.tables.append(roots_table)
  return smry
 
 def summary2(self, title=None, alpha=.05, float_format="%.4f"):
  """Experimental summary function for ARIMA Results
 
  Parameters
  -----------
  title : string, optional
   Title for the top table. If not None, then this replaces the
   default title
  alpha : float
   significance level for the confidence intervals
  float_format: string
   print format for floats in parameters summary
 
  Returns
  -------
  smry : Summary instance
   This holds the summary table and text, which can be printed or
   converted to various output formats.
 
  See Also
  --------
  statsmodels.iolib.summary2.Summary : class to hold summary
   results
 
  """
  from pandas import DataFrame
  # get sample TODO: make better sample machinery for estimation
  k_diff = getattr(self, 'k_diff', 0)
  if 'mle' in self.model.method:
   start = k_diff
  else:
   start = k_diff + self.k_ar
  if self.data.dates is not None:
   dates = self.data.dates
   sample = [dates[start].strftime('%m-%d-%Y')]
   sample += [dates[-1].strftime('%m-%d-%Y')]
  else:
   sample = str(start) + ' - ' + str(len(self.data.orig_endog))
 
  k_ar, k_ma = self.k_ar, self.k_ma
 
  # Roots table
  if k_ma and k_ar:
   arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
   mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
   stubs = arstubs + mastubs
   roots = np.r_[self.arroots, self.maroots]
   freq = np.r_[self.arfreq, self.mafreq]
  elif k_ma:
   mastubs = ["MA.%d" % i for i in range(1, k_ma + 1)]
   stubs = mastubs
   roots = self.maroots
   freq = self.mafreq
  elif k_ar:
   arstubs = ["AR.%d" % i for i in range(1, k_ar + 1)]
   stubs = arstubs
   roots = self.arroots
   freq = self.arfreq
  else: # 0, 0 order
   stubs = []
 
  if len(stubs):
   modulus = np.abs(roots)
   data = np.column_stack((roots.real, roots.imag, modulus, freq))
   data = DataFrame(data)
   data.columns = ['Real', 'Imaginary', 'Modulus', 'Frequency']
   data.index = stubs
 
  # Summary
  from statsmodels.iolib import summary2
  smry = summary2.Summary()
 
  # Model info
  model_info = summary2.summary_model(self)
  model_info['Method:'] = self.model.method
  model_info['Sample:'] = sample[0]
  model_info[' '] = sample[-1]
  model_info['S.D. of innovations:'] = "%#5.3f" % self.sigma2**.5
  model_info['HQIC:'] = "%#5.3f" % self.hqic
  model_info['No. Observations:'] = str(len(self.model.endog))
 
  # Parameters
  params = summary2.summary_params(self)
  smry.add_dict(model_info)
  smry.add_df(params, float_format=float_format)
  if len(stubs):
   smry.add_df(data, float_format="%17.4f")
  smry.add_title(results=self, title=title)
 
  return smry
 
 def plot_predict(self, start=None, end=None, exog=None, dynamic=False,
      alpha=.05, plot_insample=True, ax=None):
  from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
  _ = _import_mpl()
  fig, ax = create_mpl_ax(ax)
 
 
  # use predict so you set dates
  forecast = self.predict(start, end, exog, dynamic)
  # doing this twice. just add a plot keyword to predict?
  start = self.model._get_predict_start(start, dynamic=False)
  end, out_of_sample = self.model._get_predict_end(end, dynamic=False)
 
  if out_of_sample:
   steps = out_of_sample
   fc_error = self._forecast_error(steps)
   conf_int = self._forecast_conf_int(forecast[-steps:], fc_error,
            alpha)
 
 
  if hasattr(self.data, "predict_dates"):
   from pandas import TimeSeries
   forecast = TimeSeries(forecast, index=self.data.predict_dates)
   ax = forecast.plot(ax=ax, label='forecast')
  else:
   ax.plot(forecast)
 
  x = ax.get_lines()[-1].get_xdata()
  if out_of_sample:
   label = "{0:.0%} confidence interval".format(1 - alpha)
   ax.fill_between(x[-out_of_sample:], conf_int[:, 0], conf_int[:, 1],
       color='gray', alpha=.5, label=label)
 
  if plot_insample:
   ax.plot(x[:end + 1 - start], self.model.endog[start:end+1],
     label=self.model.endog_names)
 
  ax.legend(loc='best')
 
  return fig
 plot_predict.__doc__ = _plot_predict
 
 
class ARMAResultsWrapper(wrap.ResultsWrapper):
 _attrs = {}
 _wrap_attrs = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_attrs,
         _attrs)
 _methods = {}
 _wrap_methods = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_methods,
          _methods)
wrap.populate_wrapper(ARMAResultsWrapper, ARMAResults)
 
 
class ARIMAResults(ARMAResults):
 def predict(self, start=None, end=None, exog=None, typ='linear',
    dynamic=False):
  return self.model.predict(self.params, start, end, exog, typ, dynamic)
 predict.__doc__ = _arima_results_predict
 
 def _forecast_error(self, steps):
  sigma2 = self.sigma2
  ma_rep = arma2ma(np.r_[1, -self.arparams],
       np.r_[1, self.maparams], nobs=steps)
 
  fcerr = np.sqrt(np.cumsum(cumsum_n(ma_rep, self.k_diff)**2)*sigma2)
  return fcerr
 
 def _forecast_conf_int(self, forecast, fcerr, alpha):
  const = norm.ppf(1 - alpha/2.)
  conf_int = np.c_[forecast - const*fcerr, forecast + const*fcerr]
  return conf_int
 
 def forecast(self, steps=1, exog=None, alpha=.05):
  """
  Out-of-sample forecasts
 
  Parameters
  ----------
  steps : int
   The number of out of sample forecasts from the end of the
   sample.
  exog : array
   If the model is an ARIMAX, you must provide out of sample
   values for the exogenous variables. This should not include
   the constant.
  alpha : float
   The confidence intervals for the forecasts are (1 - alpha) %
 
  Returns
  -------
  forecast : array
   Array of out of sample forecasts
  stderr : array
   Array of the standard error of the forecasts.
  conf_int : array
   2d array of the confidence interval for the forecast
 
  Notes
  -----
  Prediction is done in the levels of the original endogenous variable.
  If you would like prediction of differences in levels use `predict`.
  """
  if exog is not None:
   if self.k_exog == 1 and exog.ndim == 1:
    exog = exog[:, None]
   if exog.shape[0] != steps:
    raise ValueError("new exog needed for each step")
   # prepend in-sample exog observations
   exog = np.vstack((self.model.exog[-self.k_ar:, self.k_trend:],
        exog))
  forecast, ct = _arma_predict_out_of_sample(self.params, steps, self.resid,
            self.k_ar, self.k_ma,
            self.k_trend, self.k_exog,
            self.model.endog,
            exog, method=self.model.method)
 
  #self.constant = ct
  d = self.k_diff
  endog = self.model.data.endog[-d:]
  forecast = unintegrate(forecast, unintegrate_levels(endog, d))[d:]
 
  # get forecast errors
  fcerr = self._forecast_error(steps)
  conf_int = self._forecast_conf_int(forecast, fcerr, alpha)
  return forecast, fcerr, conf_int
 
 def plot_predict(self, start=None, end=None, exog=None, dynamic=False,
      alpha=.05, plot_insample=True, ax=None):
  from statsmodels.graphics.utils import _import_mpl, create_mpl_ax
  _ = _import_mpl()
  fig, ax = create_mpl_ax(ax)
 
  # use predict so you set dates
  forecast = self.predict(start, end, exog, 'levels', dynamic)
  # doing this twice. just add a plot keyword to predict?
  start = self.model._get_predict_start(start, dynamic=dynamic)
  end, out_of_sample = self.model._get_predict_end(end, dynamic=dynamic)
 
  if out_of_sample:
   steps = out_of_sample
   fc_error = self._forecast_error(steps)
   conf_int = self._forecast_conf_int(forecast[-steps:], fc_error,
            alpha)
 
  if hasattr(self.data, "predict_dates"):
   from pandas import TimeSeries
   forecast = TimeSeries(forecast, index=self.data.predict_dates)
   ax = forecast.plot(ax=ax, label='forecast')
  else:
   ax.plot(forecast)
 
  x = ax.get_lines()[-1].get_xdata()
  if out_of_sample:
   label = "{0:.0%} confidence interval".format(1 - alpha)
   ax.fill_between(x[-out_of_sample:], conf_int[:, 0], conf_int[:, 1],
       color='gray', alpha=.5, label=label)
 
  if plot_insample:
   import re
   k_diff = self.k_diff
   label = re.sub("D\d*\.", "", self.model.endog_names)
   levels = unintegrate(self.model.endog,
         self.model._first_unintegrate)
   ax.plot(x[:end + 1 - start],
     levels[start + k_diff:end + k_diff + 1], label=label)
 
  ax.legend(loc='best')
 
  return fig
 
 plot_predict.__doc__ = _arima_plot_predict
 
 
class ARIMAResultsWrapper(ARMAResultsWrapper):
 pass
wrap.populate_wrapper(ARIMAResultsWrapper, ARIMAResults)
 
 
if __name__ == "__main__":
 import statsmodels.api as sm
 
 # simulate arma process
 from statsmodels.tsa.arima_process import arma_generate_sample
 y = arma_generate_sample([1., -.75], [1., .25], nsample=1000)
 arma = ARMA(y)
 res = arma.fit(trend='nc', order=(1, 1))
 
 np.random.seed(12345)
 y_arma22 = arma_generate_sample([1., -.85, .35], [1, .25, -.9],
         nsample=1000)
 arma22 = ARMA(y_arma22)
 res22 = arma22.fit(trend='nc', order=(2, 2))
 
 # test CSS
 arma22_css = ARMA(y_arma22)
 res22css = arma22_css.fit(trend='nc', order=(2, 2), method='css')
 
 data = sm.datasets.sunspots.load()
 ar = ARMA(data.endog)
 resar = ar.fit(trend='nc', order=(9, 0))
 
 y_arma31 = arma_generate_sample([1, -.75, -.35, .25], [.1],
         nsample=1000)
 
 arma31css = ARMA(y_arma31)
 res31css = arma31css.fit(order=(3, 1), method="css", trend="nc",
        transparams=True)
 
 y_arma13 = arma_generate_sample([1., -.75], [1, .25, -.5, .8],
         nsample=1000)
 arma13css = ARMA(y_arma13)
 res13css = arma13css.fit(order=(1, 3), method='css', trend='nc')
 
# check css for p < q and q < p
 y_arma41 = arma_generate_sample([1., -.75, .35, .25, -.3], [1, -.35],
         nsample=1000)
 arma41css = ARMA(y_arma41)
 res41css = arma41css.fit(order=(4, 1), trend='nc', method='css')
 
 y_arma14 = arma_generate_sample([1, -.25], [1., -.75, .35, .25, -.3],
         nsample=1000)
 arma14css = ARMA(y_arma14)
 res14css = arma14css.fit(order=(4, 1), trend='nc', method='css')
 
 # ARIMA Model
 from statsmodels.datasets import webuse
 dta = webuse('wpi1')
 wpi = dta['wpi']
 
 mod = ARIMA(wpi, (1, 1, 1)).fit()

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