Can discrete time series be non-stationary?












1












$begingroup$


We all know about time series that grow over time, but it seems like we only ever see continuous values such as the plot shown below. Example of non-stationary time series



But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?






time-series stationarity







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    We all know about time series that grow over time, but it seems like we only ever see continuous values such as the plot shown below. Example of non-stationary time series



    But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?






    time-series stationarity







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      We all know about time series that grow over time, but it seems like we only ever see continuous values such as the plot shown below. Example of non-stationary time series



      But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?






      time-series stationarity







      share|cite|improve this question









      $endgroup$




      We all know about time series that grow over time, but it seems like we only ever see continuous values such as the plot shown below. Example of non-stationary time series



      But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?






      time-series stationarity




      time-series stationarity






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 3 hours ago









      Linas KleizaLinas Kleiza

      294




      294






















          1 Answer
          1






          active

          oldest

          votes


















          3












          $begingroup$

          Per the title, certainly discrete series can be nonstationary.



          The total count of events in a Poisson process is one example - let $N_t$ be the number of events by time $t$ and consider the time series formed by looking at $N_t$ at integer times: $N_1$, $N_2$, $N_3$,...



          We have that $E(N_t)=lambda t$, $text{Var}(N_t)=lambda t$; so the above time series is not even weakly stationary.




          But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?




          Sure, you could readily construct a binary time series process that was non-stationary.



          Let's make an example. Let $Y_t$ be the random binary value at time $t$ ($t=1,2,...$). Let $S_t$ bet the number of $1$'s observed up to time $t$; i.e. $S_0=0,S_t=sum_{i=1}^t Y_i,,: t = 1,2,...$.



          Now make $p_{t+1}=P(Y_{t+1}=1|S_t)=frac{1}{1+exp(-alpha S_t)}$ for some $alpha>0$.



          So $p_1=0.5$. It will remain at $0.5$ until you observe a $1$, at which point $p$ increases. e.g. if $alpha=0.3$, then after the first $1$ is observed, $p$ increases to about $0.574$ whereas if $alpha=0.1$, $p$ only increases to about $0.525$.



          (There are many other ways to make nonstationary binary series)



          Here's an example using the above scheme with $alpha=0.005$, for the first 1000 observations. You can see that it starts out with a similar density of 0 and 1 values but the 1-strip rapidly starts to get heavier and the 0-strip gets lighter and lighter, until near the end the 0's have become quite rare.



          The conditional probability of a 1 for the last observation ($y_{1000}$) was about 0.979.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks for the example. But for continuous non-stationary time series you can take the differences to make it stationary. Is there an analogous procedure for non-stationary discrete time series?
            $endgroup$
            – Linas Kleiza
            50 mins ago






          • 1




            $begingroup$
            Not in general, no; it would work with the Poisson example I raised, but cannot work for the binary one. (For that matter, it's not always the case that differencing results in stationarity with continuous series. For some very particular sets of assumptions differencing produces stationarity, but not all)
            $endgroup$
            – Glen_b♦
            37 mins ago











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "65"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: false,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: null,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fstats.stackexchange.com%2fquestions%2f389230%2fcan-discrete-time-series-be-non-stationary%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3












          $begingroup$

          Per the title, certainly discrete series can be nonstationary.



          The total count of events in a Poisson process is one example - let $N_t$ be the number of events by time $t$ and consider the time series formed by looking at $N_t$ at integer times: $N_1$, $N_2$, $N_3$,...



          We have that $E(N_t)=lambda t$, $text{Var}(N_t)=lambda t$; so the above time series is not even weakly stationary.




          But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?




          Sure, you could readily construct a binary time series process that was non-stationary.



          Let's make an example. Let $Y_t$ be the random binary value at time $t$ ($t=1,2,...$). Let $S_t$ bet the number of $1$'s observed up to time $t$; i.e. $S_0=0,S_t=sum_{i=1}^t Y_i,,: t = 1,2,...$.



          Now make $p_{t+1}=P(Y_{t+1}=1|S_t)=frac{1}{1+exp(-alpha S_t)}$ for some $alpha>0$.



          So $p_1=0.5$. It will remain at $0.5$ until you observe a $1$, at which point $p$ increases. e.g. if $alpha=0.3$, then after the first $1$ is observed, $p$ increases to about $0.574$ whereas if $alpha=0.1$, $p$ only increases to about $0.525$.



          (There are many other ways to make nonstationary binary series)



          Here's an example using the above scheme with $alpha=0.005$, for the first 1000 observations. You can see that it starts out with a similar density of 0 and 1 values but the 1-strip rapidly starts to get heavier and the 0-strip gets lighter and lighter, until near the end the 0's have become quite rare.



          The conditional probability of a 1 for the last observation ($y_{1000}$) was about 0.979.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks for the example. But for continuous non-stationary time series you can take the differences to make it stationary. Is there an analogous procedure for non-stationary discrete time series?
            $endgroup$
            – Linas Kleiza
            50 mins ago






          • 1




            $begingroup$
            Not in general, no; it would work with the Poisson example I raised, but cannot work for the binary one. (For that matter, it's not always the case that differencing results in stationarity with continuous series. For some very particular sets of assumptions differencing produces stationarity, but not all)
            $endgroup$
            – Glen_b♦
            37 mins ago
















          3












          $begingroup$

          Per the title, certainly discrete series can be nonstationary.



          The total count of events in a Poisson process is one example - let $N_t$ be the number of events by time $t$ and consider the time series formed by looking at $N_t$ at integer times: $N_1$, $N_2$, $N_3$,...



          We have that $E(N_t)=lambda t$, $text{Var}(N_t)=lambda t$; so the above time series is not even weakly stationary.




          But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?




          Sure, you could readily construct a binary time series process that was non-stationary.



          Let's make an example. Let $Y_t$ be the random binary value at time $t$ ($t=1,2,...$). Let $S_t$ bet the number of $1$'s observed up to time $t$; i.e. $S_0=0,S_t=sum_{i=1}^t Y_i,,: t = 1,2,...$.



          Now make $p_{t+1}=P(Y_{t+1}=1|S_t)=frac{1}{1+exp(-alpha S_t)}$ for some $alpha>0$.



          So $p_1=0.5$. It will remain at $0.5$ until you observe a $1$, at which point $p$ increases. e.g. if $alpha=0.3$, then after the first $1$ is observed, $p$ increases to about $0.574$ whereas if $alpha=0.1$, $p$ only increases to about $0.525$.



          (There are many other ways to make nonstationary binary series)



          Here's an example using the above scheme with $alpha=0.005$, for the first 1000 observations. You can see that it starts out with a similar density of 0 and 1 values but the 1-strip rapidly starts to get heavier and the 0-strip gets lighter and lighter, until near the end the 0's have become quite rare.



          The conditional probability of a 1 for the last observation ($y_{1000}$) was about 0.979.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks for the example. But for continuous non-stationary time series you can take the differences to make it stationary. Is there an analogous procedure for non-stationary discrete time series?
            $endgroup$
            – Linas Kleiza
            50 mins ago






          • 1




            $begingroup$
            Not in general, no; it would work with the Poisson example I raised, but cannot work for the binary one. (For that matter, it's not always the case that differencing results in stationarity with continuous series. For some very particular sets of assumptions differencing produces stationarity, but not all)
            $endgroup$
            – Glen_b♦
            37 mins ago














          3












          3








          3





          $begingroup$

          Per the title, certainly discrete series can be nonstationary.



          The total count of events in a Poisson process is one example - let $N_t$ be the number of events by time $t$ and consider the time series formed by looking at $N_t$ at integer times: $N_1$, $N_2$, $N_3$,...



          We have that $E(N_t)=lambda t$, $text{Var}(N_t)=lambda t$; so the above time series is not even weakly stationary.




          But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?




          Sure, you could readily construct a binary time series process that was non-stationary.



          Let's make an example. Let $Y_t$ be the random binary value at time $t$ ($t=1,2,...$). Let $S_t$ bet the number of $1$'s observed up to time $t$; i.e. $S_0=0,S_t=sum_{i=1}^t Y_i,,: t = 1,2,...$.



          Now make $p_{t+1}=P(Y_{t+1}=1|S_t)=frac{1}{1+exp(-alpha S_t)}$ for some $alpha>0$.



          So $p_1=0.5$. It will remain at $0.5$ until you observe a $1$, at which point $p$ increases. e.g. if $alpha=0.3$, then after the first $1$ is observed, $p$ increases to about $0.574$ whereas if $alpha=0.1$, $p$ only increases to about $0.525$.



          (There are many other ways to make nonstationary binary series)



          Here's an example using the above scheme with $alpha=0.005$, for the first 1000 observations. You can see that it starts out with a similar density of 0 and 1 values but the 1-strip rapidly starts to get heavier and the 0-strip gets lighter and lighter, until near the end the 0's have become quite rare.



          The conditional probability of a 1 for the last observation ($y_{1000}$) was about 0.979.






          share|cite|improve this answer











          $endgroup$



          Per the title, certainly discrete series can be nonstationary.



          The total count of events in a Poisson process is one example - let $N_t$ be the number of events by time $t$ and consider the time series formed by looking at $N_t$ at integer times: $N_1$, $N_2$, $N_3$,...



          We have that $E(N_t)=lambda t$, $text{Var}(N_t)=lambda t$; so the above time series is not even weakly stationary.




          But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?




          Sure, you could readily construct a binary time series process that was non-stationary.



          Let's make an example. Let $Y_t$ be the random binary value at time $t$ ($t=1,2,...$). Let $S_t$ bet the number of $1$'s observed up to time $t$; i.e. $S_0=0,S_t=sum_{i=1}^t Y_i,,: t = 1,2,...$.



          Now make $p_{t+1}=P(Y_{t+1}=1|S_t)=frac{1}{1+exp(-alpha S_t)}$ for some $alpha>0$.



          So $p_1=0.5$. It will remain at $0.5$ until you observe a $1$, at which point $p$ increases. e.g. if $alpha=0.3$, then after the first $1$ is observed, $p$ increases to about $0.574$ whereas if $alpha=0.1$, $p$ only increases to about $0.525$.



          (There are many other ways to make nonstationary binary series)



          Here's an example using the above scheme with $alpha=0.005$, for the first 1000 observations. You can see that it starts out with a similar density of 0 and 1 values but the 1-strip rapidly starts to get heavier and the 0-strip gets lighter and lighter, until near the end the 0's have become quite rare.



          The conditional probability of a 1 for the last observation ($y_{1000}$) was about 0.979.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 2 hours ago

























          answered 3 hours ago









          Glen_b♦Glen_b

          210k22401743




          210k22401743












          • $begingroup$
            Thanks for the example. But for continuous non-stationary time series you can take the differences to make it stationary. Is there an analogous procedure for non-stationary discrete time series?
            $endgroup$
            – Linas Kleiza
            50 mins ago






          • 1




            $begingroup$
            Not in general, no; it would work with the Poisson example I raised, but cannot work for the binary one. (For that matter, it's not always the case that differencing results in stationarity with continuous series. For some very particular sets of assumptions differencing produces stationarity, but not all)
            $endgroup$
            – Glen_b♦
            37 mins ago


















          • $begingroup$
            Thanks for the example. But for continuous non-stationary time series you can take the differences to make it stationary. Is there an analogous procedure for non-stationary discrete time series?
            $endgroup$
            – Linas Kleiza
            50 mins ago






          • 1




            $begingroup$
            Not in general, no; it would work with the Poisson example I raised, but cannot work for the binary one. (For that matter, it's not always the case that differencing results in stationarity with continuous series. For some very particular sets of assumptions differencing produces stationarity, but not all)
            $endgroup$
            – Glen_b♦
            37 mins ago
















          $begingroup$
          Thanks for the example. But for continuous non-stationary time series you can take the differences to make it stationary. Is there an analogous procedure for non-stationary discrete time series?
          $endgroup$
          – Linas Kleiza
          50 mins ago




          $begingroup$
          Thanks for the example. But for continuous non-stationary time series you can take the differences to make it stationary. Is there an analogous procedure for non-stationary discrete time series?
          $endgroup$
          – Linas Kleiza
          50 mins ago




          1




          1




          $begingroup$
          Not in general, no; it would work with the Poisson example I raised, but cannot work for the binary one. (For that matter, it's not always the case that differencing results in stationarity with continuous series. For some very particular sets of assumptions differencing produces stationarity, but not all)
          $endgroup$
          – Glen_b♦
          37 mins ago




          $begingroup$
          Not in general, no; it would work with the Poisson example I raised, but cannot work for the binary one. (For that matter, it's not always the case that differencing results in stationarity with continuous series. For some very particular sets of assumptions differencing produces stationarity, but not all)
          $endgroup$
          – Glen_b♦
          37 mins ago


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Cross Validated!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fstats.stackexchange.com%2fquestions%2f389230%2fcan-discrete-time-series-be-non-stationary%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          -

          Popular posts from this blog

          サソリ

          Image
          この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。 出典を追加して記事の信頼性向上にご協力ください。 （ 2019年1月 ） 「 サソリ 」のその他の用法については「サソリ (曖昧さ回避)」をご覧ください。 「 スコーピオン 」と「 スコルピオン 」はこの項目へ転送されています。その他の用法については「スコーピオン」をご覧ください。 サソリ目 生息年代: 430–0 Ma PreЄ Є O S D C P T J K Pg N シルル紀前期-現世 ダイオウサソリ（ Pandinus Imperator ） 分類 界 : 動物界 Animalia 門 : 節足動物門 Arthropoda 亜門 : 鋏角亜門 Chelicerata 綱 : クモガタ綱 Arachnida 目 : サソリ目 Scorpiones 学名 Scorpiones C. L. Koch, 1837 英名 Scorpion 上科 Pseudochactoidea キョクトウサソリ上科 Buthoidea Chaeriloidea Chactoidea Iuroidea Scorpionoidea サソリ （ サソリ類 、 蠍 、 蝎 、 scorpion ）は、鋏角亜門・クモガタ綱の サソリ目 （ Scorpiones ）に属する節足動物の総称である。体の前端に鋏型の触肢、後端に毒針を有する捕食者である。 1700種以上を含め、最古の化石記録は4億3千万年前のシルル紀まで遡る。有毒生物として広く知られるが、人間の命にかかわる毒を持つものはごく一部の種類に過ぎない。 目次 1 形態 1.1 前体 1.2 後体 1.2.1 中体 1.2.2 終体 2 雌雄 3 生態 3.1 天敵 3.2 生殖 4 分布 5 毒 6 蛍光現象 7 分類
          Read more

          広島県道265号伴広島線

          Image
          一般県道 広島県道265号伴広島線 路線延長 km 起点 広島県広島市安佐南区伴東7丁目 終点 広島県広島市中区榎町 接続する 主な道路 （記法） 広島県道38号広島豊平線 広島県道71号広島湯来線 山陽自動車道五日市IC 国道183号（ 国道261号重複） ■テンプレート（■ノート ■使い方） ■PJ道路 広島県道265号伴広島線 （ひろしまけんどう265ごう ともひろしません）は広島県広島市安佐南区から広島県広島市中区を結ぶ一般県道である。 目次 1 路線データ 2 通過する自治体 3 見どころ 4 接続道路 4.1 本線 4.2 旧道 5 別名 6 関連項目 路線データ 本線 起点：広島市安佐南区伴東7丁目・大原下橋南詰交差点（広島県道38号広島豊平線（本線・旧道）交点） 終点：広島市中区榎町・十日市交差点（国道183号（国道261号重用）交点） 総延長： 旧道 起点：広島市安佐南区伴東4丁目・新細坂交差点（広島県道38号広島豊平線交点） 終点：広島市安佐南区伴中央4丁目・寺組橋(南)交差点（広島県道265号伴広島線（本線）交点） 総延長： 通過する自治体 広島県 広島市（安佐南区、佐伯区、西区、中区） 見どころ 広島高速交通広島新交通1号線（アストラムライン） 広島広域公園 広島広域公園陸上競技場（広島ビッグアーチ） 己斐峠 己斐橋（太田川放水路） 天満橋（天満川） 接続道路 本線 広島県道38号広島豊平線（本線・旧道）（広島市安佐南区伴東7丁目・大原下橋南詰交差点〔起点〕） 広島県道71号広島湯来線（本線）（広島市安佐南区伴中央2丁目・伴交番入口交差点 - 広島市安佐南区伴中央4丁目・寺組橋(南)交差点間で重用） 広島県道71号広島湯来線（旧道）（広島県道265号伴広島線（旧道）重用）（広島市安佐南区伴中央4丁目・寺組橋(南)交差点） 広島県道71号広島湯来線（広島市安佐南区伴中央4丁目・寺組橋(南)交差点 - 広島市佐伯区五日市町石内・五月が丘(南)交差点間で重用）
          Read more

          Setup Asymptote in Texstudio

          Image
          0 Good time of the day! I am trying to compile the following file: documentclass[12pt]{article} usepackage[inline]{asymptote} begin{document} Here is a venn diagram produced with Asymptote, drawn to width 4cm: defA{A} defB{V{B}} begin{figure} begin{center} begin{asy} size(4cm,0); pen colour1=red; pen colour2=green; pair z0=(0,0); pair z1=(-1,0); pair z2=(1,0); real r=1.5; path c1=circle(z1,r); path c2=circle(z2,r); fill(c1,colour1); fill(c2,colour2); picture intersection=new picture; fill(intersection,c1,colour1+colour2); clip(intersection,c2); add(intersection); draw(c1); draw(c2); //draw("$A$",box,z1)
          Read more