拉爾斯·馬爾克斯 伊芙熱尼婭·波布科娃

鄧成汝 李帥崢 林 戈 [譯]

1 引言：空間容量的理論

在任何城市模型中，最重要的變量都是距離和吸引力[1]?？臻g句法研究促進(jìn)了幾何關(guān)系描述新方法和距離度量方法的發(fā)展。它們開辟了新的研究領(lǐng)域，尤其是在捕獲行人運(yùn)動(dòng)軌跡方面[2]。但是，對(duì)吸引點(diǎn)的描述和度量，在該領(lǐng)域卻一直未受重點(diǎn)關(guān)注。

在更早的一篇論文中，我們將吸引力解釋為以密度和差異性的形式呈現(xiàn)的、對(duì)距離變量進(jìn)行補(bǔ)充的空間形態(tài)變量。其中，密度和建筑相關(guān)，而差異性與地塊系統(tǒng)相關(guān)。在本文中，我們將把重點(diǎn)放在較少受到研究關(guān)注的差異性變量上。早期的實(shí)證研究表明，土地劃分成地塊的程度，與社會(huì)經(jīng)濟(jì)要素（例如居民和經(jīng)濟(jì)活動(dòng)）的多樣性之間存在相關(guān)性。在此基礎(chǔ)上，我們會(huì)在此把對(duì)斯德哥爾摩廣泛實(shí)證研究的結(jié)果展現(xiàn)出來(lái)，目的在于為提出描述空間形態(tài)差異性的空間變量鋪平道路。這個(gè)變量會(huì)直接影響社會(huì)經(jīng)濟(jì)多樣性。本研究將會(huì)關(guān)注地塊系統(tǒng)布局的衡量指標(biāo)和經(jīng)濟(jì)活動(dòng)的多樣性模型之間的相關(guān)性。

顯然，關(guān)于城市多樣性眾多辯論源于簡(jiǎn)·雅各布斯（Jane Jacobs）的著作。在她所有關(guān)于城市的著作中，多樣性都是她的中心主題。其中最廣為人知的，便是《美國(guó)大城市的死與生》(The Death and Life of Great American Cities)[3]，以及她在那本書中提出的產(chǎn)生城市多樣性的4個(gè)條件—一個(gè)以上的主要功能；小街塊；不同年代的建筑物；密集人口—體現(xiàn)出了這一主題。在這4 個(gè)條件之中，正如接下來(lái)會(huì)看到的那樣，若需要為我們的模型提出一種度量差異性程度的指標(biāo)，我們會(huì)發(fā)現(xiàn)，“不同年代的建筑物”這一條件—也許有點(diǎn)讓人吃驚，是最有希望被選中的。其原因是，盡管“一個(gè)以上的主要功能”這一條件確實(shí)對(duì)城市的多樣性產(chǎn)生很大的影響，但它涉及的純粹只是多樣性的產(chǎn)生過程，而沒有與我們想在此尋找的那種空間形式有任何明顯的聯(lián)系。另一方面，“小街塊”的概念，更多涉及的是可達(dá)性，或者更準(zhǔn)確來(lái)說，是前面提到的在空間句法理論中所涉及的認(rèn)知上的距離的布局。[4]最后，密集人口的概念很顯然涉及的只是通常意義的密度的概念，而這已經(jīng)得到充分的討論?！安煌甏慕ㄖ铩钡母拍?，從另一方面來(lái)看，似乎在根源上與其他條件有所不同。而且乍一看，這一概念本身看起來(lái)既是理論上最乏味的，也是最不具可操作性的——我們?nèi)绾我?guī)劃出具有不同年代建筑物的城市，或者就此事發(fā)展出一套理論呢？但是，這正是我們將要嘗試的。

這里的陷阱是，固執(zhí)地只把建筑物視為事物，而不視作過程。向后者的轉(zhuǎn)變意味著要把建筑物營(yíng)造的空間轉(zhuǎn)換為建筑物在其中生長(zhǎng)的空間，即不同的領(lǐng)域定義了我們稱之為地塊的物權(quán)。這些空間更多的是從制度上被定義，而不是從實(shí)體空間上定義的。但即便如此，它們無(wú)疑也代表了城市空間中較為中心和常見的類別。如果我們開始考慮這點(diǎn)，我們就會(huì)意識(shí)到，將空間劃分成若干個(gè)單獨(dú)的空間，對(duì)任何建筑工作來(lái)說，都具有多么基礎(chǔ)的作用——如果沒有用墻壁劃分空間的藝術(shù)，那么建筑又是什么呢？這背后的主要原理正是產(chǎn)生多樣性的目的，即為單獨(dú)的不同類別的“事物”或“活動(dòng)”，生成各自獨(dú)立的空間。[5-6]這顯然是將建筑物劃分為獨(dú)立房間背后的主要理由。

因此，對(duì)土地的水平劃分（無(wú)論是在實(shí)體空間上，還是在制度上），作為一種基本的空間手法似乎是有充分根據(jù)的。與此同時(shí)，增加變化的方式，可以采取類似于提高密度的基本手法——在豎直方向增加樓層。如果我們將這個(gè)概念更具體地引入到城市空間形態(tài)模型中，我們可以提出下面的論點(diǎn)：地塊的特定區(qū)域，作為由一特定集合的物權(quán)所界定的一塊土地，代表了城市中一個(gè)角色的存在。這個(gè)角色存在的形式可能是這塊土地的所有者，或者是需要在城市的某一特定空間中進(jìn)行各種活動(dòng)的類似角色。只要維持在不同機(jī)構(gòu)設(shè)定的框架內(nèi)，例如，地方規(guī)劃法規(guī)，或特定土地的特定物權(quán)內(nèi)，特定角色就可以在此特定區(qū)域內(nèi)自由行動(dòng)。

此外，此類角色通常會(huì)制定一項(xiàng)特定策略，以進(jìn)一步發(fā)展和維護(hù)其物權(quán)。因此，與地塊相對(duì)較少的地區(qū)相比，地塊相對(duì)較多的地區(qū)似乎有潛力容納更多這樣的角色，從而有更多的策略來(lái)開發(fā)和維護(hù)其地塊，而且這很可能也意味著這樣的策略更具備多樣性。最后，與具有相對(duì)較少地塊，從而擁有更少角色和更少發(fā)展策略的區(qū)域相比，這種區(qū)域似乎更具備產(chǎn)生出多樣化內(nèi)容的潛力。因此，似乎正是土地的這種劃分方式，以及因此在角色和策略中產(chǎn)生出的潛在的多樣性，隨著時(shí)間的流逝，能使現(xiàn)存建筑產(chǎn)生出更大的變化，也就是雅各布斯“不同年代建筑物”的概念。

正是這種假設(shè)使我們相信，在地塊的數(shù)量、形狀、大小和布局上，我們可以識(shí)別出空間形態(tài)的一個(gè)變量。這個(gè)變量與城市多樣性有直接關(guān)系，并對(duì)這種多樣性產(chǎn)生直接的影響。此外，我們提出，將某個(gè)區(qū)域中的地塊數(shù)量稱為該區(qū)域的空間容量，即容納差異的能力。[5-6]其中，更高的容量會(huì)更有潛力產(chǎn)生異質(zhì)化的內(nèi)容，而更低的容量則更有可能帶來(lái)同質(zhì)化的內(nèi)容。顯然，其他因素（例如土地使用法規(guī)，或其他空間變量，還有街道的中心性和建筑物密度）也可以覆蓋地塊系統(tǒng)的空間形態(tài)在此所帶來(lái)的影響，但是在研究中，我們?cè)噲D隔離開來(lái)的，是這個(gè)變量給城市實(shí)體或過程所帶來(lái)的特定 影響。

在下文中，我們將論述出一項(xiàng)實(shí)證研究。該研究將進(jìn)行關(guān)聯(lián)性分析，將不同類型的多樣性作為因變量，將被衡量為地塊可達(dá)性的差異性程度作為自變量。本文的框架如下：接下來(lái)的兩節(jié)將介紹衡量社會(huì)經(jīng)濟(jì)多樣性背景理論的復(fù)雜性，其中會(huì)特別關(guān)注尺度和分類問題。這些章節(jié)將會(huì)為后文實(shí)證研究中所作出的選擇打好基礎(chǔ)。此后，我們將對(duì)實(shí)證研究的研究方法進(jìn)行概述，包括：(1)如何衡量作為因變量的社會(huì)經(jīng)濟(jì)多樣性；(2)如何衡量作為自變量的空間差異性；(3)如何能夠解決尺度問題；(4)如何構(gòu)建統(tǒng)計(jì)模型以隔離差異性變量帶來(lái)的影響；(5)對(duì)統(tǒng)計(jì)檢驗(yàn)的描述。此后將會(huì)展示本研究的研究結(jié)果。最后的結(jié)論將會(huì)結(jié)合前面介紹過的理論來(lái)對(duì)研究結(jié)果進(jìn)行討論，并給將來(lái)的研究提供建議。

2 多樣性與尺度：何為尺度

尺度這一術(shù)語(yǔ)應(yīng)用于許多領(lǐng)域，并且通常在多學(xué)科之間的解釋不盡相同。其相關(guān)概念如：等級(jí)、分辨率、范圍和層次結(jié)構(gòu)，也通常用作其替換詞或同義詞。在城市研究中，對(duì)尺度一詞的廣泛引用，也促使其產(chǎn)生了各種分類，如“本地—全球”、“微觀—中觀—宏觀”、“鄰里—地區(qū)—城市—區(qū)域”。然而，這些術(shù)語(yǔ)由于其內(nèi)在的相對(duì)性而具有模糊性。對(duì)于特定的情況，它們的定義通常具有關(guān)聯(lián)性，而對(duì)于某些術(shù)語(yǔ)，如“本地”的理解則因城市而異。這將成為一個(gè)問題，因?yàn)樵谌魏纬鞘薪；蚩臻g分析的研究中，尺度的選擇將從根本上影響研究的分析和解釋。

根據(jù)邁克·巴蒂(Mike Batty)的說法，尺度基本上是指兩件事，“我們觀察城市的分辨率水平，本質(zhì)上是地圖比例，以及不同大小的地點(diǎn)或城市的功能差異級(jí)別”。[7]也就是說，尺度可能涉及分辨率或大小，但兩者也很容易被混淆。例如，上面列出的所有分類，例如與高度相關(guān)的宏觀—中觀—微觀集，通常指分辨率，尤其是在具體城市環(huán)境中應(yīng)用時(shí)可以容易地用于大小分類。例如，鄰里作為與“區(qū)域”或“城市”尺度相關(guān)的特定尺度，在我們談到“鄰里”時(shí)，我們是在談?wù)摮鞘械奶囟ǚ直媛?，即放大一個(gè)區(qū)域，以便我們可以識(shí)別更多有關(guān)該區(qū)域情況的詳細(xì)信息，但這當(dāng)然不會(huì)使我們所討論的城市變小或變大。盡管如此，在這個(gè)過程中，我們很容易在一開始把特定的鄰里想象成一個(gè)小城市，也就是說，我們已經(jīng)改變了城市的尺度，但顯然不是這樣。

我們可能會(huì)陷入一個(gè)包羅萬(wàn)象、更宏偉或更抽象的城市，這樣做與現(xiàn)實(shí)相對(duì)應(yīng)的直接風(fēng)險(xiǎn)就是將尺度稱為等級(jí)。用艾倫·威爾遜的話說：“尺度是一種等級(jí)，在這方面清晰的構(gòu)想至關(guān)重要?！币粋€(gè)典型的例子是將城市現(xiàn)象分為微觀、中觀和宏觀尺度，這在概念上意味著有一系列的城市是相互重疊的，盡管我們知道事實(shí)并非如此。當(dāng)再次反思時(shí)我們會(huì)立即意識(shí)到這是無(wú)稽之談，但即便如此，從概念上講，這種失誤已經(jīng)形成，也很難記住我們所說并不是真的。例如，當(dāng)新空間經(jīng)濟(jì)的引入者談?wù)撾x心力和向心力可以集聚城市經(jīng)濟(jì)活動(dòng)的時(shí)候，我們被誘導(dǎo)著將此想象成某種宏觀尺度的作用力徘徊在城市中。[8]實(shí)際上，這些作用力是在綜合宏觀尺度上分析確定的，但它們肯定不是在宏觀尺度上表現(xiàn)出來(lái)的。相反，這種作用力雖然在宏觀上留有明顯痕跡，但其必然植根于人類微觀層面的日常活動(dòng)中。

此外，還需要明確我們討論的尺度的實(shí)質(zhì)是什么。繼威爾遜之后，可以說，空間分析人員在分辨率上面臨3 方面問題，“需要將相關(guān)系統(tǒng)的構(gòu)成進(jìn)行界定和分類；其中很多構(gòu)成部分需要在空間中定位；且其行為需隨著時(shí)間動(dòng)態(tài)來(lái)進(jìn)行描述”。這導(dǎo)致需要確定尺度的三個(gè)特點(diǎn)：可劃分性（分類的數(shù)量和廣度）、空間性（實(shí)體所在的區(qū)域單元大?。┖蜁r(shí)間性（為縱向描述和分析提供基礎(chǔ)的單位時(shí)間長(zhǎng)度）。時(shí)間解析本身就是一個(gè)大問題，因?yàn)榇蠖鄶?shù)空間分析簡(jiǎn)單地排除了時(shí)間維度，從而產(chǎn)生了我們所習(xí)慣的典型的城市靜態(tài)描述。原則上來(lái)說，這些靜態(tài)描述是非常不現(xiàn)實(shí)的，且對(duì)于多數(shù)使用此描述的研究得出的結(jié)論是有風(fēng)險(xiǎn)的，然而我們?cè)诖蠖鄶?shù)城市研究中用的正是靜態(tài)描述。另外，在大部分研究中，尺度劃分和尺度的空間性都存在。但是，必須跟蹤大小和分辨率分別在不同的尺度劃分和尺度空間中如何變化，這便再次造成了諸多混淆。劃分的實(shí)體，即為我們要在空間定位和分析的現(xiàn)象或活動(dòng)，例如企業(yè)可以有不同的規(guī)模，也就是說，可以有大型企業(yè)和小型企業(yè)，但這并不意味著分辨率的變化。然而，我們當(dāng)然也可以有分類數(shù)據(jù)的分辨率變化，例如，如何將企業(yè)的規(guī)模進(jìn)行分類。舉個(gè)例子，我們可以使用高分辨率來(lái)做到這一點(diǎn)，比如使用絕對(duì)員工數(shù)量；或者使用較低的分辨率，比如將員工人數(shù)進(jìn)行分組。

另一方面，如果我們想要討論空間大小，需要另一種路徑。首先，需要絕對(duì)清楚的是我們要分析的空間大小，其次是從何種角度定義空間大小。例如，如果我們想要比較兩個(gè)鄰里的大小，從兩個(gè)空間單元相比較的原則上來(lái)說，我們當(dāng)然可以簡(jiǎn)單地通過比較各個(gè)單位的大小，例如通過測(cè)量它們的面積來(lái)做到這一點(diǎn)。然而，在大多數(shù)情況下，這種比較可能不夠翔實(shí)，而且相當(dāng)武斷。更有趣的是以某種系統(tǒng)的方式衡量空間大小，尤其是考慮到在當(dāng)代研究中城市系統(tǒng)理論的主導(dǎo)地位。這將涉及以某種方式衡量單個(gè)空間單位對(duì)整個(gè)系統(tǒng)的影響大小或重要性，例如，單個(gè)街區(qū)在整個(gè)地區(qū)或城市的作用。還有，它可以涉及從單個(gè)空間單位到所有其他空間單位的相對(duì)距離，其中短距離可以被認(rèn)為是一個(gè)全局性位置，并被解釋為一種衡量個(gè)體空間單位的系統(tǒng)規(guī)模的方法。此外，這種分析可以在單個(gè)空間單位的不同距離半徑范圍內(nèi)進(jìn)行，例如按米制距離設(shè)置。反過來(lái)，這種半徑可以用作不同尺度的分析定義。但現(xiàn)在需要追尋我們?cè)谧鍪裁?，這些定義是指大小的尺度還是分辨率的尺度？最有趣的是，它們是大小的尺度的定義，因此非常有用。分辨率尺度是由空間單位的選擇定義的，而不是由我們定義的半徑。因此，這些半徑為比較不同空間尺度下個(gè)體空間單元的作用和功能提供了重要的可能性！

正如前面所述，這種不同尺度的角色和功能比較對(duì)于正確理解城市如何運(yùn)作是絕對(duì)重要的。例如，在一個(gè)尺度上被檢測(cè)為相對(duì)均勻的區(qū)域可能另一個(gè)尺度上被證明是異質(zhì)的。此外，對(duì)重疊尺度的研究是同樣重要的，即地區(qū)同時(shí)在不同尺度上運(yùn)轉(zhuǎn)會(huì)得到加強(qiáng)或削弱，這證明對(duì)我們理解城市現(xiàn)象和進(jìn)程至關(guān)重要。此外，對(duì)重疊尺度的研究，即在不同尺度上同時(shí)表現(xiàn)的地方，從而被加強(qiáng)或削弱，對(duì)我們理解城市現(xiàn)象和過程是至關(guān)重要的。關(guān)于尺度之間的這種關(guān)鍵的相互作用，例如，雅各布斯對(duì)規(guī)劃者建立自治社區(qū)的能力提出批判的立場(chǎng)，因?yàn)樗麄冾B固地關(guān)注鄰里尺度，而忽視了從其與整個(gè)城市尺度的關(guān)聯(lián)的角度來(lái)正確理解街區(qū)的運(yùn)作。正如在其他地方所詳細(xì)討論的，雅各布斯可能是第一個(gè)提出需要建立一致的城市系統(tǒng)觀點(diǎn)的人，比較著名的是她回答了自己提出的問題：“城市問題為何？：恰是組織復(fù)雜性的 問題?！盵3]

總之，在大多數(shù)情況下，對(duì)城市現(xiàn)象進(jìn)行合理的空間分析，需要采用涵蓋微觀、中觀和宏觀等多尺度研究方法，以便獲得全面的觀點(diǎn)，特別是需要在尺度之間移動(dòng)尋找相互關(guān)系和強(qiáng)化弱化手段。更具體地說，如果我們接受城市經(jīng)濟(jì)學(xué)中關(guān)于多樣性致使區(qū)域城市增長(zhǎng)的假設(shè)，那么當(dāng)我們談?wù)摮鞘械亩鄻有詴r(shí)，我們特指什么尺度就至關(guān)重要了。這種增長(zhǎng)極有可能是由于尺度之間的相互作用造成的，而非任何其他因素。[9-10]不僅如此，這對(duì)城市規(guī)劃、設(shè)計(jì)的政策和具體干預(yù)非常重要。例如，城市設(shè)計(jì)通常被認(rèn)為主要關(guān)注地方尺度，但通過更好地了解尺度之間的相互作用，城市設(shè)計(jì)或許被證明可以間接地影響其他尺度。相反，為了在地方尺度實(shí)現(xiàn)目標(biāo)，可能有必要介入其他尺度。由此，涉及鄰里、地區(qū)和城市尺度的層次分析至關(guān)重要。

為了解決上述問題，我們?cè)趲讉€(gè)尺度上測(cè)量變量，而且重要的是，我們通過出行網(wǎng)絡(luò)的可達(dá)性來(lái)測(cè)量變量，例如，通過地塊或經(jīng)濟(jì)活動(dòng)的可達(dá)性 ，這與空間句法早期發(fā)展為地方語(yǔ)法分析的路徑相一致。為了解決自變量的局部尺度問題，我們選擇測(cè)量 500m 半徑內(nèi)地塊的可達(dá)性1，這通常被認(rèn)為是步行意愿的近似距離。為了解決中觀到一定尺度甚至宏觀尺度的問題，我們還在1,000m 和2,500m 的半徑處增加了測(cè)量值。2對(duì)于因變量的局部尺度問題，可通過兩種不同的分類來(lái)解決，第一種分類旨在廣泛捕捉城市尺度的經(jīng)濟(jì)活動(dòng)多樣性，第二種分類旨在捕獲地區(qū)級(jí)更具體的經(jīng)濟(jì)活動(dòng)（零售）多樣性。

3 多樣性與分類法：分類法是何意

任何類型的空間分析基礎(chǔ)都是對(duì)一個(gè)令人滿意的分類系統(tǒng)的發(fā)展或選擇。像大多數(shù)實(shí)證研究一樣，空間分析研究呈現(xiàn)豐富的個(gè)體集合，需要以某種方式進(jìn)行分類，以便進(jìn)行充分的研究。威爾遜在上文中稱之為可劃分實(shí)體。這樣的分類取決于研究的目的，根據(jù)這些目的，相同的個(gè)體可以被歸類到完全不同的類別中。因此，開發(fā)一個(gè)充分的分類系統(tǒng)需要很高的精度。同時(shí)，這被證明為空間分析中最困難的任務(wù)之一。一般而言，分類被認(rèn)為是“我們向來(lái)自現(xiàn)實(shí)世界的大量信息流施加某種秩序和連貫性的基本過程”，并且它“被視為一種建構(gòu)現(xiàn)實(shí)以檢驗(yàn)假設(shè)的手段”。這意味著無(wú)法獨(dú)立于研究目的來(lái)評(píng)估分類系統(tǒng)的充分性。為了使理論和分類之間具有適當(dāng)?shù)南嗷リP(guān)系，一個(gè)主要問題或預(yù)設(shè)假設(shè)的重要性經(jīng)常被提起。在這一方面，正如威爾遜所強(qiáng)調(diào)的，我們應(yīng)該意識(shí)到，“沒有絕對(duì)正確的分類方法”[1]。

在我們的案例中，出于分析和衡量城市空間多樣性的目的，需要進(jìn)行明確的分類。其中類別的選擇、數(shù)量和屬性背后的原理，都將對(duì)最終的多樣性產(chǎn)生至關(guān)重要的影響。例如，當(dāng)我們衡量一個(gè)區(qū)域的多樣性時(shí)，如果考慮其主要職能，如將其分為居住與工作人口，與考慮其經(jīng)濟(jì)因素，如辦公、商業(yè)及工業(yè)因素相比，同一區(qū)域的多樣性結(jié)果將大不相同。將分類的層次結(jié)構(gòu)向下移動(dòng)，衡量商業(yè)領(lǐng)域，確切地說是零售領(lǐng)域的多樣性時(shí)，可以根據(jù)所提供商品的類型（例如衣服、鞋子和家具）對(duì)其進(jìn)行分類與排序，這也將產(chǎn)生不同的值。這意味著一個(gè)區(qū)域可以同時(shí)具有較高的多樣性和較低的多樣性，這都取決于所使用的分類方式。

本文將采用基于OpenStreetMap 和從OSM編碼系統(tǒng)提取的兩種多樣性分類方法。之所以使用OpenStreetMap，是因?yàn)榭梢员容^一個(gè)國(guó)家或多個(gè)國(guó)家的城市之間的差異。當(dāng)前我們的研究重點(diǎn)集中于分析一個(gè)城市：斯德哥爾摩，但作為更廣泛的研究項(xiàng)目的一部分，研究可能將擴(kuò)展到其他城市。OSM 的類別基于點(diǎn)數(shù)據(jù)，包括零售與服務(wù)、銀行、酒店、衛(wèi)生、教育機(jī)構(gòu)、公共設(shè)施、文化與體育設(shè)施等類別。另外對(duì)零售方面提出了更精細(xì)的分類，包括食品店、百貨商店、服裝店、保健美容店、家居用品店、家具店、電子產(chǎn)品商店、體育用品店、文具店以及書店。基于這些數(shù)據(jù)，我們提出引入一種多樣性：城市層面的多樣性（之后可稱為總體多樣性），以及地區(qū)或街道層面的零售多樣性。之所以選擇零售業(yè)作為衡量本地規(guī)模多樣性的指標(biāo)，是因?yàn)槿藗兤毡檎J(rèn)為零售業(yè)可以反映城市中與步行相關(guān)的經(jīng)濟(jì)活動(dòng)的強(qiáng)度。這兩種多樣性指數(shù)均使用城市研究中常用的辛普森多樣性指數(shù)計(jì)算的，可以很容易地轉(zhuǎn)化為可達(dá)性 評(píng)價(jià)。

4 研究方法

總體來(lái)講，方法步驟包括：(1)測(cè)定多樣性的因變量；(2)衡量差異性的自變量；(3)通過控制變量（街道中心性與建筑密度）來(lái)構(gòu)建子模型；(4)將數(shù)據(jù)中的自變量與因變量連接至一個(gè)模型；(5)最后，差異性的自變量與多樣性的因變量之間的協(xié)方差的統(tǒng)計(jì)分析。

第一步：測(cè)定多樣性的因變量。多樣性的因變量使用辛普森多樣性指數(shù)進(jìn)行測(cè)定，該指數(shù)可以被轉(zhuǎn)換為可達(dá)性評(píng)價(jià)。辛普森多樣性指數(shù)是衡量城市活動(dòng)多樣性的公認(rèn)指標(biāo)，目前的問題是應(yīng)該將哪些類別包括在內(nèi)。如前所述，基于OSM 數(shù)據(jù)，我們提出使用兩種多樣性分類，分別針對(duì)兩種城市規(guī)模：總體多樣性和零售多樣性?？傮w多樣性（下文表示為Dgeneral）包括各種基本的城市服務(wù)（不包括辦公），它們?cè)谡麄€(gè)城市中分布得更均勻，而零售多樣性（下文表示為Dretail）通常與步行相關(guān)的經(jīng)濟(jì)活動(dòng)的強(qiáng)度相關(guān)。

表1 / Table 1皮爾森相關(guān)系數(shù)的概要統(tǒng)計(jì)。對(duì)于每個(gè)子模型，更高的相關(guān)性系數(shù)會(huì)被加粗。 / Summary of Pearson's correlations. Higher correlations per sub-model are marked in bold來(lái)源: 作者提供 / Source: Provided by the author

兩個(gè)多樣性指數(shù)（Dgeneral and Dretail）用于計(jì)算并衡量為500m 半徑內(nèi)的可達(dá)性。首先，計(jì)算每個(gè)單獨(dú)類別的可達(dá)性，其次計(jì)算所有類別的可達(dá)性，然后將所得的值計(jì)算辛普森多樣性指數(shù)(D=Σ(n/N)2)3，其中n 是每個(gè)類別中活動(dòng)的量，N是所有活動(dòng)的總數(shù)。

第二步：衡量差異性的自變量。差異性變量可以用更簡(jiǎn)明的術(shù)語(yǔ)描述為地塊大小，或者在可達(dá)性相關(guān)的術(shù)語(yǔ)中描述為地塊的可達(dá)數(shù)量，因?yàn)楫?dāng)?shù)貕K較小時(shí)，它們通常會(huì)很多。因此，根據(jù)上述討論，差異性變量將按照每個(gè)地址點(diǎn)的地塊可達(dá)性絕對(duì)數(shù)量進(jìn)行度量，根據(jù)幾個(gè)尺度或半徑分為：500m、1,000m 以及2,500m 的步行距離，在下文中將被表示為地塊的可達(dá)數(shù)量或可達(dá)性（Aplot500, Aplot1000 and Aplot2500）。

第三步：通過控制變量（街道中心性與建筑密度）來(lái)構(gòu)建子模型。為了評(píng)估差異性變量與兩種不同的多樣性之間的關(guān)系，必須要控制另外兩種空間形態(tài)的變量（街道中心性與建筑密度）。早期研究已經(jīng)分別根據(jù)中心性與密度分析生成了街道與建筑，它們可以用于在我們測(cè)量變量的地塊選擇中，使這些變量保持不變。

利用基于質(zhì)心的聚類方法生成多尺度的中心性的街道類型，以基于街道路段之間的中心性根據(jù)不同的尺度4對(duì)其進(jìn)行分類。聚類分析得出了5 種類型的中心性，其中只有兩種類型（“城市”和“鄰里”）的觀察結(jié)果納入了我們的分析之中，因?yàn)樵谄渌? 種類型中幾乎沒有發(fā)現(xiàn)經(jīng)濟(jì)活動(dòng)。街道類型“城市”包括在更大尺度之間的中心性增加的街道路段，“鄰里”街道路段的之間的中心性在大多數(shù)尺度范圍內(nèi)始終較高，但在大多數(shù)本地尺度上明顯下降。

建筑密度的類型將使用聚類分析法，基于兩個(gè)輸入變量：容積率（Floor Space Index ，F(xiàn)SI） 和 建 筑 密 度（Ground Space Index，GSI），在500m 步行距離內(nèi)測(cè)量。聚類產(chǎn)生了6 種密度類型，其中選擇兩種類型進(jìn)行分析，因?yàn)榕c選中的兩種街道類型相似，只有其中兩種類型與大量的經(jīng)濟(jì)活動(dòng)相關(guān)。選中的兩種密度類型為：“密集中高層”（城市中心FSI 和GSI 值最高的組合）和“緊湊中高層”（相比之下，其FSI 與GSI 的值略低）。

這些控制變量得出了4 個(gè)子模型（兩種街道類型x 三種密度類型，見圖1），這讓我們可以評(píng)估總體或零售多樣性與不同尺度下地塊可達(dá)性之間的相互關(guān)系（見圖1）。

第四步：連接自變量與因變量的數(shù)據(jù)。斯德哥爾摩的地址點(diǎn)圖層將用于連接所有組件（街道、建筑物、地塊及活動(dòng)），并將它們的屬性鏈接到一個(gè)模型中。之所以選擇使用地址點(diǎn)是因?yàn)榈貕K或建筑物通?？梢耘c不同的街道路段相關(guān)聯(lián)。在這里，地塊和建筑物與街道相連，人們可以從街道上進(jìn)入地塊和建筑物，它們可以用地址點(diǎn)來(lái)表現(xiàn)。

第五步：統(tǒng)計(jì)分析。首先，運(yùn)行二元皮爾森相關(guān)系數(shù)分析，在三個(gè)尺度下的地塊可達(dá)性與總體多樣性和零售多樣性均相關(guān)。由于自變量的高度共線性，無(wú)法計(jì)算多元回歸模型，只有一個(gè)變量的線性回歸才能顯示出與皮爾森相關(guān)系數(shù)分析相似的結(jié)果。盡管如此，我們?nèi)匀辉谙乱徊綄?duì)相關(guān)性最高的子模型進(jìn)行線性回歸擬合，以反映殘差值。這樣，若在地塊之外還有其他空間變量未被納入本次分析之中，且這些變量會(huì)影響到特定區(qū)域過高或過低的多樣性，這些殘差值便可用于評(píng)估預(yù)測(cè)值過高或過低的值。

5 統(tǒng)計(jì)分析的結(jié)果

若對(duì)總體多樣性與地塊的可達(dá)性進(jìn)行相關(guān)性分析，在子模型“S?dermalm”以及“城市”這種街道類型中（除了子模型“城市+密集中高層”），在更大的半徑范圍（Aplot2500）內(nèi)，相關(guān)性會(huì)更高。 在街道類型“鄰里”與密度類型“緊湊中高層”的組合中，若半徑范圍較?。ˋplot500），與地塊類型的相關(guān)性就會(huì)變 ?。ū?A）。

若對(duì)零售多樣性與地塊可達(dá)性進(jìn)行相關(guān)性分析，對(duì)于所有子模型（“鄰里+緊湊中高層”子模型除外），較小半徑范圍的地方（Aplot500）通常相關(guān)性更高（表1B）。

我們可以在此得出結(jié)論：零售業(yè)的多樣性確實(shí)與從本地尺度衡量的地塊可達(dá)性更有相關(guān)性。這突出了城市中以步行為導(dǎo)向的城市中心的位置。同時(shí)，我們先前的論點(diǎn)是，總體多樣性在整個(gè)城市中分布更均勻。這一發(fā)現(xiàn)得到了支持，即在更大的半徑范圍內(nèi)，總體多樣性與地塊可達(dá)性相關(guān)性更高。但是，這里做出的任何結(jié)論仍不夠成熟，仍有必要進(jìn)一步地調(diào)查——至少要在其他城市進(jìn)行類似分析。

我們主要的關(guān)注點(diǎn)是地塊可達(dá)性與零售多樣性之間的相關(guān)性，因?yàn)槿缜八?，零售業(yè)集群通常被認(rèn)為暗示了步行友好型的城市中心的位置。因此，下一步我們將專門研究地塊可達(dá)性與零售多樣性之間具有最高相關(guān)性的子模型，并構(gòu)建線性回歸模型。這是為了查看數(shù)據(jù)中是否存在空間相關(guān)性問題，而該問題無(wú)法通過模型來(lái)得到解釋。把殘差值落位到地圖上，將便于我們觀察預(yù)測(cè)值過高或過低的樣本是否在地圖上發(fā)生了聚集——因?yàn)檫@意味著子模型中還有其他空間變量覆蓋掉地塊可達(dá)性對(duì)零售多樣性的影響。因此，我們繼續(xù)進(jìn)行研究，對(duì)子模型“鄰里+密集中高層”（R2=0.194，p＜0.05）進(jìn)行線性回歸分析，并把殘差值落位到地圖上（圖2）。

殘差值地圖（圖2）通過突出顯示預(yù)測(cè)值過高或過低的殘差值的集中程度，來(lái)展示數(shù)據(jù)中是否存在空間自相關(guān)問題。預(yù)測(cè)值過低的值（圖2，棕色），表示該區(qū)域的觀測(cè)值（零售多樣性）低于地塊可達(dá)性所預(yù)測(cè)的值。反過來(lái)，預(yù)測(cè)值過高的值（圖2，綠色）表明觀測(cè)值高于預(yù)期。地圖上的黃色區(qū)域（圖2）表示觀測(cè)值被較為準(zhǔn)確地預(yù)測(cè)出來(lái)。高估值或低估值的集聚區(qū)域，對(duì)于結(jié)果的解讀很有幫助，因?yàn)楦鶕?jù)我們的模型，突出顯示的這些區(qū)域表明，除了地塊可達(dá)性以外，可能存在其他一些因素（空間因素或其他因素），導(dǎo)致該區(qū)域出現(xiàn)了相比于預(yù)期更高或更低的零售多樣性。

其中一個(gè)最值得注意的案例是，盡管通?？梢酝ㄟ^地塊可達(dá)性來(lái)很好地預(yù)測(cè)零售業(yè)的多樣性，但其中有個(gè)令人驚訝的例外是，繁忙的購(gòu)物街Birger Jarlsgatan（地圖上的棕色），其多樣性低于預(yù)期。因此，我們可以得出結(jié)論，這條街上可能還存在其他因素影響著零售多樣化。然而，經(jīng)過仔細(xì)觀察，我們意識(shí)到這是一條高檔時(shí)裝購(gòu)物街。這再次強(qiáng)調(diào)了對(duì)尺度和類別關(guān)注的必要性。雖然Birger Jarlsgatan 擁有高檔時(shí)裝業(yè)集群，其經(jīng)濟(jì)活動(dòng)的總體多樣性不太可能沿著它出現(xiàn)，但其周圍的五金店并不多，因?yàn)榇蠖鄶?shù)商店都在出售時(shí)尚服裝。但是，如果我們能對(duì)多樣性進(jìn)行更細(xì)粒度的分析，把僅出售服裝的零售店也納入考慮范圍，Birger Jarlsgatan 將可能表現(xiàn)出很高的多樣性。討論尚為時(shí)過早，需要進(jìn)一步研究，但其強(qiáng)調(diào)了多樣性研究中尺度和分類的核心問題。

6 結(jié) 論

本文有兩個(gè)目的。首先是通過實(shí)證研究展現(xiàn)并檢驗(yàn)了城市多樣性與差異性變量（地塊）之間的潛在的聯(lián)系；其次是講述衡量多樣性方法的復(fù)雜性，這涉及尺度與分類之間相互關(guān)聯(lián)問題。為了解決這兩個(gè)問題，我們提出可以通過引入不同類型的多樣性指標(biāo)來(lái)解決尺度問題。在我們的案例中，這種多樣性對(duì)應(yīng)的是總體多樣性（全局性）和零售多樣性（本地性）。另外，我們提出將其與跨越多個(gè)空間尺度的差異性變量進(jìn)行相關(guān)性分析。為了控制實(shí)驗(yàn)中建筑密度和街道中心性對(duì)結(jié)果可能產(chǎn)生的影響，我們引入了幾個(gè)子模型。其中在每個(gè)子模型中，這兩個(gè)額外的空間變量保持不變。

正如皮爾森相關(guān)系數(shù)所表示的那樣，具有較高總體多樣性值（能獲得各種基本服務(wù)）的區(qū)域，在更大的半徑范圍內(nèi)，與地塊可達(dá)性確實(shí)具有更高的相關(guān)性；而具有較高零售多樣性值（能獲得各種零售服務(wù)）的區(qū)域，在較小的半徑范圍內(nèi)，與地塊可達(dá)性的相關(guān)性更高。這一發(fā)現(xiàn)可以進(jìn)一步被作為起點(diǎn)，在多個(gè)城市中進(jìn)行更廣泛的同類研究。

此外，當(dāng)我們更詳細(xì)地研究零售多樣性的分布時(shí)，通過關(guān)注其中一個(gè)子模型，把線性回歸模型得到的殘差值落位到地圖上之后，我們發(fā)現(xiàn)，某些特定區(qū)域的零售多樣性無(wú)法用區(qū)域內(nèi)出現(xiàn)的較多數(shù)量的地塊來(lái)解釋。然后我們建議，可能有必要通過引入一種特定類型的零售商業(yè)——時(shí)尚商店下的業(yè)態(tài)分類，來(lái)引入更精細(xì)尺度下多樣性的度量方法。

盡管還沒有定論，但這些觀察結(jié)果確實(shí)支持了我們的假設(shè)，即地塊數(shù)量（差異性變量）與城市多樣性之間存在重要的聯(lián)系。這是一個(gè)重要的發(fā)現(xiàn)。它需要得到進(jìn)一步和更全面的 研究。

圖2 / Figure 2兩個(gè)子模型的殘差值地圖：S?dermalm和“鄰里+緊湊中高層” / Map of residuals for two sub-models: S?dermalm and Neighbourhood + Dense mid-rise

ORIGINAL TEXTS IN ENGLISH

Spatial Configuration Of Plot Systems And Urban Diversity:

Empirical support for a differentiation variable in spatial morphology

Lars Marcus, Evgeniya Bobkova

1 Introduction: A Theory of Spatial Capacity

The central variables in any urban model are distance and attraction (Wilson, 2000). Space syntax research has contributed to the development of new geometric descriptions and measures of distance that have broken new ground, not least when it comes to capturing pedestrian movement . However, the description and measurement of attractions has not been central to the field.

In an earlier paper we interpreted attractions as additional variables of spatial form to distance in the form of density and diあerentiation, where the first was related to buildings and the second to plot systems. In this paper we specifically address the far less studied variable of differentiation. Earlier studies have shown strong indications that there is a correlation between the degree of land division into plots (parcels) and the diversity of socio-economic content, such as residents and economic activity. Building on this, we here present results from an extensive empirical study in Stockholm, aiming to pave the way towards a spatial variable of differentiation in spatial morphology, with direct impact on socio-economic diversity. The investigation concerns a correlation analysis between, on the one hand, measures of plot systems configuration, and on the other hand, diversity models of economic activity.

The origin of much debate on diversity in cities is of course the writings of Jane Jacobs, for whom diversity was the central theme throughout all of her texts on cities. Most famously it is spelled out in ‘the Death and Life of Great American Cities’ and the four conditions for generating diversity in cities that she famously proposed in that book: more than one primary function； short blocks； buildings of varying age； and dense concentration of people (Jacobs, 1961). Of these, as we shall see, we will actually find the condition buildings of varying age to be, perhaps a bit surprisingly, the most promising from our current perspective of the need to develop a measure of diあerentiation for our model. The reason is that while the condition more than one primary function certainly has a strong influence on the degree of diversity in cities, it rather deals with pure programming of diversity than having any obvious connection to spatial form of the kind that we are looking for here. The notion of short blocks, on the other hand, rather deals with accessibility, or quite exactly the configuration of cognitive distances dealt with in space syntax theory as discussed above. Finally, the notion of dense concentration of people clearly concerns the idea of density in general and as such has already been thoroughly discussed (Berghauser Pont & Marcus 2014). The notion of buildings of varying age, on the other hand, seems principally diあerent from the others and at first sight presents itself, perhaps, as the one of both least theoretical interesting and least practically applicable - how do we plan cities with buildings of varying age or, for that matter, build a theory around such a thing. However, that is exactly what we shall attempt.

The trap here is to cling to the idea of buildings as things rather than as processes. A shift to the latter implies a shift from the spaces that buildings create to the spaces in which buildings evolve, that is, the different domains defined property rights that we call plots. These are spaces that are institutionally defined rather than physically defined, but even so they undoubtedly represent a central and common category of spaces in cities. If we start to think about it we realise how fundamental such division of space into several separate spaces is in any architectural endeavour - what is architecture if not the art of dividing space by walls - where the primary rationale behind this exactly is the aim to generate diversity, that is, generate discrete spaces for separate and diあerent categories of ‘things’ or activities； this clearly is the major rationale behind the division of buildings into separate rooms for instance.

It therefore seems well founded to see the horizontal division of land, whether defined physically or institutionally, as a fundamental spatial technique whereby one can support an increase in diあerentiation in a similar manner to how the vertical addition of floor-space is a fundamental technique in increasing density. If we more specifically introduce this concept to a model of urban spatial form, we can make the following argument. The particular domain of the plot, as a piece of land defined by a specific set of property rights, represents the presence in the city of an actor in the form of its owner or proprietor or the like, which, furthermore, entails a very precise location of the activities of that actor in urban space (Marcus 2010； Bobkova et al., 2017). It is within this particular domain the specific actor is free to act, as long as keeping within the framework set by diあerent institutions, for instance, the local planning regulations or the particular property rights of the concerned piece of land.

Such actors, furthermore, will normally develop a particular strategy for the further development and maintenance of their property. An area with comparatively many plots therefore seems to have the potential to carry more such actors and thereby more strategies for the development and maintenance of its plots than an area with comparatively few plots, and this, most likely, will also imply a greater diversity of such strategies. In the end, such an area seems to have the potential to more easily develop a diverse content than the area with comparatively few plots and hence few actors and strategies. Consequently, it seems to be exactly this division of land and the subsequent creation of potential diversity in actors and strategies that over time can generate a greater variety in the building stock, that is, Jacobs’ notion of buildings of varying age.

It is this hypothesis that makes us believe that we in the number, shape, size and configuration of plots can identify a variable of spatial form with a direct relation and influence on urban diversity. Moreover, we propose that the number of plots in an area can be called the spatial capacity of that area, that is, the capacity to carry diあerences, where a high capacity creates a greater potential for a heterogeneous content, while a low capacity does the same for a more homogenous content (Marcus, 2000； 2003； 2010； Bobkova et al. 2017ab). Obviously other factors like landuse regulations or other spatial variables, such as street centrality and building density, certainly can override the eあect of the spatial form of plot systems here, but what we are trying to isolate in our study is the particular influence of this variable on urban entities or processes.

In the following we will present an empirical study that relates different kinds of diversity as dependent variables to the independent variable of diあerentiation measured as accessibility to plots. The outline of the paper is as follows. In the next two sections, theoretical complications behind measuring socio-economic diversity will be presented, with particular focus on the issues of scale and categorization. These sections will provide the support for the choices made for the empirical tests presented next. Thereafter, we will present a methodological overview of the empirical study, including: (1) how to measure the dependent variable of socio-economic diversity, (2) how to measure the independent variable of spatial diあerentiation, (3) how to address the issue of scale, (4) how to construct the statistical models that allow for isolation of the diあerentiation variable, and (4), the description of statistical tests. Thereafter the results from the study will be reported and, in the conclusion finally, the results will be discussed in relation to the theories earlier introduced with suggestions for future research.

2 Diversity and Scale: What We Mean By Scale

The term scale is used in many fields and is often interpreted quite diあerently in one discipline from another. There are also many related concepts like level, resolution, extent, and hierarchy, often used as replacements or synonyms. In studies on cities, not least, there is a wide range of references to scale, which also have given rise to a variety of categorisations, such as "local-global", "micromeso-macro", "neighbourhood-district-city-region". However, these terms are by rule vague due to their intrinsic relative nature. They typically are defined in relation to each other for a particular case and what is understood as, for instance, "local" varies from city to city. Together this makes scale one of the most easily confused concepts in the study of cities. This is problematic since choice of scale fundamentally influences both analysis and interpretation in any study of urban modelling or spatial analysis.

According to Mike Batty, we by scale basically mean two things: “the level of resolution at which we observe the city, which is essentially map scale, and the level of functional differentiation that takes place in different sizes of location or city” (Batty, 2005). That is, scale can concern resolution or size but these two are easily confused. All the categorisations listed above, for example, such as the highly relative macro-meso-micro set, typically refer to resolution but can easily be taken for categories of size, not least when applied in concrete urban settings. For instance, when speaking about "neighbourhood", as a particular scale in relation to "district" or "city" scale, we speak about a particular resolution of cities, that is, we zoom in on an area so that we can identify more detail about what is going on there, but this does of course not make the city where we do this smaller or larger. Still, it is an operation where it is easy to begin thinking about the particular neighbourhood as a small city, that is, that we have changed size of the city, but this is obviously not the case.

The immediate risk here of slipping into some overarching, grander or more abstract city that easily takes on life of its own, is encouraged by the fact that we often speak of scales as hierarchies. In Alan Wilson’s words: “Scale is a form of hierarchy and clarity of vision in this respect is critical” (Wilson 2000). A typical case is the categorisation of urban phenomena into micro, meso and macro scale, which conceptually imply, even though we know that it is not true, that there is a series of cities, so to speak, on top of each other. Once again, upon reflection we instantly realise that this is nonsense, but even so, conceptually the slip has already been made and it is diきcult to keep in mind that we do not really mean what we say. For instance, when the introducers of the new spatial economy talk about centrifugal and centripetal forces on the concentration of economic activity in cities (Fujita et al., 1999), we are enticed to envision these as some kind of macro scale forces hovering over cities. In reality, these are forces that analytically are identified on an aggregated macro scale, but that is certainly not where they are acted out. On the contrary, such forces, while leaving distinct traces on aggregated level, are by necessity rooted in human everyday activity on the micro scale.

Moreover, we also need to keep our vision clear concerning what entity it is that we discuss the scale of. Following Wilson (2000), we can say that spatial analysts face the question of resolution in three ways: “entities that are components of systems of interest have to be defined and categorised； many of them have to be located in space； and their behaviour has to be described over time”. This leads to three aspects of scale necessary to decide on: “sectorial (number and breadth of categories), spatial (size of area units within which entities are to be located) and temporal (length of time units which provide the basis for longitudinal description and analysis)”. Temporal resolution constitutes a big issue in itself in that most spatial analyses simply leave out the time dimension, giving rise to the typically static descriptions of cities that we have grown accustomed to. In principle, these are highly unrealistic and for most uses risky to draw conclusions on, however, that is exactly what we do in most studies of cities. The sectorial and spatial aspects, on the other hand, are in most studies there. However, it is important to keep track of how both size and resolution can vary in both, once again, creating many reasons for confusion. Sectorial entities, that is, phenomena or activities that we want to locate and analyse in space, for instance businesses, can vary in size, that is, we can have large businesses and small businesses, which does not imply variations in resolution. However, we can certainly also have variations in resolution of sectorial data, for instance, in how we categorise size in businesses. We can do that with high resolution using, for instance, absolute number of employees, or with a lower resolution where we, for instance, group the number of employees into bundles.

If we want to discuss spatial size, on the other hand, we need another way of entry. First of all, we need to be absolutely clear about what it is that we want to analyse the size of and, second, from what point of view we define size. For instance, if we want to compare the size of one neighbourhood to another, in principle comparing one spatial unit to another, we can of course do this simply by comparing the size of the individual units, for example by measuring their area. Such comparison, however, is in most cases likely to be rather uninformative and quite arbitrary. More interesting, it seems, is to measure size in some systemic way, not least given the predominance of systems views of cities in contemporary research. This would concern to somehow measure the size of impact or importance of the individual spatial unit on the system as a whole, for instance, the role of the individual neighbourhood on the whole district or city. It could, for example, concern the relative distance from the individual spatial unit to all other spatial units, where a short such distance could be argued to represent a strategic location and be interpreted as one way of measuring the individual unit’s systemic size. Moreover, such analysis could be conducted within diあerent distance radii of the individual spatial unit, for instance set by metric distance. Such radii could, in turn, be used as definitions of analysis at diあerent scales, but now we really need to keep track of what we are doing. Are these definitions of scale of size or scale of resolution? Most interestingly, they are definitions of scale of size and therefore extremely useful. The scale of resolution is defined by the choice of spatial unit, not by radii as we have defined it. These radii, therefore, open for the valuable possibility to compare the individual spatial units’ role and function at diあerent scales of size!

As already touched upon, such comparison of role and function at different scales of size is absolutely central for a proper understanding of how cities work. For instance, an area detected as relatively homogeneous at one scale may prove to be quite heterogeneous at another. Moreover, it is not least enquiries of overlapping scales, that is, where localities simultaneously perform on different scales and thereby are reinforced or weakened, that prove critical for our understanding of urban phenomena and processes. Concerning this critical interplay between scales, Jacobs, for instance, took a critical position to planners’ ability to establish self-governing neighbourhoods, given their stubborn focus on the neighbourhood scale and negligence in putting it in relation to the over-all city scale for a proper understanding of the functioning of neighbourhoods. As discussed thoroughly elsewhere, Jacobs was probably the first to argue the need of a consistent systems view of cities, famously answering her own question: What kind of a problem is a city?: “Cities happen to be problems in organized complexity” (Jacobs, 1961).

To summarise, proper spatial analyses of urban phenomena in most cases necessitate having a multi-scale approach that covers, for instance, micro-, meso- and macro-studies, in order to get the full perspective and especially there is a need to move between scales looking for interrelations and reinforcements or lack thereof. More specifically, if we accept the presumption in urban economics that diversity leads to regional urban growth (Glaeser et al., 1992； 2001), the issue of what scale we are referring to when we speak of diversity in the city is critical. Most likely such growth is due to interaction between scales more than anything. Not least, this is important for policy and concrete interventions in urban planning and design. For instance, urban design is normally understood to concern primarily the local scale but through better knowledge about the interaction between scales urban design can perhaps be proven to indirectly influence also other scales. Conversely, to achieve aims on the local scale, interventions on other scales might be necessary. In this regard, hierarchical analysis covering neighbourhood, district and city scale, suggests itself as critical.

To address the issues described above we measure our variables at several scales, and importantly, we measure them as accessibility through the movement network, for instance, as accessibility to plots or economic activity , in line with earlier developments of space syntax analysis into place syntax analysis. To address the local scale of the independent variable, we choose to measure accessibility to plots within a 500m radius, which is commonly recognized as an approximate distance for the willingness to walk (Gehl, 2010). To address the meso and, to a certain extent, the global scale, we also add measures at a radius of 1,000m and 2,500m. For the dependent variable, scale is addressed by applying two diあerent categorisations, where the first is aimed to broadly capture diversity in economic activity at the urban scale, and the second to capture more specific diversity in economic activity (retail) on the district level. The problem of categorisation in diversity is discussed in the next section.

3 Diversity And Categorisation: What We Mean By Categorisation

Fundamental for any type of spatial analysis is the development or choice of a satisfying system of classification (Harvey 1969； Wilson, 2000). As in most empirical studies, spatial analysis studies present a rich set of individualities that needs to be sorted one way or another to be accessible for adequate study, what was called sectorial entities by Wilson above. Such a classification depends on the aims of the enquiry, where the very same individuals can be sorted very diあerently depending on these aims. Hence, the development of an adequate classification system requires great precision. At the same time, this has proven to be one of the most diきcult tasks in spatial analysis. Generally speaking, classification is regarded as “the basic procedure by which we impose some sort of order and coherence upon the vast inflow of information from the real world”, and [It is] “regarded as a means for structuring reality to test hypothesis” (Harvey, 1969). This implies that the adequacy of a classification system cannot be evaluated independently of the purpose of the study. The importance of a primary question or a presupposed hypothesis is often stated, in order to have a proper interrelationship between theory and classification (Harvey, 1969). In extension of this, as emphasised by Wilson, one should be aware that: “there is no absolutely right way to do categorization” (Wilson, 2000).

In our particular case, the need of a clear classification arises from the aim of analysing and measuring diversity in urban space. The principles behind the choice of classes, the number of classes, as well as their attributes, will all have crucial eあects on the final diversity values. For example, if we measure the diversity in an area concerning primary functions, for instance, divided into residential and working populations, this will be very different from the same areas diversity concerning economic sectors, such as oきcial, commercial and industrial sectors. Moving down the hierarchy, measuring the diversity in, for example, the commercial sector, and more specifically retailing, which, for instance, can be classified and sorted based on the type of goods oあered, such as clothes, shoes and furniture, this will yield diあerent values yet. This means that an area can have a high diversity and a low diversity at the same time, all depending on the classification used.

In this paper we will use two types of diversity classification that is based on OpenStreetMap and extracted from OSM coding system. The reason of using OpenStreetMap, is justified by the possibility to compare diあerences between cities within one country or in several countries. Our current study focuses on analysing one city of Stockholm, but has the ambition to be potentially extended to other cities as a part of larger research project. OSM categories are based on point data and include such categories as retail and services, food, banks, hotels, health, education, public facilities, culture and sports. It also proposes more fine-grain categorisation of retail activities, that include food and department stores, clothes, health and beauty, households, furniture, electronics, sport and stationary and books. Based on this data we propose to introduce to kind of diversity: diversity on the city level (further referred to as general diversity), and retail diversity on district or street level. The choice of retail as a measure of local scale diversity, is justified by the fact that it is generally recognized to indicate the intensity of pedestrian-related economic activities in cities (Scoppa & Peponis, 2015； Sevtsuk, 2014； Sevtsuk, 2010； Sevtsuk, 2010； Krafta, 1996). Both diversity indices are calculated using Simpson Diversity Index, that is commonly used in urban studies (Talen, 2008), and can be easily translated into accessibility measure.

4 Methodology

The general methodological steps include measuring the dependent variable of diversity (1), measuring the independent variable of diあerentiation (2), constructing sub-models by controlling variables of street centrality and building density (3), linking data on dependent and independent variables in one model (4), and finally, statistical analysis of co-variation between independent variable of diあerentiation and dependent variable of diversity (5).

Step 1. Measuring general and retail diversity

The dependent variable of diversity is measured using Simpson Diversity Index that is translated into an accessibility measure. Simpson Diversity Index is a generally recognized indicator for measuring diversity of urban activities (Talen, 2008), and our question at hand is what categories should be included. As described earlier, it is proposed to use two categorisations of diversity based on OSM data, that address two urban scales: general diversity and retail diversity. General diversity (referred further as Dgeneral), includes all kinds of basic urban services (excluding offices) that are more evenly distributed across the city, and retail diversity (referred further as Dretail) is usually associated with the intensity of pedestrian-related economic activities in vital city centers (Scoppa & Peponis, 2015； Sevtsuk, 2014； Sevtsuk, 2010； Sevtsuk, 2010； Krafta, 1996).

Two diversity indices (Dgeneral and Dretail) are calculated and measured as accessibility within a 500m radius. First, accessibility to each separate category is calculated, second, accessibility to the total number of categories is calculated, and then the resulting numbers are used to calculate Simpson Diversity Index (D=Σ(n/N)2) , where n is the number of activities within each category, and N is the total number of all activities.

Step 2. Measuring independent variable of diあerentiation

The differentiation variable can in more simple terms be described as plot size, or, in accessibility terms, as the accessible number of plots, because if plots are smaller, they typically are many (Bobkova, 2017a). Hence, the diあerentiation variable is measured as the absolute number of plots accessible from every single address point, across several scales or radii, according to the discussion above: 500m, 1000m and 2500m walking distance, and is further referred to as the accessible number of plots or accessibility to plots (Aplot500, Aplot1000 and Aplot2500).

Step 3. Constructing sub-models by controlling for street centrality and building density

To evaluate how the variable of diあerentiation is related to the two diあerent kinds of diversity, two other variables of spatial form (street centrality and building density) have to be controlled for. Typologies of both streets according to centrality and buildings according to density has been analytically generated in earlier research that here is used in our selection of locations from which our variables are measured (Berghauser Pont et al., in review) so that these variables remain constant.

Multi-scalar centrality street types were generated using centroid-based clustering to classify street segments based on their individual betweenness centrality profile through different scales (Berghauser Pont, et al., in review). The cluster analysis resulted in five centrality types, where only observations from two of these ("City"and "Neighbourhood") have been included in our analysis, because in the other three types economic activity was hardly found at all. The street type "City" includes street segments of increasing betweenness centrality at higher scales and "Neighbourhood" has segments with consistently high betweenness across most scales, but dropping clearly at the most local scales (ibid.).

Building density types were developed using cluster analysis (Berghauser Pont et al., 2017), based on two input variables: Floor Space Index (FSI) and Ground Space Index (GSI) (Berghauser Pont & Haupt, 2010), measured within 500m walking distance. The clustering generated six density types, of which two types were selected for our analysis, because, similarly to the two selected street types, only these were associated with a substantial number of economic activities. The two selected density types were "Dense midrise" (the highest combination of FSI and GSI characteristic for city centres) and "Compact mid-rise" (slightly lower FSI and GSI values compared to "Dense mid-rise").

Including these control variables gives us 4 sub-models (two street types x three density types, see figure 1), that allows us to evaluate co-relation between general or retail diversity and accessibility to plots at diあerent scales (figure 1).

Step 4. Linking data on dependent and independent variable

A layer of address points for Stockholm is used to join all the components (streets, buildings, plots and activities), and link their properties in one model. The choice of using the address points is justified by the fact that a plot or a building typically can be associated with diあerent street segments. Here, plots and buildings are linked to the streets from where one can enter it, which is represented by the address point (Berghauser Pont et al., in review).

Step 5. Statistical analysis

First, bivariate Pearson’s correlations are run, where accessibility to plots at three scales is correlated both with general and retail diversity. Because our independent variables are highly collinear, multiple regression models are not run, since linear regression for only one variable only would show the similar results as the Pearson’s correlation results. Nevertheless, we still run linear regressions for the sub-models with the highest correlation on the next step, in order to map residual values. They allow to evaluate underpredicted or overpredicted values, if there is any other spatial variable besides plots not included in our analysis, that influences higher or lower diversity in particular area.

5 Results Of Statistical Analysis

When general diversity is correlated with accessibility to plots, correlation generally gets higher at higher radii (Aplot2500) in sub-model "S?dermalm" as well as in street type "City", except the sub-model "City + Density type Dense Mid-rise". In Street type "Neighbourhood" combined with Density type "compact mid-rise", the correlation with plot types gets lower at the lower radii (table 1A).

When retail diversity is correlated with accessibility to plots, correlation generally gets higher at lower radii (Aplot500), for all sub-models except "Neighbourhood + Compact Mid-rise" (table 1B).

We may conclude here, that retail diversity indeed corresponds better to accessibility to plots measured locally, and highlights the location of pedestrian-oriented urban centres found in the city. At the same time, our earlier argument that general diversity is more evenly distributed across the city, is supported by the finding that it is correlated better with accessibility to plots at larger radii. However, any conclusions here are still premature. There is need for further investigation, not least through similar analysis in other cities.

Our key interest is the correlation between the accessibility to plots and retail diversity, because, as mentioned before, retail clusters are often recognised as indicating the location of pedestrian-friendly urban centres. Hence, in the next step we look specifically at the sub-models with the highest correlation between accessibility to plots and retail diversity and then run linear regression models. This is done in order to see if there is a problem with spatial correlation in the data, which cannot be explained by the model. Mapping residual values then allows to see if there are concentrations of observed values on the map that are over-predicted or under-predicted, which would mean that there are other spatial variables in our sub-model that override our explanation of retail diversity through accessibility to plots. Hence, we proceed and run linear regression analysis for the sub-model ‘Neighbourhood + Dense Mid-rise’ (R2=0,194, p＜0,05) and mapping residual values (figure 2).

The map of residual values (figure 2), shows whether there are any problems of spatial auto-correlation in the data by highlighting the concentrations of under- or over-predicted residual values. Underpredicted values (figure 2, brown), mean that the observed value (retail diversity) in the area is lower than predicted by the accessibility to plots. In turn, overpredicted values (figure 2, green) show that observed values are higher than expected. Yellow areas on the maps (figure 2) show that observed values are relatively well predicted. Clustered areas of over- or underpredicted values are useful for the interpretation of the results, because they highlight the areas where it might be that some other conditions, spatial or other, besides accessibility to plots contribute to higher or lower retail diversity in the area than expected according to our model.

A case of principal interest is that even though retail diversity is generally well predicted by accessibility to plots, a surprising exception is the busy shopping street Birger Jarlsgatan (brown on the map), where diversity is found to be lower than expected. We may therefore conclude that there may be some other condition influencing retail diversity along this street. On closer scrutiny however, we realise that this is a street for high fashion retail, which again highlights the necessary concern for scale and categorisation. While general diversity of economic activity is unlikely along Birger Jarlsgatan with its high fashion cluster, neither is it very diverse in retail, there are not many hard ware stores around, since most shops carry fashion clothes. However, if we would make an even more fine-grained analysis of diversity, concerning only retail that carry clothes, Birger Jarlsgatan is likely to demonstrate a very high diversity. This discussion is yet premature and calls for further studies, but it draws attention to the central issue of scale and categorisation in studies of diversity.

6 Conclusion

The aim of this paper was twofold, first, to present and test empirically potential link between urban diversity and the variable of diあerentiation (plots) and second, to present the whole complexity of measuring diversity, that is related to the interconnected issues of scale and categorisation. In order to deal with these two problems, we proposed that scalar issue can be tackled by introducing diあerent kinds of diversity, in our case general (global) and retail diversity (local). In addition, we proposed to correlate it with the variable of differentiation across several spatial scales. In order to control our tests for possible influence of building density and street centrality on the results, we introduced several sub-models, where within each sub-model, these two other spatial variables remain constant.

As it was shown by Pearson’s correlations, areas that have higher general diversity values (access to variety of basic services) are indeed better correlated with accessibility to plots at higher radii, while areas that have higher retail diversity values (access to variety of retail services) correlate better with accessibility to plots on lower radii. This finding can further serve as a starting point, to conduct a more extensive study of similar kind, but across several cities.

In addition, when we investigated distribution of retail diversity in more detail, by focusing on one of the sub-models and mapping residual values from linear regression, it was found, that retail diversity of some particular areas cannot be explained by the higher number of plots present in the area. We suggested then, that there is possibly a necessity to introduce diversity measure at even finer scale, by introducing categorisation within one particular kind of retail: fashion stores.

While still far from conclusive, these observations do support our hypothesis that there is an important connection between the number of plots (differentiation variable) and urban diversity； an important finding that calls for further and more comprehensive investigation.