騰訊哈勃

Simple OLS Regression, Pairs Bootstrap Resampling, and Hypothesis Testing to observe the effect of Hubble’s Law in Python.

通過簡單的OLS回歸，配對Bootstrap重采樣和假設(shè)檢驗(yàn)來觀察哈勃定律在Python中的效果。

In this post, we will revisit Hubble’s Law and examine the original dataset he used by running an Ordinary Least Squares Linear Regression on the 24 measurements of distances and recessional velocities of extra-galactic nebulae. Then, we will use a pairs bootstrap resampling to calculate the RSS Minima and perform a hypothesis test on the measured effect of galactic distance on recessional velocities.

在本文中，我們將回顧哈勃定律，并通過對銀河外星云的距離和后退速度的24個(gè)測量值進(jìn)行普通最小二乘線性回歸來檢驗(yàn)他使用的原始數(shù)據(jù)集。 然后，我們將使用成對的自舉重采樣來計(jì)算RSS最小值，并對銀河距離對后退速度的測量影響進(jìn)行假設(shè)檢驗(yàn)。

Based on the results of the hypothesis test we can conclude with a high degree of statistical signficance that distance has an observed effect on the recessional velocity of galaxies. This is concrete evidence of Hubble’s Law that the universe is constantly expanding.

根據(jù)假設(shè)檢驗(yàn)的結(jié)果，我們可以得出高度的統(tǒng)計(jì)意義，即距離對星系的后退速度有明顯影響。 這是哈勃定律證明宇宙不斷膨脹的具體證據(jù)。

Before we get into that let’s familiarize ourselves with Hubble’s Law.

在開始討論之前，讓我們熟悉哈勃定律。

哈勃定律 (Hubble’s Law)

In Edwin Hubble’s famous PNAS article “A relation between distance and radial velocity among extra-galactic nebulae” (1), Hubble provided evidence for one of science’s greatest discoveries: the expanding universe. Hubble demonstrated that galaxies are moving away from Earth with a recession velocity that is correlated to their distance from Earth. In other words, galaxies that are further away from Earth move away faster than nearby galaxies. This is commonly referred to as Hubble’s Law. Hubble’s classic graph of observed velocity vs. distance for nearby galaxies (presented above) visualizes this phenomenon. This graph has become a milestone in the scientific community, as it displays the linear relationship between galactic recessional velocity (v) and distance from Earth (d):

在埃德溫·哈勃(Edwin Hubble)著名的PNAS文章“銀河外星云之間的距離與徑向速度之間的關(guān)系”(1)中，哈勃為科學(xué)上最偉大的發(fā)現(xiàn)之一：膨脹中的宇宙提供了證據(jù)。 哈勃證明，星系正在以與地球到地球的距離相關(guān)的衰退速度離開地球。 換句話說，距離地球較遠(yuǎn)的星系比附近的星系移動(dòng)得更快。 這通常稱為哈勃定律。 哈勃(Hubble)關(guān)于附近星系觀測到的速度與距離的經(jīng)典關(guān)系圖(如上所示)將這一現(xiàn)象形象化。 這張圖已成為科學(xué)界的里程碑，因?yàn)樗@示了銀河退縮速度(v)與距地球的距離(d)之間的線性關(guān)系：

v = Ho x d

v = Ho xd

Here v is the galaxy’s recessional velocity and d is the galaxy’s distance from Earth. Ho is an empirically determined constant called Hubble’s constant. Even though the expansion rate is persistent in all directions at any given time, it changes throughout the lifetime of the universe. The well-calibrated expansion rate at the present time, Ho, is about 70 kilometers per second per megaparsec (note on units used here: recession velocity is in kilometers per second and distance is in megaparsec, 1 megaparsec = 1M parsecs, 1 parsec = 3.26 light-years). (2)

這里v是星系的后退速度， d是星系到地球的距離。 Ho是根據(jù)經(jīng)驗(yàn)確定的常數(shù)，稱為哈勃常數(shù)。 即使在任何給定時(shí)間在所有方向上都具有持久的膨脹率，它在整個(gè)宇宙的生命周期中都會(huì)發(fā)生變化。 目前，經(jīng)過良好校準(zhǔn)的擴(kuò)展速度Ho約為每秒每兆帕秒70公里(此處使用的單位請注意：后退速度以千米每秒為單位，距離以兆帕秒為單位，1兆帕秒= 1M帕秒，1帕秒= 3.26光年。 (2)

Hubble used the Hooker Telescope at Mount Wilson Observatory for some of his most important discoveries. ? Emilio Segrè Visual Archives / American Institute of Physics / Photo Researchers, Inc.哈勃使用了威爾遜山天文臺的胡克望遠(yuǎn)鏡進(jìn)行了一些最重要的發(fā)現(xiàn)。 ?EmilioSegrè視覺檔案館/美國物理研究所/攝影研究公司

Hubble’s remarkable feat was obtained using a very small sample of measurements of velocities and distances for 24 nearby galaxies. The distances to these galaxies were inaccurately measured from the visible brightness of their stars. In addition to plotting all of the individual 24 galaxies in the diagram, Hubble also grouped them into 9 clusters (open circles on Hubble’s diagram) based on their closeness in direction and distance, as a means of minimizing the scatter. Hubble’s experiment was conclusive in convincing the scientific community of the existence of the expanding universe. (2)

哈勃的非凡成就是使用非常小的24個(gè)附近星系的速度和距離測量樣本獲得的。 距這些星系的距離是根據(jù)其恒星的可見亮度進(jìn)行的不準(zhǔn)確測量。 除了在圖中繪制所有24個(gè)星系之外，哈勃還根據(jù)方向和距離的緊密程度將它們分為9個(gè)簇(哈勃圖上的空心圓)，以最大程度地減少散射。 哈勃的實(shí)驗(yàn)在說服科學(xué)界相信不斷膨脹的宇宙的存在方面是結(jié)論性的。 (2)

Hubble’s diagram of galactic recessional velocity versus distance. (Hubble, Proceedings of the National Academy of Sciences, 1929, 15, 168)哈勃的銀河退縮速度與距離的關(guān)系圖。 (哈勃，《美國國家科學(xué)院院刊》，1929年，第15、168頁)

Hubble’s diagram shows a strong linear relationship between velocity and distance. What makes this graph profound is the extensive implications of the observed trend: we live in a large, dynamically evolving universe that is expanding all directions. It is not the type of universe that Albert Einstein assumed in 1917. In fact, Einstein factored in a cosmological constant into his equations to keep the universe static, as it was believed to be at the time. Contrary to Einstein’s beliefs, Hubble’s results suggested that the universe has been expanding for billions of years, from an early beginning of the “Big Bang” up until the present. (2)

哈勃圖顯示了速度和距離之間的強(qiáng)線性關(guān)系。 使該圖更深刻的是觀察到的趨勢的廣泛含義：我們生活在一個(gè)巨大的，動(dòng)態(tài)演化的宇宙中，宇宙在向各個(gè)方向擴(kuò)展。 這不是阿爾伯特·愛因斯坦(Albert Einstein)在1917年所假定的那種宇宙。實(shí)際上，愛因斯坦將宇宙學(xué)常數(shù)納入其方程式中，以保持宇宙的靜態(tài)性，這在當(dāng)時(shí)被認(rèn)為是這樣。 與愛因斯坦的看法相反，哈勃的結(jié)果表明，從“大爆炸”的早期開始到現(xiàn)在，宇宙已經(jīng)膨脹了數(shù)十億年。 (2)

Although Hubble successfully displayed the beautiful linear relationship in his diagram, Hubble’s values for his distances in 1929 were too small by a factor of ~7. The expansion rate Ho was also too large by the same factor. However, despite this large imprecision and its great ramifications for the expansion rate and age of the universe, Hubble’s discovery of the expanding universe is not affected. The underlying linear equation of v ～ d still holds true! (2)

盡管哈勃在圖表中成功顯示出漂亮的線性關(guān)系，但哈勃在1929年的距離值太小了約7倍。 出于相同的原因，膨脹率Ho也太大。 但是，盡管存在很大的不精確性，并且對宇宙的膨脹率和年齡有很大的影響，但哈勃關(guān)于宇宙膨脹的發(fā)現(xiàn)并沒有受到影響。 v的基本線性方程?d仍然適用！ (2)

Note that Einstein’s theory of relativity forecasts deviations from a strictly linear interpretation of Hubble’s law. The amount of deviation depends on the total mass of the universe. A greater understanding of Hubble’s law can inform us about the amount of total matter in the universe. It might also provide more information about dark matter… (3)

請注意，愛因斯坦的相對論預(yù)測偏離哈勃定律的嚴(yán)格線性解釋。 偏差量取決于宇宙的總質(zhì)量。 對哈勃定律有更深入的了解可以使我們了解宇宙中總物質(zhì)的數(shù)量。 它還可能提供有關(guān)暗物質(zhì)的更多信息……(3)

Hubble’s Law was the primary observational evidence in support of the Big Bang theory. Hubble was well renown for his discoveries and in 1990 NASA named the Hubble space telescope after him. (4)

哈勃定律是支持大爆炸理論的主要觀察證據(jù)。 哈勃因其發(fā)現(xiàn)而享譽(yù)世界。1990年，美國國家航空航天局(NASA)以他的名字命名了哈勃太空望遠(yuǎn)鏡。 (4)

Excellent! With that out of the way, now we can start diving into all the fun we’re going to have with hacker stats and Ordinary Least Squares (OLS) Regression. Let’s get started.

優(yōu)秀的！ 有了這種方式，現(xiàn)在我們就可以開始研究黑客統(tǒng)計(jì)數(shù)據(jù)和普通最小二乘(OLS)回歸帶來的所有樂趣。 讓我們開始吧。

實(shí)驗(yàn)設(shè)計(jì)(方法論) (Experimental Design (Methodology))

• Exploratory Data Analysis (EDA).
  探索性數(shù)據(jù)分析(EDA)。
• Adjust galactic distances by a factor of 7.
  將銀河距離調(diào)整7倍。
• OLS using the original Hubble dataset of 24 measurements of galactic distances and recession velocities.
  OLS使用最初的哈勃數(shù)據(jù)集，其中包含24個(gè)銀河距離和后退速度測量值。
• Pairs Bootstrap Resampling of 24 measurements.
  配對Bootstrap重采樣24個(gè)測量值。
• Hypothesis Test → measure the effect of distance on recession velocities.
  假設(shè)檢驗(yàn)→測量距離對衰退速度的影響。

關(guān)于數(shù)據(jù) (About the Data)

Source: “A relation between distance and radial velocity among extra-galactic nebulae” by Edwin Hubble. (1)

資料來源：埃德溫·哈勃(Edwin Hubble)的“銀河外星云之間的距離與徑向速度之間的關(guān)系”。 (1)

• Object Name: Name of the galaxy.

  對象名稱：星系的名稱。

• Distance [Mpc] (r): Distance from Earth in megaparsecs. 1 megaparsec = 1M parsecs, 1 parsec = 3.26 light-years.

  距離[Mpc](r)：距地球的距離，單位為兆帕。 1兆帕秒= 1M帕秒，1帕秒= 3.26光年。

• Velocity [Km/second] (v): Recessional velocity, how fast a galaxy is moving away from Earth. Recessional velocity was recorded in kilometers per second.

  速度[Km / second](v)：衰退速度，銀河系離開地球移動(dòng)的速度。 衰退速度以公里每秒記錄。

Hubble’s 24 measurements of galactic distances and recession velocities. (Hubble, Proceedings of the National Academy of Sciences, 1929, 15, 168)哈勃對銀河系距離和后退速度的24次測量。 (哈勃，《美國國家科學(xué)院院刊》，1929年，第15、168頁)

**Note to my technical readers: If you are interested in the Python code that I used to generate the plots, calculations, etc. feel free to check out my GitHub repo.**

**我的技術(shù)讀者注意：如果您對我用來生成繪圖，計(jì)算等的Python代碼感興趣，請隨時(shí)查看我的 GitHub存儲庫 。**

EDA (EDA)

The first thing we will do is look at the normalized deviations of distances and recessional velocities to?examine?their?relationship. The mean describes the center of the data. The standard deviation describes the spread of the data. It is convenient to normalize two variables in order to perform a fair comparison.

我們要做的第一件事是查看距離和后退速度的歸一化偏差，以檢查它們之間的關(guān)系。 平均值描述數(shù)據(jù)的中心。 標(biāo)準(zhǔn)差描述數(shù)據(jù)的傳播。 標(biāo)準(zhǔn)化兩個(gè)變量以便進(jìn)行公平比較很方便。

With the exception of the first couple of measurements, upon visual inspection of the two normalized arrays of the deviations the galactic distances and their recession velocities seem to be highly correlated. Let’s adjust the distance values?and?generate?summary?statistics?of?our?data.

除了前幾對測量值以外，在目視檢查兩個(gè)標(biāo)準(zhǔn)化的偏差陣列后，銀河距離及其后退速度似乎高度相關(guān)。 讓我們調(diào)整距離值并生成數(shù)據(jù)的摘要統(tǒng)計(jì)信息。

Adjust Distances by Factor of 7

以7的系數(shù)調(diào)整距離

Now we are going to adjust the distance values by multiplying them by a factor of 7. We can look at the result of our adjustment through descriptive statistics.

現(xiàn)在，我們將距離值乘以7來調(diào)整距離值。我們可以通過描述性統(tǒng)計(jì)數(shù)據(jù)查看調(diào)整結(jié)果。

Descriptive Statistics of Hubble’s 24 observations of galactic distances and recession velocities, including adjusted distances ie. distances7哈勃對銀河系距離和后退速度(包括調(diào)整后的距離)的24個(gè)觀測值的描述性統(tǒng)計(jì)。 距離7

Despite the increase in distance by a large factor of 7, recessional velocity and galactic distance are still highly correlated. Let’s calculate the Pearson correlation coefficient and visualize the correlation with the adjusted distance variable via a scatter plot.

盡管距離增加了7倍，但后退速度和銀河距離仍然高度相關(guān)。 讓我們計(jì)算皮爾遜相關(guān)系數(shù)，并通過散點(diǎn)圖可視化與調(diào)整后的距離變量的相關(guān)性。

Correlation

相關(guān)性

Aside from the adjustment of the x-axis, this graph doesn’t look much different than the one that Hubble created. The data exhibits a strong linear relationship with a Pearson correlation coefficient of ~0.8. Next we will perform an Ordinary Least Squares Regression to further understand this relationship as a result of a linear function.

除了調(diào)整x軸外，此圖看起來與哈勃?jiǎng)?chuàng)建的圖沒有太大不同。 數(shù)據(jù)表現(xiàn)出很強(qiáng)的線性關(guān)系，皮爾遜相關(guān)系數(shù)約為0.8。 接下來，我們將執(zhí)行普通最小二乘回歸，以進(jìn)一步了解線性函數(shù)的關(guān)系。

最小二乘 (OLS)

The regression results below were generated via the statsmodels ols() API in Python.

下面的回歸結(jié)果是通過Python中的statsmodels ols()API生成的。

statsmodels ols() results: velocities ~ distances7statsmodels ols()結(jié)果：速度?distances7

For every unit increase in distance, recessional velocity increases by 64.88 km per second.

每增加單位距離，后退速度將增加64.88 km / s。

According to the R-squared value, 62% of the variance of recession velocities are explained by distances.

根據(jù)R平方值，用距離解釋了衰退速度變化的62％。

We’ve already observed Hubble’s Law with a couple of lines of Python code. We examined correlation and concluded with enough confidence that the majority of the variance can be explained by the model. Technically, we could stop at this point and call it day. But let’s take this a step further and understand the residuals like any good scientist would.

我們已經(jīng)用幾行Python代碼觀察了哈勃定律。 我們檢查了相關(guān)性，并以足夠的信心得出結(jié)論，該模型可以解釋大部分方差。 從技術(shù)上講，我們可以在這一點(diǎn)上停下來并將其命名為“ day”。 但是，讓我們更進(jìn)一步，像任何優(yōu)秀科學(xué)家一樣理解殘差。

Residuals, RSS and RMSE

殘差，RSS和RMSE

If we interpret R-squared as the variances that can be explained by our OLS model, the residual sum of squares (RSS) represents the amount of errors that are not explained by the model.

如果我們將R平方解釋為可以由我們的OLS模型解釋的方差，則殘差平方和(RSS)表示該模型無法解釋的誤差量。

The solution of OLS regression is the set of coefficient values for which the RSS is minimal. We’ll revisit this topic when we look at bootstrap resampling in the next section.

OLS回歸的解決方案是RSS最小的一組系數(shù)值。 在下一節(jié)中，我們將在介紹引導(dǎo)程序重采樣時(shí)重新討論該主題。

Here we have Root Mean Square Error (RMSE) of ~223, which can be interpreted as the spread of prediction errors, or how concentrated the data is around the line of best fit. Let’s look at a probability plot to visualize the spread of residuals.

在這里，我們的均方根誤差(RMSE)為?223，可以解釋為預(yù)測誤差的散布，或者數(shù)據(jù)在最佳擬合線附近的集中程度。 讓我們看一下概率圖，以可視化殘差的分布。

The probability plot of the residuals of our OLS model is approximately linear, supporting the assumption that the error terms are normally distributed.

我們的OLS模型殘差的概率圖近似線性，支持誤差項(xiàng)呈正態(tài)分布的假設(shè)。

Again we could also stop right here, but we’re going to keep moving and generate some bootstrap replicates to validate some of the conclusions we’ve witnessed from OLS Regression and uncover a couple of new ones?of?our?own.

同樣，我們也可以在這里停止，但是我們將繼續(xù)前進(jìn)并生成一些引導(dǎo)程序副本，以驗(yàn)證從OLS Regression見證的一些結(jié)論，并發(fā)現(xiàn)我們自己的一些新結(jié)論。

使用雙自舉重采樣 (Resampling with Pairs Bootstraps)

Pairs bootstrap involves resampling pairs of data with replacement. Each collection of pairs fit with a regression model. We will do this again, and again, and again n number of times generating bootstrap n sample statistics from the explanatory and dependent variables, in addition to model parameter estimates after running the OLS model n number of times. We will also calculate the RSS Minima using Bootstrap Resampling to identify the linear equation that best minimizes the errors.

Pairs bootstrap涉及重新采樣數(shù)據(jù)對并進(jìn)行替換。 對的每個(gè)集合均符合回歸模型。 我們會(huì)再次做到這一點(diǎn)，又一次，又一次次從生成的解釋變量和因變量的自舉n個(gè)采樣統(tǒng)計(jì)n個(gè)，除了模型參數(shù)估計(jì)運(yùn)行時(shí)間的OLS模型n個(gè)后。 我們還將使用Bootstrap重采樣來計(jì)算RSS最小值，以識別最能使誤差最小的線性方程。

The goal is to use bootstrap resampling to compute one mean for each sample and create a distribution of sample means and then compute the standard error to quantify the uncertainty in the sample statistic as an estimator for the population average and standard deviation. This comes in very handy since we don’t know the true values for the population average or standard deviation. Instead, we will infer it using bootstrap resampling.

目標(biāo)是使用自舉重采樣為每個(gè)樣本計(jì)算一個(gè)均值，并創(chuàng)建樣本均值的分布，然后計(jì)算標(biāo)準(zhǔn)誤差以量化樣本統(tǒng)計(jì)數(shù)據(jù)中的不確定性，作為總體平均值和標(biāo)準(zhǔn)偏差的估計(jì)量。 這非常方便，因?yàn)槲覀儾恢揽傮w平均值或標(biāo)準(zhǔn)差的真實(shí)值。 相反，我們將使用引導(dǎo)重采樣來推斷它。

According to the central limit theorem, if we generate enough replicates the resampled distributions will follow a normal distribution, which is one of the assumptions for a hypothesis test. More on that in the next section. For now, let’s generate 1,000 paired replicates for each variable.

根據(jù)中心極限定理，如果我們生成足夠多的重復(fù)項(xiàng)，則重新采樣的分布將遵循正態(tài)分布，這是假設(shè)檢驗(yàn)的假設(shè)之一。 下一節(jié)將對此進(jìn)行更多介紹。 現(xiàn)在，讓我們?yōu)槊總€(gè)變量生成1,000個(gè)成對的重復(fù)。

Through way of bootstrap, we inferred that the expected average value of galactic distances is 6.47 Mpc with an uncertainty of about 1 Mpc. This is really close to the sample mean and standard deviation we generated early on. In addition, we can infer with 95% confidence that the true population average lies somewhere between 4.78 and 8.16 Mpc, based on the data provided.

通過自舉，我們推斷銀河距離的預(yù)期平均值為6.47 Mpc，不確定性約為1 Mpc。 這確實(shí)接近我們早期生成的樣本均值和標(biāo)準(zhǔn)差。 此外，根據(jù)提供的數(shù)據(jù)，我們可以以95％的置信度推斷出真正的人口平均數(shù)介于4.78和8.16 Mpc之間。

Notice we have a black line in the middle to mark the expected value. Uncertainty here is just one measure of the spread of the distribution of sample means. Moreover, notice the uncertainty we computed also fits inside the confidence interval. You can think of the uncertainty as the one-sigma confidence interval.

注意，中間有一條黑線標(biāo)記期望值。 這里的不確定度只是衡量樣本均值分布范圍的一種方法。 此外，請注意，我們計(jì)算出的不確定性也適合置信區(qū)間內(nèi)。 您可以將不確定性視為一個(gè)1西格瑪?shù)闹眯艆^(qū)間。

In addition, the vertical red lines mark the 5th (left) and 95th (right) percentiles, which denote the extent of the confidence interval or the range of values containing the inner 95% of sample means.

此外，垂直紅線標(biāo)記第5個(gè)(左)和第95個(gè)(右)百分位數(shù)，表示置信區(qū)間的范圍或包含內(nèi)部95％樣本均值的值的范圍。

Similarly for velocities, we inferred that the expected average value of velocities is about 378 km per second with an uncertainty of about 74 km per second. In addition, we can infer with 95% confidence that the true population average lies somewhere between 238 and 526 km per second, based on the data provided.

同樣，對于速度，我們推斷速度的期望平均值約為每秒378公里，不確定度約為每秒74公里。 此外，根據(jù)提供的數(shù)據(jù)，我們可以以95％的置信度推斷出真實(shí)的平均人口數(shù)量在每秒238至526 km之間。

Now we’re going to conduct a similar exercise, this time with the model slope and intercept parameters. That’s right! You can also use bootstrap resampling to compute the estimate, standard error, and confidence interval for OLS model parameters, all thanks to the central limit theorem. We’re basically going to use each pairs bootstrap replicate as an input into an OLS model to generate bootstrap slope and intercept estimates. Let’s give it a try.

現(xiàn)在，我們將使用模型斜率和截距參數(shù)進(jìn)行類似的練習(xí)。 那就對了！ 您還可以使用引導(dǎo)重采樣來計(jì)算OLS模型參數(shù)的估計(jì)值，標(biāo)準(zhǔn)誤差和置信區(qū)間，這全都?xì)w功于中心極限定理。 基本上，我們將使用每對引導(dǎo)復(fù)制作為OLS模型的輸入，以生成引導(dǎo)斜率和截距估計(jì)。 試一試吧。

We inferred that the estimate of the slope is 65.17 km per second/Mpc with a standard error of 10.33 km per second/Mpc. We are 95% confident that the true slope lies somewhere between 46.33 and 87.34 km per second/Mpc, based on the data provided.

我們推斷斜率的估計(jì)值為65.17 km / s / Mpc，標(biāo)準(zhǔn)誤差為10.33 km / s / Mpc。 根據(jù)提供的數(shù)據(jù)，我們有95％的把握是真實(shí)的斜率在46.33和87.34 km /秒/ Mpc之間。

Note that this is very close to the summary output of statsmodels ols().

請注意，這與statsmodels ols()的摘要輸出非常接近。

We inferred that the estimate of the intercept is -43.23 km per second with a standard error of 78.44 km per second. We are 95% confident that the true intercept lies somewhere between -200.99 and 104.23 km per second, based on the data provided.

我們推斷，截距的估計(jì)值為每秒-43.23 km，標(biāo)準(zhǔn)誤為每秒78.44 km。 根據(jù)提供的數(shù)據(jù)，我們有95％的信心確定真正的截距在每秒-200.99至104.23 km之間。

Now we’re going to generate the RSS Minima via Pairs Bootstrap Resampling.

現(xiàn)在，我們將通過Pairs Bootstrap重采樣來生成RSS最小值。

Visualizing the RSS Minima

可視化RSS最小值

Recall when we looked at RSS before, the solution of OLS is the set of coefficient values for which the RSS is minimal. Now we’re going to use the same replicates we generated to visualize the RSS Minima. Then we’re going to retrieve the model parameters (slope and intercept) that generated the RSS Minima.

回想一下我們以前看過RSS時(shí)，OLS的解是RSS最小的一組系數(shù)值。 現(xiàn)在，我們將使用生成的相同副本來可視化RSS Minima。 然后，我們將檢索生成RSS最小值的模型參數(shù)(坡度和截距)。

Amazing! The best slope and intercept are the ones out of arrays of slopes and intercepts that yielded the minimum RSS value. Notice that our slope value is almost equivalent to the well-calibrated expansion rate (Ho) at the present time.

驚人！ 最佳斜率和截距是那些產(chǎn)生最小RSS值的斜率和截距數(shù)組中的斜率和截距。 請注意，目前我們的斜率值幾乎等于經(jīng)過良好校準(zhǔn)的膨脹率( Ho )。

Behind the scenes, we used the 95% confidence intervals that we generated for the slope and intercept estimates to filter out model parameter values that weren’t within range.

在幕后，我們使用為斜率生成的95％置信區(qū)間并截取估計(jì)值，以過濾掉不在范圍內(nèi)的模型參數(shù)值。

Now that we have the RSS Minima and the model parameters that yielded it, we can visualize the new model with a scatter plot.

現(xiàn)在我們有了RSS Minima和產(chǎn)生它的模型參數(shù)，我們可以用散點(diǎn)圖可視化新模型了。

If we compare this scatter plot to the one we generated earlier during EDA, there’s a slight difference as the red line is a bit steeper. It doesn’t pass through the second galaxy from the top at 14 Mpc but rather is slightly above it. We can consider this an improvement in the overall fit of the model!

如果將散布圖與我們在EDA期間生成的散布圖進(jìn)行比較，則會(huì)有一點(diǎn)差異，因?yàn)榧t線更陡一些。 它沒有以14 Mpc的速度從頂部穿過第二個(gè)星系，而是略高于它。 我們可以認(rèn)為這是模型整體擬合的改進(jìn)！

In the final section, we will conduct a hypothesis test to examine the theory that the length of galactic distance from Earth has an effect on the galaxy’s recessional velocity. We’ve already used a number of tools in our hacker stats toolbox to examine Hubble’s Law. Let’s put the finishing touches on the icing of the cake!

在最后一節(jié)中，我們將進(jìn)行假設(shè)檢驗(yàn)，以檢驗(yàn)銀河系與地球之間的距離長度對銀河系的后退速度有影響的理論。 我們已經(jīng)在黑客統(tǒng)計(jì)信息工具箱中使用了許多工具來檢查哈勃定律。 讓我們?yōu)榈案忮\上添花！

假設(shè)檢驗(yàn)→銀河星云的距離是否對其后退速度有觀察到的影響？ (Hypothesis Test → Do the distances of galactic nebulae have an observed effect on their recession velocities?)

Recall that we used the assumption of the central limit theorem to generate enough replicates to obtain paired resampled normal distributions of galactic distances and recessional velocities. Data that is normally distributed is one of the assumptions required for a hypothesis test.

回想一下，我們使用中心極限定理的假設(shè)來生成足夠的重復(fù)項(xiàng)，以獲得成對的重新采樣的銀河距離和后退速度正態(tài)分布。 正態(tài)分布的數(shù)據(jù)是假設(shè)檢驗(yàn)所需的假設(shè)之一。

Now we will test whether the length of galactic distance has an observed effect on recessional velocity. We will define short and long distances of galaxies from planet Earth. Then we will resample and shuffle the velocities and take the difference in resampled means as a test statistic. In other words, if the test statistic distribution truly exhibits a difference in effect (ie. mean difference of velocities > 0 ) then we can reject the null hypothesis, and conclude with enough power that the results are statistically significant.

現(xiàn)在，我們將測試銀河距離的長度是否對后退速度有觀察到的影響。 我們將定義星系與地球之間的短距離和長距離。 然后，我們將對速度進(jìn)行重新采樣和改組，并將重新采樣的均值之差作為檢驗(yàn)統(tǒng)計(jì)量。 換句話說，如果檢驗(yàn)統(tǒng)計(jì)量分布確實(shí)顯示出效果上的差異(即，速度的平均差異> 0)，那么我們可以拒絕原假設(shè)，并以足夠的能力得出結(jié)論，該結(jié)果在統(tǒng)計(jì)上是有意義的。

See the null and alternative hypotheses below:

請參見下面的原假設(shè)和替代假設(shè)：

Null Hypothesis

零假設(shè)

• The length of distance has no effect on the recession velocity of Extra-Galactic Nebulae.

  距離的長度對銀河外星云的后退速度沒有影響。

Alternative Hypothesis

替代假設(shè)

• The length of distance has an observed effect on the recession velocity velocities of Extra-Galactic Nebulae.

  距離的長度對銀河外星云的后退速度有影響。

Assumptions

假設(shè)條件

For our experiment, we will use a 95% significance level, which will make our alpha value 0.05. We define short distances as distances less than 7 Mpc; Conversely, we define long distances as distances that are greater or equal to 7 Mpc → Note that this will be done with our adjusted values for distances.

對于我們的實(shí)驗(yàn)，我們將使用95％的顯著性水平，這將使我們的alpha值為0.05。 我們將短距離定義為小于7 Mpc的距離； 相反，我們將長距離定義為大于或等于7 Mpc的距離→請注意，這將通過我們對距離的調(diào)整值來完成。

We’re going to use a T-test since we do not know the true standard deviation of the population; We will use 1,000 bootstrap replicates of galactic recessional velocities.

因?yàn)槲覀儾恢揽傮w的真實(shí)標(biāo)準(zhǔn)偏差，所以我們將使用T檢驗(yàn)。 我們將使用1,000個(gè)銀河衰退速度的自舉復(fù)制。

The test statistic is the difference between a recession velocity drawn from shorter distances and one drawn from longer distances. The distribution of difference values is built up by subtracting each point in the shorter range with one from the longer range, to see if the mean difference is greater than zero, also known as the effect size.

檢驗(yàn)統(tǒng)計(jì)量是從較短距離得出的衰退速度與從較長距離得出的衰退速度之差。 差值的分布是通過將較短范圍內(nèi)的每個(gè)點(diǎn)減去較長范圍中的一個(gè)點(diǎn)而建立的，以查看平均差是否大于零(也稱為效果大小)。

And there we have it! The mean of the test statistic is not zero (denoted by the shaded region in gray), which tells us that there is on average an 83.76 km per second difference in velocities when comparing short and long galactic distances. Again, we refer to this as our effect size. In other words, galaxies that are closer to Earth are moving away at a much slower rate than galaxies that are a lot further away from Earth. The increase in the galactic distance from Earth had an observed effect on recessional velocity. The standard error of the test statistic distribution is also not zero, so there is uncertainty in the size of the effect.

我們終于得到它了！ 測試統(tǒng)計(jì)的平均值不為零(由灰色陰影區(qū)域表示)，這告訴我們在比較短銀河和長銀河距離時(shí)平均每秒速度相差83.76 km。 同樣，我們將此稱為效應(yīng)大小。 換句話說，離地球更近的星系的移動(dòng)速度比離地球更遠(yuǎn)的星系移動(dòng)的速度要慢得多。 到地球的銀河距離的增加對后退速度有觀察到的影響。 測試統(tǒng)計(jì)量分布的標(biāo)準(zhǔn)誤差也不是零，因此影響的大小存在不確定性。

It’s also worthwhile to mention that shuffling the resampled data points had an effect on the randomness of our experiment. We shuffled the data in order to make sure that each sample is composed of random and independent data points, which are other assumptions required for a hypothesis test. If we didn’t shuffle the data then the effect size would be much greater due to the time-ordered effect on the mean.

還值得一提的是，對重新采樣的數(shù)據(jù)點(diǎn)進(jìn)行混洗會(huì)影響我們實(shí)驗(yàn)的隨機(jī)性。 我們對數(shù)據(jù)進(jìn)行了混洗，以確保每個(gè)樣本均由隨機(jī)和獨(dú)立的數(shù)據(jù)點(diǎn)組成，這是假設(shè)檢驗(yàn)所需的其他假設(shè)。 如果我們不對數(shù)據(jù)進(jìn)行混洗，那么由于均值的時(shí)間順序影響，影響大小會(huì)更大。

Finally, our P-value is extremely small → 0.0000001

最后，我們的P值非常小→0.0000001

Thus, we can conclude with a high degree of statistical significance that the distances of galaxies from Earth have an effect on their recessional velocities, which is observational evidence of Hubble’s Law. The universe is constantly expanding all around us!

因此，我們可以得出具有高度統(tǒng)計(jì)意義的結(jié)論，即星系與地球的距離會(huì)影響它們的后退速度，這是哈勃定律的觀測證據(jù)。 宇宙不斷在我們周圍擴(kuò)展！

P.S. ~ Note that we didn’t use power analysis to determine the sample size for the hypothesis test upfront, since we weren’t privy to the standard effect size at the beginning of the experiment. According to traditional stats textbooks, one should determine the needed sample size prior to performing a hypothesis test. Instead, we opted for the hacker stats approach: we ran a hypothesis test with 1,000 samples, retrieved the effect size and standard error of the effect, and hacked the needed sample size. The result was roughly 910 observations, which worked well in our favor. Got to love hacker stats!

PS?請注意，由于我們在實(shí)驗(yàn)開始時(shí)并不熟悉標(biāo)準(zhǔn)效應(yīng)量，因此我們并未使用功效分析來預(yù)先確定假設(shè)檢驗(yàn)的樣本量。 根據(jù)傳統(tǒng)的統(tǒng)計(jì)教科書，應(yīng)該在執(zhí)行假設(shè)檢驗(yàn)之前確定所需的樣本量。 相反，我們選擇了黑客統(tǒng)計(jì)方法：我們對1,000個(gè)樣本進(jìn)行了假設(shè)檢驗(yàn)，檢索了效應(yīng)大小和效應(yīng)的標(biāo)準(zhǔn)誤，然后黑客入侵了所需的樣本量。 結(jié)果大約是910個(gè)觀測值，對我們有利。 愛上了黑客統(tǒng)計(jì)信息！

下一步 (Next Steps)

• Add 22 estimated distances for the T-test.
  為T檢驗(yàn)加上22個(gè)估計(jì)的距離。
• Identify Nebulae Clusters with KMeans.
  用KMeans識別星云團(tuán)。
• Use other data sources of galactic distances & recession velocities…
  使用銀河距離和后退速度的其他數(shù)據(jù)源…

We could have used the 22 estimated distances for our T-test as well, bringing the total number of observations up to 46.

我們也可以將22個(gè)估計(jì)的距離用于T檢驗(yàn)，從而使觀測的總數(shù)達(dá)到46個(gè)。

Hubble grouped the 24 galaxies into 9 clusters. This would be an interesting exercise to see if we get the same cluster centroids with KMeans or Agglomerative clustering.

哈勃將24個(gè)星系分為9個(gè)星團(tuán)。 看看我們是否獲得具有KMeans或聚集聚類的相同聚類質(zhì)心，這將是一個(gè)有趣的練習(xí)。

Finally, we can also use data from the Hubble telescope to examine Hubble’s law.

最后，我們還可以使用哈勃望遠(yuǎn)鏡的數(shù)據(jù)檢查哈勃定律。

As Hubble concludes in his PNAS paper, “The results establish a roughly linear relation between velocities and distances among nebulae for which velocities have been previously published, and the [relationship] appears to dominate the distribution of velocities…..New data to be expected in the near future may modify the significance of the present investigation or, if confirmatory, will lead to a solution many times the weight.” (1)

正如哈勃在其PNAS論文中總結(jié)的那樣：“結(jié)果建立了速度與星云之間的距離之間的大致線性關(guān)系，先前已經(jīng)針對該距離發(fā)布了速度，[關(guān)系]似乎主導(dǎo)了速度的分布……..新數(shù)據(jù)有望得到預(yù)料在不久的將來可能會(huì)改變當(dāng)前調(diào)查的意義，或者，如果證實(shí)這一點(diǎn)，將導(dǎo)致解決方案的重量增加很多倍。” (1)

翻譯自: https://medium.com/datadriveninvestor/revisiting-hubbles-law-with-hacker-stats-in-python-9b56604916c1

騰訊哈勃

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