From dec74e83ab2c865632a1697ef5e371513997122f Mon Sep 17 00:00:00 2001 From: Carlos Camara Date: Thu, 19 Oct 2023 15:24:47 +0100 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- ...3_1_Data_Processing_and_Summarization.html | 51 +- .../figure-html/cell-16-output-3.png | Bin 71189 -> 70575 bytes .../figure-html/cell-16-output-4.png | Bin 70627 -> 70528 bytes .../IM939_Lab_3_2_Linear_Regression.html | 539 +++++++++++------- .../figure-html/cell-10-output-1.png | Bin 86604 -> 86674 bytes .../figure-html/cell-10-output-2.png | Bin 0 -> 34956 bytes .../figure-html/cell-11-output-1.png | Bin 112431 -> 86570 bytes .../figure-html/cell-11-output-2.png | Bin 0 -> 74450 bytes .../figure-html/cell-12-output-1.png | Bin 0 -> 106107 bytes .../figure-html/cell-14-output-2.png | Bin 129969 -> 129991 bytes .../figure-html/cell-15-output-2.png | Bin 0 -> 130226 bytes .../figure-html/cell-16-output-1.png | Bin 160088 -> 160180 bytes .../figure-html/cell-16-output-2.png | Bin 0 -> 130017 bytes .../figure-html/cell-17-output-1.png | Bin 0 -> 159940 bytes .../figure-html/cell-17-output-2.png | Bin 0 -> 130047 bytes .../figure-html/cell-18-output-1.png | Bin 0 -> 160079 bytes .../figure-html/cell-19-output-1.png | Bin 0 -> 159646 bytes .../figure-html/cell-21-output-2.png | Bin 0 -> 134269 bytes .../figure-html/cell-22-output-2.png | Bin 0 -> 134269 bytes .../figure-html/cell-23-output-2.png | Bin 0 -> 96616 bytes .../figure-html/cell-6-output-2.png | Bin 0 -> 22409 bytes search.json | 36 +- sitemap.xml | 88 +-- 24 files changed, 435 insertions(+), 281 deletions(-) create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-10-output-2.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-11-output-2.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-12-output-1.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-15-output-2.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-16-output-2.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-17-output-1.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-17-output-2.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-18-output-1.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-19-output-1.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-21-output-2.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-22-output-2.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-23-output-2.png create mode 100644 content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression_files/figure-html/cell-6-output-2.png diff --git a/.nojekyll b/.nojekyll index 3905dc8..b37f113 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -7d01fd32 \ No newline at end of file +51169f94 \ No newline at end of file diff --git a/content/labs/Lab_3/IM939_Lab_3_1_Data_Processing_and_Summarization.html b/content/labs/Lab_3/IM939_Lab_3_1_Data_Processing_and_Summarization.html index 39749ee..ad202d5 100644 --- a/content/labs/Lab_3/IM939_Lab_3_1_Data_Processing_and_Summarization.html +++ b/content/labs/Lab_3/IM939_Lab_3_1_Data_Processing_and_Summarization.html @@ -1039,8 +1039,7 @@

# Getting the Data
 df_tuber = pd.read_csv('data/raw/TB_burden_countries_2014-09-29.csv')
 
-df_tuber.head()
-
+df_tuber.head()
@@ -1256,8 +1255,8 @@

# Filling missing numeric values ONLY (categorical values will be untouched)
 df_tuber = df_tuber.fillna(value=df_tuber.mean(numeric_only = True))
 
-# Inspecting missing values in dataset
-pd.isnull(df_tuber).sum()

+# Count missing values +df_tuber.isna().sum()
country                      0
 iso2                        23
@@ -1461,27 +1460,27 @@ 

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Table of contents

@@ -588,14 +595,12 @@

12  https://www.tutorialspoint.com/scikit_learn/scikit_learn_linear_regression.htm

-
-

12.1 Sea Ice Dataset

-

The script below is the based on the Sea Ice dataset from Igual and Seguí (2017). You can find the data here: ftp://sidads.colorado.edu/DATASETS/NOAA/ or read more about the project here: https://nsidc.org/data/seaice_index

-

This notebook will walk you through an example of a simple linear regression and the other notebook “IM939 Lab 3 - Linear Regression Exercise.ipynb” will include the example of a multiple linear regression, too.

-
-
-

12.2 Simple and Multiple Linear Regression

+

In this lab, inspired by Igual and Seguí (2017), we will introduce simple linear regression in Python to try to answer the question: Has there been a decrease in the amount of ice in the last years? To do so, we are going to use Scikit Learn, a machine learning library, and the Sea Ice Index Daily and Monthly Image Viewer dataset from the National Snow and Ice Data Center.

+
+

12.1 Key concepts’ refresher

+

Let’s refresh some theoretical concepts to understand what we are going to do.

+
+

12.1.1 Simple and Multiple Linear Regression

In the linear model the response \(\textbf{y}\) depends linearly from the covariates \(\textbf{x}_i\).

In the simple linear regression, with a single variable, we described the relationship between the predictor and the response with a straight line. The general linear model: \[ \textbf{y} = a_0+ a_1 \textbf{x}_1 \]

The parameter \(a_0\) is called the constant term or the intercept.

@@ -603,18 +608,19 @@

\[ \textbf{y} = a_1 \textbf{x}_1 + \dots + a_m \textbf{x}_{m} \]

The \(a_i\) are termed the parameters of the model or the coefficients.

-
-

12.3 Ordinary Least Squares

+
+

12.1.2 Ordinary Least Squares

Ordinary Least Squares (OLS) is the simplest and most common estimator in which the coefficients \(a\)’s of the simple linear regression: \(\textbf{y} = a_0+a_1 \textbf{x}\), are chosen to minimize the square of the distance between the predicted values and the actual values.

\[ ||a_0 + a_1 \textbf{x} - \textbf{y} ||^2_2 = \sum_{j=1}^n (a_0+a_1 x_{j} - y_j )^2,\]

This expression is often called sum of squared errors of prediction (SSE).

-
-

12.4 Case study: Climate Change and Sea Ice Extent

-

The question: Has there been a decrease in the amount of ice in the last years?

-
-

12.4.1 Reading Data

-

There are five steps. First, let’s load the data that is already in the data folder: SeaIce.txt. It is a text file, specifically is a Tab separated file where each Tab delimites the following columns:

+
+
+

12.2 Case study: Climate Change and Sea Ice Extent

+

Remember, we are trying to answer the following research question: Has there been a decrease in the amount of ice in the last years?

+
+

12.2.1 Data assessment

+

First, let’s load the SeaIce.txt dataset that is already in the data folder1. It is a text file, but unlike csv files, where columns are separated by commas (,), it is a Tab separated file, where each Tab delimites the following columns:

  • 1 The original dataset was downloaded from https://nsidc.org/data/g02135/versions/3, which provides useful metadata information, as well as API services.

    • Year: 4-digit year
    • mo: 1- or 2-digit month
    • @@ -623,18 +629,31 @@

      extent: Sea ice extent in millions of square km
    • area: Sea ice area in millions of square km
    -

    Once we upload the data, we can create a DataFrame1 using Pandas.

    +

    Once we upload the data, we can create a DataFrame using Pandas using the well known read_csv() function, but in this case, because columns are not separated by commas as expected, but Tabs, we will need to use the delim_whitespace=True argument.

    import pandas as pd
    -ice = pd.read_csv('data/raw/SeaIce.txt', 
    -                  delim_whitespace=True)
    -
    -print('shape:', ice.shape) #this returns number of rows and columns in a dataset
    -ice.head() 
    +ice = pd.read_csv('data/raw/SeaIce.txt', delim_whitespace=True) + +ice.info()
    -
    shape: (424, 6)
    +
    <class 'pandas.core.frame.DataFrame'>
    +RangeIndex: 424 entries, 0 to 423
    +Data columns (total 6 columns):
    + #   Column     Non-Null Count  Dtype  
    +---  ------     --------------  -----  
    + 0   year       424 non-null    int64  
    + 1   mo         424 non-null    int64  
    + 2   data_type  424 non-null    object 
    + 3   region     424 non-null    object 
    + 4   extent     424 non-null    float64
    + 5   area       424 non-null    float64
    +dtypes: float64(2), int64(2), object(2)
    +memory usage: 20.0+ KB
    -
    +
    +
    +
    ice.head() 
    +
    @@ -702,110 +721,240 @@

    -

    We can compute the mean for that interval of time (1981 through 2010), before data cleaning.

    -
    -
    ice.mean(numeric_only = True)
    -
    -
    year      1996.000000
    -mo           6.500000
    -extent     -35.443066
    -area       -37.921108
    -dtype: float64
    +

    And we can get some summary statistics from the numerical attributes:

    +
    +
    ice.describe()
    +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    yearmoextentarea
    count424.000000424.000000424.000000424.000000
    mean1996.0000006.500000-35.443066-37.921108
    std10.2147163.474323686.736905686.566381
    min1978.0000001.000000-9999.000000-9999.000000
    25%1987.0000003.0000009.2725006.347500
    50%1996.0000006.50000012.3850009.895000
    75%2005.00000010.00000014.54000012.222500
    max2014.00000012.00000016.45000013.840000
    + +
    +
    +
    +
    +
    +
    + +
    +
    +Warning +
    +
    +
    +

    Did we receive a negative mean for extent and area? What this could possibly mean? Probably, inspecting those attributes visually could give us a clue.

    -

    Did we receive a negative mean…?

    -
    -

    12.4.2 Data visualisation to explore data

    -

    Do you remember Seaborn? We will use lmplot() function from Seaborn to explore linear relationship of different forms, e.g. relationship between the month of the year (variable) and extent (responses):

    -
    +
    +

    12.2.2 Data visualisation to explore data

    +

    We will use Seaborn’s lmplot() function to explore linear relationship of different forms, e.g. relationship between the month of the year (variable) and extent (responses):

    +
    import matplotlib.pyplot as plt
     import numpy as np
     import pandas as pd
     import seaborn as sns
    -
    +
    # Visualize the data
    -x = ice.year
    -y = ice.extent
    -plt.scatter(x, y, color = 'red')
    -plt.xlabel('Year')
    -plt.ylabel('Extent')
    -
    +plt.scatter(ice.mo, ice.extent, color = 'red') +plt.xlabel('Year') +plt.ylabel('Extent')
    +
    Text(0, 0.5, 'Extent')
    -

    +

    -

    We detect some outlier or missing data. we are going to use function np.unique and find the unique elements of an array.

    -
    -
    ?np.unique
    +
    +
    +
    + +
    +
    +Warning +
    +
    +

    We detect some outlier or missing data. This might have to do with those negative mean values that we detected previously.

    +
    +
    +

    We can use numpy’s function np.unique() to find the unique elements of an array.

    -
    print ('Different values in data_type field:', np.unique(ice.data_type.values))   # there is a -9999 value!
    +
    print ('Different values in data_type field:', np.unique(ice.data_type.values))   # there is a -9999 value!
    Different values in data_type field: ['-9999' 'Goddard' 'NRTSI-G']

    Let’s see what type of data we have, other than the ones printed above

    -
    print (ice[(ice.data_type != 'Goddard')
    -          & (ice.data_type != 'NRTSI-G')])
    +
    print (ice[(ice.data_type != 'Goddard')
    +          & (ice.data_type != 'NRTSI-G')])
         year  mo data_type region  extent    area
     9    1988   1     -9999      N -9999.0 -9999.0
     397  1987  12     -9999      N -9999.0 -9999.0
    -

    Data cleaning: we checked all the values and notice -9999 values in data_type field which should contain ‘Goddard’ or ‘NRTSI-G’ (some type of input dataset). So we clean them by removing these instances

    +
    +
    +

    12.2.3 Data cleaning

    +

    We checked all the values and notice -9999 values in data_type field which should contain Goddard or NRTSI-G (some type of input dataset).

    +

    In this case, we will clean them by creating a copy of the original dataframe that does not include these instances.

    +
    +
    +
    + +
    +
    +Dataframe copies vs instances +
    +
    +
    +

    Unless we do not explicitly create a copy of a dataframe, when subsetting a dataframes we are actually creating instances. Whereas copies are totally independent objects from the original one, instances are reduced “views” from the original, meaning that if we change a value on the instance, we are also changing the value on the original data frame, which may not be what we wanted to do.

    +

    An explanation of how (and why) do we need to create copies can be found here: https://www.statology.org/pandas-copy-dataframe/

    +
    +
    -
    # We can easily clean the data now:
    -ice2 = ice[ice.data_type != '-9999']
    -print ('shape:', ice2.shape)
    -# And repeat the plot
    -x = ice2.year
    -y = ice2.extent
    -plt.scatter(x, y, color = 'red')
    -plt.xlabel('Month')
    -plt.ylabel('Extent')
    +
    # We can easily clean the data now:
    +ice2 = ice[ice.data_type != '-9999'].copy()
    +
    +print ('shape:', ice2.shape)
    shape: (422, 6)
    -
    +
    +
    +
    # And repeat the plot, without the outliers
    +plt.scatter(ice2.mo, ice2.extent, color = 'red')
    +plt.xlabel('Month')
    +plt.ylabel('Extent')
    +
    Text(0, 0.5, 'Extent')
    -

    +

    -
    -
    sns.lmplot(data = ice2, x = "mo", y = "extent")
    -
    -

    +
    +
    sns.relplot(ice2, x = "mo", y = "extent", aspect = 2)
    +
    +
    /Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
    +  if pd.api.types.is_categorical_dtype(vector):
    +/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
    +  if pd.api.types.is_categorical_dtype(vector):
    +/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
    +  if pd.api.types.is_categorical_dtype(vector):
    +/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
    +  if pd.api.types.is_categorical_dtype(vector):
    +/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
    +  if pd.api.types.is_categorical_dtype(vector):
    +/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
    +  if pd.api.types.is_categorical_dtype(vector):
    +
    +
    +

    -

    Above you can see ice extent data by month. You can see a monthly fluctuation of the sea ice extent, as would be expected for the different seasons of the year. In order to run regression, and avoid this fluctuation we can normalize data. This will let us see the evolution of the extent over the years.

    -
    -

    12.4.3 Normalization

    -

    The lmplot() function from the Seaborn module is intended for exploring linear relationships of different forms in multidimensional datesets. Input data must be in a Pandas DataFrame. To plot them, we provide the predictor and response variable names along with the dataset

    -
    -
    sns.lmplot(ice2, x = "mo", y = "extent", height = 5.2, aspect = 2);
    -
    -# Uncomment below to save the resulting plot.
    -#plt.savefig("figs/CleanedByMonth.png", dpi = 300, bbox_inches = 'tight')
    +
    +
    +

    12.3 Regression model fit

    +

    Now that we have a clean dataset, we can use Seaborn’s lmplot() function comparing month vs extent.

    +

    The lmplot() function from the Seaborn module is intended for exploring linear relationships of different forms in multidimensional datesets. Input data must be in a Pandas DataFrame. To plot them, we provide the predictor (mo) and response (extent) variable names along with the dataset (ice2).

    +
    +
    sns.lmplot(ice2, x = "mo", y = "extent", aspect=2);
    +
    +# Uncomment below to save the resulting plot.
    +#plt.savefig("figs/CleanedByMonth.png", dpi = 300, bbox_inches = 'tight')
    -

    +

    +

    Above you can see ice extent data by month. You can see a monthly fluctuation of the sea ice extent, as would be expected for the different seasons of the year. In order to run regression, and avoid this fluctuation we can normalize data. This will let us see the evolution of the extent over the years.

    +
    +

    12.3.1 Normalization

    -
    # Compute the mean for each month.
    -grouped = ice2.groupby('mo')
    -month_means = grouped.extent.mean()
    -month_variances = grouped.extent.var()
    -print ('Means:', month_means)
    -print ('Variances:',month_variances)
    +
    # Compute the mean for each month.
    +month_means = ice2.groupby('mo').extent.mean()
    +
    +# Compute the variance for each month.
    +month_variances = ice2.groupby('mo').extent.var()
    +
    +# Show the values:
    +print('Means:', month_means)
    +print('\n') # Add a new line between the two prints, so they are easily distinguishible.
    +print ('Variances:',month_variances)
    Means: mo
     1     14.479429
    @@ -821,6 +970,8 @@ 

    To capture variation per month, we can compute mean for the i-th interval of time (using 1979-2014) and subtract it from the set of extent values for that month . This can be converted to a relative pecentage difference by dividing it by the total avareage (1979-2014) and multiplying by 100.

    -
    # Data normalization
    -for i in range(12):
    -    ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -    
    -sns.lmplot(ice2 , x = "mo", y = "extent", height = 5.2, aspect = 2);
    -plt.savefig("figs/IceExtentNormalizedByMonth.png", dpi = 300, bbox_inches = 'tight')
    +
    # Data normalization
    +for i in range(12):
    +    ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +    
    +sns.lmplot(ice2 , x = "mo", y = "extent", height = 5.2, aspect = 2);
    +#plt.savefig("figs/IceExtentNormalizedByMonth.png", dpi = 300, bbox_inches = 'tight')
    -
    /var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +
    /var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    -/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_80350/751478846.py:3: SettingWithCopyWarning: 
    +  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
    +/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
     A value is trying to be set on a copy of a slice from a DataFrame
     
     See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
    -  ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean().copy()
    + ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()

    -
    print ('mean:', ice2.extent.mean())
    -print ('var:', ice2.extent.var())
    +
    print ('mean:', ice2.extent.mean())
    +print ('var:', ice2.extent.var())
    mean: -7.745252569896827e-16
     var: 31.983239774968798
    @@ -921,41 +1072,41 @@

    Let us consider the entire year

    -
    sns.lmplot(ice2, x = "year", y = "extent", height = 5.2, aspect = 2);
    -plt.savefig("figs/IceExtentAllMonthsByYearlmplot.png", dpi = 300, bbox_inches = 'tight')
    +
    sns.lmplot(ice2, x = "year", y = "extent", height = 5.2, aspect = 2);
    +#plt.savefig("figs/IceExtentAllMonthsByYearlmplot.png", dpi = 300, bbox_inches = 'tight')

    -
    -

    12.4.4 Pearson’s correlation

    +
    +

    12.3.2 Pearson’s correlation

    Let’s calculate Pearson’s correlation coefficient and the p-value for testing non-correlation.

    -
    import scipy.stats
    -scipy.stats.pearsonr(ice2.year.values, ice2.extent.values)
    -
    +
    import scipy.stats
    +scipy.stats.pearsonr(ice2.year.values, ice2.extent.values)
    +
    PearsonRResult(statistic=-0.8183500709897178, pvalue=4.4492318168687107e-103)
    -
    -

    12.4.5 Simple OLS

    +
    +

    12.3.3 Simple OLS

    We can also compute the trend as a simple linear regression (OLS) and quantitatively evaluate it.

    -

    For that we use using Scikit-learn, library that provides a variety of both supervised and unsupervised machine learning techniques. Scikit-learn provides an object-oriented interface centered around the concept of an Estimator. The Estimator.fit method sets the state of the estimator based on the training data. Usually, the data is comprised of a two-dimensional numpy array \(X\) of shape (n_samples, n_predictors) that holds the so-called feature matrix and a one-dimensional numpy array \(\textbf{y}\) that holds the responses. Some estimators allow the user to control the fitting behavior. For example, the sklearn.linear_model.LinearRegression estimator allows the user to specify whether or not to fit an intercept term. This is done by setting the corresponding constructor arguments of the estimator object. During the fitting process, the state of the estimator is stored in instance attributes that have a trailing underscore (’’). For example, the coefficients of a LinearRegression estimator are stored in the attribute coef.

    +

    For that we use using Scikit-learn, library that provides a variety of both supervised and unsupervised machine learning techniques. Scikit-learn provides an object-oriented interface centered around the concept of an Estimator. The Estimator.fit method sets the state of the estimator based on the training data. Usually, the data is comprised of a two-dimensional numpy array \(X\) of shape (n_samples, n_predictors) that holds the so-called feature matrix and a one-dimensional numpy array \(\textbf{y}\) that holds the responses. Some estimators allow the user to control the fitting behavior. For example, the sklearn.linear_model.LinearRegression estimator allows the user to specify whether or not to fit an intercept term. This is done by setting the corresponding constructor arguments of the estimator object. During the fitting process, the state of the estimator is stored in instance attributes that have a trailing underscore (_). For example, the coefficients of a LinearRegression estimator are stored in the attribute coef_.

    Estimators that can generate predictions provide a Estimator.predict method. In the case of regression, Estimator.predict will return the predicted regression values, \(\hat{\textbf{y}}\).

    -
    from sklearn.linear_model import LinearRegression
    -
    -est = LinearRegression(fit_intercept = True)
    -
    -x = ice2[['year']]
    -y = ice2[['extent']]
    -
    -est.fit(x, y)
    -
    -print("Coefficients:", est.coef_)
    -print ("Intercept:", est.intercept_)
    +
    from sklearn.linear_model import LinearRegression
    +
    +est = LinearRegression(fit_intercept = True)
    +
    +x = ice2[['year']]
    +y = ice2[['extent']]
    +
    +est.fit(x, y)
    +
    +print("Coefficients:", est.coef_)
    +print ("Intercept:", est.intercept_)
    Coefficients: [[-0.45275459]]
     Intercept: [903.71640207]
    @@ -963,22 +1114,22 @@

    We can evaluate the model fitting by computing the mean squared error (\(MSE\)) and the coefficient of determination (\(R^2\)) of the model. The coefficient \(R^2\) is defined as \((1 - \textbf{u}/\textbf{v})\), where \(\textbf{u}\) is the residual sum of squares \(\sum (\textbf{y} - \hat{\textbf{y}})^2\) and \(\textbf{v}\) is the regression sum of squares \(\sum (\textbf{y} - \bar{\textbf{y}})^2\), where \(\bar{\textbf{y}}\) is the mean. The best possible score for \(R^2\) is 1.0: lower values are worse. These measures can provide a quantitative answer to the question we are facing: Is there a negative trend in the evolution of sea ice extent over recent years?

    -
    from sklearn import metrics
    -
    -# Analysis for all months together.
    -x = ice2[['year']]
    -y = ice2[['extent']]
    -model = LinearRegression()
    -model.fit(x, y)
    -y_hat = model.predict(x)
    -plt.plot(x, y,'o', alpha = 0.5)
    -plt.plot(x, y_hat, 'r', alpha = 0.5)
    -plt.xlabel('year')
    -plt.ylabel('extent (All months)')
    -print ("MSE:", metrics.mean_squared_error(y_hat, y))
    -print ("R^2:", metrics.r2_score(y_hat, y))
    -print ("var:", y.var())
    -plt.savefig("figs/IceExtentLinearRegressionAllMonthsByYearPrediction.png", dpi = 300, bbox_inches = 'tight')
    +
    from sklearn import metrics
    +
    +# Analysis for all months together.
    +x = ice2[['year']]
    +y = ice2[['extent']]
    +model = LinearRegression()
    +model.fit(x, y)
    +y_hat = model.predict(x)
    +plt.plot(x, y,'o', alpha = 0.5)
    +plt.plot(x, y_hat, 'r', alpha = 0.5)
    +plt.xlabel('year')
    +plt.ylabel('extent (All months)')
    +print ("MSE:", metrics.mean_squared_error(y_hat, y))
    +print ("R^2:", metrics.r2_score(y_hat, y))
    +print ("var:", y.var())
    +#plt.savefig("figs/IceExtentLinearRegressionAllMonthsByYearPrediction.png", dpi = 300, bbox_inches = 'tight')
    MSE: 10.539131639803488
     R^2: 0.5067870382100226
    @@ -991,26 +1142,26 @@ 

    We can conclude that the data show a long-term negative trend in recent years.

    -
    # Analysis for a particular month.
    -#For January
    -jan = ice2[ice2.mo == 1];
    -
    -x = jan[['year']]
    -y = jan[['extent']]
    -
    -model = LinearRegression()
    -model.fit(x, y)
    -
    -y_hat = model.predict(x)
    -
    -plt.figure()
    -plt.plot(x, y,'-o', alpha = 0.5)
    -plt.plot(x, y_hat, 'r', alpha = 0.5)
    -plt.xlabel('year')
    -plt.ylabel('extent (January)')
    -
    -print ("MSE:", metrics.mean_squared_error(y_hat, y))
    -print ("R^2:", metrics.r2_score(y_hat, y))
    +
    # Analysis for a particular month.
    +#For January
    +jan = ice2[ice2.mo == 1];
    +
    +x = jan[['year']]
    +y = jan[['extent']]
    +
    +model = LinearRegression()
    +model.fit(x, y)
    +
    +y_hat = model.predict(x)
    +
    +plt.figure()
    +plt.plot(x, y,'-o', alpha = 0.5)
    +plt.plot(x, y_hat, 'r', alpha = 0.5)
    +plt.xlabel('year')
    +plt.ylabel('extent (January)')
    +
    +print ("MSE:", metrics.mean_squared_error(y_hat, y))
    +print ("R^2:", metrics.r2_score(y_hat, y))
    MSE: 3.8395160752867565
     R^2: 0.7810636041396216
    @@ -1021,12 +1172,12 @@

    We can also estimate the extent value for 2025. For that we use the function predict of the model.

    -
    X = np.array(2025) 
    -y_hat = model.predict(X.reshape(-1, 1))
    -j = 1 # January
    -# Original value (before normalization)
    -y_hat = (y_hat*month_means.mean()/100) + month_means[j]
    -print ("Prediction of extent for January 2025 (in millions of square km):", y_hat)
    +
    X = np.array(2025) 
    +y_hat = model.predict(X.reshape(-1, 1))
    +j = 1 # January
    +# Original value (before normalization)
    +y_hat = (y_hat*month_means.mean()/100) + month_means[j]
    +print ("Prediction of extent for January 2025 (in millions of square km):", y_hat)
    Prediction of extent for January 2025 (in millions of square km): [[13.14449923]]
    @@ -1035,7 +1186,12 @@

    -

    Prediction of extent for January 2025 (in millions of square km): [ 13.14449923]

    +

    Prediction of extent for January 2025 (in millions of square km): 13.14449923

    +

    +
    +
    +

    12.4 More information

    +

    You can find more information about using scikit for Linear regression here: https://www.tutorialspoint.com/scikit_learn/scikit_learn_linear_regression.htm

    -
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