Housing market analysis

We have a data set of real estate prices from Aimes, Iowa. Our goal is to provide a model to predict prices using a subset of the columns to be able to reduce the size of the database the company is required to store. We are also interested in providing descriptive statistics for the market, specifically which neighborhoods have many sales and high prices. Finally, we are interested in describing how this data has changed over time.

Results summary

The most significant predictor of housing price is the OverallQuality of the property. Additionally, there is a nonlinear relation between this quality and SalePrice, with a quadratic term added to the model showing an even stronger correlation than the linear term.

I suggest a simplified model with six parameters to predict SalePrices. The linear model with ['LotArea', 'OverallQual', 'OverallCond', 'YearBuilt', 'GrLivArea', 'OverallQualSquared'] explains the variation in the data just about as well as the best model with twice as many parameters (difference in R-squared on a test set is less than 0.01). This parameter selection is stable over repeated selections of test and training data in a 2:1 split.

NAmes and CollgCr are the largest markets (with CollgCr also having high prices). NoRidge, NridgHt, and StoneBr have the highest average sale prices. Crawfor has prices that are trending higher.

Detailed Analysis

Read in the data

First save the list of variables we are allowed to use, and read in the just the training data from those columns.

avail_columns = ["LotArea", "Utilities", "Neighborhood", "BldgType", "HouseStyle", "OverallQual" \
           ,"OverallCond", "YearBuilt", "YearRemodAdd", "RoofStyle", "RoofMatl" \
           ,"GrLivArea", "FullBath", "HalfBath", "BedroomAbvGr", "KitchenAbvGr" \
           ,"MoSold", "YrSold", "SalePrice"]


df = pd.read_csv("train.csv", usecols=avail_columns)
print df
LotArea Utilities Neighborhood BldgType HouseStyle OverallQual OverallCond YearBuilt YearRemodAdd RoofStyle RoofMatl GrLivArea FullBath HalfBath BedroomAbvGr KitchenAbvGr MoSold YrSold SalePrice
0 8450 AllPub CollgCr 1Fam 2Story 7 5 2003 2003 Gable CompShg 1710 2 1 3 1 2 2008 208500
1 9600 AllPub Veenker 1Fam 1Story 6 8 1976 1976 Gable CompShg 1262 2 0 3 1 5 2007 181500
2 11250 AllPub CollgCr 1Fam 2Story 7 5 2001 2002 Gable CompShg 1786 2 1 3 1 9 2008 223500
3 9550 AllPub Crawfor 1Fam 2Story 7 5 1915 1970 Gable CompShg 1717 1 0 3 1 2 2006 140000
4 14260 AllPub NoRidge 1Fam 2Story 8 5 2000 2000 Gable CompShg 2198 2 1 4 1 12 2008 250000
5 14115 AllPub Mitchel 1Fam 1.5Fin 5 5 1993 1995 Gable CompShg 1362 1 1 1 1 10 2009 143000
6 10084 AllPub Somerst 1Fam 1Story 8 5 2004 2005 Gable CompShg 1694 2 0 3 1 8 2007 307000
7 10382 AllPub NWAmes 1Fam 2Story 7 6 1973 1973 Gable CompShg 2090 2 1 3 1 11 2009 200000
8 6120 AllPub OldTown 1Fam 1.5Fin 7 5 1931 1950 Gable CompShg 1774 2 0 2 2 4 2008 129900
9 7420 AllPub BrkSide 2fmCon 1.5Unf 5 6 1939 1950 Gable CompShg 1077 1 0 2 2 1 2008 118000
10 11200 AllPub Sawyer 1Fam 1Story 5 5 1965 1965 Hip CompShg 1040 1 0 3 1 2 2008 129500
11 11924 AllPub NridgHt 1Fam 2Story 9 5 2005 2006 Hip CompShg 2324 3 0 4 1 7 2006 345000
12 12968 AllPub Sawyer 1Fam 1Story 5 6 1962 1962 Hip CompShg 912 1 0 2 1 9 2008 144000
13 10652 AllPub CollgCr 1Fam 1Story 7 5 2006 2007 Gable CompShg 1494 2 0 3 1 8 2007 279500
14 10920 AllPub NAmes 1Fam 1Story 6 5 1960 1960 Hip CompShg 1253 1 1 2 1 5 2008 157000
15 6120 AllPub BrkSide 1Fam 1.5Unf 7 8 1929 2001 Gable CompShg 854 1 0 2 1 7 2007 132000
16 11241 AllPub NAmes 1Fam 1Story 6 7 1970 1970 Gable CompShg 1004 1 0 2 1 3 2010 149000
17 10791 AllPub Sawyer Duplex 1Story 4 5 1967 1967 Gable CompShg 1296 2 0 2 2 10 2006 90000
18 13695 AllPub SawyerW 1Fam 1Story 5 5 2004 2004 Gable CompShg 1114 1 1 3 1 6 2008 159000
19 7560 AllPub NAmes 1Fam 1Story 5 6 1958 1965 Hip CompShg 1339 1 0 3 1 5 2009 139000
20 14215 AllPub NridgHt 1Fam 2Story 8 5 2005 2006 Gable CompShg 2376 3 1 4 1 11 2006 325300
21 7449 AllPub IDOTRR 1Fam 1.5Unf 7 7 1930 1950 Gable CompShg 1108 1 0 3 1 6 2007 139400
22 9742 AllPub CollgCr 1Fam 1Story 8 5 2002 2002 Hip CompShg 1795 2 0 3 1 9 2008 230000
23 4224 AllPub MeadowV TwnhsE 1Story 5 7 1976 1976 Gable CompShg 1060 1 0 3 1 6 2007 129900
24 8246 AllPub Sawyer 1Fam 1Story 5 8 1968 2001 Gable CompShg 1060 1 0 3 1 5 2010 154000
25 14230 AllPub NridgHt 1Fam 1Story 8 5 2007 2007 Gable CompShg 1600 2 0 3 1 7 2009 256300
26 7200 AllPub NAmes 1Fam 1Story 5 7 1951 2000 Gable CompShg 900 1 0 3 1 5 2010 134800
27 11478 AllPub NridgHt 1Fam 1Story 8 5 2007 2008 Gable CompShg 1704 2 0 3 1 5 2010 306000
28 16321 AllPub NAmes 1Fam 1Story 5 6 1957 1997 Gable CompShg 1600 1 0 2 1 12 2006 207500
29 6324 AllPub BrkSide 1Fam 1Story 4 6 1927 1950 Gable CompShg 520 1 0 1 1 5 2008 68500
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
1430 21930 AllPub Gilbert 1Fam 2Story 5 5 2005 2005 Gable CompShg 1838 2 1 4 1 7 2006 192140
1431 4928 AllPub NPkVill TwnhsE 1Story 6 6 1976 1976 Gable CompShg 958 2 0 2 1 10 2009 143750
1432 10800 AllPub OldTown 1Fam 1Story 4 6 1927 2007 Gable CompShg 968 2 0 4 1 8 2007 64500
1433 10261 AllPub Gilbert 1Fam 2Story 6 5 2000 2000 Gable CompShg 1792 2 1 3 1 5 2008 186500
1434 17400 AllPub Mitchel 1Fam 1Story 5 5 1977 1977 Gable CompShg 1126 2 0 3 1 5 2006 160000
1435 8400 AllPub NAmes 1Fam 1Story 6 9 1962 2005 Gable CompShg 1537 1 1 3 1 7 2008 174000
1436 9000 AllPub NAmes 1Fam 1Story 4 6 1971 1971 Gable CompShg 864 1 0 3 1 5 2007 120500
1437 12444 AllPub NridgHt 1Fam 1Story 8 5 2008 2008 Hip CompShg 1932 2 0 2 1 11 2008 394617
1438 7407 AllPub OldTown 1Fam 1Story 6 7 1957 1996 Gable CompShg 1236 1 0 2 1 4 2010 149700
1439 11584 AllPub NWAmes 1Fam SLvl 7 6 1979 1979 Hip CompShg 1725 2 1 3 1 11 2007 197000
1440 11526 AllPub Crawfor 1Fam 2.5Fin 6 7 1922 1994 Gable CompShg 2555 2 0 3 1 9 2008 191000
1441 4426 AllPub CollgCr TwnhsE 1Story 6 5 2004 2004 Gable CompShg 848 1 0 1 1 5 2008 149300
1442 11003 AllPub Somerst 1Fam 2Story 10 5 2008 2008 Gable CompShg 2007 2 1 3 1 4 2009 310000
1443 8854 AllPub BrkSide 1Fam 1.5Unf 6 6 1916 1950 Gable CompShg 952 1 0 2 1 5 2009 121000
1444 8500 AllPub CollgCr 1Fam 1Story 7 5 2004 2004 Gable CompShg 1422 2 0 3 1 11 2007 179600
1445 8400 AllPub Sawyer 1Fam SFoyer 6 5 1966 1966 Gable CompShg 913 1 0 3 1 5 2007 129000
1446 26142 AllPub Mitchel 1Fam 1Story 5 7 1962 1962 Gable CompShg 1188 1 0 3 1 4 2010 157900
1447 10000 AllPub CollgCr 1Fam 2Story 8 5 1995 1996 Gable CompShg 2090 2 1 3 1 12 2007 240000
1448 11767 AllPub Edwards 1Fam 2Story 4 7 1910 2000 Gable CompShg 1346 1 1 2 1 5 2007 112000
1449 1533 AllPub MeadowV Twnhs SFoyer 5 7 1970 1970 Gable CompShg 630 1 0 1 1 8 2006 92000
1450 9000 AllPub NAmes Duplex 2Story 5 5 1974 1974 Gable CompShg 1792 2 2 4 2 9 2009 136000
1451 9262 AllPub Somerst 1Fam 1Story 8 5 2008 2009 Gable CompShg 1578 2 0 3 1 5 2009 287090
1452 3675 AllPub Edwards TwnhsE SLvl 5 5 2005 2005 Gable CompShg 1072 1 0 2 1 5 2006 145000
1453 17217 AllPub Mitchel 1Fam 1Story 5 5 2006 2006 Gable CompShg 1140 1 0 3 1 7 2006 84500
1454 7500 AllPub Somerst 1Fam 1Story 7 5 2004 2005 Gable CompShg 1221 2 0 2 1 10 2009 185000
1455 7917 AllPub Gilbert 1Fam 2Story 6 5 1999 2000 Gable CompShg 1647 2 1 3 1 8 2007 175000
1456 13175 AllPub NWAmes 1Fam 1Story 6 6 1978 1988 Gable CompShg 2073 2 0 3 1 2 2010 210000
1457 9042 AllPub Crawfor 1Fam 2Story 7 9 1941 2006 Gable CompShg 2340 2 0 4 1 5 2010 266500
1458 9717 AllPub NAmes 1Fam 1Story 5 6 1950 1996 Hip CompShg 1078 1 0 2 1 4 2010 142125
1459 9937 AllPub Edwards 1Fam 1Story 5 6 1965 1965 Gable CompShg 1256 1 1 3 1 6 2008 147500

1460 rows × 19 columns

Explore each variable and correlations

Check the info on the data frame for column data types and null values. Any column with dtype of int64 is complete since a single string or NaN entry would change the dtype for the entire column. The object columns are all expected to have string values, so no worries there. Check the value_counts on the object columns for mis-spelled labels.

df['YrSold'].describe()
count    1460.000000
mean     2007.815753
std         1.328095
min      2006.000000
25%      2007.000000
50%      2008.000000
75%      2009.000000
max      2010.000000
Name: YrSold, dtype: float64
df.BldgType.value_counts()
1Fam      1220
TwnhsE     114
Duplex      52
Twnhs       43
2fmCon      31
Name: BldgType, dtype: int64

Need to decide which variables should become dummified. We do this with categorical classifications that we expect to have a strong impact on the result. Utilities and RoofMatl may not be good choices since there is so little variation… though we should inspect them to see if there is a strong correlation with the price.

correl = df.corr()**2
fig, ax = plt.subplots(figsize=(12,10))
sns.heatmap(correl);

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for i in avail_columns:
    if df[i].dtype == int:
        df.plot(x=i,y="SalePrice",kind="scatter")
        plt.show()

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Explore nonlinear relationships

Some of the features are correlated in a nonlinear fashion with the sales price, so I will check correlations with the logarithm of the sales price, and with the logarithm of the features… I can also try polynomial fits with the features. Overall Quality, YearBuilt,

Looks like some features show outliers (such as LotArea being greather than 60000 units, HalfBath ==2, or BedroomAbvGr > 5, KitchenAbvGr == 0 or 3). One can make a stron case that the estimate for these lots should be given by a different function than for the others… Perhaps these are times when the categorical variables would work well?

plt.scatter(df.OverallQual**2,df.SalePrice)
#df.corr()
<matplotlib.collections.PathCollection at 0x1178f5750>

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df["OverallQualSquared"] = df.OverallQual**2
df["YearBuiltSquared"] = df.YearBuilt**2

Training test split

features = ['LotArea', 'OverallQual', 'OverallCond', 'YearBuilt', 'YearRemodAdd',
       'GrLivArea', 'FullBath', 'HalfBath',
       'BedroomAbvGr', 'KitchenAbvGr', 'MoSold', 'YrSold',
       'OverallQualSquared', 'YearBuiltSquared']

X = df[features]
y = df['SalePrice']

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=.33)

Fitting

# REGRESSION MODELS

results = []
for n in range(1,len(features)+1):
    max_r2 = 0
    best_features = []
    for cols in itertools.combinations(features, n):
        fit_xtrain = X_train[list(cols)]
        fit_xtest = X_test[list(cols)]        

#FIT LINEAR MODEL
        line = lm.LinearRegression(normalize=True)

#for algorithm in [line, lasso, ridge]:
        for algorithm in [line]:
            ## Fit model to data
            model = algorithm.fit(fit_xtrain,y_train)

            test_r2 = algorithm.score(fit_xtest, y_test)
#             print "Test r2 score : {}".format(test_r2);
            if test_r2 > max_r2:
                max_r2 = test_r2
                coeff_sum = abs(algorithm.coef_).mean()
                mse = mean_squared_error(y_true=y_test, y_pred=algorithm.predict(fit_xtest))
                best_features = list(cols)
    results.append( (max_r2,coeff_sum, mse, best_features))
results
[(0.64493256596024184,  3738.6815185367077,  2186715449.1880145,
  ['OverallQualSquared']),
 (0.72915774869635741,  1444.2671637627586,  1668006914.8561254,
  ['GrLivArea', 'OverallQualSquared']),
 (0.75483471919634171,   939.9441940699586,  1509872930.4412842,
  ['YearBuilt', 'GrLivArea', 'OverallQualSquared']),
 (0.79589913657403477,  12011.917742913709,  1256973939.1988173,
  ['LotArea',       'OverallQual',       'OverallCond',
   'YearBuilt',       'GrLivArea',       'HalfBath',
   'KitchenAbvGr',       'OverallQualSquared']),
 (0.79640356966477699,  10782.082461790924,  1253867341.6152034,
  ['LotArea',       'OverallQual',       'OverallCond',
   'YearBuilt',       'GrLivArea',       'HalfBath',
   'BedroomAbvGr',       'KitchenAbvGr',       'OverallQualSquared']),
 (0.79670268601380689,  9925.112054464651,  1252025206.0690444,
  ['LotArea',       'OverallQual',       'OverallCond',
   'YearBuilt',       'GrLivArea',       'FullBath',
   'HalfBath',       'BedroomAbvGr',       'KitchenAbvGr',
   'OverallQualSquared']),
 (0.79624488667288296,  7607.3176169180288,  1254844605.4153514,
  ['LotArea',       'OverallQual',       'OverallCond',
   'YearBuilt',       'YearRemodAdd',       'GrLivArea',
   'FullBath',       'HalfBath',       'BedroomAbvGr',
   'KitchenAbvGr',       'MoSold',       'YrSold',
   'OverallQualSquared',       'YearBuiltSquared'])]

Plot the progression of the best coefficient of determination as the number of features selected grows.

plt.plot(range(1,len(results)+1),[r[0] for r in results])
plt.xlabel("number of features")
plt.ylabel("best R^2")

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Plot the absolute sum of the coefficients. Note that this only works when the features are normalized in some fashion before fitting. A significant increase in this sum is an indication of the model being overfit. Comparing this to the difference in R2 score would make a good cross check for overfitting.

plt.plot(range(1,len(results)+1),[r[1] for r in results])
plt.xlabel("number of features")
plt.ylabel("Abs sum of coeffs")

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Plot the best mean squared error (MSE) for each number of features corresponding to the best R2 score.

plt.plot(range(1,len(results)+1),[r[2] for r in results])
plt.xlabel("number of features")
plt.ylabel("MSE")

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Compare the be R2 score with a value from near the point of diminishing R2 returns.

print max([r[0] for r in results])
results[5]
0.796794359218


(0.79010957401318027,     12257.811452486383,     1292629492.7138808,
 ['LotArea',      'OverallQual',      'OverallCond',
  'YearBuilt',      'GrLivArea',      'OverallQualSquared'])

Selected model

The r-squared value for the test set drops plateaus around 0.83, when six features are used. The set selected is: [‘LotArea’, ‘OverallQual’, ‘OverallCond’, ‘YearBuilt’, ‘GrLivArea’, ‘OverallQualSquared’]. The linear model with these features selected from the entire training set will capture most of the trends in the data without risking overfitting or storing extraneous data (particualrly since one of the features selected is derived from another!).

The model with 13 parameters (excluding only month sold) consistently beats the model with 6 parameters, but only by a small margin. Considering the colinearity between many of the parameters (seen in the heatmap during initial data exploration) a smaller parameter set is desireable. Selecting parameters with Ridge and Lasso was not as successful as the fitting with limited parameter sets, so I have opted for the latter.

A model with 8 parameters sometimes did slightly better than the model with 6, but the feature selection was not consistent (due to colinearity), so I went for the six final parameters. The differences in R^2 between the full model and 6 parameters were someitmes as large as 0.01, while the 8 parameter fit was half that. This increase in success did not seem to justify the loss in certainty over the best feature set.

correl = df[['LotArea', 'OverallQual', 'OverallCond', 'YearBuilt', 'GrLivArea', 'OverallQualSquared','SalePrice']].corr()**2
fig, ax = plt.subplots(figsize=(12,10))
sns.heatmap(correl);

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Neighborhood summary

The first figure shows the distribution of sales prices. Anything over $170k will be toward the high end of the market. Anything over $300k is really an exclusive property.

Below that is a box plot graph showing the distribution of prices in each neighborhood. It seems NoRidge, NridgHt, and StoneBr have higher median SalePrice than the other neighborhoods. A t test on this hypothesis shows significance with a p-value less than 0.0001.

Below that are tables showing the number of homes sold and the mean sale price in each neighborhood. Interestingly, there is no correlation between those two variables (as seen by the scatter plot below the tables.

NridgHt, in particular, has a good combination of sale price and sale volume. NAmes and CollgCr are two other neighborhoods that have lower home prices, but much more sales volume. CollgCr is a good market with over 100 homes sold in the dataset, and a mean sale price of nearly $200k.

Finally, I have some charts and tables showing how sale price and volume change over time in each neighborhood in the data set. This is a fairly stable market in terms of meaningful rankings of the neighborhoods. Crawfor, however, is worth watching, as it’s mean sale price has been trending upward during the duration of the data set.

To sum up: NAmes and CollgCr are the largest markets (with CollgCr also having high prices). NoRidge, NridgHt, and StoneBr have the highest average sale prices. Crawfor has prices that are trending higher.

df.SalePrice.hist(bins=20)
<matplotlib.axes._subplots.AxesSubplot at 0x1178356d0>

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df.boxplot(column="SalePrice", by="Neighborhood",figsize=(14,8),rot=90,fontsize=16)
<matplotlib.axes._subplots.AxesSubplot at 0x1182aa550>

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mask = (df['Neighborhood'] =='NoRidge' ) \
        | (df['Neighborhood']=='NridgHt') \
        | (df['Neighborhood']=='StoneBr')
tstat, pval = stats.ttest_ind(a= df[mask]['SalePrice'], \
                b= df[[not x for x in mask]]['SalePrice'], \
                equal_var=False)
print 2* pval
6.94365699791e-37
df.pivot_table(index=["Neighborhood"], values=["SalePrice"], \
               aggfunc=[np.mean,len]).sort_values(('mean','SalePrice'),ascending=False)
mean len
SalePrice SalePrice
Neighborhood
NoRidge 335295 41
NridgHt 316270 77
StoneBr 310499 25
Timber 242247 38
Veenker 238772 11
Somerst 225379 86
ClearCr 212565 28
Crawfor 210624 51
CollgCr 197965 150
Blmngtn 194870 17
Gilbert 192854 79
NWAmes 189050 73
SawyerW 186555 59
Mitchel 156270 49
NAmes 145847 225
NPkVill 142694 9
SWISU 142591 25
Blueste 137500 2
Sawyer 136793 74
OldTown 128225 113
Edwards 128219 100
BrkSide 124834 58
BrDale 104493 16
IDOTRR 100123 37
MeadowV 98576 17
df.pivot_table(index=["Neighborhood"], values=["SalePrice"], \
               aggfunc=[np.mean,len]).sort_values(('len','SalePrice'),ascending=False)
mean len
SalePrice SalePrice
Neighborhood
NAmes 145847 225
CollgCr 197965 150
OldTown 128225 113
Edwards 128219 100
Somerst 225379 86
Gilbert 192854 79
NridgHt 316270 77
Sawyer 136793 74
NWAmes 189050 73
SawyerW 186555 59
BrkSide 124834 58
Crawfor 210624 51
Mitchel 156270 49
NoRidge 335295 41
Timber 242247 38
IDOTRR 100123 37
ClearCr 212565 28
StoneBr 310499 25
SWISU 142591 25
Blmngtn 194870 17
MeadowV 98576 17
BrDale 104493 16
Veenker 238772 11
NPkVill 142694 9
Blueste 137500 2
df.pivot_table(index=["Neighborhood"], values=["SalePrice"], \
               aggfunc=[np.mean,len]).plot.scatter(x="mean",y="len")
<matplotlib.axes._subplots.AxesSubplot at 0x118173610>

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df.pivot_table(index=["YrSold"], values=["SalePrice"], \
               columns=["Neighborhood"],aggfunc=np.mean \
              ).plot(figsize=(12,10))
<matplotlib.axes._subplots.AxesSubplot at 0x117bace50>

png

df.pivot_table(index=["Neighborhood"], values=["SalePrice"], \
               columns=["YrSold"],aggfunc=np.mean \
              ).sort_values(("SalePrice",2010),ascending=False)
SalePrice
YrSold 2006 2007 2008 2009 2010
Neighborhood
StoneBr 365046.0 279585.0 245000.0 319967.0 318886.0
NridgHt 305491.0 310833.0 332422.0 323143.0 308281.0
Crawfor 196635.0 198777.0 254411.0 180211.0 296833.0
NoRidge 322333.0 399730.0 304750.0 323875.0 289938.0
ClearCr 199166.0 236333.0 208991.0 169875.0 246850.0
Timber 264485.0 229470.0 234361.0 245437.0 245160.0
Somerst 210268.0 233248.0 225631.0 236315.0 206762.0
CollgCr 199016.0 213999.0 187718.0 192317.0 203700.0
Blmngtn 217087.0 183350.0 175447.0 176720.0 192000.0
NWAmes 199463.0 175267.0 193820.0 185133.0 187428.0
Gilbert 200250.0 181967.0 186000.0 199955.0 185500.0
SawyerW 164787.0 209300.0 184080.0 183934.0 184076.0
Mitchel 150036.0 136731.0 165280.0 167860.0 166950.0
NAmes 138985.0 142962.0 151553.0 143880.0 153665.0
SWISU 130125.0 187500.0 139612.0 141048.0 141333.0
NPkVill NaN 141500.0 140000.0 146937.0 136750.0
Sawyer 149735.0 133935.0 128900.0 136925.0 132400.0
OldTown 135963.0 114794.0 147670.0 116378.0 122464.0
Edwards 134403.0 132588.0 132473.0 123855.0 111445.0
BrkSide 112746.0 135737.0 121707.0 134994.0 96500.0
BrDale 96750.0 113833.0 95225.0 118625.0 88000.0
IDOTRR 95758.0 118933.0 91642.0 89580.0 86278.0
MeadowV 123466.0 105850.0 98000.0 88400.0 81333.0
Blueste NaN NaN 151000.0 124000.0 NaN
Veenker 273333.0 214900.0 244000.0 NaN NaN
df.pivot_table(index=["YrSold"], values=["SalePrice"], columns=["Neighborhood"],aggfunc=len \
              ).plot(figsize=(12,10))
<matplotlib.axes._subplots.AxesSubplot at 0x117da6f90>

png